automotive control systems - myunionapps – Your Union App

automotive control systems - myunionapps – Your Union App
AUTOMOTIVE CONTROL SYSTEMS
This engineering textbook is designed to introduce advanced control systems for vehicles,
including advanced automotive concepts and the next generation of vehicles for Intelligent Transportation Systems (ITS). For each automotive-control problem considered,
the authors emphasize the physics and underlying principles behind the control-system
concept and design. This is an exciting and rapidly developing field for which many articles
and reports exist but no modern unifying text. An extensive list of references is provided
at the end of each chapter for all topics covered. This is currently the only textbook,
including problems and examples, that covers and integrates the topics of automotive
powertrain control, vehicle control, and ITS. The emphasis is on fundamental concepts
and methods for automotive control systems rather than the rapidly changing specific
technologies. Many of the text examples, as well as the end-of-chapter problems, require
the use of MATLAB and/or Simulink.
A. Galip Ulsoy is the C. D. Mote Jr. Distinguished University Professor and the
William Clay Ford Professor of Manufacturing at the University of Michigan. He
served as director of the Ground Robotics Reliability Center and deputy director of the
Engineering Research Center for Reconfigurable Manufacturing Systems. He has been
on the faculty of the Department of Mechanical Engineering at Michigan since 1980
and was the founding director of the Program in Manufacturing. He served as technical
editor of the American Society of Mechanical Engineers’ (ASME) Journal of Dynamic
Systems, Measurement, and Control and is the founding technical editor of the ASME
Dynamic Systems and Control Magazine. Professor Ulsoy is a member of the National
Academy of Engineering and a Fellow of the ASME, the International Federation of
Automatic Control, and the Society of Manufacturing Engineers; a Senior Member
of IEEE; and a member of several other professional and honorary organizations. He
is the past president of the American Automatic Control Council. He co-authored,
with Warren R. DeVries, Microcomputer Applications in Manufacturing, and he is a
co-author, with Sun Yi and Patrick W. Nelson, of Time Delay Systems. He has published
more than 300 refereed technical articles in journals, conferences, and books.
Huei Peng is a Professor in the Department of Mechanical Engineering at the University
of Michigan. He served as the executive director of interdisciplinary and professional
engineering programs. His research interests include vehicle dynamics and control, electromechanical systems, optimal control, human-driver modeling, vehicle active-safety
systems, control of hybrid and fuel-cell vehicles, energy-system design, and control for
mobile robots. He has received numerous awards and honors, including the ChangJiang Scholar Award, Tsinghua University; a 2008 Fellow of the ASME; the Outstanding Achievement Award, Mechanical Engineering Department, University of Michigan
(2005); the Best Paper Award, 7th International Symposium on Advanced Vehicle Control (2004); and the CAREER Award, National Science Foundation (July 1998–June
2002). He has published more than 200 refereed technical articles in journals, conferences, and books. Professor Peng is co-editor of Advanced Automotive Technologies with
J. S. Freeman and co-author of Control of Fuel Cell Power Systems – Principles, Modeling,
Analysis and Feedback Design, with Jay T. Pukrushpan and Anna G. Stefanopoulou.
Melih Çakmakcı is a professor of Mechanical Engineering at Bilkent University in
Ankara, Turkey. His research areas include modeling, analysis and control of dynamic
systems, control systems, smart mechatronics, modeling of manufacturing systems and
their control, automotive control systems, optimal energy-management algorithms, and
design and analysis of network control systems. Prior to joining Bilkent University, he
was a senior engineer at the Ford Scientific Research Center.
Automotive Control Systems
A. Galip Ulsoy
University of Michigan
Huei Peng
University of Michigan
Melih Çakmakcı
Bilkent University
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town,
Singapore, São Paulo, Delhi, Mexico City
Cambridge University Press
32 Avenue of the Americas, New York, NY 10013-2473, USA
www.cambridge.org
Information on this title: www.cambridge.org/9781107010116
C A. Galip Ulsoy, Huei Peng, and Melih Çakmakcı 2012
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2012
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Ulsoy, Ali Galip.
Automotive control systems / A. Galip Ulsoy, University of Michigan, Huei Peng,
University of Michigan, Melih Çakmakci, Bilkent University.
p. cm.
Includes index.
ISBN 978-1-107-01011-6 (hardback)
1. Automobiles – Automatic control. 2. Adaptive control systems.
3. Automobiles – Motors – Control systems. I. Peng, Huei. II. Çakmakci,
Melih. III. Title.
TL152.8.U47 2012
629.25′ 8–dc23
2011052559
ISBN 978-1-107-01011-6 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such Web sites is, or will remain,
accurate or appropriate.
Contents
page ix
Preface
PART I INTRODUCTION AND BACKGROUND
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Motivation, Background, and Overview
1.2 Overview of Automotive Control Systems
3
7
2 Automotive Control-System Design Process . . . . . . . . . . . . . . . . . . . 21
2.1 Introduction
2.2 Identifying the Control Requirements
21
22
3 Review of Engine Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Engine Operations
3.2 Engine Control Loops
3.3 Control-Oriented Engine Modeling
33
37
42
4 Review of Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1
4.2
4.3
4.4
Coordinates and Notation for Vehicle Dynamics
Longitudinal Vehicle Motion
Lateral Vehicle Motion
Vertical Vehicle Motion
54
58
64
77
5 Human Factors and Driver Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Human Factors in Vehicle Automation
5.2 Driver Modeling
93
101
PART II POWERTRAIN CONTROL SYSTEMS
6 Air–Fuel Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1 Lambda Control
6.2 PI Control of a First-Order System with Delay
119
120
7 Control of Spark Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.1 Knock Control
124
v
vi
Contents
8 Idle-Speed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9 Transmission Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.1 Electronic Transmission Control
9.2 Clutch Control for AWD
131
133
10 Control of Hybrid Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.1
10.2
10.3
10.4
10.5
10.6
Series, Parallel, and Split Hybrid Configurations
Hybrid Vehicle-Control Hierarchy
Control Concepts for Series Hybrids
Control Concepts for Parallel Hybrids
Control Concept for Split Hybrids
Feedback-Based Supervisory Controller for PHEVs
148
152
157
166
177
178
11 Modeling and Control of Fuel Cells for Vehicles . . . . . . . . . . . . . . . . 187
11.1
11.2
11.3
11.4
11.5
Introduction
Modeling of Fuel-Cell Systems
Control of Fuel-Cell Systems
Control of Fuel-Cell Vehicles
Parametric Design Considerations
187
189
196
201
205
PART III VEHICLE CONTROL SYSTEMS
12 Cruise and Headway Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.1 Cruise-Controller Design
12.2 Autonomous Cruise Control: Speed and Headway Control
213
224
13 Antilock Brake and Traction-Control Systems . . . . . . . . . . . . . . . . . 232
13.1 Modeling
13.2 Antilock Braking Systems
13.3 Traction Control
234
236
247
14 Vehicle Stability Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
14.1
14.2
14.3
14.4
Introduction
Linear Vehicle Model
Nonlinear Vehicle Model
VSC Design Principles
258
261
263
266
15 Four-Wheel Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
15.1 Basic Properties
15.2 Goals of 4WS Algorithms
272
274
16 Active Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
16.1 Optimal Active Suspension for Single-DOF Model
16.2 Optimal Active Suspension for Two-DOF Model
16.3 Optimal Active Suspension with State Estimation
288
290
294
Contents
vii
PART IV INTELLIGENT TRANSPORTATION SYSTEMS
17 Overview of Intelligent Transportation Systems . . . . . . . . . . . . . . . . 309
17.1
17.2
17.3
17.4
Advanced Traffic Management Systems
Advanced Traveler Information Systems
Commercial Vehicle Operations
Advanced Vehicle-Control Systems
310
312
314
314
18 Preventing Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
18.1 Active Safety Technologies
18.2 Collision Detection and Avoidance
322
322
19 Longitudinal Motion Control and Platoons . . . . . . . . . . . . . . . . . . . . 332
19.1 Site-Specific Information
19.2 Platooning
19.3 String Stability
332
337
343
20 Automated Steering and Lateral Control . . . . . . . . . . . . . . . . . . . . . 348
20.1 Lane Sensing
20.2 Automated Lane-Following Control
20.3 Automated Lane-Change Control
348
352
356
APPENDICES
Appendix A: Review of Control-Theory Fundamentals . . . . . . . . . . . . . . 363
A.1 Review of Feedback Control
A.2 Mathematical Background and Design Techniques
363
370
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle
Lateral/Yaw/Roll Model . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Index
391
Preface
This textbook is organized in four major parts as follows:
I. Introduction and Background is an introduction to the topic of automotive
control systems and a review of background material on engine modeling, vehicle
dynamics, and human factors.
II. Powertrain Control Systems includes topics such as air–fuel ratio control, idlespeed control, spark-timing control, control of transmissions, control of hybridelectric vehicles, and fuel-cell vehicle control.
III. Vehicle Control Systems covers cruise control and headway-control systems,
traction-control systems (including antilock brakes), active suspensions, vehiclestability control, and four-wheel steering.
IV. Intelligent Transportation Systems (ITS) includes an overview of ITS technologies, collision detection and avoidance systems, automated highways, platooning, and automated steering.
With multiple chapters in each part, this textbook contains sufficient material for a
one-semester course on automotive control systems. The coverage of the material
is at the first-year graduate or advanced undergraduate level in engineering. It is
assumed that students have a basic undergraduate-level background in dynamics,
automatic control, and automotive engineering.
This textbook is written for engineering students who are interested in participating in the development of advanced control systems for vehicles, including
advanced automotive concepts and the next generation of vehicles for ITS. This
is an exciting and rapidly developing field for which numerous articles and reports
exist. An extensive list of references, therefore, is provided at the end of each chapter
for all topics covered. Due to the breadth of topics treated, the reference lists are
by no means comprehensive, and new studies are always appearing. However, the
lists cover many major contributions and the basic concepts in each sub-area. This
textbook is intended to provide a framework for unifying the vast literature represented by the references listed at the end of each chapter. It is currently the only
textbook, including problems and examples, that covers and integrates the topics of
automotive powertrain control, vehicle control, and ITS.
The emphasis is on fundamental concepts and methods for automotive control systems rather than the rapidly changing specific technologies. For each
ix
x
Preface
automotive-control problem considered, we emphasize the physics and underlying principles behind the control-system concept and design. Any one of the many
topics covered (e.g., engine control, vehicle-stability control, or platooning) could be
discussed in more detail. However, rather than treating a specific control problem
in its full complexity, we use each automotive control application as an opportunity
to focus on a key engineering aspect of the control-design problem. For example,
we discuss the importance of regulating the air–fuel ratio in engine control, the
benefits for vehicle dynamics of reducing the vehicle side-slip angle in four-wheel
steering and vehicle-stability control, the importance of predictive/preview action
in the material on driver modeling, the concept of string stability for platoons and
autonomous cruise-control systems, and the role of risk homeostasis in active-safetysystems design.
We also use various automotive-control applications to focus on specific control
methodologies. For example, the Smith predictor for control of time-delay systems is
introduced in air–fuel ratio control; linear quadratic optimal estimation and control
is introduced for active suspensions; adaptive control using recursive least squares
estimation is introduced in the chapter on cruise-control systems; and sliding-mode
control is introduced in the discussion of traction-control systems. However, all of
these methods can be applied to many other automotive-control problems.
End-of-chapter problems are included and many are used in our courses as
homework and/or examination problems. Throughout the text, we include examples to illustrate key points. Many of these examples, as well as the end-of-chapter
problems, require the use of MATLAB and/or Simulink. It is assumed that students are familiar with these computational engineering tools; for those who are not,
we highly recommend the Control Tutorials for MATLAB and Simulink Web site
(www.engin.umich.edu/class/ctms) for self-study.
This textbook is based on course notes originally developed by A. Galip Ulsoy
during the mid-1990s, then refined and added to by both Ulsoy and Huei Peng during
a period of fifteen years of teaching this material to beginning graduate students at
the University of Michigan, Ann Arbor. The students are primarily from mechanical
engineering disciplines, but students with a suitable background from other engineering disciplines also are included, as well as practicing engineers in the automotive
industry who take the course through distance-learning programs and short courses.
We sincerely thank all of our former students for their useful feedback, which
led to many improvements in and additions to this material. We also welcome
your comments so that we can continue to improve future versions. The current
textbook was rewritten extensively from those course notes in collaboration with
Melih Çakmakcı, who was not only a former student who took the course but also
has worked in the automotive industry as a control engineer for a decade. He brings
an additional perspective to the material from his extensive industrial experience.
A. Galip Ulsoy
Huei Peng
Melih Çakmakcı
PART I
INTRODUCTION AND
BACKGROUND
1
Introduction
The century-old automobile – the preferred mode for personal mobility throughout
the developed world – is rapidly becoming a complex electromechanical system. Various new electromechanical technologies are being added to automobiles to improve
operational safety, reduce congestion and energy consumption, and minimize environmental impact. This chapter introduces these trends and provides a brief overview
of the major automobile subsystems and the automotive control systems described
in detail in subsequent chapters.
1.1 Motivation, Background, and Overview
The main trends in automotive technology, and major automotive subsystems, are
briefly reviewed.
Trends in Automotive Control Systems
The most noteworthy trend in the development of modern automobiles in recent
decades is their rapid transformation into complex electromechanical systems. Current vehicles often include many new features that were not widely available a few
decades ago. Examples include hybrid powertrains, electronic engine and transmission controls, cruise control, antilock brakes, differential braking, and active/semiactive suspensions. Many of these functions have been achieved using only mechanical devices. The major advantages of electromechanical (or mechatronic) devices,
as opposed to their purely mechanical counterparts, include (1) the ability to embed
knowledge about the system behavior into the system design, (2) the flexibility
inherent in those systems to trade off among different goals, and (3) the potential
to coordinate the functioning of subsystems. Knowledge about system behavior –
in terms of vehicle, engine, or even driver dynamic models or constraints on physical variables – is included in the design of electromechanical systems. Flexibility
enables adaptation to the environment, thereby providing more reliable performance in a wide variety of conditions. In addition, reprogrammability implies lower
cost through exchanged and reused parts. Sharing of information makes it possible
to integrate subsystems and obtain superior performance and functionality, which
are not possible with uncoordinated systems.
3
4
Introduction
Today’s electrical and electronic devices have evolved into systems with good
reliability and relatively low cost. They feature many new benefits including
increased safety, reduced congestion and emissions, improved gas mileage, better
drivability, and greater driver satisfaction and passenger comfort. Safety is perhaps
the most important motivation for the increased use of electronics in automobiles.
On average, one person dies every minute somewhere in the world due to a car
crash. The cost of crashes totals 3 percent of the world’s gross domestic product
(GDP) and was nearly $1 trillion in 2000. Clearly, the emotional toll of accidents
and fatalities is immeasurable (Jones 2002). Data from the National Highway Transportation Safety Association (NHTSA) show that 6,335,000 accidents (with 37,081
fatalities) occurred on U.S. highways in 1998 (NHTSA 1999). In 2008, the same
statistic improved by about 10 percent to 5,811,000 accidents (with 34,017 fatalities) (NHTSA 2009). Data also indicate that although various factors contribute to
accidents, human error accounts for 90 percent of all accidents (Hedrick et al. 1994).
Delays due to congestion are a major problem in metropolitan areas, providing strong motivation for an increase in automotive electronics. Traffic-information
systems can reduce delays significantly by alerting drivers to accidents, congested
areas, and alternate routes. Automated highway systems (AHS) at on ramps and
tollbooths also can improve traffic flows. Significantly higher traffic flows can be
achieved by closely packing automatically controlled vehicles in “platoons” on special highway lanes. These AHS concepts, developed and demonstrated in California,
require automatic longitudinal and lateral control of vehicles (Rajamani et al. 2000).
In 1970, only 30 million vehicles were produced and 246 million vehicles were
registered worldwide; by 1997, these numbers had increased to 56 million and
709 million, respectively. By 2005, 65 million vehicles were produced and more than
800 million were registered (Powers and Nicastri 2000). Consequently, another major
factor that contributes to the increased use of electronics is the expanding government regulation of automotive emissions. For example, the 2005 standard for hydrocarbon (HC) emissions was less than 2 percent of the 1970 allowance; for carbon
monoxide (CO), it was 10 percent of the 1970 level; and for oxides of nitrogen
(NOx ), it was 7 percent of the 1970 level. The California requirements for ultra-low
emission vehicles (ULEV) reduced the levels approximately by half again. Spilling
5.7 liters of gasoline on a driveway produces as many HC emissions as a ULEV
vehicle driven more than 160,000 kilometers. At the same time, government regulations also require improved fuel economy. Advanced control technologies (e.g.,
fuel injection, air–fuel ratio control, spark-timing control, exhaust-gas recirculation
[EGR], and idle-speed control) are and will continue to be instrumental in reducing
emissions and improving fuel economy (e.g., hybrid-electric, all-electric, and fuel-cell
vehicles).
This evolution (some might say “revolution”) of automotive electronics also is
enabled by recent advances in relevant technologies, including solid-state electronics,
computer technology, and control theory. Table 1.1 summarizes developments in
automotive electronics from 1965 through 2010. The already-evident trend toward
increased automotive electronics can be expected to continue in the foreseeable
future (Cook et al. 2007; Ford 1986; Powers and Nicastri 2000). In the next decade,
significant advances are expected in the use of power electronics, advanced control
systems, and alternative powertrain concepts. Among others, new technologies are
1.1 Motivation, Background, and Overview
Table 1.1. Historical development of automotive electronics
Year
Examples of automotive electronics available
1965
Solid-state radio, alternator rectifier
1970
Speed control
1975
Electronic ignition, digital clock
1980
Electronic voltage regulator, electronic engine controller, electronic instrument cluster,
electronic fuel injection
1985
Clock integrated with radio, audio graphic equalizer, electronic air suspension
1990
Antilock brakes, integrated engine and speed control, cellular phones, power doors and
windows
1995
Navigation systems, advanced entertainment/information systems, active suspensions
2000
Collision avoidance, autonomous cruise control, vehicle stability enhancement, CVT
2005
Hybrid electric vehicles, driver monitoring, drive-by-wire, integrated vehicle controls
2010
Driver-assist systems (e.g., automated parallel parking), integrated telematics (i.e.,
location-aware vehicles via mobile devices), plug-in hybrid electric vehicles
being developed for fuel-efficiency management, integrated chassis control, power
management of hybrid vehicles, electrical power steering, collision warning and
prevention, automatic lane following, rollover and lane-departure warnings, and
fuel-cell vehicles.
In the near future, it is anticipated that these advancements may reach beyond
individual vehicles and eventually lead to the development of Intelligent Transportation Systems (ITS) (Jurgen 1995). Due to rapidly increasing highway congestion, it
is necessary for automotive and transportation engineers to devise ways to increase
safety and throughput on existing highways. The term ITS (previously referred to
as Intelligent Vehicle/Highway Systems [IVHS]) defines a collection of concepts,
devices, and services to combine control, sensing, and communication technologies
to improve the safety, mobility, efficiency, and environmental impacts of vehicle and
highway systems. The importance of ITS is in its potential to produce a paradigm shift
in transportation – that is, away from individual vehicles and roadways and toward
the development of those that can cooperate effectively, efficiently, and intelligently.
Major Automobile Subsystems
To provide background for subsequent chapters, this section is an introductory
overview of an automobile and its major subsystems. Refer to other sources, including Bastow et al. (2004), Bosch (2009), Dixon (1992), Ellis (1969), Gillespie (1992),
Mizutani (1992), Ribbens (2003), Segel (1986), Washine (1989) and Wong (2008),
for more in-depth discussions. The functional systems of an automobile are shown
in Figure 1.1 and are classified as follows:
Chassis or Body. This basic structure of an automobile supports many other
systems described herein, as well as passengers and loads. It is supported by
the suspension, which connects it to the axles and the wheels. The design of
the chassis also affects vehicle dynamics, aerodynamic drag, fuel efficiency, and
passenger comfort. The current trend is toward lighter body structures, including
5
6
Introduction
Figure 1.1. Vehicle subsystems.
the more efficient use of lighter-weight materials that nonetheless are durable
and crashworthy.
Engine. This component provides the power for moving a vehicle as well as
operating various subsystems. The most prevalent of the many engine designs
is the piston-type, spark-ignited, liquid-cooled, internal combustion engine with
four strokes per cycle and gasoline fuel. Engine controls, which improve engine
performance in various ways, are used widely. Many new engine technologies
(e.g., homogeneous-charge compression-ignition, electric, hybrid, and fuel-cell)
also are being developed (Ashley 2001).
Drive Train or Powertrain. This system consists of the engine, transmission,
driveshaft, differential, and driven wheels. The transmission is a gear system
that adjusts the ratio of wheel speed to engine speed to achieve near-optimum
engine performance. Automatic transmissions already are commonplace, and
electronic transmission-control systems and continuously variable transmissions
(CVT) are being introduced. A driveshaft is used in front-engine, rear-wheeldrive systems to transmit the engine power to the drive wheels. The differential
provides not only the right-angle transfer of the driveshaft rotary motion to
the wheels but also a torque increase through the gear ratio, thereby allowing
the driven wheels to turn at different speeds (e.g., when turning a corner). The
wheels and pneumatic tires provide traction between the vehicle and the road
surface. Traction-control systems have been developed to provide good traction
under a variety of road-surface conditions.
Steering. Steering allows a driver to change the orientation of a vehicle’s front
wheels to control the direction of forward motion. A rack-and-pinion steeringsystem design is typical in many modern automobiles. Power-assisted steering
is now commonplace and four-wheel-steering (4WS) vehicles are emerging.
Suspension. The two major functions of the suspension system are to (1) provide
a smooth ride inside the automobile, and (2) maintain contact between the
1.2 Overview of Automotive Control Systems
wheels and the road surface. An independent-strut–type suspension design is
common. A semirigid axle suspension system also is typical on the rear wheels of
front-wheel-drive (FWD) vehicles. The suspension design also influences vehicle
dynamics. Active and semi-active suspensions, which use electronic controls, are
currently available on some vehicles.
Brakes. The brakes are the means for bringing a vehicle to a stop. Two common
designs are drum and disk brakes. Antilock brakes, which use electronic controls
to limit wheel slip, are now common on many commercial vehicles.
Instrumentation. A modern vehicle includes many electronic sensors, actuators,
and other instrumentation. In today’s cars, there are more than two dozen sensors in the powertrain alone. Most vehicles also now include dozens of microprocessors (e.g., for electronic engine control and diagnostics). Technologies such
as the global positioning system (GPS) are starting to be used in automobiles.
The average value of automotive electronics per vehicle, which was less than
$100 in the 1960s, reached approximately $1,000 in 1990 and more than $2,000 by
2000 (Ford 1986). Due to increasing power needs, today’s 14-volt (V) electrical
systems (with a 12-V battery) eventually may be replaced with a 42-V system
(with a 36-V battery).
1.2 Overview of Automotive Control Systems
The automobile is rapidly becoming a complex electromechanical system due in part
to advances in computing and sensing technologies as well as advances in estimation
and control theory. Vehicles now include hierarchically distributed, onboard computing systems, which coordinate several distinct control functions. Among these
are control functions associated with the engine and transmission, cruise control,
traction control, and active suspensions, which are discussed in subsequent chapters
of this book.
The control functions in an automobile can be grouped as follows: (1) powertrain
control, (2) vehicle control, and (3) body control (Mizutani 1992). Before discussing
each topic in detail, we briefly introduce these control systems, the basic concepts,
and the terminology associated with control-system design.
Powertrain control consists of engine- and transmission-control systems and is
discussed in Part II. The engine-control systems may include fuel-injection
control, carburetor control, ignition or spark-timing control, idle-speed control, antiknock-control systems, and exhaust-gas recirculation (EGR) control.
The goal of engine-control systems is to ensure that an engine operates at
near-optimal conditions at all times. Electronic transmission control is used
primarily in automatic transmissions. Transmission-control systems determine
the optimal shift point for the torque converter and the lockup operation
point based on throttle-angle and vehicle-speed measurements. Often, a single
electronic control unit (ECU) handles both engine and transmission control
functions. Four-wheel drive (4WD) systems are used (1) to obtain the optimal
torque–transmission ratio, (2) in braking and acceleration, and (3) between the
front and rear wheels. This optimal ratio depends on the vehicle forward velocity.
7
8
Introduction
Table 1.2. Automotive control functions and variables
Controlled
variable
Control input
Control algorithm
Sensors and actuators
Fuel Control
Air–fuel ratio
Injected fuel
Smith Predictor
Airflow, EGO, fuel
injector
EGR Control
EGR rate
EGR valve
opening
Optimal control
Valve position, EGR
valve
Spark-Timing
Control
Spark timing
Primary current
Rule-based, optimal Crank angle, vibration
control
Idle-Speed Control
Idle speed
Airflow rate
PI, linear quadratic
regulator
PI, adaptive PI
Engine speed, idle speed
control valve, throttle
Cruise Control
Vehicle speed
Airflow rate
Transmission
Gear ratio
Pressure, current Rule-based
Vehicle speed, MAP
All-wheel drive,
four-wheel drive
Torque
distribution
Pressure, current Rule-based, P, PI,
PID
Engine speed, steering
angle, control valve
Four-wheel steering Wheel angle
Stepper motor
Vehicle speed, wheel
angle, stepper motor
ABS
Pressure, current Rule-based, sliding
mode
Slip ratio
Feed forward, PI
Vehicle speed, throttle
Vehicle speed, wheel
speed, control valve
Typically, 4WD systems have been achieved using mechanical rather than electromechanical components and are not discussed in this book.
Vehicle control systems, discussed in Part III, include suspension control, steering control (e.g., 4WS), cruise control, braking control (e.g., antilock brake
systems [ABS]), and traction control. These systems improve various vehicle
functions including response, steering stability, ride, and handling; many were
introduced in recent decades or are currently being developed.
Body control refers to systems such as automatic air conditioning, electronic
meters, multi-instrument displays, energy control systems, security systems,
communication systems, door-lock systems, power windows, and rear-obstacle
detection. The intent of these systems is to increase driving comfort and convenience and to improve the value of the automobile. These features often are
perceived immediately by drivers as a benefit and typically are introduced first
in luxury vehicles. Body-control systems are not discussed in detail in this book.
Several of the systems listed here are shown in Table 1.2, including the controlled
variable, the manipulated variable (i.e., control input), the control logic (or control
algorithm) used, the measured variables, and the actuators used to generate the
control input. These vehicle control systems can be compared to the “generic”
control-system block diagram shown in Figure 1.2. A typical feedback-control system
consists of four basic elements: (1) controller, (2) actuator, (3) controlled system,
and (4) sensor. The controller receives a reference (or set-point) input, which defines
the desired value of the controlled variable, and a feedback signal from the sensor,
which is a measurement of the controlled variable. The controller then applies a
particular control logic (or law or algorithm) to compute a control signal. The control
1.2 Overview of Automotive Control Systems
Figure 1.2. Control-system block diagrams.
signal is sent to the actuator, which supplies energy to the system by converting this
information-type input signal to a power-type input to the controlled system. The
controlled system responds to the actuator input as well as any other uncontrolled
inputs (i.e., disturbances) that act on it. The sensor provides a measurement of the
controlled variable for the purpose of feedback to the controller.
The control system illustrated in Figure 1.2 is a simple feedback loop with a
single-input/single-output (SISO) system, which attempts to control only one variable. In reality, many automotive control systems consist of several such loops that
interact in a complex manner. For example, an electronic engine-control system
includes many controlled variables, actuators, and sensors; in fact, they are multiinput/multi-output (MIMO) control systems. The detailed analysis of these control
problems can be carried out but is a complex process. Instead, the first stage of
control-system analysis or design can be performed by neglecting the interactions
among the various control tasks and treating each as an independent SISO control
system. In a typical electronic engine-control system, the following SISO control
systems can be identified (see Table 1.2):
Air–Fuel Ratio Control. The air–fuel ratio is the controlled variable and it is
controlled by fuel injection at each cylinder. A mass airflow sensor is used and
the fuel injector is the actuator. An optimal control is used to maintain the air–
fuel mixture at stoichiometry (i.e., air–fuel ratio = 14.7); this reference is selected
because in conjunction with a catalytic converter, it provides near-optimal performance in an engine. Thus, accurate air–fuel ratio control is important from
the perspective of reducing emissions as well as other performance measures.
Current systems use an EGO sensor for air–fuel ratio control.
EGR Control. The controlled variable is the EGR rate, the EGR control valve
is the actuator, and the engine temperature and speed measurements are used to
compute the proper EGR rate. EGR effectively reduces peak combustion temperature, thereby reducing emissions. The drawbacks of EGR include increased
HC emission, deteriorated fuel economy, and combustion instability at idle or
low engine speed and/or when an engine is cold. Typically, the EGR function
may be turned off when an engine is cold or when a vehicle is accelerating or
idling.
9
10
Introduction
Figure 1.3. Control-system block diagram: Mathematical representation.
Spark-Timing Control. Spark timing is adjusted to affect engine-torque output.
Moreover, the response is usually much faster than throttle-angle manipulations.
It also is used to affect emission and fuel economy and to minimize engine knock.
Because the “timing” is with respect to top dead center (TDC), the crank angle
must be measured.
Electronic-Transmission Control. The hydraulic pressure and solenoid status
can be controlled for fuel economy (i.e., shift-point control) and comfort (i.e.,
torque control during shifting). The shift point typically is regulated based on
two measurements: vehicle speed and manifold absolute pressure (MAP), or
throttle angle. The latter measurement is an indicator of engine load.
Idle-Speed Control. The purpose of the idle-speed control function is to maintain
idle speed in the presence of load disturbance as well as to minimize speed
for reduced fuel consumption and emission. An idle-speed controller typically
measures the idle speed and adjusts the airflow rate using either the throttle or
an idle-speed control valve.
Subsequent chapters describe in more detail not only these powertrain (i.e.,
engine and transmission) control functions but also vehicle-control functions such as
cruise control, traction control, active suspensions, and 4WS. For design purposes,
each function is treated as a stand-alone SISO control system. In fact, they are
interacting MIMO systems, which must be accounted for when they are integrated
in an automobile. In addition, they become the building blocks for even higher-level
control functions, such as those described in Part IV, the ITS chapters.
Control Structures and Algorithms
Figure 1.3 is a typical mathematical representation of the physical elements shown in
Figure 1.2. Note that the actuator, controlled system, and sensor blocks are combined
into a process (or plant) transfer function:
G p (s) =
where s is the Laplace transform variable.
B p (s)
A p (s)
(1.1)
1.2 Overview of Automotive Control Systems
11
In addition, the controller is represented by:
S(s)
T (s)
uc −
y
u=
R(s)
R(s)
(1.2)
where uc is the reference (or command) input and y is the controlled variable.
Depending on the particular form of the polynomials R, S, and T, particular control
laws are obtained. Simple control laws include Proportional (P) control:
(1.3)
u(t ) = KP (uc (t ) − y(t )) = KP e(t )
which corresponds to R = 1, T = S = KP in Eq. (1.2). Integral (I) control is given by:
u(t ) = KI
t
(1.4)
e(t )dt
0
which corresponds to R = s, T = S = KI in Eq. (1.2). Similarly, a Derivative (D)
controller is:
de(t )
(1.5)
u(t ) = KD
dt
which corresponds to R = 1, T = S = KD s in Eq. (1.2). Common combinations of
these basic control laws are as follows:
Proportional plus Derivative (PD) control:
u(t ) = KP e(t ) + KD
de(t )
dt
(1.6)
Proportional plus Integral (PI) control:
u(t ) = KP e(t ) + KI
t
(1.7)
e(t )dt
0
and Proportional plus Integral plus Derivative (PID) control:
u(t ) = KP e(t ) + KI
t
e(t )dt + KD
de(t )
dt
(1.8)
0
A more complete description of feedback-control concepts is provided in basic
textbooks (see also Appendix A). Of course, several other control algorithms not
mentioned herein are used in vehicle-control applications; for example, optimal
control methods are widely used. Current trends include the use of robust control
and adaptive control methods to handle process-parameter variations, which are
common in automotive control applications. In addition, many automotive systems
have nonlinear characteristics (e.g., the engine) and require the use of nonlinear
methods for effective control.
Examples of Automotive Control Systems
Air–Fuel Ratio Control. The air–fuel ratio is the controlled variable and it is
controlled by fuel injection at each cylinder. An airflow sensor is used and the
fuel injector is the actuator. An optimal control is used to maintain the air–fuel
12
Introduction
Figure 1.4. Effect of air–fuel ratio on engine performance (Ribbens 2003).
mixture at stoichiometry (i.e., air–fuel ratio = 14.7). As shown in Figure 1.4, this
value provides near-optimal performance in an engine; that is, engine torque
(T) is near maximum, brake-specific fuel consumption (BSFC) is near minimum, and hydrocarbon (HC) and carbon monoxide (CO) emissions are near
minimum. However, nitrous oxide (NOx ) emissions are maximized. The importance of the catalytic converter found on most modern vehicles is illustrated in
Figure 1.5, which shows that the conversion efficiency is also very high for HC,
CO, and NOx emissions in a very narrow band near stoichiometry. Thus, accurate
air–fuel ratio control is important from the perspective of reducing emissions as
well as other performance measures. Current systems use an EGO sensor for
air–fuel ratio control (Grizzle et al. 1991). The problem of air–fuel ratio control
is discussed in detail in Chapter 6.
Figure 1.5. Conversion efficiency of two-way catalytic converter (Ribbens 2003).
1.2 Overview of Automotive Control Systems
Figure 1.6. (a) NOx emissions versus air–fuel ratio for various EGR levels; (b) Typical variation in engine performance with EGR (Ribbens 2003).
EGR Control. The controlled variable is the EGR rate, the EGR control valve
is the actuator, and the engine temperature and speed measurements are used
to compute the proper EGR rate (Figure 1.6). EGR effectively reduces peak
combustion temperature, thereby reducing the NOx emission. The drawbacks
of EGR include increased HC emission, deteriorated fuel economy, and combustion instability at idle or low engine speed and/or when an engine is cold.
Open-loop (i.e., offline optimization) control typically is used, and the EGR
function may be turned off when an engine is cold or when a vehicle is accelerating or idling.
Spark-Timing Control. This function is a mechanism or, rather, an objective. It
has been observed that up to a certain point, engine torque increases with spark
advance. In Figure 1.7, the minimum spark advance for best torque (MBT)
denotes the spark advance for optimal torque. Therefore, spark timing can be
adjusted to affect engine torque output. Moreover, the response is usually much
faster than throttle-angle manipulations. Spark advance also can be used to
affect emissions and fuel economy and to minimize engine knock. Because the
“timing” is with respect to TDC, the crank angle must be measured.
Figure 1.7. Typical variation of performance with spark timing (Ribbens 2003).
13
14
Introduction
When engine torque or fuel economy is a concern, the peak-pressure concept
is a simple but accurate rule to follow. It was found that under varying humidity,
engine-speed, load, air–fuel ratio, and other conditions that the angle-to-peak
pressure generated by optimal spark timing is roughly constant (which does
make sense!). Therefore, if we can measure engine-cylinder pressure and adjust
spark timing to achieve constant angle-to-peak pressure (i.e., 10 to 12 degrees),
all of these perturbations will be rejected. Spark-timing control is discussed in
Chapter 7.
Automatic Transmission Control. Hydraulic pressure and solenoid status can be
controlled for fuel economy (i.e., shift-point control) and comfort (i.e., torque
control during shifting). The shift point is usually regulated based on two measurements: vehicle speed and MAP or throttle angle. The latter measurement
is an indicator of engine load. Again, the control strategy is the open-loop table
lookup type. Torque control also is concerned with ride quality during gear
shifting. The basic idea is to control the hydraulic pressures at both oncoming
and offgoing clutches to transfer smoothly the load while minimizing the torque
disturbance at the output shaft. By directly controlling clutch pressures, it is
possible to enable the use of transmissions with low mechanical and hydraulic
complexity. Transmission-control problems are discussed in Chapter 9.
Idle-Speed Control. The purpose of the idle-speed control function is to maintain
idle speed in the presence of load disturbance, as well as minimize possible speed
for fuel consumption and emission. An idle-speed controller typically uses a feedforward lookup table to handle engine loads (i.e., air conditioning compressor,
automatic transmission, power steering charging system, and vehicle speed). A
PI control algorithm in the feedback loop measures the idle speed and adjusts
the airflow rate using either the throttle or an idle-speed control valve.
The idle-speed control is important because variations can cause stalling
and affect fuel economy and are easily perceived by human drivers. Idle-speed
control is discussed in detail in Chapter 8.
Cruise Control. A cruise-control system (see Chapter 12) adjusts the throttle
angle, using a throttle actuator, to maintain a desired vehicle forward velocity.
The velocity sensor typically is driven by a flexible cable from the driveshaft,
and the process model (i.e., Gp (s) in Eq. [1.1]) must represent how the vehicle
forward velocity changes as the throttle actuator input is varied (i.e., y = vehicle forward velocity and u = throttle actuator input). A PI control algorithm
typically is used in many cruise-control applications.
Sensors, Actuators, and Controller Modules
It is clear that many sensors and actuators are required to implement the various
control systems discussed in this chapter. For engine and vehicle control, for example,
measurements of the following variables are required.
Manifold Pressure. A MAP sensor produces a voltage signal that is proportional
to the average manifold absolute pressure. A variety of sensor designs is used
that typically are based on measuring the deflection of a diaphragm, which is
deflected by the manifold pressure.
1.2 Overview of Automotive Control Systems
15
Crankshaft Angular Position and Engine Speed. This can be measured at the
crankshaft or at the camshaft. A typical sensor is of the noncontacting magnetic reluctance or hall-effect type. It produces a voltage proportional to the
distance between a fixed magnetic sensor and protruding metal tabs on a rotating disk attached to the crankshaft. For example, four tabs might be attached
with each corresponding to the TDC position for a specific cylinder on a fourcylinder engine. The number of pulses then indicates angular position and their
frequency is proportional to the engine speed. Optical techniques also can be
used to measure crankshaft angular position and speed.
Airflow Rate. The mass flow rate of air into the engine is measured using a mass
airflow sensor (MAS), which is based on the same principles as the classical
hot-wire anemometer.
Throttle Angle. A potentiometer (i.e., rotary variable resistor) can be used to
measure the throttle angle.
Exhaust gas oxygen (EGO). The EGO sensor provides an indirect measurement of the air–fuel ratio. This is a highly nonlinear sensor, that essentially
indicates either lean or rich air–fuel ratios.
Engine Knock. Engine knock typically is measured indirectly by some type of
vibration sensor mounted on the engine block. More direct measurement using
cylinder pressure is possible but not used commercially.
Vehicle Speed. An optical vehicle-speed sensor similar in operation to the
engine-speed sensor is used. It is connected by a flexible cable to the driveshaft, which rotates at an angular speed proportional to the vehicle wheel speed.
Longitudinal Slip. This is the difference – normalized relative to vehicle speed –
between the vehicle speed and the tire circumferential speed. It is difficult to
measure directly and typically is estimated based on changes in wheel speed.
The wheel speed in normal operation is limited by the inertia of the wheel but,
when slip occurs, rapid changes in wheel speed take place. These changes, as
obtained by differencing of the speed measurement during short time intervals,
can be used as a longitudinal-slip sensor. This is the key sensor for ABS and
traction-control systems.
Steering Angle. This is determined by measuring the rotation of the steering
wheel. A typical approach uses photocells, a light source, and a disk with multiple
codes. This is used, for example, in 4WS systems.
Vehicle Acceleration. Accelerometers can be used to measure vehicle acceleration in various directions. For lateral acceleration, which is important for ride
and handling, the speed sensor and a steering-angle, d, sensor can be used to
calculate the lateral acceleration from:
ay ≈ mu2 d/l
(1.9)
Suspension Stroke. This typically is measured using a linear displacement sensor,
such as a linear variable differential transformer (LVDT), and is important for
active suspension control.
Of course, there are many other sensors; this list certainly is not exhaustive. Also,
there are efforts to estimate quantities of interest (e.g., yaw rate, traction forces, and
16
Introduction
wheel slip) from indirect measurements. Following is a discussion of some important
actuation devices needed for engine and vehicle-control systems.
Fuel Metering. This is accomplished using either an electronic carburetor or a
fuel injector. A fuel injector is a solenoid-operated valve that passes fuel from a
constant-pressure source when the solenoid current is on and blocks fuel flow
when it is off. The fuel injector is either mounted in the throttle body (i.e.,
single-point fuel injection) or at each cylinder intake port near the intake valve
(i.e., multipoint fuel injection). The duty cycle of the fuel injector can be varied
to manipulate the air–fuel ratio.
Spark Ignition. This electronic ignition includes the coil, distributor, and spark
plugs as well as the associated electronics.
Exhaust Gas. The EGR actuator is a valve that connects the intake and exhaust
manifolds. This allows mixing of exhaust gases in the intake manifold.
Throttle Actuator. This adjusts the throttle angle in cruise-control applications.
Typically, a control solenoid is used to actuate a pressure-control valve, which
allows the vacuum from the intake manifold to exert a pull on a spring-loaded
throttle lever.
Brake-Pressure Modulators. These are used in the ABS. Modulators momentarily reduce the brake pressure and, consequently, the braking force to prevent
the wheels from locking.
Suspension Actuator. Semi-active suspensions use a special adjustable shock
absorber in which resistance to oil flow can be adjusted by changing the size
of the orifice. A fully active suspension includes an electrohydraulic actuator
that applies a force between the sprung and unsprung mass in response to control signals. A fully active system typically incorporates a passive air-spring–type
suspension in parallel with this actuator.
Vehicle Communication Networks
Vehicle communication networks provide the infrastructure to exchange information among vehicle electronic units. These electronic units are not only actuator
and sensor components with network capabilities but also ECUs such as the engine
controller.
Figure 1.8 is a configuration for common electronic components that communicate on the vehicle networks. For those components of the network for which the
information exchange is less critical (e.g., automatic configuration of the seat position and/or the confirmed position), a slower and less expensive network system such
as the Local Interconnect Network (LIN) (Motorola 1999) is preferred. For safety
and vehicle performance, critical information exchange high-speed communication
network protocols such as the Controller Area Network (CAN) (Tindell et al. 1995)
or FlexRay (Makovitz and Temple 2006) are preferred.
Delay of the information can be critical for the algorithms that span multiple
ECUs. In fact, characterization of the vehicle networks for a particular vehicle configuration affects algorithms run in individual control units and frequency of the
information exchange. For critical safety features such as x-by-wire (e.g., drive-bywire and steer-by-wire) applications, guaranteed delivery of the information within
1.2 Overview of Automotive Control Systems
Door
Latch
CAN
17
Seat
Controller
Window
Controller
LIN
X
Low
Speed
X Control Unit
Network Gateway
Vehicle
Stability
Controller
Engine
Controller
Brake
Controller
Transmission
Controller
High
Speed
Battery
Controller
Figure 1.8. Simple vehicle-control network.
specified time boundaries is required. To satisfy the strict communication requirements, design features such as multiple physical routes and software-based message
priorities are considered (Davis et al. 2007).
Supervisory and Distributed Control Algorithms in Automotive Applications
Figure 1.9 shows powertrain components and control-oriented features for a typical
hybrid vehicle in today’s market. Modern vehicles consist of many subsystems, and
Engine
Throttle Body
Idle Speed
Fueling
Cooling
AC
Transmission
Clutch control
Shift Scheduling
CVT
Battery System
Contactor Control
Power Limits
Cell Balancing
Braking
ESC/ABS
EHB
Regenerative Braking
Motor/Generator
Torque Control
Speed Control
Start/stop
Chassis
Active Body Control
Roll Stabilization
Active Suspension
Figure 1.9. Vehicle subsystems and features.
18
Introduction
Engine
Control
Algorithms
Vehicle Stability
Control Algorithms
Vehicle Supervisory
Control Algorithms
Transmission
Control Algorithms
Motor
Control Algorithms
Motor
Control Algorithms
Battery
Control Algorithms
Brake
Control Algorithms
Figure 1.10. Vehicle supervisory control.
overall performance of a vehicle depends on consistent and reliable operation of all
of them. Supervisory vehicle-control algorithms coordinate and monitor the operation of control algorithms located in the subsystem controller units, as shown in
Figure 1.10. A typical example is the energy-management algorithms in hybrid electric vehicles. Although engine, transmission, motor, brake, and battery controller
modules are individually commanded by their respective subsystems, a higher-level
control algorithm is required to determine the power flow to or from the battery
as well as the composition of the engine and motor torque provided to the wheels
(Powers and Nicastri 2000).
With additional performance requirements for modern vehicles, the relative
amount of supervisory-control software compared to the total amount is increasing.
The combination of this trend with the desire to maximize the computing resourceallocation requirements and modularity for cost-reduction purposes means that
supervisory control algorithms (e.g., energy management and telematics) are distributed to subsystem controllers and rely on vehicle communication networks and
external information inputs (Leen and Hefferman 2002; Navet et al. 2005). Modularizing the design of components in such a networked control system to make them
swappable (or “plug-n-play”) can reduce development time, calibration costs, and
maintenance and other costs (Çakmakcı and Ulsoy 2009; 2011).
PROBLEMS
1. From your own experience while driving and traveling in today’s vehicles, provide
examples of automotive control systems and discuss how they add value.
2. Table 1.1 summarizes developments in automotive electronics. Based on your
reading of articles, such as (Cook et al. 2007; Jones 2002; Powers and Nicastri 2000)
and others, provide additional items to those already listed. Extend the table to
speculate on technologies that may appear in the future. Please cite your sources.
REFERENCES
Ashley, S., 2001, “A Low-Pollution Engine Solution,” Scientific American, Vol. 284, June
2001, p. 90.
References
Bastow, D., G. Howard, and J. P. Whitehead, 2004, Car Suspension and Handling, SAE
International.
Bosch, R., 2009, Automotive Handbook, Bentley Publishers.
Çakmakcı, M., and A. G. Ulsoy, 2009, “Improving Component Swapping Modularity Using BiDirectional Communication in Networked Control Systems,” IEEE/ASME Transactions
on Mechatronics, Vol. 14, No. 3, June 2009, pp. 307–16.
Çakmakcı, M., and A. G. Ulsoy, 2011, “Modular Discrete Optimal MIMO Controller for a
VCT Engine,” IEEE Transactions on Control Technology, Vol. 19, No. 5, September 2011,
pp. 1168–77.
Cook, J. A., I. Kolmanovsky, D. McNamara, E. C. Nelson, and K. V. Prasad, 2007, “Control,
Computing and Communications: Technologies for the Twenty-First Century Model T,”
Proceedings of the IEEE, Vol. 95, No. 2, February 2007, pp. 334–54.
Davis, R., A. Burns, R. Bril, and J. Lukkien, 2007, “Controller Area Network (CAN) Schedulability Analysis: Refuted, Revisited and Revised,” Real-Time Systems, Vol. 35, April 2007,
pp. 239–72.
Dixon, J. C., 1992, Tyres, Suspension and Handling, Cambridge University Press.
Ellis, J., 1969, Vehicle Dynamics, Century Publishing.
Ford Motor Company, 1986, “Automotive Electronics in the Year 2000: A Ford Motor
Company Perspective,” Proceedings of the CONVERGENCE ’86 Conference, October
1986.
Gillespie, T. D., 1992, Fundamentals of Vehicle Dynamics, SAE International.
Grizzle, J. W., K. Dobbins, and J. Cook, 1991, “Individual Cylinder Air–Fuel Ratio Control
with a Single EGO Sensor,” IEEE Transactions on Vehicular Technology, Vol. 40, No. 1,
February 1991, pp. 280–6.
Hedrick, J. K., Tomizuka, M., and P. Varaiya, 1994, “Control Issues in Automated Highway
Systems,” IEEE Control Systems Magazine, December 1994.
Jones, W.D., 2002, “Building Safer Cars,” IEEE Spectrum, January 2002, pp. 82–5.
Jurgen, R., 1995, Automotive Electronics Handbook, McGraw-Hill, Inc.
Leen, G. and D. Heffernan, 2002, “Expanding Automotive Electronic Systems,” Computer,
Vol. 35, 2002, pp. 88–93.
Makovitz, R. and C. Temple, 2006, “FlexRay – A Communication Network for Automotive
Control Systems,” IEEE International Workshop on Factory Automation, 2006.
Mizutani, S., 1992, Car Electronics, Society of Automotive Engineers.
Motorola, 1999, “LIN Specification and Press Announcement,” SAE World Congress, Detroit,
1999.
National Highway Traffic Safety Administration (NHTSA), 1999, Traffic Safety Facts 1998: A
Compilation of Motor Vehicle Crash Data from the Fatality Analysis Reporting System and
the General Estimates System, National Center for Statistics and Analysis, U.S. Department
of Transportation, Washington, D.C., October 1999.
National Highway Traffic Safety Administration (NHTSA), 2009, Traffic Safety Facts 2008: A
Compilation of Motor Vehicle Crash Data from the Fatality Analysis Reporting System and
the General Estimates System, National Center for Statistics and Analysis, U.S. Department
of Transportation, Washington, D.C., October 2009.
Navet, N., Y. Song, F. Simonot-Lion and C. Wilwert, 2005, “Trends in Automotive Communication Systems, Proceedings of the IEEE, Vol. 93, 2005, pp. 1204–23.
Powers, W. F. and P. R. Nicastri, 2000, “Automotive Vehicle Control Challenges in the 21st
Century,” Control Engineering Practice, Vol. 8, No. 6, June 2000, pp. 605–18.
Rajamani, R., H. S. Tan, B. K. Law and W. B. Zhang, 2000, “Demonstration of Integrated
Longitudinal and Lateral Control for the Operation of Automated Vehicles in Platoons,”
IEEE Transactions on Control Systems Technology, Vol. 8, No. 4, July 2000, pp. 695–
708.
Ribbens, W. B., 2003, (6th edition), Understanding Automotive Electronics, ButterworthHeinemann.
Segel, L., 1990, Vehicle Dynamics, Course Notes, Department of Mechanical Engineering,
University of Michigan.
19
20
Introduction
Tindell, K. W., A. Burns and A. J. Wellings, 1995, “Calculating Controller-Area Network
(CAN) Message Response Times,” Control Engineering Practice, Vol. 3, No. 8, August
1995, pp. 1163–9.
Washino, S., 1988, Automobile Electronics, Gordon and Breach, New York.
Wong, J. Y., 2008, Theory of Ground Vehicles, Wiley.
2
Automotive Control-System Design Process
2.1 Introduction
Generally, “solving” the controller design problem means finding the proper mathematical representation of a control action that meets a set of desired performance
criteria. In reality, this is only one part of the solution (albeit an important part);
the control-systems development process also includes steps for selecting the correct
hardware – loaded with the proper software – for the controller module, which is
the real end-product of this process (Figure 2.1).
The control-system development process begins by first developing the highlevel system requirements, which are generally verbal and abstract and rarely point
to a recognizable control design problem such as those that traditional engineering
students would see in their control classes. The formal and technical requirements
documents can be described as “wish lists” regarding the overall system features
and performance. The result of the process is the controller module, which is to be
deployed in bulk to the end product. The purpose of studying the control-systems
development process is to provide a reliable, robust, and repeatable sequence of
actions to develop ECUs.
In recent years, computer-aided design and analysis tools (e.g., MATLAB and
Simulink) have improved the efficiency of design processes and increased the application of the model-based controller design and development process (Chrisofakis
et al. 2011; Mahapatra et al. 2008; Michaels et al. 2010; Powers and Nicastri 2000).
Figure 2.2 is a general outline of the model-based controller design and deployment process. The major components of this process (i.e., design, implementation,
and testing) are discussed in the next section. The process outline in Figure 2.2 is
based on development and testing portions that progress in parallel and continuously interact throughout the development cycle. This is, in fact, one of the most
important features of model-based design, which enables debugging and validation
of the current work while minimizing changes from the previous phase. Therefore,
as the control development evolves, so does the testing platform for its debugging
and validation.
Another important feature of model-based design and deployment is the ability
to reuse models and routines (e.g., testing and data analysis) from similar past and
ongoing projects. This enables engineers to compare continuously the performance
21
22
Automotive Control-System Design Process
Control
Development
Process
Controller Module
Requirement
Documents
Figure 2.1. Role of control engineering.
of their models and control algorithms. The self-documenting nature of these models
eases the workload of archiving the developed tools and results of the finished project
for future use.
2.2 Identifying the Control Requirements
The high-level requirements specified for a vehicle can be general and abstract, as
in the following examples:
Based on customer feedback, design a control system so that the vehicle is durable with
a pleasant driving feel and good fuel economy.
Vehicle shall meet 100,000-mile life requirements.
Today, part of a control engineer’s responsibility is to determine how overall vehicle
requirements affect the control problem. This may include understanding and quantifying how the vehicle-life requirements affect the way that the automatic transmission shifts gears, the engine controller operates the engine, and the battery-control
module manages battery power. The formal methods for identifying control-system
problems from vehicle requirements are performed by control-system engineers.
Systems Engineering for Control Development
The act of developing a control system from verbal requirements can be as simple
as an engineer reading the entire document and developing the matching control
design problems based on his or her experience and understanding of the document.
Although this still may be acceptable for small companies (i.e., five or fewer employees), today’s competitive environment – driven by the need for consistent quality and
reduced warranty costs – requires a more structured and traceable approach. Probably the most common feature of all of these approaches is deliberately reviewing and
documenting the results of the requirements-analysis, design, implementation, verification, and maintenance phases throughout product development. In fact, in many
Research
Papers/Reports
Models
From Previous
Projects
Test Data
From Previous
Projects
Algorithm
Development
Tools
Research
Requirements
Perf. Met.
Define
Control
Problem
Mathematical
Modeling
Develop
Candidate
Algorithm
Analytical Solution
and/or
Computer Simulations
Control Algorithm
Selection
Control
Algorithm
Develop
Algorithm
Code
Define
Realtime Execution
Procedure
Build
Real-time
Application
Design
Requirements
Perf. Met.
HW Module
Selection
Develop
Platform Based
Code
Implementation
Module Development
Open
Loop
Model-in
the Loop
Controller-in
the loop
Module-in
the loop
Testing
Figure 2.2. Model-based control design and deployment process.
Hardware in
the loop
Vehicle in
the loop
Controller
Module
23
24
Automotive Control-System Design Process
Requirements
Design
Figure 2.3. The Waterfall Model.
Implementation
Varification
Maintenance
industries, documents related to these steps are required for regulatory and legal
purposes to observe how design decisions were made. Actions taken by engineering
teams at each phase and how these phases interact can be different for each product
and company. The Waterfall and V-Diagram Models are two common approaches
used in industry.
The Waterfall Model, illustrated in Figure 2.3, is a sequential software development process in which progress is seen as flowing steadily downward (like a waterfall)
through the phases of requirements analysis, design, implementation, verification,
and maintenance (Jacobson, Booch, and Rumbaugh 1999). Although this type of
development model has been a widely used method for analyzing complex requirements since the early days of systems engineering, its sequential nature (i.e., no
feedback) throughout the design problem is considered a weakness.
The V-Diagram Model is another development-process model that can be
viewed as an extension of the Waterfall Model. Instead of moving down linearly,
the process steps are redirected upward after the coding phase to form the typical V shape (Figure 2.4). The V-Diagram Model demonstrates the relationships
Vehicle
Vehicle
Requirements
Management
Acceptance
Testing
System
Analysis
Subsystem
Compliance
Testing
Architecture
Modelling
System
Testing
High Level
Design
Subsystem
Integration
Testing
Detailed
Design
Module
Build
Subsystem
Testing
Unit
Testing
Module
Module
DESIGN
Figure 2.4. System engineering V-Diagram.
VALIDATION
2.2 Identifying the Control Requirements
25
Plate
Engine Subsystem
Spring
Servomotor
Throttle Module
Housing
Timing Belt
Timing Gear
Camshaft
Housing
Valve Timing
Module
Vehicle Powertrain
(System)
Electronic
Control
Unit
Figure 2.5. Vehicle-system decomposition.
between each phase of the development life cycle and its associated phase of testing
(Jacobson, Booch, and Rumbaugh 1999; Stahl, Voelter, and Czarnecki 2006).
Figure 2.5 is a typical decomposition of a vehicle’s powertrain system parts from
the vehicle level to the module level for the engine subsystem. A vehicle powertrain
system is composed of many subsystems, including the engine, transmission, and
brakes, which in turn are composed of smaller functional modules. For example,
an internal-combustion engine has a throttle module, valve-timing module, and oilpump module. These modules can be decomposed further into parts that typically
are supplied as off-the-shelf products or manufactured as single pieces. In modern
automotive systems, almost every subsystem has an electronic control module to
control and monitor the functional operations that communicate with the subsystem
actuators, sensors, and other modules.
Developing algorithms that successfully will perform control operations can
be challenging when considering the complexity of the performance requirements
in today’s automotive systems. The process of obtaining control-system requirements from higher-level vehicle requirements is conducted in sequential steps
that are called requirements cascade studies (Philips 2005). Figure 2.6 illustrates
the path for developing control-algorithm solutions from a generic set of vehicle
requirements.
In Figure 2.7, the method described in Figure 2.5 is shown for a specific example. The example shows how the vehicle’s 100,000-mile requirement affects specific
features (i.e., control problems) for a particular vehicle application. The effect of
this and other requirements defines the feature control problem to be solved. The
solutions obtained from all of the features represent the control algorithm for a
vehicle.
26
Automotive Control-System Design Process
Vehicle Level Requirements
(Attribute, Regulatory,
Corporate, Etc.)
Implementation of
Control Solutions
Sub-System Level
Control Problems
Module
Specifications
Px
Module
Specifications
P1
P1.1
Subsystem Level
Requirements
P1.2
Module
Specifications
P1.2.1
Module
Specifications
P2
Figure 2.6. Requirement mapping and decomposition.
Algorithm Development
In the algorithm-design step (Figure 2.8), the control design problem is first formulated based on the given performance requirements and developed mathematical
formulation. There is more than one control design approach to provide a solution
for the control problem. By using analytical methods and/or computer simulations,
the best alternative among the candidate algorithms is selected. If the control problem is similar to a previous application, development teams often prefer to start with
Control Problems
Vehicle Level Requirement
100K Miles Powertrain
Warranty
Battery SOC
Calculation
Implementation
Engine
Control
Module
Battery Observer
Model
Torque
Request
Table
Engine Power
Request Calculation
Transmission
Subsystem Level Requirement:
The control system should modify
powertrain operating points to
maintain Battery State of Charge
(SOC) within manufacturer
recommended limits so that
durability is achieved
Battery Power
Request
Determination
Control Module
Engine Torque
Monitor
Electric Motor
Motor Current
Control
Control Module
Motor Limits
Calculate Actual
Battery Power
Battery
Control Module
Battery Cooling
Battery SOC
Reported
Cell Balance
Figure 2.7. Requirement mapping and decomposition (example).
2.2 Identifying the Control Requirements
27
Test Data
From Previous
Projects
Models
From Previous
Projects
Research
Papers/Reports
Algorithm
Development
Tools
Research (Internal/External)
Requirements
Perf. Met.
Define
Control
Problem
Mathematical
Modeling
Develop
Candidate
Algorithm
Analytical Solution
and/or
Computer Simulations
Control Algorithm
Selection
Algorithm
Implementation
Design
Figure 2.8. Design-phase step.
an existing control algorithm and improve the solution by building on the existing
(and proven) solution.
Implementation
During the algorithm-implementation phase (Figure 2.9), the objective is to develop
a real-time application to be executed in the control module using the desired control algorithm. By using an automatic code-generation tool, the target computerlanguage equivalent (e.g., the language C) of the control algorithm model is
generated by a computer tool. This process also is known as autocoding. Once
the algorithm block diagram model has been autocoded, it is combined with the
hardware-platform-dependent code (e.g., device drivers). When the real-time application is generated, the control algorithm is ready to be downloaded and executed
in the hardware module.
An alternative method to autocoding is handcoding – that is, developing the
equivalent of the control-algorithm model by manually writing the computer code.
Many developers argued in the early days of autocoding that a computer-generated
code would be unnecessarily long and inefficient due to the generalized codegeneration algorithms used. However, recent studies (e.g., Hodge, Ye, and Stuart
2004) show that with the appropriate initial setup, code generation can be as effective as handwritten code while requiring a fraction of the development time. The use
Control
Algorithm
Requirements
Perf. Met.
Develop
Algorithm
Code
HW Module
Selection
Define
Realtime Execution
Procedure
Develop
Platform Based
Code
Figure 2.9. Algorithm-implementation step.
Build
Real-time
Application
Implementation
HW Module
Release
28
Automotive Control-System Design Process
of autocoding also minimizes the effects of human error during initial development
and successive modifications.
When developing the executable code, real-time constraints of the target hardware (i.e., the controller module) also should be considered. Software implementation of the algorithm should be matched to available computing resources. If there
are overruns during the real-time execution, the algorithm should be simplified
or new target hardware should be selected. In today’s modern vehicles, controller
modules also communicate with other controllers via communication networks. The
effects of the loss of or limited communication with one or more contacts should be
investigated and the necessary modifications implemented.
Testing and Validation
Testing in the model-based control development process starts as early as in the
algorithm-development step. By testing an algorithm’s open-loop (Figure 2.10a),
developers can feed in simple test vectors and analyze the test output for expected
functionality. These simple algorithms also can be tested against the simpler conceptual vehicle models, which are available in earlier stages of the program (Figure
2.10b). These models then are fortified with improvements based on component- and
vehicle-testing data, which makes them suitable for more complex testing procedures
such as module, component, and vehicle loop-type testing.
Hardware-in-the-Loop Systems Overview
During the controller module development process, combinations of hardware and
simulated elements are used to evaluate system performance.
When a developed algorithm or mathematical representation of a plant model must be verified, the quickest way that
requires the least effort is to perform open-loop testing. In an open-loop-testing
configuration (Figure 2.10a), algorithms or models are provided with test-vector
inputs, and the outputs are compared with a set of expected results. The platform for
open-loop testing can be a simulation environment (e.g., MATLAB or Simulink)
or a test-bench when the controller prototype or production module is available.
Perhaps the most common form of open-loop testing is vehicle road-testing. A distinctive challenge associated with open-loop testing is the generation of input vectors
suitable for the test purposes. This is particularly difficult when the subject being
tested is part of a larger algorithm or plant dynamics and its input/output (I/O)
relationship is counterintuitive, which requires some internal states to be “staged”
for testing the feature in question. Moreover, in most types of robustness testing,
output-based generation of the input is required.
The type of testing in which the input to the test subject is generated based on
the output from previous output is called closed-loop testing. These tests require
the actual hardware – or its emulation – to react properly to the test output. In the
simulation environment, closed-loop testing can be performed by using algorithmin-the-loop (Figure 2.10b) and/or controller-in-the-loop testing (Figure 2.10c). The
difference between the models of an algorithm and a controller is that the latter
includes the algorithm model as well as the I/O processing schemes and timing
OPEN-LOOP VERSUS CLOSED-LOOP TESTING.
2.2 Identifying the Control Requirements
Figure 2.10. Different types of testing.
constraints of the targeted implementation. Generally, algorithm models are available in the development cycle as early as in the conceptual-design phase, whereas
controller models begin to take shape toward the end of the implementation phase.
Closed-loop testing configurations in which hardware versions of one or more components exist in
the test setup are referred to as hardware-in-the-loop (HIL) testing configurations.
HARDWARE-IN-THE-LOOP TESTING AND COMMON CONFIGURATIONS.
29
30
Automotive Control-System Design Process
Engine
HIL Simulator
Dyno
or
Plant Model
Desktop Computer
Engine Controller Module
(a) Component-in-the-Loop Testing
(b) Vehicle-in-the-Loop Testing
(c) Vehicle Road Testing
Figure 2.11. Different types of testing (continued).
Primary components in an HIL system are the HIL simulator, host computer, and
hardware being tested. An HIL simulator is the real-time–oriented test computer in
which the emulation program for driving the hardware is run. The host computer is
the supporting computer that communicates with the HIL simulator during the test
through an HIL runtime graphic user interface (GUI), and it collects data related
to testing. For some tests, secondary data-acquisition hardware such as CAN cards
and calibration tools is included in the test setup for debugging and verification purposes. When physical hardware is included in the HIL setup, sensors and actuators
to support its interfacing to the emulation also are included (Isermann, Schaffnit,
and Sinsel 1999; Kendall and Jones 1999; Powell, Bailey, and Cikanek 1998).
The most common HIL for testing is the rapid-prototyping controller or
the production-level controller module. This is the configuration in which the
controller is driven by an HIL simulator that emulates the rest of the system
(Figure 2.10d). More elaborate configurations of these HIL systems exist, including
multiple-controllers-in-the-loop, powertrain-component-in-the-loop (Figure 2.11a),
and vehicle-in-the-loop (Figure 2.11b) systems. As observed by many developers
(e.g., Hatipoglu and Malik 1999; Kendall and Jones 1999), it generally is expected
that as the complexity of the HIL increases, the accuracy of the testing also increases,
References
31
Vehicle
Testing
Requirements
Management
Acceptance
Testing
System
Analysis
Vehicle in
the loop
Compliance
Testing
System
Testing
Integration
Testing
High Level
Design
Detailed
Design
Module
Build
Controller Model - in
the loop
Algorithm - in
the Loop
Subsystem
Testing
Unit
Testing
Open
Loop
Module
Testing
DESIGN
Controller HW in
the loop
Subsystem
Level Testing
Architecture
Modelling
Accuracy
VALIDATION
Figure 2.12. Accuracy of testing on different levels of integration.
as shown in Figure 2.12. It is shown in this figure that HIL testing fills the void
between testing based on simulation models (available early) and actual vehicle
testing (available later) and keeps the iterative nature of the V-Diagram Model
development process intact.
PROBLEMS
1. Consider the Waterfall Model diagram in Figure 2.3. How would you modify the
process described here to take advantage of some of the features of the V-Diagram
Model process described in Figure 2.4?
2. Consider the high-level requirement: “Vehicle shall meet 100,000-mile life
requirements.” Discuss with examples what this could mean to the design of a
braking system for a family sedan vehicle that will be driven primarily in city-traffic
conditions.
3. Using your preferred modeling and simulation tool (e.g., MATLAB, Simulink,
or SciLab):
(a) Simulate the result of the block diagram.
(b) Generate the c-code equivalent of the block diagram.
(c) Discuss possible advantages and disadvantages of manually developing
equivalent computer code.
4. Consider the component-testing setup in Figure 2.11a for engine testing. For which
other automotive components can you develop similar testing setups? Describe the
test equipment and operation of this facility using a diagram.
REFERENCES
Chrisofakis, E., et al., 2011, Simulation-Based Development of Automotive Control Software with Modelica, in Proceedings of the 8th International Modelica Conference, Dresden,
Germany.
32
Automotive Control-System Design Process
Hatipoglu, C., and A. Malik, 1999, “Simulation-Based ABS Algorithm Development,” SAE
International.
Hodge, G., J. Ye, and W. Stuart, 2004, “Multi-Target Modelling for Embedded Software
Development for Automotive Applications,” 2004 SAE World Congress, Detroit, MI.
Isermann, R., J. Schaffnit, and S. Sinsel, 1999, “Hardware-in-the-Loop Simulation for the
Design and Testing of Engine-Control Systems,” Control Engineering Practice, Vol. 7,
1999, pp. 643–53.
Jacobson, I., G. Booch, and J. Rumbaugh, 1999, The Unified Software Development Process,
Pearson Education India, 1999.
Kendall, I. R., and R. P. Jones, 1999, “An Investigation into the Use of Hardware-in-theLoop Simulation Testing for Automotive Electronic Control Systems,” Control Engineering
Practice, Vol. 7, No. 11, November 1999, pp. 1343–56.
Mahapatra, S., et al., 2008, “Model-Based Design for Hybrid Electric Vehicle Systems,” SAE
Paper 2008010085.
Michaels, L., et al., 2010, “Model-Based Systems Engineering and Control System Development via Virtual Hardware-in-the-Loop Simulation,” SAE Technical Paper No. 01-2325.
Philips, A. M., 2005, “Technical Challenges of Hybrids,” SAE Technical Symposium on
Engineering Propulsion, 2005.
Powell, B. K., K. E. Bailey, and S. R. Cikanek, 1998, “Dynamic Modeling and Control of
Hybrid Electric Vehicle Powertrain Systems,” IEEE Control Systems Magazine, Vol. 18,
October 1998, pp. 17–33.
Powers, W. F., and P. R. Nicastri, 2000, “Automotive Vehicle Control Challenges in the 21st
Century,” Control Engineering Practice, Vol. 8, June 2000, pp. 605–18.
Stahl, T., M. Voelter, and K. Czarnecki, 2006, Model-Driven Software Development: Technology, Engineering, Management, John Wiley & Sons, 2006.
3
Review of Engine Modeling
For obvious reasons, engine-control systems were among the first developed for vehicles: The engine is not only the most crucial component for automobile performance;
its emission performance also significantly affects the environment. As discussed in
Chapter 1, engine-control systems may include fuel-injection control (i.e., air–fuel
ratio control), ignition or spark-timing control, antiknock-control systems, idle-speed
control, EGR control, and transmission control. The goal of engine-control systems
is to ensure that the engine operates at near-optimal conditions at all times in terms
of drivability, fuel economy, and emissions.
Overall, engine-control systems are complex due to the nonlinearity of many of
the components and the interactions among the several related control functions:
air–fuel ratio control, idle-speed control, knock (or spark-timing) control, EGR
control, and transmission control. In this chapter, each major phase of the operation
of a spark-ignited gasoline engine and its dynamic modeling is discussed from the
control perspective. Subsequent chapters consider specific engine-control problems
(e.g., air–fuel ratio control, spark timing, EGR, and idle-speed control), as well as
control problems associated with hybrid and fuel-cell vehicles.
3.1 Engine Operations
As shown in the conceptual block diagram in Figure 3.1, engine operations can be
divided into several key phases. This discussion is specific to a four-stroke, spark
ignition (SI), Otto gasoline engine (Figure 3.2). During each crankshaft revolution,
there are two strokes of the piston and a total of four strokes, as follows (see
Figure 3.2) (Heywood 1988; Stone 1994):
(a) Induction Stroke. The intake valve is opened and the piston travels down the
cylinder and draws in a charge of air (or a charge of premixed fuel and air).
(b) Compression Stroke. Both valves are closed and the piston travels up the cylinder. As the piston approaches TDC, SI occurs.
(c) Expansion (or Power) Stroke. Combustion propagates throughout the charge,
raising the pressure and temperature, thereby forcing the piston down. At the
end of the power stroke, the exhaust valve opens and the irreversible expansion
of the exhaust gases is termed blow-down.
33
34
Review of Engine Modeling
FUEL
COMMAND
EGR
COMMAND
EGR
SPARK
COMMAND
FUEL
SYS.
ENGINE
TORQUE
˙e
m
CHARGE MASS RATE
θc
THROTTLE
BODY
˙a
m
MANIFOLD
PLENUM
ENGINE
PUMP
˙f
m
ENGINE
POWER
TORQUE LOAD
INERTIA
SPEED
MANIFOLD PRESSURE
Figure 3.1. Schematic block diagram of nonlinear engine-system elements (Cook and Powell
1988).
(d) Exhaust Stroke. The exhaust valve remains open and as the piston travels up the
cylinder. The remaining gases are expelled. At the end of the exhaust stroke,
the exhaust valve closes. Some exhaust gases remain and dilute the next charge.
Because this cycle is completed only once every two crankshaft revolutions
(i.e., 720 degrees), the valve (and fuel-injection) gear must be driven (usually by a
camshaft) at half the engine speed. In a single-cylinder engine, power is produced
only during the power stroke, which is only one quarter of the cycle. During other
parts of the cycle, crankshaft rotation is maintained by power stored in a mechanical
flywheel. In a multicylinder engine, the power strokes are staggered so that power is
Fuel injector/sparking plug
Inlet valve
Air
(+ fuel)
Exhaust valve
Products
Fuel
Piston
Figure 3.2. A four-stroke engine.
Connecting-rod
Crank case
Crankshaft
Sump
3.1 Engine Operations
Figure 3.3. Typical pressure-volume diagram for a four-stroke SI engine.
produced during a larger fraction of the cycle than for a single-cylinder engine. For
satisfactory SI and flame propagation, the air–fuel mixture must be stoichiometric
(i.e., chemically balanced). This is important for emissions, as discussed in Chapter 5.
Spark timing is important for performance, emissions, and prevention of engine
knock (i.e., spontaneous self-ignition). Figure 3.3 is a typical pressure-volume diagram for a four-stroke engine. TDC is at B and D, and bottom dead center (BDC)
is at A and C. The stroke from A to B is the compression stroke, B to C is the
power stroke, C to D is the exhaust stroke, and D to A is the intake stroke. The
valve openings and closures are marked 1 to 4, the spark occurs at 5, and the flame
extinguishes at 6.
The following section is a brief description of each phase of engine operations,
as shown in the block diagram in Figure 3.1.
Throttle Body
The mass airflow into the intake manifold is adjusted by the driver setting of the
throttle input via an accelerator-pedal command. The throttle plate acts as an airflow
control valve controlled by the accelerator pedal. The mass-airflow dependency on
the throttle angle is nonlinear; however, for small changes about an equilibrium
condition, the mass airflow rate is proportional to the change in throttle angle. The
engine requires a relatively rich mixture and increased airflow to start when it is cold.
This required enrichment may be provided by a choke valve, leading to different
throttle dynamics when the flow is choked or unchoked.
Intake Manifold
Change in pressure in the intake manifold is proportional to the mass flow in (i.e., due
to the throttle command) minus the mass flow out (i.e., from the engine pumping).
Models are derived based on conservation of mass and the ideal gas law.
35
36
Review of Engine Modeling
EGR
The EGR command adjusts the opening of an EGR valve, which directs a portion
of the exhaust gas into the charge to the cylinder. This can affect the engine performance, emissions, and fuel consumption. The primary goal of EGR is to reduce
NOx emissions.
Fuel System
The fuel system controls the amount of fuel injected into the intake manifold (or
cylinders). The main goal is to ensure that the ratio of the mass of air to the mass of
fuel is regulated at the desired level. Under normal cruising conditions, the desired
air–fuel ratio is stoichiometric (i.e., 14.7). For air–fuel ratio control, the fuel-flow
rate is proportional to the airflow rate. Therefore, it is usually necessary to measure
air mass flow rate. Various fuel-injection system designs exist. Typically, for best
control, fuel is injected into each cylinder near each intake valve. However, fuel also
can be injected by a single injector at the intake manifold.
Engine Pumping
The engine behaves like a pump to produce the airflow, EGR flow, and fuel flow out
of the intake manifold and into the cylinders. An important result of this pumping
mechanism is the induction-to-power (IP) delay (or lag), which can be treated as a
pure delay corresponding to 360 degrees of crankshaft rotation. In other words, from
the I/O perspective, the control action (e.g., change in throttle angle) takes effect
after a time delay, which is engine-speed dependent.
Spark Command
The spark command ignites the air–fuel mixture in the cylinder to produce torque.
The produced engine torque depends not only on the air–fuel ratio but also is
affected by spark timing/advance. Spark advance is the time before TDC when a
spark is initiated, and it is usually expressed in degrees of crankshaft rotation relative
to TDC. The so-called MBT is used to maximize torque while maintaining a margin
of safety to prevent knock (Figure 3.4).
Engine Power
Engine torque due to combustion leads to engine torque influenced by delayed
pressure, delayed fuel, spark advance, and engine friction.
Engine Inertia
The powertrain rotational inertia and load torques (i.e., to drive the vehicle and
accessories) must be overcome by engine torque (i.e., Newton’s second law).
3.2 Engine Control Loops
Figure 3.4. Spark-timing effect on engine performance (Ribbens 2003).
3.2 Engine Control Loops
In this section, each control loop associated with the engine operation is described.
Air–Fuel Ratio Control
As discussed previously, there are various performance metrics: emissions (i.e., CO,
NOx , and HC), fuel consumption (i.e., BSFC), and output power (i.e., torque).
Most (but not all) are optimized, in conjunction with a catalytic converter, at or
near stoichiometry (i.e., air–fuel ratio = 14.7). A key actuation mechanism for the
air–fuel ratio function is the fuel injector(s). The throttle-angle setting from the
driver determines the mass airflow rate, measured by a MAS, and the fuel flow
is proportional to this rate. The MAP also is measured. The critical sensor for
closed-loop air–fuel ratio control is the EGO sensor, which detects oxygen in the
exhaust.
EGR
The percentage of exhaust gas in the charge is controlled by the EGR valve based on
readings from the MAP sensor and engine temperature and speed. Higher percentages of EGR lower NOx ; however, other performance metrics (e.g., BSFC and HC)
deteriorate with higher EGR. The EGR and air–fuel-ratio loops are highly coupled.
Spark Timing
The combustion in the cylinder is initiated by the spark-plug firing, typically a few
degrees of crank angle before TDC. The so-called MBT is used to maximize torque
while also maintaining a margin of safety to prevent engine knock. Advancing spark
timing can increase torque and reduce fuel consumption. However, this is usually
37
38
Review of Engine Modeling
associated with increased emissions and the danger of engine knocks occurring. To
achieve good spark control, the crankshaft angle must be measured or estimated
accurately. Spark timing interacts with the idle-speed control and air–fuel-ratio control loops.
Idle-Speed Control
The goal is to measure and control engine speed at idle by adjusting airflow rate using
the throttle or an idle-speed control valve (which provides better precision compared
to the throttle). Maintaining consistent engine speed at idling despite load variations
is important for perceived vehicle quality and to ensure low emissions and improved
fuel economy. This control function requires measurement of the crankshaft angular
position and engine speed.
Transmission Control
The main purpose of a transmission is to match the engine and vehicle speeds so
that the engine can work in a more efficient region. Therefore, the gear selection
for the transmission is said to be a mechanism for “engine control.” Usually, a
“shift map” with two independent variables is constructed, which then is used to
determine up-shift and down-shift points. To implement the shift map, vehiclespeed and throttle-angle measurements are necessary. In addition to determining
the gear position, it is important to ensure that shifting from one gear ratio to the
next is executed smoothly. This entails precise coordination of the friction torques
of various clutches.
There are several engine-operation modes that influence how each control loop
operates, including the following:
(a) Engine Crank (Start). The primary goal is reliable engine start-up; less emphasis
is placed on fuel economy and emissions, and EGR is not used. Typically, the
engine speed is low, the air–fuel ratio is low, and the spark is retarded.
(b) Engine Warm-Up. The primary goal is a rapid and smooth engine warm-up. Typically, the EGR is off, the air–fuel ratio is low, and fuel economy and emissions
are not primary concerns.
(c) Open-Loop Control. The primary goal is to control the engine until the EGO
sensor reaches the correct operating temperature and produces reliable output.
(d) Closed-Loop Control. The primary goal is tight control of performance, fuel
economy, and emissions under closed-loop control using the EGO sensor.
(e) Hard Acceleration. The primary goal is high performance, with less emphasis on
fuel economy and emissions. The air–fuel ratio is rich, EGR is off, and EGO is
not in the loop.
(f) Deceleration and Idling. The primary goal is reduced fuel consumption and
emissions. The air–fuel ratio is lean, engine speed is kept low and constant, and
EGR is on.
In general, a dynamic model of the engine is a complex, nonlinear, dynamic system.
There are various models suitable for simulations of a subset of the phenomena
mentioned here. One example is discussed in the following subsection.
3.2 Engine Control Loops
39
Figure 3.5. Engine example included in MATLAB/Simulink.
A nonlinear, three-state engine
dynamics model (Crossley and Cook 1991) for a four-stroke SI engine is included
as a demonstration module in MATLAB. To start the demonstration, type
“sldemo_engine” at the MATLAB prompt:
EXAMPLE 3.1: SIMULINK/MATLAB ENGINE MODEL.
≫ sldemo engine
To run the simulation, choose Start from the Simulation menu. Figure 3.5 shows
the model in Simulink block-diagram form.
Click on the different blocks in the Simulink block diagram to see the underlying structure of each block. For example, the throttle & manifold block generates the mass airflow rate from throttle-angle, manifold-pressure, atmosphericpressure, and engine-speed inputs. This block is composed of complex throttle
and intake manifold blocks, as shown here:
40
Review of Engine Modeling
The throttle block switches based on different cases (i.e., choked versus
unchoked flow) to produce the airflow in response to the throttle-angle command. For lower manifold pressures, the flow through the throttle body is sonic
and the outflow is a function of only the throttle angle, as follows:
mdot = g(Pm)*(2.821–0.05231*TA+0.10299*TA∧ 2–0.00063*TA∧ 3)
where:
TA: throttle angle (degrees)
Pm: manifold pressure (bar)
g(Pm) = 1 for Pm <= P(amb), otherwise
g(Pm) = (2/P(amb))*sqrt(Pm*P(amb)–Pm∧ 2)
mdot: mass flow rate of air (g/s)
This is shown in the following Simulink block diagram:
The intake manifold block includes manifold-pressure dynamics and calculates the manifold pressure and produces the mass airflow rate to the cylinder.
The difference in the inflow and outflow mass rates is multiplied by a gain and
integrated to obtain the manifold pressure based on the ideal gas law and homogeneous temperatures and pressures in the air–fuel mixture. It is assumed that
there is no EGR, but it can be added easily. The mass flow rate of the air–fuel
mixture pumped into the cylinders is given by an empirically derived equation
as a function of the manifold pressure and engine speed. This is shown in the
following block diagram:
3.2 Engine Control Loops
where:
R: specific gas constant
T: temperature (K)
Vm: manifold volume (m∧ 3)
gamma: ratio of specific heats (1.4)
mdot: mass flow rate of air (g/s)
Pdot_m: rate of change of manifold pressure, Pm
N: engine speed (rad/s)
Pm: manifold pressure (bar)
The combustion block generates the engine torque using an empirical curve
fit to the input air charge, fuel flow, spark advance, and engine speed, as follows:
The vehicle dynamics block is a simple rotational inertia (J) with load and
engine-torque inputs, as follows:
There are additional blocks for intake, compression, valve timing, and drag
torque in this Simulink model. The intake-to-power-stroke delay is given by
Delay = π /N, where N is the engine speed in rad/s. This assumes that the delay
is one quarter of the total engine cycle (i.e., intake, compression, power and
exhaust strokes) for a four-stroke engine.
The output of this simulation model is a plot of the engine speed (revolutions
per minute [RPM]) versus time (seconds), for a simulation time of 10 seconds,
and a plot of the throttle input (degrees) and load torques (N · m) versus time
41
42
Review of Engine Modeling
as shown here. The first plot shows the system inputs, the throttle angle (lower
curve), and the load torque versus time in seconds:
The second plot shows the system response – that is, the engine speed in
RPM versus time in seconds.
This Simulink engine model can be used as a standalone simulation model
or combined as part of a larger powertrain simulation model. For example, this
model could be used as part of an integrated vehicle and powertrain simulation
for the development of a traction-control system.
3.3 Control-Oriented Engine Modeling
Many different models for engine control are reported in the literature (e.g., Cassidy
et al. 1980; Cho and Hedrick 1989; Dobner 1983; Hendricks 1990; Kamei et al. 1987;
and Powell 1979). In this section, the dynamic modeling of an engine (see Figure
3.1) is discussed for purposes of control-system design and evaluation. First, a simple
linearized model of engine dynamics is derived and presented. Then, a more complex
nonlinear engine model, suitable for engine diagnostics, is provided.
Linearized Engine Dynamics Model
First, we model the throttle following the block diagram shown in Figure 3.1. A
linearized throttle model, which represents the mass airflow rate as proportional to
change in throttle angle, is given by:
ṁa = Kθ θ
where θ is the throttle angle, ṁa is the air mass inflow rate, Kθ is the linearized
airflow rate sensitivity, and θ represents the change in throttle angle from the
steady condition about which the system is linearized. In general, the symbol is
used to represent incremental variables that give changes about the steady values
for which linearization was performed.
3.3 Control-Oriented Engine Modeling
43
SPARK
COMMAND
FUEL
COMMAND
∆F
∆δ
Gδ
Gf
∆θ
Kθ
AIR BYPASS
COMMAND
+
−
∆P
τpKP
τpS + 1
GP
+
+
MANIFOLD FILLING
e−2ST
DISTURBANCE
+
+
−
+
240 ° IP LAG
KRτR
τRS + 1
ROTATIONAL
DYNAMICS
KN
PUMPING
FEEDBACK
Figure 3.6. Block diagram of the linearized engine (Cook and Powell 1988).
The manifold-filling dynamics is based on the Ideal Gas Law and expresses the
change in intake-manifold pressure due to mass airflow into the manifold minus
mass airflow out of the manifold:
RT
Ṗ =
(ṁa − Ṁ)
(3.1)
V
where P is the manifold pressure, R is the ideal-gas constant, T is the air temperature,
V is the manifold volume, ṁa is the mass flow rate of air into the manifold (see
Eq. (3.1)), and Ṁ is the mass flow of air out of the manifold due to engine pumping.
After linearization, this equation becomes:
∂ ṁa
∂ Ṁ
(3.2)
−
Ṗ = KP
P + KP Kθ θ − KP KN N
∂P
∂P
Ṁ
where N is the engine speed, KP = ṁ Ṗ−Ṁ , and KN = ∂∂N
. Defining τ p =
a
∂ ṁ−1
results in the transfer function shown in Figure 3.6. Next, define AM as
∂ Ṁ
a
KP
∂P
− ∂P
an estimate of the mass flow rate, with Ṁ obtained from a speed density air-sensing
system:
AM = cPN
(3.3)
where c is a proportionality constant. Linearizing Eq. (3.3) yields:
AM = c(PoN + NoP)
(3.4)
Figure 3.7 is an engine-induction map that enables calculation of the various coefficients in Eqs. (3.1), (3.2), and (3.4) at various operating conditions.
For control to a specific air–fuel ratio, the engine fuel-flow rate is proportional
to the airflow rate. The amount of fuel injected in any one event is proportional
to the airflow rate divided by engine speed (which is assumed proportional to the
air charge). Two possible injection-timing strategies are sequential and bank to
bank. Sequential injection-timing meters fuel individually to each cylinder during
the appropriate portion of the engine cycle (e.g., immediately before the intake
valve opens in each cylinder). Thus, each cylinder receives a fuel charge delayed by
∆N
44
Review of Engine Modeling
62.0
52.0
MASS FLOW RATE-LB/HR
1200
72.0
4000 RPM
3500
LE
3000
G
1000
800
TT
LE
P
TE
LA
AN
2500
42.0
2000
O
R
TH
600
1500
32.0
1000
400
22.0
500
200
12.0
0
0
2
4
6
8
10
12
14
MANIFOLD PRESSURE-PSI
Figure 3.7. Engine induction map (Cook and Powell 1988).
the same amount from the time of injection. In bank-to-bank fuel injection (e.g.,
for a six-cylinder engine), the amount of fuel to be injected is calculated once per
engine revolution based on the speed-density airflow. For simplicity and economy,
the injectors are “slaved” in groups of three and fired alternately at 360-degree
crank-angle increments delayed by 240 degrees from the fuel-metering calculation.
An estimate of the air, fuel, and EGR mass flow rates out of the manifold and
into the cylinders can be obtained by treating the engine as a pump. A pure transport
delay, which is referred to as the IP stroke lag, is shown in Figure 3.6. Mass-flow-rate
samples for each cylinder eventually produce torque via the combustion process.
This delay is 180 to 360 degrees of crankshaft rotation. The sampling in the system is
engine-speed–based rather than time-based. Thus, the IP lag can be represented by
a transport delay of two sampling periods, T. For a six-cylinder engine, the control
sampling time T = 720/6 = 120 degrees.
Torque is produced from the combustion process and depends on several variables in the system. For the linear model, the following functional dependence is
assumed for the engine brake or output torque:
Te = F (Md , Fd , δ, N)
(3.5)
where Md is the mass charge delayed by the IP lag, Fd is the fuel delayed by the IP
lag, and δ is the ignition timing in degrees before TDC. The mass charge is a function
of manifold pressure and engine speed; therefore, a linearized relationship can be
given as:
Te = G p Pd + G f Fd + Gδ δ + FN N
(3.6)
where G p is the influence of delayed pressure on torque, G f is the influence of
delayed fuel on torque, Gδ is the influence of spark advance on torque, and FN is the
engine friction. The first three terms define what is usually called the combustion
torque, Tc .
3.3 Control-Oriented Engine Modeling
45
Table 3.1. Six-cylinder-engine model
parameters at N = 600 RPM (Cook and
Powell 1988)
Parameter
Unit
Value
Kθ
tP
KP
GP
Gδ
T
tR
KR
KN
Gf
(lb/hr)/deg
sec
lbf-h/(lbm-in2 -sec)
ft-lbf/psi
ft-lbf/deg
sec
sec
rpm/(ft-lbf-sec)
lbm/(rpm-hr)
ft-lbf/lbm
20.000
0.210
0.776
13.370
10.000
0.033
3.980
67.200
0.080
36.600
A powertrain rotational dynamics model is needed to complete the model in
Figure 3.2. Applying Newton’s second law to the crankshaft rotation gives:
Je Ṅ =
30
30
T − TL
π e
π
(3.7)
where Je is the engine rotational inertia and TL is the external (i.e., disturbance
or load) torque that the engine must overcome. For a vehicle with an automatic
transmission, TL consists of the load applied by the torque converter, and it loads
from the driven accessories (e.g., air-conditioning compressor). Linearizing Eq. (3.7)
and substituting from Eq. (3.6) gives:
30
30 2N
(G P + G f Fd + Gδ δ − Td ) (3.8)
−
F
Ṅ +
N N =
2
Je π Ki
Je π P
where the transfer function shown in Figure 3.6 is obtained by defining the time
− FN ), KR = J30π , and Td represents the incremental disconstant τR = 1/ J30π ( 2N
Ki2
e
e
turbance torque. The model parameters must be determined by testing, and typical
values of the model parameters are listed in Table 3.1 for a six-cylinder engine at
N = 600 RPM. Figure 3.8 shows results from validation studies for the linearized
engine model given in Eqs. (3.1), (3.2), (3.4), and (3.8).
Consider the linearized
engine model given in Eqs. (3.1)–(3.8) and express these equations in standardstate equation form:
EXAMPLE 3.2: LINEARIZED ENGINE DYNAMICS MODEL.
ẋ = Ax + Bu u + bv v
y = Cx + Du
(3.9)
(3.10)
where the states of the model are the mass airflow, the change in manifold
pressure, and the change in engine speed:
⎧
⎧
⎫
⎫
⎨ θ ⎬
⎨ ma ⎬
x = P
(3.11)
u = δ
and v = Td
⎩
⎩
⎭
⎭
Fd
N
46
Review of Engine Modeling
10.2
THROTTLE
ANGLE
(DEG)
7
20
MANIFOLD
PRESSURE
(″Hg)
Vehicle
15
Model
11
Model
1225
Vehicle
SPEED
(RPM)
750
0
1
2
0
1
2
TIME (SEC)
Figure 3.8. Sample transient response for validation (Cook and Powell 1988).
The inputs are the change in throttle angle, the change in spark timing, the
change in delayed fuel, and the disturbance torque. The coefficient matrices for
the state equations in Eq. (3.9) are:
⎡
⎤
⎤
⎡
⎫
⎧
0
0
Kθ
0
0
0
⎨ 0 ⎬
⎢
⎥
⎥
⎢
⎢
⎥
0
0 ⎥
A=⎢
⎦ bv = ⎩ 0 ⎭
⎣ 0 −1/τP −KP KN ⎦ Bu = ⎣ KP Kθ
−KR
0
K G K G
0 K G
−1/τ
R
P
R
R
δ
R
f
(3.12)
If we let y = x, then C = I and D = 0 in Eq. (3.10). These equations now
can be used as the basis for various open-loop analyses and simulation studies.
Parameter values needed for the linearized model of a six-cylinder engine at
N = 600 RPM are listed in Table 3.1. The value of Kθ = 20 (lb/hr)/deg can be
estimated from Figure 3.7, and the value of Gδ = 10 (ft-lbf/deg) can be estimated
from other data. Recall that the incremental fuel-flow input Fd is delayed by
approximately two sampling periods, T, which also is given in Table 3.1.
Nonlinear Engine Dynamics Model
Whereas the linearized model in the previous section may be adequate for a basic
understanding of the engine air-intake–fueling–combustion dynamics, achieving levels of control and diagnostic performance that are mandated by current and future
U.S. Environmental Protection Agency (EPA) regulations requires the use of a
considerably more precise model. The derivation of such a model is presented here,
3.3 Control-Oriented Engine Modeling
47
Table 3.2. List of symbols for engine model
No.
Symbols
Description [Units]
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
pm
pa
Ta
Cd,th
ṁa,th
R
ṁa,cyl
Vm
Vd
ηv
ṁ ff
ṁ f i
τf
X
ṁ f c
τm
tc
tt
φm
θs
intake manifold pressure [Pa]
ambient pressure [Pa]
ambient temperature [K]
throttle discharge coefficient
mass flow rate of air at throttle [Kg/sec]
ideal gas constant [J/(kg · K)]
mass flow rate of air into cylinder [kg/sec]
intake manifold volume [m3 ]
displacement volume [m3 ]
volumetric efficiency
fuel flow rate from fuel puddle [kg/sec]
fuel flow rate from injector [kg/sec]
fuel evaporation time constant [sec]
fraction of injected fuel entering puddle
mass flow rate of fuel entering the cylinder
fuel flow rate from film [kg/sec]
cycle delay [sec]
transportation delay [sec]
measured equivalence ratio
crank-angle sampling interval (radians)
based on (Krishnaswamy et al. 1996). This model is developed in the crank-angle
domain and incorporates sufficient detail to capture the essential characteristics
of powertrain-system behavior while also remaining simple enough to be implementable onboard a vehicle (e.g., for engine diagnostics).
Toward this end, a hybrid identification approach was taken for model development. That is, the basic equations describing the system dynamics were derived
from consideration of physical principles, but the equations then were parameterized
by constants (e.g., discharge coefficient for the throttle) that were identified using
empirical identification techniques from experimental data. This approach enables
the development of reasonably accurate models that still accommodate physical
intuition, which allows inferences to be made about system operation through the
monitoring of these physically based constants and system variables. Table 3.2 lists
all of the variables used in the model.
Air-Intake Model
The air-intake model incorporates a nonlinear airflow model that models the flow
past the throttle, taking into account the different dynamics under both choked and
unchoked flow conditions, the manifold-filling dynamics, and the variation in engine
volumetric efficiency as a function of the engine operating condition. The mass flow
rate of air at throttle depends on the following flow conditions:
For unchoked flow:
γ /(γ −1)
2
pm
>
pa
γ +1
(3.13)
48
Review of Engine Modeling
ṁa,th
CD,th Ath pa
= RTa
pm
pa
1/γ 1/2
2γ
pm (γ −1)/γ
1−
γ −1
pa
(3.14)
For choked flow:
γ /(γ −1)
pm
2
≤
pa
γ +1
CD,th Ath pa 1/2
γ
RTa
ṁa,th =
2
γ +1
(γ +1)/2(γ −1)
(3.15)
(3.16)
Using the conservation of mass in the intake manifold, the manifold dynamics
can be described as:
dma,m
= ṁa,th − ṁa,cyl
dt
(3.17)
Using the Ideal Gas Law, we can write:
ηV ω
dpm
RT
+ v d pm = ṁa,th a
dt
4πVm
Vm
(3.18)
In the crank-angle domain:
ω
ηV ω
dpm
RT
+ v d pm = ṁa,th a
dθ
4πVm
Vm
(3.19)
The actual mass flow rate of air into a cylinder can be computed using the
volumetric efficiency (i.e., speed-density equation):
ṁa,cyl =
ηv PmVd ω
4π RTa
(3.20)
The volumetric efficiency was modeled as a polynomial function of engine speed,
manifold pressure, and throttle opening:
ηv = a0 + a1 ω + a2 ω2 + a3 Pm + a4 α + a5 α 2 + a6 α 3
(3.21)
where the coefficients ai ’s usually are determined empirically.
Fuel Dynamics (Wall Wetting Model)
The physics of the process by which fuel that is sprayed into the cylinder by the
injectors vaporizes and participates in combustion is complex. In simplified form, it
can be considered that part of the fuel vaporizes quickly enough (or is sufficiently
finely atomized) to participate directly in combustion, whereas the rest of the fuel
spray impinges on the cylinder wall, where it combines with the fuel “puddle” from
earlier injections and then evaporates from the puddle to form part of the air–fuel
mixture. Thus, the dynamics of the fueling system can be modeled as follows:
dṁ ff
dt
=−
ṁ f c =
1
ṁ + X ṁ f i
τ f ff
1
ṁ + (1 − X )ṁ f i
τ f ff
(3.22)
3.3 Control-Oriented Engine Modeling
49
Table 3.3. Empirically determined engine parameters
No.
Parameter
Description
1.
2.
3.
4.
5.
6.
Cd,th
ηv
τf
X
θc + θt
τm
Throttle-discharge coefficient
Volumetric efficiency
Fuel-evaporation constant
Direct entry fraction of fuel
Transport + EGO sensor delay
EGO sensor time constant
where the variables are as defined in Table 3.2. Again, in the crank-angle domain,
the model becomes:
dṁ ff
1
X
=−
ṁ ff + ṁ f i
dθ
τfω
ω
(3.23)
1
ṁ f c = ṁ ff + (1 − X )ṁ f i
τf
Air–Fuel-Ratio Dynamics
Using the mass flow rate of air and fuel from Eqs. (3.20) and (3.23), the air–fuel ratio
in a cylinder can be expressed as:
ṁ f i
1 ṁ ff
+ (1 − X )
ṁac
τ f ṁac
(3.24)
ṁ f i
1
1 ṁ ff
+ (1 − X )
=
ṁac
AF
τ f ṁac
(3.25)
ṁ f c
ṁac
=
Because AF = ṁac /ṁ f c ,
which introduces the equivalence ratio φ = 1/AF ,
φ=
ṁ f i
1 ṁ ff
+ (1 − X )
ṁac
τ f ṁac
(3.26)
Exhaust Transport Delay and Sensor Dynamics
The delay of the exhaust gas is the sum of cycle delay and transport delay. Therefore,
sensor dynamics combined with delay can be expressed as:
τm
dφm
+ φm = φ(t − tc − tt )
dt
(3.27)
In the crank-angle domain:
dφm
+ φm = φ(θ − θc − θt )
(3.28)
dθ
Table 3.3 lists the parameters in these equations that were determined empirically.
Determination of the parameters is not a trivial task and involves the resolution of
many issues, including the performance of both dynamic and static engine tests –
with suitably designed inputs that excite the relevant dynamics of the engine – and
τm ω
50
Review of Engine Modeling
the means of analysis of the data (i.e., both linear and nonlinear least squares techniques). However, a discussion of the identification procedures is beyond the scope
of this chapter.
PROBLEMS
1. Run in Simulink the open-loop-engine demonstration simulation described in this
chapter. Indicate the operating conditions selected for your simulation and present
the output plots from it.
2. Open-loop engine linear dynamics simulation. Use the linearized engine model
given in Eqs. (3.1)–(3.8) and the parameter values given in Table 3.1 and Example
3.1 to simulate the dynamics of the engine to the following inputs:
(a) Unit step disturbance torque (qualitatively compare to top trace in Figure
8.4 and to the results in Example 3.1).
(b) Separate unit step inputs for θ , δ, and Fd (qualitatively compare to the
results in Example 3.1 for the input θ ).
Plot the responses y versus time.
3. Repeat the simulations in Problem 2; however, include the IP delay. That is, in
your simulation, the inputs with subscript d should be delayed by the time 2T, and
T = 0.033 seconds, as given in Table 3.1.
4. You are an engineer at an automotive company, which has determined the parameters for the open-loop engine model. The state-space model for the engine is given
by:
_
x = Ax + Bu u + bv v
y = Cx + Du
where the states of the model are the mass airflow, the change in the manifold
pressure, and the change in engine speed; the inputs to the open-loop model are
throttle angle, spark-advance command, and fuel injected to the system. That is,
⎧
⎫
⎫
⎧
⎨ θ ⎬
⎨ ma ⎬
x = P , u = δ
⎩
⎭
⎭
⎩
Fd
N
v is the torque disturbance and v = Td . The coefficient matrices for the state
equation are:
⎡
⎡
⎡
⎤
⎤
⎤
Kθ
0
0
0
0
0
0
0
0 ⎦ bv = ⎣ 0 ⎦
A = ⎣ 0 1/τP −KP KN ⎦ Bu = ⎣ KP Kθ
0
KR Gδ KR G f
0 KR GP −1/τR
−KR
Your supervisor wants you to provide a quick estimate for the step response to the
throttle angle for the closed-loop engine model. From experience, you know that the
fuel injected must be kept close to stoichiometry – thus, Fd = KAF ṁa – and that as
a general rule, spark timing must be advanced as engine speed increases. Therefore,
δ = KSA N. You may neglect further the effects of torque disturbance assuming
smooth driving conditions.
References
51
Please answer the following:
(a) Find the simplified state-space model with θ as the only input.
(b) Find the transfer functions from θ to each of the three states.
(c) Find the steady-state value of N and ma for a unit step in θ . Will both of
them be stable? Explain.
