gerard_thesis.

gerard_thesis.
Global Chassis Control
and Braking Control
using Tyre Forces Measurement
Mathieu GERARD
Cover: Neurons under the surface.
Emilie Yane Lopes
Global Chassis Control
and Braking Control
using Tyre Forces Measurement
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op
maandag 28 maart 2011 om 15:00 uur
door
Mathieu Pierre GERARD
ingénieur civil électricien, Université de Liège, België,
geboren te Charleroi, België.
Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.ir. M. Verhaegen
Samenstelling promotiecommissie:
Rector Magnificus
Prof.dr.ir. M. Verhaegen
Prof.dr.ir. E.G.M. Holweg
Prof.dr.ir. J. Sjöberg
Prof.dr.ir. S.M. Savaresi
Prof.dr. H. Nijmeijer
dr.ir. J.P. Maurice
dr.ir. M. Corno
Prof.dr.ir. R. Babuška
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft
Chalmers University of Technology
Politecnico di Milano
Eindhoven University of Technology
TNO Automotive
Technische Universiteit Delft
Technische Universiteit Delft, reservelid
This thesis has been completed in partial fulfillment of the requirements of the Dutch
Institute for Systems and Control (DISC) for graduate studies.
The research described in this thesis was supported by the Competence Centre for
Automotive Research (CCAR), TNO Automotive and SKF Automotive Division.
Published and distributed by: Mathieu GERARD
E-mail: [email protected]
ISBN 9789491211072
c 2011 by Mathieu GERARD
Copyright All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system, without
written permission of the author.
Printed in the Netherlands
To my family
Acknowledgments
A thesis is of course a personal achievement, but all this would never have been
possible without all of you; you who gave me a hand, supported me, or simply made
me smile during the last 4 years. To all of you I would like to say a huge thank you.
First of all, I would like to thank my supervisor and all my colleagues within
DCSC and in the Intelligent Automotive Systems group. Michel, I’m grateful for the
constructive talks we had together and for the freedom you allowed me to take in
my research. Edward, it was nice to work in this automotive research environment.
Edwin, I really appreciated all your help regarding vehicle dynamics and lab experimentation. It was extremely valuable to have such an expert like you next door, and
to feel always welcome. Matteo, the inspiration I got from our regular discussions
was priceless. Your arrival in our group had a very positive impact on my research.
Kitty, Ellen, Saskia, Esther, Olaf, it was always pleasant to come to you and I valued
your great support. Arjan, Kees, it was nice to be able to rely on you in the lab.
Then, I would like to thank the people I’ve been collaborating with. William, I’m
really glad we had such an interesting and productive collaboration. It was always a
pleasure to work with you, to visit you in Paris and to receive you in Delft. Bernie,
Vratislav, Jan-Pieter, Jeroen, Hanno, I always appreciated to have your industrial
point of view on the research.
Furthermore, a very warm thank you to all my friends. Gabriel, Arturo, your
friendship is priceless. I really enjoy the fun we have together and I’m grateful for
your support through the difficulties of life. I’m delighted you accepted to be my
paranymphs. Alberto, it was great to have you as neighbor and to feel that you are
always happy to listen and to help. Ronald, Hein, Theo, Ralph, Stephan, Marcel,
Bas, Patrick, let me just say RedBull. Antoine, Segi, Julien, Bishop, and all the
Villersois, I’m happy we stayed in touch despite the distance. Paolo, Justin, Stefan,
Ilhan, Rogier, DCSC would not have been the same without you. Emilie, I will never
forget the dinners, the cocktails and singstar, and thank you for your help with the
cover. Jelmer, it is really exciting to start this entrepreneurial adventure with you.
Tânia, now I know what love really means.
Finally my biggest thank you goes to my family. Maman, Papa, Anne-So, Nico,
vous avez toujours été là pour moi. Tant de bons moments, de vacances ensemble,
de rires et de pleurs aussi. Merci de m’avoir laissé partir loin, même à contre-coeur.
Mais ca reste toujours un immense bonheur de rentrer à la maison. Nous formons
une famille formidable. It’s my pleasure to dedicate this thesis to you.
Mathieu GERARD
Delft, March 2011
v
vi
Contents
1 Introduction
1.1 Global Chassis Control . . . . . . . . . . . . .
1.1.1 Approaches for Global Chassis Control
1.1.2 Scope of this thesis . . . . . . . . . . .
1.1.3 Contributions of this thesis . . . . . .
1.2 Tyre forces measurement . . . . . . . . . . . .
1.2.1 Contributions of this thesis . . . . . .
1.3 Braking control . . . . . . . . . . . . . . . . .
1.3.1 Contributions of this thesis . . . . . .
1.3.2 Tyre-in-the-loop experimental facility
1.4 List of publications . . . . . . . . . . . . . . .
1.4.1 Journal publications . . . . . . . . . .
1.4.2 Conference publications . . . . . . . .
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Global Chassis Control
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2 Global and Local Control
2.1 Tyre forces . . . . . . . . . . . . . . . . . . . . .
2.2 Actuators . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Electro-hydraulic brakes . . . . . . . . . .
2.2.2 Brake-by-wire . . . . . . . . . . . . . . . .
2.2.3 Steer-by-wire . . . . . . . . . . . . . . . .
2.2.4 Active differential . . . . . . . . . . . . .
2.2.5 In-wheel motor . . . . . . . . . . . . . . .
2.3 Sensing and estimation . . . . . . . . . . . . . . .
2.4 Controller structure . . . . . . . . . . . . . . . .
2.5 Control allocation . . . . . . . . . . . . . . . . .
2.5.1 Constraints . . . . . . . . . . . . . . . . .
2.5.2 Continuous optimization . . . . . . . . . .
2.6 Vehicle building blocks . . . . . . . . . . . . . . .
2.6.1 Corner module . . . . . . . . . . . . . . .
2.6.2 Active-steering rack with torque control .
2.6.3 Manual steering wheel with torque control
2.7 Example . . . . . . . . . . . . . . . . . . . . . . .
vii
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2.8
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38
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3 Hybrid Descent Method
3.1 Problem formulation . . . . . . . . . . . . . . . . . . .
3.2 Hybrid steepest descent solution . . . . . . . . . . . .
3.2.1 Filippov solutions and sliding modes . . . . . .
3.2.2 Stationary points of the update law (3.2)-(3.3)
3.2.3 Asymptotic stability . . . . . . . . . . . . . . .
3.3 Practical implementation . . . . . . . . . . . . . . . .
3.4 A simulation example . . . . . . . . . . . . . . . . . .
3.5 Application to Model Predictive Control . . . . . . . .
3.5.1 Model Predictive Control . . . . . . . . . . . .
3.5.2 The hybrid feedback controller . . . . . . . . .
3.5.3 Simulation . . . . . . . . . . . . . . . . . . . .
3.6 Application to Control Allocation . . . . . . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
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2.9
II
Simulation results . . . . . . .
2.8.1 Vehicle model . . . . . .
2.8.2 Simulation with inactive
2.8.3 Simulation with ABS . .
Conclusion . . . . . . . . . . .
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constraints
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Braking Control
67
4 ABS based on Wheel Acceleration
4.1 Simplified modelling . . . . . . . . . . . . . . . . .
4.1.1 Wheel speed dynamics . . . . . . . . . . . .
4.1.2 Tyre force modelling . . . . . . . . . . . . .
4.1.3 Wheel slip and wheel acceleration dynamics
4.2 Modelling practical phenomena . . . . . . . . . . .
4.2.1 Oscillations in measurements . . . . . . . .
4.2.2 Wheel acceleration . . . . . . . . . . . . . .
4.2.3 Brake pressure dynamics . . . . . . . . . . .
4.2.4 Brake efficiency . . . . . . . . . . . . . . . .
4.2.5 Relaxation length . . . . . . . . . . . . . . .
4.3 The theoretical algorithm . . . . . . . . . . . . . .
4.3.1 The five-phase hybrid control strategy . . .
4.3.2 Tuning of the algorithm . . . . . . . . . . .
4.4 Modified algorithms . . . . . . . . . . . . . . . . .
4.4.1 Pressure derivative profiles . . . . . . . . .
4.4.2 Open-loop pressure steps . . . . . . . . . .
4.4.3 Closed-loop acceleration control . . . . . . .
4.5 Experimental validation . . . . . . . . . . . . . . .
4.5.1 Pressure derivative profiles . . . . . . . . .
4.5.2 Open-loop pressure steps . . . . . . . . . .
4.5.3 Closed-loop acceleration control . . . . . . .
4.6 Comparison with the Bosch algorithm . . . . . . .
viii
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4.7
4.6.1 The original Bosch algorithm . . . . . . . . . . . . . . . . . .
4.6.2 Vehicle measurements and a modified Bosch algorithm . . . .
4.6.3 Comparison between the five-phase and the Bosch algorithms
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 ABS based on Tyre Force
5.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Switching strategy . . . . . . . . . . . . . . . . . . . .
5.2.2 Closed-loop control of the wheel acceleration . . . . .
5.2.3 Bounding the trajectory for a simple model . . . . . .
5.3 Controller tuning . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Tuning the controller gain . . . . . . . . . . . . . . . .
5.3.2 Tuning the acceleration levels . . . . . . . . . . . . . .
5.4 Single-wheel validation . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Experimental validation . . . . . . . . . . . . . . . . .
5.4.3 Comparison with acceleration based hybrid algorithm
5.5 Simulation for the two-wheel vehicle . . . . . . . . . . . . . .
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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108
6 ABS using Tyre Lateral Force
6.1 Modelling . . . . . . . . . . . . . . . . . .
6.1.1 Single-track model . . . . . . . . .
6.1.2 Nonlinear Tyre Model . . . . . . .
6.1.3 Linear Analysis of the Single-Track
6.2 Algorithm . . . . . . . . . . . . . . . . . .
6.3 Requirement on the front axle . . . . . . .
6.3.1 Simulations . . . . . . . . . . . . .
6.4 Requirement on the rear axle . . . . . . .
6.4.1 Simulations . . . . . . . . . . . . .
6.5 Conclusion . . . . . . . . . . . . . . . . .
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Model
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7 Slip Control based on a Cascaded Approach
7.1 System modelling . . . . . . . . . . . . . . . . .
7.2 Control design . . . . . . . . . . . . . . . . . .
7.2.1 An homogeneous target filter . . . . . .
7.2.2 A new time-scale . . . . . . . . . . . . .
7.2.3 Dynamic set-point . . . . . . . . . . . .
7.2.4 Control law with proportional feedback
7.2.5 Stability of ż = F1 (λ, z) . . . . . . . . .
7.2.6 Control law with feedforward . . . . . .
7.2.7 Feedforward with tyre uncertainties . .
7.3 Simulation results . . . . . . . . . . . . . . . .
7.4 Experimental validation . . . . . . . . . . . . .
7.5 Conclusion . . . . . . . . . . . . . . . . . . . .
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8 Conclusions and Recommendations
8.1 Main conclusions . . . . . . . . . . . . . . . . . . . .
8.2 Recommendations and directions for future research
8.2.1 Global Chassis Control . . . . . . . . . . . . .
8.2.2 Braking Control . . . . . . . . . . . . . . . .
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Bibliography
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Summary
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Samenvatting
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Résumé
159
Curriculum Vitae
161
x
Chapter 1
Introduction
Mobility and traffic safety is a major concern in the society today. According to the
World Health Organization, 1.2 million people died on the road in 2004 [123]. In
2002, The Netherlands was the 6th safest country in terms of the number of fatalities
per population [38]. Still, since then, road accidents have killed 900 Dutch people per
year on average [20]. An enormous human potential is so being destroyed, with also
grave social and economic consequences.
Various organizations and governments are taking measures to limit this slow but
continuous catastrophy. With its “Nota Mobiliteit”, the Dutch government is giving
itself the objective of reducing the number of victims to less than 580 in 2020 [116].
The measures taken in the past already made the situation to improve, as seen in
the constant slow decrease of the fatalities in the Dutch statistics, from 1066 in 2002
to 791 in 2007 [20]. This evolution is plotted on Figure 1.1. But more effort is still
required to reach the target. Next to the improvement of the road infrastructure,
the limitation of the speed and the fight against drunk drivers, the reduction of the
road accidents also relies on the improvement of the safety of the vehicle itself, and
in particular on active safety able to prevent accidents.
Many accidents take place because the vehicle is not following the trajectory that
the driver desires. There are two reasons why this can happen. The first one is
that, in its current configuration, the vehicle is physically not capable of following the
trajectory. New actuation mechanisms and new control algorithms can better exploit
the entire tyre potential and so extend the vehicle limits. The second reason is that the
driver is not able to apply the suitable steering wheel and pedal actions that will make
the vehicle follow the desired trajectory. The vehicle dynamics is changing depending
on many factors like the speed or the environmental conditions. A normal driver is not
able to properly control his vehicle in conditions that are not regularly encountered.
The introduction of control algorithms can change the dynamics of the vehicle in
those unusual conditions so that it remains predictable and easily controllable for the
driver. This thesis contributes to the increase of the vehicle safety by both pushing
the limits of the vehicle and facilitating the controllability for the driver.
In order to help the driver controlling his vehicle, electronic active safety systems
are added. This enables the raw driver commands to be modified at the actuator
level. In 1978, the Anti-lock Braking System (ABS) was introduced to assist drivers
1
CHAPTER 1. INTRODUCTION
Traffic fatalities in the Netherlands
1100
History
Target
1000
Fatalities
900
800
700
600
500
2000
2005
2010
Year
2015
2020
Figure 1.1: Evolution of the number of fatalities between 2002 and 2007, and target
for 2020. (Source [20])
1986: Tyre pressure monitoring
1987: 4-wheel steering
1992: ESP
1978: ABS
1975
1980
1985
1990
1995
2000: Lane keep Assist
2004: Active front steering
2008: Lane departure prevention
2000
2005
2010
1987: Electronic throttle control
2003: Park assist
1987: Traction control
2002: Electronic parking brake
1999: Adaptive cruise control
Figure 1.2: Timeline for the introduction of active safety systems in passenger cars.
Active safety in top 50 best selling cars
100
90
ABS
ESP
Traction control
80
70
%
60
50
40
30
20
10
0
1990
1995
2000
2005
2010
Figure 1.3: Percentage of the top 50 best selling cars fitted with electronic safety
equipment. (Source [20])
2
during heavy breaking. Since then, dozens of systems have been developed to assist
drivers in all kind of tasks, as illustrated in Figure 1.2. A system like Electronic
Stability Control (ESC) is capable of reducing single vehicle accidents by up to 49%
[41]. Nowadays, cars can be equipped with for example: Anti-lock Brake System,
Electronic Stability Control, Traction Control, Electronic Brakeforce Distribution,
Active Front Steering, Adaptive Cruise Control, Lane Keeping Assist, Tire Pressure
Monitoring, Emergency Brake Assist, Automatic Braking. So far, all those systems
act mostly independently, with their own set of sensors and actuators. This can
introduce performance limitation from un-modelled or unexpected interactions; at
worse such interaction can cause instability and loss of function [54]. Furthermore,
optimality is not guaranteed in the sense that the trajectory of the vehicle might
not be the closest to the driver’s desired one. In order to improve this, coordination
between all the available systems is required.
With the increase of the amount of microcontrollers able to act on the dynamics
of the vehicle, nowadays two main feedback loops can be identified for controlling the
vehicle. This concept is illustrated on Figure 1.4. The first and original loop (outerloop) is closed by the driver. He is responsible for defining the desired trajectory of
the vehicle and giving instructions through the pedals and the steering wheel. The
driving task can be decomposed in various ways [1]. The most well-known approach
is based on the level of cognition of the task [88] where the 3 following hierarchical
levels are defined:
• Strategical (route planning)
• Tactical (interaction with traffic and road)
• Operational (vehicle control through pedals and steering wheel).
Many modern Advanced Driver Assistant Systems (ADAS) can assist the driver in
his driving tasks at the strategical and tactical levels [50, 1]. This thesis is about
helping the driver at the operational level by making the operation of the car through
pedals and steering wheel as intuitive as possible, in all driving conditions.
The same framework could be used if the vehicle is partially or totally controlled
by an automatic driver. In that case, the structure makes it straightforward for the
ad
Ro
Ideal Car
Virtual Car (controlled car)
lft
TU
De
Controller
Inner-Loop
Outer-Loop
Figure 1.4: The vehicle motion is controlled via two feedback loops. The inner-loop,
closed by the controller, renders the car easier to drive. The tuning is based on
the image the driver has about his ideal car. The outer-loop, closed by the driver,
maintains the trajectory.
3
CHAPTER 1. INTRODUCTION
high level automatic trajectory planners and decision makers to interface with the
low level control of the vehicle. For all the developments of this thesis, there is no
difference between having a human or electronic driver. It is therefore assumed that
a human driver is controlling the vehicle for the sake of realism in the coming years.
The second loop (inner-loop) is composed by all the electronic systems acting on
the vehicle dynamics. The purpose of this internal loop is to change the dynamics
of (part of) the vehicle to make it easier for the driver to perform his driving task.
The advances in electronics from the last decades has lead to an increasing amount of
sensors, control units and actuators in vehicles, giving the inner-loop much more potential, but also making it more complex. To properly consider the coupling between
the actuators, coming for example from the non-linear tyre dynamics, and to seek
for a global optimum for the control of the entire vehicle, a multi-input multi-output
control strategy is necessary. In automotive, this has the name of Global Chassis
Control (GCC).
It should be noted that the inner-loop containing the electronics is usually at least
one order of magnitude faster than the outer-loop containing the driver. Therefore, it
is assumed that both loops can be separated and designed independently without risk
of bad interaction. If future results establish that coupling cannot be neglected, this
assumption should be reconsidered and both loops should be designed in an integrated
manner.
1.1
Global Chassis Control
The literature is quite vague when it comes to giving a definition of Global Chassis
Control. The only statement that could be accepted by the entire community is that
Global Chassis Contol involves a multi-input multi-output system, combining at least
2 traditionally independent vehicle control systems. In the following, a definition of
Global Chassis Control is proposed.
Global Chassis Control is a methodology to design the inner-loop and the vehicle
dynamics control systems in order to fulfill a certain number of objectives, set by the
car manufacturer. Many objectives can be considered:
Stability and safety. The most important objective is to guarantee safe driving by helping the driver maintaining the desired trajectory and the stability of the
vehicle. First it should be easy to transmit the desired trajectory through steering
and pedal actions. Human do control objects by learning a model, a representation
of how the object reacts. The simpler the model and the simpler it is to learn it and
use it. In particular, research has shown that human have a larger ability at learning
linear models than nonlinear ones. Further, first order systems are much simpler to
control for human than second or third [87]. Simplifying the dynamics of the vehicle
by rendering it linear, time-invariant and of low order will make the driver much better at controlling it’s vehicle in all conditions. Secondly, the vehicle should be able to
make an optimal use of the tyre potential. The main forces acting on the vehicle are
generated by the tyres. The force each tyre can generate is limited, depending on the
tyre state and the road conditions. Adding new actuators in the vehicle could allow
to better exploit the tyre potential, as long as all the actuators around one tyre are
synchronized.
4
1.1. GLOBAL CHASSIS CONTROL
Ease of development. To reduce the production costs and the time to market,
it is necessary to develop a controller architecture suitable for various vehicles. The
implementation on a specific vehicle should require only a very limited and methodic
tuning and not a complete redesign. It has been predicted that in 2010, 13% of the
production cost of a vehicle would be software, against only 4% in 2000 [56].
Feel-by-wire. Thanks to the inner-loop, the actions of the drivers are not directly
translated to actuator commands anymore and the controller can change the way the
vehicle reacts. In other words, it means that the feeling of the vehicle can be modified
in software. This is attractive for car manufacturers as it gives the possibility to
define feeling objectives at a high level, can make the tuning of the feeling faster and
less expensive and allow for a change of the feeling online, depending on environment
conditions or the style of the driver. More details can be found in [58], [32] and [120].
Consistent state estimation. A good control has to rely on a good measurement or estimation of the states of the vehicle. The number of sensors present in cars
is increasing so that more dynamics can be observed. Today, many subsystems are
still using their own sets of sensors, or they read the raw sensor signals from a bus and
do not benefit from the smart processing done by other subsystems. This prevents
each subsystem from using the same consistent set of measurement, which might limit
performance and be critical in case of failure of a sensor. Also, it is undesirable that
all the subsystems need to be reimplemented if the set of sensors is changed. For
example, if the vehicle get equipped with load sensing bearings enabling the measurement of the forces close to the tyre, an update of the state estimator should be
sufficient to let all the subsystems benefit from the more accurate measurement set
[127, 122].
Fault tolerant. The failure of a sensor or an actuator can always happen. If
nothing is done, this can lead to a dramatic change in the vehicle dynamics which
the driver is unable to respond to. However, it is often possible to reconfigure the
controller in order to minimize the influence of the faulty component. The state
estimation algorithm can be adapted to neglect the faulty sensor and reconstruct
the states from the other signals. The actuation can also be reconfigured to get the
remaining actuators to do as much as possible of the action that the faulty actuator
cannot perform. As two examples, let’s mention the steer-by-brake [37] and the
brake-by-steer [61] concepts. Controlled individual braking (brake only on one side)
can make the car to follow a curvy path [37], while indivitual steering (both wheels
either inwards or outwards) can make the vehicle to brake in straight line [61]. The
way the global chassis controller is designed should allow for a clear way of changing
its structure during driving if a fault is detected.
Energy efficiency. It can happen that many actuators are able to produce the
same effect on the vehicle. For example, differential braking or steering can both
produce a yaw moment on the chassis, or braking can be done using friction or
regenerative brakes. In those cases, a choice has to be made on which combination of
actuator to use. Seen the current necessity to reduce the impact of transportation on
the environment, it is wise in normal driving conditions to chose for the action which
is using the least energy. Still, in emergency situations, the considerations on the
environments will weight less than saving lives. In that case, all the available energy
should be used to maintain stability and the desired trajectory.
5
CHAPTER 1. INTRODUCTION
Physical system
Driver interpreter
Path follower
Chassis controller
Local
controller
Control
allocation
Local
controller
Actuator
Actuator
Vehicle
Actuator
Sensing and
estimation
Figure 1.5: General scheme, with 2 levels, for the hierarchical approaches for Global
Chassis Control.
1.1.1
Approaches for Global Chassis Control
Different approaches can be considered to design a global chassis controller. On
one hand, one can argue that, since many individual systems already exist, it would
be commercially more viable to keep those systems and interconnect them. This
approach could also allow proprietary code from different suppliers to work together.
However, it is still really unclear how such an interconnection can be implemented,
and in particular how to make the architecture modular. An extensive description of
the state-of-the-art in this domain and of the outstanding research problems is given
in [54].
On the other hand, many researchers consider approaches with an hierarchical
architecture [4, 75, 65, 72, 93, 89]. The controller can either be fully centralized or
many levels of controllers can be implemented, from the high-level motion control
to the low-level current control in the actuators. A general scheme is presented on
Figure 1.5. Such an approach has the advantage to have a clear structure that can
be analyzed. In all publications, the total forces and moment desired on the chassis
constitutes the input to the control allocation block. They are defined by the driver
interpreter and a path follower [4]. A common goal for defining the forces on the
chassis is to render the car similar to a linear bicycle. This objective is enforced in
[89] using a sliding mode controller. When the desired forces on the chassis are known,
they need to be distributed to the actuators and the tyres through control allocation
(see Chapter 2). For this step, two major differences are found in the literature.
• The first option is to distribute the total forces directly to the actuators using
a complete nonlinear car model [4, 72, 93]. This gives a major advantage when
it comes to expressing constraints on the actuators. However, the nonlinearity
in the model requires the distribution to use nonlinear programming, which is
complex. Workarounds are proposed, like linearizing the model at each time
step [4], or like avoiding optimization by simply using a Moore-Penrose inverse
matrix [93]. On Figure 1.5, this approach gives a non-convex control allocation
block while the local controller blocks are empty. This class is called “direct
allocation”.
6
1.1. GLOBAL CHASSIS CONTROL
• The other option is to first distribute the total forces to intermediate tyre forces
in the vehicle frame [4, 75, 65, 89]. Then the tyre forces are realized at the local
level using an inverse tyre model. This method has the advantage of making the
distribution model linear and with no uncertain parameter, which enable the
use of simpler optimization techniques. The drawback might be the expression
of the constraints, especially those reducing the number of degree of freedom,
called “restrictors” in [4]. On Figure 1.5, this approach gives a less complex and
convex control allocation block. The local controller blocks are responsible for
the tyre inversion. This class is called “indirect allocation”.
Both approaches rely on the knowledge of the tyre model. In the first case, the
model appears in the equations of the control allocation. In the second case, the
local transformation block need to do a tyre model inversion. This constitutes a large
practical problem for two major reasons. First because the tyre model is nonlinear
and cannot be inverted on its full domain; and secondly because the tyre parameters
are unknown, constantly varying and impossible to estimate accurately online.
1.1.2
Scope of this thesis
This research focuses on the two first objectives of the Global Chassis Control definition: “Stability and Safety”, and “Ease of development”. In particular, the emphasis
is set on exploiting the tyre potential and designing a modular architecture. Still, it
is kept in mind that the proposed structure should clearly be an enabler for the other
objectives. The coupling with the driver and the interpretation of his steering and
pedal actions is not addressed in this research. The complexity of a driver interpreter
is such that it does constitute a research on its own. It is here considered that a driver
interpreter is available and delivers requirements in term of forces on the chassis. It
should also be noted that mainly the dynamics in the road plane are considered (longitudinal, lateral and yaw). Experimentation on a real vehicle has not been possible
during the timespan of this thesis. Validation of the developed control algorithms is
done within a commercially available full vehicle simulation environment.
1.1.3
Contributions of this thesis
This thesis follows the direction of the hierarchical approach and proposes three major
contributions:
• First of all, a detailed analysis of the force distribution problem, as done in Chapter 2, shows that the optimization problem has some particular characteristics:
continuous optimization is required as the cost function and the constraints are
constantly changing, the optimal solution does not need to be known instantly
as long as an optimizing direction is found, and the computational complexity
should be limited to the minimum as the implementation is done on an embedded microcontroller. From this analysis, it can be understood that traditional
optimization methods might not be suitable in this case. Therefore, a new
method for continuous convex optimization is developed: the Hybrid Descent
Method. The method is described in Chapter 3.
7
CHAPTER 1. INTRODUCTION