5. An engineer was transferred to the engine-test department and he needs to model
the breathing dynamics of a four-stroke, six-cylinder SI engine. The notebook of the
previous engineer contains the manifold-filling equation:
d
P = 0.5(ṁθ − ṁcyl )
dt m
where (Pm , bar) is the intake manifold pressure, (ṁθ , g/s) is the mass airflow rate
into the manifold through the throttle body, and (ṁcyl , g/s) is the pumping mass
airflow rate into the cylinders. In the next page of the notebook, the new engineer
finds this equation:
ṁθ = kθ θ + k1 Pm
where kq is either (10/3) or −3. Please answer the following questions:
(a) Which value (i.e., 10/3 or −3) would you choose for kq ? (Hint: Recall that
the flow rate is proportional to the pressure difference and that the intakemanifold pressure typically is less than the pressure upstream in the throttle
body.)
The next page in the notebook contains the following experiments:
Experiment 1: θ = 30◦ , Pm = 0.0 bar, ṁθ = 100 g/s
Experiment 2: θ = 30◦ , Pm = 0.5 bar, ṁθ = 45 g/s
(b) Can you determine the constant k1 ?
Finally, the linear function that defines the engine-pumping rate is found to be:
N
+ k2 Pm
500
where N in RPM is the engine speed. Two sets of experiments were conducted to
identify k2 . The notebook indicates manifold-pressure measurements of 0.5 and 0.9
bar and pumping rates of 50 and 10 g/s without pairing the pressure and pumping
rate of each experiment.
ṁcyl =
(c) Can you guess the pairs of pressure and cylinder flow?
(d) Based on your answer to (c), find the value of the constant k2 .
REFERENCES
Ashley, S., 2001, “A Low-Pollution Engine Solution,” Scientific American, June 2001,
pp. 91–5.
Bidan, P., S. Boverie, and V. Chaumerliac, 1995, “Nonlinear Control of a Spark-Ignition
Engine,” IEEE Transactions on Control Systems Technology, Vol. 3, No.1, March 1995.
Bosch, R., 2008, Automotive Handbook, Wiley & Sons.
Cassidy, J., M. Athans, and W. H. Lee, 1980, “On the Design of Electronic Automotive Engine
Controls Using Linear Quadratic Control Theory,” IEEE Transactions on Automatic Control, Vol. 25, No. 5, pp. 901–12.
52
Review of Engine Modeling
Cho, D., 1991, “Research and Development Needs for Engine and Powertrain Control Systems,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), 1991, Advanced
Automotive Technologies-1991, ASME DE–Vol. 40, New York, pp. 23–34.
Cho, D., and J. K. Hedrick, 1989, “Automotive Powertrain Modeling for Control,” ASME
Journal of Dynamic Systems, Measurement and Control, Vol. 111, December 1989,
pp. 568–76.
Connoly, F., and G. Rizzoni, 1994, “Real-ime Estimation of Engine Torque for the Detection of Engine Misfires,” ASME Journal of Dynamic Systems, Measurement and Control,
pp. 675–86.
Cook, J. A., and B. K. Powell, 1988, “Modeling of an Internal Combustion Engine for Control
Analysis,” IEEE Control Systems Magazine, August 1988, pp. 20–6.
Cook, J. A., J. Sun, J. H. Buckland, I. V. Kolmanovsky, H. Peng, and J. W. Grizzle, 2006,
“Automotive Powertrain Control: A Survey,” Asian Journal of Control, Vol. 8, No. 3,
September 2006, pp. 237–60.
Crossley, P. R., and J. A. Cook, 1991, Proceedings of the IEEE International Control 91
Conference, Vol. 2, pp. 921–25, March 1991, Edinburgh, Scotland, UK.
Crouse, W. H., and D. Anglin, 1977, Automotive Emission Control, McGraw-Hill.
Crouse, W. H., and D. L. Anglin, 1986, Automotive Engines, seventh edition, McGraw-Hill.
Dobner, D. J., 1983, “Dynamic Engine Models for Control Development – Part I: Non-Linear
and Linear Model Formulation,” Application of Control Theory in the Automotive Industry,
Inderscience Publishers.
Green, J. H., and J. K. Hedrick, 1990, “Nonlinear Speed Control for Automotive Engines,”
Proceedings of the American Control Conference, San Diego, CA, May 1990, pp. 2891–7.
Guzzella, L., and A. Sciarretta, Vehicle Propulsion Systems: Introduction to Modeling and
Optimization, Springer Verlag, 2007 (second edition).
Hebbale, K. V., and Y. A. Ghoneim, 1991, “A Speed and Acceleration Estimation Algorithm
for Powertrain Control,” Proceedings of the American Control Conference, Boston, MA,
June 1991, pp. 415–20.
Hendricks, E., 1990, “Mean Value SI Engine Model for Control Studies,” Proceedings of the
American Control Conference, San Diego, CA, May 1990, pp. 1882–6.
Heywood, J., 1988, Internal Combustion Engine Fundamentals, McGraw-Hill.
Hrovat, D., and W. F. Powers, 1988, “Computer Control Systems for Automotive Power
Trains,” IEEE Control Systems Magazine, August 1988, pp. 3–10.
Jankovic, M., and I. Kolmanovsky, 2009, “Developments in Control of Time-Delay Systems
for Automotive Powertrain Applications,” in B. Balachandran et al. (eds.), Delay Differential Equations: Recent Advances and New Directions, Springer.
Kamei, E., H. Namba, K. Osaki, and M. Ohba, 1987, “Application of Reduced Order Model
to Automotive Engine Control System,” ASME Journal of Dynamic Systems, Measurement
and Control, Vol. 109, September 1987, pp. 232–7.
Kiencke, U., and L. Nielsen, 2005, Automotive Control Systems for Engine, Driveline and
Vehicle, Springer.
Knowles, D., 1989, Automotive Emission Control & Computer Systems, second edition,
Prentice-Hall.
Krishnaswamy, V., C. Siviero, F. Carbognani, G. Rizzoni, and V. Utkin, 1996, “Application of
Sliding Mode Observers to Automobile Powertrain Diagnostics,” Proceedings of the IEEE
Conference on Control Applications, Dearborn, MI, CCA 96, pp. 355–60.
Melgaard, H., E. Hendricks, and H. Madsen, 1990, “Continuous Identification of a FourStroke SI Engine,” Proceedings of the American Control Conference, San Diego, CA, May
1990, pp. 1876–81.
Moskwa, J. J., and J. K. Hedrick, 1990, “Nonlinear Algorithms for Automotive Engine Control,” IEEE Control Systems Magazine, April 1990, pp. 88–93.
Nesbit, C., and J. K. Hedrick, 1991, “Adaptive Engine Control,” Proceedings of the American
Control Conference, Boston, MA, June 1991, pp. 2072–6.
Powell, B. K., 1979, “A Dynamic Model for Automotive Engine Control Analysis,” Proceedings of the IEEE Conference on Decision and Control, pp. 120–6.
References
Ribbens, W. B., 2003, Understanding Automotive Electronics (6th edition), ButterworthHeinemann.
Shiao, Y., and J. J. Moskwa, “Cylinder Pressure and Combustion Heat Release Estimation for
SI Engine Diagnostics Using Nonlinear Sliding Observers,” IEEE Transactions on Control
Systems Technology, Vol. 3, No. 1, March 1995.
Shiga, H., and S. Mizutani, 1988, Car Electronics, Nippondenso Co.
Stone, Richard, 1994, Introduction to Internal Combustion Engines, SAE International, second
edition, 1994.
Sweet, L. M., 1981, “Control Systems for Automotive Vehicle Fuel Economy: A Literature Review,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 103,
September 1981, pp. 173–80.
Tabe, T., M. Ohba, E. Kamei, and H. Namba, 1987, “On the Application of Modern Control
Theory to Automotive Engine Control,” IEEE Transactions on Industrial Electronics,
Vol. 34, No. 1, February 1987, pp. 35–9.
Washino, S., 1989, Automobile Electronics, Gordon and Breach Science Publishers, New
York.
53
4
Review of Vehicle Dynamics
Design of control systems for ground vehicles must start from an adequate understanding of their dynamic behavior. Although a detailed discussion of vehicle dynamics is beyond the scope of this chapter, simple dynamic models suitable for controller
design are necessary for control studies and are developed and presented herein.
These simple models are used in subsequent chapters as the basis for controller
designs (e.g., cruise control, antilock brakes, traction control, steering control, and
active suspensions). More complex (i.e., nonlinear, high-order, and fully coupled)
models for vehicle dynamics often are needed to evaluate, using simulation studies,
the controllers that are designed using simple control-design models. Such complex
models are discussed in detail in the literature (Ellis 1966; Gillespie 1992; Segel 1990;
Venhovens 1993; Wong 2008) and also can be implemented in commercial dynamic
simulation software (e.g., ADAMS and CARSIM).
First, the standard notation and terminology for vehicle dynamics is introduced
with definitions of reference frames and coordinates used to describe vehicle motion.
Next, the longitudinal motion of the vehicle, including braking and acceleration,
is presented. Then, lateral-motion dynamics, or vehicle steering or handling, is
described. Finally, the vertical motion of vehicles is discussed.
4.1 Coordinates and Notation for Vehicle Dynamics
Lumped parameter models concentrate the distributed mechanical properties of
mass (kg), stiffness (N/m), and damping (Ns/m) at imagined physical locations. In
particular, many elementary analyses of vehicles are based on treating the vehicle
as one or more concentrated (lumped) masses located at the center of gravity (CG)
of their respective rigid bodies. The Society of Automotive Engineers (SAE) has
introduced standard coordinates and notation for describing vehicle dynamics that
are widely used (SAE 1976).1
1
54
The ISO coordinates are also frequently used and are the same as the SAE coordinates in Figure 4.1
except that the vehicle-fixed axis z points up rather than down; therefore, following the right-hand
rule, the y axis points to the left of the driver rather than to the right. Because this difference can
cause confusion when reading the literature on vehicle dynamics and control, readers should be sure
to establish which coordinate system is being used.
4.1 Coordinates and Notation for Vehicle Dynamics
55
Table 4.1. SAE body fixed vehicle axis system: symbols and definitions
Axis
Translational
velocity
Angular
displacement
Angular
velocity
Force
component
Moment
component
x
y
z
u (forward)
v (lateral)
w (vertical)
φ
θ
ψ
p or φ̇ (roll)
q or θ˙ (pitch)
r or ψ̇ (yaw)
Fx
Fy
Fz
Mx
My
Mz
Figure 4.1 is the vehicle-fixed x, y, z coordinate system for describing the motion
of a vehicle treated as a single lumped mass concentrated at the CG of the vehicle. This body-fixed coordinate system moves with the vehicle, which is assumed
to be rigid. This coordinate system – a right-hand rule, Cartesian coordinate system – is summarized in Table 4.1. Vehicle motion typically is described in terms of
the velocities (i.e., forward, lateral, vertical, roll, pitch, and yaw) in the vehiclefixed coordinate system as referenced to an earth-fixed (i.e., inertial) reference
frame.
Figure 4.2 shows an earth-fixed, Cartesian reference frame with coordinates X,
Y, and Z (i.e., vertical travel, positive downward) and defines the heading angle (i.e.,
between x and X in ground plane) ψ; the course angle (i.e., between vehicle-velocity
vector and X axis), ν; and the sideslip angle (i.e., between x axis and the vehicleforward-velocity vector, V = u ex + v ey ), β. Note that the course angle νψ + β.
The sideslip angle is a result of the compliance of the pneumatic tire, and it has a
significant effect on vehicle dynamics.
Figure 4.3 illustrates the tire coordinate system, with coordinates X ′ , Y ′ , and Z′ .
The forces (Fx , Fy , Fz ) and moments (Mx , My , Mz ) associated with the individual
tire are defined in the axis directions. The wheel torque is usually denoted as T and
wheel-spin velocity as ω. Therefore, the steer angle, δ, is between the direction of
the wheel heading (X ′ ) and the vehicle heading.
Equations of motion typically are obtained by the application of Newton’s second law, which for the translational motion of a rigid vehicle of mass m is:
(4.1)
F = ma
and for rotational motion (about the center of mass, a fixed point, or the instant
center) is:
(4.2)
M = Ḣ
y
lateral
pitch
Figure 4.1. Vehicle-fixed coordinate system.
x longitudinal
roll
vertical
yaw
z
56
X
Review of Vehicle Dynamics
x
ψ
Vehicle speed
β
ν
Figure 4.2. Earth-fixed and vehicle-fixed coordinate
systems.
y
Y
where F is the vector sum of the external forces acting on the vehicle mass m, a is
the acceleration vector, M is the vector sum of the external moments acting on the
vehicle, and Ḣ is the time rate of change of the moment of momentum of the rigid
body about the mass center or a fixed point in an inertial reference frame. When a
or Ḣ is zero, Eqs. (4.1) and (4.2) represent static equilibrium. Several examples of
the application of Newton’s second law to vehicle static and dynamic calculations
are provided in the following sections. Instead of using Newton’s laws, the equations
of motion also can be derived using Lagrange’s method based on energy concepts.
A two-step procedure is used when applying Newton’s laws to derive vehicledynamic equations. First, kinematics (i.e., the geometry of motion) is used to derive
the acceleration term represented by the vector a or Ḣ using the coordinates that
are defined in Table 4.1 and Figures 4.1 through 4.3. Next, the force or moment
terms, represented by the vectors F or M, must be obtained. Depending on the
Figure 4.3. Tire-axis system.
4.1 Coordinates and Notation for Vehicle Dynamics
Figure 4.4. Friction-ellipse tire model.
particular situation being modeled, they include the vehicle weight (i.e., due to
gravity), dissipative forces (e.g., aerodynamic drag and rolling resistance), and – most
important, – the forces and moments that arise due to road–tire contact (e.g., braking
and tractive and handling forces). The pneumatic tires have an important role in the
dynamic behavior of a vehicle, and a brief background on this topic is provided
here (Clark 1971; Wong 1993). Referring to the tire-axis system in Figure 4.3, the
longitudinal and lateral tire forces are obtained by integrating the longitudinal and
lateral shear stresses over the tire-contact patch.
The forces created at the tire–road interface are friction limited and depend
on a tire–road friction coefficient, μ. The tire–road friction coefficient can vary
significantly with road conditions (e.g., paved, gravel, wet, or icy). Typically, we
are interested in the tire longitudinal, Fx , and lateral, Fy , forces and the self-aligning
torque, Mz . When local shear forces are below the friction limit and the tire elements
adhere to the road surface, the tire forces and moments, based on a quasistatic
assumption, can be represented as follows:
⎧
⎫
⎪
⎬
⎨ Fx ⎪
Fy = f (rw , γ , r, λ, α)
⎪
⎪
⎩
⎭
Mz
where λ is the longitudinal slip, tan α = (v/u) is the lateral slip in terms of tire lateral
and longitudinal velocities v and u, γ is the camber angle, and rw is the wheel radius.
When the excitation is large enough to cause loss of tire–road adhesion (i.e., the
no-sliding assumption is violated), then:
⎧
⎫
⎪
⎬
⎨ Fx ⎪
Fy = f (rw , γ , r, λ, α, μ)
⎪
⎪
⎭
⎩
Mz
Tire forces and moments are highly nonlinear and difficult to model. Simplified
models such as the Brush or Pacejka “Magic Formula” models (Bakker et al. 1987,
1989) were developed for use in simulation studies (Venhovens 1993).
A useful concept for combined lateral and longitudinal slip, as shown in Figure
4.4, is the friction-ellipse tire model. The basic idea for combined lateral and longitudinal tire forces is that when the resultant shear force exceeds the local friction
limit, sliding occurs. This model is more useful conceptually than computationally.
The next section considers the longitudinal motion of the vehicle, which is
required to understand the dynamic behavior in terms of braking and acceleration.
57
58
Review of Vehicle Dynamics
DA
hA
Rhz
B
Θ
Rxf
Fzf
a
Fxf
h
W
b
Rxr
Figure 4.5. Forces acting on a vehicle.
Rhx
A
Fzr
Fxr
4.2 Longitudinal Vehicle Motion
Figure 4.5 is a free-body diagram of a vehicle with mass m on an incline with angle .
The diagram includes the significant forces acting on the vehicle: g is the gravitational
constant; DA is the aerodynamic force; Rh is a drawbar force; W = mg is the weight
of the vehicle; Fx is the tractive force; Rx is the rolling-resistance force; and max , an
equivalent inertial force, acts at the CG. The subscripts f and r refer to the front (at
B) and rear (at A) tire-reaction forces, respectively.
Application of Newton’s second law for the z direction (i.e., no vertical acceleration) gives:
0 = W cos − Fz f − Fzr + Rhz
(4.3)
and in the x direction:
max = (W/g)ax = m
du
= Fxr + Fx f − W sin − Rxr − Rx f − DA + Rhx (4.4)
dt
The aerodynamic-drag force depends on the relative velocity between the vehicle
and the surrounding air and is given by the semi-empirical relationship:
DA = 0.5ρ Cd A(u + uw )2
(4.5)
where ρ is the air density (= 1.202 kg/m3 at an altitude of 200 m), Cd is the drag
coefficient, A is the maximum vehicle cross-sectional area (≈ 0.9)(track) (height) for
passenger vehicles), u is the vehicle-forward velocity, and uw is the wind velocity (i.e.,
positive for a headwind and negative for a tailwind). The drag coefficient for vehicles
ranges from about 0.2 (i.e., streamlined passenger vehicles with underbody cover)
to 1.5 (i.e., trucks); 0.4 is a typical value for passenger cars (Bosch 2009; Gillespie
1992).
The rolling resistance arises due to the work of deformation on the tire and the
road surface, and it is roughly proportional to the normal force on the tire:
Rx = Rx f + Rxr = f (Fz f + Fzr )
(4.6)
where f is the rolling-resistance coefficient in the range of about 0.01 to 0.4, with
0.015 as a typical value for passenger vehicles.
To
determine the aerodynamic-drag and the rolling-resistance coefficients, measurements from two coast-down tests can be used: one at a high speed and the
other at a lower speed.
EXAMPLE 4.1: DETERMINE AERODYNAMIC DRAG AND ROLLING RESISTANCE.
4.2 Longitudinal Vehicle Motion
59
Table 4.2. Measurements of two coast-down tests
High-speed test
Low-speed test
Initial speed
Final speed
Time duration
Vi1
Vf 1
t1
Vi2
Vf 2
t2
Average speed
V1 =
Average deceleration
a1 =
Vi1 + V f 1
2
Vi1 − V f 1
t1
V2 =
a2 =
Vi2 + V f 2
2
Vi2 − V f 2
t2
Using the example results in Table 4.2, we obtain:
DA1 + Rx = 0.5ρCd AV12 + f mg = ma1
DA2 + Rx = 0.5ρCd AV22 + f mg = ma2
Coefficients of aerodynamic drag and rolling resistance then can be obtained:
Cd =
m(a1 − a2 )
0.5ρA V12 − V22
f =
a1V22 − a2V12
g V22 − V12
As a first step in the analysis of braking and acceleration performance, it is necessary to determine axle loads (see the free-body
diagram in Figure 4.5). The loads at each axle consist of a static component plus
load transferred from front to rear (or vice versa) due to the forces acting on the
vehicle. The load on the front axle is found by summing moments about Point
A and on the rear axle by summing moments about Point B. Assuming a constant value of ax (i.e., constant acceleration or deceleration in the longitudinal
direction), Rh = 0, and the vehicle has constant pitch, then these moments must
sum to zero:
EXAMPLE 4.2: AXLE LOADS.
Fz f L + DA hA + (W/g)ax h + W h sin − W b cos = 0
(4.7)
Fzr L − DA hA − (W/g)ax h − W h sin − W a cos = 0
(4.8)
Special cases can be obtained; for example, on level ground, the cosine term is
one and the sine term is zero. Grades usually are given as slopes in percentage and
correspond to tan . Grades on interstate highways typically are restricted to less
that 4 percent and rarely exceed 10 to 12 percent on primary and secondary roads.
Thus, in most cases, the approximations sin ≈ and cos ≈ 1 can be used.
The aerodynamic-drag force is proportional to the square of the vehicle-forward
velocity and typically is negligible for low speeds. For forward acceleration, as
shown in Figure 4.6, the load is transferred from the front axle to the rear axle in
proportion to the normalized acceleration (ax /g) and the normalized CG height
(h/L).
Next, we consider the forces Fxf and Fxr during braking. Braking forces – as long
as all of the wheels are rolling – can be described by the equation:
Fb = (Tb + Iw αw )/rw
(4.9)
60
Review of Vehicle Dynamics
μp
μ=
Fx
Fz
Figure 4.6. Braking coefficient versus slip.
0
λ
where Fb is the brake force at a particular wheel, Tb is the brake torque produced at
that wheel by application of the brake, Iw is the total rotational inertia of the wheel
and drive components, αw is the rotational deceleration of the wheel, and rw is the
wheel radius. Except when the wheel is locked up and slipping, αw = Ax /rw , which
can be used when Eq. (4.9) is combined with Eq. (4.4). The brake force in Eq. (4.9)
is limited by the frictional coupling between the tire and the road, and it depends on
the slip – caused by deformation – between the tire and the road surface. The slip
ratio λ is defined as the ratio of longitudinal-slip velocity in the contact patch (i.e.,
tire-circumferential speed–vehicle-forward speed) to vehicle-forward speed:
rw ω − u
u
rw ω − u
=
rw ω
λ=
during braking
(4.10)
during acceleration
(4.11)
where ω is the tire-rotational speed. However, it is common practice to ignore the
sign of λ when presenting tire-force characteristics. As shown in Figure 4.6, except
at very low tire–road friction, the braking force reaches a peak value at λ ≈ 0.15, or
15 percent. The braking coefficient μ is defined as the ratio between braking force
and tire normal force, and it represents a normalized measure of braking forces. The
peak value of μ = μ p is a key property because it determines the maximum braking
force for a particular tire–road combination. At higher values of slip, the coefficient
diminishes continuously.
Equation (4.4), a simple model
of the longitudinal motion of a vehicle, can be used to determine changes in
the vehicle-forward motion due to grades, braking, acceleration, and so forth.
With Rhx = 0, Fxr = 0 (FWD) , and ax = du/dt and using Eqs. (4.5) and (4.6) in
Eq. (4.4), we obtain:
EXAMPLE 4.3: VEHICLE LONGITUDINAL MOTION.
m(du/dt ) = Fx f − W sin − fW cos − 0.5ρCd A(u + uw )2
(4.12)
Equation (4.12) can be used in a variety of simple analyses. Assume that in a
braking maneuver, all of the forces on the right-hand side are constant and equal
to −Fx ; the equation then becomes:
m du = −Fx dt
(4.13)
which can be integrated to give:
ui − u f = (Fx /m)(t f − ti )
(4.14)
4.2 Longitudinal Vehicle Motion
61
Vehicle Forward Speed
12
10
8
6
Figure 4.7. Simulation results of Example 4.3.
4
2
0
-2
0
2
4
6
Time (sec)
8
where the subscripts i and f denote initial and final values, respectively, and t
denotes time. Using the fact that u = dx/dt and that for the vehicle to stop,
uf = 0, the stopping distance can be found as:
(x f − xi ) = mu2i /(2Fx )
(4.15)
(t f − ti ) = mui /Fx
(4.16)
and the stopping time as:
Based on this simplified analysis, the stopping time is proportional to the vehicleforward velocity at the time that the brakes are applied, whereas the stopping
distance is proportional to the square of that velocity. More accurate results
for the vehicle-forward velocity u(t) in various braking or acceleration cases
are obtained by numerically integrating the nonlinear ordinary differential
equation, Eq. (4.12). A simple MATLAB program for this purpose is provided
herein. Notice that the results are similar to the prediction based on Eqs.
(4.14)–(4.16). With m = 2,000kg, ui = 12 m/s, the average velocity uav = (ui +
uf )/2 = 6 m/s, and Fx = 2,000 + (0.02)(2,000)(9.8) + (0.5)(1.202)(0.4)(2)(36)
N, Eq. (4.16) predicts a stopping time of 9.89 seconds, which agrees closely with
the simulation results shown in Figure 4.7.
% Ex4_3.m
% Init. time, final time, and initial
% values of the variable x are:
ti=0.0; tf=10.0; ui = [12.0];
% Tol and trace are used
% by the integration routine ode23:
tol = 1.0E-4; trace = 1;
% Perform integration and store
% the results in x
[t,u] = ode23(‘Ex3_3a’,ti,tf,ui,tol,trace);
% Plot the results
plot(t,u,‘r’)
title(‘Vehicle Forward Speed’);
10
62
Review of Vehicle Dynamics
xlabel(‘Time (sec)’)
ylabel(‘u (m/sec)’); grid;
function udot = Ex3_3a(t,u);
% Equations of longitudinal motion
% for a vehicle.
% The parameters are:
% u - vehicle forward velocity
% m - vehicle mass
% Fx - tractive or braking force
% W - vehicle weight, W = m*g
% Theta - road grade angle in radians
% f - rolling friction coefficient
% rho - density of air
% Cd - aerodynamic drag coefficient
% A - cross sectional area of vehicle
% uw - wind velocity
%
% Parameter values):
m=2000; g=9.8; W=m*g;
Fx=-2000; Theta=0.0; f=0.02;
rho=1.202; Cd=0.4; A=2; uw=0.0;
%
if u > 0
udot = [(1/m)*(Fx-W*sin(Theta)
- f*W*cos(Theta)-0.5*rho*Cd*A*(u+uw)ˆ2)];
else
udot=0;
end
The traction
forces Fxf and/or Fxr are required to move a vehicle forward. They are produced at the tire–road interface due to rotation of the wheels by the engine and
drivetrain. The engine torque (as well as power and specific fuel consumption)
produced is a function of speed. Using two assumptions – manual transmission
and small (no) tire slip – the tractive force available from the engine to overcome
load forces and to accelerate a vehicle is shown to be as follows (Gillespie 1992):
(4.17)
Fx = Te Nt N f (ηt f /rw ) − (Ie + It )Nt2 N 2f + Id N 2f + Iw ax /rw2
EXAMPLE 4.4: POWER LIMITS ON LONGITUDINAL ACCELERATION.
where Te is the engine torque, Nt is the gear ratio of the transmission, Nf is
the gear ratio of the final drive, ηt f (<1) is a correction factor to represent the
efficiency of the overall drive system, rw is the radius of the wheel, Iw is the
rotational inertia of the wheels and axle shafts, and Id is the rotational inertia of
the driveshaft.
This tractive-force expression has two parts. The first term on the right-hand
side is the engine torque times the overall gear ratio and efficiency of the drive
4.2 Longitudinal Vehicle Motion
63
δf
αf
Figure 4.8. Schematic of a “bicycle model.”
a
R
r
αr
v
b
system. The second term on the right-hand side represents the “loss” of tractive
force due to the inertia of the drive-system components. Equation (4.17) can be
substituted in Eq. (4.4) to compute the vehicle motion under acceleration using
computational methods.
Maximum performance in longitudinal acceleration can be limited by engine
power (at high speeds) or by the traction limits of the drive wheels (e.g., at low
speeds or on a slippery road surface). The engine provides the propulsion force
needed to accelerate the vehicle and typically is characterized by torque and
power curves as a function of speed (Figure 4.8). Power is given by the product
of torque and speed, as follows:
P = T
(4.18)
where P is power in watt (or ft-lb/sec), T is engine torque in N-m (or ft-lb), and is engine speed in rad/sec. Also note that 1 horsepower (HP) = 550 ft.lb/sec, or
0.736 kW.
Neglecting all resistance forces (i.e., at low to moderate speeds), a very
rough upper limit on forward acceleration is obtained from Newton’s second
law:
max = Fx
(4.19)
Because the drive power is the product of tractive force Fx times the forward
speed u (ft/sec), the acceleration can be rewritten as:
ax = (Fx /m) = (P/um) = (Pg/uW )
(4.20)
where W is the weight of the automobile (lb) and g is the gravitational constant
(32.2 ft/sec2 ). Thus, the maximum acceleration is inversely proportional to the
forward speed and the weight of the vehicle.
A more detailed analysis of vehicle acceleration, or tractive-force limit, can
be obtained by considering the engine, torque converter, transmission, and tire
characteristics as well as the losses due to inertia and friction in the drivetrain.
At times, engine dynamics is approximately included in longitudinal dynamics
studies (e.g., cruise control and platooning) as a first-order dynamic lag between
throttle command and the generation of the engine-drive torque.
64
Review of Vehicle Dynamics
Assuming adequate power from the engine, the acceleration (or tractive
force) may be limited by the coefficient of friction between the tire and the road:
(4.21)
Fx = μFz
where Fx is the traction force, m is the coefficient of friction, and Fz is the
normal force on the drive wheels. Similar to the μ-λ curve in Figure 4.6, the
coefficient of friction and the traction force depend on the longitudinal slip
between the tire and the road surface. The calculation of Fz must consider not
only fore–aft weight shift due to acceleration and braking (see Example 4.1)
but also transverse weight shift due to cornering. The true vehicle-acceleration
performance under the combination of all of these possible limiting factors
usually is obtained from simulations. Under simplifying assumptions, we may
be able to study analytically the meaningful vehicle performance. For example,
assuming (1) small (i.e., no) lateral and yaw motions, and (2) a slippery road (i.e.,
tire friction is the performance limit), we can study the acceleration performance
of FWD versus rear-wheel drive (RWD) vehicles, or the “optimal” brake (or
drive) distribution ratio of front versus rear axles for maximum deceleration
(or acceleration) performance.
4.3 Lateral Vehicle Motion
This section presents both two and three degree-of-freedom models for vehicle
handling.
Two-Degree-of-Freedom Lateral Model (“Bicycle Model”)
in Steady Cornering
The lateral motion of a vehicle is discussed based on the so-called linear bicycle
model illustrated in Figure 4.8. The term bicycle model is a misnomer in that this
model typically is not used to model bicycle-handling dynamics; rather, it has arisen
in the automotive literature because in this model, the right and left wheels are
collapsed into one (see Figure 4.8). The steady turning (i.e., cornering) behavior,
at constant forward speed u0, due to a small and steady steer displacement, δ f ,
at the front wheels is considered first. For the analysis presented here to be valid,
the following assumptions must hold: (1) the radius of the turn, R, must be large
compared to the vehicle wheelbase, L = a + b, and the vehicle track, t; (2) the left
and right steer angles of the front wheels must be approximately the same (= δ f );
(3) the sideslip angles of the front wheels, α f , are equal, as are the sideslip angles of
the rear wheels, αr ; (4) the sideslip angle at the CG is β = tan−1 uv ≈ uv ; and (5) the
radius, R; sideslip angle, β; and the yaw velocity, r = ψ̇ are fixed in a steady turn so
that the instantaneous speed tangent to the path at the CG is u = rR.
Before applying Newton’s second law, consider first the necessary geometrical
relationships implied in Figure 4.8. Assuming a positive steer angle causing a turn to
the right and resulting in positive slip angles being established at the front and rear
tires (i.e., the tire-slip angle is defined as the angle from tire speed to tire-orientation
direction), we obtain:
αf = δf −
v + ar
u
and
αr = −
v − br
u
(4.22)
4.3 Lateral Vehicle Motion
65
1
0.9
0.8
Normalized force Fy /Fz
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
slip angle (deg)
10
12
14
Figure 4.9. Tire lateral force, Fy , versus slip angle, α.
Therefore,
L
R
(4.23)
L
+ α f − αr
R
(4.24)
α f − αr = δ f −
or
δf =
This implies that the steering angle, δ f , which is necessary to negotiate a curve,
consists of two parts: the static part is equal to L/R (also known as the Ackermann
angle) and the dynamic part, which is equal to the difference between the front- and
rear-tire slip angles. If the front tire-slip angle is larger than the rear slip angle, this
condition is termed understeer. This implies that the steering angle must be larger
than the Ackermann angle to maintain a constant-radius turn at nonzero speed. If
the rear slip angle is greater than the front slip angle, the front steering angle is
less than the Ackerman angle, a condition termed oversteer. Finally, if the front and
rear slip angles are equal, the steering angle is equal to the Ackerman angle and the
condition is termed neutral steer.
The side forces (i.e., cornering forces) acting at the vehicle tires, Fyf and Fyr , are
related to the slip angles, α f and αr , as shown in Figure 4.9. Although the relationship
between Fy and α is nonlinear, for small slip angles, we can use the approximation:
Fy = Cα α
(4.25)
where the cornering stiffness, Cα , is defined as the slope of the Fy versus α curve
at α = 0 and therefore is positive. The cornering stiffness depends on several tire
geometric and material properties; for a given tire, it depends on the vertical load,
Fz , and the inflation pressure. It should be emphasized that the force and cornering
66
Review of Vehicle Dynamics
stiffness for bicycle models are defined for vehicle axles. The cornering-stiffness
value, therefore, is about twice the value of tire-cornering stiffness encountered in
the literature when the tire force of individual tires is analyzed.
Now we can derive the equations of motion applicable to the situation shown in
Figure 4.8. Applying Newton’s second law to translation in the y direction gives:
mu2 /R = Fy f + Fyr
(4.26)
where the acceleration in the y direction, ay = ur = u2 /R, is due to the centripetal
acceleration of a vehicle in a steady turn. Also requiring the sum of the moments
about the CG to be zero in the steady turn gives:
0 = Fy f a − Fyr b
(4.27)
Thus, the cornering forces at the front and rear tires, respectively, are:
Fyr = m(a/L)(u2 /R)
2
Fy f = m(b/L)(u /R)
(4.28)
(4.29)
Substituting Eqs. (4.28) and (4.29) in Eq. (4.25) and solving for the slip angles gives:
αf =
mu2 b 1
R L Cα f
(4.30)
αr =
mu2 a 1
R L Cαr
(4.31)
Now, substitute these slip-angle expressions into Eq. (4.23) to obtain the steering
angle:
L
δf = +
R
mb
ma
−
LCα f
LCαr
u2
L
≡ + Kus · ay
R
R
(4.32)
where Kus , defined in Eq. (4.32), is termed the understeer coefficient and has the units
of (rad/(m/s2 )).
When Kus > 0, the vehicle is termed understeer; when Kus < 0, the vehicle is
termed oversteer; and when Kus = 0; the vehicle is termed neutral steer. For a neutralsteer vehicle, no change in steering angle with speed will be required to maintain
a constant-radius turn. For an understeer vehicle, the steer angle must increase
with speed (i.e., with the square of the speed) to maintain a constant-radius turn.
An oversteer vehicle in a constant-radius turn must decrease the steer angle with
increasing speed. Notice also that an understeer vehicle develops greater sideslip
angles at the front wheels than at the rear (i.e., |α f | > |αr |), whereas the opposite
is true for oversteered vehicles. For an understeer vehicle, the characteristic speed
is defined as the speed at which the steer angle is twice the Ackermann angle (i.e.,
Kus · ay = L
). Thus, the characteristic speed is given by:
R
uchar =
!
L
Kus
(4.33)
4.3 Lateral Vehicle Motion
67
Fixed radius of curvature
2L
R
Understeer
Figure 4.10. Effect of vehicle speed on steering
angle.
Neutral Speed
L
R
Oversteer
ucrit
0
Vehicle Speed
In the oversteer case, there is a critical speed above which the vehicle becomes
unstable:
!
−L
ucrit =
(4.34)
Kus
where, in the oversteer case, Kus < 0 by definition.
The effects of speed on steering angle in a constant-radius turn are summarized
in Figure 4.10. These results can be interpreted further in terms of the lateralacceleration gain:
u2
ay
= R =
Ga =
δf
δf
u2
L
δf ·
R
L
=
u2
L
1 + Kus
u2
L
=
u2
L + Kus u2
and the yaw-rate (or yaw-velocity) gain:
u
r
u
R
Gr =
=
=
δf
δf
L + Kus u2
(4.35)
(4.36)
From Eqs. (4.35) and (4.36), it is clear that when Kus < 0 (i.e., oversteer), these
gains become unbounded at the critical-speed value defined in Eq. (4.34). Also note
that the vertical forces at the front and rear axles, Fzf and Fzr , can be expressed (by
applying Newton’s second law in the z direction) as follows:
Fz f = (mg)(b/L)
(4.37)
Fzr = (mg)(a/L)
(4.38)
and
Consequently, Eq. (4.32) can be rewritten as follows:
2
Fz f
Fzr u2
L
L
′ u
δf = +
−
= + Kus
R
Cα f
Cαr Rg
R
Rg
(4.39)
where (u2 /Rg) is a nondimensional lateral acceleration in g units and the ratios
(Fz /Cα ) are the “cornering-compliance coefficients” in (Nz/Ny/deg) for the front and
rear wheels. Because the cornering stiffness, Cα , is strongly dependent on the normal (i.e., vertical) load at the tire, data often are given for the cornering-compliance
coefficient, (Fz /Cα ), or its inverse, the cornering-stiffness coefficient. The nondimen′
sional understeer gradient Kus
has units of rad/g rather than rad/(m/s2 ).
uchar
68
Review of Vehicle Dynamics
EXAMPLE 4.5: STEADY TURNING BEHAVIOR AND UNDERSTEER GRADIENT. Consider
an automobile with a weight of 8,000N on the front axle and 6,000N on the
rear axle, a wheel base of 2.5 meters, and the following tire cornering-stiffness
characteristics (i.e., for one tire):
Vertical Load
(N)
Cornering Stiffness
(N/deg)
Cornering Compliance
Coefficient (N/(N/deg))
1,500
2,000
2,500
3,000
3,500
4,000
502
656
787
903
1,014
1,100
2.987
3.050
3.178
3.323
3.450
3.636
(a) Determine the Ackermann angles for a turn-radius value of R = 200 meters.
This can be calculated easily from δ f = L
with the correct units:
R
δf =
2.5(m)
= 0.0125(rad) = 0.716(deg)
200(m)
(b) Determine the understeer gradient, Kus , using Eq. (4.39). This requires that
we know the cornering stiffness of the tires at the prevailing loads. On the
front axle, we have a load of 4,000N per tire. From the data in the table, we
obtain a cornering stiffness of 1,100 N/deg. Similarly, for the rear axle, the
load is 3,000N per tire and the cornering stiffness is 903 N/deg. Although
the units of Kus are degrees, the unit “deg/g” is written as a reminder. Thus:
′
Kus
8000N
6000N
−
1100∗ 2N/ deg 903∗ 2N/ deg
= (3.6363 − 3.3222) deg/g
= 0.3141 deg/g
(c) Determine the characteristic speed using Eq. (4.33):
′
Kus = Kus
/9.81 = 0.032 deg/(m/sec2 ) = 0.0005588 rad/(m/sec2 )
uchar = L/Kus = 66.9 m/sec
(d) Determine the lateral-acceleration gain using Eq. (4.35) for u = 55 mph =
24.56 m/sec:
m/ sec2
u2
m/ sec2
=
3.71
(4.40)
Ga =
=
212.7
L + Kus u2
rad
deg
(e) Determine the yaw-velocity gain using Eq. (4.36) for u = 55 mph =
24.56 m/sec:
u
rad/sec
deg/sec
Gr =
=
8.6586
=
8.6586
(4.41)
L + Kus u2
rad
deg
The calculated understeer gradient shows that this vehicle is close to being
neutral steer. This is a consequence of only the tire properties; the steering
4.3 Lateral Vehicle Motion
69
β
δf
Fxf
Fyf
Figure 4.11. Projection of forces onto the direction of the instant
center.
R
Fxr
Fyr
and suspension systems also contribute to the actual value of the understeer
gradient. Also note the units in these calculations, especially those in Eqs. (4.40)
and (4.41).
Finally, we consider the effect of tractive forces, Fxf and Fxr , on the understeer/oversteer behavior of a vehicle. For a vehicle in a steady turn, the application
of Newton’s second law in the lateral direction now gives (instead of Eq. [4.26]) the
following equations at each axle:
Fy f cos(δ f − β ) + Fx f sin(δ f − β ) = Fy f cos
a
b u2
+ α f + Fx f sin
+ αf = m
R
R
LR
(4.42)
a
and
Fyr cos(β ) − Fxr sin(β ) = Fyr cos
a u2
b
b
− αr − Fxr sin
− αr = m
R
R
LR
(4.43)
which can be obtained easily by projecting all of the forces onto the direction of the
instantaneous rotation center, as shown in Figure 4.11.
Assuming small slip angles such that sin α = α and cos α = 1 and a linear-tire
assumption as in Eq. (4.25), we obtain:
m
αf =
a
b u2o
− Fx f
LR
R
Cα f + Fx f
m
αr =
b
a u2o
+ Fxr
LR
R
Cαr + Fxr
A modified form of Eq. (4.32) to include the effects of tractive forces thus can be
obtained:
b u2
a
b
a u2
m
+ Fxr
L
L m L R − Fx f R
L
R
R
δ f = + (α f − αr ) = +
−
R
R
Cα f + Fx f
Cαr + Fxr
Fx f
mb
a
b
Fxr
ma
u2
L
−
+
−
(4.44)
= −
R Cα f + Fx f R Cαr + Fxr R
L(Cα f + Fx f ) L(Cαr + Fxr ) R
Although it is much more complex than Eq. (4.32), this equation also has the same
form. The Ackermann angle now is modified by the presence of the tractive forces,
as is the expression for the understeer coefficient, Kus . For FWD vehicles, the Ackermann angle is reduced and the understeer coefficient, Kus , increases (i.e., the vehicle
becomes more understeer).
70
Review of Vehicle Dynamics
Two- and Three-Degree-of-Freedom Lateral Dynamic Models
In the previous analyses based on the two-degree-of-freedom (DOF) bicycle model,
it was assumed that a vehicle is in a steady turn and the rolling motion of the vehicle
sprung mass is negligible. Transient linear models (i.e., not cornering in a steady
turn) now are considered for both the two- and three-DOF cases. The three-DOF
dynamic model includes a roll degree of freedom, φ, in addition to the lateral motion
y and the yaw motion ψ.
From Newton’s second law, we obtain:
Fy f + Fyr = may = mÿ = m(u0 r + v̇) = m(uor + uoβ̇ )
(4.45)
aFy f − bFyr = Iz ṙ
(4.46)
where tan β = uv or v ≈ βu0 was used. If we further assume the front and rear tires
0
(or, more precisely, axles) are linear, we have:
ar
v + ar
= Cα f δ − β −
Fy f = Cα f α f = Cα f δ −
uo
u
o
br − v
br
= Cαr
Fyr = Cαr αr = Cαr
−β
uo
uo
Combining with Eqs. (4.45) and (4.46), we obtain:
Cαr b − Cα f a
r + Cα f δ
muoβ̇ + muor = −(Cα f + Cαr )β +
uo
Cα f a2 + Cαr b2
r + aCα f δ
Iz ṙ = (bCαr − aCα f )β −
uo
(4.47)
(4.48)
which is the two-DOF linear model for lateral dynamics. In state-space form,
ẋ = Ax + Bu, where x = [β r]T , these two equations are as follows:
⎡ ⎤
⎤
⎡
Cα f + Cαr
aCα f − bCαr
Cα f
−
−
−
1
⎥ ⎢
⎢
⎥
muo
mu2o
⎢
⎥ β
β̇
⎢ muo ⎥
⎢
⎥
=⎢ +⎢
⎥ δ (4.49)
2
2
⎥
Cα f a + Cαr b
ṙ
⎣ aCα f ⎦
⎣ − aCα f − bCαr
⎦ r
−
Iz
Iz
Iz uo
When the relative motion of an automobile (relative to the road) is of interest,
we must define two additional state variables: y, the lateral displacement, and ψ,
the yaw angle of the automobile, relative to the road. Recall that tan β = uv or v ≈
0
βu0 , ẏ = v + uoψ and ψ̇ = r, ÿ = v̇ + uor, and with x = [y v ψ r]T , the state-space
equation is:
⎡
⎤
⎡
⎤
0
1
uo
0
0
⎥⎡ ⎤ ⎢
⎥
⎡ ⎤ ⎢
−aCα f + bCαr
C + Cαr
⎢
⎥ y
⎢ Cα f ⎥
y
⎢0 − αf
⎥
0
−
u
⎢
⎥
o⎥
muo
muo
⎢ m ⎥
⎥ ⎢
d ⎢
v⎥
⎥⎢
⎢
⎢
⎥
⎢v⎥=⎢
⎥δ
⎢
⎥⎣ ⎦ + ⎢
0
0
1
⎥ ψ
0 ⎥
dt ⎣ ψ ⎦ ⎢ 0
⎢
⎥
⎢
⎥
⎢ aC ⎥
2
2 ⎥
⎢
r
r
−aC
+
bC
a
+
C
b
C
⎣
αf ⎦
αf
αr
αf
αr
⎣
⎦
0
0 −
Iz
Iz uo
Iz uo
(4.50)
4.3 Lateral Vehicle Motion
71
When the road is curved and its orientation (i.e., yaw angle) is denoted as ψd (and
its changing rate rd ), the vehicle-dynamic equations are then:
Fy f + Fyr = m(ÿ + uord )
(4.51)
aFy f − bFyr = Iz ṙ
(4.52)
Substitute the following equations in Eqs. (4.51) and (4.52):
v + ar
ẏ
ar
= Cα f δ −
Fy f = Cα f α f = Cα f δ −
−
+ (ψ − ψd )
uo
uo uo
ẏ
br − v
br
= Cαr − +
Fyr = Cαr αr = Cαr
+ (ψ − ψd )
uo
uo uo
to obtain:
m(ÿ + uord ) = −
Iz ṙ = −
Cα f + Cαr
uo
aCα f − bCαr
uo
ẏ −
ẏ +
−Cα f a + Cαr b
uo
2
2
Cα f a + Cαr b
uo
r + Cα f δ + (Cα f + Cαr )(ψ − ψd )
(4.53)
r + aCα f δ + (aCα f − bCαr )(ψ − ψd )
(4.54)
In state-space form, these can be written as Eq. (4.55):
⎡
⎤
1
0
0
⎤ ⎢
⎡
Cα f + Cαr
−aCα f + bCαr ⎥
⎢ 0 − Cα f + Cαr
⎥
y
⎢
⎥
mu
m
mu
⎢
⎥
⎥
d ⎢
o
o
⎥
⎢ ẏ ⎥ = ⎢
⎢
⎥
⎣
⎦
0
0
1
dt ψ − ψd
⎢0
⎥
⎢
2
2
r
Cα f a + Cαr b ⎥
−aCα f + bCαr aCα f − bCαr
⎦
⎣
0
−
Iz uo
Iz
Iz uo
⎡
⎤
0
⎡
⎡
⎤ ⎢ C ⎥
⎤
y
0
⎢ αf ⎥
⎥
⎢ ẏ ⎥ ⎢
⎢ −uo ⎥
m ⎥
⎥+⎢
⎥
×⎢
δ+⎢
⎢
⎣ψ − ψ ⎦ ⎢ 0 ⎥
⎣ −1 ⎦ rd
⎥
d
⎢
⎥
⎣ aCα f ⎦
r
0
0
Iz
(4.55)
A three-DOF vehicle has a nonrolling (i.e., unsprung) mass, mNR , in the plane
and a rolling (i.e., sprung) mass, mR , which is constrained to rotate (i.e., roll) about
an axis fixed in the nonrolling mass (Figure 4.12). In Figure 4.12, note that an xyz axis
is fixed in the nonrolling mass such that the z axis is vertical and passes through the
center of mass of the two mass systems and that the x axis is located in the vertical
plane of symmetry and passes through the center of mass of the nonrolling mass.
From Figure 4.14, that:
c is the distance along the x axis between the z axis and the CG of the rolling
body.
72
Review of Vehicle Dynamics
c
X
m
R
ψ
O
e
u
v
h
Y
x
x
θ
R
m
NR
z
roll axis
y
Figure 4.12. Schematic of the three-DOF model.
e is the distance along the x axis between the z axis and the CG of the nonrolling
body.
θR is the angle downward from the horizontal of the axis (i.e., fixed in the
nonrolling mass), about which the rolling mass is constrained to roll (i.e., the
roll axis).
h is the vertical distance (in the z direction) between the CG of the rolling mass
and the roll axis.
r is the yaw velocity, ψ̇.
p is the roll velocity, φ̇.
u is the velocity in the x direction, u = Ẋ cos ψ + Ẏ sin ψ.
v is the velocity in the y direction, v = −Ẋ sin ψ + Ẏ cos ψ.
The derivation of the equations of motion is omitted here (see Appendix B).
In deriving the equations of motion, it is assumed that u represents a perturbation
from a nominal forward velocity u0 , where u0 is constant and large and u is small.
The quantities u, v, r, p, and ϕ are all assumed to be small so their products can
be neglected. Based on these assumptions and the definitions given previously, the
linearized equations of motion for the three-DOF vehicle lateral dynamics are as
follows:
⎡
⎤
⎤
⎡
−Yβ muo − Yr
0
−Yφ ⎡ β ⎤ ⎡ Y ⎤
muo
0 mR h 0 ⎡ β̇ ⎤
δ
⎢ 0
⎥ ⎢
⎢ r ⎥ ⎢ Nδ ⎥
−Nβ
−Nr
0 −Nφ ⎥
Iz Ixz 0 ⎥
ṙ
⎢
⎢
⎥⎢
⎥
⎢
⎥⎢ ⎥ = ⎢ ⎥δ
⎥⎢ ⎥ + ⎢
⎣ mR huo Ixz Ix 0 ⎦ ⎣ ṗ ⎦ ⎣ 0
mR huo −L p −Lφ ⎦ ⎣ p ⎦ ⎣ 0 ⎦
φ̇
0
φ
0
0
0 1
0
0
−1
0
(4.56)
4.3 Lateral Vehicle Motion
73
where
m = mNR + mR
Ix = Ixx|R + mR h2 − 2θR Ixz|R + θR2 Izz|R
Ixz = mR hc − Ixz|R + θR Izz|R
Iz = Izz|R + Izz|NR + mR c2 + mNR e2
The so-called stability derivatives used in Eq. (4.56) are defined as follows:
Yβ = −(Cα f + Cαr )
Yφ = Cαr
Yr =
∂γ f
∂δr
+ Cγ f
∂φ
∂φ
Nβ = −aCα f + bCαr
Nφ = aCγ f
L p = −cR
∂γ f
∂φ
− bCar
−aCα f + bCαr
Yδ = Cα f
Nr = −
∂δr
∂φ
u0
a2Cα f + b2Cαr
u0
Nδ = aCα f
Lφ = mR g h − kR
Note that Eq. (4.56) is not in standard-state equation form but rather in the form
E_x + Fx = Gδ, where x = [β r p φ]T . Furthermore, one of the combined inertia
terms (Iz ) in fact is state-dependent. In other words, Eq. (4.56) appears linear but in
fact is nonlinear. Nevertheless, if we ignore the influence of roll angle on Iz (which
is justified because the state-dependent term is of the second order), then Eq. (4.56)
can be transformed into a state-space equation ẋ = Ax + Bδ, where A = −E−1 F and
B = E−1 G. The state and input Matrices A and B can be obtained symbolically, but
they also can be calculated numerically.
A steady-turning analysis can be performed using Eq. (4.56) by setting d/dt = 0.
After some algebraic manipulations, we obtain a relationship similar to Eq. (4.32).
However, this relationship accounts for the influence on understeer gradient of the
roll DOF, φ, associated with the rolling (or sprung) mass (i.e., the third term in
brackets on the right-hand side):
ma
mR h ∂δr Cγ f ∂γ f
mb
u2
L
−
+
−
δ= +
R
Cα f L Cαr L
Lφ
∂φ
Cα f ∂φ
R
(4.57)
If the roll-related states (i.e., p and φ) are dropped, with v = βu0 , then Eq.
(4.56) reduces to the two-DOF bicycle model shown in Eq. (4.49). Using the stability
derivatives defined in the three-DOF model, the two-DOF model in Eq. (4.49)
becomes:
⎡
⎤
⎤
⎡
Yβ
Yr
Yδ
− u0 ⎥ ⎢
⎢ m ⎥
d v
⎢ mu0 m
⎥ v
⎥
(4.58)
=⎢
+⎢
⎥
⎣ Nδ ⎦ δ
⎣ Nβ
dt r
Nr ⎦ r
Iz
Iz u0
Iz
74
Review of Vehicle Dynamics
4
Yaw rate (deg/sec)
Lateral speed (m/sec)
0.4
0.2
0
-0.2
-0.4
0
2
4
time (sec)
-2
0
2
4
time (sec)
6
0
2
4
time (sec)
6
0.5
1
Steering (deg)
Lat. Accel. (m/sec 2)
0
-4
6
2
0
-1
-2
2
0
2
4
time (sec)
0
-0.5
6
Figure 4.13. Two-DOF model time response.
Although the equations
presented here are linearized, they are sufficiently accurate under small steering
inputs. In this example, the two-DOF model in Eq. (4.50) and the three-DOF
model in Eq. (4.56) are both simulated and the results are shown in Figures
m
4.13 and 4.14. The lateral acceleration, defined as ay = u0 (β̇ + r) + mR h ṗ for the
EXAMPLE 4.6: SIMULATION OF AUTOMOBILE HANDLING.
3 DOF
3 DOF
4
Yaw rate (deg/sec)
Lateral speed (m/sec)
0.4
0.2
0
-0.2
-0.4
0
2
4
time (sec)
3 DOF
-2
0
2
4
time (sec)
3 DOF
6
0
2
4
time (sec)
6
2
roll angle (rad)
Lat. Accel. (m/sec 2 )
0
-4
6
2
1
0
-1
-2
2
0
2
4
time (sec)
6
Figure 4.14. Three-DOF model response.
1
0
-1
-2
4.3 Lateral Vehicle Motion
three-DOF model, is different than the two-DOF model (where p = 0). This,
however, is not the main reason why the three-DOF response differs from that
of the two-DOF. What do you think is the main reason?
% Ex4_6a.m
% 2DOF model
a
= 1.14;
% distance c.g. to front axle (m)
L
= 2.54;
% wheel base (m)
m
= 1500;
% mass (kg)
Iz = 2420.0;
% yaw moment of inertia (kg-mˆ2)
Caf = 44000*2;
% cornering stiffness--front axle (N/rad)
Car = 47000*2;
% cornering stiffness-- rear axle (N/rad)
b=L-a; g=9.81;
Kus = m*b/(L*Caf) - m*a/(L*Car);
% (rad/(m/secˆ2))
u_char = (L/Kus)ˆ0.5;
% understeer vehicle
u = u_char;
A=[-(Caf+Car)/(m*u),
(b*Car-a*Caf)/(m*u)-u
(b*Car-a*Caf)/(Iz*u),
-(aˆ2*Caf+bˆ2*Car)/(Iz*u)];
B=[Caf/m; a*Caf/Iz];
C_lat = [1 0]; D_lat = 0;
% Lateral speed
C_yaw = [0 1]; D_yaw = 0;
% Yaw rate
C_acc=A(1,:) + u*[0,1];
D_acc = B(1);
% Lateral acceleration
C = [C_lat; C_yaw; C_acc];
D = [D_lat; D_yaw; D_acc];
t=[0:0.01:6];
U=0.5*pi/180*sin(1/3*2*pi*t);
% 0.5 degree, 0.333Hz
% sine steering
Y=lsim(A,B,C,D,U,t);
% Note small lsim
subplot(221)
plot(t,Y(:,1),‘r’); grid
xlabel(‘time (sec)’)
ylabel(‘Lateral speed (m/sec)’)
subplot(222)
plot(t,Y(:,2)*180/pi,‘r’); grid
xlabel(‘time (sec)’)
ylabel(‘Yaw rate (deg/sec)’)
subplot(223)
plot(t,Y(:,3),‘r’); grid
xlabel(‘time (sec)’)
ylabel(‘Lat. Accel.(m/secˆ2)’)
subplot(224)
plot(t,U*180/pi,‘r’); grid
75
76
Review of Vehicle Dynamics
xlabel(‘time (sec)’)
ylabel(‘Steering (deg)’)
% Ex4_6b.m
% 3DOF model
mR=1363.64; mNR=136.36;
m=(mR + mNR);
% Kg
IzzNR=220.0; IxxR=400.0; IxzR=75.0; IzzR=2200.0;
% Kg-mˆ2
c=0.14; e=1.4; h=0.35; b=1.4; a=1.14; % meters
L=a+b; g=9.81;
u=33.7256;
% vehicle speed
Theta_R=(5.0*pi/180);
% Theta_R = 5 degree
Caf=44000*2;
% cornering stiffnessfront axle (N/rad)
Car=47000*2;
% cornering stiffness-rear axle (N/rad)
Cgf=2000*2;
% camber thrust stiffness (N/rad)
dgfdf=0.8;
ddrdf=-0.095;
kR=700*180/pi;
cR=21.0*180/pi;
%
%
%
%
degree incline change per degree roll
degree rear steering per degree roll
N-m per radian of roll
N-m per rad/sec of roll rate
% Define the coefficients in the matrix equations:
Ix=IxxR + mR*(hˆ2) - 2*Theta_R*IxzR + (Theta_Rˆ2) *IzzR;
Ixz=mR*h*c - IxzR + Theta_R*IzzR;
Iz=IzzR + IzzNR + mR*(cˆ2) + mNR*(eˆ2);
Yb = -(Caf+Car); Yr = (b*Car-a*Caf)/u;
Yf = (Car*ddrdf)+(Cgf*dgfdf); Yd = Caf;
Nb = b*Car - a*Caf; Nr = -(aˆ2*Caf+bˆ2*Car)/u;
Nf=a*Cgf*dgfdf - b*Car*ddrdf; Nd= a*Caf;
Lp= -cR; Lf = (mR*g*h-kR);
% Transform into state equation form:
E=[m*u 0 mR*h 0;
0 Iz Ixz 0;
mR*h*u Ixz Ix 0;
0 0 0 1];
F=[-Yb (m*u-Yr) 0 -Yf;
-Nb -Nr 0 -Nf;
0 mR*h*u -Lp -Lf;
0 0 -1 0];
G=[Yd;Nd;0;0];
A=-(inv(E)*F); B=inv(E)*G;
% Define the outputs as
% lat speed, lat accel, yaw rate and roll angle
C=[u 0 0 0
u*(A(1,:)+[0 1 0 0])+(mR*h/m)*A(3,:)
4.4 Vertical Vehicle Motion
77
0 1 0 0
0 0 0 1];
D=[0;u*B(1)+(mR*h/m)*B(3);0;0];
t=[0:0.01:6];
U=0.5*pi/180*sin(1/3*2*pi*t);
%0.5 degree, 0.333Hz
%sine steering
Y=lsim(A,B,C,D,U,t);
subplot, subplot(221)
plot(t,Y(:,1),‘r’); title(‘3 DOF’);
xlabel(‘time (sec)’); ylabel(‘Lateral speed (m/sec)’); grid;
subplot(222)
plot(t,Y(:,3)*180/pi,‘r’); title(‘3 DOF’);
xlabel(‘time (sec)’); ylabel(‘Yaw rate (deg/sec)’); grid;
subplot(223)
plot(t,Y(:,2),‘r’); title(‘3 DOF’);
xlabel(‘time (sec)’); ylabel(‘Lat. Accel.(m/secˆ2)’); grid;
subplot(224)
plot(t,Y(:,4)*180/pi,‘r’); title(‘3 DOF’);
xlabel(‘time (sec)’); ylabel(‘roll angle (rad)’); grid;
4.4 Vertical Vehicle Motion
The vertical motion of a vehicle caused by irregular road surfaces and suspension
characteristics is important in vehicle design. The human perception and tolerance
of these vertical motions are critical factors in perceived “quality” of a vehicle.
To understand these issues, we must be familiar with each of the following factors:
(1) the excitation sources, (2) the vehicle dynamic response, and (3) the vehicle occupant ride perception. Characteristics of these factors are described in the following
subsections.
Road Model
Although there are other sources of excitation (e.g., imperfections in the tire–wheel
assembly and engine/transmission/driveline excitation), only road excitation is considered here. The road-surface profiles are stochastic in nature and can be represented by their statistical properties. A useful and compact representation is the
power spectral density (PSD) of the road profile, which is found by taking the Fourier
transform of the auto-correlation function of the measured road profile as a function
of time. The Fourier transform essentially represents the measured time signal as a
series of sinusoidal functions with varying amplitudes and phases. The PSD represents the power in the signal at a particular frequency and typically is plotted versus
frequency, ω (in rad/s) or f (in Hertz = Hz = cycles/s). Note that 2π f = ω and that
for a vehicle moving at a constant longitudinal velocity, u0 , the distance traveled,
x, and the time, t, are related by x = u0 t. Thus, we can define a spatial frequency,
ω′ = ω/u0 (in rad/m or rad/ft) or f ′ = f/u0 (in cycles/m or cycles/ft), as shown in
Figure 4.15.
78
Review of Vehicle Dynamics
SITE = 5
SITE = 8
SITE = 10
10
SPECTRAL DENSITY (IN2/CYCLE/FT)
AVERAGE BITUMINOUS
1
AVERAGE PCC
.1
Figure 4.15. PSD of road elevation
(Gillespie 1992).
.01
.001
.0001
.00001
.01
.1
WAVENUMBER (CYCLE/FT)
Sometimes it is convenient, as discussed subsequently, to express the road
input excitation as a velocity or acceleration input rather than a displacement (or
elevation). Figure 4.16 shows the PSD curves for vertical displacement, velocity,
and acceleration for a typical road, assuming a constant vehicle-forward speed of
u0 = 50 miles per hour (mph). Although the elevation PSD decreases significantly
ELEVATION
VELOCITY
10
ACCELERATION
10
.1
1
.01
Real Road PSD
Average Road PSD
1
.01
.001
PSD (g2/Hz)
PSD (in2/sec2/Hz)
PSD (in2/Hz)
.1
.1
.001
.0001
.01
.0001
.00001
.2
1.0
10
FREQUENCY (Hz)
50
.2
1.0
10
FREQUENCY (Hz)
50
.2
1.0
10
FREQUENCY (Hz)
Figure 4.16. PSD of road elevation, velocity, and acceleration (Gillespie 1992).
50
4.4 Vertical Vehicle Motion
79
Figure 4.17. Coloring filter for road-displacement profile.
random
input
road profile
G(s)
with frequency, the velocity PSD remains essentially flat and the acceleration PSD
increases significantly. A completely random signal (i.e., a “white-noise” signal) is
characterized by a flat (or constant) PSD versus frequency plot. This fact can be used
to advantage in simulating road profiles by treating a road profile as the output of a
linear coloring filter G(s = jω) (i.e., a dynamic system with transfer function G(s);
Figure 4.17) with a random input. Thus, we can utilize a random-number generator
available on a computer system as the basis for generating various road profiles.
The PSD of the road profile, Sr (w), is given by:
Sr (ω) = |G(ω)|2 (σ 2 )
(4.59)
where σ 2 is the variance (i.e., square of the standard deviation) of a zero mean,
normally distributed random input from a Gaussian (normal) random-number
generator. Assuming a simple first-order filter, G(s) = (ω0 /(s + ω0 )), we obtain
|G( jω)|2 = [(ω0 /ω)2 /((ω0 /ω)2 + 1)] and:
1/( f ′2 )
′
Sr ( f ) = S0
(4.60)
1 + ( f0′ / f ′ )2
where:
Sr (f ′ ) = PSD of road displacement (elevation).
S0 = (σ f0′ )2 is the roughness magnitude parameter
(e.g., 1.25E-5 ft for a rough road and 1.25E-6 ft for a smooth road).
′
f0 = ω0 /(2π u0 ) is the cutoff spatial frequency
(e.g., 0.05 cycle/ft for bituminous and 0.02 cycle/ft for Portland-cement roads).
It also is typical to represent the road-velocity profile – because its PSD is fairly
flat (see Figure 4.16) – as a purely random (i.e., white-noise) input with Sr( f ′ ) =
constant = (σ 2 /2π ).
A road-profile simulation can be
carried out using Eq. (4.60) and the values f0′ = 0.02, u = 80 ft/sec, and S0 =
1.25 × 10−5 ft. The simulation results are shown here and illustrate a fairly flat
PSD for the random input variable, w(t), and one that falls off with frequency
(similar to Figure 4.15) for the simulated road elevation, y(t). Unlike Figure 4.16,
the frequency here is plotted on a linear rather than a logarithmic scale.
EXAMPLE 4.7: SIMULATION OF ROAD PROFILE.
% Ex4_7.m
%
% Generate a zero-mean
% normally distributed sequence
% of 256 numbers with standard
% deviation = 1 as w
w=zeros(256,1);
Vel=80;
% ft/sec
w0 = 2.*pi*Vel*0.02;
% PCC road surface
80
Review of Vehicle Dynamics
S0 = 1.25e-5;
% rought road
sigma = (2*pi*Vel*sqrt(S0))/w0;
w = sigma*randn(size(w));
% Define the filter
T = 0.001; Fs = 1/T;
B = [1-exp(-w0*T)];
A = [1 -exp(-w0*T)];
% Obtained filter output y
% for the random input w
y=filter(B,A,w);
% Determine the power spectrum
% for y and w
P = spectrum(w,y,256);
specplot(P,Fs)
10
10
10
10
10
10
Pxx - X Power Spectral Density
0
-1
-2
-3
-4
-5
0
10
10
10
10
10
100
-2
200
300
Frequency
Pyy - Y Power Spectral Density
400
500
-4
-6
-8
-10
0
100
200
300
Frequency
400
500
4.4 Vertical Vehicle Motion
81
ms
zs
ks
Figure 4.18. Standard quarter-car model.
mus
zus
kus
z0
Vertical-Vehicle-Motion Model
Ideally, the vehicle-ride dynamics consider the pitch and roll motions of the vehicle
as well as the vertical (i.e., heave) motion. A half-car model accounts for both pitch
and heave motions and leads to a four-DOF model. However, it can be shown that
the coupling between the pitch-and-roll and the heave motions is not significant
for typical passenger vehicles, and it is adequate to consider a so-called quarter-car
model as illustrated in Figure 4.18. A two-DOF quarter-car model considers both
the vehicle sprung mass and the unsprung mass associated with the wheel/tire/axle
assembly. A one-DOF quarter-car model neglects the unsprung mass. Whereas both
models are useful in suspension design and control, the two-DOF quarter-car model
represents a good compromise between model simplicity and accuracy.
In the following analyses (which are used throughout this book), only the twoDOF quarter-car model of vertical-vehicle dynamics is considered. In the model, an
active suspension is assumed to be in parallel with a passive suspension, ks and cs .
The tire stiffness, kus , and damping, cus , also are modeled. Note that the tire damping,
cus , can sometimes be neglected. The dynamic equations that govern the motions of
these two masses are as follows:
ms z̈s + cs (żs − żus ) + ks (zs − zus ) = − f
mus z̈us + cs (żus − żs ) + ks (zus − zs ) + cus (żus − ż0 ) + kus (zus − z0 ) = f
(4.61)
(4.62)
The equations of motion in standard-state variable form are as follows:
⎡
⎤
0
1
0
0
⎡
⎤
⎤
⎡
⎢ kus
(cs + cus )
ks
cs ⎥ zus − z0
zus − z0
⎢−
⎥
−
⎢
⎢
⎥
⎥
d ⎢
mus
mus
mus ⎥
⎥ ⎢ żus ⎥
⎢ żus ⎥ = ⎢ mus
⎢ 0
⎥
⎣
⎦
z
−
z
dt ⎣ zs − zus ⎦
−1
0
1 ⎥
s
us
⎢
⎣
⎦
c
k
c
żs
żs
s
0
− s − s
ms
ms
ms
⎡
⎤
⎤
⎡
0
−1
⎢ ms ⎥
⎢ cus ⎥
⎢
⎥ f
⎥
⎢
⎢m ⎥
m ⎥
+⎢
(4.63)
⎢ us ⎥ m + ⎢ us ⎥ ż0
⎣ 0 ⎦ s ⎣ 0 ⎦
−1
0
⇒ ẋ = Ax + Bu + Gw
cs f
cus
82
Review of Vehicle Dynamics
where u(t) is the scalar control force (normalized by the sprung mass) and the state
variables are the tire deflection x1 = zus − z0 , the unsprung-mass velocity x2 = żs , the
suspension stroke x3 = zs − zus , and the sprung-mass velocity x4 = żs . The coefficients
of the state equation also can be presented in the following normalized form:
⎡
⎤
⎤
⎡
⎤
⎡
0
−1
0
1
0
0
⎢
⎥
⎢
⎥
⎢ −ω2 −2(ρζ2 ω2 + ζ1 ω1 ) ρω2 2ρζ2 ω2 ⎥
1
2
⎥ ; B = ⎢ ρ ⎥ ; G = ⎢ 2ζ1 ω1 ⎥
A=⎢
⎣ 0 ⎦
⎣ 0 ⎦
⎣ 0
−1
0
1 ⎦
0
2ζ2 ω2
−ω22
−2ζ2 ω2
0
−1
(4.64)
where typical values
of the parameters are ρ = (ms /mus ) = 10.0, ω1 = kus /mus =
20π rad/s, ω2 = ks /ms = 2π rad/s, ζ1 = cus /(2mus ω1 ) = 0.0, and ζ2 = cs /(2ms ω2 ) =
0.3. The ground velocity input, w(t) = z0 (t), is assumed to be zero mean and Gaussian
with a variance of 2π Au0 , where A is the ground-motion amplitude and u0 is the
vehicle-forward velocity; typical values are A = 1.6 × 10−5 ft and u0 = 80 ft/s. If
there is no active suspension-normalized control force, u(t) = 0, then the equations
represent the vertical motion for a system with only a passive suspension.
Using the model in
Eqs. (4.61)–(4.64) with the parameter values given immediately following those
equations, we can simulate the response of the vehicle-vertical motion to a
random ground-velocity input w(t). The simulation results are shown here. The
plots show tire deflection, suspension stroke, and vertical acceleration of the suspended mass versus time. A more useful output of the program is the frequencyresponse plots for the same variables (also shown here). The peaks at ω2 = 2π
rad/s (1 Hz) are associated with the suspension mode, whereas the peaks at
ω1 = 20π rad/s (10 Hz) are associated with the “wheel-hop” mode.
EXAMPLE 4.8: SIMULATION OF VEHICLE-VERTICAL RESPONSE.
% Ex4_8.m
% x(1) = tire deflection (zus-z0)
% x(2) = velocity of unsprung mass (d(zus)/dt)
% x(3) = suspension stroke (zs-zus)
% x(4) = sprung mass speed (d(zs)/dt)
% cs = damping of passive suspension
% ks = stiff. of passive suspension
% ms = sprung mass
% mus = unsprung mass
% cus = tire damping coefficient
% kus = tire stiffness
% road displacement input = z0(t)
% road velocity input = d(z0)/dt
% Parameters:
w1 = 20*pi; % w1 = sqrt(kus/mus)
w2 = 2.0*pi; % w2 = sqrt(ks/ms)
z1 = 0.0; % z1 = cus/(2*mus*w1)
z2 = 0.3; % z2 = cs/(2*ms*w2)
4.4 Vertical Vehicle Motion
rho = 10.; % rho = ms/mus
% Open loop system equations:
A = [0 1 0 0
-w1ˆ2 -2*(z2*w2*rho+z1*w1)
rho*w2ˆ2
2*z2*w2*rho
0 -1 0 1
0 2*z2*w2 -w2ˆ2 -2*z2*w2];
B = [0 rho 0 -1]’;
G = [-1 2*z1*w1 0 0]’;
% Define outputs of interest:
C1=[1 0 0 0]; D1= 0.0;
% output=tire deflection
[num1, den1]=ss2tf(A,G,C1,D1,1);
C2=[0 0 1 0]; D2= 0.0;
% output=suspension stroke
[num2, den2]=ss2tf(A,G,C2,D2,1);
C3=[A(4,:)]; D3= 0.0;
% output=sprung mass accel.
[num3, den3]=ss2tf(A,G,C3,D3,1);
% Generate the white noise input w(t)
t=[0:0.001:1];
Amp=1.65E-5;
Vel=80; sigma=sqrt(2.*pi*Amp*Vel);
w=sigma*randn(size(t));
% Simulate the response of interest:
y1=lsim(num1,den1,w,t);
y2=lsim(num2,den2,w,t);
y3=lsim(num3,den3,w,t);
clf; subplot(321), plot(t,y1,‘r’);
title(‘Response to Road Velocity Input’)
xlabel(‘Time, t(sec)’); ylabel(‘Tire Def’)
subplot(322), plot(t,y2,‘r’);
title(‘Response to Road Velocity Input’)
xlabel(‘Time, t(sec)’); ylabel(‘Susp Stroke’)
subplot(323), plot(t,y3,‘r’);
title(‘Response to Road Velocity Input’)
xlabel(‘Time, t(sec)’); ylabel(‘Sprung. mass accel’)
% Obtain and plot the frequency response
freq=logspace(-1,2,100);
[mag1, phase1]=bode(num1,den1,freq);
subplot(324), loglog(freq,mag1,‘r’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency, rad/s’);
ylabel(‘Tire def’); grid;
[mag2, phase2]=bode(num2,den2,freq);
subplot(325), loglog(freq,mag2,‘r’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency, rad/s’);
83
84
Review of Vehicle Dynamics
ylabel(‘Susp stroke’); grid;
[mag3, phase3]=bode(num3,den3,freq);
subplot(326), loglog(freq,mag3,‘r’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency, rad/s’);
ylabel(‘Sprung mass accel’); grid;
1
-3
x 10 Response to Road Velocity Input
2
1
Susp Stroke
Tire Def
0.5
0
-0.5
-1
0
0.2
0.4
0.6
0.8
Time, t(sec)
Response to Road Velocity Input
-1
0.2
0.4
0.6
0.8
Time, t(sec)
Frequency Response Magnitude
1
-2
Tire def
Sprung. mass accel
0
10
0
-0.1
10
-3
10
-4
0
0.2
0.4
0.6
Time, t(sec)
0.8
10
1
Frequency Response Magnitude
0
-1
10
2
0
1
0
1
10
10
Frequency, rad/s
Frequency Response Magnitude
2
10
10
Sprung mass accel
10
Susp stroke
-1
-3
1
0.1
-1
10
-2
10
-3
10
0
-2
0.2
-0.2
-3
x 10 Response to Road Velocity Input
1
10
0
10
-1
-1
10
0
1
10
10
Frequency, rad/s
2
10
10
-1
10
10
10
Frequency, rad/s
2
10
An interesting phenomenon of vehicle-suspension systems (both active and
passive) is the existence of invariant points. Because the active suspension force acts
between two masses, the overall dynamics of the suspension system is:
ms z̈s + mu z̈us + kus (zus − z0 ) = 0
(4.65)
which is independent of the passive and active design and therefore is termed the
invariant equation. In deriving Eq. (4.65), the tire damping, cus , which is usually very
4.4 Vertical Vehicle Motion
85
Figure 4.19. Constant comfort lines from various standards (Gillespie 1992).
small, is neglected. By defining three transfer functions for acceleration, rattle space,
and tire deflection, as follows:
HA (s) ≡
z̈s (s)
ż0 (s)
HRS (s) ≡
zs (s) − zus (s)
ż0 (s)
HT D (s) ≡
zus (s) − z0 (s)
ż0 (s)
(4.66)
the I/O relationship between the road excitation and the acceleration, rattle space
(i.e., suspension stroke), and tire deflection can be studied. Hedrick and Butsuen
(1988) show that because of this invariance property, a force (from a passive or
active suspension) acting between the sprung and unsprung masses cannot independently affect all three transfer functions in Eq. (4.66). Specifically, they show
that only one of these transfer functions can be independently specified (i.e., at
certain frequencies, near the “wheel-hop” frequency, and in the frequency range
of interest for suspension design). Levitt and Zorka (1991) also studied this problem, in the case in which the tire damping, Cus , is small but nonzero. They point
out that this invariance property can be altered significantly, even for small tire
damping, and they argue that tire damping should not be ignored in suspension
design.
Ride Model
The perception of the ride by a vehicle occupant is the final criterion for how a
suspension is judged. It is a subjective criterion based on the cumulative effects of
many factors, including suspension design, seat design, and vibration response of the
human body. Various researchers have attempted to quantify passenger perceptions
of comfort by relating them to measurable quantities, such as vertical acceleration and “jerk” (i.e., the time rate of change of acceleration). Figure 4.19 shows
lines of constant comfort as determined by various researchers on a plot of vertical
86
Review of Vehicle Dynamics
acceleration versus excitation frequency. Because of different interpretations of
comfort used in various studies, these results should be used only as a qualitative
guide. The results all show, however, a minimum tolerance (i.e., maximum sensitivity) to vertical acceleration in the range of 4 to 8 Hz. This sensitivity is recognized
as the result of vertical resonance of the abdominal cavity, which can lead to motion
sickness. The ISO curves in Figure 4.19(a) show that the duration of the exposure
also affects the maximum tolerable level of acceleration. The National Aeronautics
and Space Administration (NASA) results in Figure 4.19(b) show that the shape of
the curves depends on the acceleration level and tends to flatten out at low acceleration levels, and that discomfort is rather independent of frequency. Although the
reduction of vibrations to improve ride is certainly the primary goal, some level of
vibration may be desirable to provide “road feel,” which is considered an essential
element of feedback to the driver of a vehicle. In fact, suspension design – active or
passive – typically involves several important tradeoffs: (1) improve passenger comfort, (2) reduce suspension stroke for packaging reasons, and (3) improve handling
by reducing wheel hop and maintaining good tire–road contact.
EXAMPLE 4.9: TRADEOFFS IN SUSPENSION DESIGN. We assume that the suspension
design problem can be formulated as a design-optimization problem, in which
the goal is to select the suspension-stiffness parameter, ks , to minimize the
performance index (objective function):
J = ẋ24rms + r1 x21rms + r2 x23rms
where
ẋ4rms is the root mean square (rms) sprung-mass acceleration and represents
a measure of passenger-comfort goals.
x1rms is the rms tire deflection and represents a measure of road-handling
goals.
x3rms is the rms suspension stroke and represents a measure of packaging
goals.
r1 , r2 are weights that quantify the tradeoffs desired and must be selected
based on experience.
A plot of J versus ω2 = ks /ms is shown here and has a minimum near the
nominal value of ω2 = π rad/s, suggested previously.
The rms values of the variables of interest are calculated in MATLAB using
special functions available for that purpose based on the model in Eqs. (4.61)–
(4.64). For any system of the form ẋ = Ax + Gw, with a white-noise input vector
w of covariance W, the steady-state solution for x can be obtained by solving the
following algebraic matrix equation:
AXss + Xss AT = −GWGT
(4.67)
where Xss is the steady-state covariance matrix for x and W is the covariance
(“strength”) of the random disturbance (= 1 here). Equation (4.67) is termed
a Lyapunov equation and can be solved for Xss given A, G, and W. In the
following MATLAB example program, the steady-state covariance matrix (i.e.,
the solution of the previous Lyapunov equation) is obtained by calling a special
function (lyap()).
4.4 Vertical Vehicle Motion
% Ex4_9.m
%
% x(1) = tire deflection (zus-z0)
% x(2) = speed of unsprung mass
% x(3) = suspension stroke (zs-zus)
% x(4) = velocity of sprung mass
%
% cs = suspension damping
% ks = suspension stiffness
% ms = sprung mass
% mus = unsprung mass
% cus = damping coefficient of the tire
% kus = tire stiffness
% road displacement input = z0(t)
% road velocity input = d(z0)/dt
% Specify the parameter values here:
clear all;
w1 = 20*pi; % w1 = sqrt(kus/mus)
w2 = 2.0*pi; % w2 = sqrt(ks/ms)
z1 = 0.02; % z1 = cus/(2*ms*w1)
z2 = 0.3; % z2 = cs/(2*ms*w2)
rho = 10.; % rho = ms/mus
Amp=1.65E-5; Vel=80;
% Input velocity characteristics
for i=1:20,
w(i) = i*(2.0*pi)/10.0;
w2= w(i);
% Open loop system equations:
A = [0 1 0 0
-w1ˆ2 -2*(z2*w2*rho+z1*w1) rho*w2ˆ2 2*z2*w2*rho
0 -1 0 1
0 2*z2*w2 -w2ˆ2 -2*z2*w2];
B = [0 rho 0 -1]’;
G = [-1 2*z1*w1 0 0]’;
%
% calculate the rms response to
% a unit variance white
r1=5.0e4; r2=5.0E3;
% Weights for typical (T) ride case
Xss=lyap(A,G*G’);
x3_rms=sqrt(Xss(3,3));
x1_rms=sqrt(Xss(1,1));
x4dot_rms=sqrt(A(4,:)*Xss*A(4,:)’+ G(4)*G(4)’);
J(i)=(2.*pi*Amp*Vel)*(x4dot_rmsˆ2+r1*(x1_rmsˆ2)+r2*(x3_rmsˆ2));
end;
clf;
plot(w/(2*pi),J,‘r’);
xlabel(‘w2 (Hz)’), ylabel(‘Cost function J’),grid;
87
88
Review of Vehicle Dynamics
100
90
Cost function J
80
70
60
50
40
30
20
0
0.2
0.4
0.6
0.8
1
w2 (Hz)
1.2
1.4
1.6
1.8
2
In this chapter, aspects of vehicle dynamics (i.e., those most useful for subsequent chapters on vehicle control) are reviewed. For example, vehicle longitudinal
dynamics is useful when we consider the design of ABS, traction control, cruise
control, and platooning. Vehicle lateral dynamics is useful for 4WS, vehicle stability
control, active safety systems to prevent road-departure accidents, and automated
steering for AHS. Similarly, vehicle vertical dynamics is important for active suspension design. Clearly, there are many other interesting topics in vehicle dynamics
(e.g., rollover, nonconventional vehicles, articulated vehicles, and sloshing of liquid
in tankers) that are not reviewed here. Interested readers may refer to additional
information in the references provided.
PROBLEMS
1. For a passenger car with the following parameters (see Figure 4.5, m: mass; f:
rolling resistance coefficient; g: gravitational acceleration):
m = 1,000 kg
h = 0.6 m
a = 1.0 m b = 1.5 m
f = 0.02
g = 10 m/sec2
Assume for this vehicle that it is FWD, the radius of the front tires is 0.3 m, the road
is flat, the surface is dry, and therefore no excessive tire slip will occur.
Please answer the following:
(a) When a driving torque of 600N-m is applied to the front axle, calculate the
vehicle startup longitudinal acceleration (acceleration at t = 0+).
(b) When a vehicle is driving from standstill under the same conditions described
previously but now on an upslope of 5 percent (for simplicity, assume sin θ =
0.05 and cos θ = 1), calculate the normal load on the front and rear axles at
initial startup (t = 0+).
Problems 2–6
89
2. Reconsider Example 4.3. If a constant braking force of Fb = 1,000.0 N is applied
at time t = 0 seconds, estimate the stopping time and stopping distance. Explain your
assumptions and calculations.
3. Reconsider the vehicle-braking simulation in Example 4.3 (see also the MATLAB
program and the parameter values given there). Compare the stopping distances and
times that can be achieved by two slightly different vehicles. The braking force is
given by:
Fb = (Tb /rw ) + Iw /rw2 (du/dt ) if Fb < mFz
Fb = mFz if Fb ≥ mFz
The first vehicle is equipped with an ABS such that it can maintain μ ≈ μP , where
μP = 0.7. The second vehicle does not have ABS and may experience wheel lockup
shortly after braking, such that m = 0.5. Use the values Iw = 0.003 kg-m2 , rw =
0.3 m, and Fz = mg = (2,000 kg) (9.8 m/s2 ) = 19,600 N. Consider the brake-torque
level Tb = 7,000 N-m.
4. For a vehicle with ideal tires (i.e., Fy /Fz is proportional to the tire-slip angle):
(a) If we redesign the powertrain location so that the vehicle weight does not
change but the CG moves forward (i.e., new a = 90 percent of the old a,
where a is the distance from the CG to the front axle; the length l is not
changed), will the vehicle become more understeer? More oversteer? By
how much?
(b) If we compare an empty car to a car with a full load in the trunk (i.e., m
increased by 40 percent, a increased by 10 percent), will the vehicle become
more understeer? More oversteer? By how much?
5. Recall that the yaw-rate gain is the gain of the transfer function from steering
angle to vehicle yaw rate at steady-state, and that flat tires have lower cornering
stiffness than normally inflated tires. Provide equations and/or calculations as a
basis for discussing whether the following statements are true or false:
(a) “Flat rear tires make a vehicle more understeer.”
(b) “Flat front tires reduce the yaw-rate gain.”
(c) “If all vehicle parameters (including tire cornering stiffness) are fixed and
only the vehicle CG is moved backward, this vehicle becomes more understeer.”
6. In this problem, the performance of a RWD versus a FWD vehicle is considered.
Consider the longitudinal motion of an automobile traveling on a slope where the
angle of the slope is small (i.e., θ ≈ 0).
(a) Let the rolling resistance Rx = 0. Show that:
h
b
W − Fx
L
L
h
a
Fzr = W + Fx
L
L
Fz f =
where Fzf and Fzr are the axle loads on the front and rear wheels and Fx is
the total traction force.
90
Review of Vehicle Dynamics
(b) Find the maximum tractive force possible for the FWD and RWD vehicles
assuming that the coefficient of static friction is μ.
(c) If a = l/4 (i.e., the center of mass is toward the front of the vehicle because of
the large engine and transmission components mass), h = 2l/7, and assuming
μ = 0.7, compare the tractive forces of a FWD versus RWD automobile.
7. Using the two-DOF linearized handling model in Eq. (4.49), simulate the response
of a vehicle to a one-degree change in the steering angle. Plot the results for the lateral
acceleration and the yaw rate. The vehicle model to be used in the simulation is the
following:
" # " #
(mu0 D + Yβ ) (mu0 + Yr )
Yδ
β
=
δf
Nβ
r
(Iz D + Nr )
Nδ
where D denotes the derivative operator d/dt and
Ca f a
Car b
u0
Yb = Ca f + Car
Yr =
Nb = aCa f − bCar
Nr = (a2 /u0 )Ca f + (b2 /u0 )Car Nd = aCa f
u0
−
Yd = Ca f
The parameter values to be used in the simulation are as follows:
l = 2.54 m
g = 9.81 m/s2
Iz = 1,200.0 kg-m2
a = 1.14 m
u0 = 20.0 m/s
Caf = 2,400.0 N/deg
b = l − a = 1.40 m
m = 1,000 kg
Car = 2,056.0 N/deg
Is this vehicle understeer or oversteer? What is the understeer coefficient? What is
the critical or characteristic speed?
8. Consider a one-DOF suspension model (i.e., neglect the unsprung mass). Refer
to the following figure in which z0 is the vertical ground motion input and zs is the
deflection of the vehicle mass. The passive-suspension stiffness kl z + knl z3 is nonlinear, c is the passive-suspension damping, and z is the extension of the suspension
spring around its nominal position.
zs
ms
kl x + knl x 3
c
z0
(a) Write the equation of motion for the sprung mass (it is helpful to draw the
free-body diagram of the sprung mass).
(b) Use the Taylor series and linearize the equation about equilibrium (i.e.,
zs = 0 and z0 = 0). Neglect any effects due to gravity. Clearly show the steps
used to linearize the system.
References
(c) Write the transfer function from road input, z0 , to displacement of the
sprung mass, zs .
(d) Let m = 1 kg, kl = 2 N/m, knl = 1 N/m3 , and c = 2N-s/m. If the road input is
z0 = sin(8π t), what will the magnitude of zs be at steady-state?
9. Refer to the suspension design in Example 4.9. Plot the performance index J
versus ω2 as in Example 4.9 but show on the same plot the curves for each of the
following values of the mass ratio, ρ: ρ = 8, ρ = 10, and ρ = 12. How sensitive are
the value of J and the location of the minimum point to the value of ρ?
10. When a vehicle drives on a “washboard” road surface, it is generally known that
a smoother ride is possible if we drive either very slowly or very fast. Explain this
phenomenon.
REFERENCES
Asgari, J., and D. Hrovat, 1991, “Bond Graph Models of Vehicle 2D Ride and Handling
Dynamics,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced Automotive Technologies–1991, ASME DE–Vol. 40, New York, pp. 391–406.
Bakker, E., L. Nyborg, and H. B. Pacejka, 1987, “Tire Modeling for Use in Vehicle Dynamics
Studies,” SAE Paper 870421.
Bakker, E., H. B. Pacejka, and L. Linder, 1989, “A New Tyre Model with an Application in
Vehicle Dynamics Studies,” SAE Paper 89007.
Bosch, R., 2009, Automotive Handbook, Wiley & Sons, New York.
Chen, H. H., A. A. Alexandridis, and R. M. Chalasani, 1991, “Longitudinal Dynamics and
Performance Assessment of All-Wheel-Drive Vehicles,” in S. A. Velinsky, R. H. Fries, I.
Haque, and D. Wang (eds.), Advanced Automotive Technologies–1991, ASME DE–Vol.
40, New York, pp. 483–99.
Clark, S. D., 1971, Mechanics of Pneumatic Tires. National Bureau of Standards Monograph
122, November 1971.
Dixon, J. C., 1991, Tyres, Suspension and Handling, Cambridge University Press.
Ellis, J. R., 1966, Vehicle Dynamics, London.
Gillespie, T., 1992, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers.
Godthelp, J., G. J. Blaauw, and A. R. A. van der Horst, 1982, “Instrumented Car and Driving
Simulation: Measurements of Vehicle Dynamics,” Institute for Perception TNO, Report
IZF 1982–37, The Netherlands.
Hedrick, J. K. and T. Butsuen, 1988, “Invariant Properties of Automotive Suspension,”
Advanced Suspensions, Proc. of the Institute of Mechanical Engineers, Paper No. C423/88,
October 1988.
Heydinger, G. J., P. A. Grygier, and S. Lee, 1993, “Pulse Testing Techniques Applied to
Vehicle Handling Dynamics,” Proceedings of the 1993 SAE International Congress and
Exposition, SAE Paper 930828.
Kiencke, U., and L. Nielsen, 2005, Automotive Control Systems for Engine, Driveline and
Vehicle, Springer Publishing Co.
Levitt, J. A. and N. G. Zorka, 1991, “The Influence of Tire Damping in Quarter Car Active
Suspension Models,” ASME Journal of Dynamic Systems, Measurement and Control, Vol.
113, March 1991, pp. 134–7.
Lin, C. F., A. G. Ulsoy, and D. J. LeBlanc, 2000, “Vehicle Dynamics and External Disturbance
Estimation for Vehicle Path Prediction,” IEEE Transactions on Control System Technology,
Vol. 8, No. 3, May 2000, pp. 508–18.
MacAdam, C. C., 1989a, “Static Turning Analysis of Vehicles Subject to Externally Applied
Forces – A Moment Arm Ratio Formulation,” Vehicle System Dynamics, Vol. 18, No. 6,
December 1989.
91
92
Review of Vehicle Dynamics
MacAdam, C. C., 1989, “The Interaction of Aerodynamic Properties and Steering System
Characteristics on Passenger Car Handling,” Proceedings of the 11th IAVSD Symposium of
the Dynamics of Vehicles on Roads and Tracks, ed. R. Anderson, Kingston, Canada, Swets
& Zeitlinger B.V. – Lisse, 1989.
MacAdam, C. C., et al., 1990, “Crosswind Sensitivity of Passenger Cars and the Influence of
Chassis and Aerodynamic Properties on Driver Preferences,” Vehicle System Dynamics,
Vol. 19, No. 4, 1990, pp. 201–36.
Pacejka, H. B., 2006, Tyre and Vehicle Dynamics, Elsevier.
Rajamani, R., 2006, Vehicle Dynamics and Control, Springer Publishing Co.
SAE, 1976, Vehicle Dynamics Terminology: SAE J670e, Society of Automotive Engineers.
SAE, 1989, Vehicle Dynamics Related to Braking and Steering, Society of Automotive
Engineers.
Sayers, M., 1991, Introduction to AUTOSIM, University of Michigan, Ann Arbor.
Segel, L., 1966, “On the Lateral Stability and Control of the Automobile as Influenced by the
Dynamics of the Steering System,” ASME Journal of Engineering for Industry, 66, August
1966.
Segel, L., 1975, “The Tire as a Vehicle Component,” in B. Paul, K. Ullman, and H. Richardson
(eds.), Mechanics of Transportation Suspension Systems, ASME, New York, AMD–Vol.
15.
Segel, L., 1990, Vehicle Dynamics, Course Notes for ME558, University of Michigan, Ann
Arbor.
Segel, L., and C. C. MacAdam, 1987, “The Influence of the Steering System on the Directional
Response to Steering,” Proceedings of the 10th IAVSD Symposium of the Dynamics of
Vehicles on Roads and Tracks, Prague, Czech Republic, 1987.
Segel, L., and R. G. Mortimer, 1970, “Driver Braking Performance as a Function of PedalForce and Pedal-Displacement Levels,” SAE Transactions, SAE Paper 700364.
Venhovens, P. J. T., 1993, Optimal Control of Vehicle Suspensions, Doctoral Dissertation,
Delft University, The Netherlands.
Wong, J. Y., 2008, Theory of Ground Vehicles, Wiley & Sons, New York.
5
Human Factors and Driver Modeling
It often is necessary to consider the human role (i.e., drivers and passengers) in the
design of automotive systems. For example, this is evident in the discussion of passenger comfort as a key criterion for suspension design in Chapter 4. Human factors, also
known as human engineering or human-factors engineering, consist of the application
of behavioral and biological sciences to the design of machines and human–machine
systems (Sheridan 2002). The term ergonomics is used as a synonym for human
factors; however, it often is associated with narrower aspects that address anthropometry, biomechanics, and body kinematics as applied to the design of seating and
workspaces. The terms cognitive engineering and cognitive ergonomics also are used
to describe the sensory and cognitive aspects of human interactions with designed
systems. All major automotive companies, as well as many government agencies
(e.g., U.S. Department of Transportation, U.S. Department of Defense, NASA,
and Federal Aviation Administration), have research and engineering groups that
address human factors. This chapter is a brief introduction to human factors, especially as they apply to automotive control system design. The introduction is followed
by a discussion of driver models, especially for vehicle steering.
5.1 Human Factors in Vehicle Automation
Humans (i.e., drivers and passengers) clearly interact with automotive control systems in many ways. Commercial success of a new control technology for vehicles
may depend on not only the effectiveness of that technology but also acceptance by
customers. Established automotive technologies (e.g., automatic transmissions and
cruise control) are widely used in the United States but are less widely adopted in
Europe. Navigation systems are more successful in Japan than in the United States.
Customer acceptance often depends on many difficult-to-quantify factors. New automotive control technologies in which human factors must be considered carefully
include electric vehicles, hybrid electric vehicles, traction control, ABS, intelligent
(or adaptive) cruise control, cruise control, airbags, active suspensions, and navigation systems. For example, ABS design must consider not only prevention of vehicle
skidding on various road surfaces during braking but also whether to provide any
feedback to the driver (e.g., pedal vibration) about ABS activation. ABS is most
effective when a driver applies the brakes continuously rather than pumping the
93
94
Human Factors and Driver Modeling
brake pedal. Thus, the ABS design must consider how the combined human–ABS
system will perform and interact. This is a difficult task because driver behavior
is quite variable among different drivers as well as over time. For example, some
drivers prefer to have brake-pedal-vibration feedback when the ABS is activated,
whereas others are startled and actually stop the braking action in the presence of the
vibrations. Drivers also adapt to a particular technology and their behavior changes
with experience. Thus, a major issue that must be considered is driver adaptation;
that is, the presence of a new technology may cause drivers to change their behavior.
For example, once drivers adapt to the improved braking capabilities of an ABS,
they may drive more aggressively.
Human-factor issues in design also can be complicated by legal considerations.
For example, airbags are effective in reducing deaths due to car crashes among
the general public. However, they actually can increase the risk for drivers (or
passengers) of small physical stature, such as children. Thus, any provisions must
be designed so that they will be used by those who benefit while also enabling
those at higher risk of injury to deactivate them. Alternatively, smart airbags that
deploy differently must be developed. Another example of the influence of legal
considerations is intelligent (or autonomous) cruise-control systems that control
vehicle headway as well as speed. To avoid potential litigation, some of these systems
do not control the brakes, despite the potential for better performance.
As vehicles become more automated and ITS also plan to implement more
intelligence in the highway infrastructure, it is necessary to assess how helpful these
new technologies are to drivers (Barfield and Dingus 1998). For example, driver
distraction caused by using a cellphone (although it can be an important asset in
a roadside emergency) is currently a controversial topic, which has led to bans in
some localities. Making the situation more complicated is the fact that whereas some
research finds that cellphone use increases accidents, other studies have not found
any noticeable effects compared to other distractions. Vehicle navigation systems
(based on GPS technology) also have been introduced and must be carefully designed
to minimize driver distraction (Eby and Kostyniuk 1999).
With platoons of vehicles under automated lateral and longitudinal control,
AHS were demonstrated on a large scale on a special segment of highway in San
Diego, California. However, despite great technological success in the demonstrations, AHS is viewed with skepticism. This is partially due to concerns regarding
interactions between humans and the automated system – for example, transfer of
control at entrances and exits and authority/responsibility between controller and
driver, especially when failures occur.
Consequently, the design of vehicle-control systems must include consideration
of the human role. Although this is a difficult task, human-factor experiments can be
conducted using vehicle experiments on test tracks or roadways, as well as driving
simulators that include actual drivers in the system. Human-factor experiments must
be designed carefully to include the factors of interest and to exclude those that are
not a focus of a study. For example, possible factors to consider in the design of
experiments may include the driving scenario and the driver’s experience, alertness,
health, physical characteristics, age, and gender. Due to significant variations in
driver behavior (both over time and among drivers), the results of human-factor
tests typically must be evaluated statistically. It may be necessary to randomize the
5.1 Human Factors in Vehicle Automation
Figure 5.1. Single-loop driver-vehicle control-system block
diagram.
95
E
Yd
Gd
Y
Gv
+
-
order of tests (e.g., when comparing two candidate system designs) to eliminate any
bias (Charlton and O’Brien 2002; Fuller and Santos 2002).
In the following section, the crossover-model principle, which refers to an experimentally observed characteristic of all human–machine systems, is described. Driver
adaptation and risk-homeostasis theory (Wilde 1994) then are introduced. Finally,
the use of driving simulators for human-factor engineering is described briefly.
Crossover-Model Principle
This principle is based on an experimental observation from many tests performed
on a variety of human–machine systems. It also has been found empirically to be
applicable to driver steering and it is in that context that it is discussed here. Thus,
any good driver model is expected to yield results consistent with this principle. To
explain the principle, refer to the block diagram in Figure 5.1, which is a SISO model
of the driver–vehicle system. In experiments in which human operators are asked to
reduce the error, e, to zero (e.g., in computer simulations or in lane-following driving
tasks), it was observed that regardless of the characteristics of the vehicle-transfer
function, Gv (s), human operators adjust their characteristics (i.e., Gd (s)) such that
the loop-transfer function, Gd (s)Gv (s), has an invariant property in the vicinity of
the crossover frequency, ωc .
This is illustrated in Figure 5.2, which shows the magnitude of the looptransfer function, |Gd ( jω)Gv ( jω)|, versus frequency, ω. The observation, termed
the crossover model, is that this plot exhibits a slope of −20 dB/decade in the vicinity of the crossover frequency, ωc , which is the frequency at which |Gd ( jω)Gv ( jω)|
crosses the 0 dB line (i.e., where the open-loop gain equals 1). This implies that
around the frequency
point, ω = ωc , the loop-transfer function, Gd (s)Gv (s), can be
ω e−sT
approximated by c s . The crossover frequency and, consequently, the closed-loopsystem bandwidth decrease with the difficulty of the control task (i.e., the specific
Gv (s)). However, the basic crossover-model principle applies regardless of the Gv (s)
characteristics.
The crossover principle applies for frequencies near the crossover frequency but
not at much higher or lower frequencies. Nevertheless, for single-loop systems, this
principle essentially leads to correct closed-loop characteristics because the shape of
the open-loop transfer function that is far from the crossover frequency, ωc , usually
has little effect on closed-loop dynamics. In other words, the crossover model can be
Figure 5.2. Frequency-response characteristic of the crossover
model.
Gd Gv
(dB)
0
-20 dB/decade
ωc
log( ω)
96
Human Factors and Driver Modeling
60
40
20
Figure 5.3. FFT of loop-transfer function in Example 5.3.
0
-20
-40 0
10
10 1
10 2
10 3
used with good accuracy to generate the closed-loop transfer function of the system.
The phase margin ( π2 − ωc T ) of the closed-loop system also can be estimated from
the approximation but usually should be treated with caution.
We consider
here a vehicle transfer function, Gv (s), and a driver transfer function, Gd (s),
which are described in detail in Example 5.3. The MATLAB program in
that example also calculates the magnitude of the loop-transfer function (i.e.,
|Gd ( jω)Gv ( jω)|). The magnitude of the loop-transfer function can be calculated
directly from the transfer functions using the MATLAB “bode” command. In
this example, the calculation is performed from the simulation data in Example
5.3 using the “fft” (i.e., Fast Fourier Transform) function available in MATLAB.
The results, illustrated in Figure 5.3, show that the crossover frequency is
ωc ≈ 3π rad/s (1.5 Hz) and that the loop-transfer function exhibits a slope
of approximately −20 dB/decade for ω ≈ ωc . This is in accordance with
the crossover-model principle outlined herein. Thus, the simulated driver in
Example 5.3 gives results consistent with the crossover model.
EXAMPLE 5.1: ILLUSTRATION OF THE CROSSOVER-MODEL PRINCIPLE.
Risk-Homeostasis Theory
It is perhaps not surprising that as vehicles and highways are improved to increase
driver and passenger safety, drivers will push the limits of a new technology. This
hypothesis is termed risk homeostasis (Hoyes et al. 1996; MacGregor and Slovic
1989; Summala 1996) and is a controversial theory. It assumes that a driver behaves
in a manner as to maintain a certain (acceptable) level of risk, the so-called target
risk. Clearly, this level of risk varies from person to person; for example, as road
curvature varies, drivers increase or reduce their vehicle-longitudinal velocity to
maintain an acceptable level of risk. The adaptation of driver behavior to maintain
a certain level of risk can be an important factor in vehicle design as well as many
active safety systems for vehicles.
For example, ABS are designed to reduce not only stopping distances but also
the occurrence of vehicle-lateral instability during braking while steering, especially
in icy conditions. ABS have been proven in many vehicle tests to be quite effective.
When ABS was first introduced commercially as an option, the insurance premiums
5.1 Human Factors in Vehicle Automation
of owners of ABS-equipped vehicles were reduced with the expectation that the
incidence of accidents would decrease. However, an extensive study subsequently
conducted by the insurance industry showed no statistically significant reduction in
accidents of vehicles equipped with ABS. Although there was much debate about
the reasons for this surprising finding, a popular explanation is based on driver
adaptation and risk-homeostasis theory. Another possible explanation is related to
the fact that human drivers need time to adapt to the changed vehicle behavior.
From statistics, the reduction in rear-end crashes is offset by the surge in the number
of vehicle rollovers – likely due to the fact that human drivers accustomed to locked
wheels oversteer to correct vehicle motion.
It has been shown that modifying vehicle design to improve handling in a steering
maneuver does not have the desired result due to driver adaptation (Mitschke 1993).
The steering performance remains the same and the driver adapts to the vehicle
characteristics so as to reduce the effort associated with the steering task. Similar
results also were reported in studies of an active safety system to reduce the incidence
of single-vehicle roadway-departure (SVRD) accidents (Chen and Ulsoy 2002), as
discussed in Example 2 herein.
Driving Simulators
Driving simulators come in various types and are described in this section, including
a human driver and a simulated vehicle and driving scenario. They enable the safe
and inexpensive testing of new automotive control technologies as well as testing
of prototype systems during the development process. Because human factors often
are critical, driving-simulator studies can be an important tool in the development
of automotive control technologies.
In simulator experiments to test braking reactions and following distances of
drivers under various conditions, it was found that following distance correlates
surprisingly poorly with the speed of the traffic stream – to the point that most
drivers could not stop in time to avoid a collision if the driver of the vehicle ahead
were to apply the brakes suddenly (Chen et al. 1995). Due to safety concerns, such
studies are difficult to perform in actual driving experiments. Similarly, drivingsimulator studies were used to develop and evaluate active safety systems to reduce
SVRD accidents (Chen and Ulsoy 2002). The studies showed the effectiveness of
these systems but also provided early identification of issues that must be considered
in their design (e.g., driver adaptation).
The simplest driving simulators run on personal computers with joystick (or
mouse) controls and they are similar to flight simulators available commercially.
More advanced versions may have improved driver controls, such as foot pedals for
braking and acceleration, as well as a steering wheel, which also may have passive
or active force feedback to provide a realistic feel in handling. A key element of
such a simulator is vehicle dynamics, which is simulated based on models such as
those introduced in Chapter 4. Vehicle dynamics must be sufficiently high-fidelity
to provide a realistic driving experience and typically includes nonlinearities (e.g.,
tire models) as well as wind disturbances, ground-disturbance input, and so on.
Simulators also include a graphical display of what a driver sees during driving. These
displays can range from very simple to very complex, and they have an important role
97
98
Human Factors and Driver Modeling
Figure 5.4. Sample screen display of the
driving simulator.
in a driver’s perception of the driving task. Some displays include realistic depictions
of actual roadway-driving tasks of up to several hours. Basic driving simulators are
on a fixed base; however, full-scale driving simulators with a motion base also are
available (e.g., the National Advanced Driving Simulator at the University of Iowa
and large-scale driving simulators at companies including Deere, Ford, Mercedes
and Toyota). These devices have hydraulically actuated platforms that provide the
feel of acceleration and deceleration to simulate a realistic driving experience.
EXAMPLE 5.2: DRIVER ADAPTATION IN AN ACTIVE SAFETY SYSTEM FOR SVRD ACCI-
Referring to the block diagram in Figure 5.1, the proposed active safety
system can be viewed as a controller, Gs (s), placed between the driver steeringcommand output and the actual steering input applied to the vehicle. Consequently, the product of Gs (s) and Gv (s) can be viewed as an equivalent vehiclesteering dynamics model. Essentially, the Gs (s) is designed to provide more
consistent lane-keeping performance for the closed-loop driver–vehicle system,
thereby reducing the occurrence of SVRD accidents. The details of the design
of Gs (s) are not discussed here but readers are referred to Chen and Ulsoy (2001
and 2002).
A personal-computer–based driving simulator was used to evaluate the
effectiveness of this controller in reducing lane-departure accidents (Chen and
Ulsoy 2002). It has a simple graphical display (Figure 5.4) with fairly sophisticated vehicle-dynamics models. The driving simulator was used for straight-road
lane-keeping tasks, subject to lateral wind disturbances.
Two types of experiments were conducted to evaluate the controllers using
the simulator with and without the designed steering-assist controller, as follows:
DENTS.
Long Test. These experiments are designed to address the effect of fatigue.
Therefore, the six subject drivers are required to become inattentive or drowsy
during the experiments. Although systematic measures of driver fatigue have
5.1 Human Factors in Vehicle Automation
99
lane width: w
Figure 5.5. Vehicle location relative to roadway.
vehicle track: d
measured lateral
position: y
been reported in human-factors research, an accurate and quantitative measure of driver fatigue usually requires complex instrumentation. Instead, the
subject drivers are asked to follow guidelines to achieve a certain level of
fatigue (e.g., arise at 7:30 a.m. or earlier, experiments scheduled just before
the “circadian dip of alertness” in the early afternoon, and no coffee). Consequently, the six subject drivers become drowsy and performance degradation
is observed in the experiments. The driving task is at least 60 minutes long of
straight-road driving with a wind disturbance. The human driver functions as
a regulator to maintain the vehicle at the center of the lane. The data from
the long driving experiments are divided into 1-minute segments to observe
how the driving behavior varies with time.
Short Test. These experiments consist of short driving with artificially large
lateral-position error. To generate consistent lateral-position error, the ten
subject drivers are asked to steer the simulated vehicle to the left until the
center of the vehicle reaches the centerline of the road (i.e., the left edge of
the lane). They then steer the simulated vehicle back to the right until the
vehicle center reaches the right edge of the lane. After that, the ten drivers
are asked to steer the vehicle back to the center of the lane as rapidly
as possible. The total experimental time for one test run is approximately
15 seconds. The same test is repeated forty times for each driver (both with
and without the controller) to generate a large dataset for statistical analysis.
The lane-keeping performance, with and without the controller, was then
assessed using the following metrics. First, we consider two time-domain metrics described as follows.
Percentage of Road Departure (PRD). Figure 5.5 shows the relative location
of the vehicle and the roadway. Denote the lateral-position error as y(i), i =
1, 2, . . . N, where N is the number of datapoints for each data segment to be
analyzed. Road departure is defined as when y ≤ d2 or y ≥ w − d2 . These two
cases represent situations in which the vehicle edges exceed the road edges.
PRD is defined as the amount of time in which road departure occurs divided
by the overall simulation time; that is:
N
1 $
f (y(i)),
PRD =
N
i=1
where
f (y(i)) = 1, if y(i) ≤
f (y(i)) = 0, if
d
2
d
2
or y(i) ≥ w −
< y(i) < w −
d
2
d
2
100
Human Factors and Driver Modeling
450
400
350
300
250
200
150
100
50
0
–50
PM
GM
wc
PRD
STD(y)
Figure 5.6. Percentage change in lane-keeping metrics for a case with controller as compared
to a case without controller in short driving tests on the driving simulator.
For the long driving experiments, the overall simulation time is 60 seconds
because each segment of data is 1-minute long. The data show a higher PRD
observed in the later part of the long driving task, which indicates more
fatigued drivers. For the short driving experiments, the overall simulation
time is 15 seconds. The objective of this test is to verify that the PRD, or the
variation in PRD, can be reduced when the controller is implemented.
Standard
Deviation of Lateral Position (STD(y)). STD(y) is defined as
%
1 &N
1 &N
2
i=1 (y(i) − N
i=1 y(i)) . The standard deviation (or variance) of
N−1
lateral-position error is used in the literature as a performance index for
vehicle lane-keeping. With the controller, the goal is to show that STD(y) can
be maintained at a low level or that it will not vary with time (for the long
driving experiments) as much as in the case without the controller.
Also, frequency domain metrics (i.e., gain margin, phase margin, and
crossover frequency for the loop-transfer function, Gd Gs Gv ) can be used.
Using the driving-simulator data, the driver model can be computed. The
driver models then can be combined with the vehicle and controller models to
obtain the loop-transfer-function frequency response. The stability margins
and crossover frequency then are computed and used to evaluate the benefits
of the controllers. Improvement in lane-keeping performance is indicated by
increased stability (i.e., gain and phase) margins. The crossover frequency can
be viewed as a measure of driver effort, with lower values indicating less effort
to maintain a given level of lane-keeping performance.
The results obtained are interesting (Chen and Ulsoy 2002b): In the short
driving tests, all of the metrics show statistically significant improvement in
lane-keeping performance, based on a student t-test. Figure 5.6 shows the
percentage change for the system with the controller compared to the system
5.2 Driver Modeling
101
350
300
250
200
150
100
50
0
–50
–100
PM
GM
wc
PRD
STD(y)
Figure 5.7. Percentage change in lane-keeping metrics for a case with controller as compared
to a case without controller in the long driving tests on the driving simulator.
without the controller in the various metrics for the ten drivers in the short
tests. In most cases, the metrics are improved (i.e., gain and phase margins are
increased and PRD and STD(y) are reduced). Note also that the crossover
frequency is reduced, indicating less effort on the part of a driver to achieve
this level of performance.
Although these results are encouraging, the long driving tests provide a different
conclusion. Figure 5.7 shows no clear (i.e., statistically significant) change in the lanekeeping metrics (i.e., gain and phase margins, PRD, and STD(y)). In some cases,
the metrics improve; in others, they remain the same or worsen. There is only a
statistically significant reduction in crossover frequency, indicating a reduced effort
by a driver to achieve the same (statistically speaking) lane-keeping performance.
The likely explanation is based on risk-homeostasis theory. That is, drivers adapt
their behavior – by using a more relaxed driving style – to maintain the same level
of risk in terms of lane-keeping performance. In the short driving tests, there is
not enough time for this adaptation to occur; therefore, it is observed only in the
long tests. Although not shown here, these conclusions are confirmed by computing
driver models from the various test data, as described in (Chen and Ulsoy 2002).
5.2 Driver Modeling
The driving function can be decomposed into three parts: navigation, path planning,
and control. Navigation addresses the overall selection of a route. Path planning is
concerned with the recognition, decision, and selection of a path responding to traffic
situations (e.g., overtaking). The control process actuates the steering, acceleration,
and brake systems to follow the selected path determined by the two higher levels.
In this section, several driver-control models are presented.
102
Human Factors and Driver Modeling
In Chapter 4, vehicle dynamics is considered without consideration of the driver’s
role. It is obvious that the behavior of a vehicle as a closed-loop system (i.e., with
the driver closing the loop and serving simultaneously as the sensor, the controller,
and the actuator) is different than the behavior of a free vehicle. This is because
the driver is an active element of the dynamic system. For example, by steering,
the driver clearly has an important role in vehicle-lateral motion. The driver also
affects vehicle-longitudinal motions by controlling the tractive and braking forces
through the accelerator and brake pedals. For example, a good driver braking on
an icy surface often “pumps” the brakes to maintain lateral stability – much like
the ABS that is discussed in Chapter 13. Although drivers cannot affect directly the
vehicle-vertical motion, they are an important passive element in the suspensiondesign problem. Drivers also must be considered a passive element in many other
vehicle-control-system design problems (e.g., 4WS and transmission control). In this
chapter, the driver’s role as an active element in a closed-loop system is considered,
particularly in terms of vehicle steering.
To date, there have been numerous studies about developing “driver models”
for steering behavior. These studies are useful in the following ways:
r Safety Issues. How will the combined (i.e., driver and vehicle) system behave
in certain emergency situations and maneuvers (e.g., collision avoidance and
rollover)? Problems such as truck rollover and jackknifing, vehicle rollover (e.g.,
minivans and sport utility vehicles [SUVs]), and emergency handling have long
been significant issues in vehicle design.
r Vehicle Handling. How should a vehicle be designed so that the combined system
will perform well and the driver will not be easily fatigued but rather find the
vehicle pleasurable to drive? Studies in this area show that vehicles that allow
drivers to perform well are not necessarily those that are the most pleasurable
to drive.
Figure 5.8 illustrates a driver-model block diagram and delineates three types
of actions that a driver can perform while driving (Weir and McRuer 1968). This
block diagram shows that a driver may have several types of control action that can
be used, including the following:
r Precognitive behavior, in which a driver has some internally generated maneuver
commands. These usually are rote maneuvers, such as pulling into a driveway.
r Pursuit behavior, which corresponds to a feed-forward or preview action based
on knowledge of an upcoming road path. This is discussed in more detail herein.
r Compensatory behavior, which reacts to sensed errors. This is the transferfunction approach and is a subsequent discussion herein.
Various driver-steering models have been proposed in the past thirty years (Hess
and Modjtahedzadeh 1990; MacAdam 1989; Weir and McRuer 1968). We can classify these into two major categories: (1) transfer-function models, and (2) preview/predictive models. The transfer-function models mainly describe the compensatory behavior of a driver, whereas preview/predictive models focus on a driver’s
pursuit behavior. Each type of driver models is briefly described in the following
sections.
5.2 Driver Modeling
Figure 5.8. Block diagram for human driving task.
Transfer-Function Models
Consider a simple closed-loop system (see Figure 5.1) for lateral-motion control
of a vehicle through steering. The vehicle dynamics is represented by the transfer
function, Gv (s), and the driver (i.e., controller) has the transfer function, Gd (s). Here,
it is assumed implicitly that the driver acts on feedback of the vehicle-lateral position,
y. This assumption is a basic problem with the transfer-function approach; in fact,
the driver of a vehicle acts on many sensed feedback signals (e.g., lateral position
and velocity, yaw and angle and rate, roll angle and rate, and pitch angle and rate).
Thus, there are multiple feedback loops and, typically, different gains and transfer
functions must be associated with each. Such a system becomes complicated and
it is difficult to determine experimentally the control actions associated with each
feedback loop. However, the simple configuration shown in Figure 5.1 is suitable for
the experimental determination of driver control laws and gains. Another advantage
of this simple approach is that it is generalized; no assumptions must be made about
the driving strategy used by a driver. Possible approaches to determining the driver
transfer function, Gd (s), are as follows: (1) Assume that Gv (s) is completely known
from a vehicle-dynamics model and that the form of Gd (s) is known; the parameters
of Gd (s) then can be determined based on measured values of the error, e, and
the steer angle, δ. (2) Alternatively, assume that the form of the combined product
Gd (s)Gv (s) is known; then determine Gd (s) from measurements of e and y. In the
first case, the experimental data required are e and δ; in the second case, they are
e and y.
Hess and Modjtahedzadeh (1990) provided a transfer-function model of driver
steering, illustrated in Figure 5.9. The driver is represented by a low-frequency
compensation block (i.e., a PD controller, as in Example 5.3) and a high-frequency
(i.e., within one decade of the crossover frequency) compensation (or structural
model) block, which includes the driver delay, τ 0 . The block labeled GNM is a simple
second-order model of the neuromuscular system of a driver’s arms. A more detailed
103
104
Human Factors and Driver Modeling
Driver Model
−
low freq.
compensation
GL
eA
+
−
y˙ V (t)
GNM
˙
G
C
yR (t)
high-freq. compensation
(structural model)
+
u
1
Ky (S + T )
3
e −τOS
−
−
δSW
ωn2
+
vehicle
s2 + 2␨nωns + ωn2
yV (t)
K1s
K2
1 k−1
(S + T )
2
(S + 1T )
1
Gp1
Gp2
Figure 5.9. Transfer-function driver model (Hess and Modjtahedzadeh 1990).
description of the driver model, as well as typical values of the model parameters,
is in Hess and Modjtahedzadeh (1990). A simpler version of this model with only
low-frequency compensation and delay is provided in Example 5.3.
A vehicle-transfer function Gv (s)
with front-wheel-steering angle δ as the input and lateral displacement (y) as
the output signal in the straight-road following case can be obtained from this
state-space model:
⎡
⎡
⎤
⎤
0
1
uo
0
0
⎡ ⎤
y
⎢
⎥⎡ y ⎤ ⎢
⎥
Cα f + Cαr
bCαr − aCα f
⎢ Cα f ⎥
⎥
⎢ ⎥ ⎢
⎢
⎢
⎥
⎥
0
−
0
−
u
⎥
⎢v⎥ ⎢
o⎥⎢
muo
muo
⎢v⎥ ⎢
⎥ ⎢
m ⎥
d ⎢
⎢
⎥
⎥
⎥
⎢ ⎥=⎢
⎥⎢
⎥δ
⎢ ⎥+⎢
⎥ ⎢
⎢
⎥
⎥
dt ⎢
ψ
0
0
0
1
0
⎢ψ ⎥ ⎢
⎣ ⎦ ⎢
⎥
⎥
⎣ ⎦ ⎢
⎢
⎥
⎥
2
2
aC
⎣
⎣
⎦
r
bCαr − aCα f
a Cα f + b Cαr
αf ⎦
r
0
0
−
Iz
Iz uo
Iz uo
EXAMPLE 5.3: DRIVER HANDLING SIMULATION.
Consider a simple PD controller with delay as the driver model:
Gd (s) = (δ(s)/e(s)) = (Kd + Kdd s)e−sT
(5.1)
If the delay T is sufficiently small, then one of the following Padé approximations
can be used:
e−sT ≈
(T s)2 − 6(T s) + 12
;
(T s)2 + 6(T s) + 12
or
e−sT ≈
1 − 0.5T s
1 + 0.5T s
In the following discussion, the second (i.e., first-order) approximation for the time
delay is used. The vehicle parameters are assumed to be as follows:
m = 1,500 kg
l = 2.54 m
u0 = 20 m/s
IZ = 2,420 kg-m2
a = 1.14 m
Caf = 2,050 N/deg
g = 9.81 m/s2
b = l − a = 1.4 m
Car = 1,675 N/deg
5.2 Driver Modeling
105
step response
3
Figure 5.10. Response of the transfer-function model
driver–vehicle system.
lat. disp. (m)
2
1
0
-1
-2
0
1
2
3
time (sec)
4
If we consider only proportional control with neural delay (i.e., Kdd = 0), then the
system is found to be unstable (which can be verified easily by modifying the example
program presented herein). However, if we add phase lead (e.g., use PD instead of
P control such that both Kd and Kdd are nonzero), then it is possible to find gains
that stabilize the system even with a significant time delay.
The step response of a closed-loop system with a simple PD control and a 0.1second delay is shown in Figure 5.10. Apparently, the response is oscillatory and
although it controls the lateral motion of the vehicle, it does not predict accurately
the human-driver steering response. In part, this is due to neglecting the highfrequency compensation block in Figure 5.9. However, it also is known that human
drivers actually use feedback signals in addition to lateral displacement to stabilize
a vehicle, including path angle, path rate, heading angle, and heading rate.
% Ex5_3.m
a=1.14; l=2.54; b=l-a;
g=9.81; u0=20.0; m=1500; Iz=2420.0;
Caf=2050.0*57.2958; Car=1675*57.2958;
% Define the coefficients
Yb=-(Caf+Car); Yr=(Car*b/u0)-(Caf*a/u0); Yd= Caf; Nb=b*Cara*Caf; Nd= a*Caf; Nr=-(aˆ2/u0)*Caf - (bˆ2/u0)*Car;
A=[0 1 u0 0;
0 Yb/m/u0 0 Yr/m-u0;
0 0 0 1;
0 Nb/Iz/u0 0 Nr/Iz];
B=[0;Yd/m;0;Nd/Iz];
C=[1 0 0 0]; D=0;
[num,Gvden]=ss2tf(A,B,C,D,1);
Gvnum=num(3:5);
Kd = 0.02; Kdd = 0.33;
T = 0.1;
Gdnum=conv([Kdd Kd],[-T/2 1]);
Gdden=conv([1e-6 1], [T/2 1]);
Gcnum=conv(Gdnum,Gvnum);
Gcden=conv(Gdden,Gvden) + [0 0 Gcnum];
t=[0:0.01:5];
5
106
Human Factors and Driver Modeling
yd=[zeros(1,51), ones(1,300), zeros(1,150)];
y=lsim(Gcnum,Gcden,yd,t);
e=lsim(Gcden-[0 0 Gcnum],Gcden,yd,t);
e = yd’-y;
plot(t,yd,‘b’,t,y,‘r’); grid
title(‘step response’)
xlabel(‘time (sec)’)
ylabel(‘lat. disp. (m)’), pause
w=logspace(0, 2, 100);
bode(conv(Gvnum, Gdnum), conv(Gvden, Gdden),w); pause
%Calculate mean, standard deviation, and rms for e(t):
em=mean(e);
es=std(e);
erms=sqrt(mean(e.ˆ2));
% Calculation of open-loop bode plot from simulation data:
Y=fft(y); E=fft(e);
H=Y./E; magH=20*log10(abs(H)); phaseH=angle(H)*(180/pi);
semilogx(magH); grid; pause;
semilogx(phaseH); grid;
Preview/Predictive Models
Another group of models uses a combined pursuit/compensatory action and mimics
human preview and predictive behavior by incorporating a forward model; these
are known as the preview/predictive models. In these models, human drivers scan
through a future desired road path within a finite future distance when performing
a driving task. This behavior is captured in the preview/predictive block. The terms
preview and predictive, respectively, refer to a driver’s ability to see the future
desired path and to predict future vehicle response. Driver models that use preview
information over a horizon often generate approximate inverse control actions,
resulting in superior control quality when compared with transfer-function models.
This is especially true for high-lateral-acceleration path-following tasks (e.g., a sharp
curve or a double-lane-change maneuver). A well-known preview/predictive model
was proposed by MacAdam (1980) in which the driver was assumed to behave like
a preview optimal controller with delay. Because this model is widely verified and
it is closely related to the model proposed herein, the details are presented in the
following discussion.
Figure 5.11. Preview/predicted paths.
5.2 Driver Modeling
107
MacAdam’s model is based on the optimal preview control framework for SISO
linear systems. Given the state equation (e.g., bicycle model) of a vehicle:
ẋ = Ax + Bu
y = Cx
(5.2)
the control (i.e, steering) signal minimizing a quadratic cost function is solved. The
cost function proposed by MacAdam has the following form:
⎧ t+T
⎫
⎪
⎪
⎨ p
⎬
2
(5.3)
uopt (t ) = min
{[yd (η) − y(η)] δ(η − t )}dη
u ⎪
⎪
⎩
⎭
t
where yd (t) is the desired lateral displacement, y(t) is the actual lateral displacement,
and δ(t) is the weighting function over the preview window. In general, u(t) could vary
within the preview window [t, t + Tp ]. However, the solution of this problem involves
solving a partial-differential equation and may be unnecessary for approximating
human behavior. A simpler problem can be formulated by assuming u(t + τ ) = uopt (t),
∀τ ∈ [0, Tp ], which can be solved more easily. The output of the linear dynamics in
Eq. (5.2) is decomposed into zero-input response and zero-state response:
⎛ τ
⎞
y(t + τ ) = CeAτ x(t ) + C ⎝ eAη dη⎠ Bu(t ) ≡ F (τ )x(t ) + G(τ )u(t )
(5.4)
0
The optimal solution for Eq. (5.3) then can be obtained by substituting Eq. (5.4) in
Eq. (5.3) and setting the partial derivative of the cost function J with respect to u as
zero:
t+Tp
uopt (t ) =
+ t
[yd (η) − F (η − t )x(t )] G(η)δ(η − t ) dη
(5.5)
t+Tp
+
G(η)2 δ(η − t )dη
t
The optimal solution uopt (t) shown here can be viewed as a “proportional feedback”
controller operating on the error between the desired output and predicted zero
I/O over the preview window [t, t + Tp ] rather than for a single point in time. This
“previewed proportional control” was found to result in a control law that well
approximates average human-driver behavior and has been implemented in commercial software packages (i.e., CARSIM and TRUCKSIM). Both this approach
and the transfer-function approach discussed previously may need to be made adaptive to account for variations in driver behavior. Experimental studies confirm that
drivers indeed exhibit adaptive behavior in steering tasks, as discussed previously in
Example 5.3.
A simple example of the preview-driver
steering model is presented here. The preview model is not the one proposed
by MacAdam referenced previously; rather, it is a simple linear model that can
be simulated easily using MATLAB. Assume that the desired lateral position is
known at time t + Tp , where t is the current time and Tp is the preview time. In
EXAMPLE 5.4: PREVIEW DRIVER MODEL.
108
Human Factors and Driver Modeling
y d (t + Tp )
Kde
+
−sT
b2 s 2 + b1s + b0 &y&
s2 + a1s + a0
1
s
y˙
-
y
1
s
Tp
Gv' (s)
y(t + Tp )
+
+
Tp 2 /2
+
Figure 5.12. Block diagram of the preview/predictive driver–vehicle system.
other words, the value yd (t + Tp ) is a known reference input. The actual path at
time t + Tp is estimated using the simple model:
, y(t + Tp ) = y(t ) + (Tp )ẏ(t ) + Tp2 2 ÿ(t )
(5.6)
Then, the control action can be defined – for example, if we use a proportional
gain plus delay driver model with preview – as follows (with some abuse of
notation):
δ(t ) = Kd e−sT (yd (t + Tp ) − y(t + Tp ))
(5.7)
The vehicle-transfer function Gv (s) was found to be of the form:
Gv (s) =
b s2 + b1 s + bo
Y (s)
= 22 2
δ(s)
s (s + a1 s + ao )
(5.8)
Therefore, the overall closed-loop system can be presented in the block-diagram
form as shown in Figure 5.12.
Using block-diagram algebra leads to an equivalent vehicle-dynamics model
of the following form:
2 , 2
Tp 2 s + Tp s + 1)(b2 s2 + b1 s + b0 )
′
(5.9)
Gv =
s2 (s2 + a1 s + a0 )
This system can be simulated using a MATLAB program similar to the one
shown previously in Example 5.3; only the numerator of the vehicle-dynamics
transfer function is modified. Notice that the preview/predictive action adds
derivative and double-derivative terms, thereby stabilizing the closed-loop system. The simulation results, with Kd = 0.05 and T = 0.1, are presented in
Figure 5.13.
% Ex5.4.m
a=1.14; l=2.54; b=l-a;
g=9.81; u0=20.0; m=1500; Iz=2420.0;
Caf=2050.0*57.2958; Car=1675*57.2958;
% Define the coefficients
Yb=-(Caf+Car); Yr=(Car*b/u0)-(Caf*a/u0); Yd= Caf;
Nb=b*Car-a*Caf; Nd= a*Caf; Nr=-(aˆ2/u0)*Caf - (bˆ2/u0)*Car;
A=[0 1 u0 0;
0 Yb/m/u0 0 Yr/m-u0;
0 0 0 1;
0 Nb/Iz/u0 0 Nr/Iz];
5.2 Driver Modeling
109
B=[0;Yd/m;0;Nd/Iz];
C=[1 0 0 0]; D=0;
[num,Gvden]=ss2tf(A,B,C,D,1);
Gvnum=num(3:5);
Kd = 0.05; T = 0.1; Tp=0.5;
Gvnum_pv=conv(Gvnum, [Tpˆ2/2 Tp 1]);
Gdnum=Kd*[-T/2 1];
Gdden=[T/2 1];
Gcnum=conv(Gdnum,Gvnum_pv);
Gcden=conv(Gdden,Gvden) + Gcnum;
t=[0:0.01:5];
yd=[zeros(1,51), ones(1,300), zeros(1,150)];
y_p=lsim(Gcnum,Gcden,yd,t);
e=yd-y_p’;
% calculate the true lateral displacement
steer=lsim(Gdnum,Gdden,e,t);
y=lsim(Gvnum, Gvden, steer,t);
plot(t,yd, ‘-g’, t,y_p, ‘r’, t-Tp, y,‘-.b’); grid
xlabel(‘time (sec)’)
title(‘Lat. disp. (m)’)
legend(‘Yd’, ‘Y_p’, ‘Y’)
Longitudinal-Driver Models
The state of knowledge on driver models for the longitudinal control of vehicles is
poor compared with lateral control. This does not mean that there are few longitudinal driver models; in fact, there are numerous publications and a short review is
provided herein. However, the accuracy of the longitudinal models is not as good as
for the lateral models for several reasons. First, the lane-keeping task is well delineated. A driver simply must stay inside a tightly defined 12-foot lane, which means
2
Yd
Yp
1.5
Y
1
0.5
0
–0.5
–1
–1
0
1
2
3
4
5
Figure 5.13. Response of the preview/predictive driver–vehicle system.
110
Human Factors and Driver Modeling
that the maximum error on either side of the vehicle is 3 feet for passenger cars and
less than 2 feet for larger trucks! For comparison, in theory, a human driver has an
infinite margin for maintaining a large range between the controlled and the lead
vehicles. Second, vehicle longitudinal dynamics varies more among vehicles. Third,
many previous longitudinal-driver models were aimed at representing traffic-flow
behavior (i.e., macroscopic) rather than accurate vehicle-following behavior. The
research results from a macroscopic viewpoint, however, could provide an average
longitudinal model.
Since the early 1950s, many longitudinal-driving models have been proposed,
forming the basis for microscopic traffic-simulation studies. Pipes (1953) suggested
a linear “follow-the-leader” model, which assumes that the driver aims to accelerate
the vehicle in proportion to the speed difference between the lead and the following
vehicles. The proportional constant is termed the sensitivity of the driver model, and
this desired acceleration is realized after a neuromuscular delay. In mathematical
form, this model can be described as follows:
u̇n+1 (t + T ) = λ(un+1 − un )
(5.10)
This is one example of the stimulus–response model, in which the stimulus is the
relative speed and the response is the acceleration with time delay T. An apparent
drawback of this model is that the desired speed and spacing between vehicles
were not used explicitly. In other words, this equation is not a satisfactory vehiclefollowing model if what we are interested in is the vehicle-level response. Chandler
(1958) identified the parameters (i.e., sensitivity and delay) of the Pipes model based
on measured vehicle-following data. Gazis et al. (1961) extended the Pipes’s model
by assuming that the sensitivity of the follow-the-leader model is proportional to the
mth power of velocity over the lth power of range error. In other words:
ẍn (t + τ ) =
cẋn (t + τ )l
[ẋ (t ) − ẋn (t )]
[xn−1 (t ) − xn (t )]m n−1
(5.11)
Newell (1961) proposed a different model based on the assumption that a human
driver has a desired speed and a natural tendency to converge exponentially to
this desired speed. Tyler (1964) formulated the human driver as a linear optimal
controller; the cost function being optimized is a quadratic function of range error
and range-rate error. Later, Burnham et al. (1974) modified Tyler’s model to include
human reaction time and vehicle nonlinearities. Gipps (1981) proposed a switchingvehicle-speed model based on two mutually exclusive considerations: (1) to keep
a safe distance from the lead vehicle; and (2) to converge to the desired free-flow
speed. The minimum of the two speeds is used:
⎧
!
⎪
v
v (t )
(t
)
⎪
⎪
vn (t ) + 2.5an τ 1 − n
0.025 + n
⎪
⎪
⎨
Vn
Vn
vn (t + τ ) = min
.
⎪
.
vn−1 (t )2
⎪
⎪
/
2
2
⎪
b τ + bn τ − bn 2[xn−1 (t ) − sn−1 − xn (t ) − vn (t )τ ] −
⎪
⎩ n
b̂
(5.12)
Problem 1
111
To predict vehicle behavior under both free and congested traffic with a single
equation, Bando (1995) devised the “optimal velocity model” that assumes a special
basis function to describe human behavior. She also identified these parameters
using Japanese highway-traffic data and obtained good results.
Other longitudinal models proposed in the literature include the look-ahead
model (which uses the status of the three closest lead vehicles to compute acceleration command), the time-to-collision model, and the state-transition model (in
which a human driver is assumed to switch among strategies according to several
driving states).
PROBLEMS
1. Use the two-DOF vehicle-lateral-dynamics model given here. Simulate the
response of the same system in terms of lateral acceleration and lateral displacement
using a closed-loop vehicle-handling model that includes a simple (i.e., proportional
plus delay) driver model of the following form:
Gd (s) = (δ(s)/e(s)) = Kd e−sT
where Kd = 1 and T = 0.015 sec. Instead of the pure delay, use the approximation:
e−sT ≈
(T s)2 − 6(T s) + 12
;
(T s)2 + 6(T s) + 12
or
e−sT ≈
1 − 0.5T s
1 + 0.5T s
Is the response stable? If not, find a satisfactory pair (Kd ,T) for stable response. Also,
provide the simulation result for this set of (Kd ,T) parameter values and comment
on your results.
Ydes
Y
G
D
G
V
Numerical Values of Vehicle Parameters
Using the notation for parameters as listed in Chapter 4, a vehicle is defined by the
following parameter values:
l = 2.54 m
g = 9.81 m/s2
IZ = 1,200.0 kg-m2
= 0.0 rad
Cd =0.4
−mb −ma
−
K=
lCα f
lCαr
–4
= 1.98 × 10 rad/(m/s2 )
a = 1.14 m
u0 = 30.0 m/s
Cαf = −2,400.0 N/deg
f = 0.02
A = 1 m2
uchar = l/K = 113.2 m/s
b = l − a = 1.40 m
m = 1,000 kg
Cαr = −2,056.0 N/deg
ρ = 1.202 kg/m2
57.2958 deg = 1 rad
112
Human Factors and Driver Modeling
DOF Vehicle Handling Model
(mu0 D − Yβ ) (mu0 − Yr )
β
−Nβ
(Iz D − Nr )
r
=
Yδ f
Yδr
Nδ f
Nδr
δf
δr
where:
Yβ = Cαf + Cαr = −2.5531 × 105 N/rad
Yr =
Cα f a
Cαr b
= 407.9461 N · S/rad
u0
= 1.1780 × 105 N/rad
−
u0
Yδf = – Cαf = 1.3751 × 105 N/rad
Yδr = −Cαr
Nβ = a Cαf – b Cαr = 8.1589 × 103 N·m/rad
Nr = (a2 /u0 ) Cαf + (b2 /u0 ) Cαr
= −2.0480× 104 N · m · s/rad
Nδf = −a Cαf = 1.5676 × 105 N · m/rad
Nδf = bCαr = −1.6492 × 105 N · m/rad
2. From a two-DOF vehicle lateral-dynamics model, the following transfer function
is obtained:
G′ v(s) = (y(s)/δ(s)) =
1380s2 + 2400s + 34287
s2 (s2 + 29.832s + 224.523
A preview-based driver model can be developed as follows: (1) the driver model is
assumed to be Gd (s) = Kd e−sT ; (2) the error, e, is calculated as e(t + Tp ) = r(t +
Tp ) − y(t + Tp ); and (3) it is assumed that the desired path at some time in the
future, r(t + Tp ), is known and that the actual position at that same time, y(t + Tp ),
is estimated from the simple model y(t + Tp ) = y(t) + Tp ẏ(t ) + (Tp2 /2) ÿ(t ), where
Tp is preview time.
(a) Draw a more detailed version of the block diagram given here so that it also
shows the lateral velocity, vy (t) = ẏ(t ), and acceleration, ay(t) = ÿ(t ), as well
as the preview action. (Hint: This will help to obtain the answer to [b].)
(b) Refer to the block diagram here and your block diagram from (a). Determine
what Gv (s) must be for this system with preview. (Hint: G′ v(s) is given
but Gv (s) must be an equivalent vehicle-dynamics transfer function that
incorporates the preview action.)
r(t+Tp)
+
u(t)
e(t+Tp)
Gd(s)
y(t+Tp)
Gv(s)
-
3. From a two-DOF vehicle-lateral-dynamics model, the following transfer function
is obtained:
Gv (s) =
y(s)
b s2 + b1 s + b2
= 20 2
δ(s)
s (s + a1 s + a2 )
From experiments and using a Padéapproximation to the delay, a driver steering
model was found to be:
δ(s)
1 − 0.5T s
≈ Kd
Gd (s) =
e(s)
1 + 0.5T s
References
The control input, u(t), in the following figure corresponds to the steering input, δ(t).
(a) Obtain the closed-loop transfer function Gc (s) = y(s)
.
r(s)
(b) What will the steady-state response (y) to a unit step change in the reference
(i.e., desired) lateral displacement (r) be for the combined driver and vehicle
in the block diagram?
REFERENCES
Barfield, W., and T. A. Dingus, 1998, Human Factors in Intelligent Transportation Systems,
Lawrence Erlbaum Associates.
Bando, M., et al., 1995, “Phenomenological study of dynamical model of traffic flow,” J. Phys.
I France, vol. 5, pp. 1389–99.
Bekey, G. A., 1962, “The Human Operator as a Sampled-Data System,” IRE Transactions on
Human Factors in Electronics, HFE-3, September.
Brown, I. D., A. H. Tickner, and D. C. V. Simmonds, 1969, “Interference between Concurrent
Tasks of Driving and Telephoning,” Journal of Applied Psychology, 53(5), 419–24.
Burnham, G. O., J. Seo, and G. A. Bekey, 1974, “Identification of Human Driver Models in
Car Following,” IEEE Transactions on Automatic Control, Vol. AC-19, pp. 911–15.
Chandler, R. E., Herman, R. and E. W. Montroll, 1958, “Traffic Dynamics: Studies in Car
Following,” Operations Research, Vol. 6, No. 2, Mar.–Apr., 1958.
Charlton, S. G., and T. G. O’Brien (eds.), 2002, Handbook of Human Factors Testing and
Evaluation, second edition, Lawrence Erlbaum Associates, Mahwah, NJ.
Chen, S., T. B. Sheridan, H. Kusunoki, and N. Komoda, 1995, “Car-Following Measurements,
Simulations, and a Proposed Procedure for Evaluating Safety,” Proceedings of the IFAC
Symposium on Analysis, Design and Evaluation of Man-Machine Systems, Pergamon Press,
pp. 603–8.
Chen, L. K., and A. G. Ulsoy, 2001, “Identification of Driver Steering Model, and Model
Uncertainty, from Driving Simulator Data,” ASME Journal of Dynamic Systems Measurement Control, Vol. 123, No. 4, pp. 623–29, December.
Chen, L. K., and A. G. Ulsoy, 2002, “Design of a Vehicle Steering Assist Controller Using
Driver Model Uncertainty,” International Journal of Vehicle Autonomous Systems, Vol. 1,
No. 1, 2002, pp. 111–32.
Chen, L. K. and A. G. Ulsoy, 2006, “Experimental Evaluation of a Vehicle Steering Assist
Controller Using A Driving Simulator,” Vehicle System Dynamics, Vol. 44, No. 3, March,
pp. 223–45.
Davis, J. R., and C. M. Schmandt, 1989, “The Back Seat Driver: Real Time Spoken Driving
Instructions,” pp. 146–50, in D. H. M. Reekie, E. R. Case, and J. Tsai (eds), First Vehicle Navigation and Information Systems Conference (VNIS’89), New York: Institute of
Electrical and Electronics Engineers.
Dingus, T. A., J. F. Antin, M. C. Hulse, and W. W. Wierwille, 1988, “Human Factors Issues
Associated with In-Car Navigation System Usage,” Proceedings of the Human Factors
Society-32nd Annual Meeting, pp. 1448–50.
Dudek, C. L., R. D. Hutchinson, W. R. Stockton, R. J. Koppa, S. H. Richards, and T. M.
Mast, 1978, Human Factors Requirements for Real-Time Motorist Information Displays,
Volume I – Design Guide (Technical Report (FHWA-RD-78-5), Washington, DC, U.S.
Department of Transportation, Federal Highway Administration.
Eby, D. W., and L. P. Kostyniuk, 1999, “An On-the-Road Comparison of In-Vehicle Navigation Assistance Systems,” Human Factors, Vol. 41, pp. 295–311.
113
114
Human Factors and Driver Modeling
Esterberg, M. A., E. D. Sussman, and R. A. Walter, 1986, Automotive Displays and Controls –
Existing Technology and Future Trends (Report PM-45-U-NHT-86-11), Washington, DC,
U.S. Department of Transportation, Federal Highway Administration, National Highway
Traffic Safety Administration.
Fuller, R., and J. A. Santos (eds.), 2002, Human Factors for Highway Engineers, first edition,
Pergamon Press.
Gazis, D. C., R. Herman, and R. W. Rothery, 1961, “Nonlinear Follow-the-Leader Models of
Traffic Flow,” Operations Research, Vol. 9, pp. 545–66.
Gipps, P. G., 1981, “A Behavioral Car-Following Model for Computer Simulation,” Transportation Research-B, Vol. 15B, pp. 105–11.
Godthelp, H., and W. Kappler, 1988, “Effects of Vehicle Handling Characteristics on Driving
Strategy,” Human Factors, Vol. 30, No. 2, pp. 219–29.
Godthelp, H., P. Milgram, and G. J. Blaauw, 1984, “The Development of a Time-Related
Measure to Describe Driving Strategy,” Human Factors, Vol. 26, No. 3, pp. 257–68.
Green, P., 1990, Vehicle Control and Human Factors, Notes for IVHS Short Course, University
of Michigan.
Hess, R. A., and A. Modjtahedzadeh, 1990, “A Control Theoretic Model of Driver Steering
Behavior,” IEEE Control Systems Magazine, August, pp. 3–8.
Hosman, R. 1985, “Laboratory and Moving-Base Simulator Experiments on Speed and Accuracy of Visual and Whole-Body Motion Perception,” Proceedings of the IFAC Man-Machine
Systems, Varese, Italy.
Hoyes, T. W., N. A. Stanton, and R. G. Taylor, 1996, “Risk Homeostasis Theory: A Study of
Intrinsic Compensation,” Safety Science, Vol. 22, No. 1–3, pp. 77–86.
Hulse, M. C., T. A. Dingus, T. Fischer, and W. W. Wierwille, 1989, “The Influence of Roadway Parameters on Driver Perception of Attentional Demand,” Advances in Industrial
Ergonomics and Safety I, edited by Anil Mital, Taylor and Francis, pp. 451–6.
International Standards Organization, 1977, Road Vehicles – Passenger Cars – Location of
Hand Controls, Indicators and Tell-Tales (International Standard 4040), Geneva, Switzerland ISO.
Kleinman, D. L., S. Baron, and W. H. Levison, 1970, “An Optimal Control Model of Human
Response, Part I: Theory and Validation,” Automatica, Vol. 6, pp. 357–69.
Kondo, M., and A. Ajimine, 1968, “Driver’s Sight Point and Dynamics of the Driver-Vehicle
System Related to It,” SAE Automotive Engineering Congress, Detroit, MI, SAE Paper
680104.
Li, Y. T., L. R. Young, and J. L. Meiry, 1965, “Adaptive Functions of Man in Vehicle Control
Systems,” in Proceedings of the International Federation of Automatic Control Symposium,
Paddington, England.
MacAdam, C. C., 1980, “An Optimal Preview Control for Linear Systems,” ASME Journal
of Dynamic Systems, Measurement, and Control, Vol. 102, September, pp. 188–90.
MacAdam, C. C., 1981, “Application of an Optimal Preview Control for Simulation of ClosedLoop Automobile Driving,” IEEE Transactions on Systems, Man, and Cybernetics, SMC-11,
June, pp. 393–9.
MacAdam, C. C., 1983, “Frequency Domain Methods for Analyzing the Closed-Loop Directional Stability and Maneuverability of Driver/Vehicle Systems,” Proceedings of the Conference on Modern Vehicle Design Analysis, edited by M. Dorgham, London.
MacAdam, C. C., 1985, “Computer Model Predictions of the Directional Response and Stability of Driver Vehicle Systems During Anti-Skid Braking,” Proceedings of the IMech E
Conference on Antilock Braking Systems for Road Vehicles, edited by P. Newcomb, London.
MacAdam, C. C., 1988, “Development of Driver/Vehicle Steering Interaction Models for
Dynamic Analysis,” Final Technical Report, UMTRI-88-53, TACOM Contract DAAE0785-C-R069.
MacAdam, C. C., 1989, “Mathematical Modeling of Driver Steering Control at UMTRI – An
Overview,” UMTRI Research Review, July–August, pp. 1–13.
MacAdam, C. C., 2003, “Understanding and Modeling the Human Driver,” Vehicle System
Dynamics, Vol. 40, Nos. 1–3, pp. 101–34.
References
MacAdam, C. C., Sayers, M. W., Pointer, J. D. and M. Gleason , 1990, “Crosswind Sensitivity
of Passenger Cars and the Influence of Chassis and Aerodynamic Properties on Driver
Preferences,” Vehicle System Dynamics, Vol. 19, No. 4.
Macgregor, D. G. and P. Slovic, 1989, “Perception of Risk in Automotive Systems,” Human
Factors: The Journal of the Human Factors and Ergonomics Society, August 1989, Vol. 31,
No. 4 377–89.
McLean, D., T. P. Newcomb, and R. T. Spurr, 1976, “Simulation of Driver Behavior During Braking,” I. Mech. E. Conference on Braking of Road Vehicles, IME Paper C41/
176.
McLean, J. R., and E. R. Hoffman, 1971, “Analysis of Drivers’ Control Movements,” Human
Factors, Vol. 13, May, pp. 407–18.
McLean, J. R., and E. R. Hoffmann, 1972, “The Effects of Lane Width on Driver Steering Control and Performance,” Proceedings of the Sixth Australian Road Research Board
Conference, Vol. 6, No. 3, 418–40.
McLean, J. R., and E. R. Hoffmann, 1973, “The Effects of Restricted Preview on Driver
Steering Control and Performance,” Human Factors, Vol. 15, No. 6, pp. 421–30.
McRuer, D. T., Hofmann, L. G., Jex, H.R., Moore, G.P., Phatak, A.V., Weir, D.H. and
Wolkovitch, J., 1968, “New Approaches to Human-Pilot/Vehicle Dynamic Analysis,” Technical Report AFFDL-TR-67-150, WPAFB.
McRuer, D. T., R. W. Allen, and R. H. Klein, 1977, “New Results in Driver Steering Control
Models,” Human Factors, Vol. 19, No. 4, pp. 381–97.
McRuer, D. T., and R. Klein, 1975a, “Comparison of Human Driver Dynamics in Simulators
with Complex and Simple Visual Displays and in an Automobile on the Road,” 11th Annual
Conference on Manual Control.
McRuer, D. T., and R. Klein, 1975b, “Effects of Automobile Steering Characteristics on
Driver Vehicle System Dynamics in Regulation Tasks,” 11th Annual Conference on Manual
Control, pp. 408–39.
Miller, D. C. and J. I. Elkind, 1967, “The Adaptive Response of the Human Controller to
Sudden Changes in Controlled Process Dynamics,” IEEE Transactions on Human Factors
in Electronics, HFE-8, 3.
Mitschke, M., 1993, “Driver-Vehicle Lateral Dynamics under Regular Driving Conditions,”
Vehicle System Dynamics, Vol. 22, pp. 483–92.
Modjtahedzadeh, A., and R. A. Hess, 1991, “A Model of Driver Steering Dynamics for Use
in Assessing Vehicle Handling Qualities,” in S. A. Velinsky, R. H. Fries, I. Haque, and D.
Wang, D. (eds.), Advanced Automotive Technologies-1991, ASME DE-Vol. 40, New York,
pp. 41–56.
Mortimer, R. G., and C. M. Jorgeson, 1972, “Eye Fixations of Drivers as Affected by Highway
and Traffic Characteristics and Moderate Doses of Alcohol,” Proceedings of the 16th Annual
Meeting, Human Factors Society, pp. 86–92.
Nagai, M., and M. Mitschke, 1985, “Adaptive Behavior of Driver-Car Systems in Critical
Situations: Analysis by Adaptive Model,” JSAE Review, December, pp. 82–9.
Newcomb, T. P., 1981, “Driver Behavior During Braking,” SAE/I. Mech. E. Exchange Lecture. SAE Paper 810832.
Newell, G. F., 1961, “Nonlinear Effects in the Dynamics of Car Following,” Operations
Research, Vol. 9, pp. 209–29.
Pew, R. W., and S. Baron, 1982, “Perspectives on Human Performance Modeling,” Proceedings of IFAC Man-Machine Systems, Baden-Baden, Federal Republic of Germany.
Phatak, A. V., and G. A. Bekey, 1969, “Model of the Adaptive Behavior,” IEEE Transactions
on Man-Machine Systems, MMS-10, September, pp. 72–80.
Pilutti, T., U. Raschke, and Y. Koren, 1990, “Computerized Defensive Driving Rules for
Highway Maneuvers,” Proceedings of the American Control Conference, San Diego, CA,
May, pp. 809–11.
Pilutti, T., and A. G. Ulsoy, 1999, “Identification of Driver State for Lane-Keeping Tasks,”
IEEE Transactions on Systems, Man and Cybernetics, Vol. 29, No. 5, September 1999,
pp. 486–502.
115
116
Human Factors and Driver Modeling
Pipes, L. A., 1953, “An Operational Analysis of Traffic Dynamics,” Journal of Applied Physics,
Vol. 24, No. 3, pp. 274–81, March 1953.
Rashevsky, N., 1968, “Mathematical Biology of Automobile Driving,” Bulletin of Mathematical Biophysics, 30, p. 153.
Sanders, M. S., and E. J. McCormick, 1987, Human Factors in Engineering and Design (6th
edition), New York: McGraw-Hill Publishers.
Sheridan, T. B., 1964, “Control Models of Creatures Which Look Ahead,” Proceedings of the
5th National Symposium on Human Factors in Electronics.
Sheridan, T. B., 1966, “Three Models of Preview Control,” IEEE Transactions on Human
Factors in Electronics, HFE-7, June.
Sheridan, T. B., 1985, “Forty-Five Years of Man-Machine Systems: History and Trends,”
Proceedings of IFAC Man-Machine Systems, Varese, Italy.
Sheridan, T. B., 2002. Humans and Automation: System Design and Research Issues. John
Wiley & Sons, Inc., New York, NY, USA.
Snyder, H. L., and R. W. Monty, 1985, “Methodology and Results for Driver Evaluation of
Electronic Automotive Displays,” Ergonomics International ’85, 514–17.
Summala, H., 1996, “Accident Risk and Driver Behaviour,” Safety Science, Vol. 22, Nos. 1–3,
pp. 103–17.
Tyler, J. S., 1964, “The characteristics of model following systems as synthesizes by optimal
control,” IEEE Trans. Automatic Control, Vol. AC-9, pp. 485–98.
Weir, D. H., and A. V. Phatak, 1968, “Model of Human Response to Step Transitions in
Controlled Element Dynamics,” Technical Report NASA CR-671.
Weir, D., R. J. DiMarco, and D. T. McRuer, 1977, “Evaluation and Correlation of
Driver/Vehicle Data, Vol. II,” Final Technical Report, NHTSA, DOT-HS-803-246.
Weir, D. H., and D. T. McRuer, 1968, “A Theory for Driver Steering Control of Motor
Vehicles,” Highway Research Record, Vol. 247, pp. 7–28.
Wierwille, W. W., J. F. Antin, T. A. Dingus, and M. C. Hulse, 1988, “Visual Attentional
Demand of an In-Car Navigation Display,” in A. G. Gale et al. (eds.), Vision in Vehicles II,
Amsterdam: Elsevier.
Wierwille, W. W., G. A. Gagne, and J. R. Knight, 1967, “An Experimental Study of Human
Operator Models and Closed-Loop Analysis Methods for High-Speed Automobile Driving,” IEEE Transactions on Human Factors in Electronics, HFE-8, September.
Wilde, G. J. S., 1994, “Risk Homeostasis Theory and Its Promise for Improved Safety,” in
R. M. Trimpop and G. J. S. Wilde (eds.), Challenges to Accident Prevention: The Issue of
Risk Compensation Processes, Groningen, The Netherlands: Styx Publications.
Wolf, J. D., and M. F. Barrett, 1978, Driver Vehicle Effectiveness Model Volume II: Appendices, Report DOT-HS-804-338, Washington, DC, U.S. Department of Transportation.
Young, L. R. and L. Stark, 1965, “Biological Control Systems – A Critical Review and
Evaluation,” NASA Contractor Report, NASA CR-190.
PART II
POWERTRAIN CONTROL SYSTEMS
6
Air–Fuel Ratio Control
Mathematical models such as those described in Chapter 3 can be used as the basis
for engine control-system design (Kiencke and Nielsen 2005). This chapter discusses
control of the air–fuel mixture ratio that enters the cylinders (i.e., the combustion
chamber) (Grizzle et al. 1990, 1991; Kiencke 1988; Kuraoka et al. 1990; Moklegaard
et al. 2001; Ohyama 1994; Sweet 1981).
6.1 Lambda Control
Air–fuel ratio control sometimes is referred to as lambda control because the ratio
is defined as follows:
λ=
ṁa /ṁ f
λST
(6.1)
where λ is the normalized air–fuel ratio, ṁa is the mass airflow into the engine, ṁ f is
the metered-fuel mass flow rate, and λST is the air–fuel mass ratio for a stoichiometric
mixture (i.e., the air–fuel mass ratio is equal to 14.64). The driver regulates the
amount of air, ṁa , entering the cylinders, and the correct quantity of fuel, ṁ f , must
be added by the controller. Undesirable emission gases (e.g., NOx , HC, and CO) are
produced by the combustion process (see Figure 1.4). A catalytic converter reduces
the emissions but operates efficiently only if the value of λ is near 1 (see Figure
1.5). For efficient operation of the converter, the average lambda offset must be
maintained within 0.1 percent. During engine transients, short excursions of up to
3 percent may be permitted. Thus, the control specifications are quite demanding.
An EGO sensor provides the information needed for feedback control (Figure
6.1). The sensor is highly nonlinear but produces a voltage proportional to the
amount of oxygen in the exhaust gas. In turn, this indicates whether the air–fuel
mixture is rich (i.e., <14.64) or lean (i.e., >14.64). The sensor response time to a step
change in lambda is about 300 ms or less. In the air–fuel-ratio control problem, the
engine can be modeled simply by a pure delay plus a first-order lag:
G p (s) =
KM e−sTL
τM s + 1
(6.2)
119
120
Air–Fuel Ratio Control
Comparator PI-Controller
b
Unom.
+
Engine Model
+
−⑀
λ
⑀
−b
TD , TR
KM , TM
TL
Lambda-Sensor
−KL
Uλ
λ
Figure 6.1. Block diagram for lambda control (Kiencke 1988).
where the delay, TL , varies from about 100 to 1,000 ms and the lag, τM , varies from
about 50 to 500 ms.
A PI controller typically is used in this application. However, the nonlinearity of
the EGO sensor is a problem, and there is considerable interest in the development of
an inexpensive linear sensor for use with lambda control. The variation of the model
parameters also has created considerable interest in adaptive lambda controllers
that can estimate the parameters during operation (Jones et al. 1995). Also the
measurement made with the EGO sensor is an average lambda value for the engine;
research to control the lambda value in each cylinder also is of interest (Kiencke
1988).
Figure 6.1 shows a block diagram of a lambda controller. The controller includes
a nonlinear (i.e., hysteresis) element, which addresses the nonlinearity of the EGO
sensor and the pure delay TL .
6.2 PI Control of a First-Order System with Delay
In this section, we first review the control of a first-order system using PI control
and then the control of a first-order system with delay using the Smith Predictor.
Example 1 discusses the application of these methods to air–fuel ratio control.
PI Controller Design for a Process Described by a First-Order
Transfer Function
Given Gp (s) = K/(τ s + 1) and Gc (s) = KP + KI /s, the closed–loop-system transfer
function is Y(s)/Uc (s) = Gc Gp /(1 + Gc Gp ). The closed-loop system therefore has
the following characteristic equation:
s2 + ((1 + KKP )/τ )s + (KKI /τ ) = 0
(6.3)
Comparing this to a standard second-order system, we can identify:
2ζ ωn = (1 + KKP )/τ ;
and
ωn2 = (KKI /τ )
(6.4)
6.2 PI Control of a First-Order System with Delay
121
Gc (s)
uc
+
+
-
e
u
Gr (s)
e
− sT
y
G p (s)
-
(1 − e −sT )Gp (s)
Figure 6.2. Smith Predictor for control of a system with delay.
Because ωn2 > 0, for stability we must have ζ > 0. Various other performance measures can be expressed in terms of the natural frequency, ωn , and the damping ratio,
ωn ). The maximum percentage overζ . The 1 percent settling time is ts = (4.6)/(ζ
shoot (for 0 ≤ ζ ≤ 1) is Mp = exp(−π ζ / 1 − ζ 2 ) ≈ 1 − (ζ /0.6) for 0 ≤ ζ ≤ 0.6. For
example, by specifying the desirable value of Mp , we can determine ζ ; then, from ζ
and the specified value of ts , we can determine the value of ωn . Finally, from the ζ
and ωn values, we can determine the PI controller gains, KP and KI . This secondorder closed-loop system also includes a zero; therefore, the formulas given herein
for settling time and overshoot are only approximations (Franklin et al. 2010).
Controlling a System with Delay (the Smith Predictor)
Consider the control of a process with pure delay:
Y(s)/U(s) = e−sT G p (s)
(6.5)
The Smith Predictor approach is to design a controller, Gr (s), for the process without
delay, Gp (s). This can be accomplished, for example, as described in the previous
section. Then, we implement the feedback controller Gc (s) for the process with the
delay, where:
U(s)/E(s) = Gc (s) =
Gr (s)
1 + G p (s)Gr (s) − G p (s)Gr (s)e−sT
(6.6)
Then, the closed-loop system becomes:
Y(s)/Uc (s) =
Gc G p e−sT
1 + Gc G p
e−sT
= e−sT
Gr G p
1 + Gr G p
(6.7)
The closed-loop system has the response designed without the delay plus the process
delay and is illustrated in the block diagram in Figure 6.2.
In this example,
it is assumed that a linear EGO sensor is available to measure the normalized
air–fuel ratio, λ, and that the process is well modeled by a first-order lag with
delay, as in Eq. (6.2). The desired value of the air–fuel ratio is λd = 1.0 and
EXAMPLE 6.1: SMITH PREDICTOR PI CONTROL OF AIR–FUEL RATIO.
122
Air–Fuel Ratio Control
e = λd − λ. Furthermore, it is assumed that the controller is given by a PI
controller where:
K
U (s) = KP + I E(s)
(6.8)
s
With this controller, the closed-loop transfer function becomes:
KM (KP s + KI )
λ(s)
=
2
λd (s)
τM s + (1 + KM KP )s + (KM KI )
(6.9)
The characteristic equation is found by setting the denominator of this transfer
function to zero:
s2 + ((1 + KM KP )/τM )s + (KM KI /τM ) = 0
(6.10)
Using the analogy to a second-order system, as in Eq. (6.4), we can obtain the
controller gains:
ωn2 τM
KM
2ζ ωn τM − 1
KP =
KM
KI =
(6.11)
Now, to account for the process delay, TL , we can use a Smith Predictor design
based on this controller (Franklin et al. 2010). Thus, the controller is given by:
Gc (s) =
Gr (s)
1 + G p (s)Gr (s) − G p (s)Gr (s)e−sTL
(6.12)
where Gp (s) is the process model in Eq. (6.2) but without the delay:
G p (s) =
KM
τM s + 1
(6.13)
and
KI
Gr (s) = KP +
s
(6.14)
The response of the closed-loop system then is determined by the selected
values of the design parameters, ζ and ωn , and also includes the same delay,
TL , as the open-loop process. The Smith Predictor is only one approach to the
control of systems with delay, and it is effective when the delay time TL and the
process model Gp (s) are well known. Other methods for the control of systems
with delays are available and may be more effective in the presence of model
uncertainties (Yi et al. 2010).
PROBLEMS
1. For the air–fuel-ratio control problem, draw a block diagram (similar to Figure 6.2)
that shows the process model Gp , the controlled variable y, the control input u, and
the reference input r. Also list the necessary sensors and actuators.
References
2. Air–fuel ratio control can be designed as in Example 6.1 (see Figure 6.2). Assume
that a linear EGO sensor was developed so that the sensor nonlinearity can be
neglected. Use the values TL = 1.0 s, τ M = 0.5 s, and KM = 1.
(a) First, consider the process without the delay to design a PI controller Gr (s) =
KP + KI /s. This can be accomplished by trial and error or by any standard
control-design procedure. Closed-loop specifications are the settling time
ts = 0.46 seconds and the percentage overshoot Mp = 0.
(b) Next, use the Smith Predictor approach to derive the controller Gc (s) for
the process with delay.
(c) Use Simulink to simulate and compare the closed-loop systems designed in
(a) and (b), respectively.
REFERENCES
Franklin, G. F., J. D. Powell, and A. Emami-Naeni, 2010, Feedback Control of Dynamic
Systems, Pearson Publishing.
Grizzle, J. W., J. A. Cook, and K. L. Dobbins, 1990, “Individual Cylinder Air–Fuel Ratio
Control with a Single EGO Sensor,” Proceedings of the American Control Conference, San
Diego, CA, May 1990, pp. 2881–6.
Grizzle, J. W., K. Dobbins, and J. Cook, 1991, “Individual Cylinder Air–Fuel Ratio Control
with a Single EGO Sensor,” IEEE Transactions on Vehicular Technology, Vol. 40, February
1991, pp. 280–6.
Jones, V. K., B. A. Ault, G. F. Franklin, and J. D. Powell, 1995, “Identification and Air–
Fuel Ratio Control of a Spark Ignition Engine,” IEEE Transactions on Control Systems
Technology, Vol. 3, No.1, March 1995.
Kiencke, U., 1988, “A View of Automotive Control Systems,” Control Systems Magazine,
IEEE, Vol. 8, August 1988, pp. 11–19.
Kiencke, U. and L. Nielsen, 2005, Automotive Control Systems, Springer-Verlag.
Kuraoka, H., N. Ohka, M. Ohba, S. Hosoe, and F. Zhang, 1990, “Application of H-Infinity
Design to Automotive Fuel Control,” IEEE Control Systems Magazine, April 1990,
pp. 102–6.
Moklegaard L., M. Maria Druzhinina, and A. G. Stefanopoulou, 2001, “Brake Valve Timing
and Fuel Injection: A Unified Engine Torque Actuator for Heavy-Duty Vehicles,” Journal
of Vehicle System Dynamics, Vol. 36, Nos. 2–3, September 2001.
Ohyama, Y., 1994, “Air–Fuel Ratio Characteristics of Engine Drivetrain Control System
Using Electronically Controlled Throttle Valve and Air Flow Meter,” Proceedings of
AVEC’ 94, pp. 497–502.
Sweet, L. M., 1981, “Control Systems for Automotive Vehicle Fuel Economy: A Literature Review,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 103,
September 1981, pp. 173–80.
Yi, S., P. W. Nelson, and A.G. Ulsoy, 2010, Time Delay Systems: Analysis and Control Using
the Lambert W Function, World Scientific Publishing Company, Inc., Hackensack, NJ.
123
7
Control of Spark Timing
The focus of this chapter is the control of spark timing. As discussed in Chapter 3,
the spark is ignited in advance of TDC during the compression stroke. The exact
timing can influence performance, fuel economy, emissions, and knock. As discussed
in Chapter 1, advancing the spark timing can improve performance and reduce fuel
consumption. However, advanced spark timing also can lead to engine knocking
and potential engine damage. Spark-timing control can be used, for example, in
idle-speed control (see Chapter 8) with throttle control. In this chapter, we focus
on the occurrence of engine knock and the control of knock by adjustment of spark
timing.
7.1 Knock Control
Knock occurs when an unburned part of the air–fuel mixture within the combustion chamber explodes prematurely. This is called knocking because it generates
resonating gas-pressure oscillations, which are heard as a knocking sound. Knocking
can lead to serious engine damage (Heywood 1989). Historically, a low-compression
ratio or conservative spark timing was used to ensure that knocking did not occur;
however, this approach sacrifices performance and fuel economy. Knock control can
be used when a feedback sensor becomes available, which adjusts the spark timing
based on a measured variable that indicates knock. Suitable measurements include
the cylinder pressure (e.g., the 5- to 15-kHz region was found to be a good knock
indicator), engine-block vibrations, light emission within the combustion chamber,
and ion current through the gas mixture.
When the occurrence of knock is detected by one of these means, the ignition
timing is retarded by the amount αR ; when knocking does not occur, the spark
timing is advanced. With this scheme, the ignition angle is constantly adjusted about
a value, which is close to the knock limit. The control scheme here is simple but
the measurement and detection of knock are challenging, as is the selection of an
appropriate αR . Figure 7.1 illustrates the knock-control strategy described herein.
When the occurrence of knock is detected – as indicated by spikes in the top half of
Figure 7.1 – the ignition angle is retarded by the amount αR . When no knock occurs,
the ignition angle is allowed to increase with time at a rate of dαA /dt. It is obvious
124
References
125
Figure 7.1. Knock control.
that the selection of these two quantities is critical to the controller performance, as
is the development of an accurate knock-detection system.
PROBLEM
1. Write a MATLAB program to implement the simple knock-control scheme
described in this chapter. In the program, allow αR and dαA /dt to be user selectable.
Assume that a function knockdetector is available and returns a value of 1 when
knock is detected and a value of 0 otherwise. Test your results with a simulated
knock sequence qualitatively similar to the one shown in Figure 7.1, and use this
sequence to select effective values for the user-selectable control parameters.
REFERENCES
Entenmann, R., S. Unland, O. Torno, and W. Haeming, 1997, “Method for the Adaptive
Knock Control of an Internal Combustion Engine,” U.S. Patent 5,645,034.
Heywood, J. B., 1989, Internal Combustion Engine Fundamentals, McGraw-Hill, New York.
Morita, S., W. Fukui, and S. Wada, 1997, “Knock Control System for an Internal Combustion
Engine,” U.S. Patent 5,694,900.
Sawamoto, K., Y. Kawamura, T. Kita, and K. Matsushita, 1987, “Individual Cylinder Knock
Control of Detecting Cylinder Pressure,” Passenger Car Meeting and Exposition, Dearborn,
MI.
Schmillen, K. P., 1991, “Different Methods of Knock Detection and Knock Control,” SAE
International Congress and Exposition, Dearborn, MI, February 1991.
8
Idle-Speed Control
One of the most important and basic engine-control functions is idle-speed control. It requires consideration of the complete engine dynamics (as described in
Chapter 3 about engine modeling) and has been a focus of various researchers to
improve the performance of current and future engine designs (Grizzle et al. 2001;
Wang et al. 2001). This chapter discusses engine idle-speed control.
The engine speed at idle is maintained at a desired value despite changes in
engine loads (e.g., due to accessories such as an air-conditioning compressor). The
controlled variable is engine idle speed and it is measured as discussed previously.
The variables manipulated by the controller include the throttle angle and the spark
advance. An optimal control approach provides an effective framework for the
study of engine idle-speed control (Hrovat and Powers 1988). In this approach,
we consider a vector, x, of state variables and a vector, u, of control variables.
The problem then becomes finding the optimal controls, u*(t), which minimize an
objective function J(x, u) subject to constraint equations g(x, u) = 0. In general, this
is a complex problem because J and g are difficult to determine and the problem,
once formulated, is difficult to solve.
However, engineering judgment can be used to reduce the problem to a manageable but sufficiently accurate form. Assuming that during idle the EGR is not
in operation, the control inputs are considered to be u = [throttle rate, spark
rate, fuel-flow rate]. Rates of the actual control variables are used in this formulation to separate the effects of actuator dynamics. The engine model described in
Chapter 3, or a more complex version of it with about twenty state variables, typically is used. The model relating these variables must be calibrated with empirical
data. This formulation leads to a simulation model that is used to study the dynamics
of the system (the results are shown in Figure 8.1) and then to obtain a reducedorder model with five state variables (i.e., perturbations in engine speed, manifold
pressure, throttle angle, throttle-motor rate, and spark advance) and two control
inputs (i.e., perturbations in the throttle-rate command and the spark-advance command). The reduced-order model is linearized about a particular nominal operating
condition and solved as a linear quadratic (LQ) optimal-control problem. In the
LQ problem, linear-state equations describe the process model and the objective function, J(x, u), to be minimized, is quadratic in x and u. The solution to
126
Idle-Speed Control
127
1400
ENGINE SPEED, RPM
1200
Figure 8.1. Idle-speed-control simulation
results (Hrovat and Powers 1988).
1000
Throttle And
Spark Control
–5 Var Feedback
Throttle
Control
-Speed Plus
Speed Rate
Feedback
No Control
800
600
400
200
0
1.0
2.0
3.0
4.0
5.0
TIME, SECONDS
LQ problems can be obtained readily using standard numerical algorithms in
computer-aided control-system design packages such as MATLAB.
Figure 8.2 illustrates the experimental results for idle speed due to a sudden
change in engine load at time t = 0 as obtained using a throttle-only idle-speed
controller and a throttle and spark idle-speed controller. The figure compares these
results to those obtained from the simulation studies using the twenty-state simulation model. The idle-speed response with no control also is shown for comparison
purposes. Clearly, both controllers provide improvement in the idle speed over the
no-control case, and the combined throttle and spark idle-speed control strategy is
most effective. The inclusion of spark advance improves the response because the
spark command acts more rapidly than the throttle command, which has actuator
and manifold lags.
Throttle
And
Spark ISC
750
700
Data
RPM
Simulation
650
Throttle Only
ISC
600
No Control
550
0
10
20
30
40
50
60
70 75
COMBUSTION EVENTS
Figure 8.2. Idle-speed-control experimental and simulation results (Hrovat and Powers 1988).
6.0
128
Idle-Speed Control
N0
P0
+ +
C
∆AM
1 ˆ
(T = 3T )
Tˆ
Gf
FUEL GAIN
ˆ
e−2ST
AIR
BY PASS
1
T
KZ
Z−1
MANIFOLD
FILLING
−
−
DISTURBANCE
IP LAG
τP KP
τP S + 1 ∆P
1
T
GP
e−2ST
+
+
+
−
+
COMBUSTION
TORQUE
PUMPING FEEDBACK
1
T
ROTATIONAL
DYNAMICS
∆N
τR KP
τR S + 1
SPARK GAIN
GS
1
T
KN
Figure 8.3. Linearized six-cylinder engine idle-speed-control model (Hrovat and Powers
1988).
EXAMPLE 8.1: SIMPLIFIED IDLE-SPEED CONTROL. The idle-speed control problem,
with adjustment of only throttle and not spark advance, is discussed in Chapters 1
and 3. Here, a combined throttle- and spark-control system is described based on
the linearized engine model presented in Chapter 3, thereby obtaining the block
diagram shown in Figure 8.3. It is a modified version of the block diagram shown
in Figure 3.6 and it uses Eqs. (3.3) and (3.4) to estimate the mass airflow quantity
Ṁ. The control uses a pure integral action (with gain K in Figure 8.3) of airflow
through a closed throttle-bypass valve to ensure steady-state accuracy and a
proportional control (with gain Gs ) of the spark timing to enhance the speed of
response. The system in Figure 3 is given as a discrete-time system because
implementation of the controller is digital. Simulation results are shown in
Figure 8.4 for the engine parameters given in Table 3.1 and for the systems with (i.e., closed-loop) and without (i.e., open-loop) idle-speed control.
Clearly, the closed-loop control reduces the engine-speed variation and eliminates the steady-state error (i.e., offset) in engine speed due to constant torquedisturbance input.
PROBLEMS
1. Closed-loop engine dynamic simulation. Simulate the closed-loop idle-speed control system given in Example 8.1 (see Figure 8.3) to a unit-step torque-disturbance
input. In performing the simulation, neglect the IP delays and the effects of sampling;
replace the controller (Kz /(z − 1)) by the integral (I) controller (KI /s). Notice from
Figure 8.3 that the system includes fuel control based on estimation of the mass flow
Problem 1
129
5
0
−5
∆RPM
−10
−15
−20
0
.5
1
1.5
0
.5
1
1.5
2
2.5
3
3.5
4
4.5
5
2
2.5
3
3.5
4
4.5
5
5
0
−5
∆RPM
−10
−15
−20
TIME (SECONDS)
Figure 8.4. Simulation results for linearized engine idle-speed control (Hrovat and Powers
1988).
rate as in Eq. (3.4). Thus, in this controller, θ = −(KI /s) N, δ = Gs N and,
from Eq. (3.4), Fd = cN0 P + cP0 N.
(a) Use the model parameter values τ P = 0.21, KP = 0.776, GP = 13.37,
τ R = 3.98, KR = 67.2, KN = 0.08, Gf = 36.6, Kθ = 20, and Gd = 10. Fix
the controller parameter values Gs = −0.005, c = 0.000125, N0 = 600, and
P0 = 12. Try several values of the controller integral action gain, KI , until a
good response qualitatively similar to that shown in Figure 8.4 is achieved.
(b) Use Gs = 0 (i.e., no coordination with SI control); then select a KI that
results in a reasonable (i.e., stable) response.
(Hint: Using the relationships given for the controller, neglecting the IP delays and
the effects of sampling, and using Eqs. (3.1)–(3.8), obtain the following closed-loop
system equations:
ẋ = A x + bv v,
where v = Td and



0
θ 






Kθ
ma
x=
A=
K K


P
P θ




0
N


0 





0
bv =
0 





−KR
and C = [0 0 0 1].)
and
y = Cx

0
0
−KI

0
0
0


0
−1/τP
−KP KN
0 KR (GP + G f cN0 ) (KR Gδ Gs − 1/τR + KR G f cP0 )
130
Idle-Speed Control
2. Repeat the closed-loop engine simulations in Problem 1 but with the following
alternative control strategies:
(a) Try a PI control where θ = −(KI /s) N − K N with Gs = 0 and determine
whether satisfactory results can be obtained.
(b) Try feed forward plus PI control θ = −(KI /s) N − KN + KFF Td
(i.e., assume the disturbance Td is measured or can be estimated). Again,
fine-tune the gains until a good response is obtained.
(c) Use maximum control design DOF (Spark + PI + feedforward) and finetune all of the gains until the results are satisfactory.
(Hint: The A and bv matrices in Problem 1 must be modified for the PI control
algorithm and the feed-forward control actions.)
REFERENCES
Grizzle, J. W., J. Buckland, and J. Sun, 2001, “Idle Speed Control of a Direct Injection Spark
Ignition Stratified Charge Engine,” International Journal of Robust and Nonlinear Control,
Vol. 11, pp. 1043–71.
Hrovat, D., and J. Sun, 1997, “Models and Control Methodologies for IC Engine Idle Speed
Control Design,” Control Engineering Practice, Vol. 5, No. 8, August, pp. 1093–100.
Hrovat, D., and W. Powers, 1988, “Computer Control Systems for Automotive Power Trains,”
IEEE Control Systems Magazine, Vol. 8, pp. 3–10.
Wang, Y., A. Stefanopoulou, and R. Smith, 2001, “Inherent Limitations and Control Design
for Camless Engine Idle Speed Dynamics,” International Journal of Robust and Nonlinear
Control, Vol. 11, pp. 1023–42.
Ye, Z., 2007, “Modeling, Identification, Design, and Implementation of Nonlinear Automotive
Idle Speed Control Systems – An Overview,” IEEE Transactions on Systems, Man, and
Cybernetics, Part C: Applications and Reviews, Vol. 37, No. 6, November, pp. 1137–51.
9
Transmission Control
Automotive transmissions are a key element in the powertrain that connects the
power source to the drive wheels. To improve fuel economy, reduce emissions, and
enhance drivability, many new technologies have been introduced in automotive
transmissions in recent years (Sun & Hebbale 2005). Electronic control of automatic
and continuously variable transmissions (CVTs) is considered briefly in this chapter,
as well as the related topic of clutch control for AWD vehicles.
9.1 Electronic Transmission Control
Electronically controlled transmissions (ECT) are used to improve fuel economy,
performance, drivability, and shift quality. It even may be possible – because of
the flexibility provided by microcomputer software – to allow for “adaptive” shift
schedules, which can be tailored for improved fuel economy, performance, or comfort. Because of the flexibility offered in ECT, the industry is seeing an accelerated
trend away from traditional mechanical/hydraulic transmission control and a move
toward the integrated ECT/engine/traction/speed control functions. In the following
discussion, the major benefits of ECT are illustrated:
(1) Precise Lockup Control. Historically, vehicles with an automatic transmission
usually are about 10 percent less efficient than those with a manual transmission.
The efficiency loss arises mainly from the slip of the torque converter. Torqueconverter lockup has been used widely to improve fuel economy. With electronic
control, it is possible to coordinate shift point, lockup schedule, and accurate
timing of lockup release to reduce shock at gear shifting. Consequently, in recent
years, the fuel economy of an ECT is only slightly worse than for a manual
transmission.
(2) Better Shift Quality. For the same reasons, clutch pressures can be controlled for
improved shift quality. Coordinating with the engine-control unit (i.e., reducing
engine torque during gear shifting) can improve substantially shift quality and
reduce shock load on shift elements.
(3) Flexible Driving. Different gear-shift patterns (e.g., power and economy) can be
selected by a driver to better adapt to driving conditions. In general, if the information is available, the shift schedule can be programmed to consider vehicle
131
132
Transmission Control
DRIVE
PULLEY
FORWARD REVERSE POWER INPUT
SHAFT FROM
CLUTCH
CLUTCH
ENGINE
AND GEARS
METAL
BELT
DRIVE SHAFT
TO REDUCTION
GEAR AND AXLE
DRIVEN
PULLEY
a. Low Ratio
b. High Ratio
Figure 9.1. Continuously variable transmission (CVT).
status (e.g., warm-up and load), driving conditions (e.g., local or highway), and
even driver status (e.g., passive or aggressive).
(4) Weight Reduction. Both size and weight can be reduced because of reduced complexity. The number of component parts also was found to be greatly reduced,
which is translated into higher reliability and lower cost.
(5) Integrated Vehicle Control. Engine, cruise, and traction control functions can be
integrated for superior performance.
(6) Foolproof Design. Improper operation by a driver (e.g., down shifting or shifting
into reverse gear at high vehicle speed) can be detected to avoid severe damage
to the transmission.
ECTs are divided into two major categories: discrete ECTs and CVTs. The
CVT is an area of ongoing research activity, and the basic principle of operation is
illustrated in Figure 9.1. Mechanical belt/chain–type with pulley-driven CVTs are
among the most advanced with still (relatively) small-scale production. A modern
CVT system of this type usually consists of a steel belt that runs between two variablewidth pulleys. In the new push-belt design, the belt is pushed toward the output
pulley, which can transmit torques that exceed the capacity of conventional pulltype belts. The distance between pulley cones can be varied to change the gear ratio
between shafts, which effectively generates an infinite number of “gears.” A CVT
usually is less efficient than a standard discrete automatic transmission because of
the losses in the belt–pulley system. However, it enables the engine to operate in the
optimized speed range and therefore promises increased overall drivetrain efficiency.
Initial testing on a midsized engine (i.e., 3.3L) reports a 10 percent improvement in
fuel economy (Kluger and Fussner 1997). Although CVTs are not widely available
on passenger vehicles in the United States, they are installed on numerous compact
models in Europe and Japan, with an installation base of more than 1 million. We
can expect to see wider CVT applications in the next decade.
The control of CVTs is actually somewhat simpler than that of the discrete ECTs.
In the following discussion, discrete ECTs are the main consideration. There are
9.2 Clutch Control for AWD
many functions that an ECT may perform – for example, torque-converter lockup,
lube/clutch cooling, shift scheduling, control of slip in the torque-converter bypass
clutch, strategies for smooth transition from and to neutral idle, speed-ratio control
for CVTs, line-pressure control, belt-load control in CVTs, driving from a stop for
ECTs without torque converters, and shift-execution control (Hrovat and Powers
1988). In the following example, both the shift scheduling and the shift execution for
a discrete ECT are discussed. The example is for the prototype four-speed transaxle
ECT shown in Figure 9.2(a) (Hrovat and Powers 1988).
The transmission consists of a torque converter, reverse, low and high clutches,
and a single shift actuator (a so-called dog actuator). The low clutch is constantly
engaged in the first gear and the high clutch is constantly engaged in the fourth gear.
The dog actuator is used to engage second gear when displaced to the left or third
gear when displaced to the right. Power is transmitted to the front wheels through
the differential and CV-joint–equipped axles. The dog actuator essentially creates
a rigid link or engagement, thereby tolerating very small speed differentials at the
time of engagement. Considered here is a power-on upshift from first to second gear.
As illustrated in Figure 9.2(b), the shift consists of three phases: torque, inertia, and
level-holding. During the torque phase, the engine torque is transferred from the
low to the high clutch (see pressure traces in Figure 9.2[b]). When the low clutch
is unloaded, the high clutch controls the turbine speed to the new synchronous
level by following the speed-ratio ramp shown in Figure 9.2(b). This constitutes the
inertia phase of the shift. Subsequently, the turbine speed is held at the second-gear
synchronous level. This level-holding phase facilitates the dog-actuator engagement,
which is the most critical phase of the shift because of the precise speed-ratio-control
requirement. When the dog actuator has been engaged, the shift is completed by
releasing the high clutch.
Closed-loop speed-ratio control is an important function in light of the stringent
requirement for the inertia and level-holding phases. As a first step, this requires the
development of a process model for the powertrain. The model required includes
models for the engine, the torque converter, and an automatic clutch with an electrohydraulic control valve. A simplified version of the model results in a fifth-order
linear model suitable for controller design. A block diagram for the system is shown
in Figure 9.3(a), the digital implementation of the controller is shown in Figure
9.3(b), and the validation results are shown in Figure 9.4.
These results show both good ramp-following during the inertia phase and precise level-holding during the level-holding phase. Note that the initial level-holding
during engagement of the dog actuator is also quite good. The effect of making the
shift faster also is shown in Figure 9.4. Faster shifts yield improved fuel economy,
whereas slower shifts lead to improved comfort.
9.2 Clutch Control for AWD
A clutch controls the energy flow from the engine to the transmission (Figure 9.5).
Power can be transmitted from the engine to the output shaft by engaging and
disengaging the clutch. A conventional clutch is known as a compressed clutch and
consists of two basic components: the clutch cover and the disc. The clutch cover is
an outer shell that contains the friction plate and the drive block. The friction plate is
133
134
Transmission Control
LOW CLUTCH (LC)
1
OWC
R
ENGINE
I
2
T
DOG ACTUATOR
3
Te Ne
N0 T0
OUTPUT
4
HIGH CLUTCH (HC)
(a)
HC CLOSED-LOOP CONTROL
TORQUE PHASE
RAMP
LEVEL HOLDING
SPEED RATIO
ACTUAL SR
COMMANDED SR
LC PRESSURE
HC PRESSURE
TIME
(b)
Figure 9.2. (a) Example schematic for electronic transmission control. (b) One-two power-on
upshift execution phases (Hrovat and Powers 1988).
a cast piece that provides a pivot point for the diaphragm as well as a friction surface
for the steel plate and mounting surface for the drive block. The drive block, driven
by a hydraulic system or a magnetic-coil system, translates and provides pressure
between the steel plate and the friction plate. The clutch housing protects the clutch
components and also works as a heat sink for the coolant, which helps with heat
dissipation during operation.
Clutch systems for various vehicles are different in functionality and capacity.
In this section, a clutch system for an automatic AWD vehicle is studied. Automatic
9.2 Clutch Control for AWD
135
MICROCOMPUTER
Speed
Ratio
Command
Nout
(rpm)
Gf 1
Ñout
×
Gf 2
M ´(K − 1)
Ñ com
t +
−
e(K )
Gc 2
∆M(K ) +
f(K ) G
c1
(PID)
+
Kc
∆M ´ + +
M ´(K )
% Duty Cycle
∆u(K)
Multiplier
Ñt
u(K − 1)
−
Nt = Turbine Speed
Ne = Engine Speed
Nout = Output Shaft Speed
u(K )
+
Ku
Controller
–Pressure
Estimator
M´
Ñe(K )
Ñt(K)
Ñt(K)
Gf1
Ne
Gf 1
Nt
1
Nout
ACTUAL
SPEED
RATIO
PT (Clutch Pressure
w.r.t. Initial
Pressure)
(a)
ENGINE SPEED
FEEDBACK
Gef(s)
δT1
TORQUE
CONVERTER
δωj
ZOH
Ga(s)
δTt
Gtc(s)
δωt
ACTUATOR
CONTROLLER
(b)
Figure 9.3. (a) Digital shift controller block diagram. (b) Shift control system block diagram
(Hrovat and Powers 1988).
AWD (Hallowell 2005), which also is called “active AWD” or “smart AWD,” is
essentially a complex 2WD system. The torque from the engine is not transferred
to all four wheels all of the time but rather on demand. The primary shaft (either
front or rear shaft) is always engaged with the engine shaft, whereas the secondary
shaft is engaged through an AWD clutch system when necessary, thus switching the
powertrain system from 2WD to 4WD mode. For example, when slippage occurs
136
Transmission Control
SPEED RATIO
SR = 3.25
SR = 3.25
EXPERIMENT
SR = 1.93
SIMULATION
HC PRESSURE (PSI)
SR = 1.93
0
SHIFT
TERMINATED
EXPERIMENT
40
SIMULATION
40
SR = 3.25
0
400-msec
Ramp
0.0
0.2
0.4
0.6
0.8
300-msec
Ramp
200-msec
Ramp
SR = 1.93
1.0
TIME (SEC)
Figure 9.4. Simulation and experimental results for one-two power-on upshift (Hrovat and
Powers 1988).
on the primary wheels, the clutch system engages and powers the secondary shaft
to enable the vehicle to move smoothly Other situations, such as turning, climbing,
and accelerating, also require traction on the secondary wheels for improved vehicle
dynamics. The automatic AWD is implemented widely in many types of vehicles
(e.g., Honda CRV, Toyota RAV4, Land Rover Freelander, Isuzu Trooper, and Jeep
Grand Cherokee).
The clutch in automatic AWD systems provides more sophisticated functionalities compared with those in 2WD systems (Figure 9.6). The control of the clutch
Cover
Diaphragm
Throw-out bearing
Bellhousing
Pressure plate
Clutch disc
Spring
Crankshaft
Input shaft
Figure 9.5. Structure of a clutch system.
(Source: http://img528.imageshack.us/
img528/575/25838775lk1.jpg.)
Pivot ball
Flywheel
Clutch fork stop
Clutch fork
Slave cylinder
9.2 Clutch Control for AWD
137
Rear tyres
Front tyres
Transfer
Engine
Steering
wheel angle
senser
ABS/ESP®
controller
Electric
control
coupling
device
Rear
differential
Transmission
Engine
controller
4WD controller
Wheel speed
sensor
2WD⇔4WD AUTO⇔ 4WD LOCK
change switch
Figure 9.6. An automatic AWD system. (Source: http://www.autopressnews.com/2006/m03/
suzuki/sx4 iawd system.jpg.)
considers not only the impact and friction mechanism during engagement but also
the delivery of a certain amount of torque to the secondary shaft to achieve a proper
torque distribution all of the time. The clutch-control system in automatic AWD systems can be divided further into two parts. The first part, based on vehicle dynamics
and current vehicle status (e.g., longitudinal velocity, yaw rate, and lateral acceleration), calculates the desired torque distribution and the desired output torque on
the secondary shaft. The second part calculates the control effort for the actuator
to compress the plate and transfer the requested torque under the current state of
the clutch. In this book, we focus on the design of the second part of the control for
a particular type of AWD clutch (Figure 9.7). This wet-friction type of clutch does
not have an active pump system for cooling; instead, two thirds of the clutch plate is
immersed in the oil, in which the heat from the plate is dissipated. A coil system is
used as an actuator and generates electromagnetic forces to provide a compressive
force between plates.
The objective of the clutch control is to achieve fast and accurate tracking of the
reference torque signal under different operating conditions and to protect clutch
components from overheating and failure. However, there are several challenges
in designing an effective control for the clutch system. The dynamics of the system
changes with different input and operating conditions – especially the temperature
states of the components – and exhibits a highly nonlinear behavior. During engagement, the friction between the steel plate and the friction plate generates a significant
amount of heat and may raise the temperature of the components to 180 degrees C
in typical operations. Such a large increase in temperature can change significantly
many mechanical properties of the components – such as the friction coefficient
between the plate and the coil resistance of the actuator – and result in a noticeable
fluctuation in the output torque. Furthermore, even without temperature variation,
the mapping between the input (i.e., coil voltage) and the output torque is nonlinear.
Thus, designing a control that can achieve consistent performance under different
conditions is challenging for such a nonlinear system.
138
Transmission Control
Figure 9.7. An AWD clutch system. (Source: http://www.awdwiki.com/images/hyundaituscon-electromagnetic-clutch.jpg.)
Existing designs for the clutch-control system typically are an open-loop feedforward proportional control with an open-loop observer (Figure 9.8). The controller
estimates the error between the reference torque and the output torque using an
open-loop observer and then computes the control for driving the plant to yield the
desired output. There are several major drawbacks to the current designs. First, due
to the lack of feedback, it is difficult to achieve satisfactory performance with an
open-loop observer because uncertainty in the system leads to poor state estimates.
Second, the improper setting of initial conditions affects system performance for a
long period because of the slow dynamics of the thermal system. Third, the proportional control cannot achieve zero steady-state error. Fourth, a single controller
cannot offer consistent performance for different operating conditions. Fifth, there
is no analytical basis for selecting the controller gain, which is fine-tuned by trial and
error. Finally, the stability of the system, the effects of model uncertainty, and the
effects of the potential time delay in the controller have not been investigated.
In Duan et al. (2011), the design of a more effective control for the clutch
system via the piecewise affine (PWA) system framework is presented. Based on
that theoretical framework, a piecewise control is proposed to achieve consistent
performance under different operating conditions. The controller gains are selected
analytically, and the stability of the closed-loop system in the presence of uncertainty
and controller delay is guaranteed by analyzing the global stability of the system.
r
-
P Control
u
Open-Loop
Observer
Plant
Est. Torq. &
Temp.
Torq
Figure 9.8. Current control design (open-loop observer + feed forward P control).
9.2 Clutch Control for AWD
139
PWA systems are defined by dividing the state space into a finite number of
polyhedral regions in which the dynamics within the region is described via a local
linear model. The switching between these local models may depend on both inputs
and states or only on states. This structure provides a flexible and traceable framework to model a large class of nonlinear systems as well as a suitable platform for
rigorous analysis and design. In addition to the flexibility for modeling, the PWA
system framework provides a good platform for controller analysis and design. The
application of the PWA system framework to the modeling and control of a nonlinear automotive AWD clutch system is presented in Duan et al. (2011). The nonlinear
clutch system is formulated into the PWA system framework first; subsequently, a
switched control system is designed to ensure the closed-loop stability in the presence of uncertainties and delays through Lyapunov stability analysis. The results
show significant improvements in performance over current designs and demonstrate the tradeoff among performance, robustness to modeling uncertainty, and time
delays.
Model Development
Because temperature fluctuation introduces significant disturbances to the clutch
system and may lead to premature degradation of the components, estimation of
temperature states is necessary for temperature compensation and failure prevention. A model is developed to estimate the temperature states of the clutch
components. Three first-order differential equations are obtained for the thermal
system:
ẋT (t ) = AT xT (t ) + BT uw (t )
(9.1)
yT (t ) = CT xT (t )
where:
⎡
⎢−
⎢
⎢
⎢
⎢
AT = ⎢
⎢
⎢
⎢
⎢
⎣
R po + R pa + R pc
R po
Cp
Cp
R po
Co
−
R po + Roa + Roc
Co
R pc
Roc
Cc
Cc
R pc
⎤
1
⎥
⎢C
Cp
⎥
⎢ p
⎥
⎢
⎥
⎢
⎥
Roc
⎥ , B =⎢
0
⎥ T ⎢
⎢
Co
⎥
⎢
⎥
⎢
⎥
⎣
Roc + R pc + Rca ⎦
0
−
Cc
⎡
R pa
⎤
Cp ⎥
⎥
⎥
Roa ⎥
⎥
⎥,
Cs ⎥
⎥
⎥
Rca ⎦
Cc
⎤
Tp (t )
CT = [ 0 0 1 ]; xT (t ) = ⎣ To (t ) ⎦ is the temperature-state vector;
uw (t ) =
Tc (t )
yo (t )|ωo (t ) − ωi (t )|
Pmech (t )
=
is the disturbance-input vector; and yT (t ) =
Ta (t )
Ta (t )
Tc (t ) is the temperature-system output. Also, Ta is the ambient temperature; yo is
⎡
140
Transmission Control
the output torque; ωi is the input shaft speed; ωo ∈ ℜ1 is the output shaft speed; Tp ,
To, Tc are the temperature states of the clutch plate, oil, and coil, respectively; Cp ,
Co, Cc are the constant scalars representing corresponding thermal capacities; and
R po, R pa , R pc , Roa , Roc , Rca are the constant thermal resistances among the three
components. The mechanical power Pmech , generated by friction, is estimated as the
product between absolute slip speed and output torque. The inputs Ta , ωi , and ωo are
measured in the current design. The coefficients for thermal capacitance and thermal
resistance are identified using experimental data. This temperature model is used
in the design of a temperature-disturbance observer and for feed-forward control.
The resistor–inductor circuit system generates magnetic compressive force to
engage the clutch. To identify the mapping between the control input (i.e., coil
voltage) and the output torque, a first-order differential equation is used first to
describe the relationship between the coil voltage and the coil current:
d
i (t ) = Aqc (Tc )ic (t ) + Bq u(t )
dt c
(9.2)
where:
Aqc (Tc ) = Aq + Aq (Tc ) = −
c0
α(Tc (t ) − Tc0 )
− c0
Lc
Lc
(9.3)
Bq = L1 , u(t ) = Vc (t ) is the coil-voltage input, ic (t ) is the coil current, Lc is the coil
c
inductance, α is the temperature coefficient of the coil resistance, c0 is the nominal
coil resistance, and Tc0 is the nominal coil temperature.
Then, a model is developed to estimate the output torque, which is determined
by both the coil current and the plate temperature:
yo (t ) = μ(Tp )Cq (ic )
(9.4)
where yo (t ) is the output torque; μ(Tp ) is a scalar function and represents the effects
of plate temperature, Tp , on the μ factor (i.e., plate-friction coefficient); and Cq (ic ) is
the nominal output torque under nominal plate temperature Tp0 . Assuming μ(Tp )
and Cq (ic ) are independent and can be approximated by the polynomial functions in
Eqs. (9.5) and (9.6), respectively:
μ(Tp ) = 1 +
m
$
l j (Tp (t ) − Tp0 ) j
(9.5)
j=1
⎧ n
$
⎪
⎨
p i (t )k , if ic > 0
Cq (ic ) = k=1 k c
⎪
⎩
0,
if ic ≤ 0
(9.6)
where Tp0 is the nominal plate temperature and l j , pk are the coefficients that later
are identified empirically. Note that when a negative current is applied, the clutch will
disengage and no output torque will be transferred (i.e., Cq (ic ) saturates at zero when
ic ≤ 0). Thus, the temperature effects introduce nonlinearities in Eqs. (9.1) and (9.4).
Control Design and Stability Analysis
The proposed controller, as shown in Figure 9.9, contains three parts: (1) an observer
that estimates the unmeasured temperature states; (2) a feed-forward control that
9.2 Clutch Control for AWD
141
Tp
ω i ,ω o
Temp. State
Observer
To
Te
FF Cont.
r
PW PI Cont.
−
Vc
Circuit
Model
ic
Torque
Model
yo
Figure 9.9. Proposed feedback control design (disturbance observer + feed forward control +
piecewise PI feedback control).
compensates for the disturbances (i.e., temperature variation); and (3) a piecewise
PI control that realizes torque-feedback control:
u(t ) = u f f (t ) + u f b (t ) + r(t )
(9.7)
Three steps are required to develop the controller. First, an observer is designed
based on the temperature model in Eq. (9.1) to estimate the temperature disturbance. Second, a feed-forward controller is designed to remove the temperature
effects in Eqs. (9.5) and (9.6), rendering the dynamics of the nominal plant system (i.e., the plant and feed-forward system) independent of temperature states.
Third, approximating the nonlinearity in Eq. (9.4) via a piecewise linear function
leads to a PWA model for the nominal plant system. Based on the PWA model,
a piecewise PI-feedback control is developed. Compared to the original controller
in Figure 9.8, three additional sensors (i.e., coil-current sensor, output-torque sensor, and coil-temperature sensor) are proposed to be implemented to realize this
new design. The coil-temperature sensor provides temperature measurements and
helps the temperature estimates of the observer to converge to the actual states. The
torque sensor and current sensor provide signals to realize the piecewise PI feedback
control. These sensors are discussed in detail in Duan et al. (2011).
Experiments
To calibrate the parameters of the clutch model, experimental tests are conducted.
The clutch is mounted on a test bed where different operating conditions are simulated. The outside shell of the clutch is fixed on the body of the test bed and the
input and output shafts are connected to two motors, the speed of which can be
changed arbitrarily. Applying a voltage in the clutch coil generates a magnetic force
and engages the clutch. Torque sensors and angular-velocity sensors are placed on
the motors for monitoring the I/O speed and torque. Several thermistors are placed
on the clutch for measuring the temperature of the plate, oil, and coil. The measurements in these calibration experiments are Tp , To, Tc , Ta , ωi , ωo, ic , Vc , and Q.
A test profile, shown in Figure 9.10, is used for calibrating the temperature model.
It consists of a series of input and slip speeds, with a sequence of constant-power
torque steps at each combination. Coil current is applied periodically to generate
frictional energy in the system. The temperature of the components increases when
142
Transmission Control
Temparature (°C)
250
Ta
Tp
To
Tc
200
150
100
50
0
0
0.5
1
1.5
Time (sec)
2
2.5
3
x 104
4000
Input Speed
10x Clutch Differential Speed
Speed (rpm)
3000
2000
1000
0
−1000
0
0.5
1
1.5
Time (sec)
2
2.5
3
x 104
Figure 9.10. Test profile for clutch thermal system modeling.
there is an energy input and decreases when no energy is supplied. Therefore, the
temperature curves take on a zigzag pattern. Input speeds range from 125 to 3,500
RPM. At each input speed, the slip speed starts at 150 RPM and is reduced by steps.
The sampling rates for all the data are 10 Hz.
The parameters of the circuit model can be obtained from the tests. To calibrate
the torque model, two experiments with a similar setup are conducted to identify the
functions μ(Tp ) and Cq (ic ), respectively. A constant current input is used in the first
experiment to investigate the relationship between the plate temperature and the
μ factor. In the second experiment, different coil currents are applied sequentially
with sufficient cooling time in between to characterize the relationship between the
coil current and the output torque under nominal plate temperature.
Open-Loop System Results
Using the experimental data collected and the system identification toolbox from
MATLAB, the model parameters are as follows:
⎡
⎤
⎡
⎤
−0.05
1.02
0.395
0.074 0.33
0
1
AT = ⎣ 0.004 −0.026
0.096 ⎦ , BT = ⎣ 0
0.004 ⎦ , CT = 0 0 1
0.004
0.025 −0.049
0
0.01
c0 = 2.00 Ohm, Lc = 0.121 H, α = 0.004◦ C−1 .
μ(Tp ) = 1 − 0.0014(Tp − 40) − 6.8 × 10−6 (Tp − 40)2
"
−51.2585ic + 218.6445i2c − 56.9184i3c + 4.4563i4c , if
y0 (ic ) =
0, if ic ≤ 0
ic > 0
9.2 Clutch Control for AWD
143
Table 9.1. Partition over space of coil current
Region
1
2
3
4
ic
qi1 ic + qi2
<0 A
0∼0.8 A
0.8∼3.3 A
> 3.3 A
0
103ic
250ic − 117
101ic + 372
The polynomial model for μ is calibrated using the experimental data shown in
Figure 9.11. Based on the nonlinearity of yo, a PWA function with the partition in
Table 9.1 is used to approximate yo, as shown in Figure 9.12.
For these types of clutch systems, the error in the plate-temperature estimates
is typically within 15◦ C. Thus, the bounds for the uncertainty in the model can
be estimated; the step response of the open-loop system under the temperature
disturbance in Figure 9.13 is shown in Figure 9.14. Note that the variation of the
temperature states results in significant fluctuation and a decrease in the output
torque for the open-loop system.
Closed-Loop System
We now compare the existing controllers, as in Figure 9.8 with KP = 0.1 here to the
proposed controller in Duan et al. (2011). The gains of the proposed controller are
listed in Table 9.2 and vary for each region listed in Table 9.1.
1.1
Model
Data
1
Mu Factor
0.9
0.8
0.7
0.6
0.5
0.4
-50
0
50
100
Plate Temp (°C)
Figure 9.11. Mu factor versus plate temperature.
150
200
250
144
Transmission Control
Output Torque vs. Coil Current
900
Polynomial model
PWA model
800
700
Torque (Nm)
600
500
400
300
200
100
0
-1
0
1
2
Current (A)
3
4
5
Figure 9.12. Nonlinear torque model versus PWA approximation.
Temperature Disturbance
110
100
Temperature (°C)
90
80
Plate Temperature
Oil Temperature
Coil Temperature
70
60
50
40
0
1
2
3
4
5
Time (s)
Figure 9.13. Simulated temperature fluctuation.