Figure 1.6: Measurement wheel able to measure the forces and moments in the rim.
• Secondly, it is clear that relying on the inverse of the uncertain and nonlinear
tyre model is a weak point of the current approaches. Thanks to the new sensing
technologies making tyre forces measurement feasible in production vehicle, a
solution more robust to tyre parameter uncertainty can be implemented. This
thesis shows how local tyre controllers can exploit force measurement to realize the desired tyre forces without asking for any tyre parameter. The local
controllers are discussed in Chapter 2.
• Finally, a new kind of architecture is introduced based on the concept of building
blocks. All kind of vehicle with various level of actuations can be represented
with the correct number of degree of freedom, which avoid the need for difficult
constraints or restrictors. The advantage of this concept can be found in particular for vehicle having a partial level of actuation, between no action at all and
full freedom provided by e.g. autonomous corner modules [126]. The concept is
illustrated in Chapter 2.
1.2
Tyre forces measurement
A vehicle is moving thanks to the friction forces the tyres are applying on the ground.
The knowledge of those forces on each tyre is precious in order to design, tune and
control the vehicle.
Test vehicles, used to design new systems, and prototype vehicles are most often
equipped with measurement wheels, see Figure 1.6. Such device measures the strains
in the rim in order to reconstruct forces and moments in the 6 degrees of freedom. Unfortunately, such sensors are too complex, cumbersome and expensive to be installed
in production vehicle. Therefore, current control systems cannot rely on the force information. The sensors available in a car today are all motion based: accelerometers,
gyroscopes, encoders, etc. However, this might change in the near future.
8
1.2. TYRE FORCES MEASUREMENT
Figure 1.7: Load sensing bearing able to measure the forces applied to the bearing in
all directions. (Source: SKF)
Different initiatives are recently taking place in order to design tyre force sensors
that could be inserted in production vehicle. A first option would be to keep measurement rims and make them simpler and cheaper [53]. The embedding of sensors inside
the rubber of the tire is also investigated [23, 109, 108]. Still, the most promising
option is to include sensors inside the hub bearing of the wheel. Different bearing
manufacturers are investigating the possibility. At the moment, SKF is in a leading
position with working prototypes, see Figure 1.7.
It makes sense to include force sensors in the hub bearings for different reasons.
First, it is not a wear component. The bearings are now lasting long enough to
accompany the vehicle during its entire life. This is clearly at the contrary of tyres
and rims. Secondly, bearings are precision components, which are engineered at the
micron level. Measurements in bearings therefore have the potential of being precise.
Finally, force measurement in the bearing can be made on a fixed part, not rotating
with the wheel. This is a serious advantage when it comes to transmitting the sensor
signals.
Over the years, bearing manufacturers have developed extremely precise models
of their bearings. The original purpose was to predict the deformations the bearing
would undertake under a specific loading, depending on the application, see Figure 1.8.
This was precious for dimensioning and predicting the wear. The idea for estimating
the forces acting on the bearing is to use the model exactly the other way around.
Sensors like strain gauges are fitted in the bearing to measure deformations. Then,
the inverse of the model is used to reconstruct the forces and moments applied on
the bearing. This inverse model is the real key component of the technology and
constitutes the core of the intellectual property of the bearing manufacturer.
The current accuracy on the force measurement achieved today by SKF is already
good. The RMS noise is maintained under the 2% and the bias is maximum 10% in
real automotive conditions. Still, even if now only one signal is output from the sensor
with force information, it is known that there are different ways of reconstructing
the force signals from the raw sensors signals, which give different properties. For
example, some methods could offer a larger bandwidth but give a potentially larger
bias, and vice versa. One of the objectives of this research is to investigate where
9
CHAPTER 1. INTRODUCTION
Figure 1.8: Model of the deformation of a bearing under load. (Source: SKF)
force information can provide improvements compared to the current state-of-the-art.
1.2.1
Contributions of this thesis
This thesis investigates the advantages of having force measurement for chassis control, compared to the current state-of-the-art using only motion-based sensors. Different areas are considered:
• The use of force measurement for local tyre control in the Global Chassis Control framework is discussed in Chapter 2. Thanks to force measurement, the
tyre forces can be controlled in closed-loop without necessity of knowing the
uncertain current tyre model. This largely improves the robustness to road conditions compared to the traditional tyre model inversion used for example in
[4, 65, 72].
• The implementation of an Anti-lock Braking System (ABS) algorithms is shown
to be simplified and more robust to changes in tyre characteristics when using
longitudinal force measurement. A new force-based ABS algorithm is presented
in Chapter 5.
• The first objective when implementing an ABS is the need for maintaining
steerability and lateral stability during heavy braking. However, in current ABS
implementations, only the longitudinal movement of the tyre is truly considered.
Thanks to the use of lateral force measurement, a real assessment of the loss
of lateral tyre potential because of braking is becoming possible. Chapter 6
discusses this issue.
10
1.3. BRAKING CONTROL
1
0.9
Normalized force [ ]
0.8
0.7
0.6
Fx / Fz
0.5
Fy / Fz
0.4
0.3
0.2
0.1
0
0
0.2
0.4
Slip λ [ ]
0.6
0.8
1
Figure 1.9: Typical longitudinal and lateral tyre curves as a function of the longitudinal slip λ, for a constant lateral slip α = 0.1. The model used is the Magic Formula
[95].
1.3
Braking control
A particularly critical problem in vehicle dynamics control is the longitudinal control
of the tyre to get the desired driving or braking force. As long as the tyre remains in
its stable zone, this can be done with simple linear controllers, especially if one can
rely on force sensing, see Chapter 2. However, the difficulty increases when one tries
to reach a force close or exceeding the maximum tyre force, as the wheel can become
unstable. On the traction side, Traction Control [32, 43] is used. On the braking side,
an Anti-lock Braking System (ABS) needs to be implemented. In this thesis, only
the braking side is considered, especially because of the availability of lab equipment.
Still, most results can be applied on the traction side.
There are originally two main reasons for implementing an ABS system in a car.
The potential of the tyre to generate lateral forces is decreased when the longitudinal
slip λ is increased, see Figure 1.9. The first ABS objective is therefore to limit the
longitudinal slip in order to maintain steerability and lateral stability during heavy
braking. Furthermore, the brake force generated by the tyre presents a peak at a
moderate slip λ0 on most surfaces, typically between 10 and 20 %, as can be seen
on Figure 1.9. Maintaining the maximum brake force to reduce the brake distance is
the second objective. In the GCC framework, a third objective appears, as the ABS
controller needs to transmit the current tyre limit to the global controller so that
unreachable forces can be redistributed.
The ABS is the most spread active safety system for road vehicles. It is now a
standard equipment for all new passenger cars in the EU, the U.S. and Japan, see
[103, 20] and Figure 1.3. This system has been around for more than 30 years since its
introduction by Bosch in a Mercedes in 1978. Unfortunately, it is still very difficult to
find details in the literature about the ABS algorithms used in practice, as the industry
is really secretive. Furthermore both commercial and academic algorithms can still be
improved. On one hand, commercial algorithms are often based on heuristics, which
11
CHAPTER 1. INTRODUCTION
requires a lot of tuning. No theory is available to assess stability or performance
of those complex rule-based systems. On the other hand, most of the academic
research about braking control is neglecting some important practical issues, such as
actuation limitations and tyre dynamic behaviour, making the algorithms not suitable
for implementation on a real vehicle.
In the literature, one can distinguish two main classes of ABS: those based on
logic switching from wheel deceleration information (see e.g. [71, 96, 18, 79, 77, 74]),
and those based on wheel slip regulation (see e.g. [64, 98, 117, 107, 33, 26, 124]).
In the class of hybrid ABS, the wheel deceleration is used to trigger the different
phases of an hybrid automaton. The brake torque is increased, decreased or held
depending in which phase the system is, and possibly depending on the wheel speed
or acceleration. If there is a peak in the longitudinal tyre characteristic, the switching
leads to a cycling around that peak, without requiring any a priori knowledge about it.
During each cycle, the relative position of the peak is detected and tracked, ensuring
robustness against tyre-road friction variation. However, if no peak is present in the
characteristics, this class of algorithms fails to work. Those methods are not suitable
to stabilize the system around an arbitrary reference λ∗ . Furthermore, the vehicle
speed or wheel slip, which are signals difficult to measure today, see Section 2.3, are
not required. Therefore, this class of algorithms presents serious practical advantages.
Some of them have been shown to work on real vehicles.
In the class of wheel slip regulation, a target wheel slip λ∗ is given and the estimated slip λ is controlled around it. Those methods can work even if there is no peak
in the longitudinal tyre characteristics. Their usage is nevertheless confronted to two
major difficulties. Firstly, the robust measurement of the vehicle longitudinal speed,
needed to compute the slip λ, remains an open problem [73, 62, 111]. Secondly, a
non-optimal reference slip λ∗ , obtained from an uncertain tyre characteristics, can
lead to a loss of performance. Besides, most slip regulation methods comes with a
theoretical background and a stability and performance analysis. However, the design
of the algorithm is mainly based on linearization arguments. The nonlinear system is
linearized around the desired equilibrium point, and the stability analysis is thus only
valid locally (see e.g. [98] and [107]). Also, algorithms might fail to converge in the
unstable region of the tyre (see e.g. [117], where the control strategy generates a limit
cycle if the setpoint is in the unstable domain). And finally, the available approaches
are mainly based on pure feedback, without feedforward, which limits the bandwidth
of the closed-loop system.
This thesis contributes to the design, simulation, implementation and validation
of ABS algorithms, both in the class of hybrid ABS and in the class of slip regulators.
Furthermore this thesis investigates the benefits that tyre force measurement could
bring to braking control. The precise contributions are described below.
Some of the work on the topic has been done in collaboration with CNRS/Supelec,
Paris, France, and in particular with William Pasillas-Lépine. This collaboration was
started in 2008 in the framework of this research with the goal of testing a particular
hybrid ABS (5-phase, see [96]) and improving the algorithm to make it perform better
in practice. This collaboration further lead to a good understanding of ABS issues,
and to the practical implementation of a new slip regulator based on a cascaded
approach.
12
1.3. BRAKING CONTROL
1.3.1
Contributions of this thesis
• A tyre-in-the-loop laboratory setup for rapid prototyping and testing of ABS
algorithms has been developed. The setup consists of a large drum on top of
which a regular rubber tyre is rolling. Most of the mechanics was already present
before the start of this project. The contribution of this thesis lies in the update
of the electronics and software to make it suitable for closed-loop experiments.
Details on the setup can be found in Section 1.3.2.
• When tested in practice, the theoretical 5-phase algorithm could not give the
desired results. The time delay in the loop has been identified to be the main
issue. Three methods to compensate delays or to improve the algorithm by
making it more robust to delays have been developed and are described in
Chapter 4. Thanks to those methods, a hybrid ABS algorithm based on the
original 5-phase algorithm and close to the Bosch ABS has been successfully
tested in practice. This work is done in collaboration with the CNRS/Supelec
Paris.
• An ABS algorithm based on force measurement has been developed and is presented in Chapter 5. This new algorithm is more robust to varying road conditions, more intuitive to tune, and better performing than typical hybrid ABS
algorithms only based on wheel deceleration.
• The first objective when implementing ABS is maintaining steerability and lateral stability during heaving braking. However, in current ABS implementations, only the longitudinal movement of the tyre is truly considered. Thanks
to the use of lateral force measurement, a real assessment of the loss of lateral tyre potential because of braking is becoming possible. A new algorithm
exploiting lateral force measurement is introduced in Chapter 6.
• The new slip controller of [97, 49] is reworked and validated experimentally
in Chapter 7. The control strategy is based on both wheel slip and wheel
acceleration regulation through a cascaded approach; which is proven to be
globally exponentially stable in both the stable and unstable regions of the
tyre. The stability conditions are relaxed and the theory is reformulated in
order to better match the practical reality. Tests on the tyre-in-the-loop facility
show that the slip always converges to the assigned reference. This work is done
in collaboration with the CNRS/Supelec Paris.
1.3.2
Tyre-in-the-loop experimental facility
The tyre-in-the-loop experimental facility of Delft University of Technology, on which
the ABS is tested, consists of a large steel drum of 2.5 meter diameter on top of which
the tyre is rolling. An illustration can be seen on Figure 1.10. The setup has been
used for many years for tyre modelling and identification using open-loop excitation,
see [95, 125, 86]. The inertia of the drum makes it more suitable for keeping a constant
speed. Recently, the electronics was upgraded in order to allow closed-loop tests to
be performed and, in particular, rapid prototyping and testing of ABS strategies.
13
CHAPTER 1. INTRODUCTION
Figure 1.10: Illustration and picture of the tyre-in-the-loop experimental facility.
(Source illustration: [125])
The wheel rim on which the tyre is mounted is attached to an axle with a rigidly
constrained height. This axle is supported by two bearings, on both sides of the wheel.
The bearing housings are connected to the fixed frame by means of piezoelectric force
transducers. An hydraulic disk brake is mounted on one side of the axle. The pressure
in the calliper is locally controlled by a piece of analog electronics connected to a servovalve, in order to match the reference pressure. Both the wheel and the drum speeds
can be accurately measured using encoders, which allows a precise estimation of the
longitudinal slip. Currently, the wheel cannot be steered. More details about the
experimental facility can be found in [35].
1.4
List of publications
This section lists the publications written in the framework of this Ph.D. research.
1.4.1
Journal publications
• Mathieu Gerard, Michel Verhaegen and Bart De Schutter. A Hybrid Steepest
Descent Method for Constrained Convex Optimization. Automatica, vol. 45,
pg 525-531. 2009.
• Mathieu Gerard, William Pasillas-Lépine, Edwin de Vries and Michel Verhaegen. Improvements to a five-phase ABS algorithm for experimental validation.
Submitted to Vehicle Systems Dynamics.
• Mathieu Gerard, Matteo Corno, Michel Verhaegen and Edward Holweg. Hybrid
ABS Control Using Force Measurement. Submitted to IEEE Transactions on
Control Systems Technology.
14
1.4. LIST OF PUBLICATIONS
1.4.2
Conference publications
• Mathieu Gerard and Michel Verhaegen. Model Predictive Control using Hybrid
Feedback. Proceedings of the IFAC World Congress, Seoul, South Korea. 2008.
• Mathieu Gerard and Michel Verhaegen. Global and Local Chassis Control based
on Load Sensing. Proceedings of the American Control Conference, St. Louis,
MO, USA. 2009.
• Mathieu Gerard, William Pasillas-Lépine, Edwin de Vries and Michel Verhaegen. Adaptation of hybrid five-phase ABS algorithms for experimental validation. Proceedings of the IFAC Symposium Advances in Automotive Control,
Munich, Germany. 2010.
• Mathieu Gerard, Matteo Corno, Michel Verhaegen and Edward Holweg. TwoPhase Antilock Braking System using Force Measurement. Proceedings of the
10th International Symposium on Advanced Vehicle Control, Loughborough,
UK. 2010.
• Mathieu Gerard, Antonio Loria, William Pasillas-Lépine and Michel Verhaegen.
Design and experimental validation of a cascaded wheel slip control strategy.
Proceedings of the 10th International Symposium on Advanced Vehicle Control,
Loughborough, UK. 2010.
• Edo de Bruijn, Mathieu Gerard, Matteo Corno, Michel Verhaegen and Edward
Holweg. On the performance increase of wheel deceleration control through
force sensing. Proceedings of the 2010 IEEE Multi-Conference on Systems and
Control, Yokohama, Japan. 2010.
• Kimmo Eggers, Mathieu Gerard, Edwin de Vries and Michel Verhaegen. Vehicle
Side-slip Angle Estimation using Sliding Mode Observers and Lateral Forces.
Proceedings of IAVSD conference, Stockholm, Sweden. 2009.
• Diomidis Katzourakis, Edward Holweg, Mathieu Gerard and Riender Happee.
Design Issues for Haptic Steering Force Feedback on an Automotive Simulator.
Proceedings of the HAVE conference, Lecco, Italy. 2009.
15
CHAPTER 1. INTRODUCTION
16
Part I
Global Chassis Control
17
Chapter 2
Global and Local Control
The tyre forces in the road plane are the main forces defining the trajectory of the
vehicle. Furthermore, they are the only forces controllable by the driver or by means
of actuators. Other forces like gravity or air drag come as external actions that cannot
be modified and that need to be compensated by the tyre forces.
Historically, the driver could modify the tyre forces mechanically using, for example, manual steering or braking. However, nowadays, more and more actuators
allow for an electronic control of the tyre forces. Adding actuators has the potential
to provide more freedom for modifying the tyre forces and influencing the trajectory.
However, there are only 4 tyres in a car and therefore, eventually, extra actuation
or control systems will overlap with each other. A global way of controlling those
4 precious tyre forces is required in order to guarantee optimal performance of the
vehicle [54].
This research focuses on the problem of controlling all the actuators of a vehicle,
targeted at influencing the tyre forces in the road plane, such that the forces generated
at tyre level produces total forces and moment on the chassis as close as possible to the
desired ones. A few solutions have been proposed in the literature. Two main classes
of methods can be distinguished. In the first class, the total forces are distributed
directly to the actuators [17, 118, 5, 100]. This class is called “direct allocation” and
is represented on Figure 2.1. In the second class, the total forces are first distributed
to intermediate forces in the vehicle frame [4]. A local control level then takes care
of computing the actuators commands. This class is named “indirect allocation” and
is illustrated on Figure 2.2. Four approaches in the class of direct allocation are
described next.
Borrelli [17] formulates a non-linear optimization problem in the Model Predictive
Control framework. The actuator commands are directly computed by the algorithm.
This can be seen as an advantage since actuator constraints can be easily formulated.
However, the implementability on embedded microcontrollers and scalability with
respect to the number of actuators are questionable because of the computational
complexity linked to the nonlinear optimization. Moreover, non-linear programming
can lead to difficulties like local optima or slow convergence. Furthermore, many parameters and a tyre model have to be included in the model used in the optimization.
Those are often very difficult to obtain in practice during real-time operation, see
19
CHAPTER 2. GLOBAL AND LOCAL CONTROL
Physical system
Actuator
Driver interpreter
Actuator
Nonlinear
Control allocation
directly to actuators
Path follower
Vehicle
Actuator
Chassis controller
Sensing and
estimation
Figure 2.1: General scheme for the class of direct allocation Global Chassis Control.
The computation of the actuator commands is done in 1 step.
Physical system
Building block
Actuator
Local
controller
Driver interpreter
Actuator
Control
Path follower
Vehicle
allocation
Chassis controller
Local
controller
Actuator
Sensing and
estimation
Figure 2.2: General scheme for the class of indirect allocation Global Chassis Control.
The computation of the actuator commands is done in 2 steps.
Section 2.3. Methods exists to take model uncertainties into account within the MPC
framework [29]. However, the computational complexity is larger than the nominal
case.
To avoid solving non-linear programming problems on-line, Tøndel [118] proposes
to solve the optimization off-line using multiparametric programming. Solutions are
then stored in large lookup-tables. This reduces the risk of bad convergence. However, the size of the lookup-tables can become impractical if the number of variables
and parameters is large, which is the case in automotive applications. Furthermore,
capabilities to reconfigure the system, for example in case of fault, are frozen.
Andreasson [5] solves at each time step the optimization problem based on a
linearised model. This is motivated by the fact that the dynamics of the system
should not vary too much between two iterations. Unfortunately, tyres can raise
difficulties since they are known to have fast dynamics and strong non-linearities.
The computational complexity is decreased to the level of Quadratic Programming.
Plumlee [100] simplifies the problem by linearising the vehicle model and therefore
neglecting the non-linearities. Then a variant of Quadratic Programming is used.
20
2.1. TYRE FORCES
α
v
fx
β
δ
Fx
Fy
fy
Figure 2.3: The tyre frame is oriented in the direction of the wheel while the vehicle
frame is oriented towards the front of the vehicle.
In the class of indirect allocation, Andreasson [4] proposes to distribute the total
forces to the tyre forces in the vehicle frame. The low-level control of the actuators is
then done using a tyre model inversion. This approach does render the cost function of
the allocation problem convex, which considerably reduces the complexity compared
to non-linear optimization. However, if every single wheel is not completely actuated,
constraints appears between the tyre forces. Such constraints called restrictors can be
non-linear and depending on uncertain tyre parameters, which can kill the convexity
of the problem and increase the complexity. Therefore, the total problem might not
be simpler to solve than direct allocation.
This chapter proposes a new method for Global Chassis Control following the
ideas of the class of indirect allocation.
Outline
Section 2.1 introduces the basics of tyre modelling. Section 2.2 focuses on the actuators that can be used to control the vehicle in the road plane. The difficulties and
opportunities linked to sensing and estimation are quickly discussed in Section 2.3.
The controller structure is further developed in Section 2.4. The global control allocation problem is discussed in Section 2.5 while the local building blocks are introduced
in Section 2.6. Finally, an example of chassis is illustrated in Section 2.7, together
with two simulations in Section 2.8.
2.1
Tyre forces
As the purpose of the Global Chassis Controller is to control the tyre forces, understanding the basics of the tyre is of prime importance. Many researches went really
deep into modelling tyres [95, 115, 52, 82]. The objective here is to give a rapid
overview on which we can rely to motivate choices made in the rest of the chapter.
The model is based on the Magic Formula [95].
The rotational speed of the wheel ω is given by the differential equation:
J ω̇ = T − rfx
21
(2.1)
CHAPTER 2. GLOBAL AND LOCAL CONTROL
where J is the wheel inertia, T is the torque applied on the wheel, r is the wheel
radius and fx is the longitudinal tyre force.
From the state of the vehicle, the velocity vector of each wheel in the vehicle
frame (Vx , Vy ) can be computed [50]. The longitudinal and lateral velocities in the
tyre contact patch (vx , vy ) can be written
vx
Vx
cos(δ) sin(δ)
= R(δ)
(2.2)
with R(δ) =
vy
Vy
− sin(δ) cos(δ)
where R(δ) is the rotation matrix, making a rotation of an angle δ around the vertical z-axis. Note that, throughout this chapter, capital letters are used for signals
measured in the vehicle frame (x axis pointing towards the front of the vehicle), while
minuscule letters are used for signals measured in the tyre frame (x pointing in the
rolling direction of the tyre), see Figure 2.3.
The force response of the tyre to external inputs presents a lag in time. To model
the transient behavior of the tyre, a first order filtering is used for computing the
longitudinal and lateral slips, λ and α respectively:
σx λ̇ + |vx |λ
= rω − vx
(2.3)
σy α̇ + |vx |α
= vy
(2.4)
where σx and σy are parameters representing the relaxation length. In steady-state,
λ and α takes the more common expression
λ
=
α
=
rω − vx
|vx |
vy
|vx |
(2.5)
(2.6)
In pure slip conditions, meaning that either λ or α should be equal to zero, the
longitudinal and lateral tyre forces are given by:
fx0
= fz Dx sin(Cx arctan(Bx λ − Ex (Bx λ − arctan(Bx λ))))
(2.7)
fy0
= fz Dy sin(Cy arctan(By α − Ey (By α − arctan(By α))))
(2.8)
where fz is the tyre load and
• Cx , Cy are the shape factors
• Dx , Dy are the peak factors
• Ex , Ey are the curvature factors
• Kx = Bx Cx Dx , Ky = By Cy Dy are the stiffness factors
It should be noted that those parameters can change drastically depending on the
f0
f0
conditions of the tyre on the road. Figure 2.4 shows a typical curve for fxz or fyz . A
peak in the curve can clearly be observed at moderate slip. In the longitudinal case,
the left part of the curve with the positive slope is called the stable zone, while the
right part with the negative slope is called unstable zone. This denomination comes
22
2.1. TYRE FORCES
1
0
Tyre characteristic (fx /fz) [ ]
1.2
0.8
0.6
0.4
0.2
Experimental measurements
Fitted tyre curve
0
0
0.05
0.1
0.15
0.2
0.25
Wheel slip (λ) [ ]
0.3
0.35
0.4
Figure 2.4: Typical tyre characteristics based on measurements from the tyre-in-theloop facility.
from the stability analysis of the system composed of (2.1), (2.3) and (2.7). More
details are given in Chapter 5.
In case of combined slip, when λ and α are both different from zero, there is an
interaction between the longitudinal and the lateral forces modeled by:
fx
= Gx fx0
Gx
=
Bgx (λ)
(2.9)
cos(arctan(Bgx (λ)α))
(2.10)
= rx1 cos(arctan(rx2 λ))
(2.11)
fy
=
Gy fy0
Gy
=
cos(arctan(Bgy (α)λ))
(2.13)
= ry1 cos(arctan(ry2 α))
(2.14)
Bgy (α)
(2.12)
where fx0 and fy0 are the longitudinal and lateral forces in pure slip from (2.7) and
(2.8) and rx1 , rx2 , ry1 , ry2 are extra tyre parameters.
q
From this interaction, it can be noticed that the total force fx2 + fy2 that can
be generated by a tyre is limited. In particular, the total force has to remain inside
an ellipse, called friction ellipse. This limit obviously depends on a lot of factors like,
for example, the road conditions, the tyre properties and wear, and the vertical load.
The ellipses are illustrated in Figure 2.5. It is clear that no actuator can make the
tyre forces exceed the ellipse limit. However, having more freedom in controlling the
tyre forces allows to better exploit the tyre potential. One major focus of this thesis
is on the optimal use of the tyre potential up to the limit.
All tyre forces fx and fy combine at the chassis level in a longitudinal force Fcx , a
lateral force Fcy and a yaw moment Mcz , acting at the center of gravity of the vehicle.
23
CHAPTER 2. GLOBAL AND LOCAL CONTROL
a
b
fxrr
fxf r
fyf r
fyrr
Fcx
c
Mcz
Fcy
c
fxrl
fxf l
fyf l
fyrl
Figure 2.5: All tyre forces combine at the chassis level in a longitudinal force, a lateral
force and a yaw moment, acting at the center of gravity. The total force generated at
each tyre is limited by the friction ellipse.
This is illustrated on Figure 2.5. Writing down the equations we get:
Fcx
= fxf l cos δ f l + fxf r cos δ f r + fxrl cos δ rl + fxrr cos δ rr
−fyf l sin δ f l − fyf r sin δ f r − fyrl sin δ rl − fyrr sin δ rr
Fcy
Mcz
=
=
fyf l
cos δ + fyf r cos δ f r + fyrl cos δ rl + fyrr cos δ rr
+fxf l sin δ f l + fxf r sin δ f r + fxrl sin δ rl + fxrr sin δ rr
−c fxf l cos δ f l + fxrl cos δ rl − fyf l sin δ f l − fyrl sin δ rl
+c fxf r cos δ f r + fxrr cos δ rr − fyf r sin δ f r − fyrr sin δ rr
+a fyf l cos δ f l + fyf r cos δ f r + fxf l sin δ f l + fxf r sin δ f r
−b fyrl cos δ rl + fyrr cos δ rr + fxrl sin δ rl + fxrr sin δ rr
(2.15)
fl
(2.16)
(2.17)
where δ is the steering angle of the wheel and a, b and c defines the position of the
center of gravity. The superscripts f l , f r , rl , rr identify the wheels front left, front
right, rear left and rear right respectively. The vector Fc is defined as
Fc = (Fcx , Fcy , Mcz )
T
(2.18)
The equations giving Fc from f and δ are nonlinear. Computing the solution set
giving Fc = Fcd , for Fcd a given desired value for Fc , is complicated. If locally, all the
forces are expressed in the vehicle frame, see Figure 2.3, using
Fx
Fy
= RT (δ)
fx
fy
(2.19)
then the previous equations for Fcx , Fcy and Mcz can be rewritten in a linear matrix
24
2.2. ACTUATORS
form:



Fcx
1
 Fcy  =  0
−c
Mcz

0 1
1 0
a c
0 1
0
1 0
1
a −c −b




1 0

0 1 


c −b



Fxf l
Fyf l
Fxf r
Fyf r
Fxrl
Fyrl
Fxrr
Fyrr
rr T
Solving Fc = Fcd in terms of Fxf l , Fyf l , Fxf r , Fyf r , Fxrl , Fyrl , Fxrr , Fy
2.2












(2.20)
becomes trivial.
Actuators
Tens of actuators targeted at influencing the tyre forces in the road plane are available
on the market and many more are being developed. This section gives a description
of the type of actuators that are of special interest for this research. Actuators acting
on the vertical dynamics of the vehicle, like active suspensions, are not considered
here. Still, they should be added in future work as they can indirectly influence the
longitudinal and lateral dynamics.
2.2.1
Electro-hydraulic brakes
Most cars on the road today are equipped with Electro-Hydraulic Brakes (EHB).
Electrically controlled valves and a pump are added in the traditional hydraulic circuit
in order to allow a modulation of the pressure in the brake caliper [18, 27]. This
technology is an enabler for the implementation of ABS and ESP. The implementation
of ABS in Part II is based on EHB.
The dynamics are mainly characterized by the 3 following points:
• There is an actuation delay of typically 10 to 20 ms caused by the transport of
the hydraulic pressure wave in the pipes, from the valves in the control unit to
the caliper. This will largely limit the performance of the torque controller.
• The brake torque is a static linear function of the brake pressure. Still, the
brake efficiency, the gain between the torque and the pressure, is drastically
depending on the temperature and wear.
• The pressure change rate is limited because of the limited flow in the pipes and
the valve.
More details on the dynamics of the EHB can be found in Section 4.2.
The EHB is controlled by the current in the valve, which is directly related to the
derivative of the brake pressure. The output is the brake torque on the wheel.
The maximal brake pressure is limited to the pressure given by the pump. Therefore the brake torque has a maximum. Still, by design, the maximum torque is taken
larger than the torque needed to lock the wheel on any surface and therefore this
constraint is almost never active.
25
CHAPTER 2. GLOBAL AND LOCAL CONTROL
2.2.2
Brake-by-wire
From a vehicle dynamics point of view, the advantages of the brake-by-wire lie in the
faster and more precise torque response compared to EHB. Instead of hydraulics, the
brake pads are actuated by an electro motor [22]. The delay in the actuation is much
smaller than for the EHB and is often neglected. A local controller is responsible for
regulating the current in the motor to get the desired brake torque.
As for the EHB, it can be considered that the brake is properly dimensioned so
that the actuation limit is never reached.
2.2.3
Steer-by-wire
With a steer-by-wire system, the steering of one or more wheels is performed through
an electro motor. This gives the freedom to the vehicle controller to interpret, adapt
and correct driver steering actions before applying them to the wheels [22]. Steering
actuators are interesting for stabilizing the vehicle as they are the most efficient way
to generate yaw moment.
The steering can be implemented on any wheel, at the front or at the rear. Either
the entire rack is moved or each wheel is steered individually. An internal controller
is available to control the steering angle.
The steering angle is limited. It could therefore happen that the maximum tyre
force could not be reached.
2.2.4
Active differential
An active differential, or torque vectoring system, is capable of actively distributing
the driving torque between different driving wheels, e.g. between the left and right
wheel of an axle. It is referred to literature for more details on the working principles
[81].
For the applications in Global Chassis Control in this research, an active differential is always connected to a controllable engine or motor. This combination allows
any driving torque to be applied on each driven wheel individually. An internal controller is assumed to control the engine and differential such that individual wheel
torque commands can be given.
The active differential itself is most often not limited, and can produce any distribution ratio desired. However, the engine torque is limited, which gives constraints
on the maximal torque on the wheels.
2.2.5
In-wheel motor
A tendency enabled by electric driving is the introduction of in-wheel motors. The
traditional combustion engine is replaced by many decentralized small motors.
From a vehicle control point of view, any torque command can be applied on any
wheel of the vehicle. This gives complete freedom in distributing longitudinal tyre
forces between the 4 wheels. As a motor can also act as a generator, both a driving
and braking torque can be applied. Motors are known to have a fast response and a
high accuracy. However, the motor torque and the motor power are limited and are
26
2.3. SENSING AND ESTIMATION
in general not able of producing enough torque to reach the maximal tyre force on
hight-adherence surfaces.
2.3
Sensing and estimation
Sensors are becoming cheaper and more accurate every day. Modern cars are equipped
with many sensors, and in particular sensors dedicated to measuring the movements
and the conditions of the chassis. Lot of papers have been published on how to
estimate state and parameters of the vehicle from the available sensors. While some
time ago each subsystem would only use a few own sensors, there is now a trend
towards combining all available sensors to estimate the complete state of the vehicle
with large state estimators [127, 122]. The sensing and state estimation problem are
out of the scope of this thesis. Still, it is important to point out some remaining
difficulties and new opportunities.
Firstly, the robust measurement of the vehicle longitudinal speed, needed to compute the slip, remains a difficult problem where the use of complex state and parameter
estimation techniques is required [30, 73, 62, 111]. Such complexity can lead to reliability issues and high computational load. Therefore, it should not be expected that
an accurate measurement of the slip λ at each wheel is constantly available online.
Secondly, the lateral vehicle speed vy is also difficult to estimate in real-time
[9, 114, 14, 70, 3, 42]. Recent work seems to show that the use of force sensing
could improve the estimation of vy [127]. Still it is shown in [40] that lateral speed
estimation using force sensing presents severe limitations when tyre parameters are
assumed to be completely unknown.
Thirdly, it remains difficult to estimate the tyre parameters or the tyre-road
friction on-line. Lots of research has been done on tyre-road friction estimation
[48, 115, 119, 55]. But still, the presented solutions suffers from the following issues. Estimators are often based on assumptions such as unbiased sensors, known
vehicle weight distribution or identical tyre characteristics at each wheel, which are
difficult to meet in practice. Therefore such estimators only work in a restricted
number of situations. Furthermore, only a small numbers of tyre parameters are estimated, like for example assuming linear characteristics. Unfortunately, the friction
estimated during regular driving cannot properly assess how the nonlinear system will
behave at the limit. Because the system is non-linear, a part of the characteristics
cannot be identified as long as it is not excited.
Finally, this research relies on tyre force measurement, as explained in Section 1.2,
which is new for production vehicles. This will enable a new type of control loop to
be implemented, namely force-based control.
2.4
Controller structure
For the vehicle to follow a desired trajectory, a suitable total force and moment,
function of time, has to act on the chassis. Such desired total force vector Fcd (t), see
(2.18), can be computed by the global chassis controller and the driver interpreter
27
CHAPTER 2. GLOBAL AND LOCAL CONTROL
based on driver inputs, vehicle properties and other external forces like gravity and
drag.
Then the objective is to produce the tyre forces which sum up in total forces and
moment as close as possible to the desired ones, by using the available actuators. A
few major challenges can be identified:
• the system is over-actuated in the sense that there are more actuators than
degrees of freedom for the vehicle motion in the road plane,
• the system is non-linear and presents in particular saturations,
• many parameters, like the tyre-road friction at each tyre, are unknown and time
varying.
In the method presented here, the computation of the actuator commands will be
done in two steps, like in the class of indirect allocation [4]. This allow for designing
global and local controllers to tackle simpler problems with different characteristics.
At the global level, the total forces and moments desired on the chassis Fcd needs to
be distributed to the different tyres or groups of tyres. This step is called control
allocation [16]. At the local level, in what we’ll call the building blocks, see Section
2.6, the nonlinearities and the uncertainties of the tyre need to be deal with.
Separating the control of the chassis and of the tyre makes sense from a time
constant point of view. Because of the differences in mass, the chassis dynamics are
slower than the wheel dynamics, and the chassis controller can afford having a slower
sampling frequency.
Like in [4], forces in the vehicle frame are used as intermediate signals. This offers
advantages at both levels:
• The allocation problem is rendered convex and without any uncertain tyre parameters.
• The nonlinearities and uncertainties are confined in blocks of lower dimensions,
which makes them easier to handle.
The set of intermediate signals is named S. In [4], all the 8 tyre forces in the
vehicle frame are used as intermediate signals:
S=
Fxf l
Fyf l
Fxf r
Fyf r
Fxrl
Fyrl
Fxrr
Fyrr
,
independently of the actuator configuration. This is perfect if each wheel is completely
actuated, for example using Corner Modules, see Section 2.6.1. However, if the actuation is limited, restrictors will appear in order to decrease the number of degrees
of freedom. This introduces difficulties in both the global and local control. In our
method, in order to avoid the restrictors, S is defined in order to match the number
of degrees of freedom. In general, S will not contain all the 8 tyre forces like in [4].
The local control is not implemented per wheel anymore but inside vehicle building
blocks, which can contain one or more wheels depending the coupling between them.
The concept of building block is discussed in Section 2.6.
28
2.5. CONTROL ALLOCATION
2.5
Control allocation
The purpose of control allocation is to distribute the total desired control action to
a set of actuators in an over-actuated system [16]. A system is called over-actuated
when many actuators can act on the same degree of freedom. For such systems, there
are, in general, many ways of driving the actuators that will result in the same state
trajectory. In our particular case, the objective is to get Fc = Fcd , see (2.18), of
dimension 3. The set S on which Fcd needs to be distributed depends on the chassis
architecture and is assumed to have a dimension between 3 and 8. Furthermore, it is
assumed that Fc is affine in S:
Fc = GS + H
(2.21)
where G and H are matrices, possibly time-varying.
In this research, the allocation problem is formulated as an optimization problem. A cost function is used to pick the best solution out of the large feasible set.
Constraints are added to take the limitations into account.
When the objective is feasible, meaning ∃S s.t. CS + D = Fcd , the optimization
problem takes the form
min
S
subject to
E
(2.22)
Fc = Fcd
tyre constraints
actuator constraints
where E is the cost function to be defined. An interesting option is to penalize the
use of the actuators in order to reduce the energy consumption. Another option is
to strive at maintaining the tyres as far as possible from their limits by e.g. keeping
a low slip [92]. However the precise definition of such a cost function is out of the
scope of this thesis. In the simulation of Section 2.8, E is taken as the 2-norm of S:
E = kSk2 .
However, because of the constraints, it can happen that the objective is unfeasible.
The allocation should then strive at minimizing the difference between Fc and Fcd .
The following optimization problem should be considered instead
Fc − Fcd 2
min
(2.23)
W
S
subject to
R
tyre constraints
actuator constraints
where WR is a weighting factor which can give more importance to one or the other
component of Fc . As an example, more importance could be given to the lateral
dynamics of the vehicle, in order to maintain stability, compared to the longitudinal
dynamics.
2.5.1
Constraints
Both tyre and actuators are limited in what they can deliver. Such limitations appear
in the models in term of nonlinearities. But when the controller is divided in 2 levels, it
29
CHAPTER 2. GLOBAL AND LOCAL CONTROL
is desirable to make abstraction of the models in the other level to reduce complexity.
In our case, the complete model of the building blocks can be ignored by the global
controller as long as it can be assumed that the local controller can drive the block
to the desired set-point. Adding a constraint on a building block gives the allocation
a measure of what can be expected from the block.
Constraints on actuators can most of the time be anticipated as they are known
from the actuator specifications. For tyre constraints, it is the opposite. As discussed
in Section 2.3, tyre friction parameters, and in particular the maximum achievable
tyre force are extremely difficult to estimate during normal driving. In this research,
we start with the principle that the maximum friction force can be detected only when
reached. The detection of the maximum force is done by systems like ABS, presented
in Part II, or traction control systems. Only when the ABS kicks in to maintain
the tyre in its stable region, a constraint is temporarily added in the allocation at
the point where the tyre currently is in term of produced forces. This point can be
measured thanks to force sensing. When the ABS stops, the constraint is removed.
Because of the combined-slip effect [95], the total tyre force has to remain within
an ellipse. However, the parameters of this ellipse are in practice really uncertain and
no published work tackle this estimation problem in an on-line setting. Therefore,
it cannot be expected that a constraint could precisely express the tyre limitations.
Still, it is necessary, for the force distribution to evolve towards a feasible optimum,
that the optimization is given an approximation of the tangential plane to the friction
ellipse. In this work, as in [110], a rhombus is used to approximate the ellipse. While
the local controller is working to maintain the tyre on the edge of its stable region,
the rhombus will be scaled such that the current operating point of the tyre belongs
to the border of the rhombus.
2.5.2
Continuous optimization
The allocation is formulated as a constrained convex optimization problem. Compared
to traditional optimization, the allocation presents particularities:
• The desired forces on the chassis Fcd are constantly changing based on the vehicle
movement and the inputs from the driver. Therefore the force distribution needs
to be continuously updated to remain optimal.
• The constraints based on the tyre limits are rapidly popping in and out, depending if the tyre is at the limit or not.
• Since the speed of the actuators is limited, it is not necessary to always compute
the optimal force distribution right away. What is important is that the force
distribution converges fast enough towards the optimal one.
Many methods have been presented in the literature to solve allocation problems.
Good references can be found in [16] and [57]. Those methods computes the precise
solution to the allocation problem at each time step. The computational cost of such
techniques can be large as an optimization needs to take place at each time. Further,
as the constraints will change while the force distribution is updated, the computed
optimal distribution has a large chance not to be feasible.
30
2.6. VEHICLE BUILDING BLOCKS
In Chapter 3, a new Continuous Optimization method is presented, which fits
perfectly the needs of this allocation application. The force distribution is not seen as a
variable to be optimized over and over again, but as a variable that should be updated
over time to minimize the cost function while staying in the currently acceptable
region defined by the constraints. This can significantly reduce the computational
complexity while a similar level of performance is maintained.
2.6
Vehicle building blocks
The main concept behind the definition of building blocks is to group coupled actuators and mask them with a set of decoupled virtual actuators so that the control
allocation problem is simpler. In particular, we want to maintain the allocation between the forces and moment at the center of gravity and the virtual actuators convex.
This ensures that there is a unique solution to the optimization problem, and powerful
solvers are available [21].
Each building block will be assigned a desired reference and the local control loop
inside each block will make sure that the reference is achieved.
Working with blocks provides a really modular architecture. A new vehicle can
be created simply by connecting blocks to one another, like a big Lego. The new
optimization problem is created by simply summing up the forces from all blocks and
stacking all the constraints.
The precise definition of a building block is as follows. A set of actuator low-level
control signals u is a building block if and only if one can define a set of virtual
references U such that:
• there exits a bijection in steady-state between u and U , possibly time-varying,
that can be inverted by a local controller. In mathematical terms,
∃f s.t. f (u, U, t) = 0
when all derivatives of u and U are equal to zero.
• all Ui from one block are independent from each other. No degree of freedom is
lost in the block.
• all Ui from one block are independent from any other block or actuator in the
vehicle.
• U is entering the allocation problem in a convex manner.
The first condition means that, given a reference on U , only one solution for u is
allowed. In case there are many actuators to produce a similar action, a clear rule
should be available on how to distribute the total action. For example, a braking
torque could be generated both by a friction brake and an electro-motor [121].
From the actuators presented in Section 2.2, a few standard blocks can be composed:
• Corner Module (1 wheel, 3 actuators)
31
CHAPTER 2. GLOBAL AND LOCAL CONTROL
Figure 2.6: The Continental/Siemens VDO eCorner. (1) wheel rim, (2) wheel hub
motor, (3) electronic wedge brake, (4) active suspension (5) electronic steering.
• Active-steering rack with active braking and torque vectoring or in-wheel motors
(2 wheels, 5 actuators)
• Active-steering rack with active-braking (2 wheels, 3 actuators)
• Manual steering wheel with active-braking (1 wheel, 1 actuator)
2.6.1
Corner module
A Corner Module is composed of various actuators that act on one single wheel. The
idea is that a new vehicle could be constructed by simply plugging one of those blocks
at each corner. Prototypes are already available, like the Autonomous Corner Module
from Volvo [126], the eCorner from Continental, see Figure 2.6, or the HY-LIGHT
active wheel from Michelin. A large effort is spent on packaging to give the largest
freedom to the vehicle designers.
In this research, we consider the Corner Module to be composed of an individual
steering system, an in-wheel motor and a brake-by-wire. Therefore, both the longitudinal and lateral tyre force can be fully controlled. Some implementations also
provides an active suspension and a camber controller, but considering those actuators
falls outside the scope of this thesis.
Both the in-wheel motor and the brake-by-wire are used to produce the desired
total toque. It is assumed that a distribution rule is available. The actuator signals u
for the block are u = (T, δ). As noticed earlier, using u as block control input would
make the allocation problem non-convex. Therefore a new set of inputs U needs to
be defined.
Taking U as the longitudinal and longitudinal and lateral forces in the vehicle
frame (Fx , Fy ) makes the corner module to satisfy the building block definition:
• From the tyre model of Section 2.1, it can be verified that there exist a bijection
between u and U as long as the tyre is in the stable zone (part of tyre characteristics with positive slope). The local ABS system prevents the tyre from
entering the unstable zone.
32
2.6. VEHICLE BUILDING BLOCKS
• Fx and Fy can be fixed independently as long as the constraint is not reached.
• Fx and Fy do not depend on any other actuator in the car.
• Fx and Fy appears in a linear, and thus convex, way in the allocation problem.
Local control
The objective of each local tyre controller is to control one Corner Module via the
steering system and the motor/brake such that the tyre develops the desired forces,
i.e. determine the control inputs T and δ in order to drive the outputs Fx and Fy to
the reference values. The approach should be as simple as possible and as robust as
possible regarding the large uncertainty in tyre-road friction.
In the literature, The use of an inverse tyre model is proposed [4]. This open-loop
method is interesting for implementing a feedforward path in the controller. However,
seen the uncertainty on the tyre curve and the difficulty to identify the full tyre
model online, the accuracy reachable with such method is questionable. Moreover,
the computation of tyre slip is required, quantity which is still presently complicated
to estimate accurately, see Section 2.3.
In order to improve robustness with respect to tyre uncertainty, a feedback loop is
desirable. Thanks to force measurement, the implementation of such a feedback loop
is possible and efficient. Still, the complexity of the model, with the nonlinear coupling
between the state variables and the uncertainties, can make the control design difficult.
In this research, a simple approach is used, in order to illustrate the simplicity and
potential of the concept. A variation in δ will mainly affect fy . Similarly, a variation
in T will mainly affect fx . Therefore the following double decoupled integral controller
is implemented
d Fx
Ṫ
kx 0
fx
=
R(δ)
−
(2.24)
Fyd
0 ky
fy
δ̇
with kx and ky two tuning parameters. This controllers, with a straightforward
tuning, gives satisfactory results in term of stability and convergence speed in the
simulations of Section 2.8. Thanks to the integral action, the forces Fx and Fy always
converge to their reference values, for any combination of tyre parameters.
This illustrates the simplicity of implementing a local controller thanks to force
measurement. Of course, it can be expected that for a real implementation, refinement of the controller will be required to take into account the characteristics of the
actuators and other practical constraints.
If a faster system response is desired, an interesting option is to add a feedforward
part in the controller, for example based on partial tyre model inversion. Further,
if an estimate of how the longitudinal and lateral speeds of the corner module are
varying based on information about the desired trajectory of the complete vehicle is
available, this could be an interesting basis to implement compensation.
Constraints
Constraints on U = (Fx , Fy ) are originating from the tyre limits, the limited driving
torque and the limited steering angle.
33
CHAPTER 2. GLOBAL AND LOCAL CONTROL
Constraints from the tyre are indicated with a rhombus, as explained in Section
2.5.1. When the ABS, or the traction control, kicks in to maintain the tyre at the edge
of stability, the currently maximum force is computed by summing up the measured
Fx and Fy . One of the four following linear constraints is then sent to the allocation:
− Fmax ≤ Fx + Fy
≤ Fmax
(2.25)
−Fmax ≤ Fx − Fy
≤ Fmax
(2.26)
where Fmax is defined based on current force measurement. The constraint chosen
depends on the signs of Fx and Fy .
For the limit on the driving torque, a constraint of the form
fx ≤ fxmax
(2.27)
is added. fxmax is based on the actuator specifications but can also be adapted
depending the operating conditions. For being used in the allocation, (2.27) needs to
be expressed in term of U . This gives
cos(δ)Fx + sin(δ)Fy < fxmax
(2.28)
where cos(δ) and sin(δ) are seen as parameters, which values are continuously updated.
When the steering angle reaches the boundary, a similar constraint as (2.27) is
formulated but based on fy . The allocation constraint is therefore
sin(δ)Fx − cos(δ)Fy < fymax
(2.29)
where fymax is based on current measurements.
2.6.2
Active-steering rack with torque control
In case the steering angle of 2 wheels is coupled through an axle, the lateral forces on
both tyres cannot be fixed independently. If the lateral force of both tyres appeared
in the allocation problem, it would be necessary to add a non-linear equality constraint involving uncertain tyre parameters, like in [4]. The allocation problem can
be simplified if the constraint is dropped together with one degree of freedom.
Braking and driving actions on the wheels can come from actuators like active
brake, in-wheel motor or active differential. To simplify the problem at this stage,
those actuators are lumped in one torque actuator.
The block is composed of three actuators, two torque actuators and one steering
actuator, so u = (Tl , Tr , δ). The 2 longitudinal forces on the left and on the right, Fxl
and Fxr , and the total lateral force Fy in the vehicle frame is a suitable choice for U .
An equivalent choice would have been to take the total longitudinal and lateral force
together with the total yaw moment.
Fxl , Fxr and Fy can be written:
Fxl
= fxl cos(δ) − fyl sin(δ)
(2.30)
Fxr
= fxr cos(δ) − fyr sin(δ)
(2.31)
=
(2.32)
Fy
(fxl + fxr ) sin(δ) + (fyl + fyr ) cos(δ)
34
2.6. VEHICLE BUILDING BLOCKS
Local control
Similar to the Corner Module, a simple decoupled integral controller based on force
measurement can be used to locally drive the actuators. The largest influence of the
actuators, especially at large steering angles, are noticeable on the forces in the tyre
frame. However, because of the steering constraint, the system is not invertible and
therefore feedback is done directly on the errors computed in the vehicle frame. The
controller takes the form:
 