6
7
8
9
10
9.2 Clutch Control for AWD
145
Table 9.2. Controller gains for each region
Region
1
2
3
4
KPi
0.03
0.0296
0.012
0.0301
KIi
KZi
0
1.0597
0.426
1.0781
40
0
0
0
As expected, both controlled systems perform better than the open-loop system
in Figure 9.14. The comparison between the proposed design and the current design
(see Figures 9.8 and 9.9, respectively) is provided in Figure 9.15, in which the performance of the proposed design is much better than the current design, which has
a significant steady-state error.
Stability Analysis
The stability of the closed-loop system also was examined using Lyapunov stability
methods to ensure system performance (Duan et al. 2011). The stability analysis considers uncertainty caused by approximation of the nonlinear system as a PWA system
and the effects of time delays in the controller. The tradeoff among delay, uncertainty, and performance (i.e., location of the closed-loop eigenvalues) is provided in
Figure 9.16. The bottom-left area is the feasible design region with guaranteed sta-
Step Response
550
500
450
400
Sine reference
Nonlinear model
Error of nonlinear model
Torque (Nm)
350
300
250
200
150
100
50
0
0
1
2
3
4
5
Time (s)
6
Figure 9.14. Step response of the open-loop system.
7
8
9
10
146
Transmission Control
Step Response
600
500
Torque (Nm)
400
300
Step reference
Proposed Design (PW PI)
Current Design (P only)
200
100
0
0
2
4
6
8
10
Time (s)
Figure 9.15. Comparison between current design and proposed design.
bility. As shown in the figure, the increase in delay reduces the maximum tolerable
uncertainty for a particular control design. A larger control gain improves the speed
of the system but renders it less robust against uncertainty and delay. A map such as
this can assist in the selection of controller parameters to obtain a sufficiently robust
system with satisfactory performance in the presence of delay.
30
ωn=20, ζ=0.707
25
ωn=30, ζ=0.707
Delay(ms)
20
Increased Speed
15
10
Stable Region
5
0
0
10
20
30
40
Uncertainty(%)
50
60
Figure 9.16. Map of feasible design
region showing the tradeoff between
speed of response, uncertainty, and
delay.
References
PROBLEMS
1. Patent databases include a number of patents awarded for automotive ECTs.
Search for and select one and summarize (i.e., one paragraph) the innovation being
claimed in that patent.
2. Figure 9.16 shows that as the closed-loop system bandwidth (ωn ) increases, delays
and uncertainty are more likely to destabilize the system. If the delay is 15 ms and the
uncertainty is 25 percent, what is the system bandwidth at the guaranteed stability
limit?
REFERENCES
Duan, S., J. Ni, and A. G. Ulsoy, 2011, “Modeling and Control of an Automotive All-Wheel
Drive Clutch as a Piecewise Affine System,” ASME Journal of Dynamic Systems, Measures,
and Controls (in press).
Hallowell, S. J., 2005, Torque Distribution Systems and Methods for Wheeled Vehicles, U.S.
Patent 6, 909–959.
Hrovat, D., and W. F. Powers, 1988, “Computer Control Systems for Automotive Power
Trains,” IEEE Control Systems Magazine, August 1988, pp. 3–10.
Kobayashi, N., T. Tokura, K. Shiiba, T. Fukumasu, T. Asami, and A. Yoshimura, 2008,
“Development of a New Sport Shift Control System for Toyota’s Automatic Transmission,”
SAE World Congress and Exhibition, Dearborn, MI, April 2008.
Kluger, M., and D. Fussner, 1997, “An Overview of Current CVT Mechanisms, Forces and
Efficiencies,” SAE Technical Paper 970688.
Sun, Z., and K. Hebbale, 2005, “Challenges and Opportunities in Automotive Transmission
Control,” Proceedings of the American Control Conference, Portland, OR, June 2005.
Zheng, Q., K. Srinivasan, and G. Rizzoni, 1998, “Dynamic Modeling and Characterization of
Transmission Response for Controller Design,” SAE Technical Paper 981094.
147
10
Control of Hybrid Vehicles
Hybrid vehicles, especially hybrid electric vehicles (HEVs), demonstrate significant
potential in reducing fuel consumption and exhaust emissions while maintaining driving performance. Hybrid vehicles are equipped with more than one power source,
and at least one should be reversible. The reversible power source serves as an energy
storage device, whereas the other power source is either the primary power source or
a “range extender.” Most hybrid vehicles use a battery as the energy buffer, in which
case they are known as HEVs. They can be classified as series, split, and parallel
hybrids (Figure 10.1). The performance potential of the different configurations and
the associated control problems are quite different.
The lower fuel consumption of a HEV typically is the result of several design
features: (1) right sizing of the prime mover (i.e., internal combustion engine [ICE]);
(2) load-leveling and engine shutdown to avoid inefficient engine operation; (3)
regenerative braking; and (4) enhanced CVT function in certain configurations (i.e.,
series and split). Due to the multiple power sources and the complex configuration
and operation modes associated with them, the control strategy for a hybrid vehicle
is more complicated than for an engine-only vehicle. This chapter focuses on the
design of control algorithms for series, parallel, and split hybrid vehicles. However,
we first discuss the layout and pros and cons of the three configurations.
10.1 Series, Parallel, and Split Hybrid Configurations
For series hybrids, the mechanical power from the internal-combustion engine is
converted immediately to electrical power by the generator, which is properly sized
to match the engine. The electrical power from the generator then drives the motor
(or motors), which can be augmented (positively or negatively) by the battery power.
Because there is no mechanical connection between the engine and the wheels,
there is total flexibility in engine operations (i.e., the generator provides a virtual
CVT function for the engine) and the drivetrain becomes simple. The electricpower distribution provides greater packaging flexibility compared to parallel or
split hybrids – the latter two configurations both have direct mechanical connections
between the engine and the wheels. In addition, because the propulsion power is
distributed in electric form, it is possible to implement in-hub motors, which provide
AWD and enhanced vehicle-stability control (VSC) functions (see Chapter 14).
148
10.1 Series, Parallel, and Split Hybrid Configurations
Inverter
Engine
Battery
Transmission
Motor
Generator
Inverter
Engine
Battery
Motor
Vehicle
Vehicle
Vehicle
Vehicle
Configuration B
Configuration A
Generator
Engine
149
Inverter
Planetary
Gear
Battery
Motor
Power Flow
Vehicle
Vehicle
Configuration C
Figure 10.1. Hybrid vehicle configurations: A. Parallel; B. Series; C. Power-Split (Parallel/
Series).
In a typical VSC execution, braking is applied to a wheel or wheels on one side
of the vehicle to create a yaw moment. With in-hub motors, it is possible to apply
traction power to the other side, doubling the yaw-moment capacity. The power
capacity of the electric motor (or motors) must be large because the engine does not
drive the wheel directly. The most significant drawback of the series-hybrid design
is the low efficiency inherent from the multiple energy conversions. Each energy
conversion (i.e., from mechanical to/from electrical or electrical to/from chemical)
is associated with a loss of between 5 and 20 percent. The system efficiency of series
hybrid vehicles is usually considerably lower than the parallel or split counterpart.
Because of the lower efficiency, series hybrids have not been popular for lightweight
vehicles. However, they are suitable for heavy-vehicle applications and, in fact,
have been widely adopted for other transportation applications such as locomotives,
agricultural machines, and submarines.
For parallel hybrids, a secondary power source (i.e., usually an electric motor)
exists in parallel with the ICE. There are many possible configurations for parallel
hybrids (Figure 10.2). When the motor is too small for vehicle propulsion (also
known as a microhybrid), the existence of the motor (known as the belt alternator
starter or the belt starter generator) has only limited effects on the vehicle operation
(i.e., engine shutoff and startup and limited regenerative braking). It does not change
the way a vehicle is driven. Microhybrids exist only in the form of parallel hybrids
and are not of the series or the split type. An electric motor sufficiently large to
150
Control of Hybrid Vehicles
Motor
Motor
Engine
Engine
Transmission
Transmission
“P3”
Belt Alternator
Starter (BAS)
Motor
Motor
Engine
Motor
Engine
Transmission
Transmission
“P1”
Ground
coupled
Figure 10.2. Several types of parallel hybrid vehicles.
drive a vehicle, either alone or by assisting the ICE, can be placed in many different
locations. Figure 10.2 shows that the motor can be placed either pre-transmission
(“P1”), post-transmission (“P3”), or at the originally nondriven axle (i.e., ground
coupled). There is a configuration called “P2” that is the same as P1 except that a
clutch is placed between the engine and the motor, in which case the torque converter
(of the presumably automatic transmission) can be removed for better efficiency.
Because there are many types of parallel hybrids, they share several attributes,
but the pros and cons are closely dependent on the configuration. Because the overall
architecture is closest to a traditional vehicle, modifications required to obtain a
hybrid version are relatively few; thus, development time can be the shortest. The
dual power sources also provide better fail-safe characteristics than series hybrids. If
they are properly designed, both the engine and the motor can have a much smaller
power rating than the original ICE yet achieve the same or better acceleration
performance. If a traditional step-gear transmission is used, a parallel hybrid does
not have CVT-like function, which results in losses due to the coupling between
vehicle and engine speeds. The Honda Integrated-Motor-Assist (IMA) design is an
example of a P1 design (Ogawa et al. 2003).
The third type of hybrid vehicle is the power-split type. The best-known examples include the Toyota Hybrid System (Muta et al. 2004) (used in the Prius,
Estima minivan, and RX400H) and the Allison Transmission Electric Drives System
(Holmes 2001). Both hybrid systems use a planetary gear(s) as the power-summation
device as well as to provide torque ratios. Two electric machines (i.e., the motor and
the generator, although both can have either role) are used as the secondary power
sources to sustain favorable operating conditions for the ICE as well as to augment
the engine-driving torque to satisfy a driver’s demand. In the case of the original
10.1 Series, Parallel, and Split Hybrid Configurations
M
151
M
V
E
E
G
V
G
Figure 10.3. Several examples of split hybrid vehicles (Left: original Toyota Hybrid System;
Center: original General Motors Design; Right: A Timken design).
Toyota Hybrid Design, one planetary gear is used, whereas in the original General
Motors design, two planetary gears are used. Figure 10.3 shows three examples of
split hybrid designs, all of which are patented. They are represented in the leverdiagram form in which each “stick” represents a planetary gear with three nodes: sun
gear, carrier gear, and ring gear. The small icons connected to the nodes represent
the vehicle (V), engine (E), motor (M), and generator (G). From Figure 10.3, it
is clear that there are many possible split-vehicle configurations. For split vehicles
with two planetary gears, there are in theory 1,152 different ways (Liu and Peng
2008) to connect the four elements, plus possibly a ground clutch, to the 6 nodes,
with 288 having the electronic continuously variable transmission (ECVT) function
(i.e., the engine and vehicle speeds can be independent). Apparently, in addition to
component sizing and control, configuration selection is an important decision in the
design of split hybrid vehicles.
In general, split hybrids have high engine efficiency because of the enhanced
CVT (ECVT) function. The planetary gear typically is very compact (i.e., only
slightly larger than a soft-drink can), has high torque capacity, has simple and robust
mechanical design, has very high efficiency, and potentially can have multiple operation modes if clutches are used. However, because it is inherently a mechanical
transmission, it is not suitable for certain types of hybrid vehicles. For example, it
does not make sense to have a fuel-cell split hybrid. All existing split vehicles use
two electric machines and, because of the torque balance on the planetary gears,
they are inherently strong hybrids with two sizable electric machines. Therefore, it is
more challenging to design a low-cost split hybrid vehicle. Because of the existence
of three power sources, the associated sizing and control problem is more complex
than for series and parallel hybrids.
152
Control of Hybrid Vehicles
Driver
Supervisory Control System
Other
sensors/actuators
CAN bus
Engine
controller
Engine
Mot/Battery
controller
Clutch
Motor
Trans
controller
Brakes
controller
Transmission
Vehicle
Battery
Figure 10.4. An example hierarchical structure of hybrid-vehicle control system: A supervisory control unit on the top interacts with several servo-loop controllers.
10.2 Hybrid Vehicle-Control Hierarchy
Hybrid vehicles require significant modifications to the propulsion and braking systems of ground vehicles. These changes also impact the design and functioning
of other subsystems, including electric systems; heating, ventilation, and air conditioning (HVAC); and auxiliary systems. Because hybrid vehicles introduce several
additional subsystems (i.e., the traction battery, motor, generator, power converters, and clutches), the control and coordination among all subsystems become more
complex. The role of control systems for a hybrid vehicle is similar to an operating
system for a computer. Even with the best components, a vehicle with bad control
algorithms may perform poorly if the control is poorly designed. It is important to
adopt a design process that ensures guaranteed quality of the control system.
Control engineers who design algorithms for a hybrid vehicle usually need to
encompass three fundamentally different functions. At the lowest level, there are
functions such as initializations, actuator and network monitoring, and sensor diagnosis that must be performed to ensure that all hardware is functioning properly.
If the hardware is judged to be functioning properly, then the control features can
be initiated. Typically, the hybrid vehicle control system is designed hierarchically
(Figure 10.4). At the top is a supervisory control unit, interacting with the human
driver as well as other vehicle control systems such as braking, VSC, and HVAC.
Under its “supervision” are the ECU, transmission control unit (TCU), batterymanagement system, and electric motor/drive. The supervisory control unit should
be designed assuming that it is fully aware of the capabilities and limitations of the
subsystems; thus, all commands sent to the servo-loop controllers of the subsystems
10.2 Hybrid Vehicle-Control Hierarchy
are achievable. The servo-loop controllers directly regulate the output of the subsystems; for example, a gear-shift command might be sent to the TCU, which then
executes the gear shift by manipulating the clutch pressures. Similarly, a command
for engine power (or torque) might be communicated to the ECU, which then manipulates the fuel injection, valve, EGR, and so on to achieve reliably and accurately
the required engine power (or torque).
The hierarchical structure described herein simplifies the design and improves
the modularity and reusability of hybrid vehicle control systems. The supervisory
control unit typically is designed based on slow dynamics and focuses on the longhorizon performance of a vehicle (e.g., fuel economy and/or emission during a drive
cycle). The servo-loop control units (e.g., ECU and TCU) are designed to achieve
accurate regulation or tracking of the commands and must follow quickly the commands from the supervisory unit under a wide range of vehicle load, age, and environmental conditions. Their designs usually are based on classical control theories
augmented by extensive calibrations. Once designed, the servo-loop controls can be
reused easily on another hybrid vehicle. The supervisory control, conversely, must
be redesigned for each hybrid vehicle based on vehicle load and characteristics of
the powertrain subsystems. In addition, the supervisory control must strike a balance among multiple objectives, including fuel economy; electric range; emission;
and noise, vibration, harshness (NVH). The main challenge in the design of hybrid
vehicle control systems is at the supervisory-control level.
The following sections introduce design concepts for the supervisory control
functions, focusing on the design for fuel economy while regulating the battery
charge. The discussion covers the prevailing design concepts for series, parallel, and
split hybrid vehicles. Before delving into the hybrid-vehicle controls and simulations,
we first illustrate the basic concepts of drive cycles, driving-power calculation, and
fuel/battery state of charge (SOC) calculations using MATLAB examples.
The performance of traditional and hybrid vehicles is evaluated using standard drive cycles that are
selected to represent typical driving conditions, such as in an urban or a highway
setting. Drive cycles are simply speed profiles as a function of time; when the
vehicle weight, rolling resistance, and aerodynamic drag coefficient are specified, the power required to go through these cycles can be calculated. This is
illustrated in the following MATLAB example and the plots in Figure 10.5:
EXAMPLE 10.1: DRIVE CYCLES AND DRIVING POWER.
% Ex10_1.m
% Drive cycle and driving power calculation
clear all, close all
load CYC_HWFET.mat % Load highway cycle
time_highway = cyc_mph(:,1);
speed_highway = cyc_mph(:,2);
figure(1), subplot(221)
plot(time_highway, speed_highway)
xlabel(‘Time (Sec)’)
ylabel(‘Speed (mph)’)
Title(‘EPA highway cycle’)
153
154
Control of Hybrid Vehicles
EPA highway cycle
EPA Urban cycle
60
Speed (mph)
Speed (mph)
60
40
20
0
0
200
4
x 10
0
200
400
Time (Sec)
600
0
4
4
0
-5
20
0
800
Power (Watt)
Power (Watt)
5
400
600
Time (Sec)
EPA highway cycle
40
800
x 10
500
1000
Time (Sec)
EPA Urban cycle
1500
500
1000
Time (Sec)
1500
2
0
-2
-4
0
Figure 10.5. Output of Example 10.1. Driving power required for the EPA highway and urban
cycles is low, usually less than 20kw for the mid-sized vehicle simulated.
clear cyc_mph
load CYC_UDDS.mat % Load urban cycle
time_urban = cyc_mph(:,1);
speed_urban = cyc_mph(:,2);
subplot(222)
plot(time_urban, speed_urban)
xlabel(‘Time (Sec)’)
ylabel(‘Speed (mph)’)
Title(‘EPA Urban cycle’)
m = 1800;
g = 9.81;
rolling_resistance_coeff = 0.015;
aero_dynamic_drag_coeff = 0.4;
road_grad = 0;
% road grade = 0 rad
N = size(time_highway, 1);
speed_highway = speed_highway*1602/3600;
% Change the unit of speed to m/sec
distance_highway = 0;
for i = 1:N-1,
accel_highway(i) = (speed_highway(i+1)-speed_highway(i))/1.0;
% delta_T = 1 sec
F_resistant_highway(i) =
rolling_resistance_coeff*m*g*cos(road_grad) ...
+0.5*1.202*aero_dynamic_drag_coeff*speed_highway(i)ˆ2 ...
+ m*g*sin(road_grad);
10.2 Hybrid Vehicle-Control Hierarchy
power_highway(i) = (m*accel_highway(i)+
F_resistant_highway(i))*speed_highway(i);
% Power = F*V
distance_highway = distance_highway + speed_highway(i);
end
subplot(223)
plot(time_highway(1:N-1), power_highway)
xlabel(‘Time (Sec)’)
ylabel(‘Power (Watt)’)
Title(‘EPA highway cycle’)
N = size(time_urban, 1);
speed_urban = speed_urban*1602/3600;
% Change the unit of speed to m/sec
distance_urban = 0;
for i = 1:N-1,
accel_urban(i) = (speed_urban(i+1) - speed_urban(i))/1.0;
% delta_T = 1 sec
F_resistant_urban(i) =
rolling_resistance_coeff*m*g*cos(road_grad) ...
+0.5*1.202*aero_dynamic_drag_coeff*speed_urban(i)ˆ2 ...
+ m*g*sin(road_grad);
power_urban(i) = (m*accel_urban(i) +
F_resistant_urban(i))*speed_urban(i);
% Power = F*V (W)
distance_urban = distance_urban + speed_urban(i);
end
subplot(224)
plot(time_urban(1:N-1), power_urban)
xlabel(‘Time (Sec)’)
ylabel(‘Power (Watt)’)
Title(‘EPA Urban cycle’)
The ICE has been and likely will
remain the primary power source on nearly all hybrid vehicles currently on the
market. For these vehicles, efficient engine operation is a necessary (but not
quite sufficient) condition for efficient hybrid vehicles. Selection of an efficient
operation for an ICE is based on its BSFC, which typically has a “sweet spot” in
the area where the engine is most efficient. Some of the basic control concepts
start from the simple idea of operating the engine close to the sweet spot, which
is highlighted in the plots in Figure 10.6:
EXAMPLE 10.2: EFFICIENT ENGINE OPERATIONS.
% Ex10_2 Efficient Engine Operations
load engine_parameters
% Load engine torque (N-m)
% and fuel consumption (g/sec)
% Both of which is a 21 by 28 matrix
% The row (throttle) index is 0:5:100
155
156
Control of Hybrid Vehicles
BSFC g/sec/kW
100
90
80
15
70
10
60
Throttle
fuel consumption (g/sec)
20
5
50
40
0
100
30
6000
50
4000
20
10
2000
0
Throttle
0
0
Engine speed (rpm)
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Engine speed (rpm)
Engine power (kW)
100
140
90
80
70
120
Throttle
60
100
50
80
40
60
40
30
20
20
10
0
0
-20
-4
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Engine speed (rpm)
Figure 10.6. Output of Example 10.2. The engine fuel consumption and BSFC behavior
become the basis for selecting control rules for efficient engine operations.
% The column (engine speed) index is 600:200:6000 (rpm)
engine_speed = [600:200:6000]*2*pi/60;
Throttle_grid=[0:5:100];
% rad/sec
figure(2)
mesh([600:200:6000], Throttle_grid, fuel_map)
xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
zlabel(’fuel consumption (g/sec)’)
for k = 1:21,
for j = 1:28,
engine_out_power(k,j) =
engine_speed(j)*engine_torque(k,j)/1000;
% engine out power (kW)
bsfc(k,j) = fuel_map(k,j)/engine_out_power(k,j);
% BSFC = g/sec/kW
if (bsfc(k,j) < 0)
10.3 Control Concepts for Series Hybrids
157
Engine Power
Max
Power
(120kW)
Threshold
Power
(60kW)
Max power
SOC (0.25)
Low
SOC
(0.35)
High
SOC
(0.5)
1.0
SOC
Figure 10.7. “Thermostat”-control concept common for supervisory control of series hybrid
vehicles.
bsfc(k,j) = 0.3;
elseif (bsfc(k,j) > 0.3)
bsfc(k,j) = 0.3;
end
end
end
figure(3)
contour([600:200:6000], Throttle_grid, bsfc, 50)
xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
title(‘BSFC g/sec/kW’)
hold on, plot(2350, 43, ‘d’), hold off
figure(4)
[CS,H]=contour([600:200:6000], Throttle_grid, engine_out_power);
clabel(CS), xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
title(‘Engine power (kW)’)
hold on, plot(2350, 43, ‘d’), hold off
10.3 Control Concepts for Series Hybrids
The supervisory control of series hybrids is relatively straightforward – mainly
because the engine can be run independently from vehicle speed and load. This
decoupling makes it possible to determine the engine operation while completely
disregarding the vehicle states – and, because of this decoupling, it makes no sense
to demand engine torque, so a power-based control concept is more appropriate.
A commonly used design concept is to determine the engine power based only on
the battery SOC. The ICE is idling when there is adequate SOC. When the battery
SOC is low, the ICE is turned on and run at a high-efficiency level until the battery
SOC reaches a high threshold level. This “thermostat” control concept is illustrated
in Figure 10.7. The control algorithm involves only a few parameters: the high/low
bounds of the SOC dead band (i.e., 0.5 and 0.35), the threshold power (i.e., 60kW),
158
Control of Hybrid Vehicles
and the “maximum power SOC” (i.e., 0.25), which are selected based on battery
and vehicle characteristics. For modern Li-Ion batteries, the maximum power SOC
may need to be higher to improve battery life. If an engine is selected properly
based on vehicle load, the engine SOC operates mostly in the hysteresis rectangle.
If the vehicle load is very high, it may be necessary for the engine to run at a power
level higher than the threshold power. In the special case of when the thermostatic
control concept is applied to a fuel-cell–powered series hybrid, the hysteresis rectangle could disappear completely. This is because fuel cells, unlike ICEs, operate
efficiently at low power/current level because the efficiency is caused primarily by
the Ohmic loss. Therefore, lower stack current is more efficient if the ancillary loads
(e.g., compressor and humidifier) are not considered.
The thermostat concept is extremely simple and therefore was a popular concept for early hybrid-vehicle control designs. However, the concept has several
drawbacks. First, the engine will be turned on and shut down under conditions that
are completely independent of vehicle driving, causing possible confusion and/or
discomfort. Second, the engine will experience frequent on/off and deep transients,
which may need to be adjusted when considering the effect on emissions. Third, the
significant amount of energy stored in and released from the battery incur efficiency
loss, sometimes known as the “battery tax.” Round-trip efficiency loss is typically
in the range of 15 to 30 percent – which is significant and could erase most of the
efficiency gains by operating the ICE only around the efficient sweet spot.
The
engine fuel-consumption and power-generation characteristics are analyzed first.
Typical of today’s passenger automobiles in the North American market, it can
be seen that the engine is somewhat oversized. The sweet spot (denoted by
diamond symbols in Figure 10.8) corresponds to a power level of more than
60kW, which is much higher than the vehicle driving load under the EPA highway and urban cycles. Therefore, engine efficiency will be low when it operates
far from the sweet spot. In a hybrid configuration, it may be appropriate to
downsize the engine. In this example, however, we assume that the original
engine is used. Because the engine can be operated completely independently
of the vehicle state, we limit the engine to operate in a small region, denoted by
thick blue lines in Figure 10.8. From the BSFC plot, it is obvious that operating
the engine at a power level much below 40kW would not be efficient, which
should be a consideration in designing the control strategy. From the simulation results, it shows that the engine is turned on periodically to charge up the
battery, and the SOC vacillates between the selected lower and upper threshold
values.
EXAMPLE 10.3: THERMOSTATIC CONTROL CONCEPT FOR SERIES HYBRIDS.
% Ex10_3.m
% Compute fuel consumption of a thermostatic controlled
% Series huybrid vehicle for the Highway and Urban Cycles
clear all
ex10_1; % Load and run ex10_1 to compute driving power
ex10_2; % Load and run ex10_2 to compute engine map
Engine power (kW)
100
90
90
80
80
70
70
60
60
Throttle
50
30
30
20
20
10
10
20
40
0
-20
-4
0
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Engine speed (rpm)
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
Engine speed (rpm)
Results for the highway cycle
20
0
200
400
600
Time (sec)
800
0.5
SOC
2
0
0
200
400
600
Time (sec)
0.45
0.4
0.35
0
200
400
600
Time (sec)
800
60
40
20
0
800
50
Motor power (kW)
0.55
4
0
500
1000
Time (sec)
1500
0.55
0
-50
0.45
0.4
0.35
-100
0
200
400
600
Time (sec)
800
6
4
2
0
0
500
1000
Time (sec)
1500
0
500
1000
Time (sec)
1500
50
0.5
SOC
0
80
Motor power (kW)
40
Results for the Urban cycle
6
Engine power (kW)
Engine fueling rate (g/sec)
Engine power (kW)
60
100
80
40
80
120
60
50
40
0
140
Engine fueling rate (g/sec)
Throttle
BSFC g/sec/kW
100
0
500
1000
Time (sec)
1500
0
-50
-100
159
Figure 10.8. Plots generated for Example 10.3. Thick blue lines indicate efficient engine regions, which are used to selectparameters of the thermostat-control algorithm.
160
Control of Hybrid Vehicles
figure(3)
hold on, plot([600;4000], [18; 66], ‘-’, ‘LineWidth’, 3)
hold off
figure(4)
hold on, plot([600;4000], [18; 66], ‘-’, ‘LineWidth’, 3)
hold off
% The optimal operation of the engine is characterized
by the end points
% [600rpm, 18% throttle] and [4000rpm, 66% throttle].
opt_eng_speed = [600: 3400/48: 4000];
opt_throttle = [18:1:66];
% Interpolate to find fuel consumption (g/sec) and power (kW)
for each
% point
opt_fuel = interp2([600:200:6000], Throttle_grid, fuel_map,
opt_eng_speed, opt_throttle);
opt_power = interp2([600:200:6000]*2*pi/60, Throttle_grid,
engine_out_power, ...
opt_eng_speed*2*pi/60, opt_throttle);
figure(5), subplot(211)
plot(opt_eng_speed, opt_fuel), xlabel(‘Engine speed (rpm)’),
ylabel(‘fuel cnsumption (g/sec)’)
subplot(212)
plot(opt_eng_speed, opt_power), xlabel(‘Engine speed (rpm)’),
ylabel(’power (kW)’)
% Simulate for the two drive cycles, series hybrid, themodtatic
control
SOC_max_power = 0.25;
SOC_low = 0.35;
SOC_high = 0.50;
threshold_power = 60;
% kW
max_power = 120;
% kW
Battery_capacity = 2.5;
% kW-hr
% Simulate for the highway cycle
total_fuel_highway_hev = 0;
SOC(1) = 0.52;
flag = 0;
% flag tracks whether we are on the top or bottom
of the hysteresis rectangle
for i = 1: (size(time_highway, 1)-1),
if SOC(i) > SOC_high
P_eng(i) = 0; flag = 0;
10.3 Control Concepts for Series Hybrids
elseif SOC(i) > SOC_low
if flag == 0
P_eng(i) = 0;
else
P_eng(i) = threshold_power;
end
elseif SOC(i) > SOC_max_power
flag = 1;
P_eng(i) = max_power - (SOC(i)-SOC_max_power)/(SOC_lowSOC_max_power) ...
*(max_power-threshold_power);
else
flag = 1;
P_eng(i) = max_power;
end
P_batt(i) = power_highway(i)/1000 - P_eng(i);
if P_batt(i) > 0
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600/0.92/Battery_capacity;
else
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600*0.92/Battery_capacity;
end
if P_eng(i) == 0
fuel_hev(i) = 0;
else
fuel_hev(i) = interp1(opt_power, opt_fuel, P_eng(i),
‘linear’);
end
total_fuel_highway_hev = total_fuel_highway_hev + fuel_hev(i);
end
figure(6), subplot(221)
plot([1: size(time_highway,1)-1], P_eng)
xlabel(‘Time (sec)’), ylabel(‘Engine power (kW)’)
title(‘Results for the highway cycle’)
subplot(222), plot([1: size(time_highway,1)-1], fuel_hev)
xlabel(‘Time (sec)’), ylabel(‘Engine fueling rate (g/sec)’)
subplot(223), plot([1: size(time_highway,1)], SOC)
xlabel(‘Time (sec)’), ylabel(‘SOC’)
subplot(224), plot([1: size(time_highway,1)-1], P_batt)
xlabel(‘Time (sec)’), ylabel(‘Battery power (kW)’)
% Simulate for the urban cycle
total_fuel_urban_hev = 0;
SOC(1) = 0.44;
161
162
Control of Hybrid Vehicles
flag = 0;
% flag tracks whether we are on the top or bottom
of the hysteresis rectangle
for i = 1: (size(time_urban, 1)-1),
if SOC(i) > SOC_high
P_eng(i) = 0; flag = 0;
elseif SOC(i) > SOC_low
if flag == 0
P_eng(i) = 0;
else
P_eng(i) = threshold_power;
end
elseif SOC(i) > SOC_max_power
flag = 1;
P_eng(i) = max_power - (SOC(i)-SOC_max_power)/(SOC_lowSOC_max_power) ...
*(max_power-threshold_power);
else
flag = 1; P_eng(i) = max_power;
end
P_batt(i) = power_urban(i)/1000 - P_eng(i);
if P_batt(i) > 0
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600/0.92/Battery_capacity;
else
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600*0.92/Battery_capacity;
end
if P_eng(i) == 0
fuel_hev(i) = 0;
else
fuel_hev(i) = interp1(opt_power, opt_fuel, P_eng(i),
‘linear’);
end
total_fuel_urban_hev = total_fuel_urban_hev + fuel_hev(i);
end
figure(7), subplot(221)
plot([1: size(time_urban,1)-1], P_eng)
xlabel(‘Time (sec)’), ylabel(‘Engine power (kW)’)
title(‘Results for the Urban cycle’)
subplot(222), plot([1: size(time_urban,1)-1], fuel_hev)
xlabel(‘Time (sec)’), ylabel(‘Engine fueling rate (g/sec)’)
subplot(223), plot([1: size(time_urban,1)], SOC)
xlabel(‘Time (sec)’), ylabel(‘SOC’)
subplot(224), plot([1: size(time_urban,1)-1], P_batt)
xlabel(‘Time (sec)’), ylabel(‘Battery power (kW)’)
10.3 Control Concepts for Series Hybrids
163
% connvert fuel econpmy results from meter/gram to mpg
mpg_highway_HEV =
(distance_highway/1602)/(total_fuel_highway_hev/0.74/3785)
mpg_urban_HEV =
(distance_urban/1602)/(total_fuel_urban_hev/0.74/3785)
The thermostat concept can be enhanced by making the ICE power level dependent on the desired driving power and battery SOC (Abthoff et al. 1998; Hayashida
and Narusawa 1999). A major reason to have weak coupling between the engine
power and the vehicle load is to reduce efficiency losses due to energy conversion.
A simple concept is illustrated in Figure 10.9.
In Figure 10.9, the power demand is passed through a low-pass filter to create
an engine-power demand. This low-pass filter should be more complicated than a
typical low-pass filter because if the power demand is too low, it may not make sense
to run the engine at that low level of power, depending on the engine-efficiency
and the battery characteristics. If the feed-forward part of the control is achieved
properly and if the driving load is adequately high (e.g., on a long uphill), it will
reduce energy passing through the battery, thereby reducing battery-efficiency loss.
Implementation of the feed-forward algorithm is shown in Example 10.4.
EXAMPLE 10.4: MODIFIED THERMOSTATIC-CONTROL CONCEPT FOR SERIES HYBRIDS. The feed-forward enhanced thermostatic-control concept is implemented
in this example. This concept does not improve fuel economy and is not appropriate when the drive-cycle load is low because the engine would be quite inefficient
at low power. Therefore, even considering the battery tax, it may not be justified
to run the engine at low power demand. However, if the road load is high (e.g.,
significant road grade), then the new feed-forward term would help significantly
by avoiding the battery loss. In the example, we assume an optimistic battery
and power electronic loss of 8 percent (each way), which is close to the best efficiency achieved under ideal conditions. Actual loss can be significantly higher,
which means that the feed-forward enhancement may be beneficial in a wider
range of conditions.
Pd
feedforward
freq
+
Low pass filter
Pe
+
SOC
feedback
SOC
Thermostat
algorithm
Figure 10.9. A feed-forward enhanced thermostat-control concept to reduce battery
efficiency.
164
Control of Hybrid Vehicles
% Ex10_4.m
% Compute fuel consumption of a thermostatic controlled
% Series hybrid vehicle for the US06 Cycles
% With feedforward control to reduce battery efficiency loss
clear all, close all
load CYC_US06.mat
% Load the highway cycle
time_US06 = cyc_mph(:,1);
speed_US06 = cyc_mph(:,2);
figure(1), subplot(211)
plot(time_US06, speed_US06)
xlabel(‘Time (Sec)’)
ylabel(‘Speed (mph)’)
Title(‘US06 cycle’)
m = 1800;
g = 9.81;
rolling_resistance_coeff = 0.015;
aero_dynamic_drag_coeff = 0.4;
road_grad = 0;
% road grade = 0 rad
N = size(time_US06, 1);
speed_US06 = speed_US06*1602/3600; % Change the unit of speed to
m/sec
distance_US06 = 0;
for i = 1:N-1,
accel_US06(i) = (speed_US06(i+1) - speed_US06(i))/1.0;
% delta_T = 1 sec
F_resistant_US06(i) =
rolling_resistance_coeff*m*g*cos(road_grad) ...
+0.5*1.202*aero_dynamic_drag_coeff*speed_US06(i)ˆ2 ...
+ m*g*sin(road_grad);
power_US06(i) = (m*accel_US06(i) +
F_resistant_US06(i))*speed_US06(i);
% Power = F*V
distance_US06 = distance_US06 + speed_US06(i);
end
subplot(212)
plot(time_US06(1:N-1), power_US06)
xlabel(‘Time (Sec)’)
ylabel(‘Power (Watt)’)
Title(‘US06 cycle’)
ex10_2;
% Load and run ex10_2 to compute engine map
opt_eng_speed = [1500: 2500/35: 4000];
opt_throttle = [31:1:66];
opt_fuel = interp2([600:200:6000], Throttle_grid, fuel_map,
10.3 Control Concepts for Series Hybrids
opt_eng_speed, opt_throttle);
opt_power = interp2([600:200:6000]*2*pi/60, Throttle_grid,
engine_out_power, ...
opt_eng_speed*2*pi/60, opt_throttle);
[B_filter,A_filter] = butter(3, 0.02);
P_eng_FF = filtfilt(B_filter,A_filter, power_US06/1000);
% Simulate for the two drive
control
SOC_max_power = 0.25;
SOC_low = 0.35;
SOC_high = 0.50;
threshold_power = 60;
%
max_power = 120;
%
Battery_capacity = 2.5;
%
cycles, series hybrid, themodtatic
kW
kW
kW-hr
% Simulate for the US06 cycle
total_fuel_US06_hev = 0;
SOC(1) = 0.52;
flag = 0;
% flag tracks whether we are on the top or bottom
of the hysteresis rectangle
for i = 1: (size(time_US06, 1)-1),
if SOC(i) > SOC_high
P_eng_FB(i) = 0; flag = 0;
elseif SOC(i) > SOC_low
if flag == 0
P_eng_FB(i) = 0;
else
P_eng_FB(i) = threshold_power;
end
elseif SOC(i) > SOC_max_power
flag = 1;
P_eng_FB(i) = max_power - (SOC(i)-SOC_max_power)/
(SOC_low-SOC_max_power) ...
*(max_power-threshold_power);
else
flag = 1;
P_eng_FB(i) = max_power;
end
P_eng(i) = P_eng_FF(i) + P_eng_FB(i);
if P_eng(i) < threshold_power
P_eng(i) = 0;
end
P_batt(i) = power_US06(i)/1000 - P_eng(i); % unit: kW
if P_batt(i) > 0
SOC(i+1) = SOC(i) -
165
166
Control of Hybrid Vehicles
P_batt(i)*1.0/3600/0.92/Battery_capacity;
else
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600*0.92/Battery_capacity;
end
if P_eng(i) == 0
fuel_hev(i) = 0;
else
fuel_hev(i) = interp1(opt_power, opt_fuel, P_eng(i),
‘linear’);
end
total_fuel_US06_hev = total_fuel_US06_hev + fuel_hev(i);
end
figure(5), subplot(221)
plot([1: size(time_US06,1)-1], P_eng)
xlabel(‘Time (sec)’), ylabel(‘Engine power (kW)’)
title(‘Results for the US06 cycle’)
subplot(222), plot([1: size(time_US06,1)-1], fuel_hev)
xlabel(‘Time (sec)’), ylabel(‘Engine fueling rate (g/sec)’)
subplot(223), plot([1: size(time_US06,1)-1],
SOC(1:size(time_US06,1)-1))
xlabel(‘Time (sec)’), ylabel(’SOC’)
subplot(224), plot([1: size(time_US06,1)-1], P_batt)
xlabel(‘Time (sec)’), ylabel(’Battery power (kW)’)
% convert fuel economy results from meter/gram to mpg
mpg_US06_HEV =
(distance_US06/1602)/(total_fuel_US06_hev/0.74/3785)
10.4 Control Concepts for Parallel Hybrids
Parallel hybrids have powertrain configurations closest to traditional ICE-only vehicles and therefore are used in many prototype hybrid vehicles. Many different control
concepts have been introduced, with different levels of success. Three concepts are
introduced in the following sections.
Load-Leveling Concept
This concept is an “engine-centric” idea and focuses only on operating the engine
efficiently, within a band of efficient power – similar to the engine-operation analysis
described in the series-hybrid discussion. The basic concept can be explained using
Figure 10.10, which shows the BSFC of an engine – similar to the plot generated in
Example 10.2. However, the y-axis in Figure 10.10 is engine torque. If this engine is
used on a parallel hybrid vehicle, then a simple idea to avoid inefficient operations
of the engine is to select two power levels (denoted by the green and red lines) and
operate the ICE only within these two power levels. When the power requirement is
10.4 Control Concepts for Parallel Hybrids
167
Table 10.1. Example load-leveling control concept for
parallel hybrid vehicles
Engine power
SOC >= SOCt
SOC < SOCt
Pd < = Pev
0 (V < Vlow )
Pd + Pch
Pev < Pd <= Pemax
Pd
Pd + Pch
Pd > Pemax
Pemax
Pemax
lower than the “all-electric power level” (i.e., the green line), the engine will not be
used; instead, only the electric motor will be used. When the required power from
the driver is higher than the “maximum engine power line” (i.e., the red line), the
engine will operate only at that power level, and the deficit will be provided by the
battery and electric motor. When the battery SOC is too low, the electric motor
may not be able to provide the load-leveling function and the engine power will be
modified accordingly. The basic control rules are shown in Table 10.1. To improve
engine efficiency, a transmission still should be used to move the operation of the
engine along the constant-power lines.
The rules shown in Table 10.1 can be enhanced further by including other factors
in the decisions, such as emission, NVH, and vehicle speed.
The load-leveling concept
shown in Table 10.1 is implemented in this example. A noticeable change is that
the engine is downsized from the one used in the series hybrid by 30 percent. The
engine torque and fuel consumption are assumed to reduce proportionally. The
two power-threshold values are selected to be Pev = 20kW and Pemax = 70kW.
EXAMPLE 10.5: LOAD-LEVELING CONTROL CONCEPT.
Fuel Consumption (kg/kWhr)
350
0.
23
300
100
21
0. 22
0.
150
Engine only
6
250
200
assist
0.214
E ngine Torque (Nm )
400
Power
0 .2 1 2
7
0.2
5
2
0.
0.
0.
26
450
24
500
0. 24
0.23
0.25
0.26
0 .2 7
Motor only
50
800
1000
1200
1400
1600
1800
E ngine S peed (rpm )
2000
2200
2400
Figure 10.10. Load-leveling-control concept: Engine power is determined mainly based on
the desired power level.
168
Control of Hybrid Vehicles
Engine power (kW)
60
40
20
0
0
200
400
Time (sec)
600
Engine fueling rate (g/sec)
Results for the US06 cycle
80
SOC
0.6
0.55
0.5
0.45
4
2
0
0
200
400
Time (sec)
600
0
200
400
Time (sec)
600
20
Battery power (kW)
0.65
6
0
200
400
Time (sec)
600
800
0
-20
-40
-60
-80
Figure 10.11. Simulation results of a parallel hybrid vehicle based on the load-leveling-control
concept on the US06 drive cycle.
Similar to the two previous examples, it is assumed that a CVT is used to maintain
the engine operation around its optimal speed point. In this example, the US06
cycle is used, which requires higher power than the urban or the highway cycle.
It is observed (see Figure 10.11) that in a parallel-hybrid powertrain, the engine
is on most of the time (which is a major difference from a series hybrid).
% Ex10_5.m
% Compute fuel consumption of a load levelling
% Parallel hybrid vehicle for the Highway and Urban Cycles
clear all, close all
load CYC_US06.mat
% Load the US06 cycle
time_US06 = cyc_mph(:,1);
speed_US06 = cyc_mph(:,2);
figure(1), subplot(211)
plot(time_US06, speed_US06)
xlabel(‘Time (Sec)’)
ylabel(‘Speed (mph)’)
Title(‘US06 cycle’)
m = 1800;
g = 9.81;
rolling_resistance_coeff = 0.015;
aero_dynamic_drag_coeff = 0.4;
road_grad = 0;
% road grade = 0 rad
N = size(time_US06, 1);
10.4 Control Concepts for Parallel Hybrids
speed_US06 = speed_US06*1602/3600;
m/sec
distance_US06 = 0;
169
% Change the unit of speed to
for i = 1:N-1,
accel_US06(i) = (speed_US06(i+1) - speed_US06(i))/1.0;
% delta_T = 1 sec
F_resistant_US06(i) =
rolling_resistance_coeff*m*g*cos(road_grad) ...
+0.5*1.202*aero_dynamic_drag_coeff*speed_US06(i)ˆ2 ...
+ m*g*sin(road_grad);
power_US06(i) = (m*accel_US06(i) +
F_resistant_US06(i))*speed_US06(i)/1000;
% Power = F*V, in kW
distance_US06 = distance_US06 + speed_US06(i);
end
subplot(212)
plot(time_US06(1:N-1), power_US06)
xlabel(‘Time (Sec)’)
ylabel(‘Power (kW)’)
Title(‘US06 cycle’)
load engine_parameters
% Load engine torque (N-m)
% and fuel comsumption (g/sec)
% Both of which is a 21 by 28 matrix
% The row (throttle) index is 0:5:100
% The column (engine speed) index is 600:200:6000 (rpm)
engine_speed = [600:200:6000]*2*pi/60;
Throttle_grid=[0:5:100];
engine_torque = engine_torque * 0.7;
fuel_map = fuel_map * 0.7;
% rad/sec
% Scale down engine by 30%
figure(2)
mesh([600:200:6000], Throttle_grid, fuel_map)
xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
zlabel(‘fuel consumption (g/sec)’)
for k = 1:21,
for j = 1:28,
engine_out_power(k,j) =
engine_speed(j)*engine_torque(k,j)/1000;
% engine out power (kW)
bsfc(k,j) = fuel_map(k,j)/engine_out_power(k,j);
% BSFC = g/sec/kW
if (bsfc(k,j) < 0)
170
Control of Hybrid Vehicles
bsfc(k,j) = 0.3;
elseif (bsfc(k,j) > 0.3)
bsfc(k,j) = 0.3;
end
end
end
figure(3)
contour([600:200:6000], Throttle_grid, bsfc, 50)
xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
title(‘BSFC g/sec/kW’)
hold on, plot(2350, 43, ‘d’), hold off
figure(4)
[CS,H]=contour([600:200:6000], Throttle_grid, engine_out_power);
clabel(CS), xlabel(‘Engine speed (rpm)’)
ylabel(‘Throttle’)
title(‘Engine power (kW)’)
hold on, plot(2350, 43, ‘d’), hold off
figure(3)
hold on, plot([600;4000], [18; 66], ‘-’, ‘LineWidth’, 3)
hold off
figure(4)
hold on, plot([600;4000], [18; 66], ‘-’, ‘LineWidth’, 3)
hold off
opt_eng_speed = [600: 3400/48: 4000];
opt_throttle = [18:1:66];
opt_fuel = interp2([600:200:6000], Throttle_grid, fuel_map,
opt_eng_speed, opt_throttle);
opt_power = interp2([600:200:6000]*2*pi/60, Throttle_grid,
engine_out_power, ...
opt_eng_speed*2*pi/60, opt_throttle);
% Simulate for the US06 cycle, parallel hybrid, load leveling
control
P_ev = 20;
Pe_max = 70;
P_ch = 20;
SOC_t = 0.45;
Battery_capacity = 2.5;
% kW-hr
% Simulate for the US06 cycle
total_fuel_US06_hev = 0;
SOC(1) = 0.57;
10.4 Control Concepts for Parallel Hybrids
171
for i = 1: (size(time_US06, 1)-1),
if SOC(i) >= SOC_t
if power_US06(i) <= P_ev
P_eng(i) = 0;
elseif power_US06(i) <= Pe_max
P_eng(i) = power_US06(i);
else
P_eng(i) = Pe_max;
end
else
if power_US06(i) < 0
P_eng(i) = 0;
elseif power_US06(i) <= P_ev
P_eng(i) = power_US06(i) + P_ch;
elseif power_US06(i) <= Pe_max
P_eng(i) = power_US06(i) + P_ch;
else
P_eng(i) = Pe_max;
end
end
P_batt(i) = power_US06(i) - P_eng(i);
if P_batt(i) > 0
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600/0.92/Battery_capacity;
else
SOC(i+1) = SOC(i) P_batt(i)*1.0/3600*0.92/Battery_capacity;
end
% unit: kW
if P_eng(i) == 0
fuel_hev(i) = 0;
else
fuel_hev(i) = interp1(opt_power, opt_fuel, P_eng(i),
‘linear’);
end
total_fuel_US06_hev = total_fuel_US06_hev + fuel_hev(i);
end
figure(5), subplot(221)
plot([1: size(time_US06,1)-1], P_eng)
xlabel(‘Time (sec)’), ylabel(‘Engine power (kW)’)
title(‘Results for the US06 cycle’)
subplot(222), plot([1: size(time_US06,1)-1], fuel_hev)
xlabel(‘Time (sec)’), ylabel(‘Engine fueling rate (g/sec)’)
subplot(223), plot([1: size(time_US06,1)], SOC)
xlabel(‘Time (sec)’), ylabel(‘SOC’)
172
Control of Hybrid Vehicles
subplot(224), plot([1: size(time_US06,1)-1], P_batt)
xlabel(‘Time (sec)’), ylabel(‘Battery power (kW)’)
% convert fuel economy results from meter/gram to mpg
mpg_US06_HEV = (distance_US06/1602)/
(total_fuel_US06_hev/0.74/3785)
Equivalent Consumption Minimization Strategy
The ECMS concept is an instantaneous optimization concept, and the basic idea is
that the best engine/battery power split is one that achieves minimum “equivalent
fuel consumption.” In other words, the battery power is converted to an equivalent
fuel consumption, and the sum of the gasoline engine fuel and battery equivalent fuel
is minimized at every step. Because the optimal decision is made every sampling time,
it is a “greedy” or instantaneous optimization decision process. The ECMS concept
relies on the desire that instantaneous optimizations result in overall performance
that is close to horizon optimization; that is:
(10.1)
Min[ṁ f (t )]dt ≈ Min ṁ f (t )dt
In practical applications, the ECMS method usually achieves fuel-consumption performance close to algorithms that are developed based on horizon-optimization
methods, although such near-optimality is never guaranteed. In the ECMS concept,
the “fuel consumption” to be minimized includes not only the fuel consumed by
the engine but also the equivalent fuel consumption by the battery (Paganelli et al.
2000). That is,
ṁ f (t ) = ṁ f
eng
+ ṁ f
(10.2)
batt
where:
ṁ f
batt
=
SCeng · Pbatt
(10.3)
Effelectrical
In other words, the battery power is converted to an equivalent fuel consumption
by considering the efficiency of the engine (SCeng) and the electric-path efficiency
(Effelectrical ). An issue in solving an optimization problem to minimize Eq. (10.2) is
that the battery SOC may not be maintained around the desired level and could
fluctuate depending on the driving load. To mitigate this problem, Eq. (10.2) is
modified to include a multiplier that favors battery SOC around a desired region.
The modified fuel-consumption term becomes:
ṁ f
equi (t )
= ṁ f
eng
+ f (soc) · ṁ f
batt
(10.4)
where the SOC modifier f(soc) may look like the curve shown in Figure 10.12. The
characteristics of f(soc) include the following: (1) it is close to 1 around the desired
SOC level; and (2) it changes to a lower or higher magnitude to promote charge
sustenance. The equation used to generate the curve shown in Figure 10.12 is as
follows (Paganelli et al. 2002; Rodatz et al. 2005):
f (soc) = 1 − (1 − 0.7xsoc ) ·
x3soc
SOCL + SOCH
2
SOCH − SOCL
SOC −
where
xsoc =
(10.5)
Other modifier functions can be used as long as they have the same characteristics
stated previously.
10.4 Control Concepts for Parallel Hybrids
173
3
2.5
fSOC
2
1.5
1
0.5
0.4
0.45
0.5
0.55
0.6
SOC
0.65
0.7
0.75
0.8
Figure 10.12. An example SOC modifier for the ECMS-control concept.
In general, the total fuel to be minimized depends on the vehicle speed, total
desired power, and battery SOC:
ṁ f
equi (t )
= ṁ f
eng (v, Pd )
+ f (soc) · ṁ f
batt
(10.6)
In other words, the output from such an optimization problem can be represented
as a three-dimensional lookup table. When the transmission used in the vehicle has
several gears (or when a CVT is used) and the transmission is shifted properly, the
vehicle speed is no longer a factor and the engine always operates at the appropriate
engine speed for the corresponding engine-power level. In that case, the ECMS
algorithm becomes a two-dimensional lookup table, with the desired output power
and the battery SOC as the only two independent variables. This special case is
illustrated in Example 10.6.
EXAMPLE 10.6: ECMS CONTROL. The ECMS concept is implemented in this exam-
ple, with the optimal (i.e., minimum fuel consumption) point identified and
marked for a given desired output power and battery SOC level. The results are
shown in Figure 10.13.
%Ex10_6.m
% generate engine power command map using the ECMS concept
% for a parallel HEV
clear all;
close all;
load engine_parameters
Throttle_grid=[0:5:100];
engine_speed = [600:200:6000]*2*pi/60;
% rad/sec
174
Control of Hybrid Vehicles
for k = 1:21,
for j = 1:28,
engine_out_power(k,j) =
engine_speed(j)*engine_torque(k,j)/1000; % engine out power (kW)
end
end
opt_eng_speed = [600: 3400/48: 4000];
opt_throttle = [18:1:66];
opt_fuel = interp2([600:200:6000], Throttle_grid, fuel_map,
opt_eng_speed, opt_throttle);
opt_power = interp2([600:200:6000]*2*pi/60, Throttle_grid,
engine_out_power, ...
opt_eng_speed*2*pi/60, opt_throttle);
SC_eng = 0.07;
Eff_elec = 0.95;
soc_L = 0.5;
soc_H = 0.7;
soc = [0.4:0.01:0.8];
x_soc = (soc - (soc_L+soc_H)/2)/(soc_H-soc_L);
f_soc = 1-(1-0.7.*x_soc).*x_soc.ˆ3;
% 1 by41
figure(1)
plot(soc, f_soc)
xlabel(‘SOC’), ylabel(‘f_S_O_C’)
Peng = [1:1:120];
fuel_engine = interp1(opt_power, opt_fuel, Peng, ‘linear’);
for i = 1:120,
Pd(i) = i*1.0;
% Pd from 1 to 120kw
P_motor(i,:) = Pd(i)*ones(1,120) - Peng;
for j = 1:120,
if P_motor(i,j) > 0
P_batt(i,j) = P_motor(i,j)/0.92;
else
P_batt(i,j) = P_motor(i,j)*0.92;
end
end
for k = 1:41,
fuel_batt(k,i,:) = f_soc(k)*SC_eng*P_batt(i,:)/Eff_elec;
fuel_total(k,i,:) = reshape(fuel_engine,1,1,120) +
fuel_batt(k,i,:) ;
end
end
10.4 Control Concepts for Parallel Hybrids
175
% Showing example at a particular case, where Pd = 60kW,
SOC = 0.6
figure(2)
subplot(411), plot(Peng, fuel_engine)
xlabel(‘Engine Power (kW)’), ylabel(‘Fuel_e_n_g’)
title(‘Pd = 60kW, SOC = 0.6’)
subplot(412), plot(Peng, P_batt(60,:))
xlabel(‘Engine Power (kW)’), ylabel(‘Battery power (kW)’)
subplot(413), plot(Peng, squeeze(fuel_batt(21,60,:)))
xlabel(‘Engine Power (kW)’), ylabel(‘Fuel_b_a_t_t’)
subplot(414), plot(Peng, squeeze(fuel_total(21,60,:)))
xlabel(‘Engine Power (kW)’), ylabel(‘Total fuel’)
hold on
[fuel_min, index] = min(fuel_total(21,60,:));
plot(Peng(index), fuel_min, ‘r*’), hold off
Pd = 60kW, SOC = 0.6
Fueleng
10
5
Battery power (kW)
0
0
20
40
60
80
Engine Power (kW)
100
120
0
20
40
60
80
Engine Power (kW)
100
120
0
20
40
60
80
Engine Power (kW)
100
120
0
20
40
60
80
Engine Power (kW)
100
120
100
0
-100
Fuelbatt
5
0
-5
Total fuel
5.5
5
4.5
4
Figure 10.13. Results illustrating the ECMS-control concept.
176
Control of Hybrid Vehicles
w (0) u (0)
x (0 )
f (0)
w (1) u (1)
x (1)
L( 0 )
f (1)
x (2)
w(N -1) u(N-1)
x ( N - 1)
L(1)
f (N-1)
x( N )
L( N -1)
Figure 10.14. A discretized multistage decision process solved by the DP method.
Dynamic Programming
The dynamic programming (DP) method has been used to find an optimal control
strategy under either a deterministic drive cycle or a stochastic driving-power distribution. The DP problem is solved numerically, which makes it possible for nonlinear
vehicle dynamics and inequality constraints to be considered while also guaranteeing global optimality. The underlying vehicle model incorporates the dynamics,
efficiency, and constraints of subsystems to ensure that the overall vehicle performance is optimized as the constraints of the components are satisfied. The continuous
vehicle dynamics and the control actions also are discretized, and the DP solution
involves searching among discrete control actions that result in the optimal cost
function throughout the problem horizon. A multistage decision process solved by
the deterministic DP is illustrated in Figure 10.14. The dynamics are discretized so
that it has the following form:
xk+1 = fk (xk , uk , wk )
(10.7)
where the states, control inputs, and disturbance inputs are represented by
x, u, and w, and respectively. Each box in Figure 10.14 represents mapping during a sampling time, and the total dynamic process has N steps. During each step,
different control inputs result in different states and generate different transitional
cost L. The DP seeks to find the control actions u(0), u(1), . . . , u(N − 1) that minimize a cost function, which could include both transitional costs at each step and a
terminal penalty term. For example:
J=
N−1
$
(fuelk + μ · NOxk + υ · PMk ) + α(SOCN − SOC f )2
(10.8)
k=0
which penalizes fuel and emission at each step as well as a terminal penalty on battery
SOC.
The solutions from the DP optimization are the global optimal up to the grid
accuracy. For deterministic DP problems, the identified DP control solution is not
suitable for real-time implementations due to the noncausal nature and heavy computational requirement. However, it is a good design tool for benchmarking and
calibrating other real-time control strategies.
Near-optimal power-management algorithms for hybrid vehicles can be
extracted from the DP results (Lin et al. 2003a). The results were implemented
on prototype vehicles with significant fuel-economy improvement over the more
classical rule-based control-design method (Lin et al. 2003b). This fundamental
control-design concept later was implemented for several different hybrid vehicles,
including hydraulic hybrid vehicles (Wu et al. 2004) and fuel-cell hybrid vehicles
(Kim and Peng 2007; Lin et al. 2006).
10.5 Control Concept for Split Hybrids
Power Split
Engine Optimization
SOC
Engine
Engine
Vehicle Speed
Command
Command
Power Demand
Power
Speed
M/G 2
M/G 1
M/G 1
Command
Command
Command
Power
Power
Speed
Figure 10.15. Control concept for a single planetary-gear split vehicle, which includes engine
optimization, generator-speed control, and motor-torque control.
More recently, a stochastic version of the DP control design procedure was
developed (Lin et al. 2006; Tate et al. 2008). The two major benefits of the stochastic
dynamic programming (SDP) design technique, in comparison to its deterministic counterpart, are (1) the obtained control results are directly implementable as
nonlinear full-state feedback lookup tables; and (2) the control algorithm is not
“cycle beating”; instead, it has robust performance because the optimization is over
a power-demand transition probability obtained from a rich set of driving conditions.
Optimal control results based on deterministic DP and SDP have gained popularity
in the hybrid-vehicle field. Many recent hybrid-vehicle designs (Opila et al. 2009)
were presented with DP results as a benchmark or are directly motivated by DP
results.
10.5 Control Concept for Split Hybrids
As discussed previously, there are many different possible split-hybrid designs and
they all behave differently; some also have a mode shift. Therefore, except for systematic and model-based design methods such as DP, other design approaches are
likely to require ideas specific to the vehicle configuration. In the following discussion, we explain the control idea for the original Toyota Hybrid System (Harada
2000; Hermance 1999) and the concepts of how to extend that to two-planetary
split-vehicle designs.
Figure 10.15 shows a control concept explained in the literature regarding the
Toyota Hybrid Systems. First, based on a driver’s pedal positions, vehicle speed, and
battery SOC, a proper engine-power command is computed that will satisfy a driver’s
command as well as maintain the proper battery SOC. The engine characteristics
then are checked to determine the best engine speed for that engine-power level.
Although the ECU is manipulating the engine inputs (e.g., fuel, valve, and EGR) to
achieve that engine power, it is also required that the generator (i.e., M/G 1) “hold
the line” so that when the engine is trying to generate power to push the vehicle
forward, the engine speed does not rev up or down, away from its most efficient
operating speed. This desired generator speed is calculated from the vehicle speed
and the desired engine speed. A feedback control (e.g., PI control) can be designed to
177
178
Control of Hybrid Vehicles
ensure good generator speed, which involves the manipulation of generator torque.
Finally, the motor (i.e., M/G 2) is controlled to achieve power balance by ensuring
that PM/G2 = Pd −Pe −PM/G1 . In other words, regardless of what action was taken
for battery SOC sustenance and generator-speed control, the total power requested
by the driver is delivered to the wheel. This ensures that the vehicle is drivable.
The concept described here is “engine centric” and essentially ignores the efficiency losses incurred by the electric machines, power electronics, and battery. When
the efficiencies of those components are high, this approach is fine. When DP is used,
those subsystem losses are considered automatically, which leads to better efficiency.
DP for split vehicles has not been studied extensively and only limited information
is available from Liu and Peng (2008).
The control concept illustrated in Figure 10.15 can be used for power-split
hybrid vehicles using two planetary gears. However, for those vehicles, two additional factors must be considered: (1) proper timing for mode shifts must be determined, which can involve the consideration for vehicle speed and acceleration; and
(2) during the shift, coordination of the oncoming clutch and the outgoing clutch is
necessary. Similar to gear shifts in an automatic transmission, it is desirable to have
“power-on” mode shifts; that is, propulsion torque still is supplied to the wheels during a mode shift so that drivability is not impacted negatively. Proper calculations
of the clutch torque and coordination for the clutch pressure to avoid “tie-up” or
“flare,” both of which are considered “bad shifts,” are required.
10.6 Feedback-Based Supervisory Controller for PHEVs
Recently, many PHEVs have been introduced. These PHEVs have batteries that can
be charged from the electrical power grid and can operate for a limited range (i.e.,
the all-electric range [AER]) as an electric vehicle (Markel and Simpson 2006). At
present, supervisory controllers for PHEVs have centralized architectures (Wirasingha and Emadi 2011). A typical PHEV operates in a charge-depleting (CD) or
an electric-vehicle (EV) mode before the battery SOC decreases to a certain value;
then, it switches to a charge-sustaining (CS) mode and operates like a conventional HEV. The control strategies proposed for HEVs can be applied to the CS
mode-controller design for PHEVs. As discussed previously, various control-design
methods for HEVs are available. For instance, in Di Cairano et al. (2011), feedback
controllers based on model predictive control were evaluated experimentally and
showed improved fuel economy compared to two baseline strategies.
In this section, a feedback-based controller for the CS mode is presented. The
controller is designed with respect to the EPA US06 cycle; however, simulation
results demonstrate that the feedback-based controller also achieves good fuel
economy, good driving performance, and charge sustainability over other driving
cycles (e.g., the EPA UDDS and HWFET cycles).
Vehicle Model
The control-oriented vehicle model for a series PHEV is shown in Figure 10.16.
The model inputs are the wheel-power command, Pw,cmd , and the reference battery
10.6 Feedback-Based Supervisory Controller for PHEVs
179
Table 10.2. Nominal vehicle configuration
PHEV
Engine
Generator
Battery
AER (miles)
Engine Power (kW)
Generator Power (kW)
Battery Capacity (kWh)
Battery Maximum Power (kW)
Motor Power (kW)
Vehicle Weight (kg)
Motor
Vehicle
30
50
50
12
110
110
1,680
SOC, socr . The system outputs are the actual wheel power delivered, Pw ; engine fuel
consumption, fuel; and actual battery SOC, soc.
The supervisory controller generates the engine/generator-power command
and the battery-power command. The wheels are driven by the electric motor.
The battery is being charged when the battery power Pb is negative. The component sizes for the vehicle configuration used in this section are listed in Table
10.2. This PHEV is representative of current designs (e.g., the 2011 Chevrolet
Volt).
Engine
The engine is modeled using a static fuel-consumption map from ADVISOR (Markel
et al. 2002), as shown in Figure 10.16. A combustion engine can achieve a required
power (excluding the maximum power) with different combinations of torques and
speeds. However, given a required power level, there is usually a unique pair of
engine torque and speed that achieves minimum fuel consumption. The Optimal
Operating Points Line (OOP-Line) is defined as the curve on which the fuel consumption is minimized for each power level (Konev et al. 2006).
fuel
Pe,cmd
Pw,cmd
socr
EGU
SC
Pb,cmd
Pe
Pw
Pb
BAT
z
EM
soc
SC – supervisory controller
EGU – internal combustion engine and generator unit
BAT – battery
EM – electric machine
z – feedback state vector
Figure 10.16. Diagram of the PHEV: SC – supervisory controller, EGU – ICE and generator
unit, BAT – battery, EM – electric machine, z – feedback-state vector.
180
Control of Hybrid Vehicles
Fuel Converter Operation
Honda Insight 1.0L VTEC-E SI Engine from ANL Test Data
100
0.35
+
0.35
+
60
0
+
40
0.3
+
0.2
+
20
0
0
1000
Figure 10.17. Engine-fuel-consumption
map (g/W/h).
0
+
0.1
+
2000 3000 4000
Speed (rpm)
5000
6000
Using the datapoints of the fuel-consumption map in Figure 10.17, the OOPLine can be constructed as plotted in Figure 10.18. The blue dots represent the
datapoints from the engine fuel-consumption map, and the green line is the OOPLine consisting of the most efficient points corresponding to each requested power
level. For an engine to operate along the OOP-Line, it should avoid large transients;
this is based on the perspective that aggressive engine transients and engine operation
away from the OOP-Line may degrade fuel economy and emissions. The maximum
rate of requested engine-power-output change can be constrained – for example, to
3.5 kW/sec for a system considered in Konev et al. (2006) or to 11 kW/sec for a system
considered in Di Cairano et al. (2011). If the rate of engine-power-output change is
constrained, the engine can operate smoothly and efficiently along the OOP-Line
responding to power commands.
1
0.9
Fuel consumption (g/W/h)
Torque (Nm)
80
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
10
20
30
40
Engine power (kW)
50
60
Figure 10.18. Fuel consumption versus engine power along engine OOP-Line.
10.6 Feedback-Based Supervisory Controller for PHEVs
181
Battery
Battery efficiency is assumed to be ηb = 0.9. The battery is modeled as an integrator
with parameter Bs = 2.57e-5 (Di Cairano et al. 2011).
ṡoc = Bs Pb,cmd
(10.9)
where soc = socr − soc. The battery SOC is the only state of the powertrain system.
Electric Machines
The motor and generator are modeled as static with a constant mean efficiency ηm =
ηg = 0.85. This assumption can be relaxed easily by incorporating efficiency maps.
Vehicle Dynamics
The vehicle-longitudinal dynamics includes the acceleration force, the rolling resistance force, and the aerodynamic force:
F = ma + f Fz +
AρCd v2
2
(10.10)
where m is the vehicle mass, kg; a is the longitudinal acceleration, m/s2 ; f is the
rolling-resistance coefficient; Fz is the normal force on the vehicle, N; A is the
vehicle cross-sectional area, m2 ; ρ is the air density, kg/m3 ; Cd is the drag coefficient;
and v is the longitudinal velocity, m/s. The vehicle longitudinal dynamics is used to
calculate the wheel-power command for the vehicle to follow the driving cycles that
are specified with vehicle speed, as in Example 10.1.
Feedback-Based Supervisory Controller
Assume that the PHEV operates in two main modes: (1) the CD mode when the
battery SOC is larger than a certain reference value, socr ; and (2) the CS mode once
the battery SOC reaches socr . Regenerative braking is activated when the wheelpower command is negative. If the battery SOC exceeds the specified range [socmin ,
socmax ], the control-strategy priority is to drive the battery SOC back to the interval.
In the CD mode, the battery provides the propulsion energy. The engine is used
to satisfy the transient-load demand beyond the power capacity of the battery. In the
CS mode, the control strategy is to optimize fuel economy and driving performance
while sustaining battery charge.
The CD mode and regenerative-braking control are straightforward; therefore,
we focus on designing the controller for the CS mode. First, a feedback-based control
structure is proposed. Second, the controller gains are obtained through optimization. Third, the obtained controller is tested in realistic simulations during three
standard driving cycles – with negative wheel-power command and regenerative
braking included – to evaluate fuel economy, driving performance, and batterycharge sustainability.
Controller Structure
Two integrators are introduced to regulate battery SOC and to eliminate wheelpower tracking error in steady state. Recall that the powertrain system is modeled as
182
Control of Hybrid Vehicles
US06 cycle
90
Vehicle speed profile (mph)
80
70
60
50
40
30
20
10
0
0
100
200
300
Time (sec)
400
500
600
Figure 10.19. EPA US06 driving cycle.
first order with the battery SOC as the single state. The three states of the closed-loop
system are as follows:
(10.11)
z1 = soc = socr − soc
z2 = (soc)dt = (socr − soc)dt
(10.12)
(10.13)
z3 = (Pw,cmd − Pw )dt
where state z1 represents the deviation of battery SOC, soc, from the reference value,
socr ; state z2 represents the integral error of battery SOC; and state z3 represents
the integral error between power command, Pw,cmd , and actual power delivered, Pw .
The actual power equals the engine power Pe plus the battery power Pb .
The control algorithm includes state-feedback control, feed-forward control, and
the terms representing information exchange between the engine-power command
Pe,cmd and the battery-power command Pb,cmd :
Pe,cmd = K1 z + n1 Pw, cmd + ke Pb,cmd
(10.14)
Pb,cmd = K2 z + n2 Pw, cmd + kb Pe,cmd
(10.15)
where z = [z1 z2 z3 ]T is the state vector; K1 = [k1 k2 k3 ] and K2 = [k4 k5 k6 ] are
state-feedback-gain vectors; n1 and n2 are feed-forward gains; and ke and kb are
controller gains that represent information exchange.
The regulation of battery SOC should be slow to allow the battery to augment the
engine in transients, whereas the wheel-power tracking should be fast and accurate
for good driving performance and safety. To achieve different convergence rates for
different states, we use eigenstructure assignment (Magni et al. 1997) to decouple
the state z3 from the other two states, z1 and z2 , which are related to battery SOC.
The aggressive driving cycle of EPA US06 (Figure 10.19) is chosen to generate
the controller gains through optimization.
10.6 Feedback-Based Supervisory Controller for PHEVs
183
Table 10.3. Performance results for the vehicle with battery
(Bs = 2.57e-5) and corresponding optimal centralized controller
Driving cycle
MPG
errP,max
socdev,p
EPA US06
EPA HWFET
EPA UDDS
36.05
55.19
57.88
2.97e-8
4.67e-9
7.56e-9
−5.33%
−0.79%
−1.30%
In the optimization for the controller gains, the cost function J includes three
terms: engine-fuel consumption, equivalent fuel consumption from the battery at
the end of the driving cycle, and accumulated vehicle-power tracking error:
J=
Tf
0
ṁice (t )dt + Keq f |socr − soc f | + α
0
Tf
|Pw,cmd (t ) − Pw (t )|dt
(10.16)
where ṁice (t ) represents the engine-fuel consumption; socf is the battery SOC at
the end of the driving cycle; Keqf is the equivalent fuel-consumption factor from
external charge and α is the penalty weight to drive the power-tracking error to zero.
Values of α = 100 and Keqf = 375.44 are used (Li et al. 2011). The optimization
constraints include (1) stability of the closed-loop system with the linear controller
for the CS mode, which is enforced by the closed-loop-pole locations; (2) upper and
lower power limits of the engine; (3) limit on engine-power rate of change to smooth
engine power during continuous engine operation; (4) upper and lower power limits
of the battery; (5) upper and lower limits of battery SOC, soc(t); and (6) upper
and lower limits of battery SOC at the end of the driving cycle to enforce charge
sustainability.
The obtained controller was tested during three standard driving cycles to evaluate fuel economy, driving performance in terms of wheel-power-tracking error, and
battery-charge sustainability; detailed results are in Li et al. (2011).
Vehicle performance with optimal controller for the three standard driving
cycles is summarized in Table 10.3. Given in the table are the fuel economy in
miles per gallon (mpg) considering both fuel consumption from the engine and
equivalent fuel consumption from the battery at the end of the driving cycle; the
driving performance evaluated by the maximum power-tracking error, errP,max , in
kW; and the battery- CS performance evaluated by the deviation of the final SOC
from the reference SOC, socdev,p .
It is shown in Table 10.3 that the proposed controller provides good fuel economy
for each of the different driving cycles. The wheel-power-tracking error, errP,max ,
is less than 2.97e-8 kW for all simulation scenarios, which ensures good driving
performance. The maximum deviation of battery SOC at the end of the driving
cycles, socdev,p , is −5.33 percent for all simulation scenarios. For the mild EPA
UDDS and HWFET cycles, the maximum socdev,p for all battery applications is
−1.3 percent.
For the US06 driving cycle, an example power split between the engine and the
battery for the vehicle with battery parameter, Bs = 2.57e-5, and the corresponding
optimal controller are given in Figure 10.20. When the wheel-power command is
positive, the engine provides the slowly changing power and the battery provides
184
Control of Hybrid Vehicles
Wheel power command (kW)
100
50
0
–50
–100
0
100
200
300
400
500
600
500
600
500
600
Engine power (kW)
30
20
10
0
0
100
200
300
400
Battery power (kW)
100
50
0
–50
–100
0
100
200
300
400
Time (sec)
Figure 10.20. Example power split between engine and battery for a vehicle with battery 3
(Bs = 2.57e-5) and corresponding optimal controller over the US06 cycle.
the transient-power command. When the wheel-power command is negative, the
engine shuts off without delivering power and the battery is charged by regenerative
braking. The SOC trajectory for the same vehicle configuration for the US06 cycle
is plotted in Figure 10.21. The initial battery SOC in the simulation is the reference
SOC used in the control. Additional simulation results are in Li et al. (2011).
0.3
Bs = 2.57e-5
0.29
0.28
0.27
0.26
0
100
200
300
400
500
Figure 10.21. Battery SOC profile over the US06 cycle.
600
References
PROBLEMS
1. Modify the parameters in Example 10.1 to simulate the power requirements for
compact and large passenger automobiles to supplement the results already given
for a midsize automobile.
2. What enhancements to the rules in Table 10.1 should be considered for the loadleveling-control strategy for parallel hybrid vehicles?
REFERENCES
Abthoff, J. O., P. Antony, M. Kramer, and J. Seiler, 1998, “The Mercedez-Benz C-Class Series
Hybrid,” SAE Paper 981123.
Anon, 2005, “Code of Federal Regulations 474.3,” http://cfr.vlex.com/source/code-federalregulations-energy-1059.
Di Cairano, S., W. Liang, I. V. Kolmanovsky, M. L. Kuang, and A. M. Phillips, 2011, “Engine
Power Smoothing Energy Management Strategy for a Series Hybrid Electric Vehicle,”
Proceedings of the American Control Conference, San Francisco.
Harada, O., K. Yamaguchi, and Y. Shibata, 2000, “Power Output Apparatus and Method of
Controlling the Same,” U.S. Patent Number 6,067,801.
Hayashida, M., and K. Narusawa, 1999, “Optimization of Performance and Energy Consumption on Series Hybrid Electric Power System,” SAE Paper 1999-01-0922.
Hermance, D., 1999, “Toyota Hybrid System,” 1999 SAE TOPTEC Conference, Albany, NY,
May.
Holmes, A. G., D. Klemen, and M. R. Schmidt, 2001, “Electrically Variable Transmission
with Selective Input Split, Compound Split, Neutral and Reverse Modes,” U.S. Patent
No. 6,527,658.
Kim, M., and H. Peng, 2007, “Power Management and Design Optimization of Fuel Cell/
Battery Hybrid Vehicles,” Journal of Power Sources, Vol. 165, Issue 2, March, pp. 819–
32.
Konev, A., L. Lezhnev, and I. Kolmanovsky, 2006, “Control Strategy Optimization for a
Series Hybrid Vehicle,” SAE Paper 2006–01-0663.
Li, S., I. V. Kolmanovsky, and A. G. Ulsoy, 2011, “Battery Swapping Modularity Design for
Plug-in HEVs Using the Augmented Lagrangian Decomposition Method,” Proceedings of
the American Control Conference, San Francisco.
Lin, C., M. Kim, H. Peng, and J. Grizzle, 2006, “System-Level Model and Stochastic Optimal Control for a PEM Fuel Cell Hybrid Vehicle,” ASME Journal of Dynamic Systems,
Measurement and Control, Vol. 128, No. 4, pp. 878–90.
Lin, C., H. Peng, J. W. Grizzle, and J. Kang, 2003a, “Power Management Strategy for
a Parallel Hybrid Electric Truck,” IEEE Transactions on Control Systems Technology,
Vol. 11, November, pp. 839–49.
Lin, C., H. Peng, J. W. Grizzle, and M. Busdiecker, 2003b, “Control System Development
for an Advanced-Technology Medium-Duty Hybrid Electric Truck,” SAE Transactions
Journal of Commercial Vehicles, Vol. 112–2, pp. 105–13.
Liu, J., and H. Peng, 2008, “Modeling and Control of a Power-Split Hybrid Vehicle,” IEEE
Transactions on Control Systems Technology, Vol. 16, Issue 6, November, pp. 1242–51,
Digital Object Identifier 10.1109/TCST.2008.919447.
Magni, J.-F., S. Bennani, J. Terlouw, L. Faleiro, J. de la Cruz, and S. Scala, 1997, “Eigenstructure Assignment,” Robust Flight Control, Springer, pp. 22–32.
Markel, T., A. Brooker, T. Hendricks, V. Johnson, K. Kelly, B. Kramer, M. O’Keefe, S.
Sprik, and K. Wipke, 2002, “ADVISOR: A Systems Analysis Tool for Advanced Vehicle
Modeling,” Journal of Power Sources, Vol. 110, pp. 255–66.
Markel, T., and A. Simpson, 2006, “Plug-In Hybrid Electric Vehicle Energy Storage System
Design,” NREL/CP-540–39614.
185
186
Control of Hybrid Vehicles
Muta, K., M. Yamazaki, and J. Tokieda, 2004, “Development of New-Generation Hybrid
System THS II – Drastic Improvement of Power Performance and Fuel Economy,” SAE
Paper 2004-01-0064.
Ogawa, H., M. Matsuki, and T. Eguchi, 2003, “Development of a Power Train for the Hybrid
Automobile: The Civic Hybrid, SAE Transactions, Vol. 112, No. 3, pp. 373–84.
Opila, D. F., X. Wang, R. McGee, J. A. Cook and J. W. Grizzle, 2009, “Performance Comparison of Hybrid Vehicle Energy Management Controllers on Real-World Drive Cycle
Data,” Proceedings of the American Control Conference, St. Louis.
Paganelli, G., T. M. Guerra, S. Delprat, J. J. Santin, M. Delhom and E. Combes, 2000,
“Simulation and Assessment of Power Control Strategies for a Parallel Hybrid Car,”
IMechE, Vol. 214, Part D.
Paganelli, G., Y. Guezennec, and G. Rizzoni, 2002, “Optimizing Control Strategy for Hybrid
Fuel Cell Vehicle,” SAE Paper 2002-01-0102.
Rizoulis, D., J. Burl, and J. Beard, 2001, “Control Strategies for a Series-Parallel Hybrid
Electric Vehicle,” SAE Paper 2001-01-1354.
Rodatz, P., G. Paganelli, A. Sciarretta, and L. Guzzella, 2005, “Optimal Power Management of
an Experimental Fuel Cell/ Supercapacitor-Powered Hybrid Vehicle,” Control Engineering
Practice, Vol. 13, pp. 41–53.
Tate, E. D., J. W. Grizzle, and H. Peng, 2008, “Shortest Path Stochastic Control for
Hybrid Electric Vehicles,” International Journal of Robust and Nonlinear Control, Vol. 18,
Issue 14, September, pp. 1409–29.
Wirasingha, S. G., and A. Emadi, 2011, “Classification and Review of Control Strategies for
Plug-In Hybrid Electric Vehicles,” IEEE Transactions on Vehicular Technology, Vol. 60,
pp. 111–22.
Wu, B., C. Lin, Z. Filipi, H. Peng, and D. Assanis, 2004, “Optimization of Power Management
Strategies for a Hydraulic Hybrid Medium Truck,” Vehicle System Dynamics, Vol. 42,
Nos. 1–2, July–August, pp. 23–40.
11
Modeling and Control of Fuel Cells
for Vehicles
11.1 Introduction
In recent decades, various new power-plant alternatives to traditional ICEs have
emerged. Because of their packaging and efficiency properties – with typical actual
fuel-cell efficiency of approximately 50 to 60 percent compared to typical actual
engine efficiency of approximately 20 percent (i.e., gasoline) to 30 percent (i.e.,
diesel) engines – fuel cells have been studied extensively as an alternative to ICEs
(Bernardi and Verbrugge 1992; Boettner et al. 2002; Caux et al. 2005; Department
of Energy 2002; Pukrushpan et al. 2004a; Steele and Heinzel 2001; Yang et al. 1998).
Figure 11.1 is a typical configuration of the power flow in a vehicle equipped with
a fuel-cell power plant. Hydrogen as fuel is fed to the fuel-cell stack, which generates
electricity and water after chemical reactions occur. The electrical energy generated
is received by the power-conditioning circuits of the vehicle. Depending on the
driver-demand conditions, electronic control units controlling the power electronic
circuits, the battery, and the electric motor determine the correct amount of energy
that is delivered to the components; the battery is charged to the desired levels and
the electrical motor generates the desired torque for vehicle traction. For cases in
which the battery is already charged and the energy required for the electrical motor
is less than what is generated by the fuel-cell–system electrical load, dumping circuits
are used to dissipate the excess energy.
William Grove conducted the first demonstration of a fuel cell in the nineteenth
century (Grove 1838). He realized that when water is electrolyzed with a power
supply by passing an electric current through it, the output of this reaction is hydrogen
and oxygen. When the power supply is replaced with an ammeter, a small current is
obtained; that is, the electrolysis is reversed and the hydrogen and oxygen recombine,
and an electric current is produced as a result.
The reaction given in Eq. (11.1) uses hydrogen as fuel and is the fundamental
reaction for a fuel-cell system:
2H2 + O2 → 2H2 O
(11.1)
At the end of the reaction described in Eq. (11.1), electrical energy is produced. The
experiment mentioned previously reasonably describes the basic principle of the fuel
cell. To amplify the generation of electricity, certain conditions should be improved.
187
188
Modeling and Control of Fuel Cells for Vehicles
Power
Electronics
Battery
Hyrogen
Tank
Fuel Cell
Stack
Motor
Vehicle Dynamics
Figure 11.1. Typical fuel-cell hybrid-electric vehicle layout.
Primary reasons for the insufficient or small current include the low contact area and
the distance between the electrodes (i.e., resistance). As shown in Figure 11.2, the
electrodes usually are made from flat and porous material, covered with a thin layer
of electrolyte. Porosity of the material helps the process for the electrolyte and the
gas penetration to maximize the contact among the electrode, electrolyte, and gas.
Various details are different for different types of fuel cells and for the reaction
between hydrogen and oxygen, which produces an electric current. For example, to
e − e − e − e−
e − e− e− e −
e−
e−
e−
Electrolyte
Polymer
e−
Cathode, O2
O2 + 4e− + 4H+ → 2H2O
Anode, H2
H2 → 2H+ + 2e−
Diffusion &
Catalyst Layers
Figure 11.2. Basic anode-cathode construction of a fuel cell.
11.2 Modeling of Fuel-Cell Systems
189
generate the electrons, reactions at each electrode shown in Figure 11.2 should be
considered in the following two parts:
1. The hydrogen gas ionizes, releases electrons, and creates H+ ions at the anode,
as shown in Eq. (11.2):
H2 → 2H+ + 2e−
(11.2)
2. Oxygen reacts with electrons taken from the electrode and H+ ions from the
electrolyte, forming water at the cathode, as shown in Eq. (11.3).
O2 + 4e− + 4H+ → 2H2 O
(11.3)
The ionization reaction described in Eq. (11.2) releases energy. For continuous
reaction, electrons produced at the anode (i.e., Part 1) should go through an electrical
circuit to the cathode and H+ ions concurrently should pass through the electrolyte
(i.e., Part 2). Special materials such as proton exchange membranes (PEMs) can be
made to possess mobile H+ ions (i.e., protons) and are used extensively in fuel-cell
systems.
With increasing focus and attention on these activities, the fuel-cell research
community is active and has been working on many interesting topics in recent years.
One topic is fuel-cell–performance prediction so that better fuel-cell components
can be designed and optimal fuel-cell operating points can be better understood.
An example in this category is PEM fuel cells without any external humidification
(Bernardi and Verbrugge 1992) and special membranes with “self-humidification.”
There also are studies regarding the steady-state–system models that were developed for component sizing and cumulative fuel consumption or hybridization studies.
There are numerous publications on fuel-cell modeling, and a few are suitable for
control studies. The control-oriented model presented here is based on Pukrushpan et al. (2004b), which is one of the few models suitable for fuel-cell–control
design.
The remainder of this chapter is structured as follows. In the next section, a
method for obtaining the dynamic model of the fuel-cell systems is presented. Then,
suitable control methods for fuel-cell systems are studied. The chapter concludes with
a discussion of supervisory control algorithms and parametric design considerations
for fuel-cell electric vehicles.
11.2 Modeling of Fuel-Cell Systems
In this section, modeling of fuel-cell systems for developing control algorithms is
discussed. Principal ideas, terminology, and structure of the control-oriented model
presented in Pukrushpan et al. (2004b) is followed. A computer-simulation model of
a dynamic system based on mathematical equations and empirical data is an invaluable tool for control-algorithm development purposes. In these types of models,
the correct amount of physical and empirical modeling with the correct structure is
crucial for model accuracy and simulation effort.
The diagram in Figure 11.3 is an example of a fuel-cell system for automotive
applications, as presented in Pukrushpan et al. (2004a). A fuel-cell stack must be
190
Modeling and Control of Fuel Cells for Vehicles
TM
Power
Conditioning
Energy Storage
(Battery)
S
Hydrogen
Tank
Motor
Compressor
Humidifier
Fuel Cell Stack
S
Water Separator
Water Tank
Figure 11.3. A fuel-cell system for automotive applications.
integrated with several auxiliary components to form a complete fuel-cell system.
Typically, the fuel-cell stack is augmented by the following four auxiliary systems:
(1) Hydrogen supply system. This is a common approach to supply fuel to the fuelcell stack using a hydrogen tank. The hydrogen flow must be controlled using
a supply valve so that the desired flow or pressure is obtained for favorable
energy-conversion conditions.
(2) Air-supply system. The oxygen (i.e., air) is supplied by a compressor, which
is used to increase the power density of the overall system. The air-supply
system consists of an air compressor, an electric motor, and pipes or manifolds
between the components. The compressor not only achieves desired airflow, it
also increases air pressure that significantly improves the reaction rate at the
membranes – thus, the overall efficiency and power density of the stack.
(3) Cooling systems. Cooling systems generally are used to maintain the temperature of the fuel-cell stack at a desired temperature, given the outside conditions.
Because the pressurized airflow leaving the compressor is at a higher temperature, an air cooler may be needed to reduce the temperature of the air entering
the stack.
(4) Humidification system. To obtain ideal reaction conditions, external humidification systems for both anode and cathode gases also are used. Although alternatives exist, external humidification generally is preferred because it provides
more control on the fuel-cell system. The water level in the tank is maintained by
collecting water generated in the stack, which is carried out with the airflow. The
excessive heat released in the fuel-cell reaction is removed by the cooling system, which circulates de-ionized water through the fuel-cell stack and removes
the excess heat via a heat exchanger.
11.2 Modeling of Fuel-Cell Systems
191
Ist
Wcp
Supply
Manifold (SM)
H2 Tank
Wan,in
Compressor
Figure 11.4. Fuel-cell stack layout.
Cathode
(CA)
Wca,out
Wrm,out
MEMBR
(CP)
Anode
(AN )
Wan,out = 0
Return
Manifold (RM)
The power of the fuel-cell stack is a function of the current drawn from the stack
and the resulting stack voltage. The cell voltage is a function of the stack current,
reactant partial pressure inside each cell, cell temperature, and membrane humidity.
Most of the fuel-cell mathematical models used in the literature (e.g., Boettner
et al. 2002; Caux et al. 2005; Golbert and Lewin 2004; and Pukrushpan et al. 2004b)
assume that the stack is well designed and that the cells perform similarly and can
be combined as a stack. Therefore, at a single stack temperature, the same set of
polarization curves for all cells can be used.
The fuel-cell stack studied in this chapter and based on Dicks and Larminie
(2000) and Pukrushpan et al. (2004a) is shown in Figure 11.4. The inputs to this
system can be considered to be the pressurized air (i.e., oxygen) and hydrogen;
the outputs can be considered to be the electric current and the water generated.
A power-conditioning circuit usually is needed because the voltage of the fuel-cell
stack varies significantly. The nomenclature used throughout this section and in the
equations is given in Table 11.1.
For the model presented here, it is assumed that the cathode and anode volumes
of the multiple fuel cells are combined as a single-stack cathode and anode volumes.
The anode supply and the return manifold volumes are small, which allows the combination of these volumes into one “anode” volume. The model is based primarily
on physics; however, several phenomena are described using empirical equations. In
the following subsections, models for the compressor, manifolds, fuel-cell stack, air
cooler, and humidifier are presented.
Compressor and Manifolds
The rotational dynamics and a flow map are used to model the compressor. The law
of conservation of mass is used to track the gas species in each volume. The principle
of mass conservation is applied to calculate the properties of the combined gas in
the supply and return manifolds. A combined rotational model used to represent
192
Modeling and Control of Fuel Cells for Vehicles
Table 11.1. Nomenclature
Afc
AT
CD
Cp
Dw
E
F
I
J
M
P
R
T
V
W
a
c
dcp
i
m
n
nd
p
t
tm
u
ω
v
Fuel-cell active area (cm2 )
Valve-opening area (m2 )
Throttle-discharge coefficient
Specific heat (J/kg/K)
Membrane-diffusion coefficient
(cm2 /sec)
Fuel-cell open-circuit voltage
Faraday’s number
Stack current (A)
Rotational inertia (kgm2 )
Molecular mass (kg/mol)
Power (Watt)
Gas constant or electrical
resistance ()
Temperature (K)
Volume (m3 )
Mass flow rate (kg/sec)
Water activity
Water concentration (mol/cm3 )
Compressor diameter (m)
Current density (A/cm2 )
Mass (kg)
Number of cells
Electro-osmotic drag coefficient
Pressure (Pa)
Time (sec)
Membrane thickness (cm)
System input
Rotational speed (rad/sec)
Voltage (V)
x
y
γ
η
λ
τ
Mass fraction or system state vector
Mole fraction or system measurements
Ratio of the specific heats of air
Efficiency
Excess ratio or water content
Torque (Nm)
act
air
an
ca
conc
Subscripts
Activation
Air
Anode
Cathode
Concentration
cp
fc
gen
in
m
membr
ohm
out
rm
sm
st
v
w
des
dmd
regen
Compressor
Fuel cell
Generated
Inlet
Membrane
Across membrane
Ohmic loss
Outlet
Return manifold
Supply manifold
Stack
Vapor
Water
Desired
Demand
Regeneration
the dynamic behavior of the compressor is given in Eq. (11.4):
Jcp
dωcp
dt
=
1
(P − Pcp )
ωcp cm
(11.4)
where Pcm (vcm , ωcp ) is the compressor motor (cm) power and Pcp is the load power.
In addition, the compressor-motor torque can be calculated using a static-motor
equation.
The law of conservation of energy is applied to the air in the supply manifold to
account for the effect of temperature variations. The cathode supply manifold (sm )
includes pipe and stack manifold volumes between the compressor and the fuel cells.
The supply manifold pressure, psm , is governed by the mass continuity and energy
conservation equations in Eq. (11.5):
dssm
= Wcp − Wca,in
dt
γ Ra
dpsm
=
(Wcp Tcp,out − Wca,in Tsm )
dt
Vsm
(11.5)
11.2 Modeling of Fuel-Cell Systems
prm
psm
193
Ist
Ist
Wca,in
φca,in
Xca,in
Tca,in
Wca,out
Cathode Mass Flow
mO ,ca
φca
2
mN
mV,ca
2,ca
pO
φca
Wv,membr
Wv,membr
Wan,in
φan,in
pan,in
Tan,in
vst
Ist
Ist
Membrane
Hydration
pca
2,ca
λm
φan
Stack
Voltage
pH2,an
Ist
Anode Mass Flow
mH
2,an
mv,an
Figure 11.5. Dynamic model of a fuel-cell stack.
where Ra is the air-gas constant, Vsm is the supply-manifold volume, and Tsm is the
temperature of the flow inside the manifold calculated from the Ideal Gas Law. The
supply-manifold exit flow, Wsm,out , can be calculated as a function of psm and pca
using the linearized nozzle-flow equation.
Unlike the supply manifold, in which temperature changes must be considered,
the temperature in the return manifold, T, is assumed to be constant and equal to
the temperature of the flow leaving the cathode. The return manifold pressure, prm ,
is governed by the principle of mass conservation and the Ideal Gas Law through
isothermal assumptions:
R T
dsrm
= a rm (Wca,out − Wrm,out )
dt
Vrm
(11.6)
Fuel-Cell Stack Model
The electrochemical reaction at the membranes is assumed to occur instantaneously.
The fuel-cell stack (st) model contains the following four interacting submodels, as
shown in Figure 11.5:
(1)
(2)
(3)
(4)
stack-voltage model
anode-flow model
cathode-flow model
membrane-hydration model
We assume that the stack temperature is constant at T. The voltage model contains
an equation to calculate stack voltage based on fuel-cell pressure, temperature,
reactant gas partial pressures, and membrane humidity.
194
Modeling and Control of Fuel Cells for Vehicles
Cathode Mass Flow Model
This model captures the cathode airflow behavior and is developed using the mass
conservation principle and the thermodynamic and psychometric properties of air.
Several assumptions are made, as follows:
(1) All gases obey the Ideal Gas Law.
(2) The temperature of the air inside the cathode is equal to the stack temperature.
(3) The properties of the flow exiting the cathode, such as temperature, pressure,
and humidity, are assumed to be the same as those inside the cathode.
(4) When the relative humidity of the gas exceeds 100 percent, vapor condenses
into liquid form. The liquid water does not leave the stack and either evaporates
when the humidity drops below 100 percent or accumulates in the cathode.
(5) The flow channel and cathode-backing layer are combined into one volume; that
is, the spatial variations are ignored.
The mass continuity is used to balance the mass of the three elements − oxygen,
nitrogen, and water – inside the cathode volume, as shown in Eq. (11.7):
dmO
2
dt
dmN
= WO ,in − WO ,out − WO ,react
2
2
2
2
= WN ,in − WN ,out
2
2
dt
dmw,ca
= Wv,ca,in − Wv,ca,out + Wv,gen + Wv,membr
dt
(11.7)
Anode-Mass-Flow Model
This model is similar to the cathode-flow model. Hydrogen partial pressure and
anode-flow humidity are determined by balancing the mass of hydrogen, mH2 , and
water in the anode and can be calculated as shown in Eq. (11.8):
dmH
2
dt
= WH ,in − WH ,out − WH ,react
2
2
2
dmw,an
= Wv,an,in − Wv,an,out − Wv,membr
dt
(11.8)
Assumptions for this mathematical model are as follows:
(1) Pure hydrogen gas is assumed to be supplied to the anode from a hydrogen tank.
(2) It is assumed that the hydrogen flow rate can be adjusted instantaneously using
a valve while maintaining a minimum pressure difference across the membrane.
(3) The inlet hydrogen flow is assumed to have 100 percent relative humidity.
(4) The anode outlet flow represents possible hydrogen purge and is currently
assumed to be zero (W).
(5) The anode hydrogen temperature is assumed to be equal to the stack
temperature.
11.2 Modeling of Fuel-Cell Systems
195
1.2
Cell Vollage, Vfc (V)
1
Vact
0.8
Vohm
0.6
Vconc
0.4
Ohmic
Region
Activation
Region
Mass-Transport
Region
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Current Density, ifc (A/cm2)
Figure 11.6. Fuel-cell polarization curve.
Membrane-Hydration Model
The membrane-hydration model captures the effect of water transport across the
membrane. Both water content and mass flow are assumed to be uniform over the
surface area of the membrane, and they are functions of stack current and relative
humidity of the gas in the anode and cathode.
Combining the two water-transport mechanisms – the electro-osmotic drag phenomenon and the “back-diffusion” of water – the water flow across the membrane
from anode to cathode can be calculated as shown in Eq. (11.9):
Wv,membr
(cv,an − cv,ca )
i
= Mv A f c n nd − Dw
F
tm
(11.9)
The dynamically varying pressure and relative humidity of the reactant gas flow
inside the stack flow channels are calculated in the cathode- and the anode-flow
models. The process of water transfer across the membrane is governed by the
membrane-hydration model. These subsystem models are discussed in the following
subsections.
Stack-Voltage Model
Stack voltage is calculated as a function of stack current, cathode pressure, reactant partial pressures, fuel-cell temperature, and membrane humidity. The current–
voltage relationship is commonly given in the form of the polarization curve, which
is plotted as cell voltage, vfc , versus cell-current density, ifc , as shown in Figure 11.6
for a specific cell temperature and various pressures.
196
Modeling and Control of Fuel Cells for Vehicles
Because the fuel-cell stack consists of multiple fuel cells connected in series, the
stack voltage, Vst , is obtained as the sum of the individual cell voltages, and the stack
current, Ist , is equal to the cell current. The current density then is defined as stack
current per unit of cell active area, ifc . Stack voltage can be calculated by multiplying
the cell voltage, Vfc , by the number of cells – assuming that all cells are identical.
The fuel-cell voltage is calculated using a combination of physical and empirical
relationships, as shown in Eq. (11.10):
V = E − Vact − Vohm − Vconc
1
0
i c3
= E − V0 + Va 1 − e−c1 i − [iRohm ] − i c2
imax
(11.10)
The open-circuit voltage, E, is calculated from the energy balance between the
reactants and products and the Faraday Constant. The activation overvoltage, Vact ,
arises from the need to move electrons and to break and form chemical bonds at the
anode and cathode, which also depends on temperature and oxygen partial pressure.
Based on the nonlinear mathematical model developed in Eqs. (11.1) through
(11.10) and outlined in Figures 11.4 and 11.5, a dynamic model of the fuel-cell
stack can be developed with computer-aided engineering (CAE) tools such as
MATLAB/Simulink to analyze the system response. A downloadable version of
the model outlined here is available at www.springer.com/engineering/control/book/
978-1-85233-816-9 (Pukrushpan et al. 2004a). The top-level structure of the model
is shown in Figure 11.7.
For the model given in Pukrushpan et al. (2004b) and shown in Figure 11.7,
simulation results when a series of step changes in stack-current and compressormotor input voltage are applied to the stack are shown in Figures 11.8a and b,
respectively, at a nominal stack operating temperature of 80 degrees Celsius.
During the first four steps, the compressor voltage is controlled so that the optimal oxygen excess ratio (approximately 2.0) is maintained. This can be achieved
with a simple static feed-forward controller. The remaining steps then are applied
independently, resulting in different levels of oxygen-excess ratios (Figure 11.8e).
During a positive-current step, the oxygen-excess ratio drops due to the depletion
of oxygen (Figure 11.8e), which causes a significant drop in the stack voltage (Figure 11.8c). When the compressor voltage is controlled by the feed-forward algorithm, there is still a noticeable transient effect on the stack voltage (Figure 11.8c)
and oxygen partial pressure at cathode exit (Figure 11.8f). The step at t = 18 seconds
shows the response of giving a step increase in the compressor input while maintaining the constant stack current. An opposite case is shown in seconds; the response
between 18 and 22 seconds shows the effect of running the system at an excess ratio
higher than the optimum value. Although the stack voltage (i.e., power) increases,
the net power actually decreases due to the increased parasitic loss.
11.3 Control of Fuel-Cell Systems
The previous sections discussed supporting systems (i.e., hydrogen supply, air supply, cooling, and humidification) that help the fuel-cell stack system work effectively,
as shown in Figure 11.3. All of these systems have desired operating points based on
outtime
Units
Pressure - Pascal, bar
Temperature - Kelvin
Voltage - Volt
Current - Ampere
Mass Flow - kg/sec
Rotational Speed - rad/sec
Power - Watt
humidity or relative humidity - 0 to 1
Clock
outSTcurrent
1
FC, Net Power
1
Stack Current (A)
outSTvoltage
outNETpower
Stack current
outSTpower
Stack Voltage
Stack Current
Voltage
1
Net Power (W)
Anode Flow out
3
CM Power
IM Pressure
Anode Inlet Flow
Stack Voltage (V)
anode flow out = zero
Anode Flow in
Anode Pressure
Cathode Flow out
Inlet Flow
Outlet Flow
Anode Inlet Flow Control
outCMvoltage
1.01325e5
1 atm
C P Flow
2
CM Voltage (V)
1
Inlet Flow
Outlet Flow
Inlet Flow Outlet Flow
Inlet Flow
Outlet Flow
Cathode Flow in
Anode Pressure
D ownstream Pressure
Manif old Pressure
outOMpressure
Outlet Manifold
Motor Input Voltage
Magic Cooler
Turbocharger Speed (rad/s)
CP Downstream Pressure
CM Current
Downstream Pressure (Pa)
Manif old Pressure (Pa)
Inlet Manifold
0.2
Desired RH
Water injected
Static Humidifier
353
Compressor
Cathode Pressure
Stack Temperature
Temp
60/2/pi
Collect Data
O2 Partial Pressure (Pa)
rad/s to RPM
Speed RPM
OM Pressure (Pa)
O2 Excess ratio
2
O2 Excess ratio
Stack
Figure 11.7. Downloadable MATLAB/Simulink fuel-cell model.
Modeling and Control of Fuel Cells for Vehicles
80
300
Power (kW)
Stack Current (A)
198
250
200
150
60
40
100
(d)
2.5
O2 Excess Ratio
Compressor
Motor Input (V)
(a)
250
200
150
250
240
230
220
5
10
15
(c)
Time (sec)
1
(e)
260
O2 pressure (kPa)
Stack Voltage (V)
(b)
0
2
1.5
100
210
Stack
Net
20
20
25
30
35
30
25
20
15
10
0
5
10
15
(f)
20
25
30
Time (sec)
Figure 11.8. Simulation results for the fuel-cell system model.
vehicle conditions and, when regulated, they effectively help the fuel-cell stack generate energy at optimum levels. Typical operations for these systems are summarized
as follows:
(1) Hydrogen-supply system. The hydrogen flow must be controlled using a supply
valve so that the desired flow or pressure is obtained for favorable energyconversion conditions. The hydrogen supply valve is the actuator for this system,
verifying whether the proper amount of hydrogen is delivered and monitored
via the fuel-stack measurements.
(2) Air-supply system. The oxygen (i.e., air) is supplied by a compressor, which is
used to increase the power density of the overall system. The air-supply actuation
is controlled by the air-compressor motor.
(3) Cooling systems. Cooling systems generally are used to keep the temperature of
the fuel-cell stack at a desired temperature, given the outside conditions. The
cooling actuation is controlled by changing the speed of the fan motor, and the
temperature is monitored using the stack-temperature outputs.
(4) Humidification system. To obtain ideal reaction conditions, external humidification systems for both anode and cathode gases are used. The actuation of the
humidifier is controlled by changing the air-blower speed, and the humidity is
monitored by using fuel-cell stack temperature and pressure outputs.
The block diagram in Figure 11.9 shows a control system that is composed of multiple
SISO controllers. This closed-loop system uses desired stack voltage (vst,des ) as the
input and actual stack voltage (vst ) as the output. Desired stack-voltage input is
used in the electrical bus controller to determine the proper stack current, ist . It
also is mapped to a desired compressor-speed signal, ωcp,des , using a feed-forward
controller that is representative of a proper operating point for the fuel-cell system.
11.3 Control of Fuel-Cell Systems
Feedforward
Controller
199
ωcp,des
Compressor
Controller
vcp
Fuel Cell
System
Bus
Controller
vst,des
vst
ist
Figure 11.9. Possible control of the fuel-cell system with multiple SISO controllers based on
desired stack voltage.
The compressor-speed response then can be controlled by its own SISO controller,
as shown in the figure.
The controller structure proposed in Figure 11.9 is composed of a series of simple
SISO feedback and feed-forward controllers. These types of controllers usually are
preferred in control practice because of their simplicity and vast experience with the
design. However, for cases in which the plant dynamics (i.e., the fuel-cell system)
are closely coupled, implementation of the individual (or loosely coupled) SISO
controllers may not provide the most effective performance, and more complex
controller structures are recommended. Based on the nonlinear mathematical model
developed in Eqs. (11.1) through (11.10) and outlined in Figures 11.1 and 11.4, the
linear time-invariant (LTI) system analysis in the MATLAB/Simulink control system
toolbox can be used to design MIMO controllers after linearization is performed for
specific operating conditions.
A linear model based on this analysis is given in state-space form in Eq. (11.11).
For the linear system presented, the nominal operating point is chosen as Pnet = 40
kW, λO2 = 2, which corresponds to nominal inputs of Ist = 191A and vcm = 164 Volt.
ẋ = Ax + Bu
(11.11)
y = Cx + Du
where:
⎡
−6.3091
0
−10.9544
0
83.7446
0
0
⎤
24.0587
⎢
0
−161.0830
0
0
51.5292 0
−18.0261
0 ⎥
⎢
⎥
⎢
⎥
⎢−18.7858
0
−46.3136
0
275.6592 0
0
158.3471⎥
⎥
⎢
⎥
⎢
0
0
0
−17.3506 193.9373 0
0
0
⎥
⎢
A=⎢
⎥
⎥
⎢ 1.2996
0
2.9693
0.3977 −38.7024 0.1057
0
0
⎥
⎢
⎥
⎢ 16.6424
0
38.0252
5.0666 −479.3840 0
0
0
⎥
⎢
⎢
⎥
⎣
⎦
0
−450.3869
0
0
142.2084 0
−80.9472
0
2.0226
0
4.6212
0
0
0
0
−51.2108
200
Modeling and Control of Fuel Cells for Vehicles
y _1
x' = Ax+Bu
y = Cx+Du
u_1
y _2
State-Space
y _3
u_2
Scope
Figure 11.10. Simulink model to simulate the linearized MIMO model.
⎡
0
−0.0316
⎢
⎢0
⎢
⎢
⎢0
⎢
⎢
⎢3.9467
⎢
B=⎢
⎢0
⎢
⎢
⎢0
⎢
⎢
⎢0
⎣
0
⎤
⎥
−0.0040⎥
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
−0.0524⎥
⎦
0
⎡
0
0
⎢
C=⎢
0
⎣0
12.9699 10.3253
⎤
⎡
0
0
⎥
⎢
D=⎢
0 ⎥
⎦
⎣0
0 −.2966
2
and x = mO2
0
u = vcm
mH
2
1
Ist ,
mN
2
0
5.0666
−116.4460
0
0
0
1
0
−0.5693
0
0
0
psm
msm
ωcp
0
y = Wcp
psm
vst
1
mw,an
prm
3T
⎤
0 0
⎥
0 0⎥
⎦
0 0
,
In this linear model, masses, pressure, rotational speed, flow rate, power, voltage,
and current are given in grams, bar, kRPM, g/sec, kW, V, and A, respectively. Figure 11.10 shows a MATLAB/Simulink block diagram, with which the linear system
given in Eq. (11.11) can be used to perform control design studies.
Figures 11.11 and 11.12 present the dynamic response of the linear system in Eq.
(11.11) when a step input is applied to the compressor input voltage, vcm, and stack
current, Ist , respectively.
These types of studies can be used to determine the coupling between inputs
and outputs for plants to identify control-design opportunities. For example, Figure 11.12 shows that the stack-current input is weakly coupled with the supplymanifold pressure psm , whereas the other two plant outputs are meaningfully
responsive. Using results of the linearized-model analysis, simple MIMO controllers
can be designed using conventional techniques (see Problem 10.4). The design
11.4 Control of Fuel-Cell Vehicles
201
∆ W cp, [g/sec]
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
Time, [s]
3
3.5
4
4.5
5
∆ psm, [bar]
0.015
0.01
0.005
0
0.8
∆ vst
0.6
0.4
0.2
0
Figure 11.11. Response of the linearized model to a step compressor voltage input.
process for such multivariable controllers is a well-studied field in control theory, and
many results exist for the design of MIMO systems. For example, in Boettner et al.
(2002), Caux et al. (2005), Guzzella and Sciarretta (2007), and Pukrushpan et al.
(2004a), related topics such as dynamic simulation, the observability of the model
developed, and possible linearization techniques are discussed.
Figure 11.13 shows a vehicle control system controller that uses the driverdemand pedal inputs to control simultaneously the fuel-cell system and the traction
at the electric motor. Because the controller is based on a linearized model, more
than one MIMO controller should be designed and implemented. During operation
of the vehicle based on current conditions, the “state” of the vehicle can be detected
and the corresponding MIMO controller can be designed based on the linearizedsystem model at those operating conditions.
11.4 Control of Fuel-Cell Vehicles
The powertrain configuration for a typical fuel-cell–powered HEV is shown in
Figure 11.1. This configuration includes an electric motor; a battery; a fuel-cell
202
Modeling and Control of Fuel Cells for Vehicles
∆ W cp, [g/sec]
0.02
0.015
0.01
0.005
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5
Time, [s]
3
3.5
4
4.5
5
-4
∆ psm, [bar]
0
x 10
-1
-2
-3
-4
0
0
∆ vst
-0.2
-0.4
-0.6
-0.8
0
Figure 11.12. Response of the linearized model to a step stack current input.
system; and power electronics to facilitate energy flow among the battery, fuel-cell
system, and electric motor. Although it is not shown in the figure, the electric motor,
the battery, and the fuel-cell system have controller modules. The coordination of
these controllers is performed by the vehicle-supervisory control algorithm. During typical operation conditions for a fuel-cell vehicle, the driver torque demand is
obtained via the accelerator- or the brake-pedal sensor signals as well as monitored
vehicle conditions.
To design a control system for a vehicle equipped with a fuel-cell power plant,
various control-design approaches can be followed. As a first approach, we can consider each system as an individual control entity and design a single control algorithm
for each system with set points calculated from the driver’s instantaneous commands
based on the torque at the wheels’ response of the vehicle. This is similar to the
controller structure presented in Figure 11.9, with all of the vehicle functions
included. Although this approach results in simpler individual control problems to
solve, the strong interaction among these systems results in poor overall system performance (i.e., vehicle longitudinal response). As discussed in the previous section
on modeling of fuel-cell systems, the interaction between the supporting systems is
11.4 Control of Fuel-Cell Vehicles
Driver’s command
203
Measurements
Multivariable Controller
u6
Traction Motor Control
u4
u5
Humidity Control
Power Management
u3
u1
TM
Power
Conditioning
Energy Storage
(Battery)
Temperature Control
Hydrogen Flow Control
S
Hydrogen
Tank
u2
Air Flow Control
Motor
Compressor
Humidifier
S
Fuel Cell Stack
Water Separator
Water Tank
Figure 11.13. A closed-loop multivariable control system for a fuel-cell vehicle.
strong. The design of separate SISO controllers is not preferred and not used except
in the test benches and academic studies for replicating certain conditions.
In Figure 11.13, a closed-loop multivariable control system is integrated with
the fuel-cell stack system shown. Although the LTI model analysis and suitable control design methods are useful for the initial design phase, more detailed control
algorithms typically are necessary to run the complex fuel-cell systems discussed
herein. Figure 11.14 shows in detail typical inputs and outputs for such a multivariable controller, shown in Figure 11.13. This represents a case – which we can
classify as the second control-design approach – in which the fuel-cell stack supporting systems and the vehicle longitudinal dynamics are considered a single system with interacting subsystems. It is important to note that some of the inputs
in Figure 11.14 in fact can be used to decide which alternative MIMO control to
use rather than being used directly in the controller. Also, some outputs from the
controller (e.g., blower command) can be an on/off signal rather than a continuous
command.
Another approach to develop a controller for the system, using the supervisory
control development techniques through optimization, has been studied by many
researchers (Ehsani et al. 2004; Gao and Ehsani 2001; Schell et al. 2005; Yang et al.
1998). The vehicle supervisory controller controls the energy flow among the fuelcell system, the battery, and the electric motor by issuing the appropriate set-point
commands to component controllers. During a typical drive cycle, both the fuel-cell
204
Modeling and Control of Fuel Cells for Vehicles
Traction Motor
Command
Driver Demand
Vehicle Speed
Power
Management
Command
Motor Speed
Battery SOC
Stack Temperature
Stack Inlet Pressure
Air Inlet Pressure
Fuel Cell Vehicle
Multivariable
Controller
Cooling
Command
Blower
Command
Blower Speed
Hydrogen Tank
Pressure
Fuel flow
Command
Figure 11.14. More realistic fuel-cell vehicle multivariable controller I/O.
system and the battery can be used to provide the energy that is needed to meet
the power demand by the driver. The battery also can be used to store the energy
when the demand is less than the produced energy or when the electric motor is
used in generator mode for braking. The objective of controls is to utilize the energy
flow such that the fuel economy is maximized while the drivability requirements
are met and the vehicle components are operated at their recommended operating
conditions.
To meet the desired fuel-cell vehicle-control strategy objectives, the supervisory
control strategy should direct the components in the system such that the electric
motor always meets the power demanded by the driver (i.e., the drivability condition). This condition also should be accompanied by the goals of the fuel-cell system
operating at its optimum operating region and maintaining the battery power level
within an optimum range to ensure maximized charging and discharging efficiency
(i.e., the energy-efficiency condition).
When a driver gives an accelerator-pedal or brake-pedal command, it is used
to calculate a driver-power-demand command, Pdmd . For traction cases and for the
fuel-cell electric vehicle whose powertrain configuration is described in Figure 11.1,
this calculated value corresponds to power that the electric motor is commanded
to produce. Using the transmission efficiency, ηm , the electric power input to the
motor, Pm,in , may be modified as Pm,in = Pdmd /ηm at the electric motor controller
for a certain motor speed. During regenerative braking through the motor, it functions as a generator, and the electric-power output from the motor, Pm,out , may be
calculated as Pm,out = Pregen ,dmd* ηm . It is important to note that Pregen ,dmd is the
braking-power command to the motor, which may be different from the power
demanded, Pdmd , because part of it can come from the mechanical brake at the
wheels.
Based on the motor-power command and other vehicle information – such as
energy level in the battery and minimum operating power of the fuel-cell system,
below which the efficiency of the fuel cell decreases significantly – the fuel-cell
system and the battery can be requested to provide the power to the system. This
is usually accomplished by defining a power-split policy based on system conditions
11.5 Parametric Design Considerations
and various operating modes of the drivetrain, which can be described for a fuel-cell
electric vehicle as follows:
Neutral: The fuel-cell system or the battery does not supply power to the drivetrain. The fuel-cell system may operate at idle for quick start and warm-up
conditions.
Braking: The fuel-cell system operates at idle and the battery absorbs the regenerative braking energy supplied according to the brake-system operating characteristics. The amount of regenerative braking typically is calculated based on
the amount of braking reported by the brake-controller module.
Hybrid Traction: If the commanded motor-input power is greater than the rated
power of the fuel-cell system, the hybrid-traction mode is used, in which the fuelcell system operates with its rated power and the remaining power demanded
is supplied by the battery. The rated power of the fuel-cell system can be set as
the maximum level of the optimal operating region of the fuel cell.
Battery Charge: If the commanded motor-input power is smaller than the preset
minimum power of the fuel-cell system and the battery needs charging (i.e., the
energy level is less than the minimum value), the fuel-cell system operates with
its rated power, part of which goes to the drivetrain and the other part goes to
the battery.
Battery Only: If the battery does not need charging (i.e., the energy level is near
its maximum value), the fuel-cell system operates at idle and the battery alone
drives the vehicle. In the latter case, the peak power that the battery can produce
is greater than the commanded motor-input power.
Fuel Cell Only: If the load power is greater than the preset minimum power and
less than the rated power of the fuel cell, and the battery does not need charging,
the fuel-cell system alone drives the vehicle. Otherwise, if the battery does need
charging, the fuel-cell system operates with its rated power, part of which goes
to the drivetrain to drive the vehicle and the other part is used to charge the
battery.
Figure 11.15 shows a baseline supervisory control algorithm chart for a typical fuelcell vehicle and for the operating conditions described herein. This is a basic strategy
that can be used as a starting point for simulations in which Pdmd is power demanded
by the driver (i.e., braking or acceleration based on the sign); PFC is the power output
of the fuel-cell system; PFC,op is the current operating power of the fuel-cell system;
Pbatt is the battery power (i.e., charge or discharge depending on the sign); and SOC
is the battery state of the charge with maximum and minimum limits, SOCmin and
SOCmax , respectively.
11.5 Parametric Design Considerations
The parametric design of the fuel-cell–powered hybrid drivetrain includes the design
of the electric-motor power, the fuel-cell–system power, and the battery power and
energy capacity. The motor power is to meet the acceleration performance of a
vehicle, as discussed in previous chapters. Required motor power can be calculated
205
206
Modeling and Control of Fuel Cells for Vehicles
Accelerator
Pedal
Input
Brake
Pedal
Input
Pmbrk
(Mechanical
Brake Power)
Pdm d
PFC=0
Pdm d<0
yes
Pbatt=Pdm d –Pmbrk
no
BRAKING
Pdm d>PFC
PFC=PFC,op
yes
Pbatt=Pdm d –PFC
no
HYBRID TRACTION
Pdm d≤PFC,op
SOC≤SOCmin
yes
no
PFC=Pdm d
Pbatt=0
yes
yes
SOC>SOCmax
no
no
PFC=0
Pbatt=Pdm d
BATTERY ONLY
PFC=PFC,op
Pbatt=PFC–Pdm d
FUEL CELL ONLY
BATTERY CHARGE
Figure 11.15. Supervisory-control algorithm chart for a typical fuel-cell vehicle.
by looking at desired maximum acceleration and constant speed on various road
grades.
The fuel-cell system selected for a vehicle must supply sufficient power to support
it at highway speeds (e.g., high constant speeds for a long time) and to support it when
medium speeds are desired for medium time conditions (i.e., city driving) without
the help of the battery. Because it has a limited amount of energy, the battery should
be used only to supply peak power in short periods. Once the maximum power of
the motor is determined by the specified acceleration performance and the rated
power of the fuel-cell system is determined by the constant-speed driving, the power
requirements of the battery can be determined as shown in Eq. (11.12):
Pbatt = Pmotor ∗ ηm − PFC
(11.12)
where Pbatt is the rated power of the battery, Pmotor is the maximum electric-motor
power, ηm is the efficiency of the motor drive, and PFC is the operating power of
the fuel-cell system. The energy change in the battery during a driving cycle can be
expressed as given in Eq. (11.13):
E=
t=t f
t=0
Pbatt dt
(11.13)
11.5 Parametric Design Considerations
207
Vehicle Speed [m/s]
30
20
small
midsize
large
10
0
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
100
200
300
400
500
600
700
800
100
200
300
400
Time [s]
500
600
700
800
4
4
x 10
Pmot [W]
2
0
-2
SOC
0.7
0.65
0.6
0.55
4
Pbatt [W]
4
x 10
2
0
-2
0
4
4
x 10
Pfc [W]
3
2
1
0
0
Figure 11.16. Candidate fuel-cell vehicle configuration simulated for the Federal Test Procedure drive cycle.
where Pbatt is the discharge and charge power of the battery (based on the sign) and
tf is the duration of the drive cycle. The energy changes, E, in the battery depend
on the size of the fuel-cell system selected, the vehicle control strategy described in
Figure 11.15, and the drive cycle to which the vehicle is subjected.
A parametric design for these systems can be analyzed via simulation. First,
using a CS battery strategy, different drive cycles that are representative of the target
market and vehicle dynamics are run to obtain the results shown in Figures 11.16 and
11.17. By analyzing the maximum fuel-cell power and its utilization ratio (i.e., on/off
Modeling and Control of Fuel Cells for Vehicles
Vehicle Speed [m/s]
208
20
small
midsize
large
15
10
5
0
0
100
200
300
400
500
600
700
800
900
100
200
300
400
500
600
700
800
900
100
200
300
400
500
600
700
800
900
100
200
300
400
500
600
700
800
900
100
200
300
400
500
Time [s]
600
700
800
900
4
Pmot [W]
4
x 10
2
0
-2
0
0.68
SOC
0.66
0.64
0.62
0
4
Pbatt [W]
4
x 10
2
0
-2
0
4
P fc [W]
3
x 10
2
1
0
0
Figure 11.17. Candidate fuel-cell vehicle configuration simulated for the city drive cycle.
ratio) and the corresponding maximum battery power required to run the system
and its maximum energy-difference sweep, the size of the battery and the fuel-cell
system can be established. In addition to the power considerations, it is important to
consider that when a larger battery and the fuel-cell system capacities are selected,
the vehicle total weight and cost are usually heavier and more expensive. For the
conceptual design stage, various computer simulation tools can be used for sizing
and fuel-efficiency analysis (Guzzella and Amstutz 1999; Rousseau and Pasquier
2001).
Problems 1–6
209
Table 11.2. Vehicle properties
Vehicle Mass
Electric-Motor Power
Fuel-Cell Power
Battery SOC Limits
Battery Maximum Power/Energy
1,600 kg
65 kW
80 kW
0.3–0.7
40 kW/1.5 hWh
PROBLEMS
1. In Eqs. (11.1) through (11.3), the electrode reactions and charge flow for an acidelectrolyte fuel cell are presented. There also are other ways to obtain the same
reaction, such as using an alkaline-electrolyte fuel cell. For this type of fuel cell,
explain the reactions and discuss the possibility of using such systems in fuel-cell
HEVs.
2. In Figure 11.1, a typical fuel-cell electric-vehicle powertrain layout is shown.
Discuss the advantages and disadvantages of this configuration. Also compare the
operation principle of this vehicle to typical HEV applications such as the Ford
Escape Hybrid and Toyota Prius.
3. Consider the linear state-space model given in Eq. (11.11) and use the stackcurrent and compressor-voltage input data given in Pukrushpan et al. (2004a) to
obtain the plant output traces. Compare the two sets of results and explain the
differences.
4. Using the state-space linear model given in Eq. (11.11), propose an optimal MIMO
controller that will regulate the plant outputs to desired values, as shown in the following block diagram. Investigate the possibility of using a state-feedback controller;
if not, provide a controller structure accordingly. Use suitable results from Pukrushpan et al. (2004a) as necessary.
controller
y _1
r_1
In1
Out1
In2
r_2
MIMO_Controller
x' = Ax+Bu
y = Cx+Du
y _2
Fuel_Cell_Stack
y _3
r_3
Scope
plant
5. Using the control-algorithm chart in Figure 11.15, develop a supervisory control
algorithm for the fuel-cell–system model given in Eqs. (11.1) through (11.10) and in
vehicle longitudinal dynamics equations from Chapter 4. Use the vehicle parameters
listed in Table 11.2.
6. By comparing the plots in Figures 11.16 and 11.17, determine the effects of using
different battery sizes and possible supervisory control, for both highway and city
driving, to better utilize fuel-cell power.
210
Modeling and Control of Fuel Cells for Vehicles
REFERENCES
Bernardi, D. M., and M. W. Verbrugge, 1992, “A Mathematical Model of the Solid Polymer
Electrolyte Fuel Cell,” Journal of the Electrochemical Society, Vol. 139, 1992, p. 2477.
Boettner, D. D., G. Paganelli, Y. G. Guezennec, G. Rizzoni, and M. J. Moran, 2002, “Proton
Exchange Membrane Fuel Cell System Model for Automotive Vehicle Simulation and
Control,” Journal of Energy Resources Technology, Vol. 124, March, pp. 20–7.
Caux, S., J. Lachaize, M. Fadel, P. Shott, and L. Nicod, 2005, “Modelling and Control of a Fuel
Cell System and Storage Elements in Transport Applications,” Journal of Process Control,
Vol. 15, 2005, pp. 481–91.
Department of Energy (U.S.), 2002, Fuel Cell Handbook, Morgantown, PA.
Dicks, A., and J. Larminie, 2000, Fuel Cell Systems Explained, Wiley & Sons, Inc.
Ehsani, M., Y. Gao, S. E. Gay, and A. Emadi, 2004, Modern Electric, Hybrid Electric, and
Fuel Cell Vehicles: Fundamentals, Theory, and Design, CRC Press.
Gao, Y., and M. Ehsani, 2001, “Systematic Design of Fuel Cell Powered Hybrid Vehicle Drive Train,” IEEE Electric Machines and Drives Conference, 604–11, DOI:
10.1109/IEMDC.2001.939375.
Golbert, J., and D. R. Lewin, 2004, “Model-Based Control of Fuel Cells: (1) Regulatory
Control,” Journal of Power Sources, Vol. 135, September 2004, pp. 135–51.
Grove, W. R., 1838, “On a new voltaic combination.” Philosophical Magazine and Journal of
Science, Vol. 13, p. 430.
Guzzella, L., and A. Amstutz, 1999, CAE Tools for Quasi-Static Modeling and Optimization of Hybrid Powertrains, IEEE Transactions on Vehicular Technology, Vol. 48, No. 6,
pp. 1762–9.
Guzzella, Lino, and A. Sciarretta, 2007, Vehicle Propulsion Systems: Introduction to Modeling
and Optimization, Springer Publishing Co.
Pukrushpan, J. T., A. G. Stefanopoulou, and H. Peng, 2004a, Control of Fuel Cell Power
Systems: Principles, Modeling, Analysis, and Feedback Design, Springer-Verlag.
Pukrushpan, J. T., H. Peng, and A. G. Stefanopoulou, 2004b, “Control-Oriented Modeling
and Analysis for Automotive Fuel Cell Systems,” ASME Journal of Dynamic Systems,
Measurement, and Control, Vol. 126, p. 14.
Rousseau, A., and M. Pasquier, 2001, “Validation of a Hybrid Modeling Software (PSAT)
Using its Extension for Prototyping (PSAT-PRO),” Proceedings of the 2001 Global
Powertrain Congress, Detroit, MI, pp. 1–9.
Schell, Andreas, H. Peng, D. Tran, E. Stamos, C.-C. Lin, and M. J. Kim. 2005. “Modelling and
Control Strategy Development for Fuel Cell Electric Vehicles,” Annual Reviews in Control
29(1): 159–68. DOI: 10.1016/j.arcontrol.2005.02.001.
Springer-Verlag, Web site for Control of Fuel-Cell Power Systems Book, available at
www.springer.com/engineering/control/book/978–1-85233–816-9. (January 11, 2012).
Steele, B. C. H., and A. Heinzel, 2001, “Materials for Fuel-Cell Technologies,” Nature,
Vol. 414, 2001, pp. 345–52.
Yang, W., B. Bates, N. Fletcher, and R. Pow, 1998, “Control Challenges and Methodologies in Fuel Cell Vehicle Development,” Proceedings of the SAE Conference, Issue 328,
pp. 363–70.
PART III
VEHICLE CONTROL SYSTEMS
12
Cruise and Headway Control
One of the most widely adopted and visible control systems available on contemporary vehicles sold in the United States is the cruise control, which automatically
regulates the vehicle longitudinal velocity by throttle adjustments. Typically, a vehicle cruise-control system is activated by a driver who wants to maintain a constant
speed during long highway driving. This relieves the driver from having to continually adjust the throttle. The driver activates the cruise controller while driving at
a particular speed, which then is recorded as the desired (or set-point) speed to be
maintained by the controller.
Intelligent cruise control systems – also known as autonomous or adaptive cruise
control (ACC) systems – are the next-generation product for cruise control. When
no lead vehicle is within sight, an ACC vehicle behaves like a conventional cruisecontrol vehicle by maintaining a constant (i.e., target) speed. However, an ACC
vehicle also has a headway-control mode. When the vehicle, using a range sensor,
detects that it is close to the in front vehicle, the controller switches to headwaycontrol mode and adjusts the speed to maintain a desired (i.e., safe) headway. Many
field tests have been conducted to assess the real-life performance of ACC vehicles
and consumers generally are receptive to the convenience provided by them. A
rapidly growing number of luxury vehicles now offer ACC as an option.
12.1 Cruise-Controller Design
The main goal in designing the cruise-control algorithm is to maintain vehicle speed
smoothly but accurately, even under large variations in plant parameters and road
grade. In the case of cruise control for heavy trucks, the vehicle mass can change by
up to several hundred percentage points (e.g., from a tractor with no trailer to towing
a fully loaded trailer). Furthermore, a truck transmission may have numerous gears
and frequent gear shifting may occur. Therefore, powertrain behavior may vary
significantly, which causes additional complexity.
In the case of passenger automobiles or SUVs/light trucks, vehicle mass still
may change noticeably but it is within a much smaller range. Also, the cruise-control
function usually is activated only above a threshold speed (e.g., 30 mph); therefore,
gear shifting is less of a problem. In this chapter, we assume that the engine and
transmission characteristics can be inverted easily through engine-map inversion.
213
214
Cruise and Headway Control
Figure 12.1. Block diagram of linearized vehicle longitudinal dynamics.
In other words, the desired traction force can be obtained easily through table
lookup and commanding proper throttle angle or fuel-injection rate. In addition, we
assume that tire friction is not a limiting factor. Therefore, we assume that we can
directly manipulate traction force. Under these assumptions, a cruise-control design
can be based on the equation of motion for longitudinal vehicle dynamics presented
in Chapter 3 and the free-body diagram in Figure 4.5. Thus, from Eq. (4.4), the
vehicle longitudinal motion is described by:
m
du
= Fx − mg sin − f mg cos − 0.5ρACd (u + uw )2
dt
(12.1)
Equation (12.1) is nonlinear in the forward velocity, u(t), but otherwise is a simple
dynamic system: it only has one state variable. So, what are the main challenges in
cruise-control design problems? The difficulties arise mainly from two factors: (1)
plant uncertainty due to change of vehicle weight, and (2) external disturbances due
to road grade. Thus, a good cruise-control algorithm must work well under these
uncertainties. We can linearize Eq. (12.1) about a nominal operating condition. At
equilibrium (i.e., when du/dt = 0), Eq. (12.1) can be solved for:
Fx0 = mg sin 0 + f mg cos 0 + 0.5ρACd (u0 + uw )2
(12.2)
For example, using the numerical values:
g = 9.81 m/s, u0 = 20 m/s, 0 = 0, m = 1000 kg,
ρ = 1.202 kg/m3 , A = 1 m2 , Cd = 0.5, f = 0.015, uw = 2 m/s;
this yields Fx0 = 292.6 N. Linearizing Eq. (12.1) about the specified operating (i.e.,
equilibrium) state by using a Taylor series expansion yields the linearized equation:
τ u̇′ + u′ = K(F ′ + d)
(12.3)
where the incremental, or perturbed, variables are defined as:
u = u0 + u′ ;
Fx = Fx0 + F ′ ;
τ = (m/(ρCd A(u0 + uw )));
= 0 + ′
d = mg( f sin 0 − cos 0 )′ ;
K = (1/(ρCd A(u0 + uw )));
Using the parameter values given here, we obtain τ = 75.632 sec and K = 0.0756
(m/s)/N. Based on the linearized equations, the process model for control-system
design can be represented as shown in the block diagram in Figure 12.1.
Using this model, a PI controller can be designed, which has the transfer function:
Gc (s) =
F ′ (s)
= KP + KI /s
e(s)
(12.4)
12.1 Cruise-Controller Design
215
Figure 12.2. Closed-loop system with PI controller.
As shown in the block diagram in Figure 12.2, the closed-loop-system equations
become:
u′ (s) =
K(KI + KP s)
Ks
r+ 2
d
τ s2 + (KKP + 1)s + KKI
τ s + (KKP + 1)s + KKI
(12.5)
Thus, the controller gains (i.e., parameters KP and KI ) must be selected to achieve
good performance for the closed-loop system. This requires consideration of a number of factors, such as stability, steady-state errors, and transient response, which are
discussed in Example 12.1.
EXAMPLE 12.1: PI CRUISE CONTROLLER DESIGN (FIRST-ORDER PLANT). A PI cruise-
controller design requires consideration of several performance requirements
(e.g., stability, steady-state error, and transient response) in selecting the control
gains, KP and KI . Various control-design techniques, including pole placement,
root-locus, and Bode plots, can be used. These techniques are made considerably easier to apply by the availability of computer-aided control-system design
programs such as those available in MATLAB. The controller design is based
on the plant-model parameters K = 0.0756 (m/s)/N and τ = 75.632 seconds
given previously. The closed-loop system has a second-order denominator, and
its performance can be designed by manipulating the poles of the closed-loop
characteristic equation. For example, the controller gains can be selected to
achieve no overshoot (Mp = 0 or ζ = 1) and a settling time of ts = 46 seconds.
The program also generates for the fixed value of TI = (Kp /KI ) = 18.686 a rootlocus plot with KP as a parameter (see Figure 12.3). Finally, for the selected
Unit Step Disturbance Response
Unit Step Reference Response
1.4
0.06
1.2
0.05
1
Open-loop
Closed-loop
0.04
0.8
0.03
0.6
Open-loop
Closed-loop
0.4
0.02
0.01
0.2
0
0
20
40
60
Time (sec)
80
100
0
0
Figure 12.3. Response of PI control system for Example 12.1.
20
40
60
Time (sec)
80
100
216
Cruise and Headway Control
KP = 186.86 and KI = 10 values, the closed-loop system performance is simulated for step changes in reference speed and disturbance inputs, as shown
in Figure 12.3. For these selected gains, the closed-loop system exhibits zero
steady-state error and a settling time of approximately 45 seconds, as expected.
However, despite the value of ζ = 1 the response exhibits some overshoot. This
is because the settling time and overshoot criteria used here are for second-order
systems with no zeros; the closed-loop system is second-order but has a zero at
-0.0535. If the response of the closed-loop system was determined to be not fast
enough, a smaller TI could be selected. For example, the step response of the
system with TI = 9.34 and KP = 401 shows a 50 percent reduction in rising and
settling times.
% Ex12_1.m
% Define the linearized model parameters
u0=20; g=9.81; m=1000; f=0.015; Theta=0;
rho=1.202; A=1; Cd=0.5; uw=2;
% Calculate the equilibrium force, Fx0:
Fx0 = m*g*sin(Theta) + f*m*g + ...
0.5*rho*A*Cd*(u0+uw)ˆ2;
% The time constant and dc gain are:
Tau=(m/(rho*A*Cd*(u0+uw))); K=Tau/m;
% Fix the Ti gain (Ki=Kp/Ti) for the PI control:
Ti=186.86/10.0;
% The loop transfer function for PI control is
% Kp*(num/den) where num/dec = (1+1/Ti*s)*K/(1+Tau*s)
num_o=K*[Ti 1]; den_o=[Ti*Tau Ti 0];
% Calculate open loop zeros and poles:
z=roots(num_o);
p=roots(den_o);
% Obtain the root locus for Kp:
K_root=[0:10:300];
R=rlocus(num_o,den_o,K_root);
plot(R,‘*’); title(‘RootLocus Plot’);
axis([-0.2 0 -0.05 0.05]), pause;
[K_root, POLES]=rlocfind(num_o,den_o)
% Simulate the reference unit step response
% with and w/o control:
t=[0:0.5:100];
sys = tf([K],[Tau 1]);
y=step(sys,t);
Kp=186.86; KI = Kp/Ti;
numc=K*[Kp KI]; denc=[Tau K*Kp+1 K*KI];
% Calculate the closed-loop zeros and poles:
zc=roots(numc)
pc=roots(denc)
sys_c = tf(numc, denc);
yc=step(sys_c, t);
12.1 Cruise-Controller Design
plot(t,y,‘-r’,t,yc,‘--b’); grid;
xlabel(‘Time (sec)’)
title(‘Unit Step Reference Response’);
legend(‘Open-loop’, ‘Closed-loop’); pause;
% Simulate the disturbance unit step response
% with and w/o control:
yd=step(sys,t);
sys_disturbance = tf([K 0], denc);
ycd=step(sys_disturbance,t);
plot(t,yd,‘-r’,t,ycd,‘--b’); xlabel(‘Time (sec)’),grid;
title(‘Unit Step Disturbance Response’);
legend(‘Open-loop’, ‘Closed-loop’);
The simple model used for the cruise-control design in Example 12.1 has several
shortcomings. First, it was assumed that the tractive force can be changed instantaneously by the controller – that is, the throttle/engine system dynamics has been
neglected. Second, the controller design is based on linearized equations of longitudinal motion, and the time constant and gain are assumed to be constant. It reality,
the vehicle-forward dynamics are nonlinear and the corresponding linear dynamics
vary depending on operating conditions. Both issues have been addressed by various
studies in the literature (e.g., Liubakka et al. 1991; Oda et al. 1991; and Tsujii et al.
1990). Example 12.2 illustrates the design of a PID cruise controller based on a more
complex but still linear model, which accounts for the throttle/engine dynamics. It
also has been proposed to develop adaptive versions of the cruise controller to estimate the actual parameters K and τ under various operating conditions. An adaptive
version of the PI cruise-control system is discussed in Example 12.3.
The closed-loop transfer function of the vehicle shown in Figure 12.4 is:
u′ (s) =
KKa (KD s2 + KP s + KI )
r
s2 (τ s + 1)(τa s + 1) + KKa (KD s2 + KP s + KI )
+
Ks2 (τa s + 1)
d
s2 (τ s + 1)(τa s + 1) + KKa (KD s2 + KP s + KI )
EXAMPLE 12.2: PID CRUISE-CONTROLLER DESIGN (SECOND-ORDER ACTUATOR). The
process model in Eq. (12.3) neglects any dynamics associated with the actuator,
which adjusts the throttle angle. In Tsujii et al. (1990), a dc motor throttle
actuator model is given as:
Ka
Ga (s) =
s(τa s + 1)
where the input is the motor-drive duty cycle (percent) and the output is the tractive force. Parameter values used here are the same as those in Example 12.1, plus
Figure 12.4. Closed-loop system with PID controller and throttle dynamics.
217
218
Cruise and Headway Control
Unit Step Reference Response
6
1.4
1.2
5
1
4
0.8
3
0.6
2
0.4
1
0.2
0
0
0
5
10
Time (sec)
15
20
–1
× 10–4 Unit Step Disturbance Response
0
5
10
Time (sec)
15
20
Figure 12.5. Response of PID control system for Example 12.2.
Ka = 10 (Newton-sec/%) and τa = 0.05 seconds. Clearly, the actuator dynamics
usually is much faster than the vehicle dynamics (i.e., τa ≪ τ ), and it is reasonable to neglect actuator dynamics as in Example 12.1. However, in this example,
the throttle dynamics is considered and the controller is a PID controller rather
than the PI controller used in Example 12.1. The closed-loop block diagram is
shown in Figure 12.4. With the controller gains, KP = 120, KI = 12, and KD =
120, the closed-loop-system block diagram and responses to step reference and
disturbance inputs are illustrated in Figure 12.5. The response is faster than the
PI controller shown in Example 12.1 but has a larger overshoot. The P gain is
selected for a damping ratio of approximately 0.707.
% Ex12_2.m
% Define the linearized model parameters
clear
u0=20; g=9.81; m=1000; f=0.015; Theta=0;
rho=1.202; A=1; Cd=0.5; uw=2;
% Calculate the equilibrium force, Fx0:
Fx0 = m*g*sin(Theta) + f*m*g + ...
0.5*rho*A*Cd*(u0+uw)ˆ2;
% Time constant & dc gain are then:
Tau=(m/(rho*A*Cd*(u0+uw))); K=Tau/m;
% The throttle actuator is first-order with integrator:
Ka=10; TauA=0.05;
% Fix the Ti gain (Ki=Kp/Ti) for the PID control:
Ti = 100.0;
% Fix the Td gain (Td=Kd/Kp) for PID control:
Td = 1.0;
% The open loop transfer function for PID control is
% Kp*(num/den) where:
num_o=K*Ka*[Td 1 1/Ti];
den_o=[Tau*TauA Tau+TauA 1 0 0];
12.1 Cruise-Controller Design
219
% Calculate and display open loop zeros and poles:
z=roots(num_o)
p=roots(den_o)
% Obtain the root locus for Kp:
K_locus=0:5:160;
R=rlocus(num_o,den_o,K_locus);
plot(R,‘*’); title(‘Root locus’), pause
% Simulate the unit step response with PID control:
t=[0:0.5:20];
Kp=120.0;
numc=Kp*num_o;
denc=den_o+[0 0 numc];
% Calculate the closed-loop zeros and poles:
zc=roots(numc)
pc=roots(denc)
sys_c = tf(numc, denc);
yc=step(sys_c, t);
plot(t,yc,‘r’); xlabel(‘Time (sec)’), grid;
title(‘Unit Step Reference Response’); pause;
% Simulate disturbance unit step
% response with and w/o control:
sys_disturbance = tf([K*TauA K 0 0], denc);
ycd=step(sys_disturbance,t);
plot(t,ycd,‘--b’); xlabel(‘Time (sec)’),grid;
title(‘Unit Step Disturbance Response’);
As mentioned previously, the gain, K, and time constant, τ , vary with operating conditions (e.g.,
mass, speed, and wind velocity). In this example, the model in Eq. (12.3)
is used (i.e., actuator dynamics is neglected) and an estimation algorithm is
developed to identify the parameters. In other words, the parameters K and
τ are estimated on-line from the measured signals. The controller gains then
are updated by using the estimated parameters. This “adaptive” cruise-control
(ACC) system structure is illustrated in Figure 12.6, which consists of two major
EXAMPLE 12.3: ADAPTIVE PI CRUISE-CONTROLLER DESIGN.
uc
+
T(z)
-
u
1
R( z)
+
adaptation
algorithm
+
d
b
z+ a
â b̂
RLS
estimator
S(z)
Figure 12.6. Structure of the adaptive cruise control system.
y
220
Cruise and Headway Control
1.4
0.5
1.2
1
0
0.8
0.6
–0.5
0.4
0.2
–1
0
0
10
20
30
40
50
60
0
10
20
30
40
50
60
Figure 12.7. Adaptive PI control for Example 12.3.
parts: (1) a recursive least squares (RLS) parameter-estimation algorithm;
and (2) an indirect (or self-tuning) PI controller strategy. The MATLAB
program shown here is used to generate the results shown in Figure 12.7. The
implementation is digital, so a discrete-time process model is used, obtained
for a sampling rate of 1 second. The parameter estimates converge quickly
(i.e., in about 5 to 10 seconds) and the response is comparable to that shown in
Figure 12.2 (somewhat slower and with more overshoot), despite the fact that
the unknown parameters are initialized to zero. In the MATLAB program, the
RLS estimation algorithm is used to estimate the plant parameters. The basic
equations of the RLS algorithm are summarized as follows:
y(k) = φ T(k)θ
ŷ(k) = φ T(k)θˆ
θˆ (k + 1) = θˆ (k) + K(k)[y(k + 1) − φ T (k + 1)θˆ]
P(k + 1) = [I − K(k)φ T (k + 1)]P(k)/λ
K(k) = P(k)φ(k + 1)[λ + φ T (k + 1)P(k)φ(k + 1)]−1
In this example,
y(k) = −ay(k − 1) + bu(k − 1)
Therefore,
φ T (k) = [−y(k − 1) u(k − 1)]
% Ex12_3.m
K = 0.0756; Tau = 75.632;
A=-(1/Tau); B=(K/Tau); T=1.0;
[F,G]=c2d(A,B,T);
C=1; D=0; [num,den]=ss2tf(F,G,C,D,1);
b=num(2)/den(1); a=den(2)/den(1);
% initialization:
θˆ = [â b̂]T
12.1 Cruise-Controller Design
kmax=60; n=2; lambda=0.995;
theta = [-0.5;0.5]; P = 1000*eye(n);
u_old = 0.0; y_old = 0.0; uc_old = 1.0;
phi = [-y_old; u_old];
uc=1.0+0.01*randn(1);
Y = zeros(kmax+1,n+3);
Y(1,:)=[u_old uc y_old theta’];
% System with a ‘‘PI’’ controller
% designed for given z, wn
% and parameter estimation using RLS:
z=0.95; wn=0.1;
% Controller Ru = Tuc - Sy
for k=1:kmax,
uc = 1.0 + 0.02*randn(1);
y = - a*y_old + b*u_old;
[theta,P] = rls(y,theta,phi,P,lambda);
a1 = theta(1); b1=theta(2);
% we use estimated values
% (a1, b1) of the
% process for the controller
% gains and control:
s0 = (exp(-2*z*wn*T)+a1)/b1;
s1 = (1-a1-2*cos(wn*T* ...
(1-zˆ2)ˆ0.5)*exp(-z*wn*T))/b1;
t0 = s0 + s1;
u = u_old + t0*uc_old - ...
s0*y_old -s1*y;
% update the measurement vector phi
% for next iteration:
phi = [-y;u];
u_old = u; y_old = y; uc_old = uc;
Y(k+1,:)=[u uc y a1 b1];
end
% Display the results:
k=[0:kmax]’; u_uc_y_ai_bi = Y;
plot(k,Y(:,3),‘r’); title(‘Response’);
xlabel(‘Sampling Interval, k’); ylabel(‘y’); pause;
plot(k,Y(:,4:n+3)); title(‘Parameter Estimates’);
xlabel(‘Sampling Interval, k’); ylabel(‘Estimated a & b’);
% function saved in
% rls.m
function [theta, P] = rls(y, theta,
phi, P, lambda)
K=P*phi*inv(lambda+phi’*P*phi);
theta=theta+K*(y-phi’*theta);
P=(eye(size(P))-K*phi’)*P/lambda;
221
222
Cruise and Headway Control
Figure 12.8. Block diagram of the adaptive cruise control system.
EXAMPLE 12.4: ADAPTIVE PID CRUISE-CONTROLLER DESIGN. In this example, we
revisit Example 12.2, assuming that some kind of parameter-identification
method has been implemented and that we have access to accurate estimation of vehicle parameters (i.e., τ and K). How then can we take full advantage
of the three design variables (i.e., the PID gains: Kp , KI , and KD ) for superior
performance? In previous examples, we used the root-locus technique, which
helps to select one control parameter; the other two parameters were specified
in advance. Thus, a form of trial and error or engineering judgment is needed
to find a set of good control parameters. In this design, we show an alternative design method based on pole placement. First, we derive the closed-loop
transfer function:
KD Ka Ks2 + KP Ka Ks + KI Ka K
u′
=
r
τ τa s4 + (τ + τa )s3 + (1 + KD Ka K)s2 + KP Ka Ks + KI Ka K
The closed-loop poles are the roots of the denominator polynomial (i.e., the
characteristic equation), which can be rewritten in the following monic form:
s4 +
τ + τa 3 1 + KD Ka K 2 KP Ka K
KKK
s +
s +
s+ I a =0
τ τa
τ τa
τ τa
τ τa
Because we only have three design variables, we can place only three of the four
closed-loop poles. The remaining pole then is determined by the fixed coefficient
τ +τ
( τ τ a ) of the closed-loop characteristic equation. Notice that the actuator-time
a
constant is much faster than the vehicle-time constant (i.e., 0.2 versus 55∼140). If
we select three slow (i.e., dominant) closed-loop poles, the last pole, which may
move a little when τ changes, will be far from the imaginary axis and thus does
not noticeably affect the system performance. Based on the damping ratio and
bandwidth of the complex conjugate pairs, we determine that the three poles
should locate at −0.3 ± 0.1 j and −0.4. The closed-loop characteristic equation
then becomes:
τ + τa
2
−1
(s + 0.6s + 0.1)(s + 0.4) s +
τ τa
τ + τa 3
τ + τa
τ + τa
s +
− 0.66 s2 + 0.34
− 0.3 s
= s4 +
τ τa
τ τa
τ τa
τ + τa
+ 0.04
−1
τ τa
The PID gains to achieve these pole assignments are then:
τ + τa − 0.66τ τa − 1
0.34(τ + τa ) − 0.3τ τa
KP =
Ka K
Ka K
0.04(τ + τa ) − 0.04τ τa
KI =
Ka K
KD =
12.1 Cruise-Controller Design
The unplaced pole will be located at − τ1 − τ1 + 1 , which is about 10 times faster
a
than the three slower poles. In other words, the performance of the adaptive
controller is not affected much even when the vehicle-operation conditions vary.
In the following example program, we assume that the vehicle is accelerating
from 15 to 30 m/sec in 30 seconds and a 2 percent road gradient also is assumed
to be present. It can be seen that the adaptive PID control algorithm is tracking
the desired speed command with small error under these two uncertainties.
In this MATLAB program, we used the Euler approximation for integration. This may introduce some error but it is more flexible because all of the
intermediate variables (e.g., engine force F and command to actuator c) can
be examined. Sometimes it is desirable to keep track of these intermediate
variables. For example, the force (F ) in the following figure shows that the controller design is reasonable because the force command is not excessively large:
223
224
Cruise and Headway Control
% Ex12_4.m
g=9.81; m=1000; f=0.015; Theta=0;
rho=1.202; A=1; Cd=0.5; uw=0;
Ka=10; TauA=0.2; T=0.01; u(1)=15.0;
error=0.0; error_old=0.0; int_error=0.0;
F_dot=0.0; F=0.0; t=0:T:40;
r=[15*ones(1,5/T+1),15+(1:30/T)*0.5*T,30*ones(1,5/T)];
d=[zeros(1,10/T+1), -0.02*m*g*ones(1,10/T), zeros(1,20/T)];
for i=1:40/T,
Tau=(m/(rho*A*Cd*(u(i)+uw))); K=Tau/m;
Kd=(TauA+Tau-0.66*TauA*Tau-1)/(Ka*K);
Kp=(0.34*(TauA+Tau)-0.3*TauA*Tau)/(Ka*K);
Ki=(0.04*(TauA+Tau)-0.04*TauA*Tau)/(Ka*K);
error_old=error;
error=r(i)-u(i);
int_error=int_error+error*T;
c=Kd*(error-error_old)/T+Kp*error+Ki*int_error;
F_dotdot=(Ka*c-F_dot)/TauA;
F_dot=F_dot+F_dotdot*T;
F=F+F_dot*T;
u_dot=(K*(F+d(i))-u(i))/Tau;
u(i+1)=u(i)+u_dot*T;
end
plot(t(1:40/T),u(1:40/T),‘r’), grid
xlabel(‘time (sec)’); title(‘True speed (m/sec)’)
pause; plot(t(1:40/T),r(1:40/T)u(1:40/T),‘r’); grid
xlabel(‘time (sec)’); title(‘Speed error (m/sec)’)
12.2 Autonomous Cruise Control: Speed and Headway Control
A vehicle cruise-control system regulates vehicle longitudinal velocity. As an intermediate step between platooning (see Chapter 19) and cruise control, the control of
the vehicle longitudinal motion (i.e., velocity and position) is considered. The idea
is to allow the vehicle to regulate speed when there are no automobiles in front of it.
However, when a vehicle is detected in front, then the controller regulates relative
distance between the two of them (i.e., “headway”). Such a system (i.e., ACC or
intelligent cruise control) can be effective in stop-and-go traffic and on the highway
and can be used in conjunction with the auto-cruise system described in Chapter 19.
The vehicle cruise-control system uses the desired speed set by the driver (e.g., the
speed limit) as a reference input when there is no vehicle in sight. When a vehicle
is detected in front, the system switches to the headway-control mode until the lead
vehicle exceeds the reference speed or vanishes from sight, when it then switches
back to speed control. In this scenario, the driver does not need to be concerned
about longitudinal-motion control under conditions ranging from stop-and-go traffic
to highway driving. A formulation of the control-system design problem for ACC is
presented in the following example.
12.2 Autonomous Cruise Control: Speed and Headway Control
225
dl, vl
dc, vc
Figure 12.9. Headway control problem.
R
To differentiate
from the adaptive cruise control problem presented in the previous section, we
name the range-regulation enhanced cruise-control system autonomous cruise
control (ACC). In the literature, this system is commonly referred as intelligent
cruise control (ICC), adaptive cruise control (ACC), or autonomous intelligent
cruise control (AICC). When no lead vehicle is in sight, the performance of
an ACC vehicle is similar to a traditional cruise-control vehicle. When a lead
vehicle is present, we must model the interaction with the lead vehicle and keep
track of the range variable, which is one of the most important performance
variables.
It is assumed that the measurement of the vehicle longitudinal velocity (vc )
as well as of the distance (R) between the first vehicle (i.e., controlled vehicle)
and the vehicle in front (i.e., lead vehicle) are available. This is illustrated in Figure 12.9. Using a linearized description of the longitudinal velocity of each
vehicle, as discussed in a previous section, we can obtain the following state
equations:
EXAMPLE 12.5: AUTONOMOUS CRUISE CONTROL SYSTEM DESIGN.
ẋ1 = x3 − x2
ẋ2 = −(1/τc )x2 + (Kc /τc )u + (Kc /τc )wc
ẋ3 = −(1/τl )x3 + (Kl /τl )wl
where x1 = R, x2 = vc , x3 = vl and u is the control input (i.e., traction force)
applied to the controlled vehicle (i.e., Vehicle c). The inputs wc and wl represent
disturbances (e.g., grade disturbances for both vehicles and driver inputs for the
lead vehicle). The open-loop-system block diagram is shown in Figure 12.10a,
and an equivalent block diagram is shown in Figure 12.10b. The two block diagrams result in a different number of state variables because of the difference in
the selection of disturbance inputs and the fact that the vehicle displacements are
ignored in the second representation. When an ACC algorithm is implemented
using differential GPS, for example, then it may be interesting to keep vehicle
displacements as state variables. That extension is readily achieved by simply
integrating the vehicle speeds.
u +
+
Kc
1 +τc s
wc
vc
1
s
xc
-
d
+ xl
1
s
vl
Kl
1 +τl s
(a )
wl
Figure 12.10. Open-loop block diagram for ACC.
u +
+
Kc
1 +τc s
wc
vc
- +
vl
(b)
1
s
d
226
Cruise and Headway Control
r +
x&3
-
K3
s
wc
u + +
− K3 x 3 +
− K4 x4
K4
K3 s
+ -
vl
Kc
1 + τc s
vc +
-
1
s
R ≡ x1
K2
x2
K1
Figure 12.11. Closed-loop block diagram of the ACC system.
Consider the design of a controller to maintain a desired headway, r, based
on measurements of R and vc only. This requires only a range sensor mounted
on the controlled vehicle, in addition to measurement of the controlled variable
(i.e., vehicle longitudinal speed) and no information from other vehicles or from
the roadway infrastructure. The control problem is illustrated in the block diagram
shown in Figure 12.11. The reference input, r, should be computed as a function
of the vehicle forward velocity to maintain safe headway (e.g., r = th vc , where th
is the desired headway time). Also, the velocity of the lead vehicle (i.e., Vehicle l)
is not known and acts as a disturbance input. The control, u, on Vehicle c must be
calculated to maintain the desired headway. A feedback controller can be designed
for this system based on the following state equations:
ẋ1 = −x2 + vl
ẋ2 = −(1/τc )x2 + (Kc /τc )(u + wc )
ẋ3 = (R − r) = (x1 − r)
ẋ4 = x3
u = −k1 x1 − k2 x2 − k3 x3 − k4 x4
This controller feeds back the measured values x1 = R and x2 = vc ; x3 , the integral
of the error, e = d − r; and x4 , the double integral of the error. The two integrals are
used because it is assumed that the lead vehicle velocity (which acts as a disturbance
to be rejected by the control law) can be modeled as a ramp-type disturbance input.
With the double integrator built into the control algorithm, the controlled vehicle
can follow the lead vehicle even when it accelerates. The controller gains, ki , can
be determined by a variety of methods, and a pole-placement design technique is
used
in the MATLAB program. The closed-loop poles are placed at s1,2 = −ζ ωn ±
ωn 1 − ζ 2 j, and s3,4 = −αζ ωn . The values ζ = 0.9, ωn = 0.4, and α = 3.0 are used in
the MATLAB simulations for Example 12.5.
In this setup, the lead-vehicle speed (vl ) is treated as a disturbance to be rejected.
It may be desired to treat wl (instead of vl ) as the unknown disturbance and include
vl as a state variable (x5 ), which can be used for the control/warning purposes. In
this case, the dynamic equations are as follows:
ẋ1 = x5 − x2
ẋ2 = −(1/τc )x2 + (Kc /τc )u + (Kc /τc )wc
ẋ3 = (d − r) = (x1 − r)
12.2 Autonomous Cruise Control: Speed and Headway Control
227
Range
31
30
29
28
27
0
2
4
6
8
10
12
Time (sec)
14
16
18
20
Vehicle speed (m/sec)
40
Vl
Vc
30
20
10
0
0
2
4
6
8
10
12
Time (sec)
14
16
18
20
Control Force (N)
2000
1000
0
–1000
–2000
0
2
4
6
8
10
Time (sec)
12
14
16
18
20
Figure 12.12. Simulation results of the ACC system.
ẋ4 = x3
ẋ5 = −(1/τc )x5 + (Kc /τc )wl
u = −k1 x1 − k2 x2 − k3 x3 − k4 x4
which, for the closed-loop system, can be written in state-space form as:
⎡
0
−1
0
0
1
⎢
⎢ −k1 Kc /τc −(1 + k2 Kc )/τc −k3 Kc /τc −k4 Kc /τc
0
⎢
_x = ⎢
1
0
0
0
0
⎢
⎢
⎢
0
0
1
0
0
⎣
0
0
0
⎧
⎫
⎧
⎫
⎧
⎫
0 ⎪
0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪ 0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ Kc /τc ⎪
⎬
⎨ 0 ⎪
⎬
⎬
⎨ 0 ⎪
wl
+ −1 r +
wc +
0
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
0 ⎪
0 ⎪
0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭
⎭
⎩
0
Kl /τl
0
0
−1/τl
⎤
⎥
⎥
⎥
⎥
⎥x
⎥
⎥
⎦
The simulation results are shown in Figure 12.12 for a moderate deceleration case.
The controller maintains the desired range (i.e., 30 meters), despite changes in the
lead-vehicle velocity.
228
Cruise and Headway Control
% Ex12_5.m
g=9.81; uw=0.0; u0=30.0; rho =1.202;
Theta=0.0; ThetaPrime=0.0;
% Controlled vehicle parameters:
mc=1000.0; Cdc=0.5; Arc=1.5; fc=0.015;
Kc=(1/(rho*Cdc*Arc*(u0+uw)));Tc=mc*Kc;
wc0=mc*g*(fc*sin(Theta)-cos(Theta))*ThetaPrime; Fc=(u0/Kc);
% Lead vehicle parameters (typical):
ml=1500.0;Cdl=0.6;Arl=1.95;fl=0.015;
Kl=(1/(rho*Cdl*Arl*(u0+uw)));Tl=ml*Kl;
Fl=(u0/Kl);
wl0=(ml*g*(fl*sin(Theta)-cos(Theta))*ThetaPrime);
t=[0:0.1:20]’;
U0=Fc*ones(size(t));
% Nominal control force
wl= -800*(1+0.01*t);
% Ramp function
wc=wc0*ones(size(t));
disturbance=[U0 wc wl];
% 4-state system for controller-design:
Aa=[0 -1 0 0;
0 -1/Tc 0 0;
1 0 0 0;
0 0 1 0];
Ba=[0;Kc/Tc;0;0];
% Controller design:
pc=[roots([1 2*0.9*0.4 0.4ˆ2]); -1.08; -1.18];
K=place(Aa,Ba,pc);
% Closed-loop simulation (5 states, keep track of vl (x5)):
Ac=[0 -1 0 0 1;
-K(1)*Kc/Tc -(1+K(2)*Kc)/Tc -K(3)*Kc/Tc -K(4)*Kc/Tc 0;
1 0 0 0 0;
0 0 1 0 0;
0 0 0 0 -1/Tl];
Bc=[0 0 0;
0 Kc/Tc 0;
-1 0 0;
0 0 0;
0 0 Kl/Tl];
% outputs: x1 (range), vc and vl
Cc=[1 0 0 0 0;0 1 0 0 0; 0 0 0 0 1]; Dc=zeros(3,3);
r=30.0*ones(size(t));
disturbance=[r U0+wc wl]; xc0=[30 u0 0 (u0/Kc+K(1)*30+K(2)*u0)/K(4) u0];
[yc,xc]=lsim(Ac,Bc,Cc,Dc,disturbance,t,xc0);
subplot(211), plot(t,yc(:,1)); title(‘Range’);
xlabel(‘Time (sec)’); grid;
subplot(212), plot(t,yc(:,3), ‘r’,t,yc(:,2),‘-.b’);
title(‘Vehicle speed (m/sec)’);
Problems 1–3
229
xlabel(‘Time (sec)’); grid;
legend(‘Vl’, ‘Vc’); pause;
clf, subplot(211)
u= U0-K(1)*xc(:,1)-K(2)*xc(:,2)-K(3)*xc(:,3)-K(4)*xc(:,4);
plot(t, u’); title(’Control Force (N)’);
xlabel(‘Time (sec)’); grid
PROBLEMS
1. Grades on interstate highways usually are limited to about 4 percent or less
(i.e., rise over run, or the tan θ is about 0.04). On major roads, grades may occasionally
reach 10 to 20 percent. In the simulation in Example 12.1, a unit-step disturbance
input was used and caused a transient change in the speed with the PI control. Rerun
the simulation in Example 12.1 for the PI cruise controller with grade-disturbance
inputs of (a) 15 percent, and (b) −15 percent. Use the following general expression
for the grade-disturbance input:
d = mg( f sin θ0 − cos θ0 )θ ′
Use the same parameter values as in Example 12.1. Are the observed deviations
from the desired speed “acceptable” for this range of extreme disturbance inputs?
2. In this problem, carry out several calculations to assess whether an adaptive
version of the fixed-gain PI controller in Example 12.1 is desirable. Based on the
results of your calculations in Parts (a) and/or (b), do you recommend the design of
an adaptive PI cruise controller?
(a) Consider the fixed-gain PI cruise controller in Example 12.1. This was
designed for m = 1,000 kg, uw = 2 m/s, and uo = 20 m/s. Apply this controller
to the system with m = 1,250 kg, uw = 5 m/s, and uo = 30 m/s. Compared
to the performance specifications used in Example 12.1, how does this controller perform?
(b) For cruise control (see Example 12.1), linearization was used to obtain the
following time constant and gain:
τ = (m/(ρCd A(uo + uw )));
K = (1/(ρCd A(uo + uw )));
For the nominal values given in this chapter and following Eq. (12.2), plot the values
of τ and K for each of the following cases:
(1) m = [1,000:1,250] (i.e., 25 percent variation in m from curb weight to loaded
weight)
(2) uw = [−10:10] (i.e., ±10m/s [about ±22 mph] head/tail wind)
(3) uo = [20:30] (i.e., 10m/s change in forward velocity)
3. Example 12.5 considers an ACC to maintain a desired headway based on measurements of d and vc only (i.e., no inputs from other vehicles or from the roadway
infrastructure). The control problem is illustrated in block-diagram form in Figure
12.11. Note that the reference input, r, would be computed as a function of the
230
Cruise and Headway Control
vehicle forward velocity to maintain safe headway. Also note that the velocity, vl ,
of the lead vehicle is not known and acts as a disturbance input. The control u on
Vehicle c must be calculated to maintain the desired headway. A feedback controller
is designed for this system based on the following state equations:
ẋ1 = −x2 + vl
ẋ2 = −(1/τc )x2 + (Kc /τc )(u + wc )
ẋ3 = (d − r) = (x1 − r)
ẋ4 = x3
u = −k1 x1 − k2 x2 − k3 x3 − k4 x4
This controller feeds back the measured values x1 = d and x2 = vc , as well as x3 , the
integral of the error, e = d − r, and x4 , the double integral of the error. The two
integrals of error are used because it is assumed that the lead-vehicle velocity, vl ,
(which acts as a disturbance to be rejected by the control law) is well represented as a
ramp-type function in time. The controller gains, ki , can be determined by a variety of
methods, and a pole-placement design technique is used in the MATLAB
program
in Example 12.5. The closed-loop poles are placed at s1,2 = −ζ ωn ± ωn 1 − ζ 2 j and
s3,4 = −αζ ωn . The values ζ = 0.9, ωn =0.4, and α = 3.0 are used in the MATLAB
simulations in Example 12.5.
Repeat Example 12.5 but with a simpler control law that does not feed back
the double integral of the error (i.e., x4 ) and has the form u = −k1 x1 − k2 x2 − k3 x3 .
Compare the results for a ramp-disturbance input, vl , to those obtained using the
previous control law, u = −k1 x1 − k2 x2 − k3 x3 − k4 x4 .
REFERENCES
Druzhinina, M., A. G. Stefanopoulou, and L. Moklegaard, 2002a, “Speed Gradient Approach
to Longitudinal Control of Heavy-Duty Vehicles Equipped with Variable Compression Brake,” IEEE Transactions on Control System Technology, Vol. 10, No. 2, March,
pp. 209–21.
Druzhinina, M., A. G. Stefanopoulou, and L. Moklegaard, 2002b, “Adaptive Continuously
Variable Compression Braking Control for Heavy-Duty Vehicles,” ASME Journal of
Dynamic Systems, Measurement, and Control, Vol. 124, No. 3, September.
Fancher, P., H. Peng, and Z. Bareket, 1996, “Comparison of Three Control Algorithms on
Headway Control for Heavy Trucks,” Vehicle System Dynamics, Vol. 25 Suppl., pp. 139–51.
Fancher, P., H. Peng, Z. Bareket, C. Assaf, and R. Ervin, 2001, “Evaluating the Influences
of Adaptive Cruise Control Systems on the Longitudinal Dynamics of Strings of Highway
Vehicles,” Proceedings of the 2001 IAVSD Conference, Copenhagen, Denmark, August.
Himmelspach, T., and W. Ribbens, 1989, “Radar Sensor Issues for Automotive Headway
Control Applications,” Fall 1989 Summary Report for the Special Topics Course in IVHS,
Appendix G, University of Michigan.
Ioannou, P. A., and C. C. Chien, 1993, “Autonomous Intelligent Cruise Control,” IEEE
Transactions on Vehicular Technology, Vol. 42, No. 4, pp. 657–72.
Klein, R. H., and J. R. Hogue, 1980, “Effects of Crosswinds on Vehicle Response – Full-Scale
Tests and Analytical Predictions,” SAE Paper No. 800848. Detroit, MI.
Liang, C., and H. Peng, 1999, “Optimal Adaptive Cruise Control with Guaranteed String
Stability,” Vehicle System Dynamics, Vol. 32, No. 4–5, November, pp. 313–30.
Liubakka, M. K., D. S. Rhode, J. R. Winkelman, and P. V. Kokotovic, 1993, “Adaptive
Automotive Speed Control,” IEEE Transactions on Automatic Control, Vol. 38, No. 7,
July, pp. 1011–20.
References
Liubakka, M. K., J. R. Winkelman, and P. V. Kokotovic, 1991, “Adaptive Automotive Speed
Control,” Proceedings of the American Control Conference, Boston, MA, June, pp. 439–40.
Oda, K., H. Takeuchi, M. Tsujii, and M. Ohba, 1991, “Practical Estimator for Self-Tuning
Automotive Cruise Control,” Proceedings of the American Control Conference, Boston,
MA, June, pp. 2066–71.
Okuno, A., A. Kutami, and K. Fujita, 1990, “Towards Autonomous Cruising on Highways,”
SAE Paper No. 901484.
Rajamani, R., and Chunyu Zhu, “Semi-Autonomous Adaptive Cruise Control Systems,”
1999, Proceedings of the American Control Conference, Vol. 2, June, pp. 1491–5.
Swaroop, D., and K. R. Rajagopal, 1999, “Intelligent Cruise Control Systems and Traffic Flow
Stability,” Transportation Research, Part C, Vol. 7, pp. 329–52.
Tsujii, M., H. Takeuchi, K. Oda, and M. Ohba, 1990, “Application of Self-Tuning to Automotive Cruise Control,” Proceedings of the American Control Conference, San Diego, CA,
May, pp. 1843–8.
Xu, Z., and P. A. Ioannou, 1994, “Adaptive Throttle Control for Speed Tracking,” Vehicle
System Dynamics, Vol. 23, No. 4, May, pp. 293–306.
Yoshimoto, K., H. Tanabe, and M. Tanaka, 1994, “Speed Control Algorithm for an Automated
Driving Vehicle,” Proceedings of the AVEC ’94, pp. 408–13.
231
13
Antilock Brake and Traction-Control
Systems
Antilock brake systems (ABS) were first introduced on railcars at the beginning
of the 20th century. The original motivation was to avoid flat spots on the steel
wheels; however, it soon was noted that stopping distance also was reduced by the
ABS. Robert Bosch received a patent for ABS in 1936. In 1948, a Boeing B-47 was
equipped with ABS to test its effectiveness in avoiding tire blowout on dry concrete
and spinouts on icy runways. It used a “bang-bang” (i.e., dump brake pressure to
zero, then rebuild) control strategy. Fully modulating ABS control strategies were
introduced in the 1950s (e.g., Ford Lincoln, Goodyear, and HydroAire). A rearwheels-only ABS was first available in luxury automobiles in the late 1960s. The
systems used in the 1960s and 1970s were developed by Bendix, Kelsey-Hayes, and
AC Electronics, among others. Legal concerns then delayed further development in
the United States, and European companies took the lead in the next two decades.
Demand skyrocketed in the early 1990s when the benefits of ABS for vehicle-steering
control and shorter stopping distances were recognized and accepted widely. Most
new passenger vehicles sold in the United States today are equipped with ABS.
It is important to note that ABS will not work properly if the user input or road
condition varies quickly. For example, according to a recent test report by the
NHTSA (Forkenbrock et al. 1999), all of the test vehicles equipped with ABS stop
within a longer distance than those without ABS on loose-gravel roads. Therefore,
improvements still are needed in this relatively mature technology.
In 1987, Bosch combined a traction-control function with its ABS system. A
traction-control system (TCS), generally speaking, is the counterpart of ABS and
works in the case of acceleration. Both systems aim not only to avoid wheel lockup or
spin but also to maintain wheel slip near an optimal value, which achieves maximum
possible tire longitudinal forces and, in the meantime, avoids significant reduction
in tire lateral forces. TCS frequently is advertised as a device to help drivers pull off
from icy patches or slopes; however, a complete TCS actually could work under all
vehicle speeds. Unlike ABS, the major barrier to its acceptance is not legal issues but
rather competition with existing systems that achieve similar functions (e.g., limitedslip differentials and 4WD). Because the TCS function can be achieved with minimal
extra hardware requirements, and most of the work involves software modification
of ABS and engine-control programs, it maintains a good competitive position with
respect to those alternative systems.
232
13.1 Modeling
233
ABS/TCS for 2-axle commercial vehicle with TCS brake controller
1 Wheel-speed sensor, 2 Pulse ring, 3 ABS/TCS electronic control unit, 4 Pressure-control valve,
5 Shuttle valve, 6 Service-brake valve, 7 Combination brake cylinder, 8 Air reservoir, 9 Drain valve,
10 Diaphragm actuator, 11 TCS solenoid valve, 12 Relay valve.
1
Compressed-air line
Electrical line
10
2
7
4
6
3
2
1
2
1
4
12
5
11
5
4
1
2
4
10
9
9
7
8
8
Figure 13.1. ABS/TCS system (Bosch Driving Safety Systems).
ABS, shown in Figure 13.1, are commercially available on many mass-market
production vehicles, but traction control (i.e., anti-spin) systems are less widespread.
The ABS modulate the brake force (i.e., pressure), much like a good driver does
when braking on ice, to maintain braking forces near the peak of the μ − λ curve
(Figure 13.2). In ABS, only the braking force is modulated by the controller, whereas
in TCS (i.e., anti-spin), both the braking force and the traction force (through engine
control) are manipulated. Wheel speeds and brake pressures are measured, vehicle
speed and road friction are estimated, and brake pressure and engine torque are
modulated to maintain a desired slip ratio.
As shown in Figure 13.2a, the lateral force on a wheel is a maximum nearzero slip, and there is “sufficient” lateral force when the traction force is at its
maximum level, at about 10 to 30 percent slip. The lateral force goes to zero as the
slip approaches 100 percent (i.e., the wheel is fully locked). This explains the loss
of steering control or directional stability in the case of locked front or rear wheels,
respectively. These μ − λ curves depend strongly on road-surface conditions, as
Braking force
coefficient µb
Stable
Traction force
coefficient µt
1.0
Dry Pavement
Wet Asphalt
TC
ASR control range
ABS control range
Unstable
Unstable
Side force
coefficient µs
Adhesion
Coefficient (µ)
Unpacked Snow
µs
Ice
Locked Negative skip
wheel
Braking
Free rolling
wheel
Positive skip
Spinning
wheel
Accelerating
(a)
Figure 13.2. Tire-force and road-adhesion coefficients.
0
1.0
Wheel Slip (λ)
(b)
234
Antilock Brake and Traction-Control Systems
illustrated in Figure 13.2b. ABS attempt to maintain the braking force near the peak
value. When a road surface changes, however, identifying even a proper target slip
ratio can be a challenging task. The regulation of tire-slip ratio is accomplished by
modulating the brake pressure in a fast apply–hold–release type of cyclic action.
Because the road friction is unknown, the ABS cannot be guaranteed to achieve the
exact peak braking-force value. Rather, it tries to maintain a slip ratio that averages
around the target slip, which is expected to be close to where the peak force can be
obtained.
When the target slip ratio is near the optimal slip (i.e., maximum force), it is
clear from Figure 13.2 that ABS can maximize braking forces in the longitudinal
direction (i.e., minimize stopping times and distances) while also maintaining large
lateral forces for directional stability when steering in braking maneuvers. There are
other potential benefits of ABS; for example, Chapter 4 discusses the fore–aft shift
of normal forces on the tires due to acceleration or deceleration (see Example 4.2).
Without an ABS, efficient braking distribution requires proportioning the brake
pressure between the front and rear wheels. However, ABS can adjust continuously
individual wheel-brake pressures to maintain each wheel near its optimal adhesion
coefficient. This same principle applies not only for fore–aft load variations but also
for load and adhesion (i.e., μ) variations between the right and left wheels.
13.1 Modeling
A mathematical model for traction-controller design is based on a one-wheel description of that rotational dynamics and a model (see Chapter 3) for the vehicle longitudinal dynamics. The application of Newton’s second law to the rotational dynamics
of a wheel gives:
ω̇w = [(Te − Tb − rw Fx − Tw (ωw )]/Jw
(13.1)
where ωw is the wheel rotational velocity, Te is the engine torque, Tb is the braking
torque, rw is the wheel radius, Fx is the traction (or braking when negative) force, and
Tw (ωw ) is a wheel-friction torque that is a function of the wheel rotational speed. In
the following discussion, we assume dry-friction and viscous-friction models at the
wheel; that is, Tw (ωw ) = fw Fz + bw ωw . The effective wheel inertia, Jw , is the wheel
inertia, Iw , in braking. During acceleration, Jw = Iw + rg2 Ie , where Ie is the engine
inertia and rg is the gear ratio.
The wheel slip, which is due to deformation of the tread on the pneumatic tire
(Figure 13.3) during acceleration, is defined as:
λ=
ωw − ωv
ωw
when accelerating (ωw > ωv )
(13.2)
when decelerating (ωw > ωv )
(13.3)
and during deceleration as:
λ=
ωw − ωv
ωv
where ωv = u/rw and u is the vehicle longitudinal velocity. The vehicle longitudinal
dynamics is governed by Eq. (4.4), which is repeated here:
u̇ = [−0.5ρCd A(u + uw )2 − f mg cos − mg sin + Nw Fx ]/m
(13.4)
13.1 Modeling
235
Figure 13.3. Slip in a pneumatic tire. Top to
bottom: free rolling, turning, braking, and
accelerating.
The tire traction force is defined by:
Fx = μFz
(13.5)
where the surface-adhesion coefficient, μ, is a function of λ, as illustrated in Figure
13.2, and Fz is the normal force at each tire. In this book, we use positive Fz , which
is less confusing. Therefore, μ > 0 for acceleration and μ < 0 for deceleration (i.e.,
braking) for this sign convention. The number of drive wheels (for acceleration) is
Nw ; or, in the case of braking, Nw is the total number of wheels.
It is now convenient to rewrite these equations in standard state-variable form.
Define x1 = ωv and x2 = ωw to rewrite the equations of motion as:
ẋ1 = [−(0.5ρCd A)(rw x1 + uw )2 − f mg cos − mg sin + Nw Fz μ(λ)]/(mrw )
ẋ2 = [− fw Fz − bw x2 − Fz rw μ(λ) + T ]/Jw
(13.6)
where T = (Te − Tb ) and λ = (x2 − x1 )/x1 for braking = (x2 − x1 )/x2 for accelerating.
Equation (13.6) is highly nonlinear, primarily due to the μ − λ relationship but also
to the friction terms in Eqs. (13.1) and (13.4). In the following discussion, it is assumed
that the wind velocity, uw = 0, and the road-grade angle, = 0.
For simplicity, we assume that the actuator dynamics can be neglected. In actual
systems, of course, this may not be a good assumption. The brake actuator usually
needs to incorporate a time delay (on the order of 100 ms) plus a first-order time-lag
236
Antilock Brake and Traction-Control Systems
Polynomial interpolation, degree n=6
0.8
0.6
0.4
Figure 13.4. Polynomial curve fitting for tire λ − μ
curve.
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
system. The throttle actuator typically is faster. However, it is connected in series
with the engine, which requires a nonlinear manifold dynamics in the form of:
Pm
dmai (ωe , Pm )
· TC(α)
=β· f
dt
Pa
where:
mai = air mass flow rate
Pm = intake manifold pressure
f = pressure-ratio influence function
α = throttle angle
ωe = engine speed
β = maximum airflow rate
Pa = atmospheric pressure
TC = throttle-valve characteristic
function
and the engine-torque generation introduces a non-negligible lag. In this book,
however, we ignore these actuator dynamics.
13.2 Antilock Braking Systems
ω −ω
For the ABS, ωw < ωv , so λ = wω v < 0. Example 13.1 illustrates the polynomial
v
fit for the λ − μ curves, Example 13.2 introduces the so-called Magic Formula for
nonlinear tire modeling, Example 13.3 gives the nonlinear simulation for braking,
and Example 13.4 provides a linear ABS controller.
In this MATLAB program, a polynomial fit is obtained for a set of λ − μ data given at the beginning
of the example program. The polynomial curve is represented in the following
form:
EXAMPLE 13.1: CURVE FITTING OF A LAMBDA-MU CURVE.
μ=
n
$
ci λ(n+1−i)
i=1
where the coefficient values, ci, are to be identified. The results of this polynomial fit are shown in Figure 13.4, which seems to work reasonably well. Other
functional forms (i.e., nonpolynomial) could be used to represent the λ − μ
curve. For example, the Magic Formula proposed by Pacejka and co-workers
(1987, 1989) could be used to obtain a smooth fit. The Magic Formula is used
widely in the automotive industry (see Example 13.2).
13.2 Antilock Braking Systems
237
% Ex13_1.m The mu-lambda curve
% Define the mu-lambda data:
Data = ...
[ 0.0000
0.0000
0.0250
0.2250
0.0500
0.4500
0.1000
0.6500
0.1250
0.6850
0.1500
0.7050
0.1750
0.6900
0.2000
0.6800
0.2500
0.6500
0.3000
0.6350
0.3500
0.6300
0.4000
0.6275
0.4500
0.6250
0.5000
0.6225
0.5500
0.6200
0.6000
0.6175
0.6500
0.6150
0.7000
0.6125
0.7500
0.6100
0.8000
0.6075
0.8500
0.6050
0.9000
0.6000
0.9500
0.5975
1.0000
0.5950];
lambda = Data(:,1); mu = Data(:,2);
% We would like to fit the function
% mu=c(1)*lambdaˆn+c(2)*lambdaˆ(n-1)
%
+ c(3)*lambdaˆ(n-2)
%
+ ...+c(n)*lambda+c(n+1);
% to the data. In the following, n=6
[c,S]=polyfit(lambda, mu, 6);
c
pause; t = (0: 0.05: 1)’;
y = polyval(c,t);
plot(lambda,mu,‘or’,t,y,‘-b’)
title(‘Polynomial interpolation, degree n=6’)
EXAMPLE 13.2: MATLAB IMPLEMENTATION OF THE MAGIC FORMULA (PURE LONGITUDINAL SLIP). The most famous special-function–based tire model is the Magic
Formula proposed by Bakker, Nyborg, and Pacejka (1987). The special function
looks like the following:
Fx = Dx sin(Cx tan−1 (Bx φx )) + Svx
(13.7)
Fy = Dy sin(Cy tan−1 (By φy )) + Svy
(13.8)
238
Antilock Brake and Traction-Control Systems
where:
φx = (1 − Ex )(Sx + Shx ) +
φy = (1 − Ey )(α + Shy ) +
Ex
tan−1 (Bx (Sx + Shx ))
Bx
Ey
By
tan−1 (By (α + Shy ))
(13.9)
(13.10)
The meanings of the coefficients are as follows:
B = stiffness factor
C = shape factor
D = peak factor (i.e., determines peak magnitude)
E = curvature factor
Sh = horizontal offset
Sv = vertical offset
and α = slip angle (in degrees) and Sx = longitudinal slip. The special-function
form also applies to the self-aligning moment, the formulas for which are exactly
the same as those shown in Eqs. (13.7) through (13.10). In general, the coefficients are functions of the tire normal force Fz .
Raw data vs. Magic formula (Braking)
Raw data vs. Magic formula (Fraction)
0
3500
−500
Longitudinal force
Longitudinal force
3000
2500
2000
1500
1000
−1500
−2000
−2500
−3000
500
0
−1000
0
0.05
0.1
Longitudinal slip
0.15
0.2
−3500
−0.2
−0.15
−0.1
−0.05
Longitudinal slip
% Ex13_2.m
% Magic Formula empirical tire model
% Tire3 contains [slip Fx] data for Fz=3225 Newton
load tire3
s = tire3(:,1); % Longitudinal slip
fx = tire3(:,2); % Longitudinal force in Newton
% First figure: Traction case
N=4; % resample because too many data points
plot(s(1:N:size(s,1)), fx(1:N:size(fx,1)), ‘ob’)
xlabel(‘Longitudinal slip’)
ylabel(‘Longitudinal force Fx (N)’)
title(‘Raw data’)
hold on
pause
% The Magic Formula for traction case
Bx = 25;
0
13.2 Antilock Braking Systems
239
Cx = 1.35;
Dx = 3444;
Ex = -2.9;
Shx = 0;
Svx = 0;
for i=1:101,
s2(i) = (i-1)*0.002;
% slip between 0 and 0.2
phi_x = (1-Ex)*(s2(i)+Shx) + Ex/Bx*atan(Bx*(s2(i)+Shx));
fx2(i) = Dx*sin(Cx*atan(Bx*phi_x))+Svx;
end
plot(s2,fx2,‘-r’), axis([0 0.2 0 3600])
title(‘Raw data vs. Magic formula (Traction)’)
hold off
pause
% Second figure: Braking case
N=4;
plot(s(1:N:size(s,1)), fx(1:N:size(fx,1)), ‘ob’)
xlabel(‘Longitudinal slip’)
ylabel(‘Longitudinal force Fx (N)’)
hold on
% The Magic Formula for braking case
Bx = 25;
Cx = 1.15;
Dx = 3100;
Ex = -0.4;
Shx = 0;
Svx = 0;
for i=1:101,
s2(i) = (i-1)*0.002;
% slip between 0 and 0.2
phi_x = (1-Ex)*(s2(i)+Shx) + Ex/Bx*atan(Bx*(s2(i)+Shx));
fx2(i) = Dx*sin(Cx*atan(Bx*phi_x))+Svx;
end
plot(-s2,-fx2,‘-r’), axis([-0.2 0 -3600 0])
title(‘Raw data vs. Magic formula (Braking)’)
hold off
In the following discussion, the vehicle parameter values are assumed to be:
m = 1,400 kg
Fz = 3,560 N
bw = 0 kg-m/s
f = 0.01
Iw = 0.65 kg-m2
Cd = 0.5
ρ = 1.202 kg/m3
fw = 0 Nw = 4 Te = 0.0 N-m
rw = 0.31 m
A = 1.95 m2
Tb = 1,000 N-m
240
Antilock Brake and Traction-Control Systems
Vehicle Braking
20
Forward speed
Wheel speed
15
10
5
0
–5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 13.5. Simulation results of nonlinear braking.
EXAMPLE 13.3: NONLINEAR VEHICLE-BRAKING BEHAVIOR. The coefficients calculated from the μ − λ data in Example 13.1 are used in this example to simulate
the nonlinear system described by Eq. (13.6) for the parameter values listed previously. The results are shown in Figure 13.5. Notice from the figure that wheel
lock occurs rapidly (i.e., in about 0.2 seconds) and that the vehicle takes about
1 second to come to a stop. Because the wheel is locked, the lateral stability of
the vehicle can be expected to be poor.
% Ex13_3.m
ti=0.0; tf=1.0; xi=[20.0,20.0];
tol=1.0e-4; trace=1;
[t,x]=ode45(‘Ex13_3a’,ti,tf,xi,tol,trace);
plot(t,x); title(‘Vehicle Braking’);
xlabel(‘Time (sec)’); ylabel(‘x’); grid;
legend(‘Forward speed’,‘Wheel speed’)
% A separate file Ex13_3a.m
function xdot=Ex13_3a(t,x);
% Define the simulation parameters:
m=1400; rho=1.202; Cd=0.5; A=1.95; g=9.81;
Theta=0.0; bw=0.0; f=0.01; uw=0.0; fw=0.0;
Iw=2.65; rw=0.31; Nw=4; Fz=3560.0;
% Define the mu-lambda polynomial coefficients:
c=[-68.593, 238.216,-324.819,219.283,
-75.58, 12.088, -0.0068];
if x(1) >= x(2),
lambda=(x(2)-x(1))/x(1);
else
lambda=(x(2)-x(1))/x(2);
end;
al = abs(lambda);
if al > 1.0, al =1.0; end;
13.2 Antilock Braking Systems
241
k1
a21
∆Te
∆ λr
KD +
s
+
-
∆Tb
-
+
+
∆T
+
+
-
-
+
1 ∆x2
a12
s
b
1
s
+
a22
∆x1
a11
k2
d2
+
∆λ
d1
+
Figure 13.6. Integral plus state feedback form of linear ABS control.
mu=sign(lambda)*c*[alˆ6;alˆ5; ...
alˆ4;alˆ3;alˆ2;al;1];
% Define the torque input T = Te - Tb;
Te =0.0; Tb=1000.0;
T = Te - Tb;
% Define the state equations:
if x(1) < 0.0, x(1) = 0.0; end;
if x(2) < 0.0, x(2) = 0.0; end;
xdot=[(-(0.5*rho*Cd*A)*(uw+rw*x(1))ˆ2+
Nw*Fz*mu-f*m*g*cos(Theta)m*g*sin(Theta))/(m*rw);
(fw*Fz-bw*x(2)-Fz*rw*mu+T)/Iw];
if x(1) <= 0.0, xdot(1)=0.0; end;
if x(2) <= 0.0, xdot(2)=0.0; end;
A controller is designed
for an ABS based on the linearization of Eq. (13.6). The parameter values given
in the previous examples are used. The linearized controller block diagram is
shown in Figure 13.6. The controller is the integral-plus-state-feedback type,
and this controller, when applied to the linearized system, is expected to give
good performance. The linearized system performance as shown in Figure 13.7a,
however, is not very good. This is due to the presence of the zero near the origin
of the s-plane, which leads to a major overshoot in the response. We expect the
performance of this controller, when applied to the actual nonlinear system, to
be even worse due to the strong nonlinearities in Eq. (13.6).
EXAMPLE 13.4: A LINEAR CONTROLLER DESIGN FOR ABS.
a11 =
a21 =
N F
−ρCd Arw x10
+ w z
m
mrw
−Fz rw
Iw
0
∂μ
∂λ
0
∂λ
∂x1
∂μ
∂λ
0
∂λ
∂x1
a12 =
0
Nw Fz
mrw
∂μ
∂λ
0
∂λ
∂x2
0
242
Antilock Brake and Traction-Control Systems
Figure 13.7. Simulation results of linear ABS.
Fr
∂μ
∂λ
−bw
− z w
Iw
Iw
∂λ 0 ∂x2 0
1
∂λ
d2 =
=
∂x2 0
x10
a22 =
b=
1
Iw
d1 =
∂λ
∂x1
0
=−
x20
x210
This is illustrated in Figures 13.7b–d. Thus, the linear controller shown here is
not effective in practice. Furthermore, in an actual application, the μ − λ curve
characteristics are not known, as was assumed here (other parameters also are
uncertain). The results for both the linear and nonlinear models are shown in
Figure 13.7. A piecewise linear control design strategy, with gain scheduling, can
be expected to improve these results.
% Ex13_4.m
% Define the parameters:
m=1400; rho=1.202; Cd=0.5; A=1.95; g=9.81;
Theta=0.0; bw=0.0; f=0.01; uw=0.0;
Iw=0.65; Jw=Iw; rw=0.31; Nw=4; Fz=3560.0;
x10=40.0; x20=32.0; lref=(x20-x10)/x10;
al=abs(lref);
c=[-68.593,238.216,-324.819,219.2837,
-75.58, 12.0878, -0.0068];
dmdl=c*[6*alˆ5;5*alˆ4;4*alˆ3;3*alˆ2;
2*al;1;0];
dldx1=-(x20/x10ˆ2); dldx2=1/x10;
% Define the A,B,C, and D matrices:
13.2 Antilock Braking Systems
A=[(-rho*Cd*A*(rwˆ2)*x10+ ...
Nw*Fz*dmdl*dldx1)/(m*rw), ...
(Nw*Fz*dmdl*dldx2)/(m*rw),0;
(-rw*Fz*dmdl*dldx1)/Jw, ...
(-bw -rw*Fz*dmdl*dldx2)/Jw,0;
dldx1,dldx2,0;];
Bu=[0;1/Jw;0]; Br=[0;0;-1];
C=[dldx1 dldx2 0]; D=0.0;
z=0.707; wn=10.0;
p=[-z*wn+i*wn*sqrt(1-zˆ2);
-z*wn-i*wn*sqrt(1-zˆ2); -2*z*wn];
Co = det(ctrb(A,Bu));
Ob = det(obsv(A,C));
K = place(A,Bu,p);
[num,den]=ss2tf((A-Bu*K),Br,C,D,1);
zeros=roots(num);
poles=roots(den);
t=[0:0.01:1.0]; r0 = 0.05;
[y,x]=step((A-Bu*K),r0*Br,C,r0*D,1,t);
plot(t,y+lref);
title(‘Wheel Slip w/ Linear ABS’); grid;
xlabel(‘Time (sec)’); ylabel(‘lambda’);
% Ex13_4a.m
ti=0.0; tf=0.5; xi=[40.0;40.0;0.0];
tol=1.0e-4; trace=1;
[t,x]=ode45(‘Ex13_4b’,ti,tf,xi,tol,trace);
plot(t,[x(:,1) x(:,2)]);
title(‘Vehicle Simulation with ABS’);
xlabel(‘Time, t (sec)’); ylabel(‘x’); grid; pause;
K=[2.3913e+05 3.0993e+01 3.5418e+06];
u=-K*x’;
plot(t,u); title(‘vehicle ABS torque variation’);
xlabel(‘Time, t (sec)’); grid; pause;
lambda=(x(:,2)-x(:,1))./x(:,1);
plot(t,lambda);
title(‘Wheel Slip w/ ABS’);
xlabel(‘Time, t (sec)’); grid;
% A separate file Ex13_4b.m
function xdot=Ex13_4b(t,x);
% Define the simulation parameters:
m=1400; rho=1.202; Cd=0.5; A=1.95; g=9.81;
Theta=0.0; bw=0.0; f=0.01; uw=0.0;
Iw=0.65; rw=0.31; Nw=4; Fz=3560.0;
% Initialize:
Te = 0.0;
243
244
Antilock Brake and Traction-Control Systems
% Define the mu-lambda polynomial
c=[-68.593,238.216,-324.8197,219.2837,
-75.5800, 12.0878, -0.0068];
% Define wheel slip lambda:
if x(1) >= x(2),
lambda=(x(2)-x(1))/x(1);
else
lambda=(x(2)-x(1))/x(2);
end;
al = abs(lambda);
if al > 1.0, al =1.0; end;
mu=sign(lambda)*c*[alˆ6;alˆ5;alˆ4;alˆ3;alˆ2;al;1];
% Define torque input T=Te - Tb;
K=[2.3913e+05 3.0993e+01 3.5418e+06];
x10=40; x20=32; lref=(x20-x10)/x10;
Tb=7.594e+2+K*[x(1)-x10;x(2)-x20;x(3)];
if Tb<0.0, Tb=0.0; end;
T=Te-Tb;
xdot=[(-(0.5*rho*Cd*A)*(uw+rw*x(1))ˆ2+
Nw*Fz*mu-f*m*g*cos(Theta)m*g*sin(Theta))/(m*rw);
(-f*Fz-bw*x(2)-Fz*rw*mu+T)/Iw;
lambda-lref];
if x(1) <= 0.0,
x(1)=0.0; xdot(1)=0.0; end;
if x(2) <= 0.0,
x(2)=0.0; xdot(2)=0.0; end;
The previous example shows that the linear controller design is not effective for this
application because the nominal operating point is not an equilibrium (i.e., ẋ = 0). A
nonlinear control strategy is typically used in practice (e.g., Tan and Chin 1991; Tan
and Tomizuka 1990a, 1990b). One approach is to construct a rule-based or tablelookup type of control algorithm. For example, a rule to be used in a TCS might be
as follows:
T = 0.0
ωw − ωv
ω − ωv
ELSE λ = w
END
ωw
ωv
IF |λ(t )| < λ min THEN DT = sgn(λ) α1 |T(t−1)| END
IF ωw > ωv THEN λ =
IF |λ(t )| > λmax THEN DT = −sgn(λ) α2 |T(t−1)| END
T(t) = T(t−1) + T
Thus, the rule-based controller calculates a change (T) in the torque (T) for both
braking and accelerating cases. A control system of this type is intuitively appealing
but the development of the rules, tuning, and calibration depend on experience and
trial and error. For example, the values of λmin , λmax , and αi in the previous algorithm
13.2 Antilock Braking Systems
245
Vehicle Simulation with ABS
20
Vehicle speed
Wheel speed
x
10
0
–10
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Lambda
0
–0.5
–1
Torque (N-m)
–600
–700
–800
–900
Figure 13.8. Simulation results of rule-based ABS control system.
must be chosen (e.g., λmin = 0.13, λmax = 0.17, α 1 = 0.07, and α 2 = 0.09). Furthermore,
there are no guarantees regarding the behavior of such a system in terms of stability
or performance.
The previous rule-based
controller is implemented in this ABS simulation. The vehicle and wheel speeds
are shown in Figure 13.8a, and in Figure 13.8b for the slip versus time. The
slip during braking is maintained at a value near 0.15. Although the stopping
time is approximately the same as in Example 13.3 (i.e., about 0.9 seconds), it
is expected that vehicle handling should be improved over that case (see Figure
13.2a). In this simulation, the μ − λ curve is relatively flat (see Figure 13.4); thus,
we do not observe a significant difference in the stopping distance.
EXAMPLE 13.5: A RULE-BASED CONTROLLER FOR ABS.
% Ex13_5.m
% Initialize Torque (at Te=0, Tb=800 for braking and
% Te=1800, Tb=0 for accelerating)
global Torque;
Te = 0.0; Tb = 800.0; Torque = Te - Tb;
% Define the mu-lambda polynomial coefficients:
c = [-68.5937, 238.2160, -324.8197, 219.2837, ...
-75.5800, 12.0878, -0.0068];
246
Antilock Brake and Traction-Control Systems
% Perform the closed-loop simulation:
ti=0.0; tf=1.0; t1 = ti;
xi=[20.0;20.0];
% Set the rule-based control parameters:
alph1 = 0.07; alph2=0.09; lmin=0.13; lmax=0.17;
x_log(1,:)=xi’;
t_log(1)=ti;
Torque_log(1) = Torque;
OPTIONS=odeset(‘MaxStep’,0.001,‘RelTol’,1.0e-3);
for i=1:175,
i
t2=ti+i*(tf-ti)/200;
tspan=[t1:0.001:t2];
[t,x]=ode23(‘Ex13_5a’,tspan,xi,OPTIONS);
x_log(i+1,:)=x(size(x,1),:); % data log
xi=x(size(x,1),:)’;
t1 = t(size(t,1));
t_log(i+1)=t(size(t,1));
% Define wheel slip lambda:
if xi(1) >= xi(2),
lambda=(xi(2)-xi(1))/xi(1);
else
lambda=(xi(2)-xi(1))/xi(2);
end;
% Calculate the coefficient mu:
al = abs(lambda);
% if al > 1.0 al=1.0; end;
% mu = -sign(lambda)*c*[alˆ6;alˆ5;alˆ4;alˆ3;alˆ2;al;1];
% Define the change in torque delT:
if al<lmin,
delT = sign(lambda)*alph1*abs(Torque);
elseif al>lmax,
delT = -sign(lambda)*alph2*abs(Torque);
else
delT = 0.0;
end;
Torque = Torque + delT;
Torque_log(i+1) = Torque;
end
% Plot the results:
subplot(311), plot(t_log,[x_log(:,1) x_log(:,2)]);
title(‘Vehicle Simulation with ABS’);
xlabel(‘Time (sec)’); ylabel(‘x’);
13.3 Traction Control
legend(‘Vehicle speed’, ‘wheel speed’); grid;
%lambda=(x_log(:,2)-x_log(:,1))./x_log(:,2);% TCS
lambda=(x_log(:,2)-x_log(:,1))./x_log(:,1); % ABS
subplot(312), plot(t_log,lambda,‘r’);
xlabel(‘Time (sec)’); ylabel(‘Lambda’); grid;
subplot(313), plot(t_log,Torque_log,‘r’);
xlabel(‘Time (sec)’); ylabel(‘Torque (N-m)’); grid
%Ex13_5a.m
function xdot = Ex13_5a(t,x);
global Torque;
% Define the simulation parameters:
m=1400; rho=1.202; Cd=0.5; A=1.95; g=9.81;
Theta=0.0; bw=0.0; f=0.01; uw=0.0; fw=0.0;
Iw=0.65; rw=0.31; Fz=3560.0;
% For braking: Ie=0, Nw=4; and for accel: Ie=0.429, Nw=2.
rg=9.5285; Ie=0.0; Nw=4;
Jw = Iw + (rgˆ2)*Ie;
% Define the mu-lambda polynomial coefficients:
c = [-68.5937, 238.2160, -324.8197, 219.2837, ...
-75.5800, 12.0878, -0.0068];
% Define wheel slip lambda:
if x(1) >= x(2),
lambda=(x(2)-x(1))/x(1);
else
lambda=(x(2)-x(1))/x(2);
end;
% Calculate the coefficient mu:
al = abs(lambda);
if al > 1.0 al=1.0; end;
mu = sign(lambda)*c*[alˆ6;alˆ5;alˆ4;alˆ3;alˆ2;al;1];
% Define the state equations:
xdot=[(-(0.5*rho*Cd*A)*(uw+rw*x(1))ˆ2+Nw*Fz*muf*m*g*cos(Theta)-m*g*sin(Theta))/(m*rw);
(fw*Fz-bw*x(2)-Fz*rw*mu+Torque)/Jw];
if x(1) <= 0.0, x(1)=0.0; xdot(1)=0.0; end;
if x(2) <= 0.0, x(2)=0.0; xdot(2)=0.0; end;
13.3 Traction Control
The preceding section discusses manipulation of the brake torque to control the
wheel slip during deceleration. Here, the combined manipulation of the engine and
brake torques to achieve anti-slip acceleration and anti-skid braking is discussed.
For acceleration, we use the same relationships given in Eq. (13.6) except that
247
248
Antilock Brake and Traction-Control Systems
Vehicle Simulation with ABS
25
Vehicle speed
Wheel speed
x
20
15
10
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Lambda
0.4
0.2
0
Torque (N-m)
4000
2000
0
Figure 13.9. Simulation results of a rule-based TCS.
Nw is the number of drive wheels, the effective wheel inertia is Jw = Iw + rg 2 Ie ,
and the wheel slip is λ = (x2 − x1 )/x2 . Notice that during acceleration, x2 > x1 ;
therefore, λ > 0. Thus, a controller for the acceleration case can be developed in
a manner similar to the deceleration case. The two can be combined based on the
measured values of the wheel and vehicle speeds and the resulting sign of λ.
EXAMPLE 13.6: A RULE-BASED TRACTION-CONTROL SYSTEM Simulations for the
acceleration case can be performed in a manner similar to the ABS simulations
shown in Examples 13.1 through 13.5. Additional parameter values for the
acceleration case are as follows:
Ie = 0.429 kg-m2
rg = 9.5285
Nw = 2 Te = 1,800.0 N-m
A rule-based TCS simulation can be performed using the same MATLAB
program as in Example 13.5 with changes in the parameter values as indicated previously. The controller parameters were set at the values λmin = 0.13,
λmax = 0.17, α1 = 0.14, and α2 = 0.15. Also, there must be an additional limit
so that the applied torque does not exceed that of the engine torque. The simulation results are shown in Figure 13.9. The slip is maintained at a value of
approximately 0.15, which is the center of the two limits set in the controller.
The vehicle-speed measurement problem, which is necessary to determine λ, is
difficult (Anonymous 1988c). Typically, lambda is estimated based on wheel-speed
13.3 Traction Control
249
measurements and by considering the change in wheel speed (or wheel acceleration/deceleration). Notice also that actuation dynamics are neglected; that is, we
assumed that commanded changes in brake or engine torque occur instantaneously.
A complete controller design also must consider these effects.
Another more systematic nonlinear-control approach is based on the variablestructure control or the sliding-mode control approach. The basis for the slidingmode-control or variable-structure-control approach is the determination of a
switching logic, which switches between two nonlinear control functions. The switching logic depends on the sign of a switching function, S, and the system state moves
(or slides) along this function – thus, the name sliding mode control. For example,
given ẍ ± x = 0, we can have either system (S1) or (S2):
ẋ1 = x2
(S1)
ẋ2 = −x1
or
ẋ1 = x2
(S2)
ẋ2 = +x1
Define a switching surface (sliding mode) by S = ẋ + c(x − xd ); then, select the
control input u(t) to achieve S = 0. For example, if S > 0, select u to make the system
(S1); if S < 0, select u to make the system (S2). The resulting trajectories will look
like:
This method, which is equally applicable to the ABS or the TCS problem, is
briefly described here (Tan and Chin 1991). The basis for the sliding-mode control approach is the determination of a switching logic, which switches between
two nonlinear control functions. The switching logic depends on the sign of a
switching function, S, and the system-state moves (or slides) along this function –
thus, the name sliding-mode control. The switching functions for accelerating are as
follows:
S1 (λ, λ̇) = λ̇ + c1 (λ − λ1 )
and for braking:
S2 (λ, λ̇) = λ̇ + c2 (λ − λ2 )
where λ1 = 0.15 and λ2 = −0.15. The control action is calculated for accelerating
from:
If S1 < 0,
If S1 > 0,
then Ṫ = 30,000.0
then Ṫ = −H(T)c1 T – [P1 + Ec ]
250
Antilock Brake and Traction-Control Systems
and for braking from:
If S2 > 0,
If S2 < 0,
then
then
Ṫ = −30,000.0
Ṫ = −H(−T)c2 T – P2
where the reference slip λr = 0.15 and the other functions and parameters are defined
in Tan and Chin (1991).