Ṫl
kx
 Ṫr  =  0
0
δ̇

0
kx
0
 d

Fxl − Fxl
0
d
− Fxr 
0   Fxr
d
Fy − Fy
ky
(2.33)
It could be expected that the controller is not robust to large steering angles.
However, no problems have been observed in simulation, even during extreme driving
conditions. Again, this simple controller is only used for a proof of concept.
Constraints
For this block, constraints on one single tyre cannot be formulated anymore, since
the constraints have to be expressed in term of U . Still, using the same concept as
in Section 2.5.1, the tangential plane to the friction ellipse can be approximated by a
rhombus on Fxl and Fy , and on Fxl and Fy . Depending on which tyres are saturating,
some of the following constraints can be active:
− Fmax ≤
Fxl + Fy
−Fmax ≤
≤ Fmax
(2.34)
Fxl − Fy
≤ Fmax
(2.35)
−Fmax ≤ Fxr + Fy
≤ Fmax
(2.36)
−Fmax ≤ Fxr − Fy
≤ Fmax
(2.37)
where Fmax comes from the current measurement of the tyre forces while the tyre is
maintained at the limit by the ABS.
2.6.3
Manual steering wheel with torque control
In this final case, the steering angle of the wheel is fixed by the driver and cannot be
influenced by the controller. Therefore taking the longitudinal tyre force as reference
U = (fx ) does not destroy convexity in the allocation problem. The steering angle as
well as the lateral force fy will appear in the allocation formulation as parameters.
At each time step, their values are updated with the current measurement.
Local controller
The torque that needs to be applied is simply the desired force fxd times the radius
of the wheel. As the radius is approximately known, it is easy to implement a feedforward. An integral controller is added to compensate for the error in the radius
estimation. The simple controller, used for proof of concept, takes the form:
35
CHAPTER 2. GLOBAL AND LOCAL CONTROL
a
b
fxrr
δ rr
δf
fxf r
fyf r
fyrr
Fcx
c
Mcz
Fcy
c
fxrl
δf
fl
fx
fyf l
fyrl
Figure 2.7: Example of chassis with an active steering + active differential front axle,
steer-by-wire on the rear right wheel, in-wheel motors at the rear and brake-by-wire
on each wheel.
T
Ṫi
= rfxd + Ti
= −kx (fx −
(2.38)
fxd )
(2.39)
Constraints
Constraints on fx can be expressed based on tyre and actuators limitations:
fxmin < fxd < fxmax
(2.40)
with fxmin and fxmax parameters that can be predefined based on actuator specifications and adapted online based on detected tyre limits.
2.7
Example
The new Global Chassis Control methodology is applied to an example of chassis
combining some of the actuators of Section 2.2. This example does not try to be
realistic but emphasizes on the freedom available in designing new chassis.
The chassis considered is composed of 4 wheels and 10 actuators, as illustrated on
Figure 2.7. The 2 front wheels are connected to an axle with steer-by-wire. The drive
torque delivered by the controllable engine can be distributed freely between front
left and front right using the active differential. The rear-right wheel of the vehicle is
equipped with an individual steer-by-wire system and an in-wheel motor. The rearleft wheel is equipped only with an in-wheel motor, such that it can be driven but
not steered. Finally, each wheel is equipped with a brake-by-wire system.
The total forces and moment generated on the chassis coming from the tyres are
36
2.7. EXAMPLE
Fx2
Fxr1
Fy2
Fy1
Fcx
Mcz
Fcy
Fx3
Fxl1
Fy3
Figure 2.8: Example of chassis where the actuators are grouped into 3 building blocks.
given by:
Fcx
=
(fxf l + fxf r ) cos δ f − (fyf l + fyf r ) sin δ f + fxrl + fxrr cos δ rr − fyrr sin δ rr
(2.41)
Fcy
=
(fyf l + fyf r ) cos δ f + (fxf l + fxf r ) sin δ f + fyrl + fyrr cos δ rr + fxrr sin δ rr
(2.42)
Mcz
= −(fxf l cos δ f − fyf l sin δ f )c + (fxf r cos δ f − fyf r sin δ f )c + fxrl c
−(fxrr cos δ rr − fyrr sin δ rr )c + (fyf l + fyf r ) cos δ f + (fxf l + fxf r ) sin δ f a
−(fyrl + fyrr cos δ rr + fxrr sin δ r r)b
(2.43)
This chassis can be organized into building blocks. The front axle is an “Activesteering rack with torque control”, the rear right wheel is a “Corner Module” and
the rear left wheel is a “Manual steering wheel with torque control” with the steering
angle always equal to zero. Figure 2.8 illustrates the chassis with the building blocks
and the new intermediate signals S for the allocation.
The inputs for the building blocks are
S = Fxl1 Fxr1 Fy1 Fx2 Fy2 fx3
The lateral force on the rear left wheel is not controllable and is taken as a measurable disturbance.
The total forces and moment on the chassis can be easily computed from the forces
in the blocks with a linear matrix multiplication


Fxl1
  Fxr1  
 




1 1 0 1 0
1
Fcx
0
 Fy1 
  1  Fy3
 Fcy  =  0 0 1 0 1
0 
 Fx2  +


−c c a c −b −c
Mcz
−b
| {z } |
{z
}  Fy2  |
{z
}
Fc
G
H
Fx3
| {z }
S
where the underbrace notation refers to (2.21).
The allocation problem is convex and the continuous methodology of Chapter 3
can be used. The constraints in the allocation comes from the 3 building blocks.
37
CHAPTER 2. GLOBAL AND LOCAL CONTROL
Depending which tyre or actuator is saturating, constraints on the form (2.25)-(2.26),
(2.28), (2.29), (2.34)-(2.37), or (2.40) are temporarily added.
2.8
Simulation results
Two simulations are performed to illustrate the Global Chassis Controller. In the
first one, a reference maneuver is followed, for which the constraints are not hit. This
shows how the desired total forces and moment are achieved using the allocation and
the local control. In the second simulation, the vehicle is asked to brake in a splitmu situation, where the left side of the road is more slippery than the right. The
maximum tyre force is reached on the rear-left wheel and the ABS gets activated.
The steerable wheels are used to maintain the desired trajectory after redistribution
of the control action.
2.8.1
Vehicle model
The vehicle model is build in Modelica/Dymola [25, 46] using the Vehicle Dynamics
Library [6]. This commercial library provides modular complex vehicle models based
on the Modelica Multi-Body framework [94]. The standard components of the library
are extended with multi-body models of the different building blocks constituting the
example chassis.
The simulated model has 57 states and about 10000 variables. The movement of
the chassis, as well as the wheels are considered in 6 degrees of freedom. A complex
tyre model is taken from the library, which includes tyre relaxation, combined-slip
and vertical dynamics.
2.8.2
Simulation with inactive constraints
In this first simulation, the objective is to let the vehicle follow a desired trajectory,
defined by the total longitudinal and lateral forces on the chassis with a side-slip angle
constantly equal to zero. A simple PD controller is used to compute the desired yaw
moment required to maintain the side-slip at zero. The controller has the form
Mz = kvy + kd v̇y
(2.44)
where vy is the lateral speed and k and kd are tuning parameters. In the simulations,
the parameters are given the values k = 2000 and kd = 500. Such controller is easy
to implement in simulation but would be much more involved in practice, seen the
difficulty to measure or estimate vy , see Section 2.3.
The reference total forces on the chassis can be seen on the left plot of Figure 2.9.
On the same plot, the allocated forces and achieved forces are also visible. Both steps
in the control will introduce some errors and delay, but it can still be concluded that
the achieved forces precisely follow the desired ones.
The control allocation is implemented using the Hybrid Descent Method of Chapter 3. In this simulation, the constraints are not hit and therefore no constraint
handling is required. The left plot of Figure 2.9 shows the forces produced by each
block.
38
2.9. CONCLUSION
A local controller is implemented in each building block. The same values are
taken for the controller parameters in each block: kx = 10, ky = 0.001. From the left
plot of Figure 2.9, where the desired and achieved forces are very close together, it
can be concluded that the local controllers are able to properly control the actuators.
This confirms that a simple force-based linear control can deal with a complex and
uncertain tyre model.
2.8.3
Simulation with ABS
In this second simulation, the vehicle is asked to brake in straight-line on a split-mu
road. The right side of the road has a fiction coefficient of 1, while the left side is
more slippery with a friction coefficient of only 0.5. The desired total brake force is
set to 6000N, for a car of 1100kg. The total lateral force is set to zero. The total
yaw moment is controlled, with the same PD controller as in the previous situation,
in order to maintain the lateral speed at zero. At the local level, the ABS controller
presented in Chapter 5 is implemented. The resuls are shown on Figure 2.10.
At the beginning of the simulation, the tyre characteristics are still unknown and
no constraint is present in the allocation. While the desired brake force on the wheels
is increasing, the maximum tyre force is reached at the rear-left wheel. This is logical
as the rear wheels are the least loaded and the left wheels have a road with less grip.
The ABS gets activated to maintain the wheel at the limit of stability and a constraint
is added in the allocation at the currently measured tyre forces. The optimization
method in the allocation takes the constraint into account and redistribute the total
forces. Lateral forces are requested on the steerable wheels in order to compensate
for the uneven brake force between left and right. With the constraint, the desired
d
eventually becomes small enough for the ABS
brake force on the rear-left wheel Fx3
to stop. At that moment, the constraint is removed and, with the objective of using
d
increases again.
the actuators as less as possible, Fx3
2.9
Conclusion
In this chapter, a framework for Global Chassis Control is presented. This framework
extends the class of indirect allocation where the total desired forces and moment on
the chassis are first distributed to a set of intermediate forces in the vehicle frame, before being translated into actuator commands. The architecture, based on the concept
of building blocks, allows vehicles with various levels of actuation to be represented
with the correct number of degrees of freedom. This avoids the need for difficult
constraints or restrictors. Thanks to the 2 control levels, the allocation problem is
maintained convex and independent from uncertain parameters, and the tyre nonlinearities and uncertainties can be deal with in blocks of smaller dimensions. This
decreases the overall complexity compared to previously published methods.
From the analysis of the allocation problem, it can be concluded that there is a
need for a continuous optimization technique. An optimization method targeted to
this problem is developed in Chapter 3.
At the local level, it is shown that force feedback enables decoupled linear integral
controllers to track the desired reference well while being robust with respect to tyre
39
CHAPTER 2. GLOBAL AND LOCAL CONTROL
1400
2500
Fxd
FxAlloc
Fx
Fyd
FyAlloc
Fy
Mzd
MzAlloc
Mz
2000
1500
1000
Fxl1
Fxr1
Fy1
Fx2
Fy2
Fx3
1200
1000
800
600
400
500
200
0
0
−200
−500
−400
−1000
3
4
5
6
7
8
9
10
−600
3
4
5
6
7
8
9
10
Time [s]
Time [s]
Figure 2.9: Simulation of a maneuver where the tyre constraints are not hit.
1000
Fxd
FxAlloc
Fx
Fyd
FyAlloc
Fy
Mzd
MzAlloc
Mz
−1000
−2000
500
Fxl1d
Fxr1d
Fy1d
Fx2d
Fy2d
Fx3d
0
Force [N]
0
−3000
−4000
−500
−1000
−5000
−1500
−6000
−7000
3
3.5
4
4.5
5
Time [s]
5.5
6
6.5
−2000
3
7
3.5
4
4.5
5
Time [s]
5.5
6
6.5
7
25
vx
rw
Fn
0.6
20
0.4
Speed [m/s]
Normalized force [ ]
0.5
0.3
0.2
15
10
0.1
5
0
−0.1
3
3.5
4
4.5
5
Time [s]
5.5
6
6.5
7
0
3
3.5
4
4.5
5
Time [s]
5.5
6
6.5
7
Figure 2.10: Simulation of a braking on split-mu. The rear-left wheel is not able
to deliver the initial force asked by the allocation and the ABS is activated. The
temporary addition of a constraint changes the allocation and the steerable wheels
are used to maintain a straight trajectory. When the ABS is not active anymore, the
desired braking force on the rear-left wheel is increased until the maximum is reached
again.
40
2.9. CONCLUSION
characteristics.
Simulations with a complex and realistic vehicle model confirms that this approach
can be suitable for practical implementation. This method can deal with the limits of
the vehicle. When the limit of a tyre is reached, the local ABS controller is activated
and a constraint is temporarily added in the allocation. The forces are redistributed
and the demand on the saturating tyre is decreased.
So far, the framework has been developed focusing on two main Global Chassis
Control objectives, see Section 1.1: “stability and Safety”, with the handling of the
tyre limits, and “Ease of development”, with the modular architecture. Still, it can be
expected that the framework is suitable for implementing a fault-tolerant control, with
an online reconfiguration of the architecture, and adding energy efficiency objectives,
for example by improving the allocation cost function.
41
42
Chapter 3
Hybrid Descent Method
Optimization problems play a role of increasing importance in many engineering domains, and specially in control theory. While some design procedures require finding
the optimum of a complex problem only once, other real-time control techniques want
to track the optimum of a time-varying cost function with time-varying constraints.
Some applications based on continuous optimization are the following:
• Model Predictive Control [84, 47]. The cost function measures the error
between the predicted outputs of the controlled plant and the desired outputs,
over a prediction horizon. The desired outputs can be changed in real-time,
for example by a human operator, which makes the optimization problem timevarying. The variables to be continuously optimized are the future control
inputs to the plant over a control horizon.
• Control Allocation [16, 57, 63]. Here the task is to optimally distribute
some desired control input to a set of actuators, based on actuator cost and
constraints. The changing character of the generic control input makes the
problem time-varying.
• Lyapunov-based control. Stabilization is performed by making a Lyapunov
function or an objective function to decrease over time down to a minimum. A
good summary of such methods applied to the field of motion coordination can
be found in [85].
• Model Reference Adaptive Control [7, 105, 60]. The objective is to adapt
the controller parameters so that the controlled plant behaves like the reference
one. The time-varying cost function is computed from the output error between
the controlled and the reference plants.
In the literature, one can distinguish 3 main approaches to deal with these problems:
• Repeated Optimization [84] where the new optimization problem is solved
at each time step;
43
CHAPTER 3. HYBRID DESCENT METHOD
• Precomputed Optimization [11] where all the possible problems are solved
off-line and stored in a look-up table, which can become very large;
• Update Laws [7] where the optimization variables are taken as states of a
dynamical system and are given a certain dynamics.
In the field of optimization, many efficient techniques exist to solve constrained
convex optimization problems [21]. They are designed to output, after several iterations, an accurate value of the optimum. Therefore they are very well suited for
off-line optimization, where a problem should be solved accurately once. However,
the iterative character and the complexity of the algorithm do not always make it
suitable for on-line implementation. This is probably why those techniques are not
often, and very cautiously, integrated into embedded controllers.
On the other hand, update laws are very easy to implement and require very
few computation power. Since many years, research on this topic has been going on
within the field of adaptive control [7, 105, 60]. For example, one interesting method
to adapt the parameters on-line in Model Reference Adaptive Control is to use a
gradient descent method for a cost function defined as the square of the output error
between the controlled real plant and the reference. This technique works well in the
unconstrained case.
However, when it comes to adding constraints, solutions in the control field are
much more limited. One possible method found in [105] and [60] is the gradient
method with projection. The idea is to project the gradient on the active constraints
when the state is on the boundary of the feasible set. In that way, the descent
direction is always such that the state stays in the feasible set. This method works
in continuous time but the discrete-time implementation is more intricate, especially
in the case of nonlinear constraints. Moreover, the way to compute the projection is
not obvious and can be complex. Therefore it is only worked out and used for very
simple cases.
In case of discrete-time implementation, which is always the case when using
digital controllers, the only currently available method, proposed in [60], uses scaling
to project back any new value of the state into the feasible set if necessary. However,
the idea is worked out only in case of a very simple feasible set (a ball centered at the
origin) and no proof of convergence is given.
This chapter bridges those two worlds of control and optimization by developing
an update law to deal with the general case of a convex cost function and convex
constraints while enabling easy integration into traditional controllers. Moreover, the
method is simple, which allows good insight, has low computational cost and is easily
discretizable for discrete-time implementation.
The proposed technique takes the form of a traditional ordinary differential equation
ẋ = f (x)
In [24], they are designed to sort lists and to diagonalize matrices. References [67]
and [15] present plenty of examples where methods coming from the control area can
be used to synthesise and analyse numerical algorithms. In this thesis, the vector
field f will be designed to solve constrained convex optimization problems. If the
optimization problem is time-varying, f can obviously be expected to also depend
44
3.1. PROBLEM FORMULATION
on time. However, the rest of the chapter will focus on an invariant problem for
simplicity.
Outline
The formal requirements are described in Section 3.1. The proposed system is described and analyzed in Section 3.2. Section 3.3 shows that the technique can still be
used after a simple discretization. Furthermore, a simulation example is presented in
Section 3.4. Finally, the use of the method is illustrated for two applications: Model
Predictive Control in Section 3.5, and Control Allocation in Section 3.6.
3.1
Problem formulation
Let us consider the convex optimization problem
min
q(x)
x
subject to
(3.1)
g(x) ≤ 0
with x ∈ ℜn , q : ℜn → ℜ a differentiable convex function and g : ℜn → ℜm such that
each gi is a differentiable convex function for i = 1, .., m. The constraint g(x) ≤ 0
defines a convex set that we will call the feasible set, for consistency with the optimization terminology. Its complement is called the infeasible set. The two following
assumptions are made:
Assumption 1 The feasible set is not empty, i.e. ∃xf s.t. g(xf ) ≤ 0.
The optimal value of the cost function in the feasible set is then denoted q ∗ , i.e.
q ∗ = minx {q(x)|g(x) ≤ 0}
Assumption 2 q ∗ is finite
The objective is to find a vector function f : ℜn → ℜn such that the dynamical
system
ẋ(t) = f (x(t))
(3.2)
has the following properties:
• for x(t̄) outside the feasible set at some time t̄, the trajectory x(t) enters into
the feasible set, i.e. ∃tf > t̄ s.t. g(x(tf )) ≤ 0.
• the trajectory x(t) remains in the feasible set as soon as x(tf ) is in the set, i.e.
g(x(t)) ≤ 0 ∀t > tf s.t. g(x(tf )) ≤ 0,
• for x(tf ) in the feasible set, the trajectory x(t) decreases the cost function q(x(t))
at all time until q(x(t)) = q ∗ , i.e. q(x(t1 )) > q(x(t2 )) ∀(t1 , t2 ) with tf ≤ t1 < t2
s.t. q(x(t1 )) > q ∗ , and limt→∞ q(x(t)) = q ∗ .
3.2
Hybrid steepest descent solution
One efficient way to decrease an unconstrained cost function is to use a gradient
descent method, as used traditionally in adaptive control [105]. Therefore the basis
45
CHAPTER 3. HYBRID DESCENT METHOD
of this method is similar. It can also be noted that the gradient of a function is not
its only descent direction. Other directions have been proposed in the literature, like
Newton’s direction [21]. The investigation of alternative directions to improve the
convergence while limiting the increase of computation complexity is left for future
work.
The original idea of the method is based on the way the constraints are considered.
Because of the computational complexity, a direct projection of the gradient on the
constraints is discarded. But on the other hand, each constraint is seen as a kind of
barrier. More precisely, each constraint which would not be satisfied at a certain time
instant will push the trajectory toward the feasible set. In that way, the trajectory
will never leave the feasible set. Furthermore, if x(t) is on the boundary of the feasible
set, it will be pushed alternatively by both the gradient and the constraint. If they
are pushing in opposite directions, x(t) will naturally slide along the border and the
projection will appear indirectly. Compared to interior point methods, this technique
has the advantage to have descent directions defined outside the feasible set, which
can be useful in case of time-varying constraints.
The proposed hybrid feedback law is therefore:
−∇q(x)
if gj (x) ≤ 0 ∀j
P
f (x) =
(3.3)
− i∈L(x) ∇gi (x)
if ∃j : gj (x) > 0
with L(x) = {l : gl (x) ≥ 0}.
The rest of this section is dedicated to the analysis of the behavior of this system
using hybrid systems techniques and Lyapunov arguments.
3.2.1
Filippov solutions and sliding modes
The vector field f (x) is measurable and essentially locally bounded but discontinuous.
Therefore the study of the solution of the vector differential equation ẋ(t) = f (x(t))
requires the use of a particular solution concept. We make use of the Filippov solution
concept [44, 112, 78] recalled in the following definition:
Definition (Filippov) A vector function x(.) is called a solution of (3.2) on [t1 , t2 ]
if x(.) is absolutely continuous on [t1 , t2 ] and for almost all t ∈ [t1 , t2 ]: ẋ ∈ K[f ](x),
where K[f ](x) is the convex hull of the limits of f (y) for y → x while y stays out of
a set of zero Lebesgue measure where f is not defined [44, 112].
At a point x around which f (x) is continuous, K[f ](x) reduces to f (x). However,
on a switching surface, K[f ](x) will contain a set of possible values for ẋ.
So, at all time, ẋ has the following form:
ẋ = −γ0 (x)∇q(x) −
m
X
γi (x)∇gi (x)
(3.4)
i=1
for some γj (x) ≥ 0, j ∈ {0, ..., m}. Depending the situation, the values of the γj (x)
will be different:
• for x strictly in the feasible set, γ0 = 1 and γi = 0 ∀i,
• for x in the infeasible set, γj = 1 ∀j ∈ L(x) and 0 otherwise (so γ0 = 0),
46
3.2. HYBRID STEEPEST DESCENT SOLUTION
• for x on a boundary, the values of the γj will depend on a possible sliding motion
as defined by the Filippov solution concept.
Following the Filippov solution concept, for x on the switching surface between the
feasible and infeasible sets, either a sliding motion can take place, i.e. a motion along
the switching surface, or a motion toward one of the sets [44]. Since the −∇gi (x) are
always pointing toward the feasible region, a sliding mode will appear only if −∇q(x)
is pointing toward the infeasible region. In that case, the sliding motion will require
γ0 (x) > 0 in (3.4). In case there is no sliding motion, then due to (3.3), we have
f (x) = −∇q(x) and γ0 (x) = 1 in (3.4). In conclusion, γ0 (x) > 0 on the boundary
between the feasible and infeasible sets.
3.2.2
Stationary points of the update law (3.2)-(3.3)
The most interesting property of the update law (3.2)-(3.3) is that the globally stable
equilibria of the dynamical system precisely coincide with the optimal points of the
constrained optimization problem. First it will be shown that the stationary points
of the systems are optimal, and vice versa. The stability of those points is proved in
the next subsection.
Definition [44] A point x = p is called stationary if it is a trajectory, that is, if
x(t) ≡ p is a solution of (3.2).
Following the definition, it can be concluded that a point p is stationary if and only
if 0 ∈ K[f ](p), [44]. Further, a point will be called an equilibrium if it is stationary
and stable.
Theorem 3.1 below states that, if the convex optimization problem is feasible, the
stationary points lie in the feasible set. The “optimality” of the stationary points is
considered in Theorem 3.2. Further, the next section demonstrates the asymptotic
stability.
Theorem 3.1. If the functions gi (x) : ℜn → ℜ are convex (i = 1, ..., m) and if there
exists an xf such that gi (xf ) ≤ 0 ∀i ∈ {1, ..., m} then
X
∇gi (x̄) 6= 0
i∈L
for any subset L of {1, ..., m} and any x̄ such that gi (x̄) > 0 for some i ∈ L.
Proof. Let us define Tx̄gi (x) the tangent hyperplane to the function gi at x̄:
∆
Tx̄gi (x) = ∇giT (x̄) (x − x̄) + gi (x̄) = Gi (x̄)x − hi (x̄)
(3.5)
where Gi (x̄) = ∇giT (x̄) and hi (x̄) = ∇giT (x̄)x̄ − gi (x̄). Due to the convexity of gi , we
know that
(3.6)
gi (x) ≥ Tx̄gi (x) ∀x̄, ∀x
The proof is done by contradiction. Assume there exist a point x̄ and a set L such
that
gi (x̄) = Gi (x̄)x̄ − hi (x̄) > 0 ∀i ∈ L
P
(3.7)
i∈L Gi (x̄) = 0
47
CHAPTER 3. HYBRID DESCENT METHOD
This directly leads to
0=
X
Gi (x̄)x̄ >
i∈L
X
hi (x̄)
(3.8)
i∈L
Furthermore, by the hypothesis of the theorem, there exists an xf such that
gi (xf ) ≤ 0 ∀i ∈ L and therefore by (3.5) and (3.6)
X
X
X
0≥
gi (xf ) ≥
(Gi (x̄)xf − hi (x̄)) = −
hi (x̄)
(3.9)
i∈L
i∈L
i∈L
Equations (3.8) and (3.9) clearly lead to a contradiction. It can therefore be
concluded that a combination (x̄, L) does not exist , which proves the theorem.
Using this result, the “optimality” of the stationary points can be assessed.
Theorem 3.2. If (3.1) is feasible then a point p is a stationary point of (3.2)-(3.3)
if and only if it is an optimal point of (3.1).
Proof. Due to Theorem 3.1, it can be concluded that f (x) is always different from 0
for x in the infeasible set and therefore the stationary points lie in the feasible set.
For any feasible x, the dynamics takes the form of equation (3.4) with γ0 (x) > 0.
If there exists a stationary point p such that ẋ(p) = 0, and by defining
λi =
γi (p)
γ0 (p)
(3.10)
it is easy to check that the following set of equations is satisfied:
gi (p)
λi
λ
g
(p)
i i
Pm
∇q(p) + i=1 λi ∇gi (p)
≤ 0 ∀i
≥ 0 ∀i
= 0 ∀i
= 0
These equations are the well-known Karush-Kuhn-Tucker (KKT) conditions, which
prove that the stationary point p is an optimal solution of the convex problem (3.1)
while the λ’s are the Lagrange multipliers [21, 13].
Moreover, if p is optimal, it satisfies the KKT conditions. The Lagrange multipliers
λ will define a suitable dynamics of the form (3.4), which belongs to K[f ](p). Therefore, 0 belongs to K[f ](p) and, following the definition, p is a stationary point.
By Assumptions 1 and 2, there always exists at least one such stationary point p.
Furthermore, q(p) = q ∗ .
3.2.3
Asymptotic stability
Finally, it can be shown that (3.2)-(3.3) is asymptotically stable and converges toward
one of the stationary points found above. In view of the structure of (3.3), we propose
the following Lyapunov function:
V (x) = max(q(x), q ∗ ) − q ∗ + β
m
X
i=1
48
max(gi (x), 0)
(3.11)
3.2. HYBRID STEEPEST DESCENT SOLUTION
with β a strictly positive parameter.
It is obvious that this Lyapunov function is strictly positive everywhere except at
stationary points where V (p) = 0:
• in the infeasible set, we have V (x) > 0 since at least one gi (x) > 0
• in the feasible set (or at the boundary), we have V (x) > 0 since q(x) > q ∗ ,
except at the stationary points where q(x) = q ∗
Unfortunately, this Lyapunov function is not differentiable everywhere. To handle
this case, the theory developed in [112] will be used. The main stability theorem is
recalled below. But first, to go more smoothly through the technicalities, let us recall
some definitions.
Definition [59] A function f (x) is said to be essentially bounded on X if the
function is unbounded only on a set of measure zero, i.e. µ{x ∈ X : |f (x)| > a} =
0 for some real number a ≥ 0 where µ is the Lebesgue measure.
Definition [31] A function V (x) is said to be regular when the usual directional
derivative exists in any direction. Examples of regular functions include smooth functions, convex Lipschitz functions, and functions that can be written as the pointwise
maximum of a set of smooth functions.
Therefore it can be concluded that V (x) is regular.
Definition [69] A continuous function α : [0, a) → [0, ∞) is said to belong to class
K if it is strictly increasing and α(0) = 0.
Definition [31] The Clarke’s generalized gradient ∂V (x) of a locally Lipschitz
function V (x) is the convex hull of the limits of the gradients of the function around
the points where the gradient of V is not defined.
∂V (x) = co{
¯
lim
y→x,y ∈Ω
/ V
∇V (y)}
for ΩV a set of measure zero where the gradient of V is not defined.
Definition [112] The set Ṽ˙ (x), which is proved in [112] to contain
when this last quantity exists, is defined as
∆
Ṽ˙ (x) =
\
ξ T K[f ](x)
(3.12)
d
dt V
(x(t)),
(3.13)
ξ∈∂V (x)
In the smooth case, where all the functions involved are differentiable, Ṽ˙ (x) reduces to the time derivative of the function V (x(t)): Ṽ˙ (x) = ∇V T ẋ.
Definition A set S is said to be negative if all the elements of the set are negative:
S < 0 ⇔ s < 0 ∀s ∈ S
Theorem 3.3. [112] Let ẋ = f (x) be essentially locally bounded in a region Q ⊃
{x ∈ ℜn | kx − pk < r} for some real number r and 0 ∈ K[f ](p). Also let V :
ℜn → ℜ be a regular function satisfying V (p) = 0 and 0 < V1 (kx − pk) ≤ V (x) ≤
V2 (kx − pk) for x 6= p in Q for some V1 , V2 ∈ class K. Then
1. Ṽ˙ (x) ≤ 0 in Q implies that x(t) ≡ p is a uniform stable function.
49
CHAPTER 3. HYBRID DESCENT METHOD
2. If in addition, there exists a class K function w(.) in Q with the property Ṽ˙ (x) ≤
−w(x) < 0 for x 6= p then the solution x(t) ≡ p is uniformly asymptotically
stable.
Now Theorem 3.3 is used to prove the stability of (3.2)-(3.3).
Theorem 3.4. The system defined by (3.2)-(3.3) is uniformly asymptotically stable
if it has one unique finite stationary point.
Proof. The vector field f (3.3) satisfies the requirements of Theorem 3.3 for any set
Q of the form {x ∈ ℜn | kx − pk < r} with a finite r and 0 ∈ K[f ](p). Moreover, the
Lyapunov function V (3.11) is continuous, since it is the sum of continuous functions;
regular and convex, since it can be written as the pointwise maximum of a set of
smooth convex functions; and positive. Then due to the convexity of the functions
and the unique finite stationary point p such that V (p) = 0, V can be bounded from
below by the function V1 (||x − p||) with V1 of class K. With the same arguments, V
can also be bounded from above by a function V2 (||x − p||) with V2 of class K.
The stability conclusion coming from Theorem 3.3 now depends on the values of
the generalized time derivative of V . The 3 main regions are first considered before
going to a more general case including the boundaries.
• For x in the feasible set we have: V (x) = q(x) − q ∗ , ∇V = ∇q, ẋ = −∇q and
therefore
Ṽ˙ (x) = −∇q(x)T ∇q(x) ≤ 0
(3.14)
which is always negative as long as ∇q(x) 6= 0, and this can only happen at
the stationary point. Moreover, as we move away from the optimal point in
the feasible set, the norm of ∇q(x) cannot decrease, because of the convexity of
q(x), and therefore Ṽ˙ (x) cannot increase. This will be important when showing
that Ṽ˙ (x) can be bounded from below and from above by class K functions.
P
∗
• For
Pset for q(x) < q we have: V (x) = β i∈L gi (x), ∇V =
P x in the infeasible
i∈L ∇gi (x) and therefore
i∈L ∇gi (x), ẋ = −
2
X
∇gi (x) < 0
Ṽ˙ (x) = −β (3.15)
i∈L
which is always strictly negative and never increasing as we move away from
the stationary point by the same convexity arguments as before.
P
∗
∗
• For x in the infeasible
set for q(x) ≥
P
Pq we have: V (x) = q(x)−q +β i∈L gi (x),
∇V = ∇q + i∈L ∇gi (x), ẋ = − i∈L ∇gi (x) and therefore
Ṽ˙ (x) = −∇q(x)T
X
!
∇gi (x)
i∈L
Since
P
2
X
∇gi (x) < 0
−β
(3.16)
i∈L
∇gi (x) 6= 0 (see Theorem 3.1), it is always possible to find a β large
enough such that Ṽ˙ (x) is strictly negative. Moreover, again thanks to convexity,
i∈L
50
3.3. PRACTICAL IMPLEMENTATION
the norm of ∇gi (x) cannot decrease as we move away from the stationary point
in the infeasible set. Therefore, a large enough β can also render Ṽ˙ (x) nonincreasing. The interpretation of β is here to create a large enough barrier such
that the constraints dominate the cost function in the infeasible set.
• In the general case, Ṽ˙ (x) is not anymore a number but a set of values for each
x. To show that Ṽ˙ (x) < 0, the fact that a subset of a negative set is also a
negative set will be used twice.
Since K[f ](x) is a subset of the “headless” cone
X
X
X̄ = {−ϕ0 ∇q(x) −
ϕi ∇gi (x)|ϕj ≥ 0,
ϕj > 0}
j
i∈L(x)
and since ∂V contains
V̄ = {∇q(x) + β
X
γi ∇gi (x)|γj ≥ 0,
i∈L(x)
X
γj ≤ 1}
j
T
Ṽ˙ (x) is included in the set W̄ = { ξ∈V̄ ξ T X̄}. So if W̄ is negative, then Ṽ˙ is
negative as well.
If K[f ](x) does not contain 0, i.e. if x is not the stationary point, then 0 is
not in the convex “headless” cone X̄ neither and the entire set is situated in
a half-space defined by the separating hyperplane passing through 0 and with
normal vector v. Such hyperplane is not unique and therefore
P a normal vector v
can be chosen such that v belongs to X̄ with ϕ0 > 0 and j ϕj = 1. Then it is
obvious that the vector − ϕ10 v belongs to V̄ for β ≥ ϕ10 . Therefore, there exists a
vector ξ = − ϕ10 v belonging to V̄ such that ξ T X̄ < 0, which implies W̄ < 0 and
finally Ṽ˙ (x) < 0. In case K[f ](p) contains zero, the stationary point is reached
and the value of the Lyapunov function will remain zero.
Finally, Theorem 3.3 holds which proves the theorem.
In the case where the system (3.2)-(3.3) has many stationary points (for a convex
optimization problem they will represent a convex set : a sublevel set of a convex
function [21]), the equivalent of LaSalle’s theorem for non-smooth systems presented
in [112] can be used to show that the system (3.2)-(3.3) will converge to a point in the
largest invariant set of {x|0 ∈ Ṽ˙ (x)}, which in our case is the complete set of optimal
points.
3.3
Practical implementation
Because of the sliding mode, i.e. the infinite number of switches, the dynamical system
cannot be simulated directly. Two methods can be considered for practical implementation. The first method is to sample the system at a given sampling frequency [8].
51
CHAPTER 3. HYBRID DESCENT METHOD
In that way, the computation can easily be scheduled on time-driven microcontrollers
and the number of switches is limited to one every period. Obviously, because of the
sampling, the accuracy of the system is going to be reduced. Two main drawbacks
can be foreseen, whatever the sampling technique used:
• during a sliding mode, the trajectory will not stay perfectly on the boundary but
will meander slightly around it. The amplitudes of the oscillations will depend
on the sampling period and on the norms of the gradients ∇q(x) and ∇gi (x).
• the trajectory will not precisely converge toward the precise optimal point but
will oscillate around it.
In this research, the sampling is done by approximating the derivative by a forward
difference, i.e. Euler’s method. For a sampling time ∆t, the difference equation is
given by
xk+1 = xk + f (xk )∆t
(3.17)
The investigation of other sampling methods as well as the precise influence of the
sampling time on the performance is left for future work. Results from [78] can be
used to ensure that the trajectory of the discretized system remains close to the one
of the original system.
In the second method, the simulation remains in continuous-time and smooth
transitions are implemented between the feasible and the infeasible set [39]. In practice, the gradient of the constraints are weighted based on the value of the constraint
via a smoothened step s, for example:
s(g(x)) =
1
g(x)
atan(
) + 0.5
π
ǫ
(3.18)
where ǫ is a tuning parameter which should be small. Then the gradient of the cost
function is weighted by a complementary function. Finally, the smooth version of f
is given by
!
m
m
X
1 X
s(gi (x))∇gi (x)
(3.19)
s(gi (x)) ∇q(x) −
fs = − 1 −
m i=1
i=1
In the simulations of Chapter 2, the Hybrid Descent Method is implemented in
continuous-time using smooth transitions. The discretization method is illustrated in
the next section.
3.4
A simulation example
To illustrate the essence of the developed method, we consider a simple convex optimization problem in 2 dimensions. The cost function and one of the constraints are
chosen to be linear while the second constraint is taken nonlinear, although convex.
T
Let us consider the following optimization problem for x = x1 x2
:
52
3.4. A SIMULATION EXAMPLE
Trajectory and Feasible Set
Time Evolution of State
20
18
18
16
16
14
14
12
12
x
x2
10
10
8
*
←x
8
6
6
4
4
2
2
0
0
x1
x2
5
10
x1
15
0
0
20
10
20
30
40
50
t
Figure 3.1: Simulation with sampling time ∆t = 0.01. The oscillations in the sliding
mode are not visible and the trajectory converges toward the optimal point. Just
after t = 3 the trajectory reaches the feasible set; after t = 11 the trajectory starts
sliding along the nonlinear constraint; and at t = 40 the optimal point is reached.
min
x
−x1
(3.20)
2
subject to
2
(x1 − 10)
(x2 − 10)
+
−1≤0
36
81
10
x1 + x2 − 28 ≤ 0
8
We have:
∇q(x) =
∇g1 (x) =
1
18 (x1
−1
− 10)
∇g2 (x) =
10
8
0
T
2
81 (x2
T
1
(3.21)
− 10)
T
(3.22)
(3.23)
Figure 3.1 presents the results of the simulation for a small sampling time ∆t =
0.01 and initial condition x0 = (0 0)T . While x is outside the feasible set (until
t = 3), the trajectory converges toward it. Then the gradient of the cost function
is followed until reaching a constraint at t = 11. Afterwards, the trajectory slides
along the constraint to the optimal point, which is reached at t = 40. Thanks to the
small sampling time, the oscillations in the sliding mode are hardly visible. It can be
checked that the Lyapunov function (with β = 3) always decreases over time. Note
that on the trajectory picture, the dashed curves represent the zero level-sets of the
constraints and therefore the interior of the bold dashed curve is the feasible set.
Figure 3.2 presents the same results for a larger sampling time ∆t = 0.5. Here the
oscillations are very large but the trajectory is still evolving in the right direction.
In order to reduce the oscillations induced by the sliding mode, smooth transitions
are implemented between the feasible and infeasible sets. The results are shown on
Figure 3.3. The oscillations are completely removed, unfortunately at a cost of a
decrease of the accuracy.
53
CHAPTER 3. HYBRID DESCENT METHOD
Trajectory and Feasible Set
Time Evolution of State
20
18
18
16
16
14
14
12
12
x
x2
10
10
8
*
←x
8
6
6
4
4
2
2
0
0
x1
x2
5
10
x1
15
0
0
20
10
20
30
40
50
t
Figure 3.2: Simulation with sampling time ∆t = 0.5. The oscillations in the sliding mode are now visible but the trajectory still evolves in the right direction until
oscillating around the optimal point.
Trajectory and Feasible Set
Time Evolution of State
20
18
18
16
16
14
14
12
12
x
x2
10
10
8
← x*
8
6
6
4
4
2
2
0
0
x1
x2
5
10
x1
15
20
0
0
10
20
30
40
50
t
Figure 3.3: Simulation with sampling time ∆t = 0.5 and smooth transitions. The
oscillations in the sliding mode are completely removed.
54
3.5. APPLICATION TO MODEL PREDICTIVE CONTROL
3.5
Application to Model Predictive Control
A very interesting modern control method is Model Predictive Control (MPC) [28,
10, 102]. Thanks to an internal model of the plant, the controller is able to predict its
expected future outputs and therefore optimize the current input accordingly. The
main advantages of MPC are the following:
• It can handle multi-input multi-output processes, processes with large timedelay, non-minimum phase processes, and unstable processes;
• It can take constraints into account in a natural manner: input constraints as
well as output constraints or state constraints;
• It can be reconfigured by changing the internal model, for example in case of
fault detection.
MPC is formulated as an optimization problem on a receding horizon. At all time,
a cost function penalizing the difference between the desired and expected outputs
should be minimized under the constraints. Many publications already studied properties and tuning of MPC cost functions and constraints, see for example [84, 47, 104]
and references therein.
Typically, most implementations of MPC are done using discrete-time controllers
and a discrete-time model of the plant is used as an internal model. Continuous-time
models could also be used like in Receding Horizon Control, see [101]. MPC based
on discrete-time internal models are considered here.
Papers in the literature often rely on on-line optimization, where a repeated optimization method is used and the exact value of the optimum is computed at each
time step, see [84]. Other approaches compute a multiparametric optimal solution
beforehand (off-line) and store the results for on-line use, see [12]. In the first case,
such an on-line optimization can be computationally expensive and therefore too demanding for small embedded microcontrollers. In the second case, the amount of data
to be stored can become extremely large if many parameters or constraints appear
in the problem formulation. Thus, real-time implementation of MPC is not an easy
task and this is probably one reason why it is cautiously used in embedded controllers
dealing with fast dynamics processes.
By applying the Hybrid Descent Method, it becomes possible to implement a
MPC-like controller using a hybrid dynamical system.
3.5.1
Model Predictive Control
In MPC, the future evolution of the system on a certain time interval is considered. An
internal model of the process to be controlled is used to compute the predictions. To
limit the number of variables and simplify the computation, a discrete-time internal
model is used with a sampling time ∆T . Therefore, if the noise-free process has the
state-space model
ẋ(t) = Ax(t) + Bu(t)
(3.24)
y(t) = Cx(t)
55
CHAPTER 3. HYBRID DESCENT METHOD
then the equivalent internal model takes the form
xk+1 = Ãxk + B̃uk
yk = C̃xk
(3.25)
Using a prediction horizon Np , the prediction covers the time interval [t, t+Np ∆T ];
and the cost function is based on the value of the output at the Np next sampling
time instants. The variables to be optimized in the procedure are the input u at the
following Nu sampling time instants. In the following, t is the measurement time;
therefore time before t has already been measured and is known, while time after t is
expected by the controller. Using the following variables:



ũ(t) = 

u(t)
u(t + ∆T )
..
.
u(t + (Nu − 1)∆T )





 
Ãx(t) + B̃u(t)
x(t + ∆T )
 x(t + 2∆T )   Ãx(t + ∆T ) + B̃u(t + ∆T )
 

x̃(t) = 
=
..
..
 

.
.
x(t + Nu ∆T )
Ãx(t+(Nu −1)∆T )+B̃u(t+(Nu −1)∆T )

 

C̃x(t + ∆T )
y(t + ∆T )
 y(t + 2∆T )   C̃x(t + 2∆T ) 

 

ỹ(t) = 

=
..
..




.
.
y(t + Nu ∆T )
C̃x(t + Nu ∆T )

(3.26)





(3.27)
(3.28)
the traditional MPC cost function q is
q(ũ(t), x(t), ỹref (t)) = ||ỹ − ỹref ||2Qy + ||∆ũ||2Qu
(3.29)
where ỹref is the desired output at the Np next sampling time instants, ∆ũ contains
the value of the signal u(t) − u(t − ∆T ) at the next Nu sampling time instants and
Qy and Qu are weighting matrices.
This cost function can be rewritten in a standard quadratic form
q(ũ(t), x(t), ỹref (t)) =
1 T
T
ũ Qũ + xT RT ũ + ỹref
S T ũ + c(x, ỹref )
2
(3.30)
where c does not depend on ũ and Q, R and S are time-independent matrices. Note
that the cost function depends on the future values of u and yref , which are embedded
in ũ and ỹref respectively; while it is parametrized by the value of x only at the current
time t which is the “initial state” of the current horizon. For concision, the explicit
dependency on t is omitted.
The gradient is therefore
∇ũ q =
∂q
= Qũ + Rx + S ỹref
∂ ũ
(3.31)
A set of constraints can be defined. Constraints on the control input can directly
be expressed while constraints on state or outputs require the use of the internal
56
3.5. APPLICATION TO MODEL PREDICTIVE CONTROL
model. Here, the constraint set, also called feasible set, is restricted to the smooth
convex case described by
gi (ũ, x) ≤ 0 i = 1...m
(3.32)
where gi : ℜn → ℜ is a differentiable convex function. Obviously, if the internal model
is used to express constraints, the functions gi will also depend on the state x.
Furthermore, rate constraints can be expressed on the control inputs. The general
form of those constraints are the following:
|u̇(t)| < u̇max ∀t
(3.33)
This constraint is expressed in continuous time for the continuous system (3.24) and
directly used in the continuous implementation of the controller (3.35).
In traditional implementation, to give time for the optimization to be performed,
the MPC controller is implemented in discrete time. Here, we will first have a look
at the continuous-time case.
3.5.2
The hybrid feedback controller
By using the Hybrid Descent Method as continuous optimization technique, the MPC
controller can be implemented as a hybrid dynamical system. The controller scheme
is represented on Figure 3.4. The future inputs ũ are driven by a differential equation
on the form
˙
ũ(t)
= f (ũ(t), x(t), ỹref (t))
(3.34)
where f is designed according to (3.3). The cost function q is given by (3.29) and
only the constraints (3.32) are considered.
To include the rate constraints (3.33), a variation of the update law can be considered. The dynamical system takes the form
˙
ũ(t)
= sat [αf (ũ(t), x(t), ỹref (t)), u̇max ]
(3.35)
where sat is the traditional saturation function that limits the value of αf (.)i to + or
- u̇maxi and α is a tuning parameter that scales the descent direction. To fully exploit
the capabilities of the actuators in terms of rate of change, a large enough α should
be used. However, a too large α will lead to a too reactive controller that could be
difficult to discretize, as will be seen later on.
For practical implementation, as discussed in Section 3.3, the control system can
be implemented either in continuous-time with smooth transitions, or in discrete-time
with for example an Euler discretization of period ∆t. It should be noted that the
sampling frequency of the controller does not need to be related to the sampling
frequency of the internal model. Two different notations are used.
3.5.3
Simulation
Some simulations are used to show that the method is working and to analyse the
influence of the parameters ∆t and α. The first simulations focus on input constraints
while the last one introduces output constraints. A random stable non-minimum
phase linear system is taken as process to be controlled. The transfer function contains
57
CHAPTER 3. HYBRID DESCENT METHOD
ỹref
y
Process
Controller
u
ũ˙ = f (ũ, x, ỹref )
ẋ = Ax + Bu
y = Cx
x
Figure 3.4: Controller scheme
Pole−Zero Map
15
Imaginary Axis
10
5
0
−5
−10
−15
−20
−15
−10
−5
Real Axis
0
5
Figure 3.5: Pole-zero map of the process to be controlled.
58
3.5. APPLICATION TO MODEL PREDICTIVE CONTROL
Output of the controlled system
Control input
1.2
0
1
−0.2
0.8
−0.4
0.4
u
y
0.6
−0.6
0.2
0
−0.8
−0.2
−0.4
0
Traditional MPC
Dynamic feedback
2
4
6
8
Traditional MPC
Dynamic feedback
10
time
−1
0
2
4
6
8
10
time
Figure 3.6: Simulation of the dynamic feedback Model Predictive Controller discretized with a very high sampling frequency. α is also chosen large. For comparison,
an implementation of a traditional MPC is shown as dotted line.
11 poles and 2 zeros in the left half plane, and one zero in the right half plane. The
pole-zero map is shown in Figure 3.5.
For use as internal model in the MPC, the model should be discretized and transformed to a state-space form. The sampling time is here taken as ∆T = 0.1.
The control input is constrained by both a range and a rate limit, which apply for
each component of ũ:
umax = 0.8
(3.36)
u̇max = 3
The tuning of the MPC is not the primary objective of this Section. Moreover,
it might be of interest to show that the controller is able to deal with large horizons.
So the following parameters are taken: control horizon Nu = 20, prediction horizon
Np = 40, and identity matrices for Qy and Qu .
In the simulation, the initial state of the process is x0 = [0, 0] and a step output
reference is asked after 1 second. Note that in this setup, the controller is not aware
of the change in set-point before it actually takes place. This precisely happen when
a human operator set in real-time the reference of the controller.
For comparison purpose, the MPC is implemented both using traditional optimization techniques and the new hybrid feedback. The traditional version relies on
the solver QuadProg in Matlab. The response is given by the dotted lines. The feedback system (3.35) is tuned using first a large and then a small α. In the case of the
large α, 3 sampling frequencies are compared.
The first simulation gives a case that should be very close to the continuous version.
To this end, the sampling frequency is taken very high: ∆t = 0.001. In order to make
sure that the controller will be reactive enough to fully exploit the actuators, a large
value is taken for α: α = 100. The results are shown in Figure 3.6. The results
are extremely close compared to the traditional MPC. The desired steady-state is
reached in the same time and the constraints are perfectly respected. This seems
really promising.
59
CHAPTER 3. HYBRID DESCENT METHOD
Output of the controlled system
Control input
1.2
0
1
−0.2
0.8
−0.4
0.4
u
y
0.6
−0.6
0.2
0
−0.8
−0.2
−0.4
0
Traditional MPC
Dynamic feedback
2
4
6
8
Traditional MPC
Dynamic feedback
10
−1
0
time
2
4
6
8
10
time
Figure 3.7: Simulation of the dynamic feedback Model Predictive Controller discretized with a 10 times longer sampling period. α is still large. Oscillations appears
when a constraint is reached and near the steady-state.
Then the sampling frequency of the controller is decreased and a 10 times larger
sampling period is taken ∆t = 0.01. The plots are displayed in Figure 3.7. As it was
expected, oscillations appears in the control input during a sliding mode, i.e. when
a constraint is reached, and when the minimum of the cost function is reached, i.e.
near the steady-state. However, the global behaviour is still close to the traditional
optimization.
To investigate further the influence of the sampling frequency of the controller,
the sampling time is again multiplied by 10 to reach ∆t = 0.1. As can be seen in
Figure 3.8, the oscillations are larger; which had to be expected. A small steady state
error appears because the oscillations are not symmetric compared to the steadystate control input. But the trajectory remains close to the desired one. So it can
be concluded that the sampling time influences the oscillations and the steady-state
value, while it also directly influences the computation complexity.
From the equation defining the controller (3.35), it can be understood that α has a
large influence on the size of the oscillations. To show this, the same simulation with a
large sampling period ∆t = 0.1 is repeated with an α much smaller: α = 0.01. Figure
3.9 show the results. It can be noticed that the oscillations have been completely
removed and that the steady-state error has completely disappeared; but at a cost
of a slower response. Therefore, a good tuning of α or even an adapting value could
improve largely the method.
Furthermore this method allows, similarly to traditional MPC, the expression of
output constraints. This is done by including the internal model in the computation
of the constraint set. For linear systems, the predicted outputs ỹ can be computed
using two time-invariant prediction matrices Px and Pu and the equation
ỹ(t) = Px x(t) + Pu ũ(t)
(3.37)
It is then possible to guarantee that y will stay above a minimum value ymin by
60
3.5. APPLICATION TO MODEL PREDICTIVE CONTROL
Output of the controlled system
Control input
1.2
0
1
−0.2
0.8
−0.4
0.4
u
y
0.6
−0.6
0.2
0
−0.8
−0.2
−0.4
0
Traditional MPC
Dynamic feedback
2
4
6
8
Traditional MPC
Dynamic feedback
−1
0
10
2
4
time
6
8
10
time
Figure 3.8: Simulation of the dynamic feedback Model Predictive Controller discretized with an again 10 times longer sampling period. α is still large. The oscillations becomes even larger and a steady-state error appears but the behaviour of the
system is still globally correct.
Control input
0
1
−0.1
0.8
−0.2
0.6
−0.3
0.4
−0.4
u
y
Output of the controlled system
1.2
0.2
−0.5
0
−0.6
−0.2
−0.4
0
−0.7
Traditional MPC
Dynamic feedback
2
4
6
8
10
time
−0.8
0
Traditional MPC
Dynamic feedback
2
4
6
8
10
time
Figure 3.9: Simulation of the dynamic feedback Model Predictive Controller discretized with the same long sampling period; but α is now chosen very small. Oscillations and steady-state error completely disappear but the response is slower.
61
CHAPTER 3. HYBRID DESCENT METHOD
Output of the controlled system
Control input
1.2
0.6
1
0.4
0.8
0.2
0
y
u
0.6
0.4
−0.2
0.2
−0.4
0
−0.6
Traditional MPC
Dynamic feedback
−0.2
0
2
4
6
8
10
−0.8
0
Traditional MPC
Dynamic feedback
2
time
4
6
8
10
time
Figure 3.10: Simulation of the dynamic feedback Model Predictive Controller with
output constraints ymin = −0.01. The controller is discretized with a long sampling
period ∆t = 0.1. α has been chosen large at the beginning to rapidly steer the system
and respect the constraint; and is decreased after 5 seconds to cancel out the large
oscillations. The constraint is well respected and the global behaviour is really good
compared to a traditional MPC.
using the state-dependent constraints
g(ũ, x) = −Pu ũ − Px x + ymin ≤ 0
(3.38)
For the same system described by the pole-zero map of Figure 3.5, the output
constraint ymin = −0.01 is added to the previous input constraints (3.36). The results
of the simulation are shown on Figure 3.10. It can be seen that the undershoot is
removed. With a good tuning of α, the results are comparable to traditional MPC.
However, the choice of α seems to be more critical than before. In case of a too low
α, the controller is not reactive enough to steer the system properly at the beginning
to respect the constraint. On the other hand, a too large α induces large oscillations
near the steady-state. In this simulation, the value of α is decreased after 5 seconds
so that both fast response and small oscillations are achieved. This shows the need
for an adaptive scheme for α, which is left for future work. So it has been shown that
output constraints can be handled by this control method.
3.6
Application to Control Allocation
In many applications, like Global Chassis Control or Flight Control, many actuators
can act on the same degree of freedom. Such systems are called over-actuated. If the
linear state-space system
ẋ = Ax + Bu
(3.39)
y
(3.40)
= Cx
is considered, over-actuation means that B is not full-rank. In other words, there are
many input u(t) that gives the same state trajectory x(t).
62
3.6. APPLICATION TO CONTROL ALLOCATION
Traditional control methods usually have difficulties to deal with over-actuated
systems. First because of the impossibility to invert the system. And secondly because
over-actuated systems often come with tight actuation constraints, which justifies the
large number of parallel actuators. Traditional control methods are easier to apply to
a system with a limited number of virtual inputs v rendering the new virtual input
matrix Bv full rank.
The problem of finding the real control input u that matches the virtual input v:
Bu = Bv v
(3.41)
is called Control Allocation [16, 57].
Many methods have been presented in the literature to mathematically formulate
and solve control allocation problems. Good references can be found in [16] and
[57]. An interesting way of formulating the problem is to write it in the form of
an optimization [21] with a cost function q weighting the use of each actuator, and
constraints representing the actuator limitations. Most of the optimization based
allocation techniques are computing the precise solution at each time step. This can
easily be implemented using traditional optimization techniques. However, this might
not be the most suitable for this kind of problems.
As described in Section 2.5.2, allocation present particularities compared to traditional optimization:
• The desired virtual input v is constantly changing, which requires u to be continuously updated.
• Constraints might rapidly pop in and out depending on the detected limitations.
• Since the speed of the actuators is limited, it is not necessary to know the final
optimum as long as the system is converging at maximal speed towards it.
Therefore, u could be seen, not as a variable to be optimized over and over again,
but as a variable continuously updated over time. This can significantly reduce the
computational complexity while a high level of performance is maintained.
The Hybrid Descent Method is suitable for implementing such continuous optimization. Another comparable method has been developed by Johansen [63]. The
methods differ in the way they handle the constraints. The Hybrid Descent Method
introduces the equality constraints in the cost function and uses the inequality constraints to define the feasible set, while [63] does the opposite: the inequality constraints are included in the cost function using barriers while the equality constraints
define the acceptable region. Furthermore, in [63] the Lagrange multipliers are computed explicitly and need to be adapted on-line, while in the Hybrid Descent Method
they are not computed but come as a by-product of the sliding mode. Both methods
have stability and convergence proofs available.
When using the Hybrid Descent Method for control allocation, the controlled
63
CHAPTER 3. HYBRID DESCENT METHOD
system keeps a state-space form:
ẋ = Ax + Bu
(3.42)
y
= Cx
(3.43)
= Ac xc + Bc y
(3.44)
v
= Cc xc + Dc y
(3.45)
u̇
= f (u, v, umax )
(3.46)
ẋc
where (3.42)-(3.43) is the original system, (3.44)-(3.45) is the controller providing the
virtual input v and (3.46) is the allocation controlling the real input u. The notation
umax is used as a generic way to express the constraints.
As in the case of MPC, see section 3.5.2, the actuator output and rate constraints
are handled in a different way. Output constraints are used to define the feasible set
in f , which generates switching. Rate constraints are used to saturate f .
The Hybrid Descent Method is used to implement the control allocation in the
Global Chassis Control example and simulation in Chapter 2.
3.7
Conclusion
A hybrid system implementing a convex optimization algorithm has been presented.
The main idea is to follow the steepest descent direction of the objective function in
the feasible set and of the constraints in the infeasible set. The continuous hybrid system guarantees that its trajectory enters the feasible set of the related optimization
problem and next converges asymptotically to the set of optimal points. Furthermore, practical implementation, using smooth transitions or discretization, has been
discussed. Smooth transitions can slightly decrease the accuracy while discretization
introduced chattering.
In this chapter, global asymptotic convergence has been proven for the class of
time-invariant convex problems. In case of non-convex problems, the convergence
can still be assured towards a local optimum. However, nothing guarantees that the
global optimum is attained. In case of time-varying optimization problems, it can be
expected that the trajectory asymptotically tracks the time-varying optimal point.
This is confirmed in the simulations of Chapter 2.
It can also be noted that the gradient of a function is not its only descent direction.
Other directions have been proposed in the literature, like Newton’s direction [21].
The investigation of alternative directions to improve the convergence while limiting
the increase of computation complexity is left for future work. Furthermore, various
way of handling constraints exist in optimization algorithms. While interior point
methods are suitable for providing very accurate solution using logarithmic barriers,
this method has the advantage to have descent directions also defined outside the
feasible set. This is particularly useful if constraints are allowed to change over time.
The scope of application of this kind of method will not be in the off-line optimization arena. However, in particular for on-line applications, this method is of interest.
This is because of its implementation as a dynamical system, its simplicity, its low
computation cost, and its capacity to be implemented in discrete-time. In particular,
64
3.7. CONCLUSION
the method has been applied to Model Predictive Control and Control Allocation.
Possible domains where the method could also be applied include Iterative Learning
Control, Adaptive Control, on-line System Identification, and all other domains where
cost functions have to be continuously minimized under constraints.
65
CHAPTER 3. HYBRID DESCENT METHOD
66
Part II
Braking Control
67
Chapter 4
ABS based on Wheel
Acceleration
In collaboration with CNRS/Supelec, Paris, France.
In this chapter, the 5-phase hybrid ABS algorithm presented in [96] is improved
and validated in tyre-in-the-loop experiments. This algorithm belongs to the class of
hybrid ABS, see Section 1.3. In previous work, the algorithm was theoretically studied
using a basic wheel model, and the theoretical limit cycle was analyzed. Conclusions
could be drawn on how to tune the algorithm to achieve the best limit cycle and on
the resulting performances.
Unfortunately, the original algorithm is not robust to the actuation delay present
in practice and fails the experimental validation. In this work, three methods to deal
with delays are developed and validated.
The most common ABS algorithm running on production vehicles is commercialized by Bosch [18]. This algorithm also belongs to the class of hybrid ABS based on
wheel acceleration and is comparable to the 5-phase ABS. As the exact commercial
algorithm is secret, a precise comparison is impossible. However, this chapter strives
at illustrating resemblances and differences.
Outline
The simplified model used in [96] is presented in Section 4.1. The modelling of the
relevant phenomena for experimental implementation is described in Section 4.2. The
theoretical algorithm derived in [96] is recalled in Section 4.3 and the systematic tuning is addressed. The influence of the time delay on the brake actuation being very
important, the theoretical algorithm had to be adapted and three possible modifications are described in Section 4.4. Results from the experimental validation are
analysed in Section 4.5, where limit cycles close to the ones expected from theory and
simulation are obtained. In Section 4.6, the algorithm is compared to the commercial
ABS implemented by Bosch.
69
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
Tyre characteristic (−µ(λ)) [ ]
1.2
1
0.8
0.6
0.4
0.2
Experimental ABS cycles
Second order rational fraction
0
0
0.05
0.1
0.15
0.2
0.25
Wheel slip (−λ) [ ]
0.3
0.35
0.4
Figure 4.1: Measurement and fitting of the nondimentional tyre characteristic µ(·)
for the tyre-in-the-loop facility. The maximum braking force is achieved with a longitudinal slip of 10 %.
4.1
Simplified modelling
The development of the five-phase algorithm presented in Section 4.3 is based on
a simplified single-wheel model. Only the longitudinal dynamics of a single loaded
wheel is considered. Even though weight transfer and combined slip are ignored,
all the basic phenomena related to ABS already appear in this very simple model.
Moreover, the limit cycles predicted by this model (Figure 4.7) are already quite
close to those obtained with more realistic models (Figure 4.8) or in experiments
(Figures 4.9 and 4.10).
4.1.1
Wheel speed dynamics
The angular velocity ω of the wheel has the following dynamics:
I ω̇ = −RFx + T,
(4.1)
where I denotes the inertia of the wheel, R its radius, Fx the longitudinal tyre force,
and T the torque applied to the wheel.
The torque T = Te − Tb is composed of the engine torque Te and the brake
torque Tb . We will assume that during ABS braking the clutch is open and thus
neglect the engine torque. Moreover, we will assume that the brake torque is given
by
Tb = γb Pb ,
(4.2)
where Pb denotes the brake pressure and γb the brake efficiency, see Section 4.2.4.
70
4.1. SIMPLIFIED MODELLING
4.1.2
Tyre force modelling
The longitudinal tyre force Fx is often modelled by a relation
Fx (λ, Fz ) = µ(λ)Fz .
(4.3)
That is, by a function that depends linearly on vertical load Fz > 0 and nonlinearly
on wheel slip λ
Rω − vx
,
(4.4)
λ=
vx
where vx is the longitudinal speed of the vehicle. Note that the slip is positive when
driving and negative when braking. Wheel lock corresponds to a slip λ = −1. It
can be observed that this definition of slip shows a singularity at zero vehicle speed.
Other ways of defining the slip allow to deal with this issue, see Section 4.2.5.
The nondimensional tyre characteristic µ(·) is an odd function such that µ(0) = 0
and µ′ (0) > 0 (see Figure 4.1). It is important to stress that this curve presents a
peak at λ = λ∗ so that µ′ (λ∗ ) = 0. The part of the curve |λ| < λ∗ with a positive
slope µ′ > 0 is called the “stable zone” of the tyre while the other side of the peak
with |λ| > λ∗ and µ′ < 0 is called the “unstable zone”. This denomination comes
from the stability analysis of the single-wheel model.
Several mathematical formulas have been used to describe this curve. Trigonometric functions are used in [95], exponentials in [27], and second order rational fractions
in both [71] and [96]:
µ(λ) =
a1 λ − a2 λ2
,
1 − a3 λ + a4 λ2
for λ ≤ 0.
(4.5)
The important point is that all this mathematical formulas use coefficients that depend on tyre characteristics, road conditions, tyre pressure, temperature, etc. However, in order to be robust, the ABS algorithm cannot assume these coefficients to be
a priori known. Formulas on how to relate the ai coefficients of (4.5) to more common
features like the slope at the origin or the position of the peak can be found in [96].
The particular tyre characteristic identified on the test bench and used in simulation
is parametrized as follows: a1 = 36, a2 = 217, a3 = 13 and a4 = 271. Figure 4.1
shows the measurements and the fitted curve.
One advantage of using rational fractions to model the tyre characteristics is that
its slope can easily be computed analytically. In particular, we have
µ′ (λ) =
4.1.3
dµ
a1 − 2a2 λ + (a2 a3 − a1 a4 )λ2
=
dλ
(1 − a3 λ + a4 λ2 )2
(4.6)
Wheel slip and wheel acceleration dynamics
In this subsection, the equations describing the evolution of wheel slip and wheel
acceleration are derived. It should be noted that other wheel deceleration models
have been proposed in the literature [90, 91, 117].
The vehicle is supposed to brake with the maximal constant deceleration a∗x allowed by road conditions, which is a∗x = −µ(λ∗ )g. In other words
v̇x = a∗x .
71
(4.7)
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
Moreover, the tyre load Fz will be assumed to be constant.
If we define the wheel slip offset and the wheel acceleration offset by
x1
= λ − λ∗
x2
= Rω̇ − a∗x ,
we obtain the following system:
ẋ1
ẋ2
1
(x2 − (λ∗ + x1 )a∗x )
vx
a
= − µ̄′ (x1 ) (x2 − (λ∗ + x1 )a∗x ) + u,
vx
=
where
R2
Fz
I
(4.8)
(4.9)
R
Ṫ .
I
(4.10)
µ̄(x) = µ(λ∗ + x) − µ(λ∗ ),
(4.11)
a=
and u =
The function µ̄(·) is defined as
and thus represents the tyre characteristics with the peak at the origin.
4.2
Modelling practical phenomena
The simplified model, presented in Section 4.1, is not sufficient to explain the behavior
observed during experiments. The modelling of five additional effects is required
and treated in the following subsections. Since the experimental validation has been
carried out on a laboratory test rig, see Section 1.3.2, the detailed modelling focusses
on the test rig phenomena. Nevertheless, it is important to stress that most issues
appearing on the experimental setup also appear on a real vehicle.
4.2.1
Oscillations in measurements
When the tyre is rolling, oscillations can be seen in all measured signals. Longitudinal
and vertical forces, as well as wheel speed and wheel acceleration are perturbed. In
our lab, such oscillations are periodic, with a period of one wheel rotation, which can
be seen on Figure 4.2. One plausible explanation originates from the difference in
tyre properties along its circumference. Possible variations in wheel diameter, rubber
carcass/belt-stiffness, tread wear, etc. can lead to load variations. This will have a
direct impact on the longitudinal and lateral forces generated in the contact patch, and
therefore indirectly on the wheel speed and the wheel acceleration. On a real vehicle,
the road irregularity will also contribute to the load variations, resulting in a more
random pattern of oscillations. Such oscillations need to be taken into account since
they can trigger control intervention at inappropriate moments. Those disturbing
oscillations will not often introduce noise in the cycle and prevent the appearance of
a constant limit cycle. Still, a surprising conclusion is that a constant ABS cycle can
appear in case of periodic disturbing oscillations. This phenomenon is visible both in
simulation and in practice and will be theoretically studied in a future work.
72
4.2. MODELLING PRACTICAL PHENOMENA
10
Wheel angular acceleration [rad/s2]
8
6
4
2
0
−2
−4
−6
−8
−10
14
14.1
14.2
14.3
Time [s]
14.4
14.5
14.6
Figure 4.2: Oscillations measured on the wheel acceleration signal during free rolling
at about 18m/s. The vertical lines indicate one wheel rotation and a clear periodicity
can be concluded.
4.2.2
Wheel acceleration
The wheel acceleration signal is not directly available and should be computed from
the wheel angular encoder. The encoder signal is composed of a series of pulses sent
when one of the teeth of the encoder passes in front of the hall sensor.
It is common to reconstruct the wheel speed from the encoder signal. Two methods
are often used [113, 32]. At low rotational speed, the best technique is to measure
the time difference between two pulses. At high speed, it becomes more accurate to
count the number of pulses received within a small time interval. The speed at which
one should switch from one method to the other depends on the speed of the internal
clock of the electronics used to measure the time difference, the sampling rate, and
the number of teeth on the encoder.
It is well known that taking the time derivative of a noisy signal considerably
amplifies its noise level. Therefore, computing the wheel acceleration by differentiating
the reconstructed speed signal amplifies high frequency noise and should be avoided.
Instead of taking a time derivative, we use a linear regression on the raw encoder
pulsed signal in order to fit a parabola to the time/displacement measurements (see
Figure 4.3). The second time derivative can then be calculated analytically, introducing less noise than with numerical differentiation [34].
4.2.3
Brake pressure dynamics
The hydraulic line and the servo-valve used to control the pressure in the disk brake
limit the performance of the actuation. In a first approximation, the controlled valve
will introduce a second-order dynamics and a flow rate limit, and the hydraulic line a
transport time delay between the desired and the actual brake pressure. Taking those
effects into account is of first importance for designing a working ABS system.
A rough identification of the characteristics gives: a pure time delay ∆t = 7 ms,
73
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
6
5
Angle [rad]
4
3
2
Angle
Encoder signal
Tops
Tops used for fitting
Fitted parabola
1
0
0
0.5
1
1.5
2
Time [s]
2.5
3
3.5
4
Figure 4.3: Linear regression on the wheel encoder pulsed signal. Pulses are sent at
constant displacement and the incoming time is measured. Fitting a line provides
speed information while fitting a parabola also provides average acceleration.
60
Ref Pres
Meas Pres
55
Brake pressure [bar]
50
45
40
35
30
25
20
8
8.5
9
9.5
Time [s]
Figure 4.4: The measured brake pressure follows the reference pressure with a delay
and a second order dynamics.
74
4.2. MODELLING PRACTICAL PHENOMENA
25
Brake pressure
50
20
40
30
Brake efficiency [Nm/bar]
Brake pressure [bar]
60
20
Brake efficiency
7
7.5
8
8.5
9
Time [s]
9.5
10
10.5
15
11
Figure 4.5: Variation of the brake pressure and brake efficiency during braking. The
efficiency increases by about 30 % because of the temperature changes. ABS controllers need to be robust to this effect.
a pressure rate limit between rmax = +750 bar/s and rmin = −500 bar/s, and a
second-order dynamics with a cut-off frequency of 60 Hz (ω = 120π) and a damping
factor ξ = 0.33. Therefore (assuming that Pact (0) = 0 and Ṗact (0) = 0), the actual
pressure Pact is driven by the following differential equation
Ṗact (t) = max min ω
2
Z
t
(Pref (t − ∆t) − Pact (t)) dt − 2ξωPact (t), rmax , rmin .
0
(4.12)
If, for simplicity, one imposes to model the brake pressure dynamics with only a single
pure time delay then the best equivalent time delay is 10 ms. The difference between
the reference and the actual brake pressure is shown on Figure 4.4. The small constant
offset between these two variables comes from the drift in the piezoelectric pressure
sensor and from the analog pressure control.
4.2.4
Brake efficiency
The brake efficiency γb in Equation (4.2) is changing during braking depending on
the brake temperature. In general, the efficiency increases with the temperature, up
to a certain level where fading starts and friction drops rapidly. This means that
the ABS algorithm needs to be robust to changes in the brake properties. During a
heavy braking on the test bench, the gain γb will typically increase by 30 % in only 4
seconds. Figure 4.5 illustrates the evolution of the brake pressure and the brake
efficiency during one test where the brake torque is kept within the same range. It
can be noticed that the pressure required decreases over time while the brake efficiency
increases by 30 %.
75
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
4.2.5
Relaxation length
Work on tyre identification like [95] and [125] has shown that the force response of
the tyre to various external inputs shows a lag in time. Since hybrid ABS controllers
specify very abrupt references to brakes actuators, and thus generate a highly dynamic
tyre response, such an effect must necessarily be taken into account.
The fact that tyre deflections (of side walls, carcass and rubber tread) have to
build up before the force is created calls for a model containing carcass compliance.
A convenient method for modelling the transient tyre behavior [125, 86] is to filter
the wheel slip
σ(λ)λ̇ + |vx |λ = Rω − vx .
In [125], the relaxation length σ is found to decrease with the tyre slip λ, to finally
reach zero at the top of the tyre characteristics. Due to the open-loop testing of tyre
slip curves, Zegelaar obtained his results in the stable zone only.
The typical manifestation of the tyre relaxation effect can be seen on the last plot
of Figure 4.9. For the tyre in the stable zone (λ < λ∗ = 0.1), the slip-force trajectory
is not the same when increasing or decreasing the force. In particular, the force
response describes an hysteresis loop: a steeper force slope can be observed during
slip decrease and a more gradual slope for slip increase. In this work, the relaxation
length at zero slip is roughly σ(0) = 0.1 m, which is a standard value for the moderate
load applied on the test rig.
4.3
The theoretical algorithm
For ABS regulation of (4.8)-(4.9), the control objective is to keep the unmeasured
variable x1 in a small neighborhood of zero, with a control u only based on the wheel
deceleration ω̇ or an equivalent signal.
Theoretically, in the phase plane, the origin (0, 0) is optimal for having the shortest
braking distance and good steerability, as x1 = 0 gives the peak of the tyre characteristics. With the cycling, the objective is to stay as close as possible to (0, 0). Any
variation in x1 will make the average brake force to decrease, while variations in x2
will require a heavier work from the brake actuator. Therefore, the size of the limit
cycle is a measure for performance. Still, to robustly detect the position of the peak
and reject measurement noise, the size of the limit cycle can never be reduced to zero.
4.3.1
The five-phase hybrid control strategy
When a hybrid control law (based on a discrete state) is used in order to control a
continuous dynamical system, the closed-loop system results in a hybrid automaton.
We refer the reader to [83] for a general introduction to hybrid systems; and to [96]
for a detailed analysis of the particular case of ABS hybrid automata.
The hybrid automaton of the ABS regulation logic, similar to the one in [96]
and [2], is described on Figure 4.6. All parameters ǫi and ui in the automaton are
positive. Each of the five phases of the algorithm defines the control action Ṫb that
should be applied to the brake. Events based on the measure of wheel acceleration
76
4.3. THE THEORETICAL ALGORITHM
x2 < 0 and x1 < 0
u1
Ṗb = − Rω
x2 ≤ −ǫ5
Ṗb =
−u5 x2
Rω
1
x2 ≥ ǫ1
Ṗb = 0
5
2
x2 ≥ ǫ2
x2 ≤ ǫ3
x2 ≤ −ǫ4
Ṗb =
u4
Rω
x2 ≤ ǫ1
Ṗb =
4
u3
Rω
3
Figure 4.6: The five-phase ABS automaton.
100
80
2
2
Wheel acceleration offset x [m/s ]
3
60
2
2
40
20
0
1
4
−20
5
−40
−60
−80
−100
−30
−20
−10
Slip offset x [%]
0
10
1
Figure 4.7: Limit cycle obtained when the theoretical five-phase algorithm of Section 4.3 is applied to the simplified model of Section 4.1. The numbers refers to the
different phases of the automaton of Figure 4.6.
trigger phase switches. The cycling between the different phases generates a repetitive trajectory. If the tuning parameters are chosen properly, on a simplified model,
this repetitive trajectory will be a non-smooth limit cycle, like the one presented in
Figure 4.7.
A general interpretation of the algorithm is as follows. When the tyre enters the
unstable zone (x1 < 0), the resulting force drop will cause the wheel to start locking.
When the acceleration crosses −ǫ5 , phase 1 is triggered. During phase 1, the pressure
is quickly reduced to change the sign of the wheel acceleration and let the wheel spin
again. Phase 1 is so short that both the wheel slip and the tyre force can be considered
to remain constant, so the change in acceleration is only determined by the change
in brake torque. When the pressure has been decreased enough to make ẋ1 > 0,
phase 2 is triggered. During phase 2, the torque is kept constant so that the wheel
acceleration is only influenced by the changes in brake force. With ẋ1 > 0, the force
77
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
will first increase, go over the maximum at x1 = 0 and then decrease again. This last
decrease forces the acceleration to cross ǫ3 which marks the return of the tyre in the
stable zone. To make sure that the acceleration will never get too large and exceed
ǫ2 , phase 3 can be shortly triggered to increase the brake pressure a bit and reset the
acceleration to ǫ1 . When the tyre is back in the stable zone, the brake pressure should
be increased, otherwise the brake force would remain far below the maximum. This
is the purpose of phase 4, which is the opposite of phase 1. When the acceleration
is less than −ǫ4 , which ensures ẋ1 < 0, phase 5 is triggered and the torque is kept
constant until wheel lock tendency is detected. Phase 1 is then triggered again and
the cycle starts over. Note that to enable the detection of wheel lock at a lower slip,
the torque can be increased slowly during phase 5 instead of being kept constant [2].
It should be pointed out that the control law does not require the knowledge of λ∗
or the tyre parameters ai to be implemented. Furthermore, it is possible to observe x2
in practice by estimating a∗x online based on accelerometer signal and average brake
pressure. The robustness towards uncertainties on a∗x can be improved by increasing
the thresholds ǫi .
4.3.2
Tuning of the algorithm
In order to tune the five-phase ABS algorithm, nine parameters need to be chosen: the
wheel deceleration thresholds ǫi and the brake pressure derivatives ui . The theoretical
study of the limit cycles performed in [96] gives a good basis and clear rules to get an
initial parameter set. The modelling of the experimental phenomena in Section 4.2
also allows for a systematic tuning.
Theoretically, the first choice is taking symmetric thresholds: ǫ5 = ǫ1 and ǫ4 = ǫ3 .
Nevertheless, in practice, several phenomena related to the delays impose taking
slightly asymmetric thresholds in order to improve performance. The oscillations
in the measurements discussed in Section 4.2.1 can trigger control intervention at
inappropriate moments. There are two ways to deal with these oscillations in our
algorithm: the first is to choose large enough thresholds so that only significant
variations in the acceleration signal are detected; the second is to filter the acceleration
signal to remove undesired high frequent noise. During testing, a compromise was
chosen between both methods. The acceleration signal is averaged on 1/10 of the
wheel rotation. At 18 m/s, the delay introduced due to filtering is about 5 ms. With
this filtering, peak to peak oscillations of 4 m/s2 can still been observed. Therefore,
the thresholds ǫi are separated by 20 m/s2 to avoid mistriggering. In the experiments,
the values ǫ1 = 40, ǫ2 = 60, ǫ3 = 20, ǫ4 = 30 and ǫ5 = 60 m/s2 were set.
Ideally, the brake pressure variations should be as quick as possible in order to
produce the smallest possible limit cycle. Therefore, the maximum brake pressure
derivatives allowed by the brake actuator were taken for the rates ui . Thats is, we
took u1 /Rω = 500 bar/sec and ui /Rω = 750 bar/sec for u3 and u4 , which is coherent
with the actuator parameters given in Section 4.2.3.
The choice of u5 is more delicate. If too small values of u5 are taken, the cycle
amplitudes are quite big and thus the performance is bad. If too big values of u5
are taken, the algorithm becomes quite sensitive with respect to delays and the wheel
might lock or remain trapped in the stable zone of the tyre. A good compromise
78
4.4. MODIFIED ALGORITHMS
100
60
2
Wheel acceleration offset x [m/s ]
80
2
40
20
0
−20
−40
−60
−80
−100
−30
−20
−10
Slip offset x [%]
0
10
1
Figure 4.8: Limit cycle obtained when the modified algorithm of Section 4.4.2 is
applied to a model that reproduces the issues of Section 4.2.
seems to be −u5 x2 /(Rω) = 50 bar/sec, typically one order of magnitude less than
quick pressure variations.
The limit cycle obtained with this specific tuning, when the five-phase algorithm
is applied to the simplified model of Section 4.1 is shown on Figure 4.7.
4.4
Modified algorithms
When implemented on the test bench, the theoretical version of the five-phase algorithm presented in Section 4.3 fails to cycle and remains blocked in an arbitrary
phase. The delays, due to measurement filtering, tyre dynamics, and actuator limitations, have been identified to be the main cause of failure. Therefore, the theoretical
algorithm needs to be improved to become more robust with respect to delays. Three
methods are proposed, analyzed and validated. All delays are lumped in a single
global actuation delay.
4.4.1
Pressure derivative profiles
In order to reduce the influence of delays, a first approach is to slow down the pressure
increase or decrease of phases 1, 3 and 4 before the wheel acceleration x2 actually
reaches the switching threshold ǫi .
Many pressure derivative profiles could be used to this end. The following sigmoid
is proposed:
Ṗi (x2 ) =
1 a
1
(Ṗi + Ṗib ) + (Ṗib − Ṗia ) tanh(ki (x2 − ǭi )),
2
2
(4.13)
where Ṗia and Ṗib are the brake pressure derivatives at the beginning and end of the
phase and ǭi is the transition point. This sigmoid allows for a fast response of the
system at the beginning of the phase, since the maximum pressure rate allowed by the
79
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
brake is used, while having a smooth transition at the end, since a smaller pressure
rate is used.
Theoretically, with a well chosen ǭi , it could be possible to take Ṗib = 0, which
would completely compensate the influence of the delay. Such a choice might, however,
harm the robustness of the algorithm, since the regulations could remain blocked in
one phase. In practice, keeping Ṗib slightly positive or negative is a more robust
choice.
The main advantage of this approach, based on pressure derivative profiles, is
that it can compensate delays while remaining robust with respect to other effects
like changes in brake efficiency. The main disadvantage is that this modified algorithm
is more difficult to tune because it involves more parameters, which depend on the
value of the delays and on the vehicle velocity.
4.4.2
Open-loop pressure steps
A different approach, that also reduces the influence of delays, is to pre-compute (using
the wheel acceleration model) the discontinuous brake pressure variation that will
make the wheel evolve from the current acceleration to the desired one. Theoretically,
in order to obtain an acceleration step of magnitude
−
∆x2 = x2 (t+
0 ) − x2 (t0 ) at t0 ,
a break pressure step of magnitude
∆P =
I
∆x2
Rγb
(4.14)
should be applied to the system, where γb if the brake efficiency defined in Equation (4.2).
But, in practice, real brake actuators are not able to follow such a discontinuous
brake pressure variation and a rate limit must be added. Moreover, asymmetry in
the brake torque response requires small adjustments on ∆P .
The main advantages of this method is that it is intrinsically robust with respect
to time delays and that it gives limit cycles that are close to those predicted by the
theoretical algorithm. The cycles given by this controller, when practical phenomena
are simulated, are shown on Figure 4.8. The main disadvantage of the method is that
it requires a good internal model, which makes it less robust with respect to model
uncertainties, like brake efficiency variations and other changes of brake actuator
parameters.
It is important to stress that this method is better suited for brake actuators that
follow a pressure or a torque reference (instead of a pressure rate); and that it might
be cumbersome to implement it on vehicles equipped with hydraulic brakes without
using brake pressure sensors.
4.4.3
Closed-loop acceleration control
A third approach is to manage the jumps, from the current wheel acceleration to
the desired one, in closed-loop. During phases 1, 3 and 4, the torque is modified in
80
4.5. EXPERIMENTAL VALIDATION
order to obtain a wheel acceleration that reaches smoothly its target value ǫi , which
enables the algorithm to switch to the next phase. During these phases, the closedloop control operates around a precomputed trajectory xref
2 . Phases 2 and 5 are left
untouched.
The reference trajectory xref
is made such that it goes as fast as possible from the
2
current wheel acceleration x2 to the next switching threshold ǫi , but with a limited
rate that changes with x2 . There are two reasons to limit ẋref
2 . Firstly, it is important
to have a reference that remains within the physical limitations of the brake actuator
described in Section 4.2.3. Secondly, the sensitivity to the delay is proportional to
the rate of change of x2 at the end of the phase. Therefore, it is natural to reduce
ẋref
to zero when xref
approaches ǫi , in order to minimize this sensitivity.
2
2
Inspired by the velocity profiles used for servo-drives [76], we will take a reference
trajectory xref
such that
2
p
p
−sign(e) 2u0 |e|
ifp2u0 |e| < v0
ref
(4.15)
ẋ2 =
−sign(e)v0
if 2u0 |e| ≥ v0
with e = xref
− ǫi . The constant v0 is the maximal first derivative for xref
and u0 its
2
2
maximal second derivative.
Once the reference xref
fixed, the controller is made of a simple proportional
2
feedback
Ṗb = K(xref
− x2 ).
2
The gain K should be taken big enough in order to track correctly the reference, but
is limited by the delay margin of the system.
The main advantage of this method is that only a limited number of parameters
need to be tuned. Furthermore, the parameters do not need to be scheduled with the
vehicle speed. Also, the same trajectory generator and controller can be used in all
phases, with the same parameters. Thanks to the feedback, the method is robust to
any change in the environment. The authors believe that, among the three methods
they have tested, this last method is the most interesting one if we compare practical
implementation issues.
4.5
Experimental validation
The three methods rendering the algorithm more robust to time delays presented
above have been tested on the tyre-in-the-loop test facility of Section 1.3.2 and give
the expected satisfying results. Thanks to the precise modelling of the test bench
phenomena, the experimental tests are very similar to the simulations. As optimal
performance was not the final aim, only a reasonable effort was put on optimizing
the tuning, and all tests were done at a constant vehicle speed. To limit the thermal
load on the brake system, the experiments were deliberately performed with a small
vertical load of 3000 N .
4.5.1
Pressure derivative profiles
Experimental testing shows that the ABS controller based on pressure derivative
profiles works well in practice. Measurements are plotted on Figure 4.9. It can be
81
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
100
100
80
80
Wheel angular acceleration [m/s2]
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
−80
−80
−100
−40
Ref Pres [bar]
Meas Pres [bar]
Accel [m/s2]
−30
−20
−10
Slip offset [%]
0
10
−100
3.5
4
4.5
5
4.5
5
Time [s]
6
1.1
4
0.9
ABS state
Normalized Braking Force µ(λ) []
5
1
0.8
3
2
0.7
1
0.6
0.5
0
0.1
0.2
Slip λ []
0.3
0.4
0.5
0
3.5
4
Time [s]
Figure 4.9: Experimental results for the ABS algorithm based on pressure derivative
profiles.
noticed that, even with the large disturbance on the brake force and acceleration,
the controlled system reaches and maintains a consistent limit cycle, encircling the
optimal point (0, 0).
Because the reduction of the pressure derivative, when approaching the switching
threshold, is probably not big enough yet, the time delay is still causing the acceleration to go far outside the predefined thresholds. Various tests have shown that
experiencing a larger positive acceleration results in a larger excursion into the stable zone. The difficulty to maintain the acceleration within the bounds is therefore
responsible for the moderate performance of this algorithm regarding the predicted
brake distance. Of course, a more precise tuning of the pressure derivative profile
could solve this issue.
The longitudinal slip oscillates between 5 and 40 percent, which is good. It could
be desired to have an even smaller excursion in order the maintain even more lateral
stability and decrease brake distance. However, with this type of ABS methods based
on wheel acceleration, the slip excursion needed to activate the phases is not directly
controlled and depends heavily on the shape of the tyre curve. The larger the slopes
of the tyre characteristic and the smaller the slip variation. In the particular case of
the tyre-in-the-loop setup, the relatively small load on the wheel gives a tendency for
82
4.5. EXPERIMENTAL VALIDATION
100
100
80
80
Wheel angular acceleration [m/s2]
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
−80
−80
Ref Pres [bar]
Meas Pres [bar]
2
Accel [m/s ]
−100
−40
−30
−20
−10
Slip offset [%]
0
10
−100
8
8.5
9
9.5
9
9.5
Time [s]
6
5
1
4
0.95
ABS state
Normalized Braking Force µ(λ) []
1.1
1.05
0.9
0.85
0.8
3
2
0.75
0.7
1
0.65
0.6
0
0.1
0.2
Slip λ []
0.3
0.4
0.5
0
8
8.5
Time [s]
Figure 4.10: Experimental results for the ABS algorithm based on open-loop pressure
references.
having large slip oscillations.
The tuning of this algorithm is quite robust. Small changes in the parameters or in
the system do not affect the general shape of the limit cycle. Because the pressure is
always kept decreasing or increasing respectively in phases 1 and 4, the acceleration
will cycle more robustly between positive and negative and influences from vehicle
speed or brake efficiency are minimized.
4.5.2
Open-loop pressure steps
The ABS controller based on open-loop pressure reference has also been successfully
tested in practice. The measurements are plotted in Figure 4.10. Also in this case,
the limit cycles are clearly visible and reproducible despite the disturbance.
This method is much better at maintaining the acceleration within the predefined
thresholds. Intrinsically, the robustness with respect to time delays is larger. Thanks
to the maximum acceleration maintained at a lower positive value, the force drop in
the stable zone is smaller and the corresponding brake distance would also be reduced.
Unfortunately, since the pressure commands are given in open-loop, the algorithm
has difficulties to react to system changes. In particular the tuning is quite sensitive
83
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
100
100
80
Wheel angular acceleration [m/s2]
80
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
Ref Pres [bar]
Meas Pres [bar]
−80
−80
Accel [m/s ]
2
2
Accel ref [m/s ]
−100
−40
−30
−20
−10
Slip offset [%]
0
10
−100
5.2
5.4
5.6
5.8
6
6.2
Time [s]
6.4
6.6
6.8
5.2
5.4
5.6
5.8
6
6.2
Time [s]
6.4
6.6
6.8
6
1.1
4
0.9
ABS state
Normalized Braking Force µ(λ) []
5
1
0.8
2
0.7
1
0.6
0.5
0
3
0.05
0.1
0.15
0.2
0.25
Slip λ []
0.3
0.35
0
0.4
Figure 4.11: Experimental results for the ABS algorithm based on closed-loop acceleration control.
with respect to the brake efficiency γb . An overestimation of γb could prevent from
triggering the next phase, while an underestimation of γb would make the acceleration
to go far outside the thresholds, thus reducing the performance. Furthermore, the
tuning of the phases 1 and 4 need to be different in practice, compared to simulation, in
order to cope with the encountered asymmetry in the brake reaction when increasing
or decreasing the pressure. The robustness gained against the time delay seems to
harm the robustness against other phenomena.
4.5.3
Closed-loop acceleration control
The version of the ABS with closed-loop acceleration control is the last one to be
successfully validated. The results of the experimental tests are plotted on Figure 4.11.
The target acceleration is visible on the second plot. In phase 4 of the algorithm, the
acceleration needs some time to reach the target value since the tyre has to go from
its unstable region to its stable region. During the other phases, the acceleration and
the reference remains very close.
Based on simulations, the feedback gain was chosen to be K = 3 and the rate
limitations fixed to v0 = 2000 and u0 = 100000. Moreover, like for the two other
84
4.6. COMPARISON WITH THE BOSCH ALGORITHM
methods, the switching thresholds ǫi are those of Section 4.3.2.
This method is the easiest to tune and gives the best performances. The acceleration is maintained within the desired bounds. The force drop in the stable zone is
limited to 0.7 which is similar to the method of Section 4.5.2. Finally, the slip offset
x2 is maintained between 20 and 30%, which is similar to the method of Section 4.5.1.
The system is cycling around the top of the tyre characteristics. Further, the control
is robust to all changes like brake efficiency or actuation delay.
4.6
Comparison with the Bosch algorithm
The ABS algorithm proposed in this chapter can be compared with the commercial
one implemented by Bosch and described in [18]. Both algorithms are based on a
hybrid logic and use the wheel acceleration as their only measurement. In this section,
some of the similarities and the differences are highlighted. Throughout this section,
the notation Bosch ABS will refer to the algorithm of [18] and Subsection 4.6.1; while
five-phase ABS will refer to the algorithm of Section 4.3.
4.6.1
The original Bosch algorithm
The algorithm described by Bosch [18] is first recalled. When the phase of idleness
is omitted, the algorithm is composed of six phases, as illustrated on Figure 4.12(a).
On this figure, in order to facilitate the comparison between both algorithms, the
phases of the Bosch algorithm are renamed to better match the phases of the fivephase algorithm of Figure 4.12(b). Phases similar to 1, 2 and 4 with either large
torque variations or constant torque can be found and are matched. Further, it can
be noticed that a phase similar to phase 3 is not present in the Bosch ABS, while
phase 5 is split in two different phases.
The jumps between the different phases are triggered by wheel acceleration thresholds. Three thresholds can be identified: “−a”, “+a”, and “+A”. This notation is
kept here to improve coherence with the original reference [18]. It should be noted,
however, that the three thresholds have values independent of each other and “−a”
is not the opposite of “+a”. On Figure 4.12(a), the threshold for the transition between phases 1 and 2 is denoted by b. In the original version of the Bosch ABS, the
constant b is equal to the parameter “−a”.
4.6.2
Vehicle measurements and a modified Bosch algorithm
Measurements performed on a BMW 5 series equipped with an ABS system from
Bosch are presented in Figure 4.13. Similar measurements originating from a Peugeot
307 also equipped with ABS system from Bosch were also analyzed and exhibited a
characteristic behavior that was similar to that of the BMW vehicle.
The data of Figure 4.13 come from the front-left wheel and the manoeuvre is a
heavy braking on a straight line, performed on a high adherence surface. The vehicle
speed during the measurement varies from 25 to 6 m/s and it can be observed that,
even if the tuning of the algorithm is adapted, the cycles are larger at a lower speed.
The wheel deceleration thresholds +A, +a and −a can be identified to be 50, 20
85
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
x2 < −a
Ṗb = 0
0
x1 < 0
Ṗb = −v1
1
x2 < −a
Ṗb = v5
x2 ≥ b
Ṗb = 0
5b
x2 ≤ +a
2
x2 ≥ +A
Ṗb = 0
Ṗb = v4
5a
x2 < +A
4
(a)
x2 < −a
Ṗb = 0
0
x1 < 0
Ṗb = −u1
x2 ≤ −ǫ5
Ṗb = u5 (x2 )
1
x2 ≥ ǫ1
Ṗb = 0
5
2
x2 ≥ ǫ2
x2 ≤ ǫ3
x2 ≤ −ǫ4
Ṗb = u4
x2 ≤ ǫ1
Ṗb = u3
4
3
(b)
Figure 4.12: (a) The hybrid automaton associated to the Bosch algorithm. (b) A
simplified version of the five-phase algorithm, considered at a constant speed.
86
4.6. COMPARISON WITH THE BOSCH ALGORITHM
120
brake pressure fl
accel fl
80
wheel angular acceleration [m/s2]
100
60
80
40
60
40
20
20
0
0
−20
−20
−40
−60
−40
−40
−30
−20
−10
Slip offset [%]
0
10
0.1
0.2
−60
9
9.5
10
time [s]
10.5
1.1
Normalized Braking Force µ(λ) []
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
0
Slip λ []
0.3
0.4
0.5
Figure 4.13: Measurements for the Bosch ABS on a BMW car, front-left wheel,
straight braking on high mu.
and -40 m/s2 , and are indicated in as straight lines on the plots. The brake pressure
variation rates are difficult to measure precisely because of the limited accuracy of the
sensors and are varying with the vehicle speed. The following average estimates can
be made: v1 = −275 bar/s, v4 = 275 bar/s and v5 = 15 bar/s. The brake efficiency
is varying a lot during the braking manoeuvre because of thermal phenomena, from
20 N m/bar at the beginning to 40 N m/bar at the end.
One major difference between the Bosch algorithm of [18] and the vehicle measurements lies at the transition between phases 1 and 2. According to the official
description of the algorithm [18], the threshold b should be “−a”. On the vehicle
data it appears, however, that the value of b seems to be much closer to “+a”. During implementation of the original Bosch algorithm, with b = “−a”, the logics could
not be made properly functional neither in simulation nor on the test bench. Switching from phase 1 to phase 2 when the wheel acceleration is still negative, at “−a”,
is too early and does not allow the wheel to spin up again. Therefore, in order to
have a working algorithm, the Bosch algorithm was modified to have a threshold b
that is positive. This modification is based on the vehicle measurements and on the
theoretical study of the first integrals of the simplified model [96]. Observe that the
stability of the five-phase algorithm is guaranteed by stopping the pressure decrease
87
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
100
120
80
Wheel acceleration offset [m/s2]
100
60
80
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
Brake Pres [bar]
2
Accel [m/s ]
−80
−60
−50
−40
−30
−20
−10
Wheel slip offset [%]
0
−80
2
10
2.5
3
3.5
Time [s]
4
4.5
5
Figure 4.14: Simulation results for the Bosch algorithm. The wheel acceleration
thresholds are +A = 50 m/s2 , +a = 20 m/s2 , and −a = −40 m/s2 . The brake
pressure variation rates are v1 = −350 bar/s, v4 = 275 bar/s and v5 = 15 bar/s.
when the wheel acceleration is positive.
The Bosch ABS is simulated with the objective to match the general behavior
observed in the measurements. The results are shown on Figure 4.14. When possible,
the parameter values identified from the BMW are used. However, the brake pressure
variation rate v1 needed to be increased in order to have a better match of the limit
cycles. Reproducing in simulation the limit cycle observed on the vehicle is extremely
difficult, firstly because of the difficulty to tune all the tyre and vehicle parameters to
fit the reality, secondly because of the differences between the algorithms implemented
on the car and in the simulation, and thirdly because of the measurement inaccuracy
and noise.
4.6.3
Comparison between the five-phase and the Bosch ABS
algorithms
The five-phase ABS and the Bosch ABS can now be compared. The comparison
focuses on the core of the algorithms, considered at a constant speed.
A first interesting difference between the algorithms is that the Bosch ABS requires
a large delay in the system, coming for example from the hydraulic actuation. If a
more ideal brake system without delay is simulated, like a brake-by-wire system or
an in-wheel motor, the algorithm will not work properly. The reason behind this
specific point is that, in the Bosch ABS, the same threshold “+A” is used to jump
from phase 2 to phase 4, and from phase 4 to phase 5a. Without delay, the algorithm
will jump to phase 5a with a too small pressure increase. The modification required
to improve the situation is to decrease the threshold between phases 4 and 5a.
A clear advantage of the five-phase ABS is its mathematical basis. A consequence
of this theoretical background is that it is easier to predict how the five-phase logic
will react in different situations and adapt it if necessary, for example in the case
of different actuator characteristics (like those of hydraulic, electromechanical, and
88
4.7. CONCLUSION
120
120
100
Wheel acceleration offset [m/s2]
100
80
80
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
Brake Pres [bar]
2
Accel [m/s ]
−80
−60
−50
−40
−30
−20
−10
Wheel slip offset [%]
0
−80
2
10
2.5
3
3.5
Time [s]
4
4.5
5
Figure 4.15: Simulation results for the Bosch algorithm. The wheel acceleration
thresholds are +A = 50 m/s2 , +a = 20 m/s2 , and −a = −40 m/s2 . The brake
pressure variation rates are v1 = −250 bar/s, v4 = 275 bar/s and v5 = 15 bar/s.
electric brakes). The Bosch algorithm, with its heuristic conception and its set of
unprecise descriptions given in the literature, remains obscure. Tuning this algorithm
properly and making modifications on it remains delicate.
It is also important to note that the Bosch algorithm does not contain a phase
comparable to the third phase of the five-phase ABS. Such a phase is useful to limit the
amplitude of the wheel-slip excursion into the stable zone, after a pressure release. The
absence of this phase can lead to cycles that are longer than normal and to a decrease
of performance, when particular initial conditions are encountered. Figure 4.15 shows
a simulation were this kind of performance decrease is exhibited, in the case of the
Bosch algorithm. Therefore, the presence of phase 3 is an advantage of the five-phase
strategy.
Finally, it can be concluded that, for the specific conditions available in the laboratory, the level of performance achieved is comparable: the size of the limit cycles
and the average brake force are similar. Of course, the Bosch ABS has many extra
rules next to the core in order to cope with a large number of specific situations, but
this falls outside the scope of this comparison.
4.7
Conclusion
In this chapter, the five-phase algorithm proposed in [96] was extended with three
methods to deal with time delays. Time delays in measurement and actuation were
found to be the main reasons why the theoretical five-phase algorithm fails in experiments.
The use of pressure derivative profiles reduces the influence of delays, but does
not perfectly compensate for them. This method is difficult to tune because of the
large number of parameters involved, but maintains a good level of robustness with
respect to other phenomena, like changes in brake efficiency.
Giving pressure reference steps in open-loop offers a good resistance with respect
89
CHAPTER 4. ABS BASED ON WHEEL ACCELERATION
to time delays and makes the algorithm remain closer to the original theoretical
framework. But the robustness of this approach, with respect to system changes,
appears to be small. And, in particular, a good estimation of the brake efficiency is
required.
By using closed-loop acceleration control, the advantages of both previous methods
are combined. A high level of robustness is achieved with respect to all practical issues
considered in Section 4.2. Moreover, the acceleration remains close to the desired
trajectory ensuring a tight limit cycle.
These three techniques give limit cycles encircling the optimal braking point at
the top of the tyre characteristic, without any need of estimating the parameters of
this curve. Only the wheel acceleration measurement is used to implement the control
logic. The proposed simulation model, including five important practical phenomena,
is capable of reproducing the same results as in the experiments and is therefore
suitable for testing ABS strategies. Thanks to the tyre-in-the-loop testing, it can be
expected that the algorithms can rapidly be installed in a complete vehicle.
Considering the descriptions of the Bosch ABS available in the literature and the
experimental measurements, it can be concluded that the core of the five-phase ABS
presents some advantages compared to its commercial analogue. Not only the fivephase ABS is supported by a theoretical foundation, but it also gives more flexibility to
handle long and short delays. Furthermore, the presence of phase 3 better maintains
regular cycles with a smaller force drop during the pressure release. The performance
regarding the size of the limit cycles are similar in the specific conditions used for the
study. One step further has been made towards the objective of having an open ABS
algorithm comparable to the Bosch ABS. However, the commercially available Bosch
ABS comes with many extra rules to cope with all kind of situations, which are still
far from begin openly available.
90
Chapter 5
ABS based on Tyre Force
The Anti-lock Braking System (ABS) is the most famous active safety system for
road vehicles, see Section 1.3. This system has been around for more than 30 years
but, in general, ABS algorithms can still be improved. First they can be made
simpler, hence easier to understand and tune, while keeping the same performance.
Moreover, their performance can be enhanced in terms of reducing the slip variations
and decreasing the brake distance. Finally, the robustness against tyre-road friction
characteristic can be improved and new algorithms should be able to handle rapid
changes in friction without complex logic. This chapter introduces a force-based ABS
that brings improvements in those 3 directions.
A few publications about ABS algorithms exploiting tyre force measurements can
be found in the literature. Kamada [66] investigates the correlation between slip
derivative and force derivative. This technique works fine in simulation but has shown
severe limitations when implemented in practice because of the noise level. Botero
[19] implements a slip regulator (see Section 1.3) using sliding mode control where the
force estimation is required to enforce sliding. De Bruijn [36] uses force measurement
to define the upper and lover torque levels between which the controller is switching.
The algorithm developed in this chapter extends the class of hybrid ABS (see Section
1.3).
Two limitations are inherent to the family of hybrid ABS: the brake force cannot
always be at its maximum since a force drop is needed to switch phase, and problems
are encountered when the tyre-road friction characteristics does not present a clear
maximum at a limited slip, like on snow or gravel. The same limitations will be found
in this new force-based ABS.
In traditional hybrid algorithms based on wheel acceleration, the acceleration is
used both for detection and control. First, the acceleration is used to detect when the
tyre has passed the maximum of its characteristics, by indirectly observing the tyre
force when the brake torque remains constant. Secondly, the acceleration needs to be
controlled to drive the tyre to a limit cycle, by varying the brake torque. Those two
utilizations are conflicting with each other. This often requires complicated logics with
many states and the alternance between phases where the brake torque is changing
quickly or kept constant; which can make the tuning more difficult and can affect the
performances.
91
CHAPTER 5. ABS BASED ON TYRE FORCE
Thanks to direct tyre force measurement, the above limitations can be alleviated,
the detection of the friction peak can be made more sensitive and the acceleration can
permanently be controlled to provide a tight limit cycle. Possibilities for implementing tyre force measurement on production vehicles, in particular using force sensing
bearings, are discussed in Section 1.2.
In this chapter, a new ABS algorithm using force and acceleration measurements
is developed. The algorithm is simpler than other approaches in the literature as
it contains only 2 states and 3 tuning parameters, and does not require parameter
scheduling. The stability is proven and the analysis provides tuning tools. The
algorithm is validated experimentally on a tyre-in-the-loop test bench and is shown
to perform better than the deceleration hybrid algorithm of Chapter 4. No knowledge
of the tyre-road characteristics is needed for control.
Outline
The wheel and vehicle models are introduced in Section 5.1. In Section 5.2, the algorithm is described and the stability proven. The tuning is discussed in Section 5.3.
Section 5.4 concentrates on the single wheel model with simulation results, experimental validation and comparison. Finally, in Section 5.5, simulations are performed
using a two-wheel model.
5.1
Modelling
The study of the ABS system will be done in two stages, using vehicle models with
different level of complexity. The wheel and tyre model remain the same throughout
this chapter while the chassis model will go from a one-wheel model to a single-track
two-wheel model.
In the first stage, only the longitudinal dynamics of a single loaded wheel is considered. In particular, the relaxation length of the tyre and the actuation delay are
taken into account. Even though weight transfer and combined slip are ignored, all
the basic phenomena related to ABS already appear in this simple model. Also, this
level of detail is suitable to model the tyre-in-the-loop experimental setup.
In the second stage, the weight transfer from the rear to the front during braking
is added. A single-track two-wheel vehicle model is used where the longitudinal as
well as the vertical dynamics of the wheels and the chassis are considered. This model
is now suitable for representing a real car braking in straight line.
The model of each wheel is as follows:
J ω̇
= rFb − Tb
τ Ḟb
= −Fb + Fz f (λ, µ, α, ...)
v − rω
rω
λ =
=1−
v
v
Ṫb = −γu(t − ∆t)
(5.1)
(5.2)
(5.3)
(5.4)
where s is the Laplace variable, J and r are the inertia and radius of the wheel, Fb
the tyre brake force, Tb the brake torque, v the vehicle speed, τ the relaxation length,
92
5.2. THE ALGORITHM
Fz the vertical load on the tyre, f the non-linear tyre characteristics expressed for
example by the magic formula, see [95], λ the tyre slip during braking, w the speed
of the wheel, γ the brake efficiency, ∆t the total delay, and u the brake pressure
derivative control input. The partial derivative of f with respect to λ is
f ′ (λ) =
∂f
(λ)
∂λ
(5.5)
The vehicle speed as well as the tyre loads are given by the vehicle model. For the
single-wheel model, the following simple equations are added to (5.1)-(5.4):
1
Fb
m
= mg
v̇
= −
Fz
(5.6)
(5.7)
where g is the gravity acceleration.
For the two-wheel model, following an approach similar to [106], 4 states are added:
the height of the vehicle center of gravity z, the pitch angle of the vehicle p and the
height of the wheel hubs at the front and at the rear ztf and ztr . The upper indices
f and r are used to distinguish the front from the rear of the vehicle. The equations
added to (5.1)-(5.4) are the following:
v̇
= −
1 f
(F + Fbr )
m b
1 f
(F + Fsr ) − g
m s
Jc p̈ = −aFsf + bFsr + (Fbf + Fbr )z
z̈
Fsf
Fsr
zsf
zsr
mt z̈tf
mt z̈tr
Fzf
Fzr
=
=
=
f
−k (zsf − zs0
− ztf ) − df żsf
r
−k r (zsr − zs0
− ztr ) − dr żsr
f
= z − h − a sin(p)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
= z − h + b sin(p)
(5.14)
= −Fsf + Fzf
(5.15)
= −Fsr + Fzr
(5.16)
−kt (ztf
−kt (ztr
(5.17)
=
=
f
− zt0
)
r
− zt0 )
(5.18)
where Jc is the inertia of the chassis around the lateral direction, Fs is the vertical
force in the suspension, zs is the vertical displacement of the suspension, h is the
height of the center of gravity, a and b the distances between the center of gravity and
the front or respectively the rear of the vehicle, k and d are stiffnesses and damping
coefficient.
5.2
The algorithm
Each wheel of the vehicle is controlled separately. Therefore, the description and
analysis of the algorithm will be done for one wheel only, according to the singlewheel model (5.1)-(5.7).
93
CHAPTER 5. ABS BASED ON TYRE FORCE
m
v
z
Fsf
Fsr
mg
zsf
kf
ztf
h
ωf
zsr
b
a
p
m tf
m tr
ωr
ztr
ktr
ktf
Fzf
dr
kr
df
Fbf
Fzr
Fbr
Figure 5.1: Illustration of the single-track two-wheel vehicle model.
Fn < F̂n − dFn+
F̂n = Fn
1
2
−
ω̇ref = ω̇ref
+
ω̇ref = ω̇ref
F̂n = Fn
Fn < F̂n − dFn−
Figure 5.2: Hybrid Automaton of the ABS algorithm. In each phase, the acceleration
is controlled in closed-loop at the desired level ω̇ref . The switching is based on force
measurement, with Fn the normalized braking force, F̂n the local maximum over time
of Fn and dFn+ and dFn− two tuning parameters. The inequalities at the beginning
of the arrows indicate the switching condition; the equalities at the end indicate the
reset assignments.
The algorithm consists of 2 phases and a switching mechanism, as depicted on
Figure 5.2. During phase 1, the control action will be such that the slip will decrease
(λ̇ < 0). During phase 2, it is the opposite (λ̇ > 0). The switching will take place
after the tyre slip λ crosses λ∗ , meaning that after some increase, the brake force is
decreasing again. This back and forth process will ensure that the system is cycling
around the optimal slip λ∗ .
To ensure that the slip increases or decreases during respectively phase 2 or 1,
the wheel angular acceleration is controlled in closed loop. This method makes the
control robust to changes in brake efficiency or tyre radius. It is also suited for
hydraulic braking since the pressure or torque do not need to be measured. However,
the large delay in the loop coming from the measurement and the actuation limits
94
5.2. THE ALGORITHM
Fb
1
2
λ∗
1
λ
Figure 5.3: General trajectory of a hybrid two-phase ABS algorithm visualized on the
tyre friction curve.
the performance of the control.
The switching mechanism is implemented looking at the brake force measurement.
If the switching takes place after the tyre slip λ crosses λ∗ , meaning that after some
increase the brake force is decreasing again, the system will cycle around the optimal
slip λ∗ .
5.2.1
Switching strategy
The strategy is explained with the help of Figure 5.3.
At the beginning of each phase, λ goes towards λ∗ so that the tyre force evolves
towards its maximum. When λ∗ is crossed, the tyre force decreases and a switch
to the other phase needs to take place, in order to make the brake force to increase
again. The switching strategy is implemented using tyre force measurement. To get
the right switching, it is important to consider only the variations of the tyre force
that originates from a change of λ. However, variations on Fz , for example linked to
weight transfer or road excitation, can have a large influence on the tyre force. To
solve this problem, we introduce
Fb
(5.19)
Fn =
Fz
where the brake force is normalized by the vertical force acting of the tyre. Depending
on the sensing solution, Fz may or not be directly measured. In case of sensors
placed in the tyre Fz can be assumed measured at the contact patch, while with
other solutions (for example load-sensing bearings) only the force at the wheel hub is
available. For generality’s sake, we will assume that Fz is available. In Section 5.5,
it will be shown how the algorithm can be adapted in case the direct measurement is
not available.
In each phase, assumed to start at time ts , the algorithm searches for the maximal
normalized force F̂n the tyre can generate
F̂n (t) = max (Fn (ζ)).
ts ≤ζ≤t
95
(5.20)
CHAPTER 5. ABS BASED ON TYRE FORCE
When the measured force drops of dFn± under the current maximum F̂n , one can
conclude that the maximum has been exceeded and a phase switch takes place. When
entering a new phase, the maximum F̂n is reset and the detection mechanism stars all
over again. In case of large actuation delays, it might be necessary to delay the start
of the detection mechanism, but as this is a minor detail, it is not addressed here.
Computing the maximum during each phase allows the algorithm to handle changes
in tyre-road friction. The use of the parameters dFn± can be compared to some kind
of hysteresis. Therefore, it can be noted that, as long as dFn± are larger than zero,
there will be no chattering in the controller.
The definition of dFn require some considerations. A too large drop would lead to
a longer brake distance while a too small would increase the sensitivity to noise. In
practice, it can be noticed that tyres can show short but large force drops which have
nothing to do with the basic characteristic. This phenomena can be observed on the
top left plot of Figure 5.11. Force drops of up to 30% of the load have been observed
on worn tyres. In order to avoid those drops influencing the switching strategy, a
combination of large enough thresholds together with a low-pass filtering of the force
measurement is employed.
5.2.2
Closed-loop control of the wheel acceleration
A way to ensure that the slip derivative is either strictly positive or negative during
each phase is to control the wheel acceleration in closed-loop. The objective of the
closed-loop controller is to drive the error on the wheel acceleration
e = ω̇ref − ω̇
(5.21)
to zero as fast as possible. To this end, the closed-loop proportional control law is
used
u = k(ω̇ref − ω̇).
(5.22)
with ω̇ref is a tuning parameter representing the target acceleration for the current
phase. To guarantee the correct slip derivative, ω̇ref should be positive in phase 1
and negative phase 2. The tuning is addressed in Section 5.3.2.
In the next subsections, the stability properties are analysed. Model-based design
methods were considered not suitable in this case since the system (5.1)-(5.7) is nonlinear, uncertain and with time delay.
If a simple version of the model is considered, it becomes possible to show that
this control law together with the switching mechanism maintains the system into a
bounded region of the state space. This constitutes a stability proof and gives bounds
on the performances. When the more complex model with delays is considered, the
following results can be extended with a linearization approach and extensive simulations to provide practical tuning tools.
5.2.3
Bounding the trajectory for a simple model
Let us consider for now the simplified model obtained neglecting the actuation delay
and tyre relaxation. Due to large differences in inertia, the wheel dynamics and
car body dynamics evolve on significantly different time scales, with the speed v
96
5.2. THE ALGORITHM
changing much more slowly than the wheel slip λ. v can be consequently dropped as
a state variable and considered as an independent slow-varying parameter, similarly
to [98, 117]. Thanks to the above assumptions the model can be written as
r rFz f (λ) − Tb
(5.23)
λ̇ = −
v
J
rFz f (λ) − Tb
(5.24)
Ṫb = −γk ω̇ref −
J
Note that the brake torque Tb and the wheel acceleration ω̇ are linked by an algebraic
equation. This means that any of the two can equivalently be used as system state.
In this subsection, λ and Tb are used as states as it allows for a simplification of the
equations.
Theorem 5.1. Consider the system (5.23-5.24) for a given velocity v together with
the proposed switching and acceleration control, and take an ǫ strictly positive.
If the controller gain k is such that
γk +
r2
Fz f ′ (λ) > 0
v
∀λ : 0 ≤ λ ≤ 1
then
• the brake torque Tb is bounded by
rFz f (λ) − B < Tb < rFz f (λ) + B
with
B>
abs
ǫ + γk ω̇ref
γk
J
+
r2
′
vJ Fz f (λ)
∀λ : 0 ≤ λ ≤ 1
(5.25)
(5.26)
−
+
abs
where ω̇ref
= max |ω̇ref
|, |ω̇ref
| .
• the slip λ is bounded by
λ<λ<λ
(5.27)
with λ and λ implicitly given by
rFz f (λ) − B
rFz f (λ) + B
v
= rFz (Fnmax − dFn+ ) + B − γk(λ − λ+ )
r
v
−
= rFz (Fnmax − dFn ) − B − γk(λ − λ− )
r
where λ+ and λ− are defined by (5.30) and (5.31), and dFn+ and dFn− are the
two tuning parameters defining the switching strategy;
and therefore the system’s trajectory remains in a bounded region of the state space
(λ, Tb ), which ensures stability.
97
CHAPTER 5. ABS BASED ON TYRE FORCE
1600
1400
1200
λ+
λ
b
T [Nm]
1000
800
λ−
λ
600
400
200
rFz f (λ)
Tb (λ(t))
0
−200
0
rFz f (λ) ± B
Tb (λ) = rFz f (λ± ) ± B − v/2γk(λ(t) − λ± )
0.1
0.2
0.3
λ []
0.4
0.5
0.6
Figure 5.4: Bounds on the trajectory when the simple model (5.23)-(5.24) is considered. First the curves rFz f (λ) ± B defines upper and lower bounds on TB . Then the
lines Tb (λ) = rFz f (λ± ) ± B − v2 γk(λ(t) − λ± ) gives upper and lower bounds on λ.
Proof. First we will show that if (5.25) holds the trajectory is attracted toward a
bounded region of the state space, then we will find bounds on the size of that region
in terms of maximum and minimum Tb and λ.
Let us define the region R as the region contained between T b = rFz f (λ) + B and
T b = rFz f (λ) − B for all λ where B is a parameter to be defined, see Figure 5.4.
From there, the region Q and S can also be defined, respectively above R: (Tb > T b );
and under R: (Tb < T b ).
The first step is to show that R is attractive when the initial condition is outside
R. This is done using Lyapunov-like arguments.
Consider the initial condition X0 = (λ0 , Tb0 ) in Q. Thanks to continuity arguments, the trajectory will either remain in Q or will cross at some point the boundary
T b . Let the function
V (λ, Tb ) = Tb − (rFz f (λ) + B)
indicate the distance of x from R along the Tb axis; as long as V is positive the
trajectory will be in Q, as soon as V = 0 the trajectory has entered R. We will now
show that it is possible to bound the time derivative of V so that the trajectory will
enter R in finite time. Consider
V̇ (λ, Tb ) = Ṫb − rFz f ′ (λ)λ̇
rFz f (λ) − Tb
r2
′
,
= −γk ω̇ref + γk + Fz f (λ)
v
J
(5.28)
if (5.25) holds and we recall that x ∈ Q, equation (5.28) is bounded from above by
r2
B
′
V̇ (λ, Tb ) < −γk ω̇ref − γk + Fz f (λ)
.
(5.29)
v
J
98
5.2. THE ALGORITHM
Depending on the phase, ω̇ref could be either negative or positive; as we are trying
to bound V̇ (λ, Tb ) from above we will consider the negative one. If we chose B such
that
−
ǫ − γk ω̇ref
∀λ : 0 ≤ λ ≤ 1
B > γk
r2
′
J + vJ Fz f (λ)
then V̇ (λ, Tb ) < −ǫ in Q with ǫ a positive value defining the speed of convergence. By
the comparison lemma [69], if x starts in Q, it will enter R in finite time regardless
of the switching strategy.
The same results can be shown to hold for x starting in S. In this case the
Lyapunov-like function is VS (λ, Tb ) = −Tb + (rFz f (λ) − B). A similar condition on B
−
+
|), B can be made
|, |ωref
(with the positive ωref ) is obtained; by taking the max(|ωref
unique for both cases, thus obtaining a symmetric attractive region around rFz f (λ).
So far, two conditions have been introduced. The first condition (5.25) on k ensures
stability while the second condition (5.26) determines the size of the attractive and
invariant region R.
Let us now consider the bound on λ by taking into account the switching strategy.
As in this simple model there is only a static relationship between tyre slip and tyre
force, the switching strategy based on forces can be implicitly rewritten in term of
slip:
• switching from phase 1 to phase 2 takes place at
−1
+
λ+ = fλ>λ
∗ Fnmax − dFn
• switching from phase 2 to phase 1 takes place at
−1
−
λ− = fλ<λ
∗ Fnmax − dFn
(5.30)
(5.31)
where Fnmax is the maximum normalized friction force. Note that, even if f is not
invertible on its entire domain, f limited to either λ > λ∗ or λ > λ∗ is invertible. The
objective is to find a bound on the trajectory of λ after a switch.
By combining equations (5.23) and (5.24), we can get
v
Ṫb = −γk ω̇ref − λ̇γk.
r
(5.32)
This can be integrated between the last switching time ts and t:
Z t
Z t
Z t
v
γk ω̇ref dt −
Ṫb dt = −
λ̇γkdt
ts r
ts
ts
v
Tb (t) − Tb (ts ) = −γk ω̇ref (t − ts ) − γk(λ(t) − λ(ts ))
r
(5.33)
Note that since ts is the most recent switching time before t, ω̇ref is constant during
the interval of integration.
From (5.25) and (5.26) we know that x will enter R in finite time; once it has
entered R, Tb at the time of the switch can be bounded:
99
CHAPTER 5. ABS BASED ON TYRE FORCE
• During phase 1 where ω̇ref > 0, we have that Tb (ts ) < rFz f (λ+ ) + B and thus
v
Tb (t) ≤ rFz f (λ+ ) + B − γk(λ(t) − λ+ ).
r
(5.34)
• During phase 2 where ω̇ref < 0, we have that Tb (ts ) > rFz f (λ+ ) − B and thus
v
Tb (t) ≥ rFz f (λ− ) − B − γk(λ(t) − λ− ).
r
(5.35)
A bound on the maximal slip, which will occur during phase 1, can be found by
combining T b and (5.34). Equivalently, a bound on the minimal slip, which will then
occur during phase 2, can be found by combining T b and (5.35). In particular, once
the trajectory has entered R, the slip λ is bounded by
λ<λ<λ
(5.36)
with λ and λ implicitly given by
rFz f (λ) − B
rFz f (λ) + B
v
= rFz f (λ+ ) + B − γk(λ − λ+ )
r
v
−
= rFz f (λ ) − B − γk(λ − λ− )
r
The bounds λ and λ are visualized on Figure 5.4. Simulations confirm that the
system trajectory is attracted by the bounded region and that once inside the bounded
region the trajectory will cycle between a maximum and a mininum λ; although
derived using conservative arguments, the above bounds provide useful indications
for the controller tuning. The following conclusions can be drawn from Theorem 5.1.
abs
abs
• The larger ω̇ref
, the larger the cycles, since B is proportional to ω̇ref
.
• The larger dFn± and the larger the cycles, since it makes λ+ and λ− to be further
away from λ∗ .
• The larger the controller gain k, the lower the influence of the tyre characteristic
r2
′
on the cycles, as γk
J dominates vJ Fz f (λ) in (5.26).
• The larger the controller gain k, the smaller the variation of the slip λ during 1
cycle, as it will increase the slope of (5.34).
• Equation (5.25) quantifies the effect of the friction
characteristic on the stability.
∂f In particular, the critical condition is on ∂λ
. The greater the negative slope
min
the more difficult it is to stabilize. It is however possible to find a bound on
∂f that will hold for all surfaces and tune k consequently.
∂λ min
• Equation (5.25) quantifies the influence of the velocity v on the stability and
size of the cycles. It is well known in the literature [106] that ABS regulation
becomes more difficult as v decreases, and this is why most ABS stop using
100
5.3. CONTROLLER TUNING
30
Upper bound [Linearization]
Upper bound [Simulation]
Lower bound [Theorem 1]
25
k
20
unstable
15
10
5
0
0
stable
0.01
0.02
∆t [s]
0.03
0.04
0.05
Figure 5.5: Stability region in the gain k - delay ∆t space.
feedback control under a critical speed. The above results provide a useful
design tool to assess in which velocity range the closed-loop system will remain
stable. For any value of k, there will always be a critical speed under which
the system cannot be guaranteed stable anymore. Equations (5.25) and (5.27)
describe the size of the cycles as the velocity gets closer to the critical threshold.
5.3
Controller tuning
When considering the more complex system (5.1)-(5.7), the computations from Theorem 1 are not valid anymore. However, it can be shown that the main conclusions
remain valid. In the next subsection, further analysis is done on the system (5.1)-(5.7)
to confirm the conclusions.
5.3.1
Tuning the controller gain
In Theorem 5.1, (5.25), stability is guaranteed by imposing a lower bound on the
controller grain k, which is plotted on Figure 5.5. When all the dynamics of the system
are considered, including the time delay, a too large controller gain will destabilize
the system. Here two approaches to determine the stability region in the gain k delay ∆t plane are proposed.
The first possibility is to run many simulations. Many combinations of delays and
gains are tried and the stability region is drawn based on the outcome. A criteria
is necessary to distinguish between stable and unstable simulations. Here, a configuration is classified unstable if there is no cycle or if the trajectory would require a
negative torque. The stability region based on simulations is shown on Figure 5.5. It
can clearly be seen that the larger the delay and the lower the controller gain has to
be. Still, this brute-force method is not pleasant as it is computationally expensive
and it requires a complete and precise model of the system.
The second option is to use a linearization approach and study the local stability
101
CHAPTER 5. ABS BASED ON TYRE FORCE
100
ki = 1
ki = 5
80
ki = 10
Wheel acceleration
60
40
20
0
−20
−40
−60
0
0.1
0.2
0.3
0.4
slip λ [ ]
0.5
0.6
0.7
Figure 5.6: Limit cycle for different controller gains k.
of the system. Linearizing (5.1)-(5.7), together with the control law (5.22), around a
slip λ̃ gives the loop transfer function from reference acceleration to error:
Lλ̃ (s) = γk
τs + 1
−s∆t
2 e
Jτ s2 + Js + Fz f ′ (λ̃) rv
(5.37)
∂f
and k = 5,
Figure 5.7 shows the Nyquist plots of (5.37) for different values of ∂λ
τ = 0.01 m, v = 18 m/s, ∆t = 0.02 s, γ = 20 Nm/bar. It can be observed that
∂f
the stability margins are rather independent from the varying parameter ∂λ
. For
∂f
varying
between
-3000
and
10000,
like
on
Figure
5.7,
the
gain
margin
changes
by
∂λ
less than 4% around 1.32 while the phase margin takes values between 20 and 30.
The associated frequencies respectively lies in the intervals [75, 90] and [55, 70] rad/s.
Similarly, the stability margins are independent of v as it appears in the same term
as Fz f ′ (λ) in the transfer function. Therefore, it can be concluded that the largest
influence on the stability margin comes from the time delay.
Based on the gain margin of (5.37) for various time delays, the stability region is
defined and plotted on Figure 5.5. Although the linearization method is valid only
in the neighborhood of the linearization point, the difference between the stability
regions given by both methods is extremely small. Therefore it makes sense to use a
linearization approach, as it is much faster and easier to compute, and it allows for
a better stability analysis. In particular, from the sensitivity analysis of the stability
margins of (5.37), it can be concluded that the stability region is robust to changes
in tyre parameters and in forward velocity.
The influence of k on the shape of the limit cycle is visualized on Figure 5.6. If the
gain is low, the system will be quite slow and the slip excursion will be large, which
means low performances. If the gain is high, the system will become oscillatory and
eventually unstable.
As a conclusion, the best tuning is to take the largest controller gain k maintaining
the system stable. Given the system delay, this maximum gain can easily be computed
using the proposed linearization approach.
102
5.4. SINGLE-WHEEL VALIDATION
5.3.2
Tuning the acceleration levels
Two important parameters to tune in the algorithm are the reference acceleration
+
−
levels ω̇ref
and ω̇ref
. For the algorithm to work, the following conditions need to be
imposed:
λ̇ =
λ̇ =
+
−rω̇ref
+ rω v̇
v2
−
−rω̇ref + rω v̇
v2
< 0
(5.38)
> 0
(5.39)
Taking into account that 0 < rω < v, the previsous inequalities can be fullfilled using
the sufficient conditions
+
ω̇ref
−
ω̇ref
> 0
<
v̇min
r
(5.40)
(5.41)
where v̇min is the maximum deceleration achievable with a road vehicle, typically
1.2g.
From Theorem 5.1, we can recall that the larger the ω̇ref , the larger the limit
cycle in term of slip variation. This has a large influence on performance, as will be
shown using simulations. The limit cycles for different values of ω̇ref can be seen on
Figure 5.8.
The overall performance in term of braking distance can be assessed using the
simulation results presented in Figure 5.9. The case with the lowest reference levels
(ω̇ref = 10 m/s2 ) at the highest speed (v = 150 km/h) is taken as reference (100 %)
and the integral of the braking force, which is inversely proportional to the braking
+
−
distance, is normalized. The value of ω̇ref
is displayed on the x-axis while ω̇ref
is taken
symmetric with respect to the vehicle acceleration. As a conclusion, the reference
acceleration levels should be taken as small as possible. In practice, tyre oscillations,
measurement noise and wish for robustness will prevent us from taking arbitrarily
+
=
small acceleration levels. The reference levels taken for experimental validation ω̇ref
−
30, ω̇ref = −40 decreases the performance by less than 5 % compared to the reference
case with the lowest reference levels per speed.
5.4
5.4.1
Single-wheel validation
Simulation
The ABS algorithm is simulated using the model (5.1)-(5.7) where the relaxation effect
of the tyre as well as the delay in actuation are taken into account. The following
values are taken for the vehicle parameters: r = 0.3m , J = 1.2kgm2 , ∆t = 0.02s and
γ = 20N m/bar. The tyre curve f (λ) is taken to match the shape of the experimental
tyre curve (see Figures 5.11) with a peak force of 2800 N at a slip λ∗ = 0.1. The
+
−
tuning of the controller is as follows: ω̇ref
= 30, ω̇ref
= −40, k = 5, dFn+ = 10%,
dFn− = 7%.
103
CHAPTER 5. ABS BASED ON TYRE FORCE
4
∂f/∂λ = −3000
∂f/∂λ = −1000
∂f/∂λ = 2000
∂f/∂λ = 5000
∂f/∂λ = 10000
3
Imaginary Axis
2
1
0
−1
−2
−3
−4
−3
−2
−1
0
1
Real Axis
2
3
4
5
Figure 5.7: Nyquist plot of the linearized closed-loop system (5.1)-(5.7) for different
∂f
.
values of ∂λ
100
+
ω̇ref
= 10
80
+
ω̇ref
= 20
Wheel acceleration
60
+
ω̇ref
= 30
40
+
ω̇ref
= 40
20
+
ω̇ref
= 50
0
+
ω̇ref
= 60
−20
−40
−60
−80
−100
0
0.1
0.2
0.3
slip λ
0.4
0.5
0.6
0.7
Figure 5.8: Limit cycle for different values of ω̇ref
104
5.4. SINGLE-WHEEL VALIDATION
In the simulation presented on Figure 5.10, the single-wheel starts braking at 55
m/s and goes nearly until a full stop. The ABS cycles are clearly visible. After 5
seconds, the friction coefficient drops from 1 to 0.7; and 3 seconds after, the friction
comes back to normal. The algorithm has no difficulty at all to deal with such a
situation. Only the basic 2 phases are used and no extra logic is required. As the
speed decreases, the ABS cycles becomes shorter. Also the influence of the actuation
delay becomes relatively more important at low speed and therefore the force drop
becomes slightly larger. Note that the same controller parameters are used along the
whole simulation.
5.4.2
Experimental validation
The proposed algorithm has been tested on the tyre-in-the-loop experimental facility
of Section 1.3.2.
The tuning is exactly like in simulation. So far, tests are performed at the constant
speed of 18 m/s. In order to get a clean force signal, suitable for being used in
switching mechanism, the force measurement signal is filtered with a second order
filter with a cut-off frequency of 25 Hz. That frequency needs to be adapted with
wheel speed. Then, the switching levels dFn+ and dFn− can easily be determined in
order to contain the remaining noise level.
The results are shown on the left side of Figure 5.11. The algorithm works precisely
like expected from the simulation. The brake force remains within 16% from the
maximum of 3000 N.
5.4.3
Comparison with acceleration based hybrid algorithm
The deceleration-based 5-phase hybrid algorithm of Chapther 4 can serve as benchmark to compare this new ABS algorithm to. The method of Section 4.4.3 is used
to deal with the actuation delay, since it leads to the best performances compared
to the other methods of Chapter 4. Experimental tests of the 5-phase algorithm are
recalled on the right side of Figure 5.11. The tests are done in similar conditions as
for the force-based ABS: same test bench, same tyre, same loading and same speed.
From the measurements, and comparing with the left side of Figure 5.11, it can
be noticed that the force-based algorithm presents some advantages:
• The force-based ABS is better at maintaining a low slip. The maximum slip
during one cycle is λ = 0.3 for the force-based ABS while it is λ = 0.4 for the
5-phase ABS.
• The average brake force is larger in the case of the force-based algorithm, leading
to a shorter braking distance. In particular, the 5-phase ABS let the force drop
more (35%) in the stable zone than the force-based ABS (23%).
• In general, the force-based algorithm is able to detect force drops at an earlier
stage and trigger more precisely the switch between phases.
• In the force-based algorithm, only 3 constant parameters need to be tuned,
compared to 8 velocity-dependent parameters in the 5-phase ABS.
105
CHAPTER 5. ABS BASED ON TYRE FORCE
100
Normalized mean Fb [%]
90
80
70
60
v = 150 km/h
v = 100 km/h
v = 80 km/h
v = 50 km/h
v = 30 km/h
50
40
0
50
100
ω̇ref
150
Figure 5.9: The choice of large ω̇ref leads to a decrease in performance, especially at
lower speeds, and therefore to an increase of the braking distance.
55
2500
45
40
Brake force [N]
Vehicle and Wheel speed [m/s]
3000
Wheel speed
Vehicle speed
50
35
30
25
2000
1500
1000
20
15
500
10
5
0
2
4
6
Time [s]
8
10
12
0
0
2
4
6
Time [s]
8
10
12
40
30
Wheel acceleration [m/s2]
20
10
0
−10
−20
−30
−40
−50
−60
0.05
0.1
0.15
0.2
0.25
Slip λ []
0.3
0.35
0.4
Figure 5.10: Wheel and vehicle speed, brake force and state trajectory during ABS
regulation. Between the 5th and 8th second, the friction drops and the algorithm can
easily handle it.
106
5.4. SINGLE-WHEEL VALIDATION
3200
3200
F
F
b
b
Average F
Fb filtered
3000
b
3000
ABS state
Average Fb
ABS state
2800
F [N]
2600
2400
2400
2200
2200
2000
2000
4.9
5
5.1
Time [s]
5.2
5.3
6
5.4
80
60
60
40
40
20
0
−20
−40
−60
−80
0
6.4
6.6
6.8
7
20
0
−20
−40
−60
0.1
0.2
Slip λ []
0.3
0.4
−80
0
0.5
1.15
0.1
0.2
Slip λ []
0.3
0.4
0.5
1.15
1.1
1.05
1.05
Normalized brake force Fb/Fz []
1.1
1
0.95
0.9
0.85
0.8
0.75
Fb/Fz
0.7
0.65
0
6.2
Time [s]
80
Wheel acceleration [m/s2]
Wheel acceleration [m/s2]
4.8
Normalized brake force Fb/Fz []
2600
b
Fb [N]
2800
0.2
Slip λ []
0.3
0.4
0.9
0.85
0.8
0.75
Fb/Fz
0.7
Average Fb/Fz
0.1
1
0.95
0.5
0.65
0
Average Fb/Fz
0.1
0.2
Slip λ []
0.3
0.4
0.5
Figure 5.11: Experimental validation of the force-based ABS and comparison with
the acceleration-based 5-phase algorithm. LEFT: force-based ABS. RIGHT: 5-phase
ABS.
107
CHAPTER 5. ABS BASED ON TYRE FORCE
5.5
Simulation for the two-wheel vehicle
The ABS strategy is also simulated using the single-track two-wheel model (5.1)-(5.4)
and (5.8)-(5.18). This scenario is realistic for a real vehicle braking in straight line
as it also account for load transfer. The ABS controllers on each wheel are running
separately. Forces are assumed measured via load sensing bearings. As a consequence
in the algorithm implementation, equation 5.19 is modified using Fs (the vertical load
on the wheel hub) instead of Fz . Both the longitudinal and vertical forces are affected
by a measurement noise of 2%.
Because of the load repartition and the weight transfer, the load on the front tyre is
much larger than on the rear tyre. To maintain the same performances, it is necessary
to increase the brake efficiency at the front wheel, to make sure that the torque can
be changed fast enough. The front brake efficiency is taken γ = 40 Nm/bar, twice as
large as in the nominal case. Accordingly, the controller gain needs to be reduced to
guarantee that (5.37) remains stable. The front controller gain is taken k = 3.
Simulation results are shown on Figure 5.12. The vehicle starts braking at 55
m/s on dry asphalt with a friction coefficient of 1. Since the longitudinal force is
normalized by the load for ABS regulation, the algorithm has no problem to handle
weight transfer.
The thresholds dFn+ and dFn− used to detect force drops are large enought not to
be mistriggered by measurement noise, making the algorithm robust. The difference
between the vertical load Fz at the contact patch and Fs in the wheel hub remains
under the 3% and it is mainly concentrated around the unsprung mass resonance
(10 Hz); the difference is small enough to allow the use of one for the other in the
switching strategy without affecting the overall performance.
5.6
Conclusion
A two-phase ABS algorithm has been presented. Thanks to tyre force measurement, the acceleration can be controlled in closed-loop to give a tight cycle and the
switching strategy between the two phases is very precise even if simple. Stability
is proven for a simple model by finding bounds on the limit cycle. When the more
complex model with delays is considered, the stability results are extended with a
linearization approach to provide practical and quantitative tuning tools. The best
performances are achieved by using the two following tuning rules. First, the controller gain should be taken as large as possible while maintaining the stability in each
phase. A linearization approach is given in order to easily compute this maximum
gain for a given delay. Secondly, the reference wheel acceleration in each phase should
be taken as small as possible, with a lower limit depending on the oscillations in the
measurements. Furthermore, the algorithm is shown to be robust against changes in
friction characteristic, vehicle speed and wheel load, without asking for extra logic or
gain scheduling. An experimental tyre-in-the-loop test confirms that the algorithm is
working in practice, exactly like predicted by the simulation. The algorithm is shown
to perform better, while being easier to tune, compared to the 5-phase algorithm
based on wheel acceleration only, implemented in Chapter 4.
108
5.6. CONCLUSION
55
6000
45
5000
40
35
Force [N]
Vehicle and Wheel speeds [m/s]
7000
Vehicle
Front wheel
Rear wheel
50
30
4000
3000
25
f
b
r
b
f
F
z
r
F
z
F
2000
20
F
1000
15
10
0
2
4
6
8
0
0
10
2
4
Time [s]
6
8
10
Time [s]
2
Wheel accel [m/s ]
Front wheel
50
0
−50
−100
−30
−20
−10
0
10
0
10
50
2
Wheel accel [m/s ]
Rear wheel
0
−50
−30
−20
−10
Slip offset [%]
Figure 5.12: Simulation of the force-based ABS on a single-track two-wheel vehicle.
109
110
Chapter 6
ABS using Tyre Lateral Force
The main reason for implementing ABS, more important than maximizing the brake
force, is to maintain a proper lateral behaviour for the vehicle during heavy braking,
see Section 1.3. On one hand, the vehicle should be able to steer to avoid an obstacle,
and on the other hand, the vehicle should have a stable yaw dynamics and not spin on
itself. Because of the combined-slip effect, the lateral forces that the tyre is capable of
producing are largely reduced when the wheel is locking, which endanger the lateral
behaviour. By keeping the wheels unlocked, the ABS largely contributes to maintaining a lateral tyre potential, meaning the ability to generate lateral forces. Still,
in all implementations of ABS available today, this main objective is never addressed
directly.
For the class of hybrid controllers, see Section 1.3, the direct objective is to cycle
around the peak of the longitudinal tyre characteristics. This maximizes the brake
force and therefore minimizes the brake distance. On most surfaces, it can be expected that such a peak in the longitudinal characteristics will be present at a small
slip value λ ≈ 0.1, see Figure 6.1. By cycling around λ ≈ 0.1, it is expected that,
indirectly, a large enough tyre potential is maintained. However, there is no certainty.
Furthermore, it can happen in certain conditions that the longitudinal tyre-road friction characteristics does not display a peak, see Figure 6.1. This makes all hybrid
algorithms to be simply not working. In the commercial version of the Bosch algorithm, the hybrid controller is probably coupled with a slip regulator to deal with this
problem.
For the class of slip regulators, see Section 1.3, the direct objective is to maintain
the longitudinal slip λ at a given target λ∗ . Next to the precise measurement of λ,
the major difficulty resides in defining λ∗ . Tyre models including non-linearities and
combined-slip are clearly impossible to estimate online fast enough for ABS applications. Therefore, λ∗ is always defined using rules and heuristics, with the hope
that the chosen value will be a good compromise between lateral tyre potential and
brake force. However, there is no guarantee, and this is one of the reason why current
stability control systems have problems on road surfaces like snow.
The objective of this research is to extend the hybrid ABS algorithm developed
in Chapter 5, such that the objective of maintaining a good lateral behaviour is
considered directly. Requirements are set on the lateral behaviour of the front and
111
CHAPTER 6. ABS USING TYRE LATERAL FORCE
Tyre characteristics with and without peak
1
Normalized tyre force Fx / Fz [ ]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
with peak
without peak
0.2
0.4
0.6
Longitudinal slip λ [ ]
0.8
1
Figure 6.1: Tyre characteristics on different grounds, one with and one without a
peak at a small slip.
the rear axle, and the ABS algorithm makes sure that the brake force is maximized
while the lateral requirements are fulfilled. This is enabled thanks to the measurement
of the lateral tyre forces.
Many different requirements could be implemented for the front and the rear axle.
Sections 6.3 and 6.4 proposes 2 interesting illustrative options. As the tuning of the
feel of the vehicle is extremely important for the branding, the final tuning of the
algorithm is left for the car manufacturers. In both cases, a minimum lateral tyre
force is computed for each axle. If the measured tyre force is lower than the minimum,
actions are taken by the ABS.
Outline
The single-track vehicle model with nonlinear combined-slip tyres is described in
Section 6.1. Afterwards, a simpler linear single-track model is analyzed to extract
basic concepts. Section 6.2 focusses on the brake control algorithm. Finally, Sections
6.3 and 6.4 presents possible requirements for the front and rear axle, together with
simulations.
6.1
Modelling
So far, only a single-wheel model was considered for ABS. Here, a vehicle model and
a tyre model including the lateral dynamics are necessary. The vehicle is modelled
using a single-track model. This simple model is sufficient to develop the basics of
the controller. The tyres are modelled using the Magic Formula [95] that includes the
combined-slip effect.
112
6.1. MODELLING
a
b
vx
δ
fxf
fyf
r
vy
fxr
fyr
Figure 6.2: Scheme of the bicycle model.
6.1.1
Single-track model
Vehicles can be modelled with various levels of details. Large vehicle models with
lots of features are definitely better in describing how a real vehicle would behave.
However, for control or observation, it is definitely best practice to take reduced size
models with fewer parameters that will include the main dynamics of the studied
systems and lump many characteristics in a small set of general parameters.
In this research, we consider a single-track model (bicycle model) with 3 states: the
longitudinal speed vx , the lateral speed vy and the yaw rate r. The main longitudinal
and lateral dynamics are well represented. The tyre models are taken non-linear
and with combined-slip effects and will be described in the next section. Effects like
suspension kinematics or load transfer are not directly modelled but can be taken into
account by adapting the tyre parameter. The bicycle scheme is shown on figure 6.2.
The equations of movement of the vehicle are written in the vehicle frame. This
reference frame is attached to the centre of mass of the vehicle. The longitudinal axis
lies in the symmetry plane of the vehicle and points forward while the lateral axis is
directed to the left, perpendicular to the symmetry plane. Further, the tyre forces
are expressed in the tyre frame. This reference frame is attached to the pivot point
of the tyre. The longitudinal axis lies parallel to the free rolling direction of the tyre,
while the lateral axis is directed to the left, perpendicular to the longitudinal axis.
The vehicle states evolves according to the following differential equations:
v̇x − vy r
=
v̇y + vx r
=
ṙ
=
1
fxf cos(δ) − fyf sin(δ) + fxr
m
1
fyf cos(δ) + fxf sin(δ) + fyr
m
1
afyf cos(δ) + afxf sin(δ) − bfyr
Izz
(6.1)
(6.2)
(6.3)
where fij are the tyre longitudinal (i = x) or lateral (i = y) forces, at the front
(j = f ) or at the rear (j = r), m is the mass of the vehicle, Izz is the inertia around
the vertical axis and a and b are the distances between the centre of gravity and the
front and rear axle respectively. The values of the vehicle parameters are given in
Table 6.1.
Based on the vehicle states, the velocities of the front and rear tyres at the contact
113
CHAPTER 6. ABS USING TYRE LATERAL FORCE
m
Izz
a
b
J
r
1000 Kg
1000 Kgm
1m
1.5 m
1 kg m2
0.3 m
Table
Vehicle mass
Vehicle yaw inertia
Distance between CoG and front
Distance between CoG and rear
Wheel inertia
Wheel radius
6.1: Vehicle parameters
patch can be written:
vxf
= vx cos(δ) + (vy + ar) sin(δ)
(6.4)
vxr
= vx
(6.5)
αf
αr
vy f
vxf
vy + ar
= δ − arctan
= arctan
vx
vy r
vy − br
= arctan
= − arctan
vxr
vx
(6.6)
(6.7)
(6.8)
The lateral slip α is used to represent the lateral velocity, since it is used by the Magic
Formula.
6.1.2
Nonlinear Tyre Model
Tyres are extremely complex elements and in practice, a large amount of disturbing
effects can be observed, which make the tyre model to deviate from a simple static
friction curve. In this research, it is important to consider the longitudinal and
lateral directions and include the combined-slip effect. For the sake of simplicity, the
relaxation length and the vertical dynamics are omitted here. The tyre model is based
on the Magic Formula [95].
The longitudinal and rotational movement of the tyre is described by the following
equations:
J ω̇j
λj
= −rfxj − Tbj
vxj − rωj
rωj
=
=1−
vxj
vxj
(6.9)
(6.10)
where j represents either front (j = f ) or rear (j = r), ωj is the angular speed of the
wheel, vxj is the forward speed of the wheel in the tyre frame, λj is the longitudinal
slip and Tbj is the brake torque.The values of the parameters are given in Table 6.1.
The brake torque Tbj are the control inputs of the system.
In pure slip conditions, meaning that either λ or α should be equal to zero, the
longitudinal and lateral tyre forces are given by:
fx0j
= Dxj sin(Cxj arctan(Bxj λj − Exj (Bxj λj − arctan(Bxj λj ))))
(6.11)
fy0j
= Dyj sin(Cyj arctan(Byj αj − Eyj (Byj αj − arctan(Byj αj ))))
(6.12)
where
114
6.1. MODELLING
Symbol
Description
Front
Rear
Rear
without peak with peak without peak
Kx
Stiffness factor (long.)
20
20
20
Cx
Shape factor (long.)
0.9
1.4
0.9
Dx
Peak factor (long.)
1
1
1
Curvature factor (long.)
0
0
0
Ex
Ky
Stiffness factor (lat.)
15
25
25
Cy
Shape factor (lat.)
1.2
1.2
1.2
Dy
Peak factor (lat.)
1
1
1
Ey
Curvature factor (lat.)
0
0
0
combined-slip param
15
15
15
rx1
rx2
combined-slip param
15
15
15
ry1
combined-slip param
15
15
15
ry2
combined-slip param
15
15
15
Table 6.2: Tyre parameters for front and rear wheels, with or without peak in the
characteristics, see Figure 6.1.
• C is the shape factor
• D is the peak factor
• E is the curvature factor
• K = BCD is the stiffness factor
The values used for simulation are given in Table 6.2.
In case of combined-slip, when λ and α are both different from zero, there is an
interaction between the longitudinal and the lateral forces modelled by:
fxj
= Gxj fx0j
Gxj
=
(6.13)
cos(arctan(Bgxj (λj )αj ))
(6.14)
= rx1j cos(arctan(rx2j λj ))
(6.15)
fyj
= Gyj fy0j
(6.16)
Gyj
=
Bgxj (λj )
Bgyj (αj )
cos(arctan(Bgyj (αj )λj ))
(6.17)
= ry1j cos(arctan(ry2j αj ))
(6.18)
where fx0j and fy0j are the longitudinal and lateral forces in pure slip and rx1j , rx2j ,
ry1j , ry2j are parameters, for which values are given in Table 6.2.
The changes in tyre characteristics due to combined-slip can be visualized on
Figure 6.3. In particular, a large longitudinal slip λ will considerably reduce the
potential for generating lateral forces. Also, the effective stiffness is largely reduced,
which can lead to instability of the vehicle, as will be seen in next section.
6.1.3
Linear Analysis of the Single-Track Model
The nonlinearity of the vehicle and the tyre plays an important role for the analysis
of the ABS. Still, the analysis of a simplified linear vehicle model can help quickly
115
CHAPTER 6. ABS USING TYRE LATERAL FORCE
Longitudinal tyre curves (front tyre)
0.9
0.9
0.8
0.8
z
1
Normalized tyre force F / F
0.7
x
Normalized tyre force Fy / Fz
Lateral tyre curves (front tyre)
1
0.6
0.5
0.4
λ=0
λ = 0.1
λ = 0.3
λ = 0.5
λ = 0.7
λ=1
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
Lateral slip α
1
1.2
0.7
0.6
0.5
0.4
α=0
α = 0.1
α = 0.3
α = 0.5
α = 0.7
α=1
0.3
0.2
0.1
1.4
0
0
0.2
0.4
0.6
Longitudinal slip λ
0.8
1
Figure 6.3: Tyre characteristics with combined-slip, lateral and longitudinal.
understanding some underlying concepts. This analysis is based on [95].
The previous nonlinear model is simplified using the following assumptions:
• The longitudinal velocity vx is considered slowly varying and therefore taken as
a parameter instead of a state. The model order is reduced to 2.
• The steering angle δ is assumed to be small.
• The lateral slip angle α is assumed to be small.
• The tyre model in the lateral direction is taken linear: Fyj = Cyj αj where Cyj
is the cornering stiffness.
The equations of movement become
(6.19)
m(v̇y + vx r) = fyf + fyr
Izz ṙ
(6.20)
= afyf − bfyr
The slip angles are expressed by
αf
αr
vy + ar
vx
vy − br
= −
vx
(6.21)
= δ−
(6.22)
The state equations can be written in state-space form:
v̇y
ṙ

=
|
−Cyf −Cyr
mvx
−aCyf +bCyr
Izz vx
−Cyf a+bCyr
− vx
mvx
−a2 Cyf −b2 Cyr
Izz vx
{z


}
A(Cyr ,vx ,δ,Cyf )
116
vy
r
+
Cyf
m
aCyf
Izz
!
δ
(6.23)
6.2. ALGORITHM
Linear Steady-State Cornering Solutions
When the vehicle is cornering with a constant steer angle at a constant speed, the
following equations characterizes the steady-state solution.
The path curvature ( R1 ≈ vrx ) can be computed as function of the steering angle
Cyf Cyr l
1
δ
=
R
Cyf Cyr l2 − mvx2 (aCyf − bCyr )
(6.24)
Introducing the understeer gradient η
η=−
mg aCyf − bCyr
l
Cyf Cyr
where g is the acceleration due to gravity, we can write
l
vx2
δ=
1+η
R
gl
(6.25)
(6.26)
It is referred to [51, 95] for details about the understeer gradient. A neutral car (η = 0)
is often taken as a target for vehicle design. It can be noticed that, for a neutral car,
the steady-state path curvature (1/R) is directly proportional to the steering angle δ.
Furthermore, the total lateral force is proportional to the path curvature
Fy =
mvx2
R
(6.27)
Considering that, in steady-state, the front and rear forces need to balance each other
afyf = bfyr
(6.28)
b vx2
l R
(6.29)
we can write
fyf = m
Stability
The local stability of the locally linearized system depending on the value of the rear
cornering stiffness Cyr can be assessed looking at the eigenvalues of the system matrix
A(Cyr , vx , δ, Cyf ). The maximum real part of the eigenvalues, depending on the rear
cornering stiffness Cyr and on the longitudinal vehicle speed vx is shown on Figure
6.4. It can be observed that there exist a minimum cornering stiffness under which the
vehicle is unstable. The larger the vehicle speed, the larger the minimum cornering
stiffness need to be to ensure stability. This trend can be verified with extensive
simulations on the non-linear model.
6.2
Algorithm
The algorithm provides the necessary braking action on each wheel in order to obtain
the maximal brake force while ensuring that the minimum required lateral force is
117
CHAPTER 6. ABS USING TYRE LATERAL FORCE
10
0
Re(max(λ))
−10
vx= 5
v = 10
−20
x
vx= 15
vx= 20
−30
vx= 25
−40
−50
−60
0
2
4
6
8
Rear cornering stiffness Cyr []
10
4
x 10
Figure 6.4: Maximum real part of the poles of the simplified and linearised model
depending on the rear cornering stiffness Cyr and on the longitudinal vehicle speed
vx . The larger the vehicle speed, the larger the minimum cornering stiffness needs to
be to ensure stability.
achieved. The algorithm is similar on both the front and the rear tyre and is running
independently. Therefore references to a specific tyre are omitted in this section.
The ABS is a 3 phase hybrid algorithm extending the algorithm of Chapter 5. The
automaton is depicted in Figure 6.5. Phase 1 is responsible for having the longitudinal
slip to increase by providing a large enough brake torque. While in phase 1, two events
can happen with the tyre:
• The longitudinal force can drop rapidly, indicating that the slip is larger than
the slip providing the largest braking force.
• The lateral force can become smaller than the minimum required.
In both cases, it is necessary to decrease the longitudinal slip, either to increase the
brake force or to increase the lateral force. Phases 2 and 3 are responsible for that.
The different switching instants and phases are visualised on Figure 6.6.
If the longitudinal force drops of more than dFb under it’s maximum over time
during the current phase F̂b , phase 2 is triggered and the slip is decreased. During
each phase, F̂b at time instant k is computed as follows
F̂b (k) = max(F̂b (k − 1), Fb (k))
At the beginning of each phase, the value of F̂b is reset. This ensures that the maximal
brake force is regularly adapted and that the algorithm is robust to changes in the
tyre-road friction characteristic. The parameter dFb defines how large the force drop
needs to be in order to fire a phase switch. Defining the best value for this parameter
is not an easy task. A too large drop will lead to a conservative algorithm giving a
longer brake distance, while a too small value increases the sensitivity to noise. In
practice, this parameter is often about a few hundred newtons. When the longitudinal
118
6.3. REQUIREMENT ON THE FRONT AXLE
Tb > rFb −
Fb < F̂b − dFb
Jω v̇
v
1
Fy > Fymin + dFy
Fy < Fymin
Tb < rFb −
Jω v̇
v
2
Fb < F̂b − dFb
Tb < rFb −
Jω v̇
v
3
Figure 6.5: Hybrid Automaton of the 3-phase ABS algorithm. In phase 1 the slip
is increased and the wheel goes towards wheel lock. In phase 2 and 3, the brake
is released and the slip is decreased. A distinction is made between phase 2 and 3
depending on what reason triggered a slip decrease: drop in longitudinal force or too
low lateral force; so that the algorithm knows what to look at for triggering a new
slip increase.
force drops again, the maximum of the tyre curve has been passed again and phase 1
is triggered once more. This part of the algorithm is similar to Chapter 5.
If the measured lateral force Fy becomes smaller than Fymin , the longitudinal slip
has to be reduced and phase 3 is triggered. As soon as Fy becomes large enough,
phase 1 could be triggered again. In order to avoid chattering, the parameter dFy is
added in the switching law. The choice of this parameter is not critical as it cannot
make the controller to fail. It should be taken as small as possible, still maintaining
a low level of chattering even in the presence of noise.
During each phase, the brake torque need to be controlled in order to get the
longitudinal slip λ to evolve in the right direction. For this purpose, a method based
on closed-loop control of the wheel acceleration has been developed in Chapter 5. Such
method is suitable for this algorithm and could be applied. However, to reduce the
complexity, a simpler method is implemented here. The brake torque Tb is assumed
to be continuously controlled and, in this case, the slip derivative λ̇ can be made
positive or negative by setting the brake torque larger or smaller than rFb − Jωv v̇ .
6.3
Requirement on the front axle
The objective, regarding the front axle, is to maintain a desired level of steerability.
This means that the response to steering cannot be too low. Different ways to express
such objective might exist. In this research, the objective is expressed in term of
desired path curvature.
The path curvature (1/R) clearly defines the response of the car to steering action.
Furthermore, it is expected that the path curvature is an instinctive measure for the
driver. Therefore a minimum path curvature is required as a function of the steering
angle. For any given desired understeer gradient η, the minimum path curvature
119
CHAPTER 6. ABS USING TYRE LATERAL FORCE
F
F̂b
dFb
2
Fb
1
3
Fymin
dFy
Fy (α)
1
λ
Figure 6.6: Visualization of the hybrid automaton on the longitudinal tyre curve.
The lightnings represent the switching instants between the phases, while the arrows
shows the direction of evolution of the longitudinal slip. Fb and Fy are the tyre forces,
measured in real time by the force sensors. F̂b and Fymin are variable generated online
by the controller.
can be computed using (6.26). As an example, the case of a neutral car (η = 0) is
considered here and the relation is taken linear
1 = ϕδ
(6.30)
R
min
From (6.29), the required minimum lateral force on the front axle is
for
6.3.1
fyfmin = γvx2 δ
(6.31)
b
γ=m ϕ
l
(6.32)
Simulations
A simulation is performed to illustrate the functioning of the ABS. The results are
shown on Figure 6.7. The vehicle is modelled using the equations of Sections 6.1.1
and 6.1.2. The vehicle is started in straight line at 30 m/s. After 2 seconds, heavy
braking is performed on the front wheel only, with a requested braque torque of -2000
Nm. Because the front tyre characteristics does not present a peak, the wheel locks.
At the 3rd second, a steering is initiated such that, at time 3.5 s, the steering angle
is 0.1 rad. Since the wheel is locked, the lateral force is lower that the minimum
required to offer the desired path curvature R1 = γδl
mb = 0.0083. Therefore the ABS
intervene and reduces the brake torque. fy is increased while λ and fx are decreased.
Switching takes place to maintain fy around the target value. It can be observed that
the path curvature is just above the desired one.
Seemingly, the uncontrolled path curvature is largely depending on the longitudinal speed: the lower the speed and the larger the uncontrolled path curvature. Therefore, the control action required to maintain the desired path curvature is larger at
120
6.3. REQUIREMENT ON THE FRONT AXLE
Path curvature
Lateral force at the front wheel
0.03
2500
F
y
Minimum F
y
0.025
2000
1 / R [1/m]
y
Lateral force F [N]
0.02
1500
1000
500
0.015
0.01
0.005
0
0
0
1
2
3
4
5
6
2
7
3
4
5
6
7
t [s]
Time [s]
Longitudinal slip at the front wheel
Braking force at the front wheel
6000
1
Longitudinal braking force Fb [N]
5000
f
Longitudinal slip λ []
0.8
0.6
0.4
0.2
4000
3000
2000
1000
0
0
1
2
3
4
5
6
0
0
7
1
2
3
Time [s]
4
5
6
7
Time [s]
Vehicle velocities
Brake torque on the front wheel
vx
30
0
vy
−200
25
−400
Velocity [m/s]
−600
T [Nm]
−800
−1000
−1200
20
15
10
−1400
−1600
5
−1800
0
−2000
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Time [s]
t [s]
Figure 6.7: Simulation of heavy braking on the front wheel. When steering is initiated,
the wheel is unlocked so that the desired cornering radius is achieved.
121
CHAPTER 6. ABS USING TYRE LATERAL FORCE
Vehicle trajectory
20
18
Without control
With control γ = 50
With control γ = 100
Displacement Y [m]
16
14
12
10
8
6
4
2
0
80
90
100
110
120
Displacement X [m]
130
140
150
Figure 6.8: Comparison of the vehicle trajectory, with and without controller, for two
different controller tuning.
high speed. In the considered simulation, for a longitudinal speed lower than 10m/s,
the uncontrolled path curvature is larger than the minimum and the controller remains
inactive.
Figure 6.8 shows a comparison of the vehicle trajectory, with and without controller, and for two different controller tuning. More simulations are performed to
study the influence of the tuning parameter γ. The results are shown on Figure 6.9.
The left plot of Figure 6.9 shows the effective front tyre characteristics. The
dotted lines illustrated the tyre characteristics in combined-slip for fixed values of
λ, between 0 and 1. It can be observed that for low α, the lowest characteristics is
followed, corresponding to a locked wheel; while for larger α, the characteristics is
effectively given a larger slope by decreasing λ.
The right plot of Figure 6.9 shows the effective handling diagram for the vehicle.
The handling diagram plots the difference between the front and the rear axle against
the lateral acceleration. It is an interesting tool to evaluate the nonlinear steering
∂δ
behaviour of a vehicle. The local understeer/oversteer behaviour given by ∂a
can be
y
read from the diagram. Details about handling diagrams for a nonlinear vehicle model
can be found in [95]. It can be observed that initially, the system is understeer: from
(0, 0) the line goes to the left. But as soon as the controller gets active, the line goes
straight towards the top. This is characteristic of a locally neutral behaviour. Note
that the handling diagrams are originating from simulation data where the system is
not precisely in steady-state.
6.4
Requirement on the rear axle
When it comes to the rear axle, the objective is to maintain the yaw stability of the
vehicle to avoid spinning. As discussed in Section 6.1.3, the vehicle is stable as long
as the cornering stiffness at the rear axle is larger than a certain minimum. The effect
122
6.4. REQUIREMENT ON THE REAR AXLE
Handling diagram
Tyre characteristics for the front wheel
1
γ = 10
γ = 30
γ = 50
γ = 70
γ = 100
0.9
0.8
0.7
0.5
ay / g
0.6
Fy / Fz
γ = 10
γ = 30
γ = 50
γ = 70
γ = 100
0.6
0.5
0.4
0.3
0.4
0.2
0.3
0.2
0.1
0.1
0
0
0.05
0.1
α
0.15
0.2
0
−0.1
−0.08
−0.06
−0.04
αf − αr
−0.02
0
Figure 6.9: Effective front tyre characteristics and handling diagram for different
values of the tuning parameter γ. Heavy braking is applied on the front wheel only.
of wheel lock is to reduce the effective cornering stiffness, which is the reason for the
loss of stability. Therefore, the control objective is to maintain a minimum cornering
stiffness on the rear axle.
The minimum required lateral force on the rear axle is then
fyrmin = Cyrmin αr
(6.33)
Unfortunately, unlike for the front axle, the required lateral force cannot be implemented using only the measurement of the steering angle and an approximation of
the vehicle speed. Here the lateral speed, or equivalently the side-slip angle, need to
be estimated, which remains a delicate problem as discussed in Section 2.3. This constitutes a limitation regarding practical implementation, which should be addressed
in future work. For example, the use of the yaw rate could be considered, similarly
as in ESP systems. If an error on the minimum cornering stiffness maintained at the
rear axle is tolerated, which is reasonable to assume in a certain measure, then this
method can be made robust to a range of errors on the estimation of the side-slip.
6.4.1
Simulations
To illustrate the main principles, 2 simulations are performed. In both cases, the
manoeuvre is the same and similar to the ones of Section 6.3.1. The initial speed of
the vehicle is 15 m/s. After 3 seconds, a large brake torque reference of 1500 Nm
is applied to the rear wheel only. Such brake torque is capable of making the wheel
to lock. After 5 seconds, a steering angle of 6 degrees (0.1 rad) is applied on the
front wheel while the braking action on the rear wheel is maintained. The manoeuvre
stops when the vehicle speed has reached zero. The steering action is used here only
as a way to potentially induce yaw instability, and the precise steering behavior is not
looked at.
In the first simulation, the longitudinal tyre curve presents a maximum at a slip
λ = 0.25. The results can be see on Figure 6.10. As soon as the braking action is
triggered, the controller takes action to maintain the maximum braking force and
123
CHAPTER 6. ABS USING TYRE LATERAL FORCE
avoid wheel lock. As soon as the steering action is initiated, the controller realizes
that the slip excursion given by the control of the brake force is too large to maintain
the desired minimum cornering stiffness. Therefore, an oscillation between phases 1
and 3 will take place instead.
It can be observed that the oscillations between phases 1 and 3 are much faster than
between phases 1 and 2. This has to do with the tyre dynamics and the parameters
dFb and dFy . Fy is much more sensitive to changes in λ than Fb . Also, the parameter
dFy can be much smaller than dFb . It should be noted that in practice, because of
delays in the system and limited actuator and sensor bandwidth, the switching will
be slower.
In the second simulation, the longitudinal tyre curve do not present any maximum,
meaning that the maximal braking force is achieved at wheel lock. The results are
presented on Figure 6.11. As soon as the braking is initiated, the wheels are locked by
purpose to reach the largest braking force. At that time the vehicle is going straight
and therefore no lateral potential is required from the tyre. However, as soon as the
vehicle starts steering, the brake is released to come back to a more moderate slip
value. Lateral potential is regained and stability is ensured.
In both simulations, the vehicle would become unstable and start spinning on itself
if the controller would be disabled.
6.5
Conclusion
In this chapter, the hybrid force-based ABS of Chapter 5 has been extended to directly
consider the lateral tyre behaviour during heavy braking. Only one extra phase was
needed, together with the lateral tyre force measurement. The general control concept
is that if the measured lateral force is smaller than a desired minimum, the longitudinal
slip is reduced to regain lateral tyre potential.
The desired minimum lateral force is computed at each axle. For the front axle,
this is based on a desired path curvature. For the rear axle, this should guarantee
yaw stability, and could be based on a desired cornering stiffness. The investigation
of other ways of computing the minimal lateral force is left for future research.
Various simulations performed on a nonlinear single track model confirms that
the controller can maintain the desired steering behaviour and vehicle yaw stability
in case of heavy braking. In the simulations, heavy braking is applied only to either
the front or the rear axle, in order to clearly illustrate the different requirements
at each axle. Combining heavy braking on all wheels at once does not bring extra
complexity and no simulation is therefore shown here.
124
6.5. CONCLUSION
Brake force at the rear wheel
Lateral force at the rear wheel
1000
4000
F
y
Minimum F
3500
y
800
3000
600
F [N]
2000
y
b
F [N]
2500
400
1500
200
1000
0
500
0
0
1
2
3
4
Time [s]
5
6
7
−200
0
8
1
2
3
4
5
6
7
Time [s]
Controller state at the rear wheel
Longitudinal slip at the rear wheel
3
1
0.9
2.5
0.8
0.7
2
State
λ []
0.6
0.5
1.5
0.4
1
0.3
0.2
0.5
0.1
0
0
1
2
3
4
Time [s]
5
6
7
8
0
0
1
2
3
4
Time (s)
5
6
7
8
Vehicle velocities
16
vx
vy
14
Velocity [m/s]
12
10
8
6
4
2
0
0
1
2
3
4
Time [s]
5
6
7
8
Figure 6.10: Simulation with a longitudinal tyre curve presenting a maximum at
λ = 0.25. After the first 3 seconds, heavy braking is initiated at the rear axle and
the controller oscillates between phases 1 and 2 to maintain the largest braking force.
After 5 seconds, the front wheels are steered and a minimum cornering stiffness is
imposed by the controller to maintain lateral stability.
125
CHAPTER 6. ABS USING TYRE LATERAL FORCE
Brake force at the rear wheel
Lateral force at the rear wheel
3000
4000
F
y
3500
Minimum F
y
2500
3000
2000
F [N]
2000
1500
y
b
F [N]
2500
1000
1500
500
1000
500
0
0
0
1
2
3
4
Time [s]
5
6
7
−500
0
8
1
2
3
4
5
6
7
Time [s]
Controller state at the rear wheel
Longitudinal slip at the rear wheel
3
1
0.9
2.5
0.8
0.7
2
State
λ []
0.6
0.5
1.5
0.4
1
0.3
0.2
0.5
0.1
0
0
1
2
3
4
Time [s]
5
6
7
8
0
0
1
2
3
4
Time (s)
5
6
7
8
Vehicle velocities
16
vx
vy
14
Velocity [m/s]
12
10
8
6
4
2
0
0
1
2
3
4
Time [s]
5
6
7
8
Figure 6.11: Simulation with a longitudinal tyre curve where the maximum is achieved
at wheel lock. After the first 3 seconds, heavy braking is initiated at the rear axle
and the wheels are locked by purpose to reach the largest braking force. After 5
seconds, the front wheel are steered and a minimum cornering stiffness is imposed by
the controller to maintain lateral stability.
126
Chapter 7
Slip Control based on a
Cascaded Approach
In collaboration with CNRS/Supelec, Paris, France.
Controlling the tyre slip λ around a given target λ∗ is a popular problem, see
Section 1.3. Even if the implementation of slip regulators requires the measurement of
λ, which is still an open problem, see Section 2.3, they have properties complementary
to hybrid ABS, which make them potentially of practical interest.
The control algorithm considered in this chapter has been introduced in [97]. This
algorithm uses both wheel-slip and wheel acceleration measurement in a cascaded
approach. Compared to existing approaches, the main interest of this new control
algorithm are:
• it provides global exponential stability to the desired target (independently of
its location in the stable or unstable region of the tyre), by taking into account
the non-linear nature of the system;
• it gives precise bounds on the gains of the control law for which stability is
proved mathematically;
• it contains a feedforward term, which improves the bandwidth of the closed-loop
system.
The aim of this chapter is twofold. On one hand, the algorithm proposed in [97]
is validated experimentally on the tyre-in-the-loop experimental facility of Section
1.3.2. On the other hand, the stability conditions proposed in [97] are relaxed, using
LMI’s, in order to better match with the requirements on the controller gains experienced in practice. Moreover, the theoretical results of [97] are reformulated to more
conveniently fit with the control laws used in the experimental framework.
Outline
The model used for design and simulation is presented in Section 7.1. The control
design and the stability proofs are discussed in Section 7.2. Section 7.3 presents
127
CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
various simulations assessing performance and robustness. Finally, the tyre-in-theloop experimental validation is performed in Section 7.4.
7.1
System modelling
The design of the control law is based on the simple single-wheel model of Section 4.1.
Only the longitudinal dynamics of a single loaded wheel is considered. This model
is realistic enough to include the basic phenomena related to ABS, like the nonlinear
tyre characteristics and the wheel rotation dynamics, and is simple enough to allow
for analytical computations, like control design and stability proof. Actuation delay
and tyre relaxation are omitted in the design phase but simulations are used later on
to assess the robustness, see Section 7.3.
For the states x1 and x2 defined as
x1
= λ
x2
= Rω̇ − a∗x (t),
with a∗x is the constant vehicle deceleration as defined in (4.7), state equations similar
to (4.8) and (4.9) can be written
dx1
dt
dx2
dt
1
(−a∗x (t)x1 + x2 )
vx (t)
aµ′ (x1 )
u
= −
(−a∗x (t)x1 + x2 ) +
,
vx (t)
vx (t)
=
(7.1)
(7.2)
2
dT
where a = RI Fz and u = vx (t) R
I dt .
Observe that the derivative of the torque is considered as control input. Depending
the kind of technology used by the brake actuator, it might be necessary to integrate
the control input in order to have a brake torque reference.
7.2
Control design
The objective is to define a control law u that drives x1 towards a given timedependent wheel-slip target λ∗ (t). Several steps are taken to obtain u. Firstly, the
time-scale is normalized by the vehicle speed, such that the dependence on speed
disappears. This is enabled by using a filtering on the target λ∗ . Secondly, the
convergence of x1 is translated into a convergence of x2 using a dynamic set-point.
Two control laws are considered, one with feedback only and one with feedback and
feedforward. Closed-loop stability is assessed based on Lyapunov arguments.
7.2.1
An homogeneous target filter
The following second order filter is defined for the target λ∗ :
dλ1
dt
dλ2
dt
=
=
λ2
vx (t)
−γ1 (λ1 − λ∗ ) − γ2 λ2
.
vx (t)
128
7.2. CONTROL DESIGN
where γ1 and γ2 are tuning parameters.
The aim of this target filter is twofold. On one hand, it allows to have a smooth
target (that one can differentiate twice) even if the original target is discontinuous
(for example, piecewise constant). On the other hand, it allows to have a system for
which all equations are divided by the vehicle speed. This homogeneity allows us to
analyse the system in a new (nonlinear) time-scale in which the dependence on speed
disappears.
Like in traditional second-order filters, the parameters γ1 and γ2 defines the bandwidth and the damping of the system. It is clearly necessary to choose values giving a
stable filter, and desirable to have enough damping to avoid oscillations [45]. Furthermore, a large bandwidth means high performances, with a smooth target λ1 quickly
converging to the real target λ∗ . However, it can be expected that a large bandwidth
will ask for a large control input u.
7.2.2
A new time-scale
Since the system is homogeneous in the vehicle speed, a change of time-scale similar
as in [96] can be applied. Let
Z t
dτ
s(t) :=
v
(τ )
x
0
hence, dt = vx (t)ds and consequently, for any function ϕ : ℜ → ℜn we have
dϕ
dϕ dt
=
.
ds
dt ds
Therefore, defining ϕ̇(s) =
ẋ1
dϕ(s)
ds
we have
= −a∗x x1 + x2
′
ẋ2
= −aµ
λ̇1
= λ2
λ̇2
(x1 )(−a∗x x1
(7.3)
+ x2 ) + u
(7.4)
(7.5)
∗
= −γ1 (λ1 − λ ) − γ2 λ2 .
(7.6)
Note that, in this chapter, the dot notation refers to the derivative with respect
to the new time-scale.
7.2.3
Dynamic set-point
The error on the tracking of the filtered slip target is
z1 = x1 − x∗1
where x∗1 = λ1 . From (7.3), the error dynamics is
ż1 = −a∗x x1 + x2 − λ2
Let us first assume that x2 is a virtual control input. Then the control law
x2 = λ2 + a∗x x1 − αz1 + z2
129
CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
with α > 0 gives
ż1 = −αz1 + z2
which is exponentially stable if z2 = 0. Here z2 can be seen as the error on the wheel
acceleration tracking
z2 = x2 − x∗2
where
x∗2 = a∗x x1 + λ2 − αz1
Observe that while x∗1 is only based on λ1 (t), the set-point x∗2 is dynamic. The
steady state part is a∗x x1 , while the two other terms are used to decrease the error on
z1 , both using cascaded feedback −αz1 and feedforward λ2 . Thanks to this dynamic
set-point, the system converges to exactly the desired wheel slip irrespectively from
the tyre characteristic, unlike some other mixed slip-acceleration approaches, like for
example [107].
The dynamics of z2 is given by
ż2
= ẋ2 − ẋ∗2
= −aµ′ (λ)(−a∗x x1 + x2 ) + u − (a∗x ẋ1 + ẍ∗1 − αż1 )
= −aµ′ (λ)(−a∗x x1 + x2 ) + u − a∗x (−a∗x x1 + x2 ) − λ̇2 + α(−αz1 + z2 )
= −(aµ′ (λ) + a∗x )(−a∗x x1 + x2 ) − α2 z1 + αz2 − λ̇2 + u
[using −a∗x x1 + x2 = −αz1 + z2 + λ2 ]
= αz1 (aµ′ (λ) + a∗x − α) − z2 (aµ′ (λ) + a∗x − α) − (aµ′ (λ) + a∗x )λ2 − λ̇2 + u.
Hence, defining
η(λ) := aµ′ (λ) + a∗x − α
(7.7)
ψ(λ) := −aµ′ (λ) − a∗x
(7.8)
ż2 = η(λ) (αz1 − z2 ) + ψ(λ)λ2 − λ̇2 + u
(7.9)
and
we obtain
The purpose of the control law u is now to drive z2 to zero, which thanks to the
dynamic set-point, ensures that z1 converges to zero as well. Two control laws are
proposed, one with feedback only and one with feedback and feedforward.
The control scheme is illustrated on Figure 7.1. The system (S) of equations (7.1)
and (7.2) is controlled by both a feedback (FB) and a feedforward (FF). The dynamic
set-point x∗2 is computed based on the error z1 . The feedforward could make use of
x1 or z1 , but this is not required.
7.2.4
Control law with proportional feedback
The first proposed control law is a linear proportional feedback on the form
u = −k1 z1 − k2 z2
(7.10)
where k1 and k2 are two tuning parameters. This control law can be shown to stabilize
the closed-loop system for suitable values of the parameters α, k1 , k2 , γ1 , γ2 . Sufficient
conditions for stability are stated in the next proposition.
130
7.2. CONTROL DESIGN
FF
x1
λ1
λ2
λ∗
z1
dynamic
set-point
x∗2
z2
FB
u
S
x2
Figure 7.1: Control scheme of the cascaded wheel slip controller.
Proposition 7.1. For a constant target λ∗ , the time-varying closed-loop system obtained by introducing the control law
u = −k1 z1 − k2 z2
into the system defined by equations (7.3)-(7.6) is globally exponentially stable and
converges to (z1 = 0, z2 = 0) if
−αz1 + z2
1. the system ż = F1 (λ, z) with F1 =
, ob−(k1 − αη(λ))z1 − (k2 + η(λ))z2
tained when λ2 = λ̇2 = 0, is globally exponentially stable,
2. the gains γ1 and γ2 are positive.
′
M
3. there exist a number cM
µ such that |µ (λ)| < cµ
∀λ ∈ ℜ,
Proof. Define λ̃1 := λ1 − λ∗ . Then, with λ∗ constant, the closed-loop equations are
ż1
= −αz1 + z2
(7.11)
ż2
˙
λ̃1
= − (k1 − αη(λ)) z1 − (k2 + η(λ)) z2 + ψ(λ)λ2 − λ̇2
(7.12)
= λ2
(7.13)
λ̇2
= −γ1 λ̃1 − γ2 λ2
(7.14)
Defining z := [z1 ; z2 ] and Λ := [λ̃1 ; λ2 ] we see that the closed-loop system has a
so-called cascaded form
where
F1 (λ, z) :=
ż
= F1 (t, z) + G(λ)Λ
(7.15)
Λ̇
=
(7.16)
F2 (Λ)
−αz1 + z2
−(k1 − αη(λ))z1 − (k2 + η(λ))z2
0
1
F2 (Λ) :=
−γ1 −γ2
131
CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
G(λ) :=
0
γ1
0
ψ(λ) + γ2
According to results for cascaded systems [80] the system (7.15)-(7.16) is globally
exponentially stable at the origin if:
(a) the origin of ż = F1 (λ, z) is globally exponentially stable;
(b) the origin of Λ̇ = F2 (Λ) is globally exponentially stable;
(c) the solutions are globally bounded uniformly in the initial conditions.
Those three conditions are ensured by the hypotheses of the theorem. Firstly, (a)
is similar to the first condition of the theorem. Secondly, (b) evidently holds for any
positive values of γ1 , γ2 as the system is linear time-invariant and of second order,
which is ensured by the second condition of the theorem. Thirdly, (c) holds if G(λ(t))
is bounded uniformly for all t, see [80]. The latter holds, since, thanks to the third
condition of the theorem, ψ(λ) is bounded for all λ ∈ ℜ.
7.2.5
Stability of ż = F1 (λ, z)
It is left to show that the system ż = F1 (λ, z) is globally exponentially stable, and to
find conditions on the parameters α, k1 and k2 .
A first result is that, for any positive value of α and k1 , the system ż = F1 (λ, z)
can be made stable using a large enough gain k2 . This is shown using Lyapunov
arguments.
Consider a Lyapunov function V on the form
V (z1 , z2 ) =
1 2 1 2
ǫz + z
2 1 2 2
(7.17)
where ǫ > 0.
The total time derivative along the trajectories of ż = F1 (λ, z) yields
V̇
= ǫz1 ż1 + z2 ż2
= ǫz1 (−αz1 + z2 ) + z2 (−(k1 − αη(λ))z1 − (k2 + η(λ))z2 )
!
T
−ǫ+k1 −αη(λ)
z1
z1
ǫα
2
= −
−ǫ+k1 −αη(λ)
z2
z2
k2 + η(λ)
2
Using Sylvester’s criteria, the matrix is positive definite if and only if
ǫα
> 0
(−ǫ + k1 − αη(λ))2
ǫα(k2 + η(λ)) >
4
(7.18)
(7.19)
where the latter condition can be rewritten
k2 >
ǫ2 + k12 − 2ǫk1 + α2 η(λ)2 − 2α(ǫ + k1 )η(λ)
4ǫα
132
(7.20)
7.2. CONTROL DESIGN
Since µ′ (λ) is bounded, there exist two numbers η|M | and ηm such that
|η(λ)| < η|M |
∀λ ∈ ℜ
and
η(λ) > ηm
∀λ ∈ ℜ
Therefore, the condition
k2 >
2
ǫ2 + k12 − 2ǫk1 + α2 η|M
| − 2α(ǫ + k1 )ηm
4ǫα
(7.21)
robustly ensures that V̇ < 0 when (z1 , z2 ) 6= (0, 0), which proves global exponential
stability.
However, in practice, it can be observed that this bound on k2 is conservative,
since a Lyapunov function with few degrees of freedom was considered. A less conservative method for checking the stability of the system is to use a Linear Matrix
Inequality (LMI) feasibility test. The system ż = F1 (λ, z) can be considered linear
with parametric uncertainties. With ηm < η(λ) < ηM ∀λ, the vertices state matrices
can be written
−α
1
(7.22)
Am =
−(k1 − αηm ) −(k2 + ηm )
−α
1
(7.23)
AM =
−(k1 − αηM ) −(k2 + ηM )
The LMI feasibility test is to try to find a symmetric matrix X such that
X
ATm X + XAm
ATM X + XAM
>
0
(7.24)
< 0
(7.25)
< 0
(7.26)
T
If X can be found, then V = z Xz is a suitable Lyapunov function that guarantees
the robust stability of the system.
7.2.6
Control law with feedforward
The linear feedback controller has been shown to be able to ensure stability. However,
it may be expected that it yields a relatively poor transiant performance. This can
be observed in the simulations of Section 7.3. Furthermore, the stability proof is not
valid if the target slip λ∗ is time-varying. To solve those two problems, an extended
control law can be considered:
u = −k1 z1 − k2 z2 − ψ(λ)λ2 + λ̇2
(7.27)
The term −ψ(λ)λ2 is not strictly speaking a feedforward as it is depending on the state
variable λ. Still, as the main purpose is to feed changes on the target slip λ∗ via λ2 ,
the terminology feedforward is used. Furthermore, as ψ(λ) is mostly uncertain, it is
common in practical implementation to replace ψ(λ) by a constant, see Section 7.2.7.
In that case, the added terms in the control law are strictly speaking feedforward
terms.
The stability of the closed-loop system is addressed in the following proposition.
133
CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
Proposition 7.2. For any arbitrary piecewise continuous target λ∗ (t), the timevarying closed-loop system obtained by introducing the control law
u = −γ1 (λ1 − λ∗ ) + (−γ2 + a∗x + aµ′ (λ)) λ2 − k1 z1 − k2 z2
(7.28)
into the system defined by equations (7.3)-(7.6) is globally exponentially stable and
converges to (z1 = 0, z2 = 0) if
• the system ż = F1 (λ, z) is globally exponentially stable,
′
M
• there exist a number cM
µ such that |µ (λ)| < cµ
∀λ ∈ ℜ,
• the gains γ1 and γ2 are positive.
Proof. With the control law, the closed-loop system writes
ż1
= −αz1 + z2
(7.29)
ż2
= − (k1 − αη(λ)) z1 − (k2 + η(λ)) z2
(7.30)
λ̇1
= λ2
(7.31)
λ̇2
∗
= −γ1 (λ1 − λ ) − γ2 λ2
(7.32)
where the dynamics of (z1 , z2 ) and (λ1 , λ2 ) are decoupled. The stability of (z1 , z2 ) is
the first condition of the proposition, while the stability of (λ1 , λ2 ) is ensured when
γ1 and γ2 are positive.
7.2.7
Feedforward with tyre uncertainties
The implementation of the feedforward requires the knowledge of the tyre characteristics µ′ (λ), which is not the case during practical implementations. Still, it can be
shown that the feedback term can maintain the system stable even if the estimates
µ̂′ (λ) of µ′ (λ) used in the feedforward is not correct.
Proposition 7.3. For a constant target λ∗ , the time-varying closed-loop system obtained by introducing the control law
u = −γ1 (λ1 − λ∗ ) + (−γ2 + a∗x + aµ̂′ (λ)) λ2 − k1 z1 − k2 z2
(7.33)
into the system defined by equations (7.3)-(7.6) is globally exponentially stable and
converges to (z1 = 0, z2 = 0) if
• the system ż = F1 (λ, z) is globally exponentially stable,
• |µ′ (λ)| and |µ̂′ (λ)| are bounded for λ ∈ ℜ,
• the gains γ1 and γ2 are positive.
Proof. With the control law, and using λ̃1 like in Proposition 7.1, the closed-loop
system writes
ż1
= −αz1 + z2
(7.34)
ż2
˙
λ̃1
= − (k1 − αη(λ)) z1 − (k2 + η(λ)) z2 + a(µ̂′ (λ) − µ′ (λ))λ2
(7.35)
= λ2
(7.36)
λ̇2
= −γ1 λ̃1 − γ2 λ2
(7.37)
134
7.3. SIMULATION RESULTS
The closed-loop system has a cascaded form and the same arguments as in Proposition
7.1 can be used.
This result shows that the stability of the closed-loop system is robust with respect
to tyre parameter uncertainties. However, the performance can be expected to be
reduced if the estimate is not correct, as observed in the simulations of Section 7.3.
An adaptive scheme able to adapt λ̂′ (x1 ) is presented in [97]. This more advanced
scheme is more challenging to implement in practice and has therefore not been tested
yet.
7.3
Simulation results
Simulations are performed based on the simple model of Section 7.1.
The first aim of the simulations is to observe the effects of the feedback and feedforward terms. In accordance with both intuition and theoretical study, the following
phenomena can be observed :
1. When the feedback gains are equal to zero (that is, in the case of pure feedforward control), the system tracks the desired wheel slip reference only if this
reference is in the tyre’s stable zone ; otherwise the purely open-loop system is
unstable (Figure 7.4).
2. When the feedback gains satisfy the conditions given in the proof of Proposition
7.1 and the feedforward is not used (that is, in the case of pure feedback control),
the system tracks the desired wheel slip reference, but with a poorer performance
during transients compared to the case with feedforward (Figure 7.5).
3. When both feedback and feedforward terms are included, with perfect system
knowledge, the system follows exactly the filtered reference (Figure 7.6).
The second aim of the simulations is to observe the effects of perturbations, in order to evaluate the robustness of the control laws when both feedback and feedforward
terms are used. Three cases are considered:
1. When a pure actuation delay is introduced in the control loop (take, for example,
the case of a typical hydraulic actuator delay of 15 ms) the performance remains
good and the system remains stable, provided that the delay is not too big
(Figure 7.7).
2. When the system’s parameters used in the control law do not match those of
the true system (like, for exemple, a change of tyre characteristics) the system
remains stable, but the performance is reduced (Figure 7.8).
3. When both pure delay and parameter uncertainties are considered, the results
are quite close to the case of pure parameter uncertainties (Figure 7.9).
In order to prepare the experimental validation, the controller has also been simulated on the more complex model developed in Chapter 4. The results of the simulations are comparable to the experimental results of the next section, and are therefore
skipped here.
135
CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
60
0.25
Measurement
Reference
50
Brake pressure [bar]
0.2
Slip λ [ ]
0.15
0.1
40
30
20
0.05
10
0
0
5
10
15
Time [s]
20
25
30
0
0
5
10
15
Time [s]
20
25
30
Figure 7.2: Experimental validation of the controller, when only the feedback is enabled (no feedforward). The target and measured slips are shown on the left plot
while the brake pressure is shown on the right plot.
7.4
Experimental validation
The controller was implemented and tested on the tyre-in-the-loop experimental facility presented in Section 1.3.2. Two experiments have been performed on the setup,
with the drum rolling at a constant speed of 18 m/s. The controller is tuned as follows: α = 250, k1 = 250, k2 = 1000, γ1 = 1.6e6 and γ2 = 2.5e3. α, k1 , γ1 and γ2 were
defined experimentally in order to get the desired performances. k2 was computed in
order to ensure stability. The chosen value does not satisfy the conservative condition
(7.21), but still it passes the LMI test.
During the first test, the slip reference is increased or decreased by steps of 4%
from 0 up to 20% which is already in the unstable region of the tyre. The control
law (7.10) with feedback only is used. The results are presented on Figure 7.2. It can
be observed that the controller drives the slip precisely towards the reference value,
both in the stable and unstable region of the tyre. It is interesting to note that the
oscillations in the slip are larger when the slip is higher. This is linked to the fact that
the damping provided by the tyre is decreasing when the brake force is approaching
the saturation. The same phenomena leads to a decrease of the relaxation length at
high slip, as observed in [125].
During the second test, performances with and without feedforward are compared.
During the first 10 seconds, only the feedback is enabled by using the control law
(7.10). Then the feedforward is turned on by switching to the control law (7.33).
On Figure 7.3, it can be observed that the convergence to a new reference is much
faster with the feedforward on. This is particularly noticeable at low slip, when the
controller with feedback only is particularly slow.
136
7.5. CONCLUSION
60
0.14
Measurement
Reference
0.12
50
Brake pressure [bar]
0.1
Slip λ [ ]
0.08
0.06
0.04
0.02
40
30
20
0
10
−0.02
−0.04
0
Feedback only
5
Feedback + Feedforward
10
Time [s]
15
20
0
0
5
10
Time [s]
15
20
Figure 7.3: Comparison of the performances with and without feedforward. The
target and measured slips are shown on the left plot while the brake pressure is
shown on the right plot.
7.5
Conclusion
In this chapter, a new cascaded wheel slip control strategy based on wheel slip and
wheel acceleration measurements was presented. Both a feedback and a feedforward
are designed. The control laws are proven to stabilize globally and asymptotically the
wheel slip around any prescribed target, both in the stable and unstable regions of
the tyre, based on a simple model. Stability in case of tyre uncertainties is also shown
mathematically. Simulations support the theory and assess the performance and
robustness to time delays and model uncertainties. The control law is finally validated
in tyre-in-the-loop experiments. The first experimental test shows the convergence of
the slip towards the reference, also in the unstable zone. The second test concludes
that the feedforward is significantly increasing the bandwidth, in particular in the
stable zone.
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CHAPTER 7. SLIP CONTROL BASED ON A CASCADED APPROACH
0.05
0.05
Wheel slip reference
Filtered reference
Wheel slip
Wheel slip reference
Filtered reference
Wheel slip
]
0
]
0
1
1
2
2
0
1
2
3
Time [s]
4
5
6
0
Figure 7.4: Pure feedforward control
(k1 = 0 and k2 = 0).
1
2
3
Time [s]
4
5
6
Figure 7.5: Pure feedback control.
0.05
0.05
Wheel slip reference
Filtered reference
Wheel slip
Wheel slip reference
Filtered reference
Wheel slip
]
0
]
0
1
1
2
2
0
1
2
3
Time [s]
4
5
6
0
Figure 7.6: Combined feedback and
feedforward control (without perturbations).
1
2
3
Time [s]
4
5
6
Figure 7.7: Combined control, with a
delay of 15 ms.
0.05
0.05
Wheel slip reference
Filtered reference
Wheel slip
Wheel slip reference
Filtered reference
Wheel slip
]
0
]
0
1
1
2
2
0
1
2
3
Time [s]
4
5
6
0
Figure 7.8: Combined control with a
perturbation of µ(·).
1
2
3
Time [s]
4
5
6
Figure 7.9: Combined control, with a
delay of 15 ms and a perturbation of
µ(·).
138
Chapter 8
Conclusions and
Recommendations
In this thesis, steps have been made towards simpler and more robust Global Chassis
Control and Braking Control algorithms thanks to the use of force sensing, compared
to currently publicly available algorithms.
In the Global Chassis Control part, a new framework extending the class of indirect
allocation has been proposed. Thanks to force measurement, it becomes possible
to implement simple local tyre force controllers that are intrinsically robust to tyre
parameter uncertainty. This is exploited when dividing the complete problem in two
levels, such that the allocation becomes convex and independent from tyre parameters.
The new architecture with building blocks introduces the right number of degrees of
freedom in the allocation when actuators or wheels are coupled, thus avoiding the
appearance of difficult constraints.
From the analysis of the allocation problem, it can be concluded that there is a
need for a continuous optimization technique. The new Hybrid Descent Method has
been developed for this purpose. The force distribution is not optimized at every
time step, but instead it is driven with an hybrid differential equation, similarly to a
dynamical system. The main idea is to follow the steepest descent direction for the
objective function in the feasible set and for the constraints in the unfeasible set. The
continuous hybrid system guarantees that its trajectory enters the feasible set of the
related optimization problem and next converges asymptotically to the set of optimal
points.
Tyre constraints are handled in a realistic way, without the assumption that the
whole tyre characteristics is known long in advance. Only when the tyre reaches its
limits, a constraint, located at the currently measured operating point, is added in
the allocation. While the force distribution is adapted, it is necessary to maintain the
tyre at its maximal force. This necessity has driven us to further developments into
ABS braking control.
In the Braking Control part, contributions in the class of hybrid controllers based
on wheel deceleration and in the class of slip regulators have been made. Those are
the two main classes of algorithms found in the literature. Firstly, the theoretical
139
CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS
hybrid 5-phase algorithm has been extended to meet the requirements for practical
implementation. Secondly, a new wheel slip controller based on a cascaded approach
has been reworked to better fit the practical needs. This piece of work results from a
collaboration with the CNRS/Supelec, Paris, France.
Furthermore, the use of force measurement for braking control has been investigated and has lead to the development of a new algorithm. The simplest version
contains only 2 phases and is based on the measurement of the tyre longitudinal force
and wheel acceleration. The algorithm can track the peak of the tyre characteristics
without knowledge of the tyre parameters, and can handle abrupt changes in friction.
When one phase is added, together with the measurement of the lateral force, the
algorithm becomes able to control how much the lateral force is reduced because of
combined-slip. This is the first ABS algorithm that directly maintain the lateral tyre
potential during heavy braking. This should be the first objective when implementing ABS. By applying the algorithm to the front axle, a minimal path curvature is
guaranteed, while by applying the algorithm to the rear axle, lateral yaw stability is
maintained.
All algorithms for straight-line braking have been tested in tyre-in-the-loop experiments and have shown satisfying results. The update of the laboratory tyre
measurement equipment for closed-loop ABS testing is also an achievement of this
thesis.
8.1
Main conclusions
The overall conclusions of the research presented in this thesis are the following:
• Tyre force measurement allows to make the control of the tyres simpler and
more robust to uncertain characteristics. However, force measurement appears
to be less useful when it comes to estimating the motion states of the vehicle,
since this would require the inversion of the nonlinear uncertain tyre model.
Therefore, the wheel acceleration is still needed for ABS control, and the yawrate or the side-slip angle are still required for detecting lateral instability. It is
not expected that the addition of tyre force sensors in a vehicle will allow more
traditional motion sensors to be removed.
• When the right number of degrees of freedom is maintained, for example by using
the building block architecture, indirect allocation can be advantageous compared to direct allocation. The allocation can be made convex and independent
from uncertain tyre parameters, while the tyre uncertainties and nonlinearities
can be dealt with in specific blocks. The main difficulty is the expression of the
constraints. The use of roughly approximated constraints, constantly updated
depending the currently measured operating point, is shown to give good results
in a simulation study.
• It is possible to implement a dynamical system performing constrained continuous optimization. The Hybrid Descent Method is a suitable option for that.
Control allocation can largely benefit from such method, since the allocation
140
8.2. RECOMMENDATIONS AND DIRECTIONS FOR FUTURE RESEARCH
can be merged with the rest of the controller and the computational complexity
can be reduced.
• The delay in the braking control loop, originating for example from the brake
actuator, the measurement or the lag in the tyre response, is the main performance limiter for ABS. Any new braking controller should be proven to be
robust to a delay of 20ms to be of practical interest.
8.2
Recommendations and directions for future research
Based on the experience acquired during this Ph.D. research, recommendations and
possible directions can be proposed for future research in global chassis control and
braking control.
8.2.1
Global Chassis Control
• The Global Chassis Control framework has been developed focusing on two
main objectives: “Stability and Safety” and “Ease of development”. Future
research should extend the framework toward, in particular, two other important objectives: “Fault tolerance” and “Energy efficiency”. For fault tolerance,
the main challenge is the detection and isolation of actuator or sensor failure.
At a global level, it can be expected that actuator failures can be handled by
adding constraints and reconfiguring the allocation problem online. Furthermore, making the Global Chassis Control energy efficient is of first importance
today, where the necessity for limiting the world’s energy consumption is well
present. The energy cost for using the different actuators should be included
in the allocation cost function. Also, a good trade off should be made between
energy consumption and performance. Still, it is believed that the energy factor
should be considered if and only if the life of the occupant of the car is not
in danger. In critical situations, the only purpose of each actuator should be
avoiding an accident, and not saving or recovering energy.
• The local controllers inside the building blocks have been designed with the
objective to show that a simple force feedback can perform well. Only limited
tyre dynamics and no actuator dynamics were considered. This constitutes a
shortcoming for practical implementation. Future research should improve the
local controllers, for example using model-based design, to get a more practical
and realistic solution.
• This research has shown that roughly approximated constraints are already good
enough to give a first proof of concept of the method. Still, the Global Chassis Controller might benefit from a more precise expression of the constraints.
Future research could investigate the opportunities in this direction.
• When humans are involved in the control loop, the performances of a controller
cannot only be measured using traditional mathematical criteria, but the human
141
CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS
perception and acceptation should be investigated. In particular for vehicle
control, the driver’s feeling is a major importance. The first step for future
research is the design of a driver interpreter, which can give the total forces
and moment references to the allocation. This is probably complex enough
for a Ph.D. research on its own. The second step is to consider the controller
response, like for example the bandwidth, the delay or the overshoot, and see
if it is acceptable for the driver. The use of driving simulators should be the
first choice for such future research. Contributions to the development of a
moving-base driving simulator have been made in the framework of this thesis
[68]. Unfortunately, no tests could be performed during the time span of this
thesis.
• A major difficulty related to research about Global Chassis Control is the really vague definition of the problem. By looking for a general framework, one
might loose the capability of quickly applying it to a specific case. It is believed
that future research about GCC could benefit from having a predefined benchmark, where all the vehicle and actuator characteristics are known. The use
of the TNO Moving-Bases [99] for that purpose has been investigated during
this thesis, and it is still believed that this would constitute a perfect platform.
Unfortunately, tests on the Moving-Bases could not be performed during the
time span of this thesis.
8.2.2
Braking Control
• The brake actuator considered in this research is an Electro-Hydraulic Brake
(EHB). Future research could consider other types of brakes, and for example
brake-by-wire systems. It might be of interest to modify the algorithms if the
brake torque can be controlled precisely. With EHB, the brake efficiency is
uncertain and the pressure rate is limited, therefore only torque increase or decrease commands are given, preferably in feedback with the wheel acceleration.
If the brake torque can be controlled precisely, open-loop commands based on
the measured force is expected to increase the performances.
• With the trend towards hybrid and electric vehicles, it can be expected that
an electromotor will be connected to each wheel, in parallel to the brake. During normal driving, the motor can be used for regenerative braking. However,
when the ABS is triggered, the motor could be used to improve the braking
performances. Electromotors usually have a much larger bandwidth, a shorter
actuation delay and a better accuracy than friction brakes. As time delay is
the main performance limiter for ABS, the improvements brought by such motor can be substantial. However, two important issues needs to be considered.
Firstly, the total desired braking torque can usually not be provided by the
motor alone. Secondly, the motor inertia cannot be neglected compared to the
wheel inertia. Future research should consider the opportunity to apply the
high-frequency component of the torque reference signal to the electromotor,
and the low-frequency component to the friction brake. As a second step, ABS
algorithms should be modified according to the previous point, as an electro142
8.2. RECOMMENDATIONS AND DIRECTIONS FOR FUTURE RESEARCH
motor can precisely control its torque. Finally, the algorithms should be made
fault-tolerant to deal with potential failure of the actuators.
• The consideration of the suspension dynamics during ABS design might be
important. Unfortunately, this could not be investigated during the time span
of this thesis. Future research should look at this issue in more details. It
is believed that the tyre-in-the-loop experimental facility could be a nice test
environment for this, provided that the rigid frame is replaced by a conventional
suspension.
• The ABS algorithm using the lateral force measurement can be considered as a
first step towards the implementation of force-based ESP, in the specific case of
heavy braking. The basic principle for maintaining a good lateral behavior is