Example 13.7 illustrates results of traction control from laboratory and road
tests. It was mentioned previously that maintaining a value of λ = 0.15–0.20 is
desirable not only for good traction in deceleration and acceleration but also for
maintaining high lateral forces during such maneuvers. Example 13.7 presents the
performance of a sliding-mode traction controller and gives results from Tan and
Chin (1991) and from Tan and Tomizuka (1990a, 1990b).This example includes an
adaptive sliding-mode controller based on estimation of the unknown μ − λ characteristics. Estimation of the tire–road friction coefficient is an important research area
not only for traction control but also for vehicle-stability control and other active
safety systems (Rajamani et al. 2010). Researchers also studied the effects of ABS
on combined longitudinal and lateral vehicle motions (Kimbrough 1991; Taheri and
Law 1991).
A nonlinear tractioncontroller design is described in a series of papers by Tan and co-authors (Tan
and Tomizuka 1990a, 1990b; Tan and Chin 1991). These papers describe a
variable-structure control (or sliding-mode control) approach to TCS problems
(Tan and Chin 1991). An adaptive version of the approach is described in Tan
and Tomizuka (1990b) and includes estimation of the μ − λ characteristics in
addition to a sliding-mode controller. Finally, in Tan and Tomizuka (1990a),
the digital (or discrete-time) implementation of this controller is described. A
detailed discussion of these control strategies is beyond the scope of this chapter and readers are referred to the references for further details. The adaptive
version, which estimates the μ − λ relationship from measurements, uses online
identification much like the ACC example in Chapter 12. However, the function to be identified here is nonlinear, as are the controlled system and the
controller. Finally, the digital implementation must consider the effects of sampling and the actuator dynamics (i.e., application of brake torque in ABS and
combined braking and engine torque for traction control).
EXAMPLE 13.7: SLIDING-MODE TRACTION CONTROLLER.
Trajectories from simulation results for anti-spin acceleration are illustrated in
Figure 13.10, which shows the convergence of the trajectory in the λ − λ̇ plane to
the sliding-mode surface, S. The control torque is shown for this same case in Figure
13.10b. Figure 13.10c shows the resulting simulated wheel and vehicle speeds during
this acceleration maneuver. Figure 13.11 illustrates a dynamometer test-cell setup
and evaluation test results for this controller for ABS applications. Results in Figures
13.11b–d show the effects of sampling time and drum surface (i.e., wet or dry). Shown
in Figure 13.12 are the results of dynamometer tests using the adaptive version of the
ABS sliding-mode controller. Figures 13.12a–c show results from tests for various slip
values and drum-surface conditions. Although the surface conditions change (i.e.,
from wet to dry or from dry to wet), the controller performs a reasonably good job of
13.3 Traction Control
251
(a)
Figure 13.10. Simulated response in λ − λ̇
plane: (a) trajectories for anti-spin control, (b) torque curves for anti-spin control, and (c) vehicle/wheel speeds for antispin control (Tan and Tomizuka 1990a).
(b)
(c)
252
Antilock Brake and Traction-Control Systems
Figure 13.11. Experimental setup and test results (Tan and Tomizuka 1990a).
maintaining the desired slip values by estimating the actual μ − λ relationship during
operation.
One of the most important variables in realizing the TCS/ABS control algorithms
presented herein is the wheel slip ratio λ. Because the vehicle speed usually is not
available and the estimation may be inaccurate, a reference vehicle speed may be
generated. The reference speed is equal to the wheel speed at the beginning of
the ABS application and decreases at a constant rate, which is determined by the
13.3 Traction Control
Figure 13.12. Vehicle test results (Tan and Tomizuka 1990a).
253
254
Antilock Brake and Traction-Control Systems
road-surface condition. ABS control algorithms implemented on production vehicles
may have the following form:
Significant tuning usually is necessary to determine a good set of rules. When
implementing on vehicles, customer feedback can be as important a factor as engineering judgment.
PROBLEMS
1. The nonlinear simulation in Example 13.3 uses the tire model in Example 13.1.
Redo the simulation in Example 13.3 using the Magic Formula tire model in Example
13.2.
2. Attempt to develop a “rule-based” ABS or traction controller similar to that
suggested in Examples 13.5 and 13.6. Suggest some modification to the rules in those
examples that you think will improve the performance of the controller. First, explain
the rule or rules that you base the controller on and then try it out in simulations
using a program such as those shown in Example 13.5 and 13.6. Compare your results
to those in the examples.
REFERENCES
Anonymous, 1986, “Traction Aids at Three Levels,” Automotive Engineering, January,
pp. 39–41.
Anonymous, 1987, “Traction Control: An ABS Extension,” Automotive Engineering, August,
pp. 75–80.
Anonymous, 1988a, “Electronic Traction Control for Luxury Car,” Automotive Engineering,
February, p. 164.
Anonymous, 1988a, “Total Traction Four-Wheel Drive and Anti-Skid Brakes in a Mid-Size
Car,” Automotive Engineering, March, pp. 82–5.
Anonymous, 1988c, “Wheel Speed Sensing Developments,” Automotive Engineering, June,
pp. 67–73.
Anonymous, 1990a, “Traction Control for Front-Wheel Drive,” Automotive Engineering,
January, pp. 89–90.
Anonymous, 1990a, “Improved Electronics Aid ABS,” Automotive Engineering, September,
pp. 30–5.
Austin, L., and D. Morrey, 2000, “Recent Advances in Anti-Lock Braking Systems and
Traction Control Systems,” Proceedings of the IMechE, Vol. 214, Part D, pp. 625–38.
Bakker, E., H. B. Pacejka, and L. Lidner, 1989, “A New Tire Model with an Application in
Vehicle Dynamics Studies,” SAE Technical Paper 890087, 1989, doi:10.4271/890087.
References
Bakker, E., L. Nyborg and H. B. Pacejka, 1987, Tyre modelling for use in vehicle dynamics
studies, SAE.
Borrelli, F., A. Bemporad, M. Fodor, and D. Hrovat, 2001, “A Hybrid Approach to Traction
Control,” Lecture Notes in Computer Science, Springer Publishing Co.
Brewer, H. K., 1983, “Design and Performance Aspects of Antilock Brake Control Systems,”
in W. E. Meyer and J. D. Walters (eds.), Frictional Interaction of Tire and Pavement, ASTM
STP 793, American Society for Testing and Materials, pp. 79–115.
Canudas de Wit, C., and P. Tsiotras, 1999, “Dynamic Tire Friction Models for Vehicle Traction Control,” Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 4,
pp. 3746–51.
Elgeskog, E., and S. Brodd, 1976, “The Influence of Wheel Slip Control Dynamics on Vehicle Stability During Braking and Steering,” Proceedings of the Institution of Mechanical
Engineers.
Forkenbrock, G. J., M. F. Flick and W. R. Garrott, 1999, NHTSA Light Vehicle Antilock Brake
System Research Program Task 4: A test Track Study of Light Vehicle ABS Performance
Over a Broad Range of Surfaces and Maneuvers, Final Report DOT HS 808 875, National
Highway Traffic Safety Administration (NHTSA), January 1999.
Gohring, E., E. C. von Glasner, and C. Bremer, 1989, “The Impact of Different ABSPhilosophies on the Directional Behavior of Commercial Vehicles,” Vehicle Dynamics
Related to Braking and Steering, SAE SP-801, SAE Paper 892500.
Herb, E., K. Krusche, E. Schwartz, and H. Wallentowitz, 1988, “Stability Control and Traction
Control at Four-Wheel Drive Cars,” Automotive Systems Technology, The Future, Vol. 1.
Hori, Y., Y. Toyoda, and Y Tsuruoka, 1998, “Traction Control of Electric Vehicle,” IEEE
Transactions on Industry Applications.
Jurgen, R., 1995, Automotive Electronics Handbook, McGraw-Hill Publishers.
Kimbrough, S., 1991, “Coordinated Braking and Steering Control for Emergency Stops and
Accelerations,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced
Automotive Technologies–1991, ASME DE-Vol. 40, New York, pp. 229–44.
Lin, W.C., D. J. Dobner, and R. D. Fruechte, 1993, “Design and Analysis of an Antilock Brake
Control System with Electric Brake Actuator,” International Journal of Vehicle Design,
Vol. 14, No. 1, pp. 13–43.
Maretzke, J., and B. Richter, 1987, Traction and Direction Control of 4WD Passenger Cars,
Volkswagenwerk AG, Wolfsburg, Germany.
Mills, V., B. Samuels, and J. Wagner, 2002, “Modeling and Analysis of Automotive Antilock
Brake Systems Subject to Vehicle Payload Shifting, Vehicle System Dynamics, Vol. 37,
No. 4, April, pp. 283–310.
Rajamani, R., D. Piyabongkarn, J. Y. Lew, , K. Yi, and G. Phanomchoeng, 2010, “Tire-Road
Friction-Coefficient Estimation,” IEEE Control Systems Magazine, Vol. 30, No. 4, August,
pp. 54–69.
Rittmannsberger, N., 1988, “Antilock Braking System and Traction Control,” Proceedings of
the International Congress on Transportation Electronics, October, pp. 195–202.
Robinson, B. J., and B. S. Riley, 1989, “Braking and Stability Performance of Cars Fitted
with Various Types of Anti-Lock Braking Systems,” Proceedings of the 12th International
Conference on Experimental Safety Vehicles, NHTSA, Sweden, pp. 836–46.
Rompe, K., A. Schindler, and M. Wallrich, 1987, “Comparison of the Braking Performance
Achieved by Average Drivers in Vehicles with Standard and Anti-Wheel Lock Brake
Systems,” SAE International Congress and Exposition. Detroit, MI, SAE Paper No. 870335.
Sado, H., S. Sakai, and Y. Hori, 1999, “Road Condition Estimation for Traction Control in
Electric Vehicles,” Proceedings of the ISIE’99 – Industrial Electronics.
SAE, 1989, Vehicle Dynamics Related to Braking and Steering, Society of Automotive Engineers.
Segel, L., 1975, “The Tire as a Vehicle Component,” in B. Paul, K. Ullman, and H. Richardson
(eds.), Mechanics of Transportation Suspension Systems, ASME, New York, AMD-Vol. 15.
Sigl, A., and H. Demel, 1990, “ASR-Traction Control, State of the Art and Some Prospects,”
ABS Traction Control and Brake Components, SAE ICE, SAE Paper No. 900204.
255
256
Antilock Brake and Traction-Control Systems
Taheri, S., and E. H. Law, 1991, “Slip Control Braking of an Automobile During Combined
Braking and Steering Maneuvers,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang
(eds.), Advanced Automotive Technologies–1991, ASME DE-Vol. 40, New York, pp. 209–
27.
Tan, H. S., and Y. K. Chin, 1991, “Vehicle Traction Control: Variable-Structure Control
Approach,” ASME Journal of Dynamic Systems, Measures, and Control, June, Vol. 113,
pp. 223–30.
Tan, H. S., and M. Tomizuka, 1990a, “Discrete-Time Controller Design for Robust Vehicle
Traction,” IEEE Control System Magazine, April, pp. 107–13.
Tan, H. S., and M. Tomizuka, 1990b, “An Adaptive Sliding-Mode Vehicle Traction Controller
Design,” Proceedings of the American Control Conference, San Diego, CA, May, pp. 1856–
61.
Watanabe, M., and N. Noguchi, 1990, “A New Algorithm for ABS to Compensate for RoadDisturbance,” ABS Traction Control and Brake Components, SAE Paper 900205.
Yeh, E.C., and G. C. Day, 1992, “A Parametric Study of Anti-Skid Brake Systems Using
Poincare Map Concept,” International Journal of Vehicle Design, Vol. 13, No. 3, pp. 210–
32.
14
Vehicle Stability Control
The concept of vehicle stability control (VSC), variously known as vehicle dynamics
control (VDC) and electronic stability program (ESP), was first introduced in 1995
and has been studied extensively as an active safety device to improve vehicle stability
and handling. The concepts are natural extensions of the ABS and TCS discussed
in Chapter 13. When “differential braking” is applied – that is, braking forces of
the left- and right-hand-side tires are different – a yaw moment is generated. This
yaw moment then slows down the vehicle and influences the vehicle lateral/yaw/roll
motion. By taking advantage of the widely available and mature ABS hardware, this
differential braking function is added on with minimal additional cost. Therefore,
this vehicle control system has enjoyed rapid market acceptance.
The vehicle yaw moment also can be generated through the manipulation of
tire-traction forces – for example, through the control of differentials or powersplit devices in AWD vehicles (Piyabongkarn et al. 2010). In such systems, the
objectives of VSC can be achieved without reducing the longitudinal velocity – but
at the cost of additional hardware. Vehicle-handling performance can be influenced
by many different actuations. In addition to differential braking and traction, allwheel steering and active/semi-active suspensions (including antiroll systems) can be
used. Hac and Bodie (2002) discussed methods for improving vehicle stability and
emergency handling using electronically controlled chassis systems. Small changes
in the balance of tire forces between the front and rear axles may affect vehicle
yaw moment and stability. They discuss methods of affecting vehicle-yaw dynamics
using controllable brakes, steering, and suspension. Brake-steering techniques are
discussed in detail in Pilutti et al. (1998).
Matsumoto et al. (1992) developed a braking-force-distribution control strategy
using a steering-wheel-angle feed-forward and yaw-rate feedback design. A target
yaw rate was calculated based on steering-wheel angle and actual yaw rate, and left–
right brake-force-distribution control provided a corrective yaw moment as required.
The vehicle model had 11 DOF, and the tire model generated forces as a function of
side-slip angle, normal load, longitudinal-slip ratio, and coefficient of friction of the
road surface. Simulations showed significant reduction in yaw-rate error for braking
during a lane change. This work is one of the earliest known publications on the
concept later known as VSC or VDC.
257
258
Vehicle Stability Control
Bosch developed a VDC system (Zanten et al. 1995) that derives the desired
vehicle motion from the steering angle, accelerator-pedal position, and brake pressure. Actual vehicle motion is determined from measurement of the yaw rate and
lateral acceleration. Differences between actual and desired motion are minimized
by regulating engine torque and brake pressures using TCS components.
Many automobile manufacturers have implemented VDC systems or have plans
to make VDC systems available in future models. Inputs to the General Motors
system-control strategy are vehicle speed, steering-wheel angle, yaw rate, brake
pressure, and lateral acceleration. The system has been shown to reduce yaw-velocity
rise time, overshoot, and settling time (Hoffman and Rizzo 1998).
14.1 Introduction
The VSC system is a relatively new, active safety concept introduced to control
vehicle lateral and yaw motions in emergency conditions. The control system that
achieves this function is known in the literature as direct yaw-moment control
(Shibahata et al. 1993), VDC (Zanten et al. 1995), brake-force-distribution control
(Matsumoto et al. 1992), differential braking (Kraft and Leffler 1990), ESP (Zanten
2000), and VSC (Koibuchi et al. 1996). In the previous decade, this vehicle-control
concept has been studied by almost all major automobile manufacturers, and many
models already offer VSC (or one of the equivalent alternative names) as an option.
The VSC function may be slightly different among designs but, essentially,
it involves the following tasks. First, the desired vehicle motion is inferred from
driver action (i.e., steering angle, acceleration-pedal position, or brake pressure).
Second, the actual vehicle motion, measured by a set of sensors (i.e., yaw rate,
lateral acceleration, and vehicle speed), is compared to the desired motion. Third,
control action (i.e., wheel-brake pressures) is applied to regulate vehicle motion to
follow the desired motion under certain stability constraints (e.g., vehicle side-slip
angle and roll motion).
The motivation for VSC is understood best by examining the difficulty for a
human driver (or control system) to control vehicle lateral dynamics under extreme
conditions. Briefly, when the vehicle side-slip angle is large, the authority of the
vehicle steering angle in generating a yaw moment becomes significantly reduced due
to tire-force saturation. This fact was first illustrated by the “β-method” proposed in
Shibahata et al. (1993). This method envisioned a vehicle driving straight (i.e., rd = 0
in Eq. (4.55)). Using the bicycle model (see Eqs. (4.45) and (4.46)), we have:
Fy f + Fyr = m(v̇ + ur)
Fy f · a − Fyr · b = Iz · ṙ
(14.1)
where all the variables are as defined in Chapter 4. The lateral tire forces are functions
of tire slip angles, where the front- and rear-axle slip angles are calculated from:
v + ar
αf = δf −
u
(14.2)
v − br
αr = −
u
The tire forces then are calculated from the Magic Formula model (see Chapter 13).
The key idea of the β-method is that when the vehicle side-slip angle β ≈ uv is large,
14.1 Introduction
259
8000
Front axle
Rear axle
7000
6000
Fy (N)
5000
4000
3000
2000
1000
0
0
5
10
15
Tire slip angle (deg)
Figure 14.1. Tire lateral-force diagram (Fy versus α).
the yaw moment aFy f − bFyr and total lateral force Fy f + Fyr become almost constant,
regardless of the steering angle, δ f – that is, a driver’s authority in controlling the
vehicle lateral/yaw motion is reduced significantly. This fact is shown in the following
example.
In this MATLAB example, three plots were generated. First, the
tire lateral-force diagram (Figure 14.1) shows the relationship between tire slip
angle α and lateral force. Notice that the tire forces are almost constant when
the slip angle is larger than 5 degrees.
Second, this example program involves the calculation of vehicle control
authority. Imagine a situation in which the vehicle side-slip angle varies between
0 and 15 degrees; under zero vehicle yaw rate (r = 0), the yaw moment aFy f − bFyr
and total lateral force Fy f + Fyr can be calculated using the Magic Formula tire
model. It can be seen in Figure 14.2 that the authority of front-axle steering
is reduced greatly when the vehicle side-slip angle is large. In other words,
regardless of whether the human driver (or automatic controller) is steering the
front wheels to the left or to the right, similar amounts of lateral force and yaw
moment are generated due to tire saturation.
EXAMPLE 14.1.
Example 14.1 illustrates the problem of a large vehicle side-slip angle, which
has been the key argument for regulating it. In early designs, many VSC systems
were designed to limit vehicle side slip to be less than a larger threshold (e.g., 5
degrees). Recently, with the ability to estimate more accurately the side-slip angle,
some companies have pushed for a lower threshold value (∼3 degrees), especially
260
Vehicle Stability Control
x 10
4
1
Yaw moment (N-m)
x 10
0.5
0.5
delta = 0 deg
0
-0.5
delta = 5 deg
0
-0.5
-1
delta = -4 deg
delta = -4 deg
-1
4
delta = 5 deg
1
Total lateral force (N)
1.5
0
2
4
6
Vehicle side Slip angle (deg)
8
-1.5
0
2
4
6
Vehicle side Slip angle (deg)
8
Figure 14.2. Control authority reduction.
on slippery (i.e., icy) road surfaces. Figures 14.1 and 14.2 are produced from tire data
obtained on high-μ surfaces and they should not be used to generalize conclusions for
low-μ cases. However, reduced steering authority also is observed for low-μ cases.
Another reason that it is a good idea to limit the vehicle side-slip angle is the fact
that human drivers usually do not have much experience driving under limiting (i.e.,
emergency) situations. It is believed that they typically drive in situations when the
vehicle side-slip angle is less than 2 degrees (Zanten 2000). Therefore, drivers usually
are taken by surprise by the nonlinear behavior such as that shown in Example 14.1,
and they commonly overreact to a situation. To avoid this problem, the proper
control of vehicle dynamics so that the vehicle never enters the highly nonlinear
region is an effective preemptive measure.
% Ex14_1.m
clear
% The Magic Formula for Fz = 3225N
By = 0.27; Cy = 1.2; Dy = 2921;
Ey = -1.6; Shy = 0; Svy = 0;
% Vehicle parameters
a = 1.14; L = 2.54;
b = L - a;
m = 1500; Iz = 2420.0; g = 9.81;
Fz_f = m*g*b/L; Fz_r = m*g*a/L;
Dy_f = Dy*Fz_f/3225; Dy_r = Dy*Fz_r/3225;
for i=1:101,
alpha(i) = (i-1)*0.14;
% slip angle 0 to 14 deg
phi_y = (1-Ey)*(alpha(i)+Shy) + Ey/By* ...
atan(By*(alpha(i)+Shy));
fy_f(i) = Dy_f*sin(Cy*atan(By*phi_y))+Svy;
fy_r(i) = Dy_r*sin(Cy*atan(By*phi_y))+Svy;
end
plot(alpha, fy_f, ‘-r’, alpha, fy_r,‘-.b’)
xlabel(‘Tire slip angle (deg)’); ylabel(‘Fy (N)’)
14.2 Linear Vehicle Model
261
legend(‘Front axle’, ‘Rear axle’); pause
% Part II, calculate Yaw moment & lateral force
r_over_u = 0;
% yaw rate = 0
for i = 1:10,
delta = i-5;
% steering angle -4 to 5 deg
for j = 1:9,
beta(j) = j-1;
% side slip angle 0 to 8 deg
alpha_f = delta - (beta(j) + 180/pi*a*r_over_u);
alpha_r = - (beta(j) - 180/pi*b*r_over_u);
phi_yf = (1-Ey)*(alpha_f+Shy) + Ey/By ...
*atan(By*(alpha_f+Shy));
Fyf
= Dy_f*sin(Cy*atan(By*phi_yf))+Svy;
phi_yr = (1-Ey)*(alpha_r+Shy) + Ey/By ...
*atan(By*(alpha_r+Shy));
Fyr
= Dy_r*sin(Cy*atan(By*phi_yr))+Svy;
yaw_moment(j, i)
= a * Fyf - b * Fyr;
lateral_force(j,i) = Fyf + Fyr;
end
end
plot(beta, yaw_moment)
xlabel(‘Vehicle side Slip angle (deg)’)
ylabel(‘Yaw moment (N-m)’); pause
plot(beta, lateral_force)
xlabel(‘Vehicle side Slip angle (deg)’)
ylabel(‘Total lateral force (N)’)
14.2 Linear Vehicle Model
We first derive a vehicle model that is suitable for VSC simulations. Apparently, if
we must simulate a vehicle roll motion (e.g., to study the effect of rollover reduction
by the VSC system), the minimum number of vehicle DOFs needed is four (i.e.,
lateral, yaw, roll, and longitudinal). This model can be built easily on the basis of
the three-DOF lateral/yaw/roll model introduced in Chapter 4 and described in
Appendix B. Because the longitudinal dynamics of the vehicle is much slower than
the yaw/roll dynamics, the simulation model does not need to be nonlinear. Rather,
the yaw/roll model can be updated at each time step, assuming that the vehicle
forward speed is constant. This separation of dynamics makes it possible to use the
linear simulation command lsim() in MATLAB to perform the VSC simulation.
Another factor to consider is actuator dynamics. We can assume that the control
signal is individual brake pressure; however, this obviously adds complexity to the
simulations. Assuming that the control signal is the yaw moment MVSC , the vehicle
lateral/yaw/roll model is then:
Yβ β + Yr r + Yφ φ + Yδ δ f = m(uβ̇ + ur) + mR h ṗ
Nβ β + Nr r + Nφ φ + Nδ δ f + MV SC = Iz ṙ + Ixz ṗ
Lφ φ + L p p = mR h(uβ̇ + ur) + Ix ṗ + Ixz ṙ
(14.3)
262
Vehicle Stability Control
and the longitudinal dynamics model is:
4
4
4
4M
− 44 V SC 44 = mu̇
T/2
(14.4)
where the vehicle-stability derivatives are defined as:
bCαr − aCα f
Yβ ≡ −(Cα f + Cαr )
Yr ≡
∂γ f
∂δ
Yφ ≡ Cαr r + Cγ f
∂φ
∂φ
Yδ ≡ Cα f
Nβ ≡ bCαr − aCα f
Nφ ≡ aCγ f
∂γ f
∂δ
− bCαr r
∂φ
∂φ
Nr ≡ −
u
a2Cα f + b2Cαr
(14.5)
u
Nδ ≡ aCα f
The control signal MV SC is assumed to be generated by an electronic braking system; thus, it always slows down the vehicle regardless of the sign of MV SC . The
lateral/yaw/roll part of the vehicle dynamics (Eq. (14.3)) is not in the standard statespace form and therefore must be modified slightly. If we treat the front-wheel
steering angle as a disturbance input and MV SC as the control signal, Eq. (14.3) can
be rearranged to:
⎡
⎤⎡ ⎤ ⎡
⎤
−Yβ mu − Yr
β̇
0
−Yφ ⎡ β ⎤
mu
0
mR h 0
⎢
⎥⎢ ⎥ ⎢
⎥
⎢ 0
Iz
Ixz
0 ⎥ ⎢ ṙ ⎥ ⎢ −Nβ
−Nr
0
−Nφ ⎥ ⎢ r ⎥
⎢
⎥⎢ ⎥ + ⎢
⎥⎢ ⎥
⎢ m hu I
⎥ ⎢ ṗ ⎥ ⎢ 0
⎥⎣ p⎦
I
0
m
hu
−L
−L
⎣
⎣ R
⎦⎣ ⎦
xz
x
R
p
φ⎦
φ
0
0
0
1
φ̇
0
0
−1
0
⎡ ⎤
⎡ ⎤
Yδ
0
⎢ ⎥
⎢ ⎥
⎢ Nδ ⎥
⎢1⎥
⎢ ⎥
⎥
=⎢
⎢ 0 ⎥ MV SC + ⎢ 0 ⎥ δ f
⎣ ⎦
⎣ ⎦
0
0
≡ E ẋ + F x = GMV SC + Hδ f
(14.6)
or:
ẋ = Ax + Bu MV SC + Bw δ f
(14.7)
where A = −E −1 F, Bu = E −1 G, Bw = E −1 H.
In this example, the vehicle model described in Eqs. (14.4) and
(14.6) is simulated. We can imagine a simple VSC algorithm: MV SC = Kx =
[ kβ 0 0 −kφ ]x, where the state vector x is defined in Eq. (14.6) and the two
control gains (i.e., kβ and kφ ) are both assumed to be positive. The previously
mentioned control law is a simple proportional control law: A positive VSC yaw
moment is requested when the vehicle side-slip angle (β) is positive or when the
vehicle roll angle (φ) is negative.
Figure 14.3 demonstrates that when road friction is high and linear tire
models are used, the need for VSC is not obvious. Notice that the simulation represents a severe “fishhook” maneuver, with the yaw rate as high as
EXAMPLE 14.2.
14.3 Nonlinear Vehicle Model
263
20
Yaw rate (deg/sec)
Side slip angle (deg)
2
1
0
-1
-2
0
2
4
Time (sec)
-40
Front steering angle (deg)
Roll angle (deg)
5
0
0
2
4
Time (sec)
6
-5
10.5
0.5
9.5
9
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
1
10
0
5
11
M V SC (N-m)
vehicle speed (m/sec)
-20
6
10
-5
0
0
-0.5
0
2
4
Time (sec)
6
-1
Figure 14.3. Linear vehicle response (VSC off).
20 degrees/second However, in this case, the vehicle slip angle is quite small;
thus, there is no apparent need for VSC. This example demonstrates the need for
a nonlinear simulation model in the simulation and validation of VSC designs.
14.3 Nonlinear Vehicle Model
Before we present the nonlinear vehicle model, it is important to point out that a
combined-slip tire model must be used because of the possibility of simultaneous
occurrence of high tire slip angle and large slip ratios. For combined-slip tire models,
it is necessary to keep track of both tire slip angle and slip ratio to calculate tire forces
(both lateral and longitudinal). To make it easier to perform the simulations, we
assume that the slip ratios of the four tires are directly controllable or, equivalently,
the wheel speeds are regulated by servo-loop controllers. This assumption makes it
possible to simulate the vehicle response using a five-state model, which is similar
to but somewhat more complicated than the one described in Section 14.2. This
assumption is reasonable because the wheel dynamics is significantly faster than
the vehicle dynamics. A good servo-loop, therefore, can regulate wheel-slip ratios
around desired values fast enough. There is an added benefit of the assumption
mentioned here: It is consistent with the evolution of VSC hardware. Most VSC
systems were developed to reside on top of ABS/TCS; thus, setting the desired
wheel-slip ratio is possible if the underlying ABS/TCS is known to work well in
regulating wheel slips.
264
Vehicle Stability Control
In the following discussion, we assume that the Magic Formula tire model with
combined-slip correction is used to compute simultaneously the lateral and longitudinal tire forces. The calculated tire forces then become the inputs to the vehicledynamic equations. When the inputs to the vehicle yaw/roll model are lateral forces
at the front and rear axles, the dynamic equations become:
Fy f + Fyr = m(ur + v̇) + mR h ṗ
Fy f · a − Fyr · b = Iz ṙ + Ixz ṗ + MV SC
Lφ φ + L p p = mR h(v̇ + ur) + Ix ṗ + Ixz ṙ
or, in the state-space form:
⎤⎡ ⎤ ⎡
⎡
0
v̇
m
0
mR h 0
⎥
⎢
⎥
⎢
⎢ 0
0
ṙ
I
I
0
z
xz
⎥⎢ ⎥ + ⎢
⎢
⎣m h I
I
0 ⎦ ⎣ ṗ ⎦ ⎣ 0
R
xz
x
φ̇
0
0
0
1
0
⎡ ⎤
⎡
⎤
0
1 1 ⎢1⎥
⎢ a −b ⎥ Fy f
⎢
⎥
⎥
=⎢
⎣ 0 ⎦ MV SC + ⎣ 0 0 ⎦ F
yr
0 0
0
Fy f
≡ E ẋ + F x = GMV SC + H
Fyr
mu
0
mR hu
0
0
0
−L p
−1
(14.8)
⎤⎡ ⎤
0
v
⎢r⎥
0 ⎥
⎥⎢ ⎥
−Lφ ⎦ ⎣ p ⎦
φ
0
(14.9)
or:
ẋ = Ax + Bu MV SC + Bw
Fy f
Fyr
(14.10)
where A = −E −1 F, Bu = E −1 G, Bw = E −1 H. Eq. (14.9) illustrates that there are
three inputs to the model: lateral forces at the front and rear axles and the yaw
moment generated from the longitudinal tire forces. From a servo-loop perspective,
there are five inputs: the front-axle steering angle (i.e., disturbance) and the slip ratios
at the four tires. Given the steering angle and slip ratios, we can calculate all four
tire slip angles and thereby compute the tire forces. Once lateral and longitudinal
tire forces are known, the three inputs to the vehicle dynamics (i.e., MV SC , Fy f , and
Fyr ) are calculated.
In this example, the vehicle model described in Eqs. (14.4) and
(14.9) is simulated, in which the tire forces are calculated from the combinedslip Magic Formula tire model. A simple VSC algorithm again is used in this
example – a proportional control law. The control action is turned on only if the
magnitude of the vehicle side-slip angle is higher than a certain threshold. Figure
14.4 illustrates the case in which the VSC is not turned on. It shows that because
of the nonlinear tire model used in this simulation, a very high vehicle slip angle
is generated, whereas the vehicle yaw rate is much lower than that from the
linear vehicle model. The vehicle gradually loses stability, with a slow increase
of yaw rate and continuously increasing side-slip angle. This phenomenon is
similar to what was observed in actual fishhook testing. When a simple proportional algorithm is applied, the vehicle side-slip angle is significantly reduced
(Figure 14.5).
EXAMPLE 14.3.
14.3 Nonlinear Vehicle Model
Yaw rate (deg/sec)
10
0
-10
0
2
4
Time (sec)
6
0
vehicle speed (m/sec)
0
2
4
-20
-40
6
2
4
Time (sec)
2
4
Time (sec)
6
0
2
6
0
-5
0
-1
6
4
Time (sec)
1
15
0
0
5
Time (sec)
16
14
0
MVSC (N-m)
Roll angle (deg)
5
-5
20
Front steering angle (deg)
Side slip angle (deg)
20
265
0
2
4
Time (sec)
6
Yaw rate (deg/sec)
5
0
-5
0
2
4
Time (sec)
6
Roll angle (deg)
5
0
0
2
4
Time (sec)
6
15.2
15
14.8
14.6
0
20
0
-20
-40
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
5
0
-5
600
MVSC (N-m)
vehicle speed (m/sec)
-5
Front steering angle (deg)
Side slip angle (deg)
Figure 14.4. Nonlinear vehicle response (VSC off).
2
4
Time (sec)
6
400
200
0
Figure 14.5. Nonlinear vehicle response (VSC on, front axle).
266
Vehicle Stability Control
14.4 VSC Design Principles
In the previous section, we successfully develop a nonlinear vehicle model that is
suitable for VSC studies. In this section, we discuss the design of VSC in more
detail. At the most fundamental level, VSC aims to improve vehicle yaw and
sometimes roll stability. Generally speaking, the design principle of the VSC control algorithm can be divided into three parts: (1) estimation of desired vehicle
(i.e., yaw) motion and tracking; (2) vehicle side-slip estimation and regulation; and
(3) (optional) rollover/wheel-liftoff prevention.
Yaw-Rate Following
The desired yaw motion of a vehicle usually is calculated from the human steeringwheel angle and vehicle states, the most important of which is the forward speed. In
some cases, a human driver might steer suddenly to perform an obstacle-avoidance
maneuver. In this situation, it is important to use differential braking to achieve
the driver’s “desired” yaw rate as quickly as possible. When the vehicle side slip
increases, the controller then must examine the situation and achieve a good balance
between yaw-rate following and limiting side-slip angle.
So, how can we determine a driver’s desired yaw motion? Usually, the steadystate yaw rate corresponding to the current driver steering input is assumed to be a
good approximation of what the driver wants to accomplish. If a linear (e.g., bicycle)
model is used, this steady-state yaw rate can be computed from either the state-space
equation or the transfer function from steering to yaw rate. Commonly, the bicycle
model is used to figure out the desired yaw rate. Because the transfer function from
the front steering angle to yaw rate is:
u
r
u
= R =
Gr =
δf
δf
L + Kus u2
(14.11)
where u is vehicle forward speed (m/sec), L is the wheel base (m), and Kus ≡
mb
LCα f
−
ma
LCαr
is the vehicle understeer coefficient rad/(m/ sec2 ). It is common to measure the
steering angle and then multiply by this speed-dependent yaw-rate gain to determine
the desired vehicle yaw rate. A simple control algorithm then can be used to reduce
the difference between the desired and the actual vehicle yaw rates. Frequently,
dead-band thresholds are implemented; for example:
rd = ψ̇d = Gr δ f
4
4
4ψ̇ − ψ̇ 4
4
4
d
≥ ψ̇thresold
If 4ψ̇ − ψ̇d 4 ≥ ψ̇thresold and
ψ̇d
Then MV SC
Else MV SC
yaw
yaw
= Kyaw (ψ̇d − ψ̇ )
=0
(14.12)
perct
(14.13)
14.4 VSC Design Principles
267
where two thresholds (i.e., absolute and percentage) are used and MV SC
VSC control torque due to the yaw-rate-following consideration.
yaw
is the
Slip-Angle Estimation and Regulation
Yaw-rate following is an important consideration for vehicle-handling purposes.
However, it is equally if not more important to consider vehicle stability. In other
words, the vehicle slip angle and/or roll angle should not become excessively large.
The accurate estimation of the vehicle side-slip angle is not a trivial task because of
the lack of an affordable sensor for vehicle lateral-speed measurement. In an actual
implementation, the vehicle side-slip angle usually is estimated from a bicycle model
with the available measurements: steering angle, yaw rate, lateral acceleration, and
forward speed (Üngören et al. 2002). If we assume that the vehicle side-slip angle
can be either measured or estimated, a simple control law again can be used. A
common algorithm uses a “sliding-surface” concept, which for this simple problem
is equivalent to PD control:
and
If |β| ≥ βthreshold
Then MV SC
Else MV SC
β
β
β · β > 0
= Kβ p · β + Kβd · β
(14.14)
=0
where the constraint β · β > 0 ensures that the control law is turned on only when
the magnitude of the side-slip angle is increasing. MV SC β is the VSC control torque
for slip-angle regulation.
Rollover or Wheel Liftoff Prevention
In addition to the yaw and handling performance considerations discussed previously, recent attention has turned to the possibility of using a VSC system for
rollover-prevention purposes. This function can be achieved by modifying the
desired yaw-rate calculation shown in Eqs. (14.11) and (14.12). For example:
ψ̇d = sat(Gr δ f , ψ̇d
threshold )
(14.15)
where sat() is the saturation function and its magnitude limit can be computed
according to the vehicle static stability factor (SSF). In other words, the desired yaw
rate can be limited so that the corresponding lateral acceleration never exceeds a
certain level that would cause rollover concerns. However, the idea shown herein is
essentially “static-model–based” and therefore is vulnerable to external disturbances
not considered in the model (e.g., a large road-bank angle). A feedback-based algorithm that uses vehicle response (i.e., roll angle, load transfer, or lateral acceleration)
is more robust and could be implemented without or in addition to Eq. (14.15). An
example is shown here:
If |φ| ≥ φthreshold
Then MV SC
Else MV SC
φ
φ
= −Kay · ay
=0
(14.16)
268
Vehicle Stability Control
Table 14.1. Parameter values used in Example 14.4 simulations
Variable
Value
Unit
ψ̇threshold
ψ̇threshold
Kus
Kyaw
βthreshold
Kβ p
Kβd
φthreshold
Kay
0.5
2
0.0 (assumed)
0.2 (or 0 when off)
3
2.5 (or 0 when off)
0.1 (or 0 when off)
4
0.0 (off)
deg/sec
%
rad/(m/sec2 )
slip per rad/sec
deg
slip per rad
slip per rad
deg
slip per (m/sec2 )
perct
where ay is the vehicle lateral acceleration and MV SC φ is the VSC control torque for
rollover prevention.
The overall VSC action then can be “combined.” A simple way to combine these
three separate control actions is to total them:
MV SC = MV SC
yaw
+ MV SC
β
+ MV SC
φ
(14.17)
which, of course, is not necessarily the best way to accomplish this. An alternative,
for example, is to emphasize stability when the slip angle or roll angle is high. Only
when there are no yaw/roll stability concerns is the yaw rate applied following the
control action. This logic can be constructed easily and is an exercise for readers. In
the following example, Eq. (14.17) is used as the combined VSC algorithm, which
clearly shows the tradeoff among (sometimes) conflicting goals.
The vehicle model used in this example is identical to that shown
in Example 14.3. The VSC algorithm, however, is based on Eqs. (14.13) through
(14.17). There is another slight change: Rather than calculating desired yawcontrol moments (i.e., MV SC = MV SC yaw + MV SC β + MV SC φ ), desired wheelslip ratios (i.e., sV SC = sV SC yaw + sV SC β + sV SC φ ) are obtained. Depending on
the sign of the final control action (i.e., sV SC = sV SC yaw + sV SC β + sV SC φ ), either
the right- or left-front tire is actuated. This modification is necessary because tire
forces are calculated from the combined-slip tire model. The tradeoff between
yaw-rate following and slip-angle regulation is demonstrated in this example.
Because the vehicle parameters used in this example correspond to a passenger
vehicle, rollover is not a concern; therefore, rollover-prevention control is turned
off to make it easier to see the tradeoff between the remaining two objectives.
For an SUV or light truck, the control logic must be more elaborate. When
all three objectives are considered, careful examination of the behavior and
timing of each control action is needed to ensure that they do not undermine
one another.
The control parameters in the simulations are summarized in Table 14.1.
All of the control gains have units for “desired slip” rather than “desired yaw
moment,” as explained previously. Four sets of data are shown here: no control,
yaw only, slip angle, and yaw/slip.
EXAMPLE 14.4.
14.4 VSC Design Principles
269
0
0
2
4
Time (sec)
16
6
0
2
4
Time (sec)
-5
2
4
Time (sec)
0
2
4
Time (sec)
0
5
6
2
4
Time (sec)
6
0
6
0
-1
6
-20
6
-5
0
Yaw rate (deg/sec)
Side slip angle (deg)
2
4
Time (sec)
0
1
15
14
0
5
MVSC (N-m)
vehicle speed (m/sec)
-5
-50
0
0
16
2
4
Time (sec)
6
-50
0
2
4
Time (sec)
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
5
0
-5
1
15
14
0
Front steering angle (deg)
Roll angle (deg)
5
6
20
50
MVSC (N-m)
2
4
Time (sec)
0
Roll angle (deg)
0
40
vehicle speed (m/sec)
0
-5
50
Yaw rate (deg/sec)
5
Front steering angle (deg)
Side slip angle (deg)
No control:
0
-1
6
Roll angle (deg)
0
-5
0
2
4
Time (sec)
16
6
12
2
4
Time (sec)
6
-5
0
2
4
Time (sec)
0
2
4
Time (sec)
6
-2000
0
2
4
Time (sec)
6
2
4
Time (sec)
6
0
-5
0
6
-1000
0
-20
0
5
0
Yaw rate (deg/sec)
0
5
1000
14
10
-50
MVSC (N-m)
vehicle speed (m/sec)
6
0
15
2
4
Time (sec)
6
0
-50
14
2
4
Time (sec)
5
0
-5
2000
14.5
13.5
0
50
Front steering angle (deg)
2
4
Time (sec)
20
MVSC (N-m)
0
5
Side slip angle (deg)
-5
0
40
Roll angle (deg)
0
50
vehicle speed (m/sec)
Yaw rate (deg/sec)
5
Front steering angle (deg)
Side slip angle (deg)
Yaw only:
0
-2000
6
0
-5
vehicle speed (m/sec)
6
0
15
2
4
Time (sec)
6
0
2
4
Time (sec)
5
6
2
4
Time (sec)
2
4
Time (sec)
0
2
4
Time (sec)
6
6
2
4
Time (sec)
6
0
-5
0
0
5
0
-2000
-5
6
0
-5
Yaw rate (deg/sec)
Side slip angle (deg)
0
2000
14.5
14
-50
0
0
15
2
4
Time (sec)
6
0
2
4
Time (sec)
0
-50
5
0
-5
2000
14.5
14
50
Front steering angle (deg)
Roll angle (deg)
5
2
4
Time (sec)
5
MVSC (N-m)
0
Front steering angle (deg)
-5
0
Roll angle (deg)
0
50
vehicle speed (m/sec)
Yaw rate (deg/sec)
5
MVSC (N-m)
Side slip angle (deg)
Slip only:
6
0
-2000
270
Vehicle Stability Control
0
16
2
4
Time (sec)
6
12
2
4
Time (sec)
6
0
2
4
Time (sec)
2
4
Time (sec)
6
0
-2000
0
0
2
4
Time (sec)
6
2
4
Time (sec)
6
0
-5
6
-1000
0
-5
5
0
-5
Yaw rate (deg/sec)
0
5
1000
14
10
-50
0
0
15
2
4
Time (sec)
6
13
0
2
4
Time (sec)
0
-50
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
0
2
4
Time (sec)
6
5
0
-5
2000
14
12
50
Front steering angle (deg)
0
-5
vehicle speed (m/sec)
6
MVSC (N-m)
Roll angle (deg)
5
2
4
Time (sec)
5
MVSC (N-m)
0
Side slip angle (deg)
-5
0
Roll angle (deg)
0
50
vehicle speed (m/sec)
Yaw rate (deg/sec)
5
Front steering angle (deg)
Side slip angle (deg)
Yaw/Slip:
6
0
-2000
PROBLEMS
1. Use the parameter values given in Example 14.1 and change the Magic Formula
parameters By , Dy , and Ey by ± 5 percent from their nominal values to show how
these parameters affect the calculated forces. Plot the results.
2. Based on results obtained from Example 14.1, explain why one of the objectives
of VSC is to maintain a small side-slip angle, β.
3. Equation 14.3 describes the effect of control yaw-moment MVSC on the threeDOF vehicle lateral/yaw/roll model. Show the effect of the control yaw-moment
MVSC on the two-DOF vehicle lateral/yaw model developed in Equation 4.49 (see
Chapter 4).
4. Repeat the simulation in Example 14.2 with non-zero values of the control gains
kb and kφ . Initially select very small gains and then gradually increase them to show
the effect of including the control.
5. Using the nonlinear model (see Example 14.3) and comparing your results to
Figures 14.4 and 14.5, determine (by trial and error) appropriate control gains kb
and kφ . The control action is turned on only if the magnitude of the vehicle side-slip
angle is higher than a certain threshold; determine an appropriate threshold.
6. Repeat the simulation in Example 14.4 but also include – based on the simple
strategy in Eq. (14.17) – the control action for preventing rollover.
7. Using simulations based on Example 14.4, propose and demonstrate an alternative
to Eq. (14.17) for combining the various elements of a VSC logic.
REFERENCES
Allen, R. W., D. H. Klyde, T. J. Rosenthal and D. M. Smith 1988, “Analytical Modeling
of Driver Response,” in Crash Avoidance Maneuvering, Vol. I: Technical Background,
DOT-HS-807-270, April.
References
Hac, A., and M. O. Bodie, 2002, “Improvements in Vehicle Handling Through Integrated
Control of Chassis Systems,” International Journal of Vehicle Design, Vol. 29, No. 1–2,
pp. 23–50.
Hoffman, D. D., and M. D. Rizzo, 1998, “Chevrolet C5 Corvette Vehicle Dynamic Control
System,” SAE Paper No. 980233, Warrendale, PA: Society of Automotive Engineers.
Koibuchi, K., M. Yamamoto, Y. Fukada, and S. Inagaki, 1996, “Vehicle Stability Control in
Limit Cornering By Active Brake,” SAE Paper No. 960487, SAE Congress.
Kraft, H. J., and H. Leffler, 1990, “The Integrated Brake and Stability Control System of the
New BMW 850i,” SAE Paper No. 900209, SAE Congress.
Lie, Anders, C. Tingvall, M. Krafft, and A. Kullgren, 2006, “The Effectiveness of Electronic
Stability Control (ESC) in Reducing Real-Life Crashes and Injuries,” Traffic Injury Prevention, Vol. 7, No. 1, pp. 38–43.
Ma, W-H., 1998, Worst-Case Evaluation Methods for Vehicles and Vehicle Control Systems,
Ph.D. Thesis (Mechanical Engineering), University of Michigan.
Matsumoto, S., H. Yamaguchi, H. Inoue, and Y. Yasuno, 1992, “Improvement of Vehicle
Dynamics Through Braking Force Distribution Control,” SAE Paper No. 920645, SAE
Congress.
Mills, V., B. Samuels, and J. Wagner, 2002, “Modeling and Analysis of Automotive Antilock
Brake Systems Subject to Vehicle Payload Shifting,” Vehicle System Dynamics, Vol. 37,
No. 4, April, pp. 283–310.
Nagai, M., M. Shino, and F. Gao, 2002, “Study on Integrated Control of Active Front Steer
Angle and Direct Yaw Moment,” JSAE Review, Vol. 23, No. 3, July, pp. 309–15.
Pilutti, T., A. G. Ulsoy, and D. Hrovat, 1998, “Vehicle Steering Intervention Through Differential Braking,” ASME Journal of Dynamic Systems, Measurement and Control, Vol. 120,
No. 3, September, pp. 314–21.
Piyabongkarn, D., J. Y. Lew, R. Rajamani, and J. A. Grogg, 2010, “Active Driveline Torque
Management Systems,” IEEE Control Systems Magazine, Vol. 30, No. 4, August, pp. 86–
102.
Shibahata, Y., K. Shimada, and T. Tomari, 1993, “Improvement of Vehicle Maneuverability
by Direct Yaw Moment Control,” Vehicle System Dynamics, Vol. 22, pp. 465–81.
Tseng, H. E., B. Ashrafi, D. Madau, T. A. Brown, and D. Recker, 1999, “The Development
of Vehicle Stability Control at Ford,” IEEE/ASME Transactions on Mechatronics, Vol. 4,
No. 3, September, pp. 223–34.
Üngören, A.Y., H. Peng, and H. Tseng, 2002, “Experimental Verification of Lateral Speed
Estimation Methods,” Proceedings of the Advanced Vehicle Control Conference, Hiroshima,
Japan, September.
Zanten, A. V., R. Erhardt, and G. Pfaff, 1995, “VDC, the Vehicle Dynamics Control System
of Bosch,” SAE Paper No. 950759, SAE Congress.
Zanten, A. V., 2000, “Bosch ESP Systems: 5 Years of Experience,” SAE Paper No. 2000-011633, SAE Congress.
271
15
Four-Wheel Steering
15.1 Basic Properties
In Chapter 4, we review the lateral dynamics of a front-wheel steering (FWS) vehicle.
Following the same procedure, the dynamic equations for a four-wheel-steering
(4WS) “bicycle” vehicle can be derived. The state-space model of the 4WS vehicle
model is as follows:
⎡
⎤
⎡ C
⎤
−(Cα f + Cαr ) bCαr − aCα f
Cαr
αf
−
u
o⎥
⎢
v̇
⎢ m
muo
muo
δf
m ⎥
⎥ v
⎢
⎢
⎥
+ ⎣ aC
=⎢
2
2 ⎥
⎦
−bC
r
bC
−
aC
−(C
a
+
C
b
)
δr
⎦
⎣
α
f
αr
αr
αf
αf
αr
ṙ
Iz
Iz
Iz uo
Iz uo
(15.1)
If we include the lateral displacement and the yaw angle as the two extra state
variables, then time-domain performance of 4WS vehicles (e.g., lane following) also
can be simulated. The fourth-order state-space model is obtained by adding two
trivial dynamic equations, ẏ = v + uoψ and ψ̇ = r, to the two-state model:
⎡
⎤
⎡
⎤
0
1
uo
0
0
0
⎢
⎥
⎡ ⎤ ⎢C
⎡ ⎤
bCαr − aCα f
⎢ −(Cα f + Cαr )
⎥
Cαr ⎥
y
⎢ αf
⎥ 0
− uo ⎥ y
⎢0
⎢
⎥
⎥⎢ v ⎥ ⎢ m
⎥ ⎢
muo
muo
d ⎢
m ⎥ δf
⎢v⎥ = ⎢
⎥⎢ ⎥ + ⎢
⎥
⎥ ⎣ψ ⎦ ⎢ 0
0 ⎥ δr
dt ⎣ψ ⎦ ⎢
0
0
1
⎢0
⎥
⎢
⎥
⎢
⎥
⎣ aCα f −bCαr ⎦
r
bCαr − aCα f
−(Cα f a2 + Cαr b2 ) ⎦ r
⎣
0
0
Iz
Iz
Iz uo
Iz uo
(15.2)
Before introducing the 4WS control strategy, we examine basic characteristics
of FWS and rear-wheel steering (RWS) vehicles, which helps us to understand what
we gain by steering the rear wheels. In the following discussion, we first prove by
using Eq. (15.1) that the rear wheels are as effective as the front wheels in terms
of yaw-rate generation. This fact then is verified using the MATLAB program in
Example 15.1. At steady-state, we obtain the following equations from Eq. (15.1):
0=−
0=
272
Cα f + Cαr
uo
bCαr − aCα f
uo
v+
v−
bCαr − aCα f − mu2o
uo
a2Cα f + b2Cαr
uo
r + Cα f δ f + Cαr δr
r + aCα f δ f − bCαr δr
15.1 Basic Properties
273
Eliminating the terms containing the lateral speed, v, we obtain:
0=
−(bCαr − aCα f )mu2o − Cα f Cαr (a + b)2
uo
r + (a + b)Cα f Cαr δ f − (a + b)Cα f Cαr δr
which shows that steering angles of the same magnitude but opposite sign at the front
and rear axles generate the same yaw-rate response (at steady-state). However, the
following sketch clearly shows that the transient-force generation at the rear axle is in
the opposite direction of steady-state force. In other words, there is a non-minimumphase (NMP) response in the vehicle lateral acceleration for a RWS vehicle, which is
not the case for a FWS vehicle. The NMP characteristic makes it difficult for human
drivers to control a RWS vehicle at high vehicle speed, which is a primary reason
why they are not as popular as FWS vehicles.
force
acceleration
acceleration
force
force
Initial
steady-state
transient
EXAMPLE 15.1: COMPARISON OF FWS AND RWS VEHICLES. The frequency responses
of the FWS and RWS cases show that the steady-state yaw responses are indeed
identical, shifted only by 180 degrees in phase. The acceleration responses, however, are very different. The 180-degree phase change of the lateral-acceleration
response for a RWS vehicle also shows that it is NMP.
10
Yaw rate gain
2
Y aw rate phas e (deg)
0
FWS
RWS
FWS
RWS
-20
-40
10
1
-60
-80
0
10
-1
10
10
0
10
1
10
2
-100
-1
10
10
Freq (rad/sec)
10
10
10
A c c phase (deg)
50
FWS
RWS
0
-50
10
1
Freq (rad/sec)
A cc gain
3
0
FWS
RWS
2
-100
-150
1
10
-1
10
10
0
Freq (rad/sec)
10
1
10
2
-200
-1
10
10
0
Freq (rad/sec )
10
1
10
2
2
274
Four-Wheel Steering
%Ex15_1.m
a=1.14; l=2.54; b=l-a;
g=9.81; u0=30.0;
m=1000; Iz=1400.0;
Caf=2400.0*57.2958; Car=2000*57.2958;
A=[ 0,
1,
u0,
0;
0, -(Caf+Car)/(m*u0), 0, (b*Car-a*Caf)/(m*u0)-u0;
0,
0,
0,
1;
0, (b*Car-a*Caf)/(Iz*u0), 0, -(a*a*Caf+b*b*Car)/(Iz*u0)];
B_2ws=[0; Caf/m; 0; a*Caf/Iz];
B_rws=[0; Car/m; 0; -b*Car/Iz];
C_r=[0,0,0,1];
C_acc=A(2,:)+u0*[0,0,0,1];
% Bode plots
w=logspace(-1,2,50);
[m_2ws_r, p_2ws_r]=bode(A,B_2ws,C_r,0,1,w);
[m_rws_r, p_rws_r]=bode(A,B_rws,C_r,0,1,w);
[m_2ws_acc, p_2ws_acc]=bode(A,B_2ws,C_acc,B_2ws(2),1,w);
[m_rws_acc, p_rws_acc]=bode(A,B_rws,C_acc,B_rws(2),1,w);
loglog(w,m_2ws_r, w, m_rws_r,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Yaw rate gain’);
legend(‘FWS’,‘RWS’); pause
% reverse phase of RWS vehicle for comparison
semilogx(w,p_2ws_r, w, p_rws_r-180,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Yaw rate phase (deg)’);
legend(‘FWS’,‘RWS’); pause
loglog(w,m_2ws_acc, w, m_rws_acc,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Acc gain’);
legend(‘FWS’,‘RWS’); pause
semilogx(w,p_2ws_acc, w, p_rws_acc-180,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Acc phase (deg)’);
legend(‘FWS’,‘RWS’);
15.2 Goals of 4WS Algorithms
When the average operating speed of motor vehicles is not very high, the need
for 4WS is not obvious. When higher performance at higher speeds is needed, the
possibility of utilizing rear steering is explored.
The 4WS control strategies used in the past are mostly feed-forward in nature.
In other words, the rear wheels usually are steered as a fixed function of the frontsteering angle, regardless of vehicle yaw rate and lateral-acceleration response. The
4WS strategies were designed to achieve one of several of the following desirable
characteristics:
r
r
r
r
r
reduced vehicle side-slip angle
reduced yaw rate and lateral-acceleration phase difference
reduced low-speed turning radius
consistent steering response (model matching)
increased tire-force reserve (especially on a low-friction surface)
15.2 Goals of 4WS Algorithms
275
The most common goal of 4WS systems proposed in the past was to reduce the
steady-state side-slip angle, β (or, equivalently, lateral speed v) to zero by adjusting
the ratio of the RWS angle to the FWS angle. This has both ergonomic and dynamic
aspects that must be considered. In terms of ergonomics, the vehicle becomes easier
to steer because it proceeds in the forward direction without any side slip. In terms
of dynamics, less energy is required when a vehicle enters a turn from a straight-line
course because with 4WS, it is necessary to turn only the tires, which have low inertia.
In two-wheel steering, in which a side-slip angle develops, the vehicle must be turned
and it has a high inertia. If we set v = 0 in Eq. (15.1), eliminate r from the resulting
equations, and solve for the ratio of the RWS angle to the FWS angle, we obtain the
following condition for zero side-slip angle at steady-state:
Kr =
δr
mu2oa − b(a + b)Cαr Cα f
=
·
δf
mu2ob + a(a + b)Cα f Cαr
(15.3)
Clearly, the steering ratio Kr to achieve the zero side-slip angle is a function of
longitudinal speed (Figure 15.1a). Equation (15.3) shows that at high speeds, the rear
wheel should be turned in the same direction but somewhat less than the front wheel.
At low speeds, the rear wheel should be turned in the direction opposite to that of
the front wheel. Steering the wheels according to Eq. (15.3) achieves good stability
in the intermediate to high speed ranges. However, it does not always provide a
favorable steering sensation for the driver because it can produce a strong understeer
characteristic (from the driver’s perspective), which causes the yaw response to
deteriorate. Steering the rear wheels in the opposite direction (i.e., phase) from the
front wheels at low speed improves vehicle maneuverability. From this basic steadystate analysis, we observe the desirability of steering the rear wheels in proportion
to the front wheels as a function of vehicle forward velocity. Development of an
appropriate control strategy, however, is not as straightforward. Some authors have
shown disadvantages of 4WS (Nalecz and Bidermann 1988) and argued against
the desirability of a zero side-slip angle as the basis for 4WS control (Abe 1990).
Examples 15.2 and 15.3 show simulation results for various 4WS control strategies.
If the fixed-gain 4WS algorithm
described in Eq. (15.3) is applied, we can see that the tire slip angle at both
front and rear axles increases compared to 2WS vehicles in the transient (i.e., at
steady-state, they are the same). In other words, direct implementation of the
zero-vehicle-slip algorithm may cause undesirable side effects.
EXAMPLE 15.2: TIRE SLIP-ANGLE COMPARISON.
1.6
1
4WS
2WS
4WS
2WS
alpha (deg)
0.8
1.2
0.6
r
f
alpha (deg)
1.4
1
0.8
0.4
0.2
0.6
0
0
0.5
1
tim e (s ec)
1.5
2
0
0.5
1
time (s ec )
1.5
2
276
Four-Wheel Steering
0.15
7
4WS
2WS
6
0.05
Yaw rate (deg/s ec)
Lat. speed (m /sec)
0.1
0
-0.05
-0.1
-0.15
5
4WS
2WS
4
3
2
1
-0.2
0
0
0.5
1
tim e (sec)
1.5
2
0
0.5
1
tim e (s ec )
1.5
% Ex15_2 Matlab program
% Comparison of wheel slip angle
a=1.5; l=3.2; b=l-a;
g=9.81; u0=20.0;
m=1855; Iz=3419.0;
Caf=2224.0*57.2958;
Car=1956*57.2958;
Kr=(-m*u0*u0*a/Car+b*l)/(-m*u0*u0*b/Caf-a*l);
A=[ -(Caf+Car)/(m*u0),
(b*Car-a*Caf)/(m*u0)-u0;
(b*Car-a*Caf)/(Iz*u0), -(a*a*Caf+b*b*Car)/(Iz*u0)];
B_4ws=[Caf/m+Car*Kr/m; a*Caf/Iz-b*Car*Kr/Iz];
B_2ws=[Caf/m; a*Caf/Iz];
% step response of 1 degree for 2WS
amp_2ws=1.0*pi/180.0;
amp_4ws=1.0/(1-Kr)*pi/180.0;
t=0:0.01:2.0;
v_4ws_sys=ss(A,
v_2ws_sys=ss(A,
r_4ws_sys=ss(A,
r_2ws_sys=ss(A,
B_4ws*amp_4ws,
B_2ws*amp_2ws,
B_4ws*amp_4ws,
B_2ws*amp_2ws,
v_4ws_step=step(v_4ws_sys,
v_2ws_step=step(v_2ws_sys,
r_4ws_step=step(r_4ws_sys,
r_2ws_step=step(r_2ws_sys,
[1,0],
[1,0],
[0,1],
[0,1],
0);
0);
0);
0);
t);
t);
t);
t);
alpha_f_4ws= amp_4ws*ones(size(v_4ws_step))
-(v_4ws_step+a*r_4ws_step)/u0;
alpha_f_2ws= amp_2ws*ones(size(v_2ws_step))
-(v_2ws_step+a*r_2ws_step)/u0;
alpha_r_4ws= amp_4ws*Kr*ones(size(v_4ws_step))
-(v_4ws_step-b*r_4ws_step)/u0;
2
15.2 Goals of 4WS Algorithms
alpha_r_2ws=-(v_2ws_step-b*r_2ws_step)/u0;
plot(t,alpha_f_4ws*180/pi,‘-.r’, t,alpha_f_2ws*180/pi,‘b’)
xlabel(‘time (sec)’)
ylabel(‘alpha_f (deg)’); legend(‘4WS’,‘2WS’)
pause
plot(t,alpha_r_4ws*180/pi,‘-.r’, t,alpha_r_2ws*180/pi,‘b’)
xlabel(‘time (sec)’),
ylabel(‘alpha_r (deg)’); legend(‘4WS’,‘2WS’)
pause
plot(t,v_4ws_step,‘-.r’, t,v_2ws_step,‘b’)
xlabel(‘time (sec)’)
ylabel(‘Lat. speed (m/sec)’); legend(‘4WS’,‘2WS’)
pause
plot(t,r_4ws_step*180/pi,‘-.r’, t,r_2ws_step*180/pi,‘b’)
xlabel(‘time (sec)’)
ylabel(‘Yaw rate (deg/sec)’)
legend(‘4WS’,‘2WS’)
Shown in Figure 15.1 is a block diagram of a simulation model for 4WS. Here, Af and Ar represent candidate
control algorithms, as shown in Figure 15.1c. The transfer functions Hf , Hr , Gf ,
and Gr represent the two-DOF vehicle dynamics in transfer-function form. Simulation and experimental results, comparing the candidate control strategies,
are provided in Figure 15.2. Controller A is the FWS vehicle that is considered
here as a basis for comparison. Controller B is simply a proportional controller
with gains Ar = Kr (calculated from Eq. (15.3)) and Af = 0. Controllers C and
D are similar to Controller B but include a mechanism to generate phase lag.
Controller E includes a lead action for the front wheels and a lag for the rear
wheels. The modification of the FWS angle has the effect of more quickly steering the wheels to a certain side-slip angle. These results show that both response
and stability can be improved substantially by steering both wheels. However,
reducing the side-slip angle to zero (as in Controller B) causes the yaw-rate gain
to drop, resulting in the deterioration of cornering performance. Also, the sideslip angle, although zero at steady-state, becomes negative during the transient,
which can be disconcerting to a driver. Thus, setting the side-slip angle to zero
causes problems for the practical use of the vehicle. The introduction of lead
and/or lag actions can improve these characteristics.
EXAMPLE 15.3: A PROTOTYPE 4WS SYSTEM.
The results of Example 15.3 also show a dilemma for 4WS system designs. On the
one hand, consumers (and engineers) are reluctant to accept steer-by-wire vehicles;
on the other hand, modifying only the RWS angle reduces the yaw and lateralacceleration characteristics of the vehicle. The latter fact has drawn some critics
toward 4WS vehicles, based on the human-factors perspective. The major obstacle
to the wide implementation of 4WS vehicles, however, is actually the cost.
277
Opposite phase
Same phase
Four-Wheel Steering
Rear wheel steer angle
relative to that of front wheels
278
0.5
0
50
100
150 (km/h)
Vehicle velocity
0.5
1.0
(a)
Vehicle
characteristics
.
ψ
δf
θ
1/N
Hf
Af
Hr
α
θ : Steering wheel angle
N : Steering gear ratio
δf : Front wheel steer angle
δr : Rear wheel steer angle
.
ψ : Yaw rate
α : Lateral G
Gf
δr
Ar
Gr
(b)
Control system
Functions
Af
Ar
(A) Front - wheel steering (
)
0
0
(B) Proportional
(
)
0
Kr
(C) 1st - order delay
(
)
0
Kr/(1 + Tr • S)
(D) 1st - order advance
(
)
0
Kr − Tr • S
(E) Front and rear wheel
(
)
Kr + Tr • S)
Kr − Tr • S
(c)
Figure 15.1. (a) Rear Wheel Steer angle relative to that of front wheels for zero sideslip angle.
(b) Block diagram of simulation model. (c) Control functions.
In this example, we compare the frequency-response plots of the yaw rate and lateralacceleration signals of 2WS and 4WS vehicles. By simply applying the
proportional-gain 4WS strategy, we see that the phase difference becomes
greatly reduced. This is somewhat expected because lateral acceleration determines how quickly the vehicle lateral speed builds up, and yaw rate determines
how quickly a vehicle is reoriented. If the phase difference between these two
signals is reduced, the side-slip angle also reduces (and vice versa).
EXAMPLE 15.4: YAW RATE AND LATERAL-ACCELERATION RESPONSE.
15.2 Goals of 4WS Algorithms
279
Front wheel steer angle sensor
Front wheel actuator
Steer angle sensor
Vehicle velocity
sensor
P
Controller
Rear wheel actuator
Rear wheel steer angle sensor
(a)
Front wheel
steer angle
(deg)
Front wheel steer angle
Rear wheel steer angle
0.5
0
Rear wheel
steer angle
(deg)
A
E
C
B Yaw rate
0.2
0
−0.2
A
Yaw rate
(deg/sec)
D
8
E
4
C
B
0
Lateral acceleration
D
Lateral
acceleration
(G)
0.4
Sideslip angle
0.2
0
0
0.5
(sec)
1
0
0.5
(sec)
(b)
Figure 15.2. (a) Four Wheel Steer system configuration. (b) Simulated (left) and experimental
(right) step response results for 4WS.
280
Four-Wheel Steering
P has e diff. (deg)
20
0
-20
4WS
FWS
-40
-60
-80
-100
-1
10
10
0
10
1
10
2
Freq (rad/s ec )
% Ex15_4.m
a=1.14; l=2.54; b=l-a;
g=9.81; u0=30.0;
m=1000; Iz=1400.0;
Caf=2400.0*57.2958; Car=2000.0*57.2958;
Kr=(-m*u0*u0*a/Car+b*l)/(-m*u0*u0*b/Caf-a*l);
A=[ 0,
1,
u0,
0;
0, -(Caf+Car)/(m*u0), 0, (b*Car-a*Caf)/(m*u0)-u0;
0,
0,
0,
1;
0, (b*Car-a*Caf)/(Iz*u0), 0, -(a*a*Caf+b*b*Car)/(Iz*u0)];
B_4ws=[0; Caf/m+Car*Kr/m; 0; a*Caf/Iz-b*Car*Kr/Iz];
B_2ws=[0; Caf/m; 0; a*Caf/Iz];
B_rws=[0; Car/m; 0; -b*Car/Iz];
C_r=[0,0,0,1];
C_acc=A(2,:)+u0*[0,0,0,1];
% Bode plots
w=logspace(-1,2,50);
[m_2ws_r, p_2ws_r]=bode(A,B_2ws,C_r,0,1,w);
[m_rws_r, p_rws_r]=bode(A,B_rws,C_r,0,1,w);
[m_4ws_r, p_4ws_r]=bode(A,B_4ws,C_r,0,1,w);
[m_2ws_acc, p_2ws_acc]=bode(A,B_2ws,C_acc,B_2ws(2),1,w);
[m_rws_acc, p_rws_acc]=bode(A,B_rws,C_acc,B_rws(2),1,w);
[m_4ws_acc, p_4ws_acc]=bode(A,B_4ws,C_acc,B_4ws(2),1,w);
loglog(w,m_2ws_r, w, m_rws_r,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Yaw rate gain’);
legend(‘FWS’,‘RWS’); pause
% reverse phase of RWS vehicle for comparison
semilogx(w,p_2ws_r, w, p_rws_r-180,‘-.’);
15.2 Goals of 4WS Algorithms
281
xlabel(‘Freq (rad/sec)’); title(‘Yaw rate phase (deg)’);
legend(‘FWS’,‘RWS’); pause
loglog(w,m_2ws_acc, w, m_rws_acc,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Acc gain’);
legend(‘FWS’,‘RWS’); pause
semilogx(w,p_2ws_acc, w, p_rws_acc-180,‘-.’);
xlabel(‘Freq (rad/sec)’); title(‘Acc phase (deg)’);
legend(‘FWS’,‘RWS’); pause
semilogx(w,p_4ws_r-p_4ws_acc,w,p_2ws_r-p_2ws_acc,‘-.’)
xlabel(‘Freq (rad/sec)’); title(‘Phase diff. (deg)’);
legend(‘4WS’,‘FWS’)
In this example, we
modify the simulation program discussed in Chapter 5 (i.e., driver modeling) to
compare the 2WS and 4WS vehicles under the same double-lane–change maneuver. It is observed that the 4WS vehicle achieves a smaller tracking error (except
at the first peak, when the vehicle is instantaneously changing orientation) and
a smaller yaw rate and acceleration. Moreover, the improved performance is
achieved with a smaller steering command from the driver. Therefore, it is fair
to say that this 4WS vehicle is easier to handle than a 2WS vehicle.
EXAMPLE 15.5: LANE-CHANGE MANEUVER: 2WS VERSUS 4WS.
lat. dis p. (m )
lat. dis p. error (m )
5
0.6
4
0.4
0.2
3
0
2
-0.2
Des ired
4WS
2WS
1
-0.4
0
4WS
2WS
-0.6
-1
-0.8
0
2
4
6
tim e (s ec )
8
10
0
2
4
6
tim e (sec)
s teering angle (deg)
8
10
Lat. A cc el. (m /s ec 2 )
1
5
0.5
0
0
-0.5
-1
-5
0
2
4
6
tim e (s ec )
8
10
0
2
4
6
tim e (sec)
8
10
282
Four-Wheel Steering
Yaw rate (deg/s ec )
10
5
0
-5
-10
-15
0
2
4
6
tim e (s ec )
8
10
% Ex15_5.m
a=1.14; l=2.54; b=l-a;
g=9.81; u0=30.0;
m=1000; Iz=1400.0;
Caf=2400.0*57.2958; Car=2000*57.2958;
Kr=(-m*u0*u0*a/Car+b*l)/(-m*u0*u0*b/Caf-a*l);
A=[ 0,
1,
u0,
0;
0, -(Caf+Car)/(m*u0), 0, (b*Car-a*Caf)/(m*u0)-u0;
0,
0,
0,
1;
0, (b*Car-a*Caf)/(Iz*u0), 0, -(a*a*Caf+b*b*Car)/(Iz*u0)];
B_4ws=[0; Caf/m+Car*Kr/m; 0; a*Caf/Iz-b*Car*Kr/Iz];
B_2ws=[0; Caf/m; 0; a*Caf/Iz];
C=[1,0,0,0]; D=0;
[num_4ws,Gvden]=ss2tf(A,B_4ws,C,D,1);
[num_2ws,Gvden]=ss2tf(A,B_2ws,C,D,1);
Gvnum_4ws=num_4ws(3:5);
Gvnum_2ws=num_2ws(3:5);
Kd = 0.03;
T = 0.1; Tp=0.3;
Gvnum_4ws=conv(Gvnum_4ws, [Tpˆ2/2 Tp 1]);
Gvnum_2ws=conv(Gvnum_2ws, [Tpˆ2/2 Tp 1]);
Gdnum=Kd*[-T/2 1];
Gdden=[T/2 1];
Gcnum_4ws=conv(Gdnum,Gvnum_4ws);
Gcnum_2ws=conv(Gdnum,Gvnum_2ws);
Gcden_4ws=conv(Gdden,Gvden) + Gcnum_4ws;
Gcden_2ws=conv(Gdden,Gvden) + Gcnum_2ws;
t=[0:0.05:10];
yd=[zeros(1,20), 0:0.06:3.6, 3.6*ones(1,39), 3.6:0.06:0, zeros(1,20)];
y_4ws=lsim(Gcnum_4ws,Gcden_4ws,yd,t);
y_2ws=lsim(Gcnum_2ws,Gcden_2ws,yd,t);
Problem 1
283
e_4ws=yd-y_4ws’;
e_2ws=yd-y_2ws’;
plot(t,yd,t,y_4ws‘,‘-.’,t,y_2ws,‘:’); grid
xlabel(‘time (sec)’), legend(‘Desired’,‘4WS’,‘2WS’)
title(‘lat. disp. (m)’), pause
plot(t,yd-y_4ws’,t,yd-y_2ws’,‘-.’); grid
xlabel(‘time (sec)’), legend(‘4WS’,‘2WS’)
title(‘lat. disp. error (m)’), pause
% Steering angle is the output of the driver block Gd
% under the input signal e=yd-y
steer_4ws=lsim(Gdnum,Gdden,e_4ws,t);
steer_2ws=lsim(Gdnum,Gdden,e_2ws,t);
plot(t,steer_4ws*180/pi, t,steer_2ws*180/pi,‘-.’); grid
xlabel(‘time (sec)’)
title(‘steering angle (deg)’), pause
% Lateral acceleration = v_dot + u0*r
C_acc=A(2,:)+u0*[0 0 0 1];
acc_4ws=lsim(A,B_4ws,C_acc,B_4ws(2),steer_4ws,t);
acc_2ws=lsim(A,B_2ws,C_acc,B_2ws(2),steer_2ws,t);
plot(t,acc_4ws,t,acc_2ws,‘-.’); grid
xlabel(‘time (sec)’)
title(‘Lat. Accel. (m/secˆ2)’)
pause
% Yaw rate
C_r=[0,0,0,1];
r_4ws=lsim(A,B_4ws,C_r,0,steer_4ws,t);
r_2ws=lsim(A,B_2ws,C_r,0,steer_2ws,t);
plot(t,r_4ws*180/pi,t,r_2ws*180/pi,‘-.’); grid
xlabel(‘time (sec)’)
title(‘Yaw rate (deg/sec)’)
PROBLEMS
1. The state equation for the 4WS vehicle under curved-road following was found
to be:
⎡
⎤
⎡
0
y
⎢
⎥ ⎢0
⎢
⎢
⎥
⎢
v
d ⎢
⎥ ⎢
⎥=⎢
⎢
dt ⎢ ψ − ψd ⎥ ⎢ 0
⎦ ⎢
⎣
⎣
r
0
−
1
Cα f + Cαr
uo
0
muo
0
bCαr − aCα f
0
0
Iz uo
0
⎢ Cα f
⎢
⎢ m
+⎢
⎢ 0
⎢
⎣ aCα f
⎡
Iz
0
Cαr
m
0
−bCαr
Iz
0
bCαr − aCα f
⎤
−
⎤
⎡
⎥
⎥
− uo ⎥ ⎢
muo
⎥⎢
⎥⎢
1
⎥⎣ψ
⎥
a2Cα f + b2Cαr ⎦
Iz uo
⎤
⎡
0
⎥ ⎥
⎢ 0 ⎥
⎥ δf
⎥
⎢
⎥
+
⎥ rd
⎢
⎥ δ
⎦
⎣
−1
⎥ r
⎦
0
⎤
y
v ⎥
⎥
⎥
− ψd ⎦
r
284
Four-Wheel Steering
For a vehicle with the parameters:
l = 2.54 m
g = 9.81 m/s2
Iz = 2,500.0 kg-m2
a = 1.14 m
u0 = 25.0 m/s
Caf = 2,400.0 N/deg
b = l – a = 1.40 m
m = 1,400 kg
Car = 2,000.0 N/deg
plot the steering gain Kr as a function of vehicle longitudinal velocity, u0 .
2. For the same vehicle model and parameters given in Problem 1, use the preview
driver program in Example 15.4 (using the parameters listed previously) to simulate
the vehicle and driver. Simulate 2WS (FWS), 4WS (i.e., with zero-slip proportional
RWS δr = Kr δ f , the FWS angle is controlled by the human driver), and RWS vehicles
under the same road-curvature conditions. The goal is to compare simulation results
and assess the handling of the three vehicles (e.g., 4WS is easier to handle than
2WS). Adjust the preview driver gain, preview time, and neuromuscular time delay
for the three cases separately, if necessary (i.e., to stabilize the closed-loop system,
if possible, and generate responses that aid in the comparison). Submit simulation
plots for:
•
•
•
•
lateral displacement (i.e., tracking error)
FWS angle (i.e., human control effort)
lateral acceleration
yaw rate
Also, if using different driver gains and time delays, make sure that these numbers
are shown clearly on the graphs.
Hint: The input Matrix B for the 4WS case should be in MATLAB format, as
follows:
B_4ws=[0; Caf/m+Car*Kr/m; 0; a*Caf/Iz-b*Car*Kr/Iz];
3. You work as an engineer for an automotive supplier that produces 4WS systems,
and your customer asks you to compare the performance of a proposed 4WS system
to a traditional 2WS system through simulations that include both a driver and a
vehicle model. For the 2WS-vehicle model, you utilize the two-DOF vehicle lateralyaw-dynamics model in Chapter 4 to obtain the following vehicle-dynamics transfer
function:
Gv (s) =
(b s2 + b1 s + b0 )
y(s)
= 22 2
δ(s)
s (s + a1 s + a0 )
where δ is the FWS angle input and y is the vehicle lateral position. Additionally,
you develop a preview-based driver model as follows: (1) the basic driver model is
assumed to be proportional plus delay, Gd (s) = Kd exp(−sT); (2) the error, e, is
calculated as e(t + Tp ) = r(t + Tp ) − y(t + Tp ); and (3) it is assumed that the desired
lateral position at some preview time, Tp , in the future, r(t + Tp ), is known and
that the actual position at that same time, y(t + Tp ), is estimated from the simple
rectilinear motion model:
yT p = y(t + Tp ) = y(t ) + Tp ẏ(t ) + Tp2/2 ÿ(t )
The parameter values to be used for the models are a = 1.14 m, b = 1.40 m, m =
1,500 kg, Iz = 2,420 (N.m.s2 )/rad, Caf = (2,050)(57.3) N/rad, Car = (1,675)(57.3) N/rad,
References
285
T = 0.05 sec, and Tp = 0.5 sec. Also, select the parameter value Kd to tune the
response and specify which value was used.
δ(t)
r(t+T p ) e(t+T p)
y(t+T p)
+
Gd (s)
G’v (s)
-
Based on the previous vehicle-dynamics transfer function and preview model,
complete the following tasks:
(a) Simulate the 2WS driver–vehicle system for the reference input trajectory,
r(t), shown in the following figure and use three values of longitudinal vehicle
speed u0 : low, medium, and high.
(b) Repeat the simulation in (a) but for the same vehicle with 4WS. Assume
that the 4WS is based on the adjustment of the RWS angle in proportion to
the FWS angle generated by the driver. That is,
Cα f
mu20 a − b(a + b)Cαr
δr = Kr δ =
δ
Cαr
mu20 b + a(a + b)Cα f
Compare the results obtained in (a) and (b) at three different speeds (i.e., three
values of u0 ) designated as low, medium, and high. Justify how the three speeds used
in these comparisons were selected.
REFERENCES
Abe, M., 1990, “Analysis on Free Control Stability of a Four-Wheel-Active-Steer Vehicle,”
JSAE Review, Vol. 11, No. 3, p. 28.
286
Four-Wheel Steering
Fukunaga, Y., N. Irie, J. Kuroki, and F. Sugasawa, 1987, “Improved Handling and Stability
Using Four-Wheel Steering,” 11th International Technical Conference on Experimental
Safety Vehicles, Washington, DC, Section 4, pp. 415–25.
Furukawa, Y., N. Yuhara, S. Sano, H. Takeda, and Y. Matsushita, 1989, “A Review of FourWheel Steering Studies from the Viewpoint of Vehicle Dynamics and Control,” Vehicle
System Dynamics: International Journal of Vehicle Mechanics and Mobility, Vol. 18, No. 1,
pp. 151–86.
Nalecz, A. G., and A. C. Bidermann, 1988, “Investigation into the Stability of Four-WheelSteering Vehicles,” International Journal of Vehicle Design, Vol. 9, No. 2, pp. 159–78.
16
Active Suspensions
Automotive suspensions are discussed in Chapter 4 in connection with the vertical
motion and ride properties of vehicles. A two-DOF quarter-car model was used,
which is simple but sufficiently detailed to capture many of the key suspensionperformance tradeoffs, such as ride quality (represented by sprung-mass acceleration); handling (represented by tire deflection); and packaging (represented by
suspension stroke, also known as the rattle space). The performance index (see
Chapter 4, Example 4.9) combines these three performance measures by assigning
adjustable weights to the three performance terms.
Studies show that passive suspensions frequently are tuned to achieve good
tradeoffs. Any improvement in one aspect of performance always is achieved at
the expense of the deteriorated performance in another. The extra DOF offered
by an active suspension could provide improved performance compared with a
strictly passive suspension. The optimal design of a suspension for a quarter-car oneDOF model, as shown in Figure 16.1a (i.e., no unsprung-mass [wheel] dynamics),
and the performance index, J1 = x21rms + r u2rms , has the structure shown in Figure
16.1b. Clearly, this structure, which includes a so-called skyhook damper, cannot be
realized by the passive-suspension configuration shown in Figure 16.1c. Note that
x1 in this one-DOF model represents the suspension stroke, r is a weight on control
signal, and u is the control force, which also is directly proportional to sprung-mass
acceleration. Clearly, an active suspension can provide performance benefits that
cannot be achieved by using a strictly passive design (Figure 16.2). Furthermore, an
active design can allow the performance to be user-selectable. For example, if a softer
or a firmer ride characteristic is preferred by a user, the weights in the performance
index used in the controller design can be changed (Hrovat 1988), leading to different
controller gains and, consequently, different performance characteristics.
Several types of “active” suspensions have been developed (Bastow 1988; Sharp
and Crolla 1987): (1) semi-active suspensions (Hac and Youn 1991; Redfield 1991);
(2) high-bandwidth, fully active suspensions (Chalasani 1986; Chalasani and Alexandridis 1986); and (3) low-bandwidth, fully active suspensions (Sharp and Hassan
1987). Although they have been available for decades, active suspensions have not
found widespread commercial application. This is due mainly to their large power
requirements (e.g., 3 times the power required by the air-conditioning compressor
and 2.5 times the peak power required by the starter), especially the high-bandwidth
287
288
Active Suspensions
“Skyhook damper”
x
m
m
Figure 16.1. (a) One-DOF vehicle model
with active suspension; (b) corresponding LQG-optimal structure; and (c) passive one-DOF model (Hrovat 1988).
m
U
k
S
w
(a)
w
w
(b)
(c)
versions. Thus, commercial implementations are primarily for low-bandwidth active
suspensions to emphasize attitude-holding performance during maneuvers and for
semi-active suspensions. Active stabilizer bars for roll control also have been implemented. In semi-active suspensions, the damping forces can be adjusted by control
of the damping coefficient, for example, by using electro-rheological fluids.
In the following sections, the design of optimal, fully active, high-bandwidth
suspensions is described first based on a single DOF model, then on a two-DOF
model, and finally the optimal active suspension for the two-DOF model with state
estimation.
16.1 Optimal Active Suspension for Single-DOF Model
Consider the design of an active suspension based on the single-DOF model, as in
Figures 16.1a–c. As shown in Figure 16.3, we can define the two states x1 = suspension
stroke (positive in extension) and x2 = sprung-mass velocity (positive downwards).
The sprung mass is denoted by ms , the suspension force by u(t), and the groundvelocity input by w(t). It is assumed that the ground-velocity input can be well
modeled as a zero-mean white-noise input, w, with variance W.
Normalized r.m.s. acceleration
100
ζ
−5.15 + ∫5.15
10
1
0.1
0.01
2 DOF
−2.29 ± ∫2.29
1 DOF
1
Normalized r.m.s. suspension stroke
Figure 16.2. Comparison between performance of conventional passive suspension
for two-DOF vehicle model and optimal
one-DOF active suspension (with representative eigenvalues) (Hrovat 1988).
16.1 Optimal Active Suspension for Single-DOF Model
289
x2
ms
Figure 16.3. Single-DOF quarter-car model for active-suspension design.
u
x1
w
The equations of motion then are written as:
ẋ1 = w − x2
u
ẋ2 = −
ms
In standard state equation form, these can be written as:
⎤
⎡
0
d x1
1
x1
0 −1
w
=
+ ⎣− 1 ⎦ u +
0
0 0
x2
dt x2
ms
or
ẋ = Ax + bu + gw
As discussed in Chapter 4, the control-design objective can be represented as a
quadratic form in the states and control input. For example, if we consider a weighted
sum with weight r of the suspension stroke and control effort, we can write:
1
0
1
0
J = E x21 (t ) + ru2 (t ) = E x(t )T Rxx x(t ) + ru2 (t )
From the Certainty Equivalence Principle, the optimal control gains are known to
be the same as for the corresponding deterministic LQ problem:
J=
∞
0
0
1
x(t )T Rxx x(t ) + ru2 (t ) dt
The optimal control is given by:
u∗ (t ) = −r−1 bT Px(t ) = −kTr x(t )
where P is the symmetric, positive-definite solution of the following algebraic Ricatti
equation:
AT P + PA − r−1 PbbT P + Rxx = 0
For our problem, we solve the previous equation with:
⎫
⎧
⎪
⎨ 0 ⎪
⎬
0 −1
P1
; b=
A=
;
P
=
1
⎪
⎪
0 0
P2
−
⎭
⎩
ms
P2
P3
290
Active Suspensions
c=
1
2 m s2 r
−
1
4
Figure 16.4. Optimal “skyhook damper” active suspension.
ms
k =r
−
1
2
which yields:
P1 =
%
1
1
2ms2 r 4 ;
1
P2 = −ms r 2 ;
P3 =
√ 3 4
2ms2 r 3
and we then can obtain the optimal control:
u∗ (t ) = −r−1 bT Px(t ) = −kTr x(t )
⎤
⎡
1
√ 12 41
2
−m
2m
r
r
1
s
s
⎣
= −r−1 0 −
√ 3 3 ⎦ x(t )
1
ms
−ms r 2
2ms2 r 4
3
2
1
1 √
1
= −r− 2 2ms2 r− 4 x(t ) = [k1 k2 ] x(t )
Thus, the optimal control is a state-feedback controller:
2
1
u∗ (t ) = −r− 2
√
1 13
x1
2ms2 r 4 ·
x2
which feeds back – with optimal gains that depend on r and ms – the suspension stroke
x1 and the sprung-mass velocity x2 . The physical interpretation of this control, as
shown in Figure 16.1b, is that of a passive suspension with a skyhook damper, which
is illustrated in Figure 16.4.
Consequently, the optimal active suspension cannot be implemented through
strictly passive means. In other words, it is not only the gains of the optimal controller
but also its structure, which is different from the passive suspension. As shown in
Figure 16.1c, the passive-suspension structure feeds back not x1 and x2 as in the
active suspension but rather x1 and ẋ1 .
16.2 Optimal Active Suspension for Two-DOF Model
Active suspension systems for automobiles also can be designed based on the twoDOF quarter-car model for vertical motion (see Chapter 4) by using optimal control methods to achieve the desired tradeoffs among passenger comfort, packaging
requirements, and vehicle-handling requirements. This typically is accomplished by
using a quadratic performance criterion (similar to the one discussed in Example 4.9)
and a linear model of the vertical vehicle dynamics. The linear model was derived
previously in Chapter 4 and is given in Eqs. (4.63) and (4.64).
16.2 Optimal Active Suspension for Two-DOF Model
291
The controller design is based on the minimization of a quadratic performance
index including weighted combinations of the squares of the rms values of sprungmass acceleration, wheel hop, rattle space, and applied-force terms:
J = E ẋ24 + r1 x21 + r2 x23 + r3 u2
= ẋ24rms + r1 x21rms + r2 x23rms + r3 u2rms
= xTrms Rxx xrms + 2uxTrms Rxu + Ruu u2rms
where Ruu = (1 + r3 ):
⎡
r1
0
⎢0
2
(2ζ
⎢
2 ω2 )
Rxx = ⎢
⎢0
−2ζ2 ω23
⎣
0 −(2ζ2 ω2 )2
⎤
0
0
−2ζ2 ω23 −(2ζ2 ω2 )2 ⎥
⎥
⎥
4
3
r2 + ω2
2ζ2 ω2 ⎥
⎦
2ζ2 ω23
Rxu
(2ζ2 ω2 )2
and the expectation operator, E, in Eq. (16.1) is defined by:
1 T
x(τ )dτ
E{x(t )} = lim
T →∞ T 0
(16.1)
⎡
0
⎤
⎢
⎥
⎢ −2ζ2 ω2 ⎥
⎢
⎥
=⎢
⎥
⎣ ω22 ⎦
(16.2)
2ζ2 ω2
(16.3)
Other formulations of the performance index have been used – for example, including jerk (i.e., time rate of change of the acceleration) as part of the passenger-comfort
criterion (Hrovat and Hubbard 1987).
Consider the design of an LQ optimal active suspension based on the model
shown in Chapter 4 and minimization of the performance index defined by Eqs.
(16.1) and (16.2) with respect to the control variable u(t). Initially, it is assumed
that all of the states, x(t), are measurable and can be used directly in the controller
implementation. Later, the implementation of the controller with measurement of
only some state variables is discussed. The LQ optimal controller is given by the
state-feedback law:
(16.4)
u(t )= −Kr x(t )
where the optimal controller gain Kr is given by:
′
′
Kr = R−1
uu [B P + Rxu ]
(16.5)
and P is obtained from the solution to the algebraic Ricatti equation:
−1
′
′
−1 ′
′
−1 ′
P(A − BR−1
uu Rxu ) + (A − Rxu Ruu B )P − PBRuu B P + Rxx − Rxu Ruu Rxu = 0
(16.6)
Equation (16.6) can be solved numerically using computer-aided-design software,
as illustrated in Example 16.1.
An LQ active-suspension design
is illustrated in this example. The LQ controller design consists of selecting
weights for use in the performance index, then determining the controller gain
Kr . The MATLAB program also generates Bode Plots showing the response of
the system to a white-noise ground-velocity input, w(t). The Bode Plot includes
EXAMPLE 16.1: LQ ACTIVE-SUSPENSION DESIGN.
292
10
Active Suspensions
Frequency Response Magnitude
−1
10
Suspension stroke
10
10
10
10
−3
10
0
10
1
10
Frequency (rad/s)
2
Frequency Response Magnitude
0
Passive
Active
10−2
10
3
10
−1
−2
−3
−4
10
0
10
1
10
Frequency (rad/s)
2
10
3
Frequency Response Magnitude
10
2
Passive
Active
Sprung mass accel.
Tire def
Passive
Active
101
10
10
0
−1
10
0
10
1
10
2
10
3
Frequency (rad/s)
Figure 16.5. Bode Plots of passive and active suspension systems.
frequency responses for the tire displacement, x1 (t); the suspension stroke, x3 (t);
and the sprung-mass acceleration, ẋ4 (t ). These results are in Figure 16.5, where
it can be seen that the active-suspension system achieves lower tire deflection
and sprung-mass acceleration in the low-frequency region, and its suspension
deflection is somewhat worse. The active suspension was found to improve
overall performance by about 17 percent.
% Ex16_1.m
clear
% Normalized vehicle parameters
w1 = 20*pi; % w1 = sqrt(kus/mus)
w2 = 2.0*pi; % w2 = sqrt(ks/ms)
z1 = 0.0; % z1 = cus/(2*ms*w1)
z2 = 0.3; % z2 = cs/(2*ms*w2)
rho = 10.0; % rho = ms/mus
% Passive system equations:
A = [0 1 0 0
-w1ˆ2 -2*(z2*w2*rho+z1*w1) rho*w2ˆ2 2*z2*w2*rho
16.2 Optimal Active Suspension for Two-DOF Model
0 -1 0 1
0 2*z2*w2 -w2ˆ2 -2*z2*w2];
B = [0 rho 0 -1]’; G = [-1 2*z1*w1 0 0]’;
% Define outputs for plotting results:
C= [1 0 0 0; % tire displacement
0 0 1 0; % suspension stroke
A(4,:)]; % sprung mass acceleration
Du = [0.0; 0.0; B(4)]; Dw = [0.0; 0.0; 0.0];
% Select weights for use in performance index:
% r1=1.1E3; r2=100.; r3=0.0; % Soft (S) ride case
r1=5.0e4; r2=5.0E3; r3=0.0; % Typical (T) ride case
% r1=1.0E6; r2=1.0E5; r3=0.0; % Harsh (H) ride case
Rxx = [r1 0 0 0
0 (2*z2*w2)ˆ2 -2*z2*w2ˆ3 -(2*z2*w2)ˆ2
0 -2*z2*w2ˆ3 (r2+w2ˆ4) 2*z2*w2ˆ3
0 -(2*z2*w2)ˆ2 2*z2*w2ˆ3 (2*z2*w2)ˆ2];
Rxu = [0 -2*z2*w2 w2ˆ2 2*z2*w2]’; Ruu = (1+r3);
% Calculate the LQ optimal gain Kr:
[Kr,S] = lqr(A,B,Rxx,Ruu,Rxu);
Ac=(A-B*Kr); Cc=(C-Du*Kr);
% Frequency response curves for the closed-loop and
% open-loop (passive) systems:
w=logspace(0,2.4,100);
[mag_p_tire, phase_p_tire] = bode(A,G,C(1,:),Dw(1),1,w);
[mag_a_tire, phase_a_tire] = bode(Ac,G,Cc(1,:),Dw(1),1,w);
[mag_p_susp, phase_p_susp] = bode(A,G,C(2,:),Dw(2),1,w);
[mag_a_susp, phase_a_susp] = bode(Ac,G,Cc(2,:),Dw(2),1,w);
[mag_p_ride, phase_p_ride] = bode(A,G,C(3,:),Dw(3),1,w);
[mag_a_ride, phase_a_ride] = bode(Ac,G,Cc(3,:),Dw(3),1,w);
loglog(w,mag_p_tire,‘r’,w,mag_a_tire,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
ylabel(‘Tire def’);
legend(‘Passive’, ‘Active’); pause
loglog(w,mag_p_susp,‘r’,w,mag_a_susp,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
ylabel(‘Suspension stroke’);
legend(‘Passive’, ‘Active’); pause
loglog(w,mag_p_ride,‘r’,w,mag_a_ride,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
293
294
Active Suspensions
ylabel(‘Sprung mass accel.’);
legend(‘Passive’, ‘Active’)
% calculate the performance index
% of the system with and without control
Xss=lyap(A,G*G’);
x3barrms=sqrt(Xss(3,3));
x1barrms=sqrt(Xss(1,1));
x4dotbarrms=sqrt([A(4,:)]*Xss*[A(4,:)]’+ [G(4)]*[G(4)]’);
Pindex=x4dotbarrmsˆ2+r1*(x1barrmsˆ2)+ r2*(x3barrmsˆ2);
Xss_act=lyap(Ac,G*G’);
x3barrms=sqrt(Xss_act(3,3));
x1barrms=sqrt(Xss_act(1,1));
x4dotbarrms=sqrt([Ac(4,:)]*Xss_act*[Ac(4,:)]’+ [G(4)]*[G(4)]’);
ubarrms=sqrt(Kr*Xss_act*Kr’);
% control signal
Pindex_act=x4dotbarrmsˆ2+r1*(x1barrmsˆ2)+
r2*(x3barrmsˆ2)+r3*(ubarrmsˆ2);
% Ratio of active performance/passive performance
Pindex_act/Pindex
16.3 Optimal Active Suspension with State Estimation
A challenge in implementing state-feedback control algorithms, including the LQ
optimal-control approach, is that all states of the system must be measurable. From
a cost perspective, it is desirable to minimize the measurements needed for activesuspension implementation. In such cases, some of the states required for the
feedback-control strategy are estimated from available measurements. This leads
to a so-called Linear Quadratic Gaussian (LQG) optimal control problem. The
unmeasured states are estimated using an optimal filter, known as the Kalman filter. The LQG optimal active-suspension design also is based on the quarter-car
model (i.e., Eqs. (4.63) and (4.64)); the performance index given in Eqs. (16.1)
and (16.2); and an output equation, which defines the measurable outputs of the
system:
(16.7)
y(t) =C x(t ) + Du(t )
where the coefficients C and D must be selected to define the measurable signals
that can be used in the controller. For example, if the suspension stroke, x3 (t), is the
only measurable variable, then C and D in Eq. (16.7) become C = [0 0 1 0], and
D = 0. Similarly, if the measured variables are the suspension stroke, x3 (t), and the
sprung-mass acceleration, ẋ4 (t ), then C and D in Eq. (16.7) become:
C=
0
0
0 2ζ2 ω2
1
−ω22
0
;
−2ζ2 ω2
D=
" #
0
0
(16.8)
16.3 Optimal Active Suspension with State Estimation
295
where the second entry of the D matrix is obtained from the last entry of the input
matrix corresponding to the G matrix, which is the matrix for the road-velocity input.
The optimal LQG controller has the form:
⌢
u(t ) = −Kr x(t )
(16.9)
and Kr is calculated exactly from the same procedure as in the LQ control case. The
⌢
state estimates, x(t ), are calculated from the equations:
d⌢
⌢
⌢
x(t ) = Ax(t ) + Bu(t ) + Ke (y(t ) − y(t ))
dt
(16.10)
and
⌢
⌢
y(t ) = Cx(t ) + Du(t )
(16.11)
The optimal estimator gain, Ke , is calculated from:
Ke = −Pe CV−1
and Pe is computed from the solution of another algebraic Ricatti equation:
Pe AT + APe − Pe CV−1 CT Pe + W = 0
(16.12)
where V is the measurement-noise covariance matrix and W is the processdisturbance covariance matrix. Like the weights Rxx , Ruu , and Rxu , these matrices
must be selected before the optimal estimator can be designed. The optimal estimator relies more on the measurement by producing a large gain, Ke , when the
confidence in the measurement is high relative to the model (i.e., when V is small
compared to W). Similarly, when V is large compared to W, the estimator gain is
small and the estimates rely more on the model than the measurement.
For the closed-loop system, the state-space model has twice the number of state
variables as the open-loop system. When the plant model is completely known and
measurement noise is small, the state-space model of the augmented system is:
" #
d x
x
G
A
−BKr
w
(16.13)
+
=
⌢
⌢
Ke C A − BKr − Ke C
0
x
dt x
When the “perceived” plant state and input matrices are different from those of the
true plant and measurement noise is included, the state-space model is:
" #
A
−BKr
ẋ
x
w
G 0
⌢
+
(16.14)
⌢
⌢
⌢
⌢
⌢
˙ =
0
K
v
x
x
Ke C A − BKr − Ke C − Ke (D − D)Kr
e
⌢
⌢
⌢
⌢
where A, B, C, and D are the perceived model matrices. Typically, the C and D
matrices are the same as the actual ones. In a case in which the feedback signal contains acceleration terms, however, they might be different from the actual matrices.
It is important to note that because the augmented system has twice the number
of state variables as the original model, the output matrices all must be adjusted
accordingly. For example, when we want to obtain the actual tire-deflection signal,
the corresponding C matrix is C = [1 0 0 0; zeros(1,4)]. The estimated tire deflection,
conversely, is obtained from a C matrix that is C = [zeros(1,4); 1 0 0 0].
Yue, Butsuen, and Hedrick (1989) consider the LQG design of active suspensions using only the suspension-stroke measurement. This LQG design approach
296
10
Active Suspensions
Frequency Response Magnitude
−1
10
Frequency Response Magnitude
0
x3 only
x3 and x4 dot
x3 only
x3 and x4 dot
Suspension stroke
10
−2
10
10
10
−3
0
10
1
10
10
Frequency (rad/s)
2
3
10
10
−1
−2
−3
−4
10
0
10
1
10
Frequency (rad/s)
2
10
3
Frequency Response Magnitude
x3 only
x3 and x4 dot
Sprung mass accel.
Tire def
10
10
0
10
0
1
2
10
10
Frequency (rad/s)
3
10
Figure 16.6. Bode Plots of LQG active-suspension systems.
is illustrated in Example 16.2. Several possible measurement sets are considered in
Ulsoy et al. (1994. They also investigate the robustness of LQ and LQG controllers
with respect to unmodeled sensor and actuator dynamics and their sensitivity to
variations in the parameters of a parallel passive suspension.
An LQG active-suspension
design, based only on suspension-stroke measurement, is illustrated in this example. The LQG controller design consists of selecting the weights for use in the
performance index, selecting the covariance matrices for use in the Kalman filter, determining the controller gain Kr , and determining the estimator gain Ke .
The program also generates the Bode Plots showing the response of the closedloop system to a white-noise ground-velocity input, w(t). The results shown in
Figure 16.6 can be compared to those in Figure 16.5.
EXAMPLE 16.2: LQG ACTIVE-SUSPENSION DESIGN.
% Ex16_2.m
clear
% Specify model parameter values:
w1 = 20*pi; % w1 = sqrt(kus/mus); w2 = 2.0*pi;
% w2 = sqrt(ks/ms)
z1 = 0.0; % z1 = cus/(2*ms*w1); z2 = 0.3; % z2 = cs/(2*ms*w2)
rho = 10.; % rho = ms/mus
16.3 Optimal Active Suspension with State Estimation
% Open loop system equations:
A = [0 1 0 0
-w1ˆ2 -2*(z2*w2*rho+z1*w1) rho*w2ˆ2 2*z2*w2*rho
0 -1 0 1
0 2*z2*w2 -w2ˆ2 -2*z2*w2];
B = [0 rho 0 -1]’; G = [-1 2*z1*w1 0 0]’;
% Select weights for use in performance index:
% r1=1.1E3; r2=100.; r3=0.0; % Soft (S) ride case
r1=5.0e4; r2=5.0E3; r3=0.0; % Typical (T) ride case
% r1=1.0E6; r2=1.0E5; r3=0.0; % Harsh (H) ride case
Rxx = [r1 0 0 0
0 (2*z2*w2)ˆ2 -2*z2*w2ˆ3 -(2*z2*w2)ˆ2
0 -2*z2*w2ˆ3 (r2+w2ˆ4) 2*z2*w2ˆ3
0 -(2*z2*w2)ˆ2 2*z2*w2ˆ3 (2*z2*w2)ˆ2];
Rxu = [0 -2*z2*w2 w2ˆ2 2*z2*w2]’; Ruu = (1+r3);
% Calculate the LQ optimal gain Kr:
[Kr,S] = lqr(A,B,Rxx,Ruu,Rxu);
% Define C1 and D1 for suspension stroke
% Define C2 and D2 for sprung mass acceleration
C1=[0 0 1 0]; D1=0;
C2=[0 0 1 0;A(4,:)]; D2=[0;G(4)];
% parameters of the noise model:
Amp=1.65E-5; Vel=80; p=0.01;
% calculation of the covariances used in the KF design
Xss=lyap(A,G*G’);
x3barrms=sqrt(Xss(3,3));
x1barrms=sqrt(Xss(1,1));
x4dotbarrms=sqrt([A(4,:)]*Xss*[A(4,:)]’+ [G(4)]*[G(4)]’);
Pindex=x4dotbarrmsˆ2+r1*(x1barrmsˆ2)+ r2*(x3barrmsˆ2);
W=(2.0*pi*Amp*Vel);
V1=(pˆ2)*(2.0*pi*Amp*Vel)*(x3barrmsˆ2);
V2=(pˆ2)*(2.0*pi*Amp*Vel)*[(x3barrmsˆ2) 0; 0 (x4dotbarrmsˆ2)];
% calculation of the steady state KF gains
Ke1=lqe(A,G,C1,W,V1);
Ke2=lqe(A,G,C2,W,V2);
% Compute the state matrices for LQG systems
Ac1 = [A, -B*Kr; Ke1*C1, A-B*Kr-Ke1*C1];
Ac2 = [A, -B*Kr; Ke2*C2, A-B*Kr-Ke2*C2];
% Define various outputs for plotting results:
Cc_lqg = [1 0 0 0 0 0 0 0; % tire displacement
0 0 1 0 0 0 0 0; % suspension stroke
A(4,:) 0 0 0 0]; % sprung mass acceleration
Dw = [0.0; 0.0; G(4)]; Du = [0.0;0.0;B(4)]; Gc = [G;0;0;0;0];
% Frequency response
w=logspace(0,2.4,100);
[mag_p_tire, phase_p_tire] = bode(A,G,[1 0 0 0],0.0,1,w);
[mag_a1_tire, phase_a1_tire] = bode(Ac1,Gc,Cc_lqg(1,:),
Dw(1),1,w);
297
298
Active Suspensions
[mag_a2_tire, phase_a2_tire] = bode(Ac2,Gc,Cc_lqg(1,:),
Dw(1),1,w);
[mag_p_susp, phase_p_susp] = bode(A,G,[0 0 1 0],0,1,w);
[mag_a1_susp, phase_a1_susp] = bode(Ac1,Gc,Cc_lqg(2,:),
Dw(2),1,w);
[mag_a2_susp, phase_a2_susp] = bode(Ac2,Gc,Cc_lqg(2,:),
Dw(2),1,w);
[mag_p_ride, phase_p_ride] = bode(A,G, A(4,:), G(4),1,w);
[mag_a1_ride, phase_a1_ride] = bode(Ac1,Gc,Cc_lqg(3,:),
Dw(3),1,w);
[mag_a2_ride, phase_a2_ride] = bode(Ac2,Gc,Cc_lqg(3,:),
Dw(3),1,w);
loglog(w,mag_a1_tire,‘r’,w,mag_a2_tire,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
ylabel(‘Tire def’);
legend(‘x3 only’, ‘x3 and x4 dot’); pause
loglog(w,mag_a1_susp,‘r’,w,mag_a2_susp,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
ylabel(‘Suspension stroke’);
legend(‘x3 only’, ‘x3 and x4 dot’); pause
loglog(w,mag_a1_ride,‘r’,w,mag_a2_ride,‘b-.’);
title(‘Frequency Response Magnitude’);
xlabel(‘Frequency (rad/s)’);
ylabel(‘Sprung mass accel.’);
legend(‘x3 only’, ‘x3 and x4 dot’);
% calculate the rms response to a unit variance white
% noise input of the system with LQG control
Xss_a1=lyap(Ac1,[G;0;0;0;0]*[G;0;0;0;0]’);
x3barrms1=sqrt([0 0 1 0 0 0 0 0]*Xss_a1*[0 0 1 0 0 0 0 0]’);
x4dotbarrms1=sqrt([A(4,:) -B(4)*Kr]*Xss_a1*[A(4,:) B(4)*Kr]‘+[G(4)]*[G(4)]’);
x1barrms1=sqrt([1 0 0 0 0 0 0 0]*Xss_a1*[1 0 0 0 0 0 0 0]’);
ubarrms1=sqrt([0 0 0 0 Kr]*Xss_a1*[0 0 0 0 Kr]’);
Pindex_a1=x4dotbarrms1ˆ2+r1*(x1barrms1ˆ2)+r2*(x3barrms1ˆ2)
+r3*(ubarrms1ˆ2);
Xss_a2=lyap(Ac2,[G;0;0;0;0]*[G;0;0;0;0]’);
x3barrms2=sqrt([C1 0 0 0 0]*Xss_a2*[C1 0 0 0 0]’);
x4dotbarrms2=sqrt([A(4,:) -B(4)*Kr]*Xss_a2*[A(4,:) B(4)*Kr]‘+[G(4)]*[G(4)]’);
x1barrms2=sqrt([1 0 0 0 0 0 0 0]*Xss_a2*[1 0 0 0 0 0 0 0]’);
ubarrms2=sqrt([0 0 0 0 Kr]*Xss_a2*[0 0 0 0 Kr]’);
Pindex_a2=x4dotbarrms2ˆ2+r1*(x1barrms2ˆ2)+r2*(x3barrms2ˆ2)
+r3*(ubarrms2ˆ2);
Pindex_a1/Pindex
Pindex_a2/Pindex
16.3 Optimal Active Suspension with State Estimation
EXAMPLE 16.3: COMPARE PASSIVE, ACTIVE LQ, AND ACTIVE LQG SUSPENSIONS. The
responses of three suspension systems under the excitation of the same groundvelocity input are compared in this example. The three suspensions compared
are the passive suspension, the active LQ suspension designed in Example 16.1,
and the active LQG suspension designed in Example 16.2. The responses for tire
deflection are compared in Figure 16.7a, for suspension stroke in Figure 16.7b,
and for sprung-mass acceleration in Figure 16.7c. The LQ and LQG results
are virtually indistinguishable. The active suspensions show some improvement
over the passive suspensions. These results also show the importance of working
with the rms values in the performance index because the time responses are
difficult to interpret in terms of performance.
% Ex16_3.m
clear;
% Specify model parameter values:
w1 = 20*pi; % w1 = sqrt(kus/mus); w2 = 2.0*pi;
% w2 = sqrt(ks/ms)
z1 = 0.0; % z1 = cus/(2*ms*w1); z2 = 0.3; % z2 = cs/(2*ms*w2)
rho = 10.; % rho = ms/mus
% Open loop system equations:
A = [0 1 0 0
-w1ˆ2 -2*(z2*w2*rho+z1*w1) rho*w2ˆ2 2*z2*w2*rho
0 -1 0 1
0 2*z2*w2 -w2ˆ2 -2*z2*w2];
B = [0 rho 0 -1]’; G = [-1 2*z1*w1 0 0]’;
C= [1 0 0 0; % tire displacement
0 0 1 0; % suspension stroke
A(4,:)]; % sprung mass acceleration
Dw = [0.0; 0.0; G(4)]; Du = [0.0;0.0;B(4)];
% Select weights for use in performance index:
% r1=1.1E3; r2=100.; r3=0.0; % Soft (S) ride case
r1=5.0e4; r2=5.0E3; r3=0.0; % Typical (T) ride case
% r1=1.0E6; r2=1.0E5; r3=0.0; % Harsh (H) ride case
Rxx = [r1 0 0 0
0 (2*z2*w2)ˆ2 -2*z2*w2ˆ3 -(2*z2*w2)ˆ2
0 -2*z2*w2ˆ3 (r2+w2ˆ4) 2*z2*w2ˆ3
0 -(2*z2*w2)ˆ2 2*z2*w2ˆ3 (2*z2*w2)ˆ2];
Rxu = [0 -2*z2*w2 w2ˆ2 2*z2*w2]’; Ruu = (1+r3);
% Calculate the LQ optimal gain Kr:
[Kr,S] = lqr(A,B,Rxx,Ruu,Rxu);
% LQ results
Ac=(A-B*Kr); Cc=(C-Du*Kr);
% Define C1 and D1 for suspension stroke
% Define C2 and D2 for sprung mass acceleration
C1=[0 0 1 0]; D1=0;
C2=[0 0 1 0;A(4,:)]; D2=[0;G(4)];
% parameters of the noise model:
299
300
Active Suspensions
x 10−3
x1 (m)
5
Tire Deflection
0
−5
0
1
2
3
4
5
Time (sec)
6
7
8
6
7
8
9
10
Susp Stroke
0.04
0.03
x3 (m)
0.02
0.01
0
−0.01
−0.02
−0.03
0
1
2
3
4
5
Time (sec)
9
10
Acceleration
1.5
x4 dot x2 (m/sec)
1
0.5
0
−0.5
−1
−1.5
−2
0
1
2
3
4
5
Time (sec)
6
Figure 16.7. Simulation results of three suspension systems.
7
8
9
10
Problem 1
Amp=1.65E-5; Vel=80; p=0.01;
% calculation of the covariances used in the KF design
Xss=lyap(A,G*G’);
x3barrms=sqrt(Xss(3,3));
x1barrms=sqrt(Xss(1,1));
x4dotbarrms=sqrt([A(4,:)]*Xss*[A(4,:)]’+ [G(4)]*[G(4)]’);
Pindex=x4dotbarrmsˆ2+r1*(x1barrmsˆ2)+ r2*(x3barrmsˆ2);
W=(2.0*pi*Amp*Vel);
V1=(pˆ2)*(2.0*pi*Amp*Vel)*(x3barrmsˆ2);
V2=(pˆ2)*(2.0*pi*Amp*Vel)*[(x3barrmsˆ2) 0; 0 (x4dotbarrmsˆ2)];
% calculation of the steady state KF gains
Ke1=lqe(A,G,C1,W,V1);
Ke2=lqe(A,G,C2,W,V2);
% Compute the state matrices for LQG systems
Ac1 = [A, -B*Kr; Ke1*C1, A-B*Kr-Ke1*C1];
Ac2 = [A, -B*Kr; Ke2*C2, A-B*Kr-Ke2*C2];
% Define various outputs for plotting results:
Cc_lqg = [1 0 0 0 0 0 0 0;
% tire displacement
0 0 1 0 0 0 0 0;
% suspension stroke
A(4,:) 0 0 0 0];
% sprung mass acceleration
Gc = [G;0;0;0;0];
t=[0:0.1:10];
w=sqrt(2*pi*Amp*Vel)*randn(size(t));
yp=lsim(A,G,C,Dw,w,t);
ylq=lsim(Ac,G,Cc,Dw,w,t);
ylqg=lsim(Ac1,Gc,Cc_lqg-Du*[0 0 0 0 Kr],Dw,w,t);
clf; plot(t,[yp(:,1) ylq(:,1) ylqg(:,1)]);
title(‘Tire Deflection’);
xlabel(‘Time (sec)’); ylabel(’x1 (m)’); grid; pause;
plot(t,[yp(:,2) ylq(:,2) ylqg(:,2)]);
title(‘Susp Stroke’);
xlabel(‘Time (sec)’); ylabel(‘x3 (m)’); grid; pause;
plot(t,[yp(:,3) ylq(:,3) ylqg(:,3)]);
title(‘Acceleration’);
xlabel(‘Time (sec)’); ylabel(‘x4 dot (m/secˆ2)’); grid;
PROBLEMS
1. Rerun the MATLAB program in Example 16.1 to generate frequency-response
plots for the LQ active suspensions with soft, typical, and harsh ride characteristics.
Compare these plots and discuss why these particular combinations of the performance index weights r1 and r2 are referred to as “soft,” “typical,” and “harsh.” For
these three LQ active suspensions and the passive-only suspension, create a table
that contains the following information: (a) the value of the performance index;
(b) the rms values of the individual terms in the performance index (e.g., suspension
301
302
Active Suspensions
stroke, tire deflection, and sprung-mass acceleration); and (c) the poles (or eigenvalues). Comment on how the various active designs change the values of these
quantities as compared to the passive case.
2. Use a MATLAB program similar to the one in Example 16.2 to design an LQG
active suspension that uses the “typical” ride weights, suspension-stroke measurement, and both sprung-mass and unsprung-mass accelerations. Provide (a) frequency
response plots, (b) values of the performance index, and (c) closed-loop eigenvalues.
3. Your supervisor asks you to use a simple one-DOF quarter-car model to develop
an adaptive version of an active suspension based on RLS estimation. She wants
you to compare the performance of the adaptive active suspension to a nonadaptive
version in simulations in which the vehicle sprung mass can vary by ±50 percent
around its nominal value and to recommend whether adaptation is needed. The
model is given by:
x2
J=
∞
0
ms
s
x1
u
d
dt
x1
x2
0 2
1
x1 + ru2 (t ) dt
0
=
0
−1
0
⎤
⎡
0
1
x1
⎣
⎦
w
u+
+
1
0
x2
−
ms
w
where x1 is the suspension stroke, x2 is the sprung-mass velocity, and the output
y = the suspension stroke. Thus:
" #
x1
y = [1 0]
= x1
x2
The nonadaptive controller is√an optimal controller
√
√ that minimizes J, given by u =
k1 x1 + k2 x2 , where k1 = −(1/ r) and k2 = (2ms / r). The model parameters have
nominal values of r = 0.0001 and ms = 1,500 kg. Thus, k1 = −100 and k2 = 547.723,
and the closed-loop eigenvalues are at s1,2 = −0.1826 ± 0.1826j for the nominal
conditions. However, because the sprung mass can vary, the actual closed-loop
performance also varies. Consequently, an adaptive, RLS-based, active suspension
based on this one-DOF quarter-car model also can be designed.
Note that for the plant-transfer function:
G(s) =
1
Y (s)
=
U (s)
ms s2
with a sampling period of h and a zero-order hold, the equivalent pulse-transfer
function is:
2 z+1
z+1
h
=
α
H(z) =
2ms
(z − 1)2
z2 − 2z + 1
References
303
Consequently, the discrete-time plant can be represented as:
y(k) = [ 2 −1
α
⎧
⎫
y(k − 1) ⎪
⎪
⎪
⎪
⎨
⎬
y(k − 2)
α]
⎪
⎪ u(k − 1) ⎪
⎪
⎩
⎭
u(k − 2)
Formulate this as z(k) = θ T f(k) in terms of unknown parameters θ . Then, the
RLS algorithm can be used to estimate the unknown parameters θ online from
measurements of y and u. The control gains can be updated accordingly. Assume a
sampling period of h = 1 second.
(a) Simulate the nonadaptive version of the active suspension for a unit-step
input w(t). Display the results for both suspension stroke and sprung-mass
acceleration for the nominal value of mass (i.e., typical vehicle load), the
maximum value of mass (i.e., fully loaded vehicle), and the minimum value
of mass (i.e., empty vehicle).
(b) Repeat the simulations in (a) but with an adaptive version of the activesuspension design; compare the results to the nonadaptive version in (a).
What will you recommend to your supervisor?
REFERENCES
Aburaya, T., M. Kawanishi, H. Kondo, T. Hamada, 1990, “Development of an Electronic
Control System for Active Suspension,” Proceedings of the Conference on Decision and
Control, Honolulu, HI, pp. 2220–5.
Alexandridis, A. A., and T. R. Weber, 1984, “Active Vibration Isolation of Truck Cabs,”
Proceedings of the American Control Conference, San Diego, CA, pp. 1199–208.
Alleyne, A., and J. K. Hedrick, 1995, “Nonlinear Adaptive Control of Active Suspensions,”
IEEE Transactions on Control Systems Technology, Vol. 3, No. 1, March.
Bastow, D., 1988, Car Suspension and Handling, second edition, Pentech Press, London.
Ben Mrad, R., J. A. Levitt, and S. D. Fassois, 1991, “A Nonlinear Model of an Automobile
Hydraulic Active Suspension System,” in S. A. Velinsky, R. H. Fries, I. Haque, and D.
Wang (eds.), Advanced Automotive Technologies–1991, ASME DE-Vol. 40, New York,
pp. 347–59.
Bender, E. K., 1967, “Optimization of the Random Vibration Characteristics of Vehicle
Suspensions Using Random Process Theory,” Sc.D. Thesis, MIT, Cambridge, MA.
Berman, A., and A. J. Hannibal (eds.), 1975, Passenger Vibration in Transportation Vehicles,
ASME AMD-Vol. 24, New York.
Chalasani, R. M., 1986, “Ride Performance Potential of Active Suspension Systems – Part I,”
ASME Monograph, AMD-Vol. 80, New York, December.
Chalasani, R. M., and A. A. Alexandridis, 1986, “Ride Performance Potential of Active
Suspension Systems – Part II,” ASME Monograph, AMD-Vol. 80, New York, December.
DeBenito, C., and S. J. Eckert, 1988, “Control of an Active Suspension System Subject to
Random Component Failures,” presented at the ASME Winter Annual Meeting, Chicago,
IL, November, Paper No. 88-WA/DSC-33.
Fabien, B. C., 1991, “Controller Gain Selection for an Electromagnetic Suspension Under
Random Excitation,” Proceedings of the American Control Conference, Boston, MA,
June.
Hac, A., 1985, “Suspension Optimization of a 2-DOF Vehicle Model Using a Stochastic
Optimal Control Technique,” Journal of Sound and Vibration, Vol. 100, No. 3, pp. 343–57.
304
Active Suspensions
Hac, A., and I. Youn, 1991, “Optimal Semi-Active Suspension with Preview Based on a
Quarter-Car Model,” Proceedings of the American Control Conference, Pittsburg, PA,
June.
Hammond, J. K., and R. F. Harrison, 1981, “Nonstationary Response of Vehicles on Rough
Ground – A State Space Approach,” ASME Journal of Dynamic Systems, Measurement,
and Control, Vol. 103, September, pp. 245–50.
Hayakawa, K., K. Matsumoto, M. Yamashita, Y. Suzuki, K. Fujimori, and H. Kimura, 1999,
“Robust H∞ -output Feedback Control of Decoupled Automobile Active Suspension Systems,” Automatic Control, IEEE Transactions on, Vol. 44, No. 2, February, pp. 392–6.
Hedrick, J. K., and T. Butsuen, 1988, “Invariant Properties of Automotive Suspension,”
Advanced Suspensions, Proceedings of the Institute of Mechanical Engineers, Paper No.
C423/88, October.
Hrovat, D., 1988, “Influence of Unsprung Weight on Vehicle Ride Quality,” Journal of Sound
and Vibration, Vol. 124, No. 3, pp. 497–516.
Hrovat, D., 1991, “Optimal Active Suspensions for 3D Vehicle Models,” Proceedings of the
American Control Conference, Boston, MA, June.
Hrovat, D., 1997, “Survey of Advanced Suspension Developments and Related Optimal
Control Applications,” Automatica, Vol. 33, No. 10, pp. 1781–817.
Hrovat, D., and M. Hubbard, 1987, “A Comparison Between Jerk Optimal and Acceleration
Optimal Vibration Isolation,” Journal of Sound and Vibration, Vol. 112, No. 2, pp. 201–10.
Ivers, D. E., and L. R. Miller, 1991, “Semi-Active Suspension Technology: An Evolutionary
View,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang, (eds.), Advanced Automotive
Technologies–1991, ASME DE-Vol. 40, New York, pp. 327–46.
Karnopp, D., 1983, “Active Damping in Road Vehicle Suspension Systems,” Vehicle System
Dynamics, Vol. 12, No. 6, pp. 291–311.
Karnopp, D. C., 1985, “Two Contrasting Versions of the Optimal Active Vehicle Suspension,”
ASME Monograph, DSC-Vol. 1, pp. 341–6.
Kasprzak, J. L., 1991, “Research and Development Needs for Road Vehicle Suspension Systems,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced Automotive
Technologies–1991, ASME DE-Vol. 40, New York, pp. 35–6.
Khulief, Y. A., and S. P. Sun, 1989, “Finite Element Modeling and Semi-Active Control
of Vibrations in Road Vehicles,” ASME Journal of Dynamic Systems, Measurement, and
Control, Vol. 111, September, pp. 521–7.
Krtolica, R., and D. Hrovat, 1992, “Optimal Active Suspension Control Based on a Half-Car
Model: An Analytical Solution,” IEEE Transactions on Automatic Control, Vol. 37, No. 4,
April, pp. 528–32.
Levitt, J. A., and N. G. Zorka, 1991, “The Influence of Tire Damping in Quarter-Car ActiveSuspension Models,” ASME Journal of Dynamic Systems, Measurement, and Control,
Vol. 113, March, pp. 134–7.
Lieh, J., 1991, “Modeling and Simulation of an Elastic Vehicle with Semi-Active Suspensions,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced Automotive
Technologies–1991, ASME DE-Vol. 40, New York, pp. 315–26.
Lin, J. S., and I. Kanellakopoulos, 1997, “Nonlinear Design of Active Suspensions,” IEEE
Control Systems Magazine, Vol. 17, No. 3, pp. 45–59.
Michelberger, P., L. Palkovics, and J. Bojkor, 1993, “Robust Design of Active-Suspension
System,” International Journal of Vehicle Design, Vol. 14, Nos. 2/3, pp. 145–65.
Patten, W. N., R. M. Chalasani, D. Allsup, and J. Blanks, 1990, “Analysis of Control Issues
for a Flexible One-Half Car Suspension Model,” Proceedings of the American Control
Conference, San Diego, CA, May.
Patten, W. N., P. Kedar, and E. Abboud, 1991, “A Variable Damper Suspension Design for
Phase-Related Road Inputs,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.),
Advanced Automotive Technologies–1991, ASME DE-Vol. 40, New York, pp. 361–74.
Rajamani, R., and J. K. Hedrick, 1991, “Semi-Active Suspensions – A Comparison between
Theory and Experiments,” Proceedings of the 12th IAVSD Symposium, August.
References
Rajamani, R., and J. K. Hedrick, 1995, “Adaptive Observers for Active Automotive Suspensions: Theory and Experiment,” IEEE Transactions on Control Systems Technology,
Vol. 3, No. 1, March.
Ray, L. R., 1991, “Robust Linear-Optimal Control Laws for Active Suspension Systems,”
in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced Automotive
Technologies–1991, ASME DE-Vol. 40, New York, pp. 291–302.
Ray, L. R., 1992, “Robust Linear Optimal Control Laws for Active Suspension Systems,”
ASME Journal of Dynamic Systems, Measurement and Control, Vol. 114, No. 4, December,
pp. 592–8.
Redfield, R. C., 1990, “Low-Bandwidth Semi-Active Damping for Suspension Control,” Proceedings of the American Control Conference, San Diego, CA, May, pp. 1357–62.
Redfield, R. C., 1991, “Performance of Low-Bandwidth, Semi-Active Damping Concepts for
Suspension Control,” Vehicle System Dynamics, Vol. 20, pp. 245–67.
Redfield, R. C., and D. C. Karnopp, 1988, “Optimal Performance of Variable Component
Suspensions,” Vehicle System Dynamics, Vol. 17, No. 5.
Satoh, M., N. Fukushima, Y. Akatsu, I. Fujimura and K. Fukuyama, 1990, “An Active Suspension Employing an Electrohydraulic Pressure Control System,” Proceedings of the Conference on Decision and Control, Honolulu, HI, pp. 2226–31.
Sharp, R. S., and D. A. Crolla, 1987, “Road Vehicle Suspension Design – A Review,” Vehicle
System Dynamics, Vol. 16, pp. 167–92.
Sharp, R. S., and S. A. Hassan, 1987, “On the Performance Capabilities of Active Automobile
Suspension Systems of Limited Bandwidth,” Vehicle System Dynamics, Vol. 16, pp. 213–25.
Sunwoo, M., and K. C. Cheok, 1990, “An Application of Explicit Self-Tuning Controller
to Vehicle Active Suspension Systems,” Proceedings of the Conference on Decision and
Control, Honolulu, HI, pp. 2251–7.
Sunwoo, M., and K. C. Cheok, 1991, “Investigation of Adaptive Control Approaches for
Vehicle Active Suspension Systems,” Proceedings of the American Control Conference,
Boston, MA, June.
Sunwoo, M., K. C. Cheok, and N. J. Huang, 1990, “Application of Model Reference Adaptive
Control to Active Suspension Systems,” Proceedings of the American Control Conference,
San Diego, CA, May, pp. 1340–6.
Thompson, A. G., 1976, “An Active Suspension with Optimal Linear State Feedback,” Vehicle
System Dynamics, Vol. 5, pp. 187–203.
Ullman, P. B., and H. Richardson (eds.), 1975, Mechanics of Transportation Suspension Systems, ASME AMD-Vol. 15, New York.
Ulsoy, A. G., D. Hrovat, and T. Tseng, 1994, “Stability Robustness of LQ and LQG Active
Suspensions,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 116,
No. 1, March, pp. 123–31.
Veillette, R. J., 1991, “Projective Controls for 2-DOF Quarter-Car Suspension,” Proceedings
of the American Control Conference, Boston, MA, June.
Williams, R. A., 1997a, “Automotive Active Suspensions – Part 1: Basic Principles,” Proceedings of the Institution of Mechanical Engineers, Vol. 211 Part D, pp. 415–26.
Williams, R. A., 1997a, “Automotive Active Suspensions – Part 2: Practical Considerations,”
Proceedings of the Institution of Mechanical Engineers, Vol. 211 Part D, pp. 427–44.
Wilson, D. A., R. S. Sharp, and S. A. Hassan, 1986, “The Application of Linear Optimal Control Theory to the Design of Active Automotive Suspensions,” Vehicle System Dynamics,
Vol. 15, pp. 105–18.
Yamashita, M., K. Fujimori, C. Uhlik, P. Kawatani and H. Kimura, 1990, “H∞ Control of an
Automotive Active Suspension,” Proceedings of the Conference on Decision and Control,
Honolulu, HI, pp. 2244–50.
Yue, C., T. Butsuen, and J. K. Hedrick, 1989, “Alternative Control Laws for Automotive
Active Suspensions,” ASME Journal of Dynamic Systems, Measurement, and Control,
Vol. 111, No. 2, pp. 286–91.
305
PART IV
INTELLIGENT TRANSPORTATION
SYSTEMS
17
Overview of Intelligent
Transportation Systems
Mobility is essential to the economic growth of any modern country and the wellbeing of its population. This is especially true for the United States because of its size
and diffuse population. Without the efficient transport of people and goods, U.S.
industries cannot compete effectively with overseas producers. However, the rapid
growth in demand and the slower growth in the capacity of highway systems have
led to congestion that is estimated to cost more than $40 billion annually. In 1970,
motorists in the United States drove approximately 1 trillion vehicle-miles; by 1985,
this had increased to 1.8 trillion vehicle-miles; and, by 2000, to 2.8 trillion vehiclemiles. These increases have led to serious congestion problems. For example, peakhour traffic operating in congested conditions on urban Interstate highways increased
from 40 percent in 1970 to nearly 70 percent in 1990. From 1982 to 2002, the vehiclemiles traveled increased by 79 percent, whereas highway-lane miles increased by
only 3 percent. The number of roadways considered congested grew from 34 to
58 percent. However, construction of the more than 40,000 miles of the multilane,
controlled-access Interstate Highway System essentially is completed. Major new
construction, especially in dense urban areas, generally is not feasible and definitely
cannot keep up with future traffic demand. Although some growth of the highway
system is inevitable, the more efficient use of the existing system is essential.
In recent decades, there also have been tremendous changes in the areas of information technology, electronics, computers, and communications. The pace of these
developments is simply astounding, and electronic devices have infiltrated every
aspect of life, including vehicles (see Chapter 1). Intelligent Transportation Systems
(ITS) (formerly known as Intelligent Vehicle Highway Systems [IVHS]), however,
represent more than simply advances in automotive electronics. ITS incorporate a
wide variety of electronic-based technologies, both on the vehicle and as part of the
highway infrastructure, which collectively are moving the world into the next generation of highway operations. These technologies offer the promise of increased
throughput on existing highways at reduced congestion levels as well as improved
safety and convenience. The ITS vision encompasses smart (i.e., control, sensing,
and communications) automobiles and highways collaborating for improved safety,
mobility, trip quality, and productivity while also reducing congestion and environmental impact.
309
310
Overview of Intelligent Transportation Systems
ITS represents a long-term vision and an evolving concept of how the personal
automobile can continue to be a primary means of transportation despite the near
saturation of current highway capacities. Some argue that this is a dangerous vision
to meet transportation needs into the 21st century and that other high-speed publictransportation systems must become an increasingly important part of the mix.
Nevertheless, it is clear that ITS technologies will have an important – even if not an
exclusive or primary – role in meeting transportation needs in the next few decades.
In many respects, the traffic-congestion problems faced today in the United
States are more severe in other parts of the world (e.g., Europe and Japan). Perhaps
it is not surprising then that Europe and Japan have taken a leadership role in the
development of ITS. A survey of ITS being tested worldwide is in Jurgen (1991),
and the situation relative to ITS in Japan is summarized in Ervin (1991). At the
Mobility 2000 Workshop (1990), ITS activities were grouped in the following four
major areas:
r
r
r
r
Advanced Traffic Management Systems (ATMS)
Advanced Traveler Information Systems (ATIS)
Commercial Vehicle Operation (CVO)
Advanced Vehicle Control Systems (AVCS)
Each of these areas is discussed briefly in the sections that follow (Special Issue on
IVHS 1991).
17.1 Advanced Traffic Management Systems
ATMS permit the real-time adjustment of traffic-control systems and variablemessage signs for drivers. Their application in selected corridors has reduced delay,
travel time, and accidents. Traffic-control systems have used control and communications technology for more than a half-century but have been slow to incorporate
the latest developments (e.g., distributed-system architecture, fiber-optic communications, and microprocessor-based equipment). The first traffic-control system using
analog technology to optimize a flow of traffic through a series of intersections was
installed in Chicago in 1926. It provided a fixed signal-timing sequence and did not
allow for variations in traffic-flow characteristics. The first digital, computer-based
system was installed in 1963 in Toronto; this system provided for more flexible signal
control by monitoring detectors as well as operation of the traffic-signal controllers
and detectors. Stored signal-timing plans were selected based on detector data or a
time-of-day basis. With the availability of reliable and inexpensive microprocessor
equipment in the 1980s, a form of signal control known as a “closed-loop system”
became available (Figure 17.1). A closed-loop system uses a central personal computer that communicates with an intermediate level of control known as “master controllers” or “local area masters.” These, in turn, communicate with the intersection
controllers. The master controllers reduce the processing load on the central computer as well as the need to communicate directly with local intersection controllers.
ATMS require sensors in traffic lanes to identify the presence of vehicles and to
communicate that information to distributed or central processors. They depend
on real-time analysis of traffic data and systems for the control of traffic- and
ramp-metering signals. ATMS inform drivers about the status of traffic through
17.1 Advanced Traffic Management Systems
311
Information
Service Provider
traffic
information
Emergency
Management
Emergency Response
Management
incident notification
incident information
Traffic
Management
TMC Incident Dispatch
Coord/Communication
TMC Basic Signal
Control
emergency vehicle
driver status update
Traffic
Maintenance
Emergency
Vehicle
incident data
Roadway
signage data
Roadway
Signal Controls
signal control data
signal control status
TMC
Coord
Other TM
Figure 17.1. Information flow in closed-loop traffic-management systems.
variable-message signs. Ultimately, they indicate alternative routes to be followed
in case of a traffic incident. ATMS incorporate incident-management procedures,
which may reap the largest benefits. For example, an accident blocking one of three
lanes reduces highway capacity by 50 percent and a 20-minute blockage wastes
2,100 vehicle-hours. This can lead to a queue that is almost 2 miles long and can
take 2.5 hours to clear. During peak-traffic periods, the time wasted and delays
associated with accidents are 50 times worse. To be effective, systems require cooperation among adjacent jurisdictions to permit switching traffic from throughways
to arterials as required by incidents. Competent staff and maintenance crews are
required to ensure high-reliability systems. ATMS are being introduced with current
technology and will benefit from advanced technology. Wherever they have been
installed, they are reducing congestion by improving traffic flow and reducing accidents and emissions (Table 17.1). The following lists the benefits of ramp-metering
Table 17.1. Benefits realized from implementation of ATMS
Location
Brooklyn, NY
Philadelphia, PA
San Antonio, TX
Japan
Houston, TX
Denver, CO
Atlanta, GA
Minnesota
Reduced
incidentclearance
time (%)
Reduced
response
time (%)
Accident
reduction
(%)
40.0
20.0
35.0
Secondary
accident
reduction
(%)
Reduced
accident
rates
Cost
savings/yr
($M)
41.0%
1.65
255,500
8.40
0.95
572,095
95,000
2,000,000
Delay
savings
(hrs/yr)
66.0
30.0
50.0
1.40
312
Overview of Intelligent Transportation Systems
and incident-management systems (Proper 1999):
r Portland, Oregon: 58 ramp meters, 43 percent accident reduction, 39 percent
travel-time reduction, 25 percent demand increase, 60 percent increase in speed
r Minneapolis/St. Paul, MN: 6 ramp meters, 8 km of freeway, 24 percent accident
reduction, 38 percent accident-rate reduction, 16 percent increase in speed
r Minneapolis, MN: 39 ramp meters, 27 km of freeway, 27 percent accident reduction, 38 percent demand increase, 35 percent increase in speed
r Seattle, WA: 22 ramp meters, 52 percent travel-time reduction, 39 percent
accident-rate reduction, 86 percent demand increase
r Denver, CO: 5 ramp meters, 50 percent accident reduction, 18.5 percent demand
increase
r Detroit, MI: 28 ramp meters, 50 percent accident reduction, 8 percent increase
in speed, 12.5 percent demand increase
r Austin, TX: 3 ramp meters, 4.2 km of freeway, 60 percent increase in speed, 7.9
percent demand increase
r Long Island, NY: 70 ramp meters, 207 km of freeway, 15 percent accident
reduction, 9 percent increase in speed
Thus, ATMS represent the highway side of the cooperative ITS concept. They
are being implemented in many major urban areas using existing technology that
can evolve as advanced technologies mature. An important feature in the future
development of ATMS is the ability to communicate with individual vehicles. This
discussion continues in the next two sections.
17.2 Advanced Traveler Information Systems
ATIS allow drivers and passengers to know the current location of their and other
vehicles and to find desired services. ATIS permit communication between the driver
and ATMS for continuous advice regarding traffic conditions, alternate routes, and
safety issues.
ATIS equipment can show vehicle location and movement on an electronic map.
This enables route planning from origin to destination, identification of businesses
or services that a driver may need, and the route to those sites. When ATIS and
ATMS are in full communication, drivers are informed of incidents and alternate
routes to avoid congestion. ATIS includes the following features:
vehicle location (via GPS) and map-matching navigation system
traffic-information receiver
route planning for minimum distance of travel
color video displays for maps, traffic information, and route guidance
onboard database with detailed maps, business directory, specific location of
services, hospitals, and tourist information
r information from traffic-management centers about congestion, incidents, and
other problems
r electronic vehicle identification for toll debiting
r
r
r
r
r
Thus, ATIS represents the vehicle side of the cooperative ITS concept. Figure 17.2
shows information that was collected and used in ATIS services in the United States
17.2 Advanced Traveler Information Systems
313
50%
45%
40%
35%
30%
Freeways
Arterials
25%
20%
15%
10%
5%
0%
Traffic speeds
Road conditions
Incidents
Current work
zones
Scheduled work
zones
Weather
conditions
Types of information
Figure 17.2. 1999 transfer of information in ATIS services in the United States (Radin et al.
2000).
in 1999. ATIS is envisioned in the following three general stages of development in
North America:
r Information Stage. The primary emphasis is providing drivers with information
to improve individual planning and decision making.
r Advisory Stage. The static onboard information available to drivers is supplemented with dynamic traffic information collected and transmitted by the infrastructure.
r Coordination Stage. Vehicles and infrastructure automatically exchange information to optimize the flow and safety of traffic throughout the entire network.
Vehicle-navigation systems use the following three main technologies to determine
the location of a vehicle:
r GPS
r inertial navigation system (i.e., dead reckoning)
r map databases
Each has advantages and disadvantages that often are used in conjunction with one
another for best results. These systems can be used for in-vehicle route guidance,
route optimization, fleet management, and emergencies.
Various technologies are used to collect the data in ATIS, including video cameras, surveillance systems, loop detectors, cellular phones, police patrol, and GPS,
as well as communications from transit authorities (e.g., road repairs) and updates
from private-sector firms (e.g., hotels and restaurants). The information can be presented to a driver using various technologies such as a cellular phone, cable TV,
314
Overview of Intelligent Transportation Systems
the Internet, e-mail, and an in-vehicle personal digital assistant (PDA). Commercially available systems such as the GPS navigation systems, OnStar and Sync, are
examples of various ATIS capabilities.
17.3 Commercial Vehicle Operations
CVO select from ATIS those features critical to commercial and emergency vehicles.
They expedite deliveries, improve operational efficiency, and increase safety. CVO
are designed to interact with ATMS as both become fully developed.
Global competition is changing the way that companies conduct business. Carriers are expected to provide faster, more reliable, and more cost-effective services.
ITS technologies are emerging as a key to reducing costs and improving productivity.
Commercial and emergency vehicles will adopt ATIS and link them to ATMS as soon
as it is feasible to do so. Additional ITS technologies (some already have been developed) including weigh-in-motion sensors, automated vehicle-identification transponders, and automated vehicle-classification devices will reduce time spent in weigh
stations, reduce labor costs to states, and minimize red tape for commercial operators.
Commercial vehicles are leading the way in the development and implementation
of ITS technologies. They already are using automatic vehicle location, tracking,
and two-way communications; routing algorithms for dispatch; and in-vehicle text
and map displays. ITS technologies being used in commercial vehicles include the
following:
r
r
r
r
r
r
automatic vehicle identification
weigh-in-motion
automatic vehicle classification
onboard computer
two-way, real-time communication
automatic clearance sensing
Thus, CVO represents the vehicle side of the cooperative ITS concept for the special
needs of commercial and emergency vehicles. Figure 17.3 is a list of current and nearfuture CVO devices on a commercial truck (Capps et al. 2001).
17.4 Advanced Vehicle-Control Systems
AVCS apply additional technology to vehicles to identify obstacles and adjacent
vehicles, thereby assisting in the prevention of collisions and resulting in safer operation at higher speeds. AVCS also may interact with the fully developed ATMS to
provide automated vehicle operations. A vision of the future of the automobile, in
which they drive themselves while the drivers relax, was first introduced in the General Motors pavilion at the 1939 World’s Fair. Since then, there have been relatively
few efforts aimed at realizing this vision. In 1979–1981, General Motors conducted
a systems study of highway automation (Bender 1991). In 1964–1980, studies on
vehicle guidance and control were conducted at Ohio State University (Fenton and
Mayhan 1991). In the same year, the University of California’s PATH program conducted extensive studies on vehicle longitudinal and lateral control (Shladover et al.
1991).
17.4 Advanced Vehicle-Control Systems
315
Mobile Communications (voice and/or data)
Company information and examples of products
for the above devices can be seen in FHWA’s
Technology Truck, on the World Wide Web at
Automatic Vehicle Location
Automatic Toll
Collection &
Screening
Transponders
http://www.ornl.gov/dp111/index.htm
- and at the Trucking Technology Magazine at
http://www.truckingtechnologymag.com
On Board
Computers
Electronic scales
Automatic
Equipment
Identification
Tire Pressure Monitoring
Vehicle Systems Monitoring
Engine Control Module, Fuel, Brake,
Transmission Monitors
Accident Avoidance Systems
K
E
Y
Common technology
Recently deployed technology
Emerging technology
Figure 17.3. CVO devices on a commercial truck.
This section, an overview of AVCS technologies, identifies important classes of
problems and studies the potential benefits and impacts. Chapters 18 through 20
provide more detail on the following AVCS technologies: collision detection and
avoidance, longitudinal motion control and platooning, and lateral motion control
and automated steering.
The goal of AVCS is to enhance vehicle control by facilitating and augmenting
driver performance. Ultimately, AVCS aim to relieve drivers of most driving tasks.
Three levels of development are foreseen for AVCS technologies, as follows:
r ACVS-I: Individual vehicle control includes only vehicle-based systems that
detect the presence of obstacles or other vehicles. Studies show that half of
all rear-end collisions and as many as a third of intersection accidents can be
prevented if drivers have an additional half-second of warning. AVCS-I can
provide the additional warning time by sensing the presence of vehicles and
obstacles in blind spots. They also can warn drivers when their alertness starts
to wane.
r AVCS-II: Cooperative driver/vehicle/highway systems implement lateral and
longitudinal vehicle control functions in specific applications, such as highoccupancy vehicle lanes. Vehicles enter the lane voluntarily using manual control
but then are under full or partial automatic control while in the lane. The advantages include increased speed and safety. “Platooning,” the linking of cadre vehicles, also is possible. For private vehicles, AVCS-II provides vehicle-to-vehicle
communication about travel paths, which will reduce collisions.
r AVCS-III: Automated vehicle-highway systems include complete automation
of the driving function for vehicles operating on specially equipped freeway
facilities. It builds on AVCS-I and AVCS-II technologies to provide “automated
chauffeuring” of vehicles from on-ramp arrival to off-ramp departure.
316
Overview of Intelligent Transportation Systems
Considerable research and development are needed before many AVCS technologies can be deployed commercially. Important areas for further research include the
following:
r availability and reliability of devices (i.e., ultrasonic, infrared, radar, and vision)
to detect spatial relationships of a vehicle to obstacles and other vehicles, and
the use of this information in the automatic control of vehicles
r the change in speed of a vehicle under automatic control to be compatible with
the limitations of human occupants and available equipment, and a facility for
extensive full-scale testing, including human-factor considerations
r special traffic lanes for AVCS-equipped vehicles and automatic inspection procedures to ensure that the AVCS are functioning before equipped vehicles can
enter them
It is anticipated that AVCS will reduce accidents and increase traffic flow; they
are predicted to double traffic flow on current freeways. These goals may seem
ambitious; however, development of the Interstate Highway System in the 1950s
and 1960s doubled lane capacity and reduced accidents by 60 percent by gradeseparating intersections and controlling access. AVCS technologies are described in
further detail in subsequent chapters.
As discussed previously, several stages of the evolution of AVCS technologies
are envisioned: beginning with in-vehicle control systems, which can detect the
presence of obstacles or other vehicles and warn the driver (i.e., AVCS-I); evolving
to partially automated driving systems on special high-occupancy vehicle lanes (i.e.,
AVCS-II); to fully cooperative vehicle–highway automation on major highways (i.e.,
AVCS-III). The human-factors aspect of the research is important to the successful
development of these proposed AVCS systems. Each phase is discussed in more
detail in the following subsections.
ACVS-I: Individual Vehicle Control
The goal of ACVS-I is to develop vehicle-based systems that detect the presence of
obstacles or other vehicles and that provide a warning to drivers. AVCS-I includes
only those advanced systems that are vehicle-based and that do not require intervehicle or vehicle–highway communications and coordination. The principal benefits
of this technology are expected in the area of safety – that is, reductions in the
annual toll of crashes, fatalities, and injuries as well as the resulting economic costs.
However, AVCS-I technologies comprise the basis for the evolution to the AVCS-II
and AVCS-III phases.
In recent years, considerable advances have been made in sensing-, warning-, and
control-systems technologies with potential application to AVCS-I. The potential
for improved safety will be realized as follows:
r reduction in driver exposure to high-risk environments
r reduction in the incidence of high-risk driver behavior
r facilitation of earlier driver response to an imminent crash by providing additional seconds of warning time
17.4 Advanced Vehicle-Control Systems
r improvement in the overall speed and quality of driver–vehicle response in a
likely crash scenario
AVCS-I systems can be classified by the manner in which they aid a driver. Systems
that provide an enhanced image of the driving scene are referred to as “perceptual
enhancement” systems (e.g., night-vision systems). A driver is expected to interpret
the enhanced images and to control the vehicle in a manner that improves mobility
and safety. “Warnings” go one step further by providing an interpretation of sensor
signals (e.g., lane-departure warning). AVCS-I systems that alter control actions to
supplement those provided by a driver are referred to as “control enhancement”
systems (e.g., headway control in ACC systems).
AVCS-II: Cooperative Driver/Vehicle/Highway Systems
The goal of AVCS-II is to implement lateral and longitudinal vehicle control functions in specific applications, such as high-occupancy vehicle lanes. Vehicles enter
lanes voluntarily under manual control but then are under full or partial automatic
control. The advantages are increased speed and safety; platooning also is possible. For private vehicles, AVCS-II offers vehicle-to-vehicle communication of travel
paths, which will reduce collisions.
AVCS-II requires both vehicle- and highway-based equipment and utilizes
vehicle-to-vehicle and roadway-to-vehicle communications systems developed in
ATMS and/or ATIS. Vehicle lateral and longitudinal position is controlled when
suitably equipped vehicles operate in dedicated instrumented lanes. Vehicles voluntarily enter and exit these lanes and under manual control but are under full
or partial system control while in them. The benefits include enhanced safety and
increased travel speed through bottleneck locations at a modest cost compared to
achieving the same benefit by increasing the number of parallel lanes. The specific
technological elements of AVCS-II, which provides a bridge between AVCS-I and
AVCS-III, include the following:
r
r
r
r
r
r
automatic lateral control
automatic longitudinal control
vehicle-to-vehicle communications (e.g., for merge or demerge)
system integration of AVCS-II technologies (e.g., platooning)
intersection-hazard warning
electric propulsion
Platooning is a principal focus of AVCS-II system development. When the first three
component technologies are combined, true driver/vehicle/highway platoon systems
can be developed. In platooning, vehicles are linked electronically into “platoons”
on one lane of a freeway. The first platooning facility likely consists of 20 to 25
miles of a two-lane freeway separated from other lanes by a barrier; a realistic
operation involves 5,000 to 10,000 vehicles. Systems required include a vehicle-tovehicle headway-control system, accurate vehicle-speed control, platoon-to-platoon
spacing control, and automated entrance diagnostics. Intersection-hazard warning
systems entail vehicle-to-vehicle communication about intended travel paths and
extend beyond the capabilities of obstacle-detection systems developed in AVCS-I.
317
318
Overview of Intelligent Transportation Systems
To reduce pollution in congested urban areas, roadway electrification and the use of
roadway-powered electric vehicles also is envisioned as a part of AVCS-II systems.
A short demonstration facility for an electrified roadway already operates, with a
single bus being used to test the concept at low speeds. High-occupancy vehicle lanes
that combine the use of electric vehicles and platooning are anticipated in congested
urban areas.
AVCS-III: Automated Vehicle–Highway Systems
The goal of AVCS-III is to achieve complete automation of the driving task on
limited-access highways. A limited demonstration of the concept has been successful
in California. The benefits of this system are derived from the application of the
following technologies as a complete package:
r drive-by-wire
r steer-by-wire
r automatic onboard diagnostics (which must be interrogated and found acceptable before entry to AVCS-II facilities is permitted)
r automatic lateral control
r automatic longitudinal control
r vehicle-to-vehicle and vehicle-wayside communication for control
r human interfaces for transitions to and from control
r integration of automated roadways with arterials and local streets
r automatic traffic-merging control
r automatic lane-changing control
r automatic trip routing and scheduling
r automatic obstacle detection and avoidance
r reliability and safety enhancement features for all functions (e.g., real-time
condition monitoring, fault detection, and separate degraded performance and
emergency operating modes)
For both technical and environmental reasons, it also would be beneficial for vehicles
to be equipped with electric powertrains. Only a few elements of AVCS-III technology are commercially available today, such as drive-by-wire (which is available on
only a few luxury vehicles).
There are no major technological barriers to the development of AVCS systems;
however, public acceptance is a major concern. Who will benefit from the technology? How will the costs be allocated? Will AVCS systems be perceived as a threat to
contemporary lifestyles, privacy, or individual autonomy? What are the education
and training requirements for AVCS drivers? Safety is a major potential benefit
of AVCS systems but also increases risk if the systems do not work as intended.
Substantial effort in system design must be directed to minimizing the probability of
failures, as well as the consequences when failures do occur. It is difficult to anticipate how the public will respond to AVCS, in which vehicles may be operating
in closer proximity and higher speeds than today. Much attention must be given
to the public’s emotional responses to this form of travel and its perceptions of its
safety.
References
PROBLEMS
1. The field of ITS is developing rapidly in terms of new technologies. Find a recent
article about one of the new developments and summarize the main points.
2. Find a Web site (e.g., through the local or regional department of transportation)
that provides information about how ITS technologies are being used in your region
and summarize the main points.
REFERENCES
Anonymous, 1990, “Final Report of the Working Group on Advanced Vehicle Control Systems (AVCS),” Mobility 2000 Workshop on Intelligent Vehicles and Highway Systems,
March.
Anonymous, 2001, “Digital Driving,” New York Times, June 13.
Anonymous, 2002, “Autos: A New Industry,” Business Week, July 15 (cover story).
Barfield, W., and T. A. Dingus, 1998, Human Factors in Intelligent Transportation Systems,
Lawrence Erlbaum Associates.
Bender, J. G., 1991, “An Overview of Systems Studies of Automated Highway Systems,”
IEEE Transactions on Vehicular Technology, February 1991, Vol. 40, No. 1, pp. 82–99.
Betsold, R. J. and J. H. Rillings, 1990, Final Report of the Working Group on Advanced Driver
Information Systems (ADIS), Dallas, TX: Mobility 2000, March.
Burns, L. D., J. B. McCormick, and C. E. Borroni-Bird, 2002, “Vehicle of Change,” Scientific
American, October, pp. 64–73.
Buxton, J. L., S. K. Honey, W. E. Suchowerskyj, and A. Tempelhof, 1991, “The Travel Pilot:
A Second-Generation Automotive Navigation System,” IEEE Transactions on Vehicular
Technology, February, Vol. 40, No. 1, pp. 41–4.
Capps, G. J., K. P. Gambrell, K. J. Johnson, 2001, Demonstration Project 111 ITS/CVO
Technology Truck, Final Project Report ORNL/TM-2001/277.
Chang, K. S., W. Li, A. Shaikhbahai, F. Assaderaghi, and P. Varaiya, 1991, “A Preliminary
Implementation for Vehicle Platoon Control System,” Proceedings of the American Control
Conference, Boston, MA, June, pp. 3078–83.
Chee, W., and M. Tomizuka, 1995, “Lane Change Maneuvers for Automated Highway Systems: Control, Estimation Algorithm and Preliminary Experimental Study,” Proceedings
of the International Mechanical Engineering Congress and Exposition, San Francisco, CA,
November.
Davis, J. R., 1989, Back Seat Driver: Voice-Assisted Automobile Navigation, Ph.D. dissertation,
Cambridge, Massachusetts Institute of Technology.
Davis, J. R., and C. M. Schmandt, 1989, “The Back Seat Driver: Real-Time Spoken Driving
Instructions,” pp. 146–50 in D. H. M. Reekie, E. R. Case, and J. Tsai (eds.), First Vehicle
Navigation and Information Systems Conference (VNIS’89), IEEE.
Dudek, C. L., R. D. Huchingson, W. R. Stockton, R. J. Koppa, S. H. Richards, and T. M. Mast,
1978, Human Factors Requirements for Real-Time Motorist Information Displays, Volume
I – Design Guide (Technical Report FHWA-RD-78–5), Washington, DC, U.S. Department
of Transportation, Federal Highway Administration.
Eby, D. W., and L. P. Kostyniuk, 1999, “An On-The-Road Comparison of In-Vehicle Navigation Assistance Systems,” Human Factors, Vol. 41, pp. 295–311.
Ervin, R. D., 1991, An American Observation of IVHS in Japan, University of Michigan IVHS
Program Report.
Esterberg, M. A., E. D. Sussman, and R. A. Walter, 1986, Automotive Displays and Controls –
Existing Technology and Future Trends (Report PM-45-U-NHT-86-11), Washington, DC,
U.S. Department of Transportation, Federal Highway Administration, National Highway
Traffic Safety Administration.
Fenton, R. E., and R. J. Mayhan, 1991, “Automated Highway Studies at the Ohio State
University,” IEEE Transactions on Vehicular Technology, Vol. 40, No. 1, pp. 100–113.
319
320
Overview of Intelligent Transportation Systems
Fontaine, H., G. Malaterre, and P. Van Elslande, 1989, “Evaluation of the Potential Efficiency
of Driving Aids,” Vehicle Navigation and Information Systems Conference ProceedingsVNIS’89, pp. 454–9, IEEE.
Ford Motor Company, 1986, “Automotive Electronics in the Year 2000: A Ford Motor
Company Perspective,” Proceedings of the CONVERGENCE ’86 Conference, Dearborn,
MI, October.
Furukawa, Y., 1992, “The Direction of the Future Automotive Safety Technology,” Proceedings of the International Congress on Transportation Electronics, SAE Paper No. 92C013.
Harris, W. J., and G. S. Bridges, 1989, Proceedings of the Workshop on Intelligent Vehicle/
Highway Systems by Mobility 2000, College Station, Texas Transportation Institute, Texas
A&M University.
Haver, D. A., and P. J. Tarnoff, 1991, “Future Directions for Traffic Management Systems,”
IEEE Transactions on Vehicular Technology, February, Vol. 40, No. 1, pp. 4–10.
Hedrick, J. K., M. Tomizuka, and P. Varaiya, 1994, “Control Issues in Automated Highway
Systems,” IEEE Control Systems Magazine, December.
Himmelspach, T., and W. Ribbens, 1989, “Radar Sensor Issues for Automotive Headway
Control Applications,” Fall 1989 Summary Report for the Special Topics Course in IVHS,
Appendix G, University of Michigan.
Jacobs, I. M., A. Salmasi, and T. J. Bernard, 1991, “The Application of a Novel TwoWay Mobile Satellite Communications and Vehicle Tracking System to the Transportation Industry,” IEEE Transactions on Vehicular Technology, February, Vol. 40, No. 1,
pp. 57–63.
Jurgen, R. K., 1991, “Smart Cars and Highways Go Global,” IEEE Spectrum, May, pp. 26–36.
Labiale, G., 1989, Influence of in Car Navigation Map Displays on Drivers Performances [sic],
SAE Paper No. 891683, Warrendale, PA: Society of Automotive Engineers.
LeBlanc, D. J., R. D. Ervin, G. E. Johnson, P. J. Th. Venhovens, G. Gerber, R. DeSonia, C.-F.
Lin, T. Pilutti, and A. G. Ulsoy, 1996, “CAPC: An Implementation of a Road-Departure
Warning System,” IEEE Control Systems Magazine, Vol. 16, No. 6, December, pp. 61–71.
LeBlanc, D. J., P. J. Th. Venhovens, C.-F. Lin, T. Pilutti, R. D. Ervin, A. G. Ulsoy, C.
MacAdam, and G. E. Johnson, 1996, “Warning and Intervention System to Prevent RoadDeparture Accidents,” Vehicle System Dynamics, Vol. 25, Suppl., pp. 383–96.
McMahon, D. H., J. K. Hedrick, and S. E. Shladover, 1990, “Vehicle Modeling for Automated
Highway Systems,” Proceedings of the American Control Conference, San Diego, CA, May,
pp. 297–303.
Michigan Department of Transportation, 1981, Scandi Project, Freeway Operations Division,
November.
Okuno, A., A. Kutami, and K. Fujita, 1990, “Towards Autonomous Cruising on Highways,”
SAE Paper No. 901484.
Oshizawa, H., and C. Collier, 1990, “Description and Performance of NAVMATE, and InVehicle Route Guidance System,” Proceedings of the American Control Conference, San
Diego, CA, May, pp. 782–7.
Proper, A. T. 1999, Intelligent Transportation Systems Benefits: 1999 Update, Federal Highway
Administration Report FHWA-OP-99-012.
Radin, S., B. Sen, J. Lappin, 2000, “Advanced Traveler Information Service (ATIS): Private
Sector Perceptions and Public Sector Activities,” Volpe Center, Boston, MA, January.
Rajamani, R., 2005, Vehicle Dynamics and Control, Springer-Verlag.
Rajamani, R., H. S. Tan, B. K. Law, and W. B. Zhang, 2000, “Demonstration of Integrated
Longitudinal and Lateral Control for the Operation of Automated Vehicles in Platoons,”
IEEE Transactions on Control Systems Technology, Vol. 8, No. 4, July, pp. 695–708.
Shladover, S., 1991, “Research and Development Issues in Intelligent Vehicle/Highway Systems (IVHS),” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced
Automotive Technologies-1991, ASME DE-Vol. 40, New York, pp. 5–9.
Shladover, S., C. A. Desoer, J. K. Hedrick, M. Tomizuka, J. Walrand, W. B. Zhang, D. H.
McMahon, H. Peng, S. Sheikholeslam, and N. McKeown, 1991, “Automatic Vehicle
References
Control Developments in the PATH Program,” IEEE Transactions on Vehicular Technology, February, Vol. 40, No. 1, pp. 114–30.
Special Issue on IVHS, 1991, IEEE Transactions on Vehicular Technology, February, Vol. 40,
No. 1.
Stehr, R. A., 1990, “Motorist Information Systems in Minnesota,” Proceedings of the American
Control Conference, San Diego, CA, May, pp. 287–90.
Trabold, W. G., and T. A. Prewitt, 1969, “A Design for an Experimental Route Guidance
System,” Highway Research Record, No. 265 (Route Guidance), pp. 50–61, Washington,
DC, Highway Research Board, National Research Council (also released as GM Research
Publication GMR-828, January 1968).
Walker, J., C. Sedney, E. Alicandri, and K. Roberts, 1989, In-Vehicle Navigation Devices:
Effects on Driver Performance (Draft Technical Report FHWA/RD-89-xxx), Washington,
DC, U.S. Department of Transportation, Federal Highway Administration.
Zwahlen, H. T., and D. P. DeBald, 1986, “Safety Aspects of Sophisticated In-Vehicle Information Displays and Controls,” Proceedings of the Human Factors Society – 30th Annual
Meeting, pp. 256–60, Santa Monica, CA: The Human Factors Society.
321
18
Preventing Collisions
18.1 Active Safety Technologies
An important motivation for AVCS technologies is safety, and a key safety technology is collision detection and avoidance systems. This type of safety enhancement
is termed “active safety,” which is different from the traditional passive-safety concept (i.e., crashworthiness) (Sun and Chen 2010). The goal is to prevent collisions,
not simply mitigate their effects. There are two major driving forces behind recent
progress in the active-safety area, as follows:
1. The continuous progress in passive-safety systems has pushed the technology
into a return/cost plateau. For example, a recent study shows that 42 percent of
fatal-crash occupants can be saved by safety belts, and 47 percent can be saved
with safety belts plus an air bag (Figure 18.1). For the remainder of accidents, the
impact energy level is simply too high to be managed by reasonable engineering
means using current technology. Most of these high-impact energy impacts,
however, can be avoided altogether by active safety technologies (ASTs).
2. Recent changes in the standards for Corporate Average Fuel Economy (CAFE)
continue to move toward reduced petroleum consumption in the United States.
An important engineering approach for higher fuel efficiency is to lower vehicle
weight; however, this solution is likely to raise safety concerns. It has been verified consistently that vehicle weight is the third-most important safety attribute
for automobiles (i.e., after safety belts and air bags). Again, a possible solution
to this safety concern is to apply ASTs.
Many enabling technologies and subsystems, which are useful for AST, have
been widely available on passenger vehicles since 2005 (Table 18.1). Therefore,
the add-on complexity and cost of introducing AST are greatly reduced. This fact,
together with the obvious diminishing returns from passive-safety devices, has made
active-safety systems increasingly attractive.
18.2 Collision Detection and Avoidance
Human drivers do not always pay close attention to traffic conditions because of distractions (e.g., looking at the mirror, adjusting the radio, or using a cellular phone) or
322
18.2 Collision Detection and Avoidance
323
Table 18.1. Vehicle control functions and active safety technology (AST) enablement
Control function
Installation base
AST function
Cruise control
ABS
TCS
Electronically controlled transmissions
Multiplex
ACC
90%
85%
20%
85%
35%
10%
Throttle actuator
Automatic braking and yaw
Small deceleration and yaw
Down-shifting
Information and control
Range sensor
cannot react in a timely manner because of weather, lighting, or physical disabilities.
The value of collision detection and warning systems is in detecting hazardous situations and providing warning or intervention actions to avoid a collision or mitigate
damage. These systems require the sensing of objects in a vehicle’s near field and the
appropriate interpretation of the sensor signals for purposes of warning or control.
Also, this function must be carried out while a vehicle is traveling at a high speed. To
date, this difficult and important area has not been researched sufficiently. However,
there is considerable related research in areas such as obstacle avoidance in robot
trajectory planning and mobile robots. The problems, although clearly related, are
not identical because robots may not operate at high speeds and are not likely to
encounter other obstacles that may be moving at high speeds. Therefore, the vehicle dynamics is important, whereas the robotic problems usually can be treated as
kinematic or even static problems. Also, robots must act autonomously on the information obtained about obstacles. AVCS systems – at least AVCS-I and AVCS-II
systems – are expected to provide information on obstacles to a driver, or perhaps
warnings, but not to take over control of the vehicle. Studies show that many accidents can be avoided if a driver has an additional 0.5 to 1 second in which to take
evasive action. Thus, the solution to the obstacle-detection and warning problem is
important and is expected to provide significant safety benefits.
One of the key enabling technologies for a collision detection and avoidance
system is the development of sensors. Ultrasonic, infrared, laser, video imaging, and
Fatality Prevention Effectiveness
200
46%
+1190 lbs.
+1050 lbs.
42%
Equal to
Added
Vehicle
Weight
+380 lbs.
0
–200 –140 lbs.
–400
–600
–800
18%
Weight
Effects
with
Belt and
Air Bag
Unbelted
Air Bag Only
Lap/
Shoulder
Belted
Belted
with
Air Bag
Figure 18.1. Diminishing returns for passive safety systems.
324
Preventing Collisions
radar have been suggested as the most plausible sensing devices, some of which have
been demonstrated in applications The collision-detection functions are divided into
the following two general groups:
1. Near-obstacle detection: The application scenarios are low speed and/or low
range (i.e., rear and blind-spot detection). The detecting signals are usually wide
beam.
2. Forward-looking sensors and detection: For high-speed applications, the range
needs to be much higher than those for near-obstacle (e.g., approximately 300
versus 30 feet). The beam width is usually much narrower so that vehicles in
neighboring lanes are not constantly detected. For control-design purposes, it
is desirable to have both range and range-rate information. The same sensing
requirement can be used for ACC, which is one reason that the development of
forward-looking sensors has received significant attention.
Figure 18.2 illustrates a typical scenario for obstacle detection of a vehicle in a
highway environment. A long-range forward-looking detection system (e.g., radar)
could be coupled with a shorter range forward- and rear-obstacle detection system
(e.g., ultrasonic) or even a short-range side-detection system. Some blind spots may
remain depending on sensor placement and range. The presence of many stationary
roadside objects (e.g., trees and shrubs, guardrails, and signposts) increases the complexity of the problem of processing sensor information. The use of computer vision
also has been considered but can be limited by computation times necessary for
image processing. In an ITS setting, some of these sensors may be on the highway
infrastructure – for example, induction coils or other vehicle detectors (Michigan
Department of Transportation 1981) – as well as on vehicles. The information available on a vehicle may be from its own near-field sensors and/or transmitted to it
from other vehicles and the roadway. For example, for merging maneuvers onto a
freeway, installing overhead vision systems at the on-ramp site used to signal vehicles
when it is safe to merge has been proposed (Pilutti et al. 1990). Such an overhead
vision system is being used in a SMART CRUISE demonstration project in Tsukuba
City, Japan, to monitor traffic flows for use in vehicle-navigation systems.
Ultrasonic sensors, already used in many applications such as autofocus cameras
and mobile robots, typically are limited to an approximate 10-meter range and are
not suitable for many highway applications. However, they may be useful (and have
been used) in vehicles for near-field obstacle detection at low speeds – for example,
to detect the curb in parallel parking and obstacles in a driveway when a driver is
backing out. The sensors can operate in conditions such as fog, smoke, and darkness
when visibility is poor. They also can be used in nonhighway vehicles, such as the
autonomous guided vehicles (AGVs) used in manufacturing plants and warehouses.
Radar is considered a leading obstacle-sensing technology for possible use in
automotive applications but there are important issues, including the appropriate
frequency band to be used, possible interference between similarly equipped vehicles, and reflected radiation from road surfaces. Infrared sensors can be used to
supply range information. Alternatively, radar can be used to enhance a driver’s
perception – allowing for better visibility and longer reaction times – in night driving
and other conditions when visibility is poor. Prototype systems have been developed
and demonstrated for use with vehicles.
Lane
width
3.75m
Lane
width
3.75m
0
50
100
[m]
200
Distance
Figure 18.2. Representative forward-looking sensor azimuth coverage.
0
50
100
[m]
200
325
326
Preventing Collisions
When the range and possibly the range-rate information are obtained, the
collision-warning algorithm typically is triggered based on the following two types
of indexes, both of which are popular and used widely:
Time to Collision (TTC): The simple use of range over range rate gives a reasonable index for collision warning. Depending on engineering judgment, this
index usually is selected to be between 5 and 8 seconds.
Safe Distance: Based on physics, a “safe following distance” d can be constructed,
which has the form:
d = Vctd +
V2
Vc2
− l
2ac
2al
where Vc is the controlled (i.e., host) vehicle speed and td is the delay time,
which consists of human neuromuscular delay plus judgment time. Vl is the
lead vehicle speed and ac and al are deceleration limits, which usually are
assumed to be the same but can change with the environment (e.g., 4.5 m/sec2
for a dry road and 3.3 m/sec2 for a wet road).
Most collision-warning systems have multiple warning levels: the first level usually leaves judgment time for a driver to take action; the second level is triggered if a
threat becomes imminent. The second level may involve intervention – for example,
automated braking or differential braking (Pilutti et al. 1998) – as well as warnings
to the driver. Nevertheless, it is important that the driver maintain ultimate control
and be able to override any automated intervention.
Collision-avoidance systems consider and are developed for various scenarios,
such as longitudinal collisions, road-departure accidents, vehicle rollover, and
jackknifing of articulated vehicles. The TTC and safe-distance metrics described
previously are suitable for longitudinal (i.e., accelerating or braking) collision
scenarios. For single-vehicle road-departure (SVRD) accidents, which account for
nearly a fourth of all highway accidents and a third of all fatalities, other metrics such
as the time to lane crossing (TLC) are described in LeBlanc et al. (1996) and Lin and
Ulsoy (1996). Such systems also may require additional sensors – for example, to
measure yaw rate (Sivashankar and Ulsoy 1998) – and may include systems to assess
the driver state (Pilutti and Ulsoy 1999). Other active-safety systems address serious
scenarios such as vehicle rollover and jackknifing of trucks (Chen et al. 2010; Chen
and Peng 1999; Ma and Peng 1999b). The next generation of active-safety systems
for vehicles also is expected to include inter-vehicle communications, allowing for
the sharing of information among vehicles to achieve cooperative active-safety
systems (Caveney 2010).
EXAMPLE 18.1: PREVENTING RUN-OFF-ROAD ACCIDENTS. In this example (revisited
in Chapter 20), we consider an active-safety system to prevent SVRD accidents
(LeBlanc et al. 1996). The system is based on the TLC metric, which is computed
from two key elements (Lin and Ulsoy 1996): (1) a projection of the vehicle path;
and (2) an estimate of the upcoming lane geometry (Lin et al. 1999). A vision system is used to look head and provide information about the upcoming roadway
geometry. On-vehicle sensors also are used to provide information about vehicle motion, which then is used with a vehicle model and disturbance estimation
(Lin et al. 2000), such as in Eq. (4.49), to determine the path projection. Both
18.2 Collision Detection and Avoidance
327
File Model Simulation Disturbances Roadway Controllers Scenarios Window
CAPC Animation
Near-field Camera
Far-field Camera
CAPC Animation
0
t = 18.20
100
TLC = 1.83
WARNING !!
Figure 18.3. Simulation environment for design and evaluation of the SVRD active safety
system.
the path projection and the roadway-geometry estimation use Kalman filters
to best combine the sensor information with model information and to reduce
uncertainty in the TLC calculations.
When the TLC is below a certain threshold value, the system issues a warning
to the driver. This can be thought of as an “electronic rumble strip,” similar to
a physical rumble strip located along the edge of a highway that generates a
sound audible to the driver when the wheels are outside the lane edge (Pilutti
and Ulsoy 2003. These electronic rumble-strip systems have been implemented
in some long-range trucking vehicles, in which SVRD due to driver drowsiness
is a serious concern.
The system has the capability to not only warn the driver but, in some
cases, also to intervene or even fully control the vehicle when the driver is not
responsive. Therefore, the system includes the capability to assess driver status
(i.e., alert, drowsy, or unresponsive), which is accomplished by utilizing a driver
model. Input to the driver is assumed to be the view captured by the onboard
vision system and the output of the driver is the measured steering-wheel angle
(Pilutti and Ulsoy 1999). For example, if the camera detects a steady right turn
ahead and the driver does not respond by steering to the right, this indicates
that the driver is inattentive and may be drowsy or even asleep. In this case,
the system uses differential braking to generate a yaw moment (i.e., to steer the
vehicle to the right) (Pilutti et al. 1998).
As shown in Figure 18.3, a detailed simulation environment was used to develop
and evaluate this active safety system for preventing SVRD accidents. In addition,
328
Preventing Collisions
driving simulators and road tests were used to evaluate and refine the proposed
system.
PROBLEMS
1. A dead-reckoning system for vehicle navigation relies on integration of a velocity
signal to determine vehicle position:
x(t ) =
t
v(τ )dτ + x(0)
0
If discrete-time measurements are made at time intervals, T, this can be approximated by:
x(t ) =
t=NT
$
v(i)T + x(0)
i=0
The dead-reckoning approach can lead to position errors when there are errors in
the measured velocity values. Assume that the vehicle-measured velocity signal is
v(t) = sin(2π t) + e(t) and that T = 0.02. Consider the following cases and, for each
one, determine x(t = 0.5) given x(0) = 0:
(a) e(t) = 0 (i.e., no measurement error)
(b) e(t) = normal random error with zero mean and a standard deviation
of 1
(c) e(t) = 1.0 is a constant
Use MATLAB to compare and discuss the differences in the value of x(0.5) in each
case.
2. Consider a vehicle with velocity vf = 31 m/s that follows a lead vehicle with
velocity vl = 30 m/s at a distance of 10 m. The total delay time, td = 1 sec, and
acceleration/deceleration limits for both vehicles are af = al = 3 m. What are the
TTC and the safe distance? How do you interpret these metrics in the context of this
scenario?
3. Consider a simple scenario in which a vehicle is driving at constant longitudinal
velocity, u, along the centerline of a lane in a straight section of roadway (i.e.,
no curvature) with a constant heading angle, ψ. Assuming that u and ψ are both
measured exactly, show that the TLC can be calculated as:
TLC = (/ sin ψ )/u
References
329
where x is the distance along the lane, as shown in the following figure:
∆x
ψ
REFERENCES
Caveney, D., 2010, “Cooperative Vehicular Safety Applications,” IEEE Control Systems
Magazine, Vol. 30, No. 4, August, pp. 38–53.
Chen, B-C., and H. Peng, 1999, “Rollover Warning of Articulated Vehicles Based on a
Time-to-Rollover Metric,” American Society of Mechanical Engineers, Dynamic Systems
and Control Division (Publication) DSC, Vol. 67, pp. 247–54, Proceedings of the ASME
International Mechanical Engineering Congress and Exposition, Nashville, TN.
Chen, L. K., and A. G. Ulsoy, 2002, “Design of a Vehicle Steering Assist Controller Using
Driver Model Uncertainty,” International Journal of Vehicle Autonomous Systems, Vol. 1,
No. 1, pp. 111–32.
Chen, L. K., and A. G. Ulsoy, 2006, “Experimental Evaluation of Steering Assist Controllers
on a Driving Simulator,” Vehicle System Dynamics, Vol. 44, No. 3, March, pp. 223–45.
Chen, S., T. B. Sheridan, H. Kusunoki, and N. Komoda, 1995, “Car Following Measurements,
Simulations, and a Proposed Procedure for Evaluating Safety,” Proceedings of the IFAC
Symposium on Analysis, Design and Evaluation of Man-Machine Systems, Pergamon Press,
pp. 603–8.
Chen, S. K., N. Moshchuk, F. Nardi, and J. Ryu, 2010, “Vehicle Rollover Avoidance,” IEEE
Control Systems Magazine, Vol. 30, No. 4, August, pp. 70–85.
Davis, J. R., 1989, Back Seat Driver: Voice Assisted Automobile Navigation, Ph.D. Dissertation,
Cambridge, Massachusetts Institute of Technology.
Davis, J. R., and C. M. Schmandt, 1989, “The Back Seat Driver: Real Time Spoken Driving
Instructions,” pp. 146–50 in D. H. M. Reekie, E. R. Case, and J. Tsai (eds.), Proceedings of
the IEEE Vehicle Navigation and Information Systems Conference (VNIS’89).
Fontaine, H., G. Malaterre, and P. Van Elslande, 1989, “Evaluation of the Potential Efficiency
of Driving Aids,” Proceedings of the IEEE Vehicle Navigation and Information Systems
Conference (VNIS’89).
Hatsopoulos, N., and J. A. Anderson, 1992, “Collision-Avoidance System Based on Optical
Flow,” Proceedings of the Intelligent Vehicles ’92 Symposium, Detroit, June–July.
Himmelspach, T., and W. Ribbens, 1989, “Radar Sensor Issues for Automotive Headway
Control Applications,” Fall 1989 Summary Report for the Special Topics Course in IVHS,
Appendix G, University of Michigan.
Hirano, M., 1993, “Development of Vehicle-Following Distance Warning System for Trucks
and Buses,” Proceedings of the IEEE-IEE Vehicle Navigation and Information Systems
Conference, Ottawa, October.
330
Preventing Collisions
Jones, W. D., 2002, “Building Safer Cars,” IEEE Spectrum, January, pp. 82–5.
Kaempchen, N., K. C. Fuerstenberg, A. G. Skibicki, and K. C. J. Dietmayer, 2004, “Sensor
Fusion for Multiple Automotive and Safety Comfort Applications,” in Advanced Microsystems for Automotive Applications, pp. 137–63, Springer, Berlin.
Kehtarnavaz, N., J. S. Lee, and N. C. Griswold, 1990, “Vision-Based Convoy Following by
Recursive Filtering,” Proceedings of the American Control Conference, San Diego, CA,
May, pp. 268–72.
Kehtarnavaz, N., and W. Sohn, 1991, “Steering Control of Autonomous Vehicle by Neural
Networks,” Proceedings of the American Control Conference, Boston, MA, June, pp. 3096–
101.
Kimbrough, S., 1991, “Coordinated Braking and Steering Control for Emergency Stops and
Accelerations,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced
Automotive Technologies–1991, ASME DE-Vol. 40, New York, pp. 229–44.
Kimbrough, S., and C. Chiu, 1990, “Automatic Steering System for Utility Trailers to Enhance
Stability and Maneuverability,” Proceedings of the American Control Conference, San
Diego, CA, May, pp. 2924–9.
Kimbrough, S., and C. Chiu, 1991, “A Brake Control Algorithm for Emergency Stops (Which
May Involve Steering) of Tow-Vehicle/Trailer Combinations,” Proceedings of the American
Control Conference, Boston, MA, June, pp. 409–14.
LeBlanc, D. J., R. D. Ervin, G. E. Johnson, P. J. Th. Venhovens, G. Gerber, R. DeSonia, C.-F.
Lin, T. Pilutti, and A. G. Ulsoy, 1996, “CAPC: An Implementation of a Road-Departure
Warning System,” IEEE Control Systems Magazine, Vol. 16, No. 6, December, pp. 61–71.
LeBlanc, D. J., P. J. Th. Venhovens, C.-F. Lin, T. Pilutti, R. D. Ervin, A. G. Ulsoy, C.
MacAdam, and G. E. Johnson, 1996, “Warning and Intervention System to Prevent RoadDeparture Accidents,” Vehicle System Dynamics, Vol. 25, Suppl., pp. 383–96.
Lin, C. F., and A. G. Ulsoy, 1996, “Time to Lane Crossing Calculation and Characterization
of Its Associated Uncertainty,” ITS Journal, Vol. 3, No. 2, pp. 85–98.
Lin, C. F., A. G. Ulsoy, and D. J. LeBlanc, 1999, “Lane Geometry Perception and the Characterization of Its Associated Uncertainty,” ASME Journal of Dynamic Systems, Measurement
and Control, Vol. 121, No. 1, March, pp. 1–9.
Lin, C.F., A. G. Ulsoy, and D. J. LeBlanc, 2000, “Vehicle Dynamics and External Disturbance
Estimation for Vehicle Path Prediction,” IEEE Transactions on Control System Technology,
Vol. 8, No. 3, May, pp. 508–18.
Ma, W.-H., and H. Peng, 1999a, “Worst-Case Evaluation Method for Dynamic Systems,” ASME Journal of Dynamic Systems, Measurement and Control, Vol. 121, No. 2,
pp. 191–9.
Ma, W.-H., and H. Peng, 1999b, “Worst-Case Vehicle Evaluation Methodology – Examples on
Truck Rollover/Jackknifing and Active Yaw Control Systems,” Vehicle System Dynamics,
Vol. 32, No. 4, pp. 389–408.
Michigan Department of Transportation, 1981, Scandi Project, Freeway Operations Division,
November.
Nguyen, H. G., and J. Y. Laisne, 1992, “Obstacle Detection Using Bi-Spectrum CCD Camera
and Image Processing,” Proceedings of the Intelligent Vehicles ’92 Symposium, Detroit,
June–July.
Niehaus, A., and R. F. Stengel, 1991, “An Expert System for Automated Highway Driving,”
IEEE Control Systems Magazine, Vol. 11, No. 3, April, pp. 53–61.
Pilutti, T., U. Raschke, and Y. Koren, 1990, “Computerized Defensive Driving Rules for
Highway Maneuvers,” Proceedings of the American Control Conference, San Diego, CA,
May.
Pilutti, T., and A.G. Ulsoy, 1999, “Identification of Driver State for Lane Keeping Tasks,”
IEEE Transactions on Systems, Man and Cybernetics, Vol. 29, No. 5, September, pp. 486–
502.
Pilutti, T., and A. G. Ulsoy, 2003, “Fuzzy Logic Based Virtual Rumble Strip for Road Departure Warning Systems,” IEEE Transactions on Intelligent Transportation Systems, Vol. 4,
No. 1, March, pp. 1–12.
References
Pilutti, T., A. G . Ulsoy, and D. Hrovat, 1998, “Vehicle Steering Intervention Through Differential Braking,” ASME Journal of Dynamic Systems, Measurement and Control, Vol. 120,
No. 3, September, pp. 314–21.
Schneider, M., 2005, “Automotive Radar – Status and Trends,” Proceedings of the German
Microwave Conference, Ulm, Germany, April, pp. 144–7.
Sivashankar, N., and A. G. Ulsoy, 1998, “Yaw Rate Estimation for Vehicle Control Applications,” ASME Journal of Dynamics, Measurement, and Control, Vol. 120, No. 2, June,
pp. 267–74.
Sun, Z., and S. K. Chen, 2010, “Automotive Active Safety Systems,” IEEE Control Systems
Magazine, Vol. 30, No. 4, August, pp. 36–7.
Thorpe, C., M. H. Hebert, T. Kanade, S. A. Shafer, 1988, “Vision and Navigation for the
Carnegie-Mellon NAVLAB,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 10, No. 3, May, pp. 362–73.
Truett, R., 1992, “The Rides of Their Lives,” Orlando Sentinel, March 25, p. B-8.
Ulke, W., R. Adomat, K. Butscher, and W. Lauer, 1994, “Radar-Based Automotive Obstacle
Detection System,” SAE Paper No. 940904.
Yasui, Y., and D. L Margolis, 1992, “Lateral Control of Automobiles Using a Looking-Ahead
Sensor,” AVEC ’92, pp. 292–7.
Zwahlen, H. T., and D. P. DeBald, 1986, “Safety Aspects of Sophisticated In-Vehicle Information Displays and Controls,” Proceedings of the Human Factors Society Annual Meeting,
pp. 256–60, Santa Monica, CA: The Human Factors Society.
331
19
Longitudinal Motion Control and Platoons
This chapter discusses the longitudinal control of vehicle motion in the context of
AVCS for ITS. It begins with a slightly refined version of a cruise-control system
with preview, which uses site-specific information available through the highway
infrastructure. It then builds on the ACC topic of Chapter 12 as an intermediate step
to the control of vehicles operating in platoons.
19.1 Site-Specific Information
It is assumed that highway infrastructure can provide information to drivers and vehicles, which can be useful in numerous ways; for example, drivers can be warned of
accidents, roadwork, inclement weather or road-surface conditions, congestion, and
roadway characteristics such as grade and curvature. This already is being implemented in many areas using programmable road signs on busy urban Interstate
highways (see Chapter 17). Here, we consider specifically the use of site-specific
information to improve the performance of vehicle control systems discussed in previous chapters. The manner in which such site-specific information can be provided
to a vehicle is discussed first and then the use of such information by vehicle control
systems.
Site-specific information can be provided to drivers and vehicles in a variety of
ways. For example, the Prometheus project in Europe uses transmitters on signposts
located along certain roadways in the Autocruise project to provide references to the
cruise-control system. A driver can choose to set the cruise-control system to pick up
speed-limit information transmitted from roadside signposts and then use this information as the reference speed for the cruise-control system. Another concept for
providing site-specific roadside information, illustrated in Figure 19.1, involves the
use of message chips embedded in pavement and placed along a roadside or on roadway bridges, overpasses, and signposts. The chips are similar to those already in use in
manufacturing plants and warehouses for inventory control and parts identification:
erasable programmable read only memory (EPROM) devices with a transmitter,
which can be reprogrammed remotely by authorized vehicles to update the information they contain and transmit. These systems have been tested in under-thepavement and roadside use and demonstrated to be feasible. In both systems, there
are issues such as placement, range, durability, tamper-resistance, and power sources.
332
19.1 Site-Specific Information
Figure 19.1. Use of message chips to provide site-specific information.
Other key issues include: What kind of information should they contain and transmit? How frequently should they be spaced? How “accurate” does the information
they transmit need to be? How often does the information need to be updated?
These issues require an understanding of the way in which the information may be
used and the potential benefits to be derived from its use.
Here, we consider use of site-specific information by the vehicle control systems. If it is available, can this information be used to improve the capabilities and
performance of in-vehicle control systems? The Autocruise system mentioned previously indicates that the answer may be affirmative. In that system, speed-limit
information is provided to the vehicle as a reference for the cruise-control system
and potentially relieves drivers of the need to reset the desired speed with changing
road conditions. Similarly, if road-roughness information is available, it may be used
by an active-suspension system to change automatically ride characteristics (e.g.,
soft versus typical versus harsh) rather than requiring drivers to make the selection.
Additional possibilities are as follows:
r Lane and road-curvature information could be provided for automated vehicle
lateral control, supplementing or alleviating the need for lane marking with
magnets or an onboard lane-sensing capability.
r Information on road-surface conditions (e.g., an icy bridge) could be provided
for use by TCS for antilock braking and anti-spin acceleration.
r Road-roughness information could be provided for use by the active-suspension
system not only to select ride characteristics but also to improve the control
algorithm by providing an estimate of the previously unknown road input.
r Altitude information could be useful for engine-control functions such as the
air–fuel ratio control.
r In addition to speed-limit information, road-grade information could be
useful for improving cruise-control-system performance, which is discussed in
Example 19.1.
EXAMPLE 19.1: CRUISE CONTROL WITH PREVIEW BASED ON SITE-SPECIFIC INFOR-
We first consider the use of site-specific information to improve the
performance of a simple cruise-control system (see Chapter 12). This assumes
that the highway infrastructure provides information to vehicles, specifically
road-grade information. Thus, if a vehicle can be warned of an upcoming grade,
the cruise control should be able to take advantage of this information to improve
speed-regulation performance. Figure 19.2 illustrates the open-loop system and
the closed-loop system (with PI control) block diagrams for a cruise-control
MATION.
333
334
Longitudinal Motion Control and Platoons
θ
θ
Openloop
− mgsin(.)
− mgsin(.)
d
F
+
+
vr +
1/ c
(m/c)s + 1
Feedback
(PI control)
v
-
Ki
Kp + s
d
F
+
+
1/ c
(m/c)s + 1
v
Figure 19.2. Open-loop and example closed-loop cruise-control systems.
system (see Chapter 12). As a basis for comparison, the response of each system
to an uphill grade of θ = 10◦ is shown in Figure 19.3.
% Ex19_1.m
m=1000; c=85; g=9.8;
theta=pi/18;
K=1/c;
tau=m/c;
disturb=-m*g*sin(theta);
Vd=25.0; Fo=Vd/K;
Ki=100.0;
Kp=500; Ti=Kp/Ki;
vo(1)=Vd;
vc(1)=Vd;
t(1)=0.0;
integ=Vd/(K*Ki);
d=0.0;
tstep=0.1;
for i=1:999,
t(i)=i*tstep;
if i>300, d=disturb; end
vdot_o=[K*(Fo+d)-vo(i)]/tau;
vo(i+1)=vo(i)+vdot_o*tstep;
integ = integ + (Vd-vc(i))*tstep;
Fc=Kp*(Vd-vc(i))+Ki*integ;
vdot_c=[K*(Fc+d)-vc(i)]/tau;
vc(i+1)=vc(i)+vdot_c*tstep;
end
t(1000)=100.0;
plot(t,vo,t,vc), xlabel(‘Time (sec)’)
gtext(‘Openloop’)
gtext(‘PI control’)
30
25
PI control
20
Figure 19.3. Vehicle speed under gradient
disturbance.
15
Openloop
10
5
0
20
40
60
Time (sec)
80
100
19.1 Site-Specific Information
335
θˆ
θ
−sT
ˆ
−mgsin(.)e
vr +
Ki
Kp + s
-
−mgsin(.)
dˆ +
d +
F
+
+
Figure 19.4. Feed-forward and preview
control systems.
θˆ
Ki
Kp + s
-
dˆ +
−mgsin(.)
d
F
+
Feedback +
Preview
Control
v
1/c
(m / c)s +1
+
+
v
1/ c
(m / c) s + 1
θ
ˆ θˆ , T, p)
f(m,
vr +
Feedback +
Feedforward
Control
Figure 19.4 illustrates two distinct ways in which the site-specific grade information can be used (1) the use of feed-forward control together with the previous
PI feedback controller; and (2) a similar approach that uses a feed-forward action
based on preview (p) (i.e., advance knowledge of an upcoming grade). If our knowledge of the vehicle parameters (m and c), grade disturbance (), and timing of
the grade (T) was exact, then the effect of the grade could be canceled exactly by
the feed-forward action, as shown in Figure 19.5a. However, these variables usually
are not exactly known, and a more realistic performance is shown in Figure 19.5b
for the feed-forward control strategy. Here, it is assumed that d is underestimated
by 10 percent and that there is a 1-second delay (T = 1) in the application of the
feed-forward compensation.
26
(a) PI + exact FF control
25.5
25
24.5
24
Figure 19.5. Simulation results of PI + FF
cruise-control systems.
0
20
40
25.5
60
80
100
(b) PI + inexact FF control
25
24.5
24
23.5
0
20
40
60
80
100
336
Longitudinal Motion Control and Platoons
25.4
PI + inexact preview control
25.2
25
Figure 19.6. Simulation results of PI + inexact preview cruise-control system.
24.8
24.6
0
20
40
60
Time (sec)
80
100
% Ex19_1a.m
m=1000.0; c=85.0; g=9.8;
theta=pi/18.0;
K=1/c;
tau=m/c;
disturb=-m*g*sin(theta);
disturb_h=disturb*0.9;
Vd=25.0;
Ki=100.0;
Kp=500.0; Ti=Kp/Ki;
vc(1)=Vd;
t(1)=0.0;
integ=Vd/(K*Ki);
d=0.0; d_h=0.0;
tstep=0.1;
for i=1:499,
t(i)=i*tstep;
if i>200, d=disturb; end
if i>210, d_h=disturb_h; end
integ = integ + (Vd-vc(i))*tstep;
Fc=Kp*(Vd-vc(i))+Ki*integ-d_h;
vdot_c=[K*(Fc+d)-vc(i)]/tau;
vc(i+1)=vc(i)+vdot_c*tstep;
end
t(500)=100.0;
plot(t,vc), xlabel(‘Time (sec)’)
title(‘PI + inexact FF control’)
With preview control, the driver anticipates that a grade is coming and begins
to increase vehicle speed before the grade is encountered. The preview is especially
important when the control signal does not affect the dynamics of the vehicle at the
same point, which is true in most cases. Figure 19.6 illustrates this strategy in which a
ramp preview signal is applied, starting 0.5 second before and ending 0.5 second after
the disturbance occurs. Again, the same inaccuracies used previously (i.e., 10 percent
underestimation) are used to obtain the results shown in Figure 19.6. It is clear
from these results that the preview control provides better performance in the case
of inexact knowledge. However, the improvements obtained by using site-specific
information are minor in this application and probably would not justify its use.
19.2 Platooning
However, these small improvements might become more attractive in applications
such as ACC and platooning, in which the performance requirements are more
demanding.
A vehicle cruise-control system (see Chapter 12) regulates the vehicle longitudinal velocity. As an intermediate step between platooning and cruise control, the
control of the vehicle longitudinal motion (i.e., velocity and position) was considered in Chapter 12. The idea of ACC is to allow a vehicle to regulate speed when
there are no automobiles in front of it. However, when a vehicle is detected, the
controller regulates the relative distance between the two vehicles (i.e., “headway”).
Such a system (i.e., autonomous or intelligent cruise control) could be effective, for
example, in stop-and-go traffic as well as on the highway and could be used in conjunction with the Autocruise system described previously. The next section discusses
the control of vehicle platoons.
%
%
%
%
%
Exactly the same program except that
the preview action uses the following
smoothed signal instead of the
step change signal in the previous
example
if (i>195) & (i<=205),
d_h=disturb_h*(i-195)/10;
end
19.2 Platooning
The goal in platooning is to improve throughput on congested highways by allowing
groups of vehicles (e.g., 10 to 20) to travel together in tightly spaced platoons (e.g.,
1-m intervals) at high speeds (e.g., 30 m/s). The spacing between platoons may be
somewhat larger than current vehicle headway on the highway (e.g., 30 m). The
potential advantages in terms of increased throughput are shown to be significant;
however, there also are serious potential problems in terms of safety and human
factors. In addition to how drivers and passengers will react to traveling at such
high speeds in such close formation, there are concerns about how they will react
in any type of emergency situation. In terms of safety, the vehicles in a platoon
are too closely spaced to prevent collision in the case of a failure – which causes
rapid deceleration of a car! However, it also can be argued that many human drivers
already are practicing platooning on current highways. It is claimed that close intraplatoon spacing actually may improve safety over current highways spacings because
the relative speed at impact in most cases will be small.
Platooning requires another level of control beyond individual vehicles (Figure
19.7). Two fundamentally different approaches to platooning have been suggested:
(1) point-following control, in which each vehicle is assigned a particular moving
slot on the highway and maintains that position; and (2) vehicle-following control, in
which each vehicle in the platoon regulates its position relative to the vehicle in front
of it based on information about the lead vehicle motion and locally measured variables (i.e., its own motion and headway to the vehicle in front). The two approaches
are compared in Table 19.1. Here, we discuss the vehicle-following control approach,
337
338
Longitudinal Motion Control and Platoons
Table 19.1. Comparison of longitudinal control approaches
Vehicle-follower control
Point-follower control
Advantages
Extremely flexible in accommodating trains or
platoons of vehicles of diverse length.
Advantages
Simple vehicle control implementation – each
vehicle follows a well-defined target.
Spacing between vehicles can be adjusted as speeds
change.
Only one communication path needed (to
roadway).
Speeds can be adjusted easily to adapt to demand
shifts or incidents.
Merging, routing, and scheduling simplified by
fixed, discrete spacing increments.
Communication burden of roadway is
minimized.
Decouples vehicle control from system
management.
Some operations can continue even when wayside
equipment fails (some fault tolerance).
Consistent with very decentralized routing and
scheduling.
Normal and emergency control modes are very similar.
Disadvantages
Control system on vehicle is complicated: sensing
spacing and speed (and maybe acceleration) relative to
predecessor; communicating information about other
vehicles in platoon; must design to minimize
interactions among vehicles (string stability); separate
mode of operation for leading vehicle in platoon.
Disadvantages
Slot length must be long enough for worst-case
condition, limiting capacity.
Vehicles need multiple communication paths – to and
from other vehicles (for control) and the roadway (for
routing and scheduling).
Not easily adaptable to sudden demand shifts.
Flow instabilities are possible if something goes wrong.
Failure of a vehicle or a wayside system
produces shutdown.
Separate communication and control means
needed to handle anomalies and failures.
Not suitable for trains or platoons of variable
length.
Figure 19.7. Platooning requires another upper level of control
(of the platoon) beyond the control of the individual vehicle.
19.2 Platooning
339
Leader Acc.
Pulse
1/s
vf
1/s
a1
v
vf
a2
a
af ao
vf
x
gvehicle vo
a
xo
a3
af ao
v1
a
v
gvehicle vo
x
xo
x1
a4
ao
a
vf
af ao
a
v3
v
gvehicle vo
x
xo
x2
vf
v gvehicle vo
a5
a5
v5
v5
af ao
v4
v gvehicle vo
x3
Vehicle 2
Vehicle 1
vf af
v2
x5
x4
x
Vehicle 3
xo
x
Vehicle 4
x5
xo
Vehicle 5
(a)
vf
Noise
CV
DA
a
Prop.Lag
DT
da1
+
−1.0
CA
S/H
+
Noise
JLim
ALim
j1
1/S
AUINV
+
a0
−1.0
Noise
DU
v
DT
1/s
da1
+
CVE
S/H
CV
v0
−1.0
Noise
DX
x
+
DT
1/s
dx1
S/H
CX
x0
−1.0
(b)
Figure 19.8. (a) Model of a platoon of five vehicles. (b) Model of an individual vehicle in the
platoon (Shladover 1991a).
which is the focus of most current research and development work in the area. We
note that platooning requires another level of control beyond the individual vehicle
(i.e., control of the platoon).
EXAMPLE 19.2: SIMULATION OF PLATOONS. Figure 19.8a is a model for a platoon
of five vehicles and a model for a single vehicle in the platoon. As shown in
340
Longitudinal Motion Control and Platoons
Figure 19.8b, each vehicle is represented by a third-order nonlinear model.
This arises from representation of the vehicle as a mass plus a propulsion system
(which has a first-order lag between commanded and actual thrust). Acceleration
and jerk limits are explicitly represented (i.e., Alim and Jlim) as saturation elements, and the first-order propulsion lag is represented by the block TAUINV.
Thus, each vehicle is represented by equations of motion of the following
form:
ȧ0 = −(1/τ )a0 + (1/τ )u
v̇0 = a0
ẋ0 = v0
where we ignore nonlinearities (i.e., saturation and quantization) and noise.
Studies show that to maintain asymptotic stability of the platoon, each vehicle
requires direct feedback of information from the platoon leader as well as from
the immediate predecessor. Figure 19.8a shows that each vehicle in the platoon
is supplied with the following input data: (1) acceleration, velocity, and position
(a, v, x) of the preceding vehicle; and (2) acceleration and velocity of the lead
vehicle (af , vf ). Each vehicle then generates as output its own acceleration,
velocity, and position (a0 , v0 , x0 ). The controller gains are CX (for vehicle
spacing errors), CVE (for velocity errors), CA (for acceleration), and CV (for
difference in velocity from the commanded value for the platoon). Thus, the
controller algorithm for each vehicle can be represented by:
u = cx (x − x0 ) + cv (v f − v0 ) + cve (v − v0 ) + ca (a − a0 ) + ca f (a f − a0 )
Note that blocks also are included in the simulation model for quantization
errors, errors due to sampling, and noise. Enhancements to and variations of this
basic model are discussed in Shladover (1991). Simulation results in Figure 19.9 show
various studies for a platoon velocity-change command (shown as Pulse in Figure
19.8a), which is an acceleration pulse of magnitude 1.0 m/s2 and a duration of 2 s; this
produces a commanded speed increase of 2.0 m/s. These results were obtained with
position and velocity feedback but no acceleration feedback. Notice that the errors
in spacing between vehicles remain less than 0.3 m for all vehicles. These results
are for a propulsion lag time constant of 0.1 s, which is achievable with an electric
powertrain but not with an ICE. Using the same controller with a propulsion time lag
more representative of an ICE does not yield acceptable performance. Figure 19.10
shows results obtained with a time constant of 0.5 s, which is more representative of
an ICE, and a controller, which uses acceleration feedback in addition to position
and velocity. The control gains for the two cases (i.e., results in Figures 19.9 and
19.10) are summarized in Table 19.2.
It has been shown that maintaining the stability of a platoon of vehicles at
high speeds and tight spacing requires inter-vehicle communication (Swaroop and
Hedrick, 1996). A large-scale demonstration of vehicle platoons, under automatic
lateral and longitudinal control, was successful, as described in Rajamani et al.
(2000).
19.2 Platooning
341
2.5
3
2
4
3
5
1
2.0
1
VELOCITY, M/S
2
JERK, M/S • • 3
1.5
1.0
0.5
2
1
0
5
3
−1
0.0
4
−2
−0.5
−3
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
(a)
7
8
9
10
9
10
(b)
1.4
0.3
VELOCITY DIFFERENCES, M/S
2
1
1.2
ACCELERATIONS, M/S2
6
1.0
0.8
0.6
0.4
0.2
0.0
−0.2
5
4
3
1
0.2
2
3
4
0.1
5
0
−.1
−.2
−.3
−0.4
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
(c)
5
6
7
8
(d)
0.30
1
SPACING DIFFERENCES, M
0.25
2
3
4
0.20
5
0.15
0.10
0.05
0.00
−.05
0
1
2
3
4
5
6
7
8
9
10
(e)
Figure 19.9. Five platoon simulation results with τ = 0.1 second: (a) velocity time history, (b)
jerk time history, (c) acceleration time history, (d) velocity difference time history, (e) spacing
time history (Shladover 1991a).
342
Longitudinal Motion Control and Platoons
4
2.5
3
2.0
2
1.5
JERKS
1
1.0
0
−1
0.5
−2
0.0
−3
−4
−0.5
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
10
9
(b)
(a)
0.20
1.5
VELOCITY DIFFERENCES
0.15
0.5
0.0
0.10
0.05
0.0
−.05
−.10
−.15
−.20
−0.5
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
(d)
(c)
0.10
0.08
SPACING DIFFERENCES
ACCELERATIONS
1.0
0.06
0.04
0.02
0.0
−.02
−.04
0
1
2
3
4
5
6
7
8
9
10
(e)
Figure 19.10. Five vehicle platoon simulation results with τ = 0.5 second and acceleration
feedback: (a) velocity time history, (b) jerk time history, (c) acceleration time history,
(d) velocity difference time history, and (e) spacing difference time history (Shladover 1991a).
19.3 String Stability
343
Table 19.2. Control configurations for the two cases
Gains on
Base vehicle
model τ = 0.1
(electric)
Alternative vehicle model
τ = 0.5 (internal
combustion)
Spacing to predecessor CX (s−2 )
Velocity difference from predecessor CVE (s−1 )
Velocity difference from platoon leader CV (s−1 )
Acceleration difference from predecessor CA (1)
Acceleration difference from platoon leader CAF (1)
3.6
0.9
2.4
0
0
18.0
4.5
12.0
2.0
2.0
19.3 String Stability
For a platoon of vehicles (Figure 19.11), we must consider individual vehicle stability
and string stability of the platoon, which is an important new concept. If the preceding
vehicle is accelerating or decelerating, then the spacing error could be nonzero; we
must ensure that the spacing error attenuates as it propagates along the string of
vehicles because it propagates upstream toward the tail of the string. If the position
of each vehicle is denoted by xi , as shown in Figure 19.11, and the desired spacing
between vehicles is denoted by L, then the spacing error is given by:
εi = xi − xi−1 + L
If the preceding vehicle is not accelerating or decelerating, the spacing error should
converge to zero; that is:
ẍi = 0
⇒
εi → 0 as t → ∞
Thus, for a constant-spacing policy (L = constant) with a proportional plus derivative
controller for the vehicle level, we can ensure the stability of the individual vehicle,
as desired:
ẍi = −k p εi − kd ε̇i
εi = xi − xi−1 + L
⇒
ε̈i = ẍi − ẍi−1 = ẍi
Next, consider the design of the upper-level (platoon) controller, which is:
⇒ ε̈i + kd ε̇i + k p εi = 0
It is designed using the simple model:
ẍ = ẍdes
xi+1
xi
xi–1
Figure 19.11. Vehicle spacing and spacing error in a platoon.
344
Longitudinal Motion Control and Platoons
101
100
10−1
10−2
10−3
10−2
100
10−1
101
Figure 19.12. The magnitude of G(s) for PD control with L = constant.
whereas a more complete model can be used for robustness analysis; for example:
τ ẍ˙ + ẍ = ẍ˙des
The spacing-error transfer function can be determined as:
εi (s) = G(s)εi−1 (s)
Conditions for the string stability of the platoon are given in Swaroop and Hendrick
(1996) as follows:
G(s) =
εi (s)
εi−1 (s)
G(s) ∞ ≤ 1
and
and
g(t ) = L−1 [G(s)]
g(t ) > 0
∀t
If, as previously, we consider a PD controller with a constant value of L, we obtain:
εi = xi − xi−1 + L
ẍi = −k p εi − kd ε̇i
⇒ ε̈i = −k p εi − kd ε̇i + k p εi−1 + kd ε̇i−1
The resulting transfer function is:
εi (s) = G(s)εi−1 (s)
G(s) =
kd s + k p
s2
+ kd s + k p
Thus, the magnitude always exceeds 1 (Figure 19.12) for all values of kp and kd , and
we cannot have string stability for L = constant and autonomous operation with PD
control.
Problems 1–2
345
Now consider, again with L = constant, a more complex control algorithm:
εi = xi − xi−1 + L
ẍi
des
= ki ẍi−1 + k2 ẍℓ + k3 ε̇i + k4 εi + k5 (ẋi − ẋℓ )
This controller requires inter-vehicle communication and can be designed to achieve
string stability. Rajamani et al. (2000) provide an example design based on slidingmode control.
Although not shown here, it has been demonstrated in the literature that when
the constant L is replaced by a constant time-gap policy (i.e., desired spacing = L +
hv) and a larger distance is maintained at higher speeds, then string stability can be
ensured without inter-vehicle communication (i.e., a properly designed ACC should
not be a problem!).
To summarize:
1.
2.
3.
4.
Platoons may be formed spontaneously as in ACC.
In AHS, platoons may be formed purposely to improve flow capacity.
Both ACC systems and AHS platoons must be designed to ensure string stability.
String stability is influenced by the headway policy. A constant-spacing policy
requires lead-vehicle acceleration (through communication) to become string
stable.
5. For a constant-time headway policy, inter-vehicle communication is not necessary (e.g., for a properly designed ACC).
PROBLEMS
1. Consider a platoon of vehicles, each with a PD controller (see Figure 19.11) with:
εi = xi − xi−1 + L
ẍi = −k p εi − kd ε̇i
Consider a constant-spacing policy and discuss the following:
(a) stability conditions for each vehicle
(b) string stability of a platoon of vehicles with no inter-vehicle communication
(c) string stability of a platoon of vehicles with inter-vehicle communication
2. Consider a platoon of two vehicles (Vehicle 1: lead vehicle; Vehicle 2: following
vehicle) in which each vehicle (i = 1 or 2) is described by the following equations:
ȧi = −(1/τi )ai + (1/τi )ui
v̇i = ai
d˙i = vi
where a is acceleration, v is velocity, and d is displacement.
(a) Rewrite the open-loop equations in standard-state variable form:
ẋ = Ax + Bu
where the states are defined as x1 = d1 , x2 = v1 , x3 = a1 , x4 = d2 , x5 = v2 , and x6 = a2 .
346
Longitudinal Motion Control and Platoons
(b) Next, assume that the control, ui , for each vehicle is given by:
u1 = u10 (t )
u2 = cv (v1 − v2 ) + cd (d1 − d2 )
where u10 (t) is a prescribed command input to the lead vehicle and u2 is a feedback
control law. Determine the closed-loop system equations in standard-state variable
form using the same definition of the state variables as in (a).
REFERENCES
Chandler, F. E., R. Herman, and E. W. Montroll, 1958, “Traffic Dynamics: Studies in Car
Following,” Operations Research, Vol. 6, pp. 165–84.
Chang, K. S., W. Li, A. Shaikhbahai, F. Assaderaghi, and P. Varaiya, 1991, “A Preliminary
Implementation for Vehicle Platoon Control System,” Proceedings of the American Control
Conference, Boston, MA, June, pp. 3078–83.
Chee, W., and M. Tomizuka, 1995, “Lane Change Maneuvers for Automated Highway Systems: Control, Estimation Algorithm and Preliminary Experimental Study,” Proceedings
of the International Mechanical Engineering Congress and Exposition, San Francisco, CA,
November.
Chen, S., T. B. Sheridan, H. Kusunoki, and N. Komoda, 1995, “Car Following Measurements,
Simulations, and a Proposed Procedure For Evaluating Safety,” Proceedings of the IFAC
Symposium on Analysis, Design and Evaluation of Man-Machine Systems, Pergamon Press,
pp. 603–8.
Fancher, P., H. Peng, Z. Bareket, C. Assaf, and R. Ervin, 2001, “Evaluating the Influences
of Adaptive Cruise Control Systems on the Longitudinal Dynamics of Strings of Highway
Vehicles,” Proceedings of the IAVSD Conference, Copenhagen, Denmark, August.
Fenton, R. E., and R. J. Mayhan, 1991, “Automated Highway Studies at The Ohio State University – An Overview,” IEEE Transactions on Vehicular Technology, February, Vol. 40,
No. 1, pp. 100–113.
Frank, A. A., S. J. Liu, and S. C. Liang, 1989, “Longitudinal Control Concepts for Automated Automobiles and Trucks Operating on a Cooperative Highway,” Proceedings of the
Future Transportation Technology Conference and Exposition, SAE SP-791, SAE Paper
No. 891708.
Hedrick, J. K., D. McMahon, V. Narendran, and D. Swaroop, 1991, “Longitudinal Vehicle
Controller Design for IVHS Systems,” Proceedings of the American Control Conference,
Boston, MA, June, pp. 3107–12.
Hedrick, J. K., M. Tomizuka, and P. Varaiya, 1994, “Control Issues in Automated Highway
Systems,” IEEE Control Systems Magazine, December.
Himmelspach, T., and W. Ribbens, 1989, “Radar Sensor Issues for Automotive Headway
Control Applications,” Fall 1989 Summary Report for the Special Topics Course in IVHS,
Appendix G, University of Michigan.
Jurgen, R. K., 1991, “Smart Cars and Highways Go Global,” IEEE Spectrum, May, pp. 26–36.
Kehtarnavaz, N., J. S. Lee, and N. C. Griswold, 1990, “Vision Based Convoy Following by
Recursive Filtering,” Proceedings of the American Control Conference, San Diego, CA,
May, pp. 268–72.
McMahon, D. H., J. K. Hedrick, and S. E. Shladover, 1990, “Vehicle Modeling for Automated
Highway Systems,” Proceedings of the American Control Conference, San Diego, CA, May,
pp. 297–303.
Mobility 2000 Workshop on Intelligent Vehicles and Highway Systems, 1990, “Final Report of
the Working Group on Advanced Vehicle Control Systems (AVCS),” Dallas, TX, March.
Okuno, A., A. Kutami, and K. Fujita, 1990. “Towards Autonomous Cruising on Highways,”
SAE Paper No. 901484.
Rajamani, R., 2006, Vehicle Dynamics and Control, Springer-Verlag.
References
Rajamani, R., S. B. Choi, B. K. Law, J. K. Hedrick, R. Prohaska, and P. Kretz, 2000, “Design
and Experimental Implementation of Longitudinal Control for a Platoon of Automated
Vehicles,” ASME Journal of Dynamic Systems, Measurement and Control, Vol. 122, No. 3,
September, pp. 470–6.
Rajamani, R., H.-S. Tan, B. K. Law, and W.-B. Zhang, 2000, “Demonstration of Integrated
Longitudinal and Lateral Control for the Operation of Automated Vehicles in Platoons,”
IEEE Transactions on Control Systems Technology, Vol. 8, No. 4, July, pp. 695–708.
Schwarzinger, M., T. Zielke, D. Noll, M. Brauckmann and W. von Seelen, 1992, “Vision-Based
Car-Following: Detection, Tracking, and Identification,” Proceedings of the Intelligent Vehicles ’92 Symposium, Detroit, June–July.
Shladover, S., 1991a, “Longitudinal Control of Automotive Vehicles in Close-Formation
Platoons,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 113, June,
pp. 231–41.
Shladover, S., 1991b, “Research and Development Issues in Intelligent Vehicle/Highway
Systems (IVHS),” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced
Automotive Technologies – 1991, ASME DE-Vol. 40, New York, pp. 5–9.
Shladover, S., C. A. Desoer, J. K. Hedrick, M. Tomizuka, J. Walrand, W. B. Zhang, D. H.
McMahon, H. Peng, S. Sheikholeslam, and N. McKeown, 1991, “Automatic Vehicle Control Developments in the Path Program,” IEEE Transactions on Vehicular Technology,
February, Vol. 40, No. 1, pp. 114–30.
Special Issue on IVHS, 1991, IEEE Transactions on Vehicular Technology, February, Vol. 40,
No. 1.
Swaroop, D., and J. K. Hedrick, 1996, “String Stability of Interconnected Systems,” IEEE
Transactions on Automatic Control, Vol. 41, No. 3, March, pp. 349–57.
Swaroop, D., and J. K. Hedrick, 1999, “Constant Spacing Strategies for Platooning in Automated Highway Systems,” ASME Journal of Dynamic Systems, Measurement and Control,
Vol. 121, No. 3, September, pp. 462–70.
347
20
Automated Steering and Lateral Control
This chapter focuses on AVCS for ITS, which are concerned with vehicle lateral
motion control. The first part of the discussion is about a potential AVCS-II or
AVCS-III technology for lane sensing and lane tracking in a lane-departure warning
system. It is an intermediate step to fully automated steering, which is a potential
AVCS-III technology.
20.1 Lane Sensing
To automate the lateral control of vehicles, it is necessary to measure the position
of a vehicle in the lane. Various schemes have been proposed and tested for this
purpose, including the use of (1) an on-board vision system that detects the painted
lane markings on a highway; (2) continuous magnetic wires imbedded in the center
of the lane; (3) radar or ultrasonic waves to measure the distance to sidewalls; (4) a
forward-looking laser to detect reflective markers; and (5) discrete magnetic markers imbedded in the roadway. Clearly, the onboard vision systems are more difficult
to develop but offer the potential capability of autonomous operation. The latter
four schemes require the necessary highway infrastructure to be in place. These
cooperative systems, however, are likely to provide more accurate lane-position
information at higher speeds with simpler and more inexpensive devices in a vehicle.
Magnetic wires and markers also are more reliable under adverse weather conditions
(e.g., fog, rain, and snow). Prototype systems based on the other sensing schemes
(i.e., vision, laser, and radar) have not been shown to work in inclement weather.
Recently, GPS-based systems have been proposed for lateral and longitudinal positioning; however, the accuracy is still not good enough, even when differential GPS
is used. Furthermore, GPS may not be available in all areas (e.g., urban canyons);
therefore, systems based on GPS measurement and road maps are still under
development.
Because vision-based lane-sensing systems are the dominant form of sensing
used in lane-detection and lateral-control projects, it certainly makes sense to
describe – at least briefly – the principle of lane detection using vision systems. Of
course, there are many different approaches, including edge detection, deformable
template, and B-snake, which almost always use two-dimensional visible light images.
348
20.1 Lane Sensing
349
c
y
y ′ = mx ′ + c
Figure 20.1. Hough transformation.
c = y ′ − mx ′
(x ′, y ′)
x
In the case of edge-detection–based methods, the operations typically involve the
following steps:
1.
2.
3.
4.
5.
6.
7.
8.
Acquire image
Digitize image (e.g., 512 × 512 × 8)
Detect edge (Sobel filter, basically a high-pass filter)
Thresholding (generate a binary image)
Noise cleaning
Hough transformation
Identify lane-marker candidates
Decide the lane markings
In the fourth step, dynamic thresholding can be implemented to partially solve the
problem of lighting changes in the environment. Noise cleaning is the step in which
human know-how and heuristics enhance the performance of the vision system. For
example, because we know that lane edges should extend through to the horizon, the
edges of small objects (e.g., trees and signs on the roadside) can be eliminated easily.
The lane locations from a previous image also could be used to distinguish lane
edges from noise. The Hough transformation in Step 6 is basically a mathematical
transformation to represent curves (in particular, straight lines y = mx + c) in the
(x,y)-coordinate in the (m,c) plane. The Hough transformation was found to be a
robust method for representing straight lines and is less susceptible to noise and
uncertainty.
The main goal for lane-sensing systems is to supply vehicle location and response
signals to calculate appropriate steering signals (i.e., automated steering). However,
intermediate systems also can be envisioned in which the driver steers the vehicle and
the lane-sensing information is used for warnings, driver assist, or even intervention.
A driver-monitoring system also can be imagined in which a probationary driver’s
steering performance is monitored, recorded, and reported. These systems could be
used in driver training and evaluation or by police to monitor those with a history
of alcohol or drug abuse. In one sense, these systems are less ambitious than fully
automated steering. However, in terms of control-system design, they are perhaps
more challenging because the vehicle–driver system dynamics must be modeled as
the basis for designing a warning or intervention system; whereas in fully automated
steering, the driver is not in the loop and the control design can be based on the
vehicle lateral dynamics model.
m
350
Automated Steering and Lateral Control
Projected Route
of the Vehicle
Actual
Lane Edge
Uncertainty in
Lane Edge Sensing
Lane
Crossing
Downrange
Distance to
Lane-Crossing
or
Time-to-Lane
Crossing
Sensing
Range
Sensing
Arc
Current
Vehicle Position
Lateral
Clearance
or
Lateral
Acceleration
Figure 20.2. TLC metric for road-departure prevention.
Example 20.1 discusses a system for preventing run-off-road accidents (also
called SVRD accidents) due to driver drowsiness and inattention, which account for
nearly 40 percent of all highway fatalities in the United States each year. This system
also can be categorized in ASTs that potentially could avoid the legal concerns that
comprise a major obstacle for AVCS, especially AHS concepts.
EXAMPLE 20.1: PREVENTING RUN-OFF-ROAD ACCIDENTS. Here, we revisit Example
18.1 for a more in-depth discussion. In many cases, run-off-road accidents can be
avoided by means of technologies that either warn the driver of an impending
departure from the roadway or provide some level of control assistance to
prevent an off-road trajectory. Such a system would require the means to (1)
sense the immediate proximity of the vehicle to the lane edge, and (2) project
the vehicle’s path and the road layout for a specific distance down range from
the current position. Figure 20.2 illustrates the concept of a lane-tracking margin
(i.e., TLC) based on lane sensing and vehicle-path projection. It is expected that
both the lane-sensing information and the vehicle-path projection will become
more uncertain with range and speed. The lane sensing could be accomplished
with onboard sensors, as shown in the figure, or using sensors installed in the
highway, as discussed previously.
Figure 20.3 (i.e., the shaded portion) illustrates how the proposed AST for preventing run-off-road accidents could be realized. Sensed-lane information (from various possible sensing technologies) would be combined with measurements of vehicle
20.1 Lane Sensing
351
Warning / Intervention / Control
Internal Feedback-loop
of Human Driver
Actual Lane Layout
(Lane Markers)
Driver
Steer
Angle
Vehicle
Lateral Position
Vehicle Speed
Yaw Rate
Motion Sensor
Driver Status
Model
Vehicle Model
Lane Sensor
Projected
Route
Measured
Data
TLC Algorithm
TLC
Lane Sensing
Previewed
Model
Roadway
Decision Rule
Warning / Intervention / Control
Proposed Active Safety
Control System
Figure 20.3. Structure of an active system to prevent SVRD accidents.
motion (e.g., vehicle velocity, acceleration, and steering-wheel displacements) to calculate the lane-tracking margin (i.e., TLC). Based on results of these calculations,
the system would take one of the following actions: (1) issue a warning to the driver,
(2) intervene in the vehicle steering in such a way as to complement the driver
actions, or (3) assume full control of steering. The vehicle-path projection requires
a model of the driver–vehicle system, not only the vehicle dynamics. The driver–
vehicle model might include an adaptive driver model so that deterioration in driver
response due to drowsiness and inattention can be detected. Detailed descriptions of
the entire system and a prototype vehicle that uses computer vision for lane sensing
are in LeBlanc et al. (1996a and 1996b).
The problem of estimating lane geometry from computer-vision images is
described in detail in Lin et al. (1999). It is accomplished by using a Kalman filter to estimate the roadway geometry based on a model of the roadway and the
vision measurements. Combining the measurements with a roadway model provides
smooth estimates despite occasional gaps in the lane markings and outliers in the
data obtained from image processing. The method also provides an estimate of
352
Automated Steering and Lateral Control
lane-geometry uncertainty (e.g., uncertainty increases with range). Measurements
of vehicle pitch and roll are used to compensate for camera motion and improve the
estimates.
The problem of path projection, based on measurements of the vehicle dynamics,
is described in Lin et al. (2000). Path projection uses a vehicle-model–based Kalman
filter, together with measurements of vehicle motion (i.e., acceleration, velocity, and
yaw), to project forward the vehicle path. The combination of these two (i.e., lane
geometry and projected path) then can be used to obtain the TLC, as described in
Lin and Ulsoy (1996). The TLC then can be used as a metric for issuing warnings to
the driver, taking control actions, or emergency-steering intervention. For example,
if TLC is less than an experimentally determined threshold, then a warning to the
driver that road-departure is imminent is issued. These decisions (i.e., whether to
warn, control, or intervene) can be enhanced if we also have information about the
state of the driver (Pilutti and Ulsoy 1999, 2003). Online estimation methods can
be developed based on measurement of the driver steering angle and the lateralposition information from the vision system to estimate the driver’s state (i.e., alert,
fatigued, or asleep). Essentially, the lane-keeping error, as seen by the vision system,
is assumed to be the input to the driver block and the steering angle is the driver
block output. As described in Pilutti and Ulsoy (1999), a series of driving-simulator
studies was used to develop methods to identify a driver’s state. Fuzzy logic then can
provide a mechanism for combining the TLC and driver-state information to make
improved decisions (Pilutti and Ulsoy 2003). One approach for emergency-steering
intervention is differential braking, as described in Pilutti et al. (1998). As described
in Chapter 5, a steering-assist control can be designed to provide for more consistent
lane-keeping performance and to reduce the likelihood of SVRDs.
The overall active-safety system to prevent SVRD accidents described herein is
fairly complex. However, elements of the system can be used as stand-alone systems
to provide information useful in other applications. For example, a warning-only
system, without including control and intervention functions, can be implemented.
A simple example of this is an “electronic rumble strip” in which the vision system
can be used to detect the lane edge and issue an audible warning to the driver. If
an estimate of the TLC or a driver-state assessment is available, it can be used to
improve the effectiveness of the simple electronic rumble strip (Pilutti and Ulsoy,
2003. A reduction in SVRDs can be achieved even with a basic system, and further
reductions will be achieved as the system is enhanced. NHTSA shows that 6,335,000
accidents (with 37,081 fatalities) occurred on U.S. highways in 1998 (NHTSA 1999),
of which about 30 percent were SVRDs, for a total of nearly 2 million accidents.
Approximately 40 percent of the fatalities (i.e., approximately 15,000) were due
to SVRDs. Even a small percentage reduction can have a significant human and
economic impact.
20.2 Automated Lane-Following Control
In the fully automated steering of vehicles, it is assumed that the driver is not in
the loop and that lane-sensing information is available. Referring to Figure 20.4,
20.2 Automated Lane-Following Control
353
ψ
δ
ρ
ds
ψd
a
uo
yr
ψ
b
3
2
0
(b)
Complex
Complex
50
Simplified
ramp
5
10
0
0
5
0
0
10
100
200
time
time
300
400
0
0
100
X (m)
200
X (m)
150
4
(c)
3
Simplified
(c)
100
Y (m)
Steering (deg)
100
20
traperoidal
−4
0
(a)
40
1
−2
150
Simplified
60
Y (m)
2
80
(b)
(a)
Y (m)
Steering (deg)
4
2
50
1
traperoidal
0
0
5
Complex
0
0
10
100
time
200
300
X (m)
Figure 20.4. The simplified two-DOF model and results comparing simulation results for the
two-DOF linear and 10-DOF nonlinear models.
which is based on the two-DOF vehicle-handling model, we can define the following
set of linear-state equations for use in lateral-control design (Peng and Tomizuka
1990):
⎡
0
1
⎢
⎢
⎥ ⎢0
⎢
⎢
⎥
⎢
ẏ
d ⎢
r
⎥ ⎢
⎥=⎢
⎢
dt ⎢ ψ − ψ d ⎥ ⎢ 0
⎦ ⎢
⎣
⎢
⎣
ψ̇ − ψ̇ d
0
⎡
yr
⎤
0
0
A1
u0
−A1
A2
u0
0
0
A3
u0
−A3
⎤
⎡
⎥
⎥
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎢
1 ⎥
⎥⎣ψ
⎥
A4 ⎦ ψ̇
u0
yr
⎤
⎡
0
⎤
⎡
0
⎤
⎢
⎥ ⎢ ⎥
⎥
⎢ A − u2 ⎥ 1
⎥ ⎢B ⎥
0⎥
⎢ 2
⎥ ⎢ 1⎥
⎥ + ⎢ ⎥δ + ⎢
⎥
⎢ 0 ⎥
⎢
⎥R
− ψd ⎥
0
⎣
⎦ ⎣ ⎦
⎦
B2
A4
− ψ̇ d
ẏr
(20.1)
300
354
Automated Steering and Lateral Control
Table 20.1. Parameter values
Parameter
Nominal value
Minimum value
Maximum value
m (kg)
Iz (kg-m2 )
Ca (N/rad)
u0 (miles/hour)
a,b (m)
ds (m)
1,550
3,100
84,000
70
1.15, 1.51
1.0
0.85 (1,550)
0.85 (3,100)
0.2 (84,000)
25
–
–
1.15 (1,550)
1.15 (3,100)
2.0 (84,000)
85
–
–
where yr is the lateral deviation of the mass center of the vehicle from the reference,
y is the yaw angle, yd is the desired yaw angle obtained from the road curve (which
is assumed to be known), R is the radius of curvature of the road, and:
A1 = −
A4 = −
(Cα f + Cαr )
A2 = −
m
(Cα f a2 + Cαr b2 )
Iz
(Cαr a − Cα f b)
B1 =
m
Fx + Cα f
m
A3 = −
B2 =
(Cαr a − Cα f b)
Iz
a(Fx + Cα f )
(20.2)
Iz
The output equation is:
0
y = yr + ds (ψ − ψd ) = 1 0
ds
1
0 x
which represents the lateral deviation as obtained from a sensor at a distance, ds ,
ahead of the mass center of the vehicle. The transfer function from the front-wheel
steer angle, d, to the output y, is given by:
ds (B1 A3 − B2 A1 ) + B2 A2 − B1 A4
y(s)
1
2
s + B1 A3 − B2 A1
(ds B2 + B1 )s +
=
δ(s)
(s)
u0
where
(20.3)
(s) = s2 s2 −
A1 + A4
(A1 A4 − A2 A3 )
s+
+ A3
u0
u20
Figure 20.4 also shows results comparing this two-DOF model (simplified) to a
nonlinear 10-DOF (i.e., more complex) model (Figure 20.5). The results show that
the two-DOF linear model is suitable for controller design in lateral control studies and show the steering commands used in the simulation comparison. Listed in
Table 20.1 are nominal, minimum, and maximum values of the parameters for the
previous model.
EXAMPLE 20.2: CONTROLLER DESIGN FOR AUTOMATED STEERING. A block diagram
of the proposed lateral control system, shown in Figure 20.6, has feed-forward
and feedback parts. It also includes a parameter estimator for the tire-cornering
stiffness, Ca , and a state estimator. The feed-forward controller is based on the
fact that freeways are designed to have fixed curvatures; therefore, if the radius
20.2 Automated Lane-Following Control
Fy3
Fx3
δ3
355
Fy1
Y
Fx1
δ1
Ys
O
V
Os
ψ
β
Xs
Fy4
Fy2
Fx2
Fx4
Zs
X
δ2
δ4
φ
h2
θ
Zu
z
Yu
Ou
γ
Xu
FB3
FB1
FA3
FA1
Fwy
Fwx
FB4
FB2
FA2
FA4
h5
h4
∆
FP3
sb2
l2
FP4
l1
FP2 sb1 FP1
Figure 20.5. The 10-DOF nonlinear model.
of curvature R is known, then the corresponding steady-state steering angle is
(from Eq. (4.44) Ca = Caf = Car , Fx = Fxf = Fxr , and L = (a + b):
δ=
mu20 (b − a) + Cα (a + b)2
R(Cα + Fx )(a + b)
(20.4)
An improved feed-forward controller can be designed using preview information (Peng and Tomizuka 1991). The feedback controller uses state-feedback
estimates and is designed using the frequency-shaped linear quadratic control
design approach (Peng and Tomizuka 1990). The details of the design procedure
are omitted here. A least-squares algorithm is used to estimate the tire-cornering
stiffness, Ca .
356
Automated Steering and Lateral Control
n1
ε
Yaw rate
sensor
n2
δF
ρ
FF
controller
road profile
δd
Actuator
+
+
yr
ÿr
Cs
G1
V
Vehicle
G2
z1
z2
δB
Parameter
Identifier
m
n3
ys
ε−εd
G3
G4
Accelerometer
z3
n4
z4
FB
controller
Îz
Î1 Î2
ys
Magnetic
sensor
state
variables
Speedometer
V
Observer/
State estimator
ÿa
Figure 20.6. Block diagram of lateral control system in Example 20.2.
Figure 20.7 shows simulation results for a wind gust of −500 N and a moment
of −200 N-m in the lateral direction for 3 seconds. Figure 20.8 shows simulations
using a discrete marker scheme, in which the markers are placed at 4-meter intervals
and all other measurements are made at 100 Hz. Figure 20.9 shows simulation results
in which a portion of the road is icy so that the value of Ca decreases to the minimum
value for 1 second in the curved section of the road. Experimental results are in Peng
et al. (1993).
20.3 Automated Lane-Change Control
If the reference/sensing system can supply continuous vehicle lateral displacement
and yaw measurement between lanes, the control algorithm developed for lane
following also can be used for lane change. Because there is no fixed path to be
followed for a lane-change maneuver, a trajectory-planning task must be performed
first to define a smooth path. Chee and Tomizuka (1995) found that a virtual desired
trajectory (VDT) defined by a trapezoidal acceleration profile easily can satisfy ridequality constraints (described by jerk and acceleration limits) and time optimality.
When the effective range of the reference/sensing systems is less than the full-lane
width, the tracking error and other vehicle states must be estimated using the state
observer. Other than these two extra components (i.e., VDT and state observer), the
system architecture and hard requirement are almost identical to a lane-following
system.
20.3 Automated Lane-Change Control
357
4
steering (deg)
lateral deviation (m)
0.2
0.1
V = 10
0
V = 32
2
V = 32
0
V = 40
−0.1
0
10
20
−2
0
30
V = 10
time (sec)
V = 40
10
20
30
time (sec)
lateral accel. (g)
0.3
V = 10
0.2
0.1
V = 40
0
−0.1
−0.2
0
10
20
30
time (sec)
Figure 20.7. Simulation results for wind-gust input.
4
0.1
steering (deg)
lateral deviation (m)
0.2
0
10 ft
13 ft
−0.1
−0.2
0
10
20
30
2
0
−2
0
lateral accel. (g)
0.3
0.2
0.1
0
−0.1
0
10
20
20
time (sec)
time (sec)
−0.2
10
30
time (sec)
Figure 20.8. Simulation results with discrete marker scheme.
30
358
Automated Steering and Lateral Control
0.1
4
V = 10
0
V = 32
V = 40
−0.1
fixed
gain
−0.2
−0.3
steering (deg)
lateral deviation (m)
0.2
0
2
V = 32
0
30
0
20
10
30
time (sec)
time (sec)
lateral accel. (g)
V = 40
−2
20
10
V = 10
0.4
V = 10
0.2
V = 40
0
−0.2
0
10
20
30
time (sec)
Figure 20.9. Simulation results with icy road section.
PROBLEMS
1. A simple method for calculating the TLC was given in Problem 18.3 for a straight
roadway. Determine the error in TLC that results from using this method on a
segment of roadway with constant curvature, R.
2. Derive the feed-forward control-steering input given in Eq. (20.4) and its numerical value for the nominal parameters given in Table 20.1. How does this value change
with the road curvature, R, with the vehicle characteristics (i.e., a, b, Ca , and m) and
with the driving characteristics u0 and Fx ?
3. Show that using the nominal parameter values in Table 20.1, the transfer function
in Eq. (20.3) is:
8.5355s2 + 3.1331s + 39.061
Y (s)
= 2 2
δ(s)
s (s + 0.65831s + 1.0816)
and design a PID feedback controller based on this transfer function.
4. Combine the feed-forward controller in Problem 2 with the feedback controller
in Problem 3. Compare the closed-loop performance of your system via simulation
to the results presented in Figure 20.7.
REFERENCES
Ackermann, J., and W. Sienel, 1990, “Robust Control for Automatic Steering,” Proceedings
of the American Control Conference, San Diego, CA, May, pp. 795–800.
Chee, W., and M. Tomizuka, 1995, “Lane Change Maneuvers for Automated Highway Systems: Control, Estimation Algorithm and Preliminary Experimental Study,” Proceedings
References
of the International Mechanical Engineering Congress and Exposition, San Francisco, CA,
November.
Fenton, R. E., and R. J. Mayhan, 1991, “Automated Highway Studies at the Ohio State
University,” IEEE Transactions on Vehicular Technology, Vol. 40, No. 1, pp. 100–113.
Kimbrough, S., 1991, “Coordinated Braking and Steering Control for Emergency Stops and
Accelerations,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.), Advanced
Automotive Technologies-1991, ASME DE-Vol. 40, New York, pp. 229–44.
LeBlanc, D. J., R. D. Ervin, G. E. Johnson, P. J. Th. Venhovens, G. Gerber, R. DeSonia, C.-F.
Lin, T. Pilutti, and A. G. Ulsoy, 1996a, “CAPC: An Implementation of a Road-Departure
Warning System,” IEEE Control Systems Magazine, Vol. 16, No. 6, December, pp. 61–71.
LeBlanc, D. J., P. J. Th. Venhovens, C.-F. Lin, T. Pilutti, R. D. Ervin, A. G. Ulsoy, C.
MacAdam, and G. E. Johnson, 1996a, “Warning and Intervention System to Prevent RoadDeparture Accidents,” Vehicle System Dynamics, Vol. 25, Suppl., pp. 383–96.
Lin, C. F., and A. G. Ulsoy, 1996, “Time to Lane Crossing Calculation and Characterization
of Its Associated Uncertainty,” ITS Journal, Vol. 3, No. 2, pp. 85–98.
Lin, C. F., A. G. Ulsoy, and D. J. LeBlanc, 1999, “Lane Geometry Perception and the Characterization of Its Associated Uncertainty,” ASME Journal of Dynamic Systems, Measurement
and Control, Vol. 121, No. 1, March, pp. 1–9.
Lin, C. F., A. G. Ulsoy, and D. J. LeBlanc, 2000, “Vehicle Dynamics and External Disturbance
Estimation for Vehicle Path Prediction,” IEEE Transactions on Control System Technology,
Vol. 8, No. 3, May, pp. 508–18.
Margolis, D. L., M. Tran, and M. Ando, 1991, “Integrated Torque and Steering Control for
Improved Vehicle Handling,” in S. A. Velinsky, R. H. Fries, I. Haque, and D. Wang (eds.),
Advanced Automotive Technologies-1991, ASME DE-Vol. 40, New York, pp. 267–90.
Matsumoto, N., and M. Tomizuka, 1990, “Vehicle Lateral Velocity and Yaw Rate Control
with Two Independent Control Inputs,” Proceedings of the American Control Conference,
San Diego, CA, May, pp. 1868–75.
NHTSA, 1999, National Highway Traffic Safety Administration, Traffic safety facts 1998.
Report DOT HS-808-983. US Department of Transportation, Washington, DC.
Peng, H., W. Zhang, S. Shladover, M. Tomizuka and A. Aral, 1993, “Magnetic-Marker Based
Lane Keeping: A Robustness Experimental Study,” Proceedings of the SAE International
Congress and Exposition, Detroit, MI, SAE Technical Paper 930556.
Peng, H., and M. Tomizuka, 1990, “Vehicle Lateral Control for Highway Automation,”
Proceedings of the American Control Conference, San Diego, CA, May, pp. 788–94.
Peng, H., and M. Tomizuka, 1991, “Preview Control for Vehicle Lateral Guidance in Highway Automation,” Proceedings of the American Control Conference, Boston, MA, June,
pp. 3090–5.
Pilutti, T., and A. G. Ulsoy, 1999, “Identification of Driver State for Lane Keeping Tasks,”
IEEE Transactions on Systems, Man and Cybernetics, Vol. 29, No. 5, September, pp. 486–
502.
Pilutti, T., and A. G. Ulsoy, 2003, “Fuzzy Logic Based Virtual Rumble Strip for Road Departure Warning Systems,” IEEE Transactions on Intelligent Transportation Systems, Vol. 4,
No. 1, March, pp. 1–12.
Pilutti, T., A. G. Ulsoy, and D. Hrovat, 1998, “Vehicle Steering Intervention Through Differential Braking,” ASME Journal of Dynamic Systems, Measurement and Control, Vol. 120,
No. 3, September, pp. 314–21.
Taheri, S., and H. Law, 1990, “Investigation of a Combined Slip Control Braking and ClosedLoop Four-Wheel Steering System for an Automobile During Combined Hard Braking
and Severe Steering,” Proceedings of the American Control Conference, San Diego, CA,
May, pp. 1862–7.
359
APPENDICES
APPENDIX A
Review of Control-Theory Fundamentals
A.1 Review of Feedback Control
This appendix, a brief review of important concepts and methods from control
theory, is intended to help readers review that material through self-study. A background in dynamic systems and control is essential for an understanding of the
material presented in this textbook on automotive control systems. Furthermore,
it is assumed that readers are familiar with the computational tools available for
simulation and control in the MATLAB/Simulink environment. An excellent online
tutorial is available at the Web site www.engin.umich.edu/class/ctms/.
Definitions and Motivation
System. A group of objects that are combined to function as an integrated part for a
specific objective (e.g., an engine, a car, or a group of vehicles).
Inputs of a system are the means by which the state of the system can be changed.
Outputs of a system are the means by which the state of the system is manifested (e.g.,
in the speed control of a car, throttle [input] and speed [output] and in the directional
control of a car, steering [input] and yaw rate [output]).
Control. This directs a system’s inputs so that the outputs behave in the manner desired.
As an example, consider the idealized vehicle-speed control problem shown in
Figure A.1, in which the goal is to control the speed of an automobile by adjustment of the throttle only.
Block-Diagram Representation of Control Systems
Block-diagram form is a simple way to present the interconnection of subsystems in
a control system. In block-diagram form, components of the system are connected
by signal flows.
The block diagram in Figure A.2, which includes physical elements and signals commonly seen on control systems, can be simplified to the form shown in
Figure A.3 for controller-design purposes.
363
364
Appendix A: Review of Control-Theory Fundamentals
Control the speed of an automobile (throttle only)
throttle
angle
voltage
control
algorithm
speed
throttle
actuator
vehicle
Output
Input
feedback
The control system
Figure A.1. An idealized control system.
Benefits and Drawbacks of Feedback
In this section, we perform several straightforward analyses to illustrate the benefits
and drawbacks of using feedback. Mathematically, if we have perfect knowledge of
the plant to be controlled and there are no external disturbances, we could “invert”
the plant and obtain the open-loop controller for perfect tracking (i.e., y(t) = r(t)
for all t). In the following discussion, this simple idea is illustrated by an algebraic
example.
Suppose we are designing a cruisecontrol system considering only steady-state response (i.e., disregard transient
response). Assuming that after extensive experiments on a flat test track we
found that for each 1 degree of throttle-pedal displacement, the steady-state
vehicle speed increases by 4 miles/hour. The open-loop control design is then:
EX. A.1.1: FEED-FORWARD CONTROL DESIGN.
r
desired
vehicle speed
in mph
y
u
F
P
throttle
angle in
deg
vehicle speed
in mph
disturbance
d
ref
e
controller
+
u
power
amplifier
actuator
sensor
noise
n
Figure A.2. Block diagram of typical control systems.
u
y
plant
A.1 Review of Feedback Control
365
d
r
Figure A.3. Simplified block diagram of
a control system.
e
controller
u
y
plant
+
-
+
+
Now, because P = 4, we should have F = 0.25 (= 1/P); indeed, if we invert the
plant and obtain the controller F, the vehicle speed y will be equal to the desired
vehicle speed at steady-state. This is why some people refer to the open-loop-control
design problem as the “plant-inversion” problem. This also can be understood from
the following general equation:
y = Pu = P(F r) = PF r
Because the control-design purpose is to make y = r, it is obvious that we should let
PF = 1 or F = P1 . However, the open-loop, or feed-forward, control-design method
however is vulnerable to external disturbances and plant uncertainties.
For the same problem illustrated in Ex.
A.1.1, suppose there exist external disturbances (e.g., road gradient and wind
gust) that were not considered in the control-design process and the roadgradient angle was found to reduce the vehicle speed at 1.5 mile/hour/degree.
The vehicle response then should be represented by the following diagram.
EX. A.1.2: OPEN-LOOP, DISTURBANCE.
road gradient
in degree
d
-1.5
r
F
desired
vehicle speed
in mph
u
P
throttle
angle in
deg
+
+
y
vehicle speed
in mph
Suppose P = 4 and F = 0.25; then we have y = r − 1.5d. In other words, the
controller does not achieve the design objective when a slope is present, and the
effect of slope is as large as the uncontrolled case. This is because there is no
feedback. There are at least two ways to solve the external disturbance problem, as
follows:
1. Measure the disturbance d: This usually is not practical or too expensive. Sometimes “disturbance observers” are constructed to estimate the magnitude of
disturbance from its effect on plant output.
2. Feedback: Because the control-design objective is to make y follow r, we intuitively should adjust the control signal u according to the observed error signal
(r-y). When (r-y) becomes larger, the control u also should be increased (assuming that we know the larger control signal u will increase the output signal y).
n
366
Appendix A: Review of Control-Theory Fundamentals
EX. A.1.3: PURE FEEDBACK, DISTURBANCE. The following figure shows that
increased feedback action (larger C) reduces the effect of disturbances, which is
evident from the following examples:
C=1 ⇒ y=
1.5
4
r−
d
5
5
400
1.5
r−
d
401
401
C = 100 ⇒ y =
C→∞
⇒ y→r
road gradient
in degree
d
-1.5
r
u
C
+
P
+
+
y
-
y = 4C(r − y) − 1.5d
⇒ y=
1.5
4C
r−
d
4C + 1
4C + 1
There is a limit, however, on the size of feedback gain that can be used due to
both practical (e.g., actuator or other hardware limits) and stability considerations.
The stability limit is explained further in Ex. A.1.9.
EX. A.1.4: FEEDBACK + FEED-FORWARD, DISTURBANCE.
d
0.25
-1.5
+
r
C
+
+
u
4
+
+
y
-
y = 4[0.25r + C(r − y)] − 1.5d
⇒ y=r−
1.5
d
4C + 1
It is shown in this equation that we can make y = r in the case when there is
no external disturbance by adding the feed-forward term. Comparing the results of
Ex. A.1.3 and Ex. A.1.4, it can be seen that the feed-forward control term cannot
reduce the deviation caused by the disturbance. The reason is that the feed-forward
control signal is not aware of the existence and magnitude of the disturbance. In
general, if we invert the plant to obtain the feed-forward controller, we can obtain a
perfect match between the reference and output signals in the case in which there is
no external disturbance. This is obvious from the following example.
A.1 Review of Feedback Control
367
EX. A.1.5: FEEDBACK + FEED-FORWARD, DISTURBANCE, GENERAL CASE.
d
1
P
W
+
r
+
C
+
u
+
P
+
y
-
y=P
2r
P
3
+ C(r − y) + W d
⇒ y=r+
W
d
1 + PC
For the same problem illustrated in
Ex. A.1.1, suppose there is a mismatch between the perceived plant P̂ and the
true plant P = P̂(1 + ). How does this affect the control-system output?
EX. A.1.6: OPEN-LOOP, PLANT UNCERTAINTY.
r
y
u
F
P̂(1+ Δ)
Because the open-loop design simply inverts the perceived plant, we have
F = 1 . Therefore:
P̂
1
· r = (1 + )r
P̂
The difference between the desired output and true output is: e = r − y =
y = P̂(1 + ) ·
r.
EX. A.1.7: CLOSED LOOP, PLANT UNCERTAINTY.
r
y
u
C
P̂(1+ Δ)
+
-
y=
CP̂(1 + )
1 + CP̂(1 + )
r
⇒ e=r−y=
1
1 + CP̂(1 + )
r
The error signal can be reduced if we increase C.
The results of Ex. A.1.1 through Ex. A.1.7 are summarized in the following
conclusions. A large loop gain CP, also known as the loop-transfer function, in a
feedback system ensures that:
1. The output closely follows the reference signal.
2. Desensitization of the output y to the disturbance, d.
3. Desensitization of the I/O map from r to y due to plant variations, .
Pitfalls of Feedback
Analysis results in the previous two sections seem to suggest that we should use
a feedback-control configuration with as large a control gain as possible. Large
368
Appendix A: Review of Control-Theory Fundamentals
feedback gains, however, can cause problems, which are illustrated in the following
examples.
EX. A.1.8: MEASUREMENT NOISE. When the output signal is contaminated by mea-
surement noise (which is true for almost all practical cases), the error signal can
be obtained as in the following equation:
d
r
C
+
u
P
+
+
y
+
-
+
n
y = PC(r − n − y) + d
⇒ e = r − y = r − PC(r − y − n) − d = r − d − PCe + PCn
⇒ e=
1
1
PC
r−
d+
n
1 + PC
1 + PC
1 + PC
A major control-design objective is to reduce the error signal as much as possible.
We are not too concerned about the first term because it can be eliminated easily by
a feed-forward controller (if there is one). The second and third terms clearly show
the dilemma posed by nature: To reduce the error signal, we would like to make the
following:
1.
1
1+PC
2.
PC
1+PC
≡ S (sensitivity function) small, to reduce the effect of the disturbance
≡ T (complementary sensitivity function) small, to reduce the effect of
measurement noise
However, we have:
S + T= 1
In other words, there is a tradeoff between disturbance-rejection and noiserejection capabilities in the design of feedback controllers: We cannot make both S
and T small (at all frequencies).
Examples A.1.1 through A.1.8 illustrate the nature of
control-design problems based on static (i.e., algebraic) plant and control systems; however, the real physical systems that we are dealing with are mostly
dynamic. Dynamic control systems pose another pitfall that is not present
for static systems: the instability problem. In the following discussion, we use
a MATLAB example to illustrate this phenomenon. The plant to be controlled in this example is assumed to be governed by the following dynamic
equation:
EX. A.1.9: INSTABILITY.
ÿ + 3ÿ + 3ẏ + y = u
A.1 Review of Feedback Control
369
Suppose the controller designed is of simple proportional form; then, the
block diagram is:
r
u
K
+
1
3
(s + 1)
y
-
where the design task is to find the appropriate K that gives satisfactory results.
The following MATLAB program simulates the step response of the closed-loop
system:
>>
>>
>>
>>
>>
>>
>>
>>
>>
K=1;
den = [1 3 3 1+K];
num = K;
t=0:0.1:30;
y=step(num,den,t);
plot(t,y);
title([‘K=’ num2str(K)]);
xlabel(‘Time (sec)’);
ylabel(‘y’);
The following figures show that as K increases, the response becomes
increasingly more oscillatory. The system finally becomes unstable when K > 8.
K=3
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
10
20
30
K=8
2
K=5
1.4
0
0
5
1
0
0.5
-5
0
10
20
30
-10
20
30
20
30
K=10
10
1.5
0
10
0
10
State-Space Realization
Dynamic equations of the plant in ordinary differential equation form can be put
into a special form – “state-equation” form – which is suitable for control-design
purposes. We first introduce the following two definitions.
370
Appendix A: Review of Control-Theory Fundamentals
State variables: A minimum set of variables, x1 (t ), . . . , xn (t ), such that knowledge
of these variables at any time to, plus information about the input signals u(τ ),
τ ≥ to, and the system dynamic equations is sufficient to determine the state of
the system at any time t > to.
State equations: A collection of first-order ordinary differential equations that
represents the relationship between input and state variables. In general, these
equations can be nonlinear and/or time varying, as shown here:
dx1 (t )
= f1 (x1 , · · · xn , u1 , · · · , um )
dt
..
.
dxn (t )
= fn (x1 , · · · xn , u1 , · · · , um )
dt
For linear systems, we can group the state equations into matrix form:
dx(t )
= Ax(t ) + Bu(t )
dt
where matrices A and B are constant or time-varying. However, in any case, they
should not depend on x and u. Assuming that the number of state variables is n,
then the size of matrix A is n × n and B is n × m. Again, nonlinear systems cannot
be represented in the matrix form shown previously.
A.2 Mathematical Background and Design Techniques
Laplace Transforms
The major reasons to use the Laplace Transform and examine the dynamic system
response in the s-domain (s: the Laplace operator) are as follows:
1. Calculus operations (i.e., differentiation and integration) can be replaced by
simpler algebraic operations (+, −, *, /).
2. Free response and forced response of the dynamic system can be solved simultaneously.
Definition: Given a function f (t ) that satisfies:
∞
4
4
4 f (t )e−σ t 4dt < ∞
0−
for some positive real σ . The Laplace Transform of f (t ) is defined as:
∞
F (s) ≡
f (t )e−st dt = L[ f (t )]
0−
As long as the time function f(t) does not grow faster than exponentially,
its Laplace Transform always will exist. This condition is satisfied by all impulse
response signals of physical systems.
EX. A.2.1:
Find the Laplace Transform of the unit-step function:
∞
−1 −st 44∞
1
F (s) =
e 0 =
1 · e−st dt =
−
s
s
0
A.2 Mathematical Background and Design Techniques
371
EX. A.2.2:
Find the Laplace Transform of the function f (t ) = e−at :
∞
−1 −(s+a)t 44∞
1
F (s) =
e
e−at · e−st dt =
=
0
s+a
s+a
0−
EX. A.2.3:
Find the Laplace Transform of the function f (t ) = sin ωt:
∞
∞
1 jωt
(e − e− jωt ) · e−st dt
F (s) =
sin ωt · e−st dt =
−
−
2
j
0
0
ω
1
1
1
= 2
−
=
2 j s − jω s + jω
s + ω2
Important Properties of the Laplace Transform
dn
1
Time domain operators: n ↔ sn ,
···
↔ n
dt
s
5 67 8
n
L
"
df (t )
dt
#
=
0
∞
4∞
df −st
e dt = f (t ) e−st 40 + s
dt
∞
f (t )e−st dt = − f (0) + sF (s) (A.1)
0
#
" #
d2 f (t )
df
− f ′ (0) = s2 F (s) − s f (0) − f ′ (0)
= sL
L
2
dt
dt
" t
# ∞ t
L
f (τ )dτ =
f (τ )dτ e−st dt
0
0
0
4∞ 1 ∞
F (s)
−1 t
f (τ )dτ e−st 40 +
f (t )e−st dt =
=
s 0
s 0
s
"
Time shift: Time delay τd in time domain ⇒ e−sτd in s-domain
∞
∞
f (t − τd )e−st dt =
L{ f (t − τd )} =
f (t − τd )e−st dt
0
=
τd
∞
f (σ )e−s(σ +τd ) dσ = e−sτd F (s)
0
Initial Value Theorem (use Eq. (A.1)):
∞
df −st
e dt = lim [− f (0) + sF (s)]
lim
s→∞ 0
s→∞
dt
⇒ 0 = lim [− f (0) + sF (s)] ⇒ lim f (t ) = lim [sF (s)]
s→∞
s→∞
t→0
Final Value Theorem (use Eq. (A.1)):
∞
df −st
lim
e dt = lim[− f (0) + sF (s)]
s→0 0
s→0
dt
⇒ f (∞) − f (0) = lim[− f (0) + sF (s)] ⇒
s→0
lim f (t ) = lim[sF (s)]
t→∞
s→0
372
t
Appendix A: Review of Control-Theory Fundamentals
t=τ
Figure A.4. Double integration in different orders.
τ
Convolution Integral
Given the I/O relationship of a linear time invariant system:
t
y(t ) =
u(t − τ )g(τ )dτ
0
L[y(t )] =
∞
0
σ =t −τ =
=
t
u(t − τ )g(τ )dτ e−st dt =
0
∞
0
0
∞
−st
u(t − τ )g(τ )e
dtdτ =
t
u(t − τ )g(τ )e−st dτ dt
0
∞
g(τ )
0
τ
∞
∞
g(τ )
0
∞
∞
u(t − τ )e−st dtdτ
τ
u(σ )e−sσ dσ e−sτ dτ = G(s)U (s)
(A.2)
0
where the change of integration limits can be understood from Figure A.4:
Eq. (A.2) is very important. The time response of a linear time invariant system
is governed by the complex convolution operation between input and output signals.
In the s-domain, however, the I/O relationship becomes a simple multiplication. In
other words, it is challenging to do any serious analysis and design work in the time
domain but it is straightforward in the s-domain.
We usually refer to G(s) as the transfer function because it operates on the
input signal to get the output signal (Figure A.5). For zero initial conditions, the
transfer function represents the ratio of output to input in the Laplace domain (i.e.,
s-domain).
This example is intended
to show why it is important to use s-domain representation rather than time
domain.
EX. A.2.4: I/O RELATIONSHIP OF CLOSED-LOOP SYSTEM.
r
C
+
u
y
P
_
To determine the output y(t) under a certain input signal r(t), we could:
1. Integrate over time; this can be achieved using MATLAB.
2. Construct the time-domain I/O relationship using convolution.
t
y(t ) =
u(t − τ1 )p(τ1 )dτ1
0
u(t ) =
0
⇒
y(t ) =
t
[r(t − τ2 ) − y(t − τ2 )]c(τ2 )dτ2
t
0
0
t−τ1
[r(t − τ1 − τ2 ) − y(t − τ1 − τ2 )]c(τ2 )dτ2 p(τ1 )dτ1
A.2 Mathematical Background and Design Techniques
373
U(s)
Figure A.5. I/O relationship in s-domain.
Y(s)=G(s)U(s)
G(s)
Because y(t) is involved on both sides of the equation, analytical solutions usually
cannot be obtained.
3. Obtain the s-domain relationship.
Y (s) = P(s)U (s)
U (s) = C(s)(R(s) − Y (s))
⇒ Y (s) =
(A.3)
P(s)C(s)
R(s)
1 + P(s)C(s)
The time domain function y(t ) then can be easily obtained by taking the inverse
Laplace Transform of the right-hand side of Eq. (A.3).
Transfer Function of Linear Time Invariant Systems
By using the Laplace Transform, differential equations can be written as algebraic
equations. This is achieved by replacing the time-domain differentiation operator dtd
by the Laplace Transform operator s. In general, if the dynamic equation of a linear
time invariant system is:
dn
dn−1
dm
y(t
)
+
a
y(t
)
+
·
·
·
+
a
y(t
)
=
b
u(t ) + · · · + bm u(t )
1
n
0
dt n
dt n−1
dt m
then we have in the s-domain ( dtd → s)
snY (s) + a1 sn−1Y (s) + · · · + anY (s) = b0 smU (s) + · · · + bmU (s)
15pt] ⇒ Y (s) =
b0 sm + b1 sm−1 + · · · + bm
U (s)
+ a1 sn−1 + · · · + an−1 s + an
5
67
8
sn
(A.4)
transfer function
Eq. (A.4) showsthe relationship between the system dynamic equation and the
transfer function for one given high-order ordinary differential equation (ODE).
In many cases, we have several lower-order ODEs. It may be easier to obtain the
state-equation form of the system first. The transfer function then can be obtained
from the state-space form ẋ(t ) = Ax(t ) + Bu(t ) and y(t ) = Cx(t ) + Du(t ) .
An important characteristic of the system to be obtained from the transfer
function is the DC gain. For the system shown in Eq. (A.4), if the input u(t ) is a
unit-step input, then the output y(t ) at steady-state has the value bm /an .
374
e
Appendix A: Review of Control-Theory Fundamentals
s1t
u(t)
s t
G(s1 )
G(s1 )e 1
y(t)
Figure A.6. I/O relationship of the transfer function.
FACT A.2.5: A PHYSICAL INTERPRETATION OF THE TRANSFER FUNCTION. When a
physical system is described by the transfer function G(s) (i.e., the impulse
response ≡ g(t), G(s) = L[g(t )]) and when the input signal (i.e., assumed to be
slow compared to the dynamics of G(s)) is in the form u(t ) = es1 t , then the output
signal (under zero initial conditions and assuming that the system modes settle
quickly) also is an exponential signal, amplified in magnitude by the systemtransfer function (i.e., y(t ) = G(s1 )es1 t ).
Proof:
y(t ) =
=
∞
g(τ )u(t − τ )dτ =
0 ∞
−s1 τ
g(τ )e
s1 t
dτ e
0
∞
g(τ )es1 (t−τ ) dτ
0
= G(s1 )es1 t
(A.5)
This relationship is illustrated in Figure A.6.
For a stable system G(s), if we apply a sinusoidal input sin(ωt ), then the steady-state output y(t) (after the transient settles)
is G( jω) sin(ωt ) = |G( jω)| sin(ωt + ∠G( jω)).
FACT A.2.6: FREQUENCY RESPONSE.
Proof: Because sin(ωt ) =
y(t ) =
1
(e jωt
2j
− e− jωt ), applying Eq. (A.5),
1
1
[G( jω)e jωt − G(− jω)e− jωt ] = [|G( jω)| e j(ωt+∠G( jω))
2j
2j
− |G(− jω)| e j(−ωt+∠G(− jω)) ] = |G( jω)| sin(ωt + ∠G( jω))
In other words, the frequency response is a special case of the relationship
described in Eq. (A.5) by letting the real part of the complex number s be zero.
The interpretation of the system frequency response is the following: When
the input signal to a linear system is sinusoidal, the output also is a sinusoidal
signal, amplified by |G( jω)| (i.e., gain of the system at this frequency ω), with a
phase shift of ∠G( jω) (i.e., phase of the system at this frequency ω). To obtain
the frequency response, we simply substitute s = jω into the transfer function
G(s). The magnitude (i.e., gain) and phase of the resulting function G( jω) then
determine the frequency response of the system.
Consider the modeling of
a mass-spring-damper system in terms of both state equations and a transfer
function.
ẋ = Ax + Bu
EX. A.2.7: MODELING A MASS-SPRING-DAMPER SYSTEM.
y = Cx + Du
y = G(s)u
A.2 Mathematical Background and Design Techniques
375
4
G( jω )
Figure A.7. Graphical interpretation of
system frequency response.
2
y
0
u = sin(ω t)
∠G( jω )
-2
-4
0
0.5
1
1.5
The state equations for x1 = x, x2 = v can be written as:
⎧ ⎫
⎪ 0 ⎪
⎨ ⎬
⎥ x
+
⎦
1 F
b
⎪
⎩ ⎪
⎭
v
−
m
m
" #
x
y = [1 0]
+ 0F
v
⎡
0
d x
⎢
=⎣ k
dt v
−
m
1
⎤
The transfer function is given by:
Y (s)/U (s) = G(s) =
1
ms2 + bs + k
Block Diagram
The block diagram of dynamic systems (in the s-domain) has the same simple algebraic relationship as the static systems introduced in Section A.1. Therefore, simplification of the block diagram in s-domain becomes straightforward.
EX. A.2.8: THREE BASIC OPERATIONS: SERIES, PARALLEL, AND FEEDBACK FORM
u
G2
G1
u
G1
G2
r
G
+
+
y
y
Y (s) = G2 (s)G1 (s)U (s)
Y (s) = [G1 (s) + G2 (s)]U (s)
+
y
-
Y (s) =
G(s)
R(s)
1 + G(s)H(s)
H
We can combine signal loops and shift connection points using the three basic operations shown in Ex. A.2.7. Examples
include the following:
EX. A.2.9: BLOCK-DIAGRAM ALGEBRA.
2
376
Appendix A: Review of Control-Theory Fundamentals
y
r
+
+
G1
+
-
-
y
r
G1
=
G2
-
G2 + G3
G3
y
r
G1
+
y
r
G2
=
G1
+
-
-
G3
G2
G2 G3
Using the basic relationship presented in Ex. A.2.8 and Ex. A.2.9, we easily can
simplify complicated systems and obtain the transfer function of a closed-loop
system.
Poles and Zeros
Given a transfer function:
b sm + b1 sm−1 + · · · + bm
Y (s)
b(s)
=
= n 0
U (s)
s + a1 sn−1 + · · · + an−1 s + an
a(s)
If b(s) and a(s) are co-prime (which usually is true for most physical systems), then
the poles and zeros are defined as:
Poles
Zeros
∀si , a(si ) = 0
∀si , b(si ) = 0
are
For example, the finite poles and zeros of the transfer function s2 2s+1
+3s+2
(−1,−2) and (−0.5), respectively. The equation a(s) = s2 + 3s + 2 = 0 is referred
to as the characteristic equation of the system, the roots of which are the poles of the
system. The poles represent the “modes” of the system under impulse input (i.e.,
impulse response, also known as natural response). There is a close relationship
between the location of poles and time-domain response.
EX. A.2.10: TIME RESPONSE, REAL POLES.
G(s) =
2s+1
s2 +3s+2
The impulse response of the system
is:
2s + 1
·1
Y (s) = G(s)U (s) = 2
s + 3s + 2
2s + 1
−1
3
−1
−1
⇒ y(t ) = L
=L
= −e−t + 3e−2t
+
s2 + 3s + 2
s+1 s+2
where the inverse Laplace Transform uses tables that are available in many
automatic-control textbooks. It is clear that the poles correspond to the “modes”
of system response. A real pole located at s = −a corresponds to the time-domain
mode of e−at . What about complex poles? Complex poles always exist in complex
A.2 Mathematical Background and Design Techniques
conjugate pairs. The reason is that differential equations of physical systems have
only real parameters.
EX. A.2.11: TIME RESPONSE, COMPLEX POLES. Find the inverse Laplace Transform
of the function G(s) =
5s+3
s2 +2s+10
5(s + 1)
5s + 3
−2
+
=
s2 + 2s + 10
(s + 1)2 + 32
(s + 1)2 + 32
2
−
·
3
5(s
+
1)
2
3
⇒ g(t ) = L−1
+
= 5e−t cos 3t − e−t sin 3t
(s + 1)2 + 32
(s + 1)2 + 32
3
G(s) =
Ziegler–Nichols Tuning Rules
Two empirical rules were proposed by Ziegler and Nichols (Franklin et al. 2010):
(1) the step-response method, which requires that the open-loop step response of the
system is roughly S-shaped; and (2) the ultimate-sensitivity method, which examines
the closed-loop response of the plant. These two methods are summarized in the
following discussion.
Step-Response Method
Examine the step response of the open-loop system. Draw the steepest slope line
and define two quantities, R and L.
R
L
Intuitively, L represents a “delay” of the system; when L is large, the control
should be conservative (i.e., smaller PI gains, larger D gains). The steepest slope
R represents the “bandwidth” of a system. When R is large, the control gain again
should be conservative. Based on extensive experiments, Ziegler and Nichols recommend the following gains:
P control
PI control
PID control
1
RL
0.9
K=
RL
1.2
K=
RL
K=
TI = 3.3L
TI = 2L TD = 0.5L
Ultimate-Sensitivity Method
Examine the closed-loop response of the P-controlled system (i.e., controller C consists of only proportional gain). Increase control gain C until the plant is marginally
stable, with constant amplitude oscillatory motion. If we denote the value of this
gain (i.e., ultimate P-gain) Ku and the oscillation period Pu , then the recommended
PID gains are as follows:
377
378
Appendix A: Review of Control-Theory Fundamentals
r
C
+
Plant
-
P control
PI control
PID control
K = 0.5Ku
K = 0.45Ku TI = 0.83Pu
K = 0.6Ku TI = 0.5Pu TD = 0.125Pu
These recommended values should be viewed as the “initial starting point.”
They usually have smaller damping than generally is desired, and fine-tuning these
values to suit specific needs typically is necessary.
EX. A.2.12:
Design a PID controller for the plant:
P(s) =
1000
(s + 1)(s + 4)(s + 10)
Obtain the open-loop step response of the plant from MATLAB. From the
plot, we have R = 16 and L = 0.23. Thus, the PID controller is obtained from the
Ziegler–Nichols rules (i.e., the step-response method)
where the control parameters are Kp = 0.326, Ki = 0.709, and Kd = 0.0375. This
design has too much overshoot, a long rise time, and a long settling time; fine-tuning
may be needed.
% Example program to plot closed% loop response of PID controlled % system
R=16;
L=0.23;
K=1.2/R/L;
A.2 Mathematical Background and Design Techniques
379
Ti=2*L;
Td=0.5*L;
Ki=K/Ti;
Kd=K*Td;
num_c=[Kd K Ki];
den_c=[1e-10 1 0]; % to avoid causality problem
num_p=1000;
den_p=[1 15 54 40];
[num_op, den_op] = series(num_c, den_c, num_p, den_p);
[num_cl, den_cl] = cloop(num_op, den_op, -1);
t=0:0.05:4;
y_p=step(num_p, den_p, t);
y_cl=step(num_cl, den_cl, t);
plot(t,y_p)
xlabel(‘Time (sec)’)
grid
pause
plot(t(1:61),y_cl(1:61)) % plot up to 3 sec
xlabel(‘Time (sec)’)
grid
Root-Locus Methods
Root locus is a technique to study the locations of roots of algebraic equations when
one parameter is varying. One application of the root-locus method to control design
examines the effect of controller gains on the locations of closed-loop poles. The
canonical form for the root-locus method looks like this:
a(s) + Kb(s) = 0,
or equivalently 1 + K
b(s)
=0
a(s)
where K is the parameter that is varying and a(s) and b(s) are monic polynomials
of s. When we study the closed-loop pole locations, the first step is to rewrite the
characteristic equation of the closed-loop transfer function into canonical form; this
process usually is straightforward.
EX. A.2.13: CANONICAL FORM FOR ROOT-LOCUS ANALYSIS.
For the mass-spring-
damper system:
k
u(t)
m
b
y(t)
If a PID control law is used, the unity-feedback closed-loop transfer function is:
Gc (s) =
KD s2 + Kp s + KI
Y (s)
=
Yd (s)
ms3 + (b + KD )s2 + (k + Kp )s + KI
380
Appendix A: Review of Control-Theory Fundamentals
The characteristic equation is then:
ms3 + (b + KD )s2 + (k + Kp )s + KI = 0
(A.6)
If we want to check the locations of the closed-loop poles as Kp varies (whereas
KI and KD are fixed), Eq. (A.6) should be rewritten first in the following canonical
form:
a(s) + Kb(s) = 0
where for this example, we have:
a(s) = s3 +
EX. A.2.14:
K
(b + KD ) 2
k
s + s+ I
m
m
m
b(s) = s and
K=
Kp
m
Design a PID controller for the plant:
P(s) =
1000
(s + 1)(s + 4)(s + 10)
PID design was presented previously using the Ziegler–Nichols rules. Because these
rules are empirical, fine-tuning usually is necessary. Trial and error can be used, but
a more systematic way can be based on the root-locus method. We could fix the
Ki and Kd gains and tune Kp for better response. Then, we fix Kp, Ki, and adjust
Kd, etc. The following MATLAB program illustrates this process. Notice that the
characteristic equation of the closed-loop system is:
K
(s + 1)(s + 4)(s + 10) + Kp + I + Kd s 1000 = 0
s
Therefore, the corresponding K, a(s), and b(s) of the canonical form is:
vary Kp
vary Ki
vary Kd
a(s)
s(s + 1)(s + 4)(s + 10) + 1000(KI + Kd s2 )
s(s + 1)(s + 4)(s + 10) + 1000(Kp s + Kd s2 )
s(s + 1)(s + 4)(s + 10) + 1000(Kp s + KI )
Kd
Kp
10
10
5
5
0
0
-5
-5
-10
-10
-5
0
-10
-10
b(s)
s
1
s2
K
1000Kp
1000KI
1000KD
Ki
5
0
-5
% Example to plot closed-loop
% response of PID controlled system
R=16;
L=0.23;
K=1.2/R/L;
Ti=2*L;
Td=0.5*L;
0
-5
-6
-4
-2
A.2 Mathematical Background and Design Techniques
Kp=K; % 0.3261
Ki=K/Ti; % 0.7089
Kd=K*Td; % 0.0375
% tuning Kp
num_kp=[1 0];
den_kp=[1 15 54 40 0]+[0 0 1000*Kd 0 1000*Ki];
K=700*Kp:100*Kp:1300*Kp;
R=rlocus(num_kp, den_kp, K);
plot(R,‘+’), title(‘Kp’)
Kp=Kp*1.1; %Select new Kp
pause;
% tuning Kd
num_kd=[1 0 0];
den_kd=[1 15 54 40 0]+[0 0 0 1000*Kp 1000*Ki];
K=800*Kd:100*Kd:1400*Kd;
R=rlocus(num_kd, den_kd, K);
plot(R,‘+’), title(‘Kd’)
Kd=Kd*1.4; %Select New Kd
pause;
% tuning Ki
num_ki=1;
den_ki=[1 15 54 40 0]+[0 0 1000*Kd 1000*Kp 0];
K=800*Ki:50*Ki:1200*Ki;
R=rlocus(num_ki, den_ki, K);
plot(R,‘+’), title(‘Ki’)
Ki=Ki*0.95; %Select New Ki
After one round of root locus, a new set of control gains that should perform
better than the original design is obtained, as shown on the following plot.
% Example program to plot closed-loop
% response of PID controlled system
R=16;
L=0.23;
K=1.2/R/L;
381
382
Appendix A: Review of Control-Theory Fundamentals
Ti=2*L;
Td=0.5*L;
Kp=K; % 0.3261
Ki=K/Ti; % 0.7089
Kd=K*Td; % 0.0375
num_c1=[Kd Kp Ki];
num_c2=[Kd*1.4 Kp*1.1 Ki*0.95];
den_c=[1e-5 1 0]; % to avoid casuality problem
num_p=1000;
den_p=[1 15 54 40];
[num_op1, den_op1] = series(num_c1, den_c, num_p, den_p);
[num_op2, den_op2] = series(num_c2, den_c, num_p, den_p);
[num_cl1, den_cl1] = cloop(num_op1, den_op1, -1);
[num_cl2, den_cl2] = cloop(num_op2, den_op2, -1);
t=0:0.05:4;
y_cl1=step(num_cl1, den_cl1, t);
y_cl2=step(num_cl2, den_cl2, t);
plot(t(1:61),y_cl1(1:61), t(1:61),y_cl2(1:61),‘-.’)
xlabel(‘time (sec)’)
legend(‘old’,‘new’)
grid
State-Feedback Control
Consider a plant modeled in terms of linear-state equations, as follows:
ẋ = Ax + Bu
(A.7)
with a state-feedback control law of the form:
u = −Kx
(A.8)
Substituting Eq. (A.8) into Eq. (A.7) yields the closed-loop system equations:
ẋ = (A − BK)x = Ac x
(A.9)
Thus, the closed-loop system dynamics is governed by the closed-loop coefficient
matrix Ac . The control gains, K, can be selected to achieve the desired closed-loop
pole locations. This is facilitated by the use of commands such as acker (or place) in
MATLAB, which are illustrated in the following example.
EX. A.2.15: Consider the following state-equation representation of an undamped
second-order system with a natural frequency of 1 rad/s and a state-feedback
controller:
" # " # ẋ1
x1
0 1
0
u
=
+
ẋ2
x2
−1 0
1
" #
x
u = −Kx = −[K1 K2 ] 1 = −K1 x1 − K2 x2
x2
A.2 Mathematical Background and Design Techniques
Find the state-feedback control law that places both of the closed-loop poles at −2. In
other words, by selecting these poles, we double the natural frequency and increase
the damping ratio from 0 to 1. The desired poles are at (s + 2)(s + 2) = s2 + 4s + 4
and the closed-loop characteristic equation is obtained from:
det[sI − (A − BK)] = s2 + K2 s + (1 + K1 ) = 0
Consequently, we obtain K1 = 3 and K2 = 4. The same result is obtained using the
following series of MATLAB commands and Simulink model:
>>A=[0,1;-1,0]; B=[0;1];
>> p=[-2;-2];
>>K=acker(A,B,p)
The response y = [x1 , x2 ] of the closed-loop system to initial conditions x(0) =
[1 0] is shown here.
Limitations of the state-feedback control approach, as presented previously, are
as follows:
1.
2.
3.
4.
The system must be controllable.
All of the states must be measureable.
No tracking of a reference input is possible.
No integral action to increase system type is included.
383
384
Appendix A: Review of Control-Theory Fundamentals
All of these limitations, except the requirement that the plant be controllable,
can be addressed by extensions of the basic state-feedback method presented here
(Franklin et al. 2010). These extensions include the use of an observer (or state
estimator), the inclusion of a reference input in the control u, and augmentation of the
state equations of the plant to include integral action. Also, optimal control methods
can be used to determine the controller gains, K, rather than using pole-placement
methods. The concepts of controllability and observability are fundamental to the
state-feedback controller-design approach and are described in Franklin et al. (2010).
REFERENCE
Franklin, G. F., J. D. Powell, and A. Emami-Naeini, 2010, Feedback Control of Dynamic
Systems, Pearson Prentice-Hall.
APPENDIX B
Two-Mass Three-Degree-of-Freedom Vehicle
Lateral/Yaw/Roll Model
One factor that greatly affects the handling behavior of vehicles is the vehicle roll
motion (i.e., rotation φ about the x-axis). In the development of the bicycle model
(see Chapter 4), it was assumed that a vehicle consists of one rigid body and that
the roll DOF was neglected. This assumption is valid only when the vehicle body
does not roll excessively. In reality, suspension systems separate vehicles into rolling
and nonrolling masses. The rolling (i.e., sprung) mass rolls relative to the nonrolling
(i.e., unsprung) mass, and considerable weight shift may occur. The change in tire
normal forces on the two sides of the vehicle can significantly change the tire lateral
force generated. Furthermore, the existence of the extra DOF (i.e., roll) significantly
changes the dynamic equations. Due to these reasons, it becomes necessary to include
the roll DOF to accurately predict vehicle handling response under high lateralacceleration level. In this appendix, a three-DOF model is developed that includes
lateral, yaw, and roll DOF.
A side view of the vehicle is shown in Figure B.1. The vehicle is divided into
rolling (mR ) and nonrolling (mNR ) masses. Point O is assumed to be the overall CG
of the vehicle (combining mR and mNR ). A coordinate system xyz is assumed to be
fixed on the vehicle, with the following features:
r The x-axis is horizontal and passes through the CG of m .
NR
r The z-axis is vertical and passes through the CG of vehicle (Point O).
r The roll-axis connects the roll center at the rear axle and front axle and is
pointing somewhat downward with an angle θR .
From the definition of vehicle CG, we have mNR · e = mR · c.
The equations of motion can be derived using Lagrange’s approach (i.e., from the
energy viewpoint). For each DOF, we obtain a second-order ODE from Lagrange’s
equation:
∂T
d ∂T
(B.1)
−
= Qr
dt ∂ q̇r
∂qr
where T is the system kinetic energy and Qr is the generalized force that includes both
conservative and nonconservative forces. In other words, the conservative forcing
terms associated with potential energy are absorbed in the Qr term. Assume that
the generalized inertial coordinate XY (with origin fixed at the Point O) and the
385
386
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle Lateral/Yaw/Roll Model
Figure B.1. Side view of the three-DOF vehicle model.
vehicle-fixed local coordinate xy defined previously are related by the yaw angle;
their relationship is described by the following equations (Figure B.2):
u = Ẋ cos ψ + Ẏ sin ψ
(B.2)
v = −Ẋ sin ψ + Ẏ cos ψ
(B.3)
In this discussion, assume that there is a series of inertial coordinates defined on
the moving Point O and that the angle ψ is always small.
The kinetic energy of the vehicle consists of two parts: the energy stored in the
rolling and the nonrolling masses. In equation form, we have T = TR + TNR or, more
precisely:
⇀
⇀
1
1
(B.4)
mNRV NR · V NR + (Izz )NR r2
2
2
because there are no pitch and roll motions for the nonrolling unit. From the fact
TNR =
⇀
⇀
⇀
that V NR = u i + (v − er) j, we have:
TNR =
1
1
mNR [u2 + (v − er)2 ] + (Izz )NR r2
2
2
⇀
⇀
(B.5)
⇀
Similarly, for the rolling mass, we have TR = 12 mRV R · V R + 12 ωR · H R . The
velocity and inertia terms, respectively, are:
⇀
⇀
⇀
⇀
V R = (u − hφr) i + v + hp + cr) j
⇀
⇀
(B.6)
⇀
ωR = p i + (r + θR p)k
(B.7)
⎤ ⎡
⎤
⎡
⎤ ⎡
⎤ ⎡
(Ixx )R p − (Ixz )R (r + θR p)
HRx
p
Ixx 0 −Ixz
⎥ ⎢
⎥
⎥ ⎢
⎥ ⎢
⎢
0
0
⎦=⎣
⎦ (B.8)
⎣ HRy ⎦ = ⎣ 0 Iyy 0 ⎦ ⎣
HRz
−Ixz 0 Izz R r + θR p
−(Ixz )R p + (Izz )R (r + θR p)
Therefore:
TR =
1
m [(u − hφr)2 + (v + hp + cr)2 ]
2 R
1
1
+ (Ixx )R p2 − (Ixz )R (r + θR p)p + (Izz )R (r + θR p)2
2
2
X
ψ
v
u
Y
y
x
Figure B.2. Relationship of the two coordinate systems.
(B.9)
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle Lateral/Yaw/Roll Model
The total energy of the vehicle is the sum of the energy of the two masses – that
is, Eqs. (B.5) and (B.9). Now that the total energy is precisely defined, the dynamic
equations can be obtained by applying Lagrange’s method to each of the three DOF.
For the lateral DOF, we obtain:
∂T
∂ Ẏ
=
∂T ∂u
∂T ∂v
+
∂u ∂ Ẏ
∂v ∂ Ẏ
(B.10)
= mNR u sin ψ + mNR (v − er) cos ψ + mR (u − hφr) sin ψ
+ mR (v + hp + cr) cos ψ
= (mNR + mR )[u sin ψ + v cos ψ] + (cmR − emNR ) r cos ψ
+ mR [hp cos ψ + hφr sin ψ]
and
∂T
= 0
∂Y
(B.11)
By neglecting second- and higher-order angular terms and using the fact that mNR ·
e = mR · c, we obtain:
$
d ∂T
Fy =
= m(v̇ + ur) + mR h ṗ
(B.12)
dt ∂ Ẏ
For the vehicle yaw direction (note that r = ψ̇), we obtain:
∂T
= −emNR (v − er) + (Izz )NR r − hφmR (u − hφr) + cmR (v + hp + cr)
∂r
− (Ixz )R p + (Izz )R (r + θR p)
= (cmR − emNR )v + [(Izz )NR + (Izz )R + mNR e2 + mR c2 + h2 φ 2 mR ]r
+ [−(Ixz )R + (Izz )R θR + cmR h] p − hφmR u ≡ Iz r + Ixz p − hφmR u
(B.13)
∂T
∂T ∂u
∂T ∂v
=
+
∂ψ
∂u ∂ψ
∂v ∂ψ
∂T
∂T
=
(−Ẋ sin ψ + Ẏ cos ψ ) +
(−Ẋ cos ψ − Ẏ sin ψ )
∂u
∂v
∂T
∂T
=
v−
u = mNR [uv − (v − er)u]
∂u
∂v
+ mR [(u − hφr)v − (v + hp + cr) u]
= (emNR − cmR )ur − mR [hpu + hφrv]
(B.14)
Therefore:
$
Mz = Iz ṙ + Ixz ṗ
(B.15)
Last but not least, the vehicle roll equation is shown here. Because the nonrolling
mass is not rolling (i.e., has no contribution to the dynamics), the subscript R is
387
388
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle Lateral/Yaw/Roll Model
added to the kinetic energy T to clearly show this point. The partial derivatives are
as follows:
∂TR
= mR h(v + hp + cr) + (Ixx )R p − (Ixx )R (r + 2θR p) + (Izz )R (r + θR p)θR
∂p
0
1
= h2 mR + (Ixx )R − 2θR (Ixz )R + θR2 (Izz )R p
0
1
+ −(Ixz )R + θR (Izz )R + mR hc r + mR hv
(B.16)
≡ Ix p + Ixz r + mR hv
∂TR
= −hrmR (u − hφr)
∂φ
(B.17)
The roll-dynamic equation is then:
$
Mx |rollaxis = Ix ṗ + Ixz ṙ + hmR v̇ + hmR ru = mR h(v̇ + ur) + Ix ṗ + Ixz ṙ (B.18)
In summary, the dynamic equations for the three-DOF (i.e., lateral, yaw, and roll)
are:
&
Fy = m(v̇ + ur) + mR h ṗ
&
Mz = Iz ṙ + Ixz ṗ
&
Mx|rollais = Ix ṗ + mR h(v̇ + ur) + Ixz ṙ
Comparing this to the two-DOF model in Eqs. (4.45)and (4.46), the extra terms
(i.e., mR h ṗ in the lateral dynamics and Ixz ṗ in the yaw dynamics) clearly show
the effect of the roll DOF on the vehicle dynamic response. The more important
influence, however, lies on the excitation side (i.e., the “F” side of F = ma) of the
equation. The external forces acting on the vehicle (or the rolling mass for the roll
equation) are as follows:
$
$
Fy = Cα f α f + Cαr αr + Cγ f
= Cα f
∂γ f
φ
∂
∂γ f
ar
∂δr
br
+ Cαr
+ Cγ f
φ−β +
φ
δf − β −
u
∂φ
u
∂φ
(B.19)
∂γ f
∂Mz
∂Mz
αf +
α
Mz = a Cα f α f + Cγ f
φ − bCαr αr +
∂φ
∂α f
∂αr r
∂γ f
∂δr
ar br
≈ aCα f δ f − β −
+ aCγ f
(B.20)
φ − bCαr
φ−β +
u
∂φ
∂φ
u
$
Mx = (mR gh − KR )φ − cR φ̇
(B.21)
where in Eq. (B.21), KR is the overall suspension stiffness and CR is the overall suspension damping. The approximation of Eq. (B.20) is justified because the aligning
moment generated from a tire is approximately two orders of magnitude smaller
than the yaw moment generated by its corresponding lateral force.
Comparing Eqs. (B.19) and (B.20) with the corresponding equations for the twoDOF vehicle model, it can be seen that two extra types of variables were introduced:
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle Lateral/Yaw/Roll Model
(1) force and moment variables due to roll-induced camber thrust changes; and
(2) rear-wheel steering angle generated by the roll motion of the sprung mass.
Commonly, stability derivatives are defined based on these force equations. The
major benefits are (1) sources of these forces are clear, and (2) the notation is
unified. Essentially, the stability derivatives are nothing but partial derivatives of the
external forcing terms with respect to the state and input variables. The definitions
of the stability derivatives are highlighted by the following two equations:
&
&
&
&
$
∂ Fy
∂ Fy
∂ Fy
∂ Fy
β+
r+
φ+
δ
Fy =
∂β
∂r
∂φ
∂δ f f
(B.22)
≡ Yβ β + Yr r + Yφ φ + Yδ δ f
$
Mz =
∂
&
&
&
&
Mz
∂ Mz
∂ Mz
∂ Mz
δf
β+
r+
φ+
∂β
∂r
∂φ
∂δ f
(B.23)
≡ Nβ β + Nr r + Nφ φ + Nδ δ f
From Eqs. (B.19) and (B.20), the stability derivatives are then:
Yβ = −(Cα f + Cαr )
Yr =
∂γ f
∂δ
Yφ = Cαr r + Cγ f
∂φ
∂φ
Yδ = Cα f
Nβ = −aCα f + bCαr
Nφ = aCγ f
∂γ f
∂φ
− bCαr
−aCα f + bCαr
Nr = −
∂δr
∂φ
u0
a2Cα f + b2Cαr
u0
Nδ = aCα f
where we now have denoted the constant longitudinal velocity u = u0 . For the roll
dynamics, the equation can be rearranged as:
$
(B.24)
Mx = (mR gh − KR )φ − cR φ̇ ≡ Lφ φ + L p p
Based on the newly defined variables, the dynamic equations are then:
Yβ β + Yr r + Yφ φ + Yδ δ = m (uβ̇ + u0 r) + mR h ṗ
(B.25)
Nβ β + Nr r + Nφ φ + Nδ δ = Iz ṙ + Ixz ṗ
(B.26)
Lφ φ + L p p = mR h (v̇ + u0 r) + Ix ṗ + Ixz ṙ
(B.27)
In matrix form, we obtain:
⎡
⎤⎡ ⎤ ⎡
⎤⎡ ⎤ ⎡ ⎤
−Yβ mu0 − Yr 0
−Yφ
muo
β̇
Yδ
0 mR h 0
β
⎥
⎥
⎢
⎥
⎢
⎥
⎢
⎢0
⎥
⎢
−N
−N
0
−N
I
0
ṙ
I
r
β
r
φ⎥ ⎢ ⎥
Z
xz
⎥⎢ ⎥ + ⎢
⎢
⎢Nδ ⎥
⎣m hu I
⎦ ⎣ ṗ⎦ ⎣0
⎦ ⎣ p ⎦ = ⎣0 ⎦ δ
I
0
m
hu
−L
−L
R
o
xz
x
R
0
p
φ
0
0
0 1
0
φ̇
φ
0
0
−1
0
(B.28)
If the roll-related states (i.e., p and φ) are dropped, with v = βu0 , Eq. (B.28)
reduces to the two-DOF bicycle model shown in Eq. (4.58).
389
390
Appendix B: Two-Mass Three-Degree-of-Freedom Vehicle Lateral/Yaw/Roll Model
Equation (B.28) is not in standard-state-equation form but rather the form Eẋ +
Fx = Gδ, where x = [β r pφ]T . In fact, one of the combined inertia terms (Iz ) is statedependent. In other words, Eq. (B.28) appears to be linear but actually is nonlinear.
Nevertheless, if we ignore the influence of roll angle on Iz (which introduces only
minimal error), Eq. (B.28) can be transformed into a state-space equation ẋ = Ax +
Bδ, where A = −E−1 F and B = E−1 G. The state and input matrices A and B can be
obtained symbolically but they also can be calculated numerically.
Index
accidents, 4, 94, 97–98, 310–311, 315–316, 322–323,
326–327, 350–352
Ackermann angle, 65–69
active safety technologies (AST), 322–323,
350
actuator, 7–9, 16, 25, 30, 133, 137, 152, 198, 217
adaptive cruise control. See autonomous cruise
control
adhesion coefficient. See tire-road friction
coefficient
advanced traffic management systems (ATMS),
310–312
advanced traveler information systems (ATIS),
312–314
advanced vehicle control systems (AVCS),
314–318
aerodynamic drag force, 5, 57–59, 153
air supply system, 39–41, 43, 190, 198
air-fuel ratio, 9, 11–16, 33–38, 49, 119
airbags, 93–94
algebraic Ricatti equation, 289, 291, 295
algorithm development, 26–27
algorithm implementation, 27–28
algorithm in the loop testing, 29
all-electric range, 178
all-wheel drive (AWD), 8, 131–138, 148, 257
anode, 188, 191–195
anthropometry, 93
anti-lock braking system (ABS), 8, 15–17, 93–97,
102, 232–234, 236, 241–254, 257, 263, 323
anti-skid braking. See anti-lock braking system
anti-spin acceleration, 233, 250–251. See also
control, traction
autocoding, 27–28
Autocruise, 332–333, 337
automated highway system (AHS), 4, 94, 345,
350
automated lane following, 352–356
automated steering, 315, 352–358
autonomous cruise control (ACC), 213, 219,
224–229, 332, 337, 345
autonomous guided vehicles, 323–324
battery charge sustainability, 183
battery tax, 158, 163
beta (β) method, 258
bicycle model, 63–64
biomechanics, 93
block diagram, 363, 375–376
body. See chassis
bottom dead center (BDC), 35
brake force, 60, 233, 257–258
brake pressure, 16, 232–234, 258, 261
brake specific fuel consumption (BSFC), 12, 37,
155–159, 166, 169–170
brake steer, 257
brakes, 7
camber angle, 57, 389
camshaft, 15, 34
CAN bus, 16–17, 30, 152
cascade studies, 25
catalytic converter. See two-way catalytic
converter (TWC)
cathode, 188–198
cell temperature, 191, 195
center of gravity (CG), 54–55, 385
certainty equivalence principle, 289
characteristic speed, 66–67
charge depleting (CD) mode, 178, 181
charge sustaining (CS) mode, 178, 181,
183
chassis, 6
choked flow, 40, 47–48
crashworthiness, 322
circadian dip of alertness, 99
closed-loop testing, 28–29
closed-loop. See control, feedback
cognitive engineering, 93
collision avoidance, 102, 314–315, 322–328
combined-slip tire model, 263–268
combustion, 6, 9, 13, 36–37, 41, 44, 48, 119,
124
commercial vehicle operation (CVO), 314–315
communication network, 16–18
391
392
Index
communication
intervehicle, 317–318, 338, 340, 345
vehicle-to-infrastructure, 317–318, 338
compensatory behavior, 102–103, 106
compression stroke, 33–35
compressor, 190–192, 196–201
congestion, 3–5, 309–312, 332
constant time-gap headway policy, 345
constant-spacing headway policy, 345
control, 363
adaptive, 11, 219–223, 302
air-fuel ratio, 8–9, 11–12, 37, 119–122
body, 7–8
clutch, 131–146
cruise, 5, 8, 14, 213–219, 332–337
derivative (D), 11
electronic transmission, 8–9, 14, 38
engine, 5–7, 33, 126, 232–233, 333
exhaust gas recirculation, 4, 7–9, 13, 16, 33–38,
40, 44, 126, 153, 177
feedback, 8, 11, 107, 119, 141, 178, 226, 290, 294,
335, 355, 364–369
feedforward, 14, 138, 163, 274, 335, 365
fuel injection, 7, 33
headway, 213, 224–229, 317
idle speed, 8–9, 14, 38, 126–128
Integral (I), 11
integrated vehicle, 132
lambda, 119
line pressure, 133
load-leveling, 148, 166–168
lockup, 131–133
optimal. See optimal control
piecewise, 138–146
point-follower, 338
powertrain, 7–8
preview, 107, 333–337
proportional (P), 8, 11, 369, 377–381
proportional plus derivative (PD), 11
proportional plus integral (PI), 8, 11, 14,
120–122, 141, 214–222, 333–335, 377–381
proportional plus integral plus derivative
(PID), 8, 11, 217–219, 222–223, 377–381
roll. See rollover prevention
rule-based, 176, 244–248, 254
self-tuning, 220
sideslip, 267
sliding mode, 249–250
spark timing, 8–9, 13–14, 124–125, 126–128
state-feedback, 226–229, 290–291, 294,
382–384
supervisory, 17, 152–153, 157, 178–179
thermostat, 157–164
traction, 6, 232–234, 247–254
vehicle stability (VSC), 257–261, 266–268
vehicle-follower, 338
vehicle, 7–8
yaw, 266–267
controlled system, 8
controlled variable, 8
controlled vehicle, 225–228
controller, 8, 11
in the loop testing, 29
module, 21, 28
convolution, 372
cooling, 190, 198, 204
cornering forces, 65
cornering stiffness, 65–66
corporate average fuel economy (CAFE), 322
course angle, 55
crank angle domain, 47–49
crankshaft, 15, 33–34
critical speed, 66–67
crossover frequency, 95, 100
crossover model principle, 95
cycle beating, 177
damping ratio, 121, 218, 222, 383
dead reckoning. See inertial navigation
degree-of-freedom (DOF), 70, 385
delay, 36, 43–44, 49, 103–110, 119–122, 138–146
development cycle, 22–24
difference equations, 210–221
differential, 6, 257
differential braking, 3, 257–258, 266, 326–327, 352
distributed computing, 17
disturbance input, 8–9, 364
dog actuator, 133–134
drag torque. See torque, load
drawbar force, 58
drivability, 4, 33, 131, 178, 204
drive cycle, 153, 160, 163, 165, 168, 203, 207–208
driver adaptation, 94–95, 97–98
driver model, longitudinal, 109–111
MacAdam’s, 102, 106–107
predictive or preview, 106–109
transfer function, 102–104
driveshaft, 6, 14–15, 62
drivetrain, 6, 62–63, 132, 148, 205
driving simulators, 94–95, 97–98, 100–101, 352
earth-fixed reference frame. See inertial reference
frame
eigenvalue assignment. See pole placement
electro-rheological fluid, 288
electrolysis, 187
electronic control unit (ECU), 7, 16, 21, 152–153,
177
electronic rumble strip, 327, 352
Electronic Stability Program (ESP). See control,
vehicle stability
electronic transmission control, 131–132
emissions, 4, 9, 12–13, 35–38, 119, 124, 131, 148,
158, 180, 311
energy management algorithms, 18
energy storage device, 148
engine, 6, 33–37
block vibrations, 124
control unit (ECU). See electronic control unit
crank, 38
Index
cycle, 33–35
damage, 124
diagnostics, 46–47
four stroke, 33, 39
friction, 36
induction map, 44
inertia, 36, 41, 45
internal combustion (ICE), 6, 33–37, 39,
148–150, 155–158, 163, 166
load, 41, 45, 126
pumping, 36, 43
spark ignition (SI), 6, 33–37, 39
sweet spot, 155, 158
volumetric efficiency, 47–48
warm-up, 38
Environmental Protection Agency (EPA) Cycle.
See drive cycle
equilibrium, 214–218
equivalent consumption minimization strategy
(ECMS), 173–175
equivalent fuel consumption, 172
erasable programmable read only memory
(EPROM), 332
ergonomics, 93
error signal, 365
exhaust gas recirculation (EGR), 7–9, 13, 16,
33–38
exhaust stroke, 33–35
expansion stroke. See power stroke
fail-safe characteristics, 150
final value theorem, 371
fishhook maneuver, 262, 264
flexible driving, 131–132
Flexray, 16
foolproof design, 132
four-wheel drive (4WD), 7–8, 135–137,
232
four-wheel-steering (4WS), 6, 8, 272–283
frequency response, 100, 273, 278, 292, 374
friction ellipse tire model, 57–58
friction plate, 133–137
front wheel drive (FWD), 7, 64, 69, 89
front wheel steering (FWS), 272–283
fuel
economy, 4, 153, 163–166, 172–178,
180–183
efficiency, 5, 208, 322
injection, 4, 16, 36, 44
system, 36, 48–49
fuel-cell stack, 187–193
fuel-cell vehicle, 4–5, 187–188, 201–208
fuel-cell, 151, 158, 176, 187–188
generalized coordinate, 385
generalized force, 385
generator, 148–152, 177–181
global positioning system (GPS), 7, 94, 225,
312–314, 348
grade, 59–62, 163, 213–214, 225, 333–336
393
handcoding, 27
hardware in the loop (HIL), 28–31
heading angle, 55, 105, 328
heave, 81
Hough transformation, 349
human factors engineering, 93
humidification, 189–190, 196, 198
hybrid electric vehicle (HEV), 5, 148, 201
hybrid electric vehicle, micro, 149
parallel, 148–153, 166–172
power split, 148–153, 177–178
series, 148–153, 157–166
hybrid electric vehicle, plug-in (PHEV), 178–183
hybrid vehicle, 5, 17, 148–152
hydrogen supply system, 190, 196, 198
hysteresis, 120, 158
ideal gas law, 35, 40, 43, 48
ignition timing. See spark timing
in-hub motors, 148–149
incident management, 311–312
induction stroke, 33–35
induction-to-power (IP) delay, 36, 43–44, 138–146
inertial navigation, 313, 328
inertial reference frame, 55–56, 385–386
initial value theorem, 371
inputs, 8–9, 363
instability, 368–369
instrumentation, 7
integrated motor assist (IMA), 150
intelligent cruise control. See autonomous cruise
control
intelligent transportation system (ITS), 4–5, 94,
309–318, 324, 332, 348
intelligent vehicle-highway system (IVHS).
See intelligent transportation system
invariant equation, 85
ionization reaction, 189
Kalman filter, 294–296, 351–352
kinetic energy, 385–388
knock, 36–37, 124–125
Lagrange’s method, 56, 385–387
lambda-mu curve, 236–237
lane
change maneuver, 281–283
geometry, 351–352
markers, 349, 351
sensing, 348–352
tracking, 348, 350–351
lane-departure accidents. See single-vehicle
road-departure (SVRD) accidents
Laplace transform, 10, 370–372
Laplace transform, properties, 371
lateral acceleration, 15, 67–68, 74, 258, 267–268,
273–274, 284
lead vehicle, 213, 224–230
level-holding phase, 133
LIN bus, 16
394
Index
linear quadratic (LQ) control. See optimal control
linearization, 42, 72, 90, 214
loop gain, 367
lumped parameter models, 54
Lyapunov equation, 86
Lyapunov stability, 139, 145
magic formula. See Pacejka tire model
manifold air pressure (MAP), 8, 10, 14,
37
manifold filling dynamics, 41, 43
manifold, intake, 35, 41, 43
supply and return, 191–192
map matching, 312
mass airflow, 42–43, 47–48, 128
maximum percentage overshoot, 121
measurement noise, 368
mechatronics, 3–5
membrane hydration model, 193, 195
message chips, 332–333
minimum spark advance for best torque (MBT),
12–13, 36–37
mobility, 3–5
model-based design, 21–22
motion sickness, 86
multi-input multi-output (MIMO), 9–10,
199–203
National Advanced Driving Simulator, 98
National Highway Transportation Safety
Association (NHTSA), 4, 232, 352
natural frequency, 121, 382–383
navigation, 93–94, 101, 312–313, 324
near obstacle detection, 324
networked control system, 18
neutral steer, 65–67
Newton’s second law, 36, 45, 54–56, 58, 63–64,
66–67, 69–70, 234
noise vibration harshness (NVH), 153, 167
non-minimum phase (NMP), 273
observer, 138–141, 356
disturbance, 141, 365
open-loop, 138
open-loop. See control, feedforward
optimal control, 8, 11, 289–290
dynamic programming (DP), 176–178
frequency shaped linear quadratic (FSLQ),
355
linear quadratic (LQ), 126–127, 289, 291
linear quadratic Gaussian (LQG), 294–295
optimal operating points line (OOP-Line),
179–180
Otto gasoline engine, 33
outputs, 8–9, 363
oversteer, 65–67
oxygen, 187–189
Pacejka tire model, 57, 236–238, 258–260, 264
Padé approximation, 104
path planning, 101
path projection, 350–352
percentage of road departure (PRD), 99–101
performance index, 291
piecewise affine system (PWA), 138–141
pitch, 55, 386
planetary gear, 149–151, 177–178
plant, 10, 364
plate friction coefficient, 140
platooning, 224, 315, 317–318, 337–343
plug-n-play, 18
pneumatic tire, 55, 57, 234–235
pole placement, 222, 226, 230, 382–384
poles, 376–377
power management, 176, 203–204
power spectral density (PSD), 77–79
power stroke, 33–35
powertrain, 6
precognitive behavior, 102
primary power source, 148, 155
proton exchange membrane (PEM), 189
pursuit behavior, 102
quantization, 340
quarter-car model, 81
ramp following, 133
ramp-metering, 310–311
range, 110, 213, 225–227, 324–326
extender, 148
rate, 110, 324–326
rattle space. See suspension stroke
reactant partial pressure, 191, 195
rear-wheel drive (RWD), 64, 89
rear-wheel steering (RWS), 272–283
recursive least squares (RLS), 50, 219–221,
355
reference (set-point) input, 8–9, 108, 203, 213, 224,
226, 364, 383
regenerative braking, 148–149, 181, 184
requirements, 21–25
reversible power source, 148
ride model, 81–86
ride quality, 85–86, 287
risk homeostasis theory, 95–97, 101
road condition, 232, 250
road departure accidents. See single-vehicle
road-departure (SVRD) accidents
road model, 77–79
roll, 55, 385
rolling resistance force, 57–59, 153–155, 181
rollover prevention, 261, 266–268
root locus method, 215, 379–382
route planning, 312
run-off-road. See single-vehicle road departure
safe distance, 326
safety, 4, 309–318
active, 322–323
passive, 322–323
Index
saturation, 340
sensitivity, 368
sensor, 8, 14, 323–326
acceleration, 15
crankshaft angular position and speed, 15
exhaust gas oxygen (EGO), 8, 12, 15, 37,
119–121
forward-looking, 324
infrared, 324
knock, 15
laser, 323
linear variable differential transformer
(LVDT), 15
manifold absolute pressure (MAP), 8, 10, 14,
37
mass airflow (MAS), 15, 37
radar, 324
slip, 15
throttle angle, 15, 126
ultrasonic, 323–324
vehicle speed, 15
vision, 323, 351
settling time, 121, 215–216, 258, 378
shift map, 38
shift quality, 131
side-slip angle, 55, 257–260, 262, 264, 266–267
single-input single-output (SISO), 9–10, 95, 107,
198–199
single-vehicle-road-departure (SVRD), 97–98,
326–327, 350–352
site-specific information, 332–337
skyhook damper, 287–290
sliding surface, 267
slip, 57
angle, 57, 267
lateral, 57
longitudinal, 60, 237–239
optimal, 234
slip ratio, 233–234
Smith predictor, 120–122
Society of Automotive Engineers (SAE),
54–55
spacing, 337–345
spark command, 36
spark ignition (SI), 16, 33–35, 39, 124, 129
spark timing, 13, 37–38, 124–125
advanced, 36–37, 124–125
retarded, 124–125
speed ratio, 133–136
sprung mass acceleration, 85–86, 287–290
sprung mass, 81–82
stability derivatives, 73, 389
stack current, 191–192, 196, 200–202
stack voltage, 191, 193–199
state equation, 45–46, 82, 107, 126, 225–230, 289,
353, 369–370
state estimation. See Kalman filter and observer,
state
state space, 369–370
state variable, 370
395
state-of-charge (SOC), 153, 157–166, 172–173,
176–184
static stability factor (SSF), 267
steer angle, 55, 64, 103
steering, 6
stoichiometric, 9, 12, 35–37, 119
stopping distance and time, 61
string stability, 343–345
suspension, 6–7, 287–288
active, 7, 81–82, 287–303
passive, 81–82, 86–88, 287–290
semi-active, 7, 287–288
strut type, 7
suspension actuator, 16
suspension stroke, 81–82, 86, 290, 292, 294–296,
299
swappable. See plug-n-play
switching function, 249
system, 363
target hardware. See controller module
Taylor series, 90, 214
testing and validation, 21, 28–31
thermal capacitance, 140
thermal resistance, 140
throttle, 39–40, 126–128
actuator, 16, 126–128
angle, 39–40, 42, 126–128
control, 126–128
motor, 126
time delay. See delay
time to collision (TTC), 326
time to lane crossing (TLC), 326–327, 350–352
tire axis system, 55–56
tire deflection. See wheel hop
tire-road friction coefficient, 57, 233–235,
250
toll debiting, 312, 315
top dead center (TDC), 10, 13, 35–36, 124
torque converter, 7, 45, 63, 131–133, 150
torque, load, 41, 45, 128
brake, 60
engine, 41, 44–45, 128
self-aligning, 56–57
Toyota Hybrid System (THS), 150–151, 177
traction control system (TCS). See control,
traction
tractive force, 56, 62
transfer function, 372–374
transmission control unit (TCU), 152–153
transmission, automatic, 6, 131–132
continuously variable (CVT), 6, 131–133, 151
electronically controlled (ECT), 6, 131–133
manual, 6, 131
two-way catalytic converter (TWC), 9, 12, 37,
119
two-wheel drive (2WD), 135–137
ultra low emission vehicles (ULEV), 4
uncertainty, 365–367, 122, 138–146
396
Index
understeer coefficient, 66
understeer, 65–67
unsprung mass, 81–82, 287
upshift, 133
wall-wetting, 48–49
waterfall model, 24
wet-friction clutch, 137
wheel hop, 81–82, 86, 291–292, 295, 299
v-diagram model, 24–25
vehicle coordinate system, 54–58,
385–386
vehicle dynamics control (VDC). See control,
vehicle stability
vehicle dynamics, lateral, 64–77
longitudinal, 58–64
vertical, 77–88
virtual desired trajectory (VDT), 356
x-by-wire, 16
yaw moment, 257–261, 264, 268
yaw rate following, 266–268
yaw-roll model, 261–264
yaw, 55, 386
zeros, 376–377
Ziegler-Nichols tuning rules, 377–379
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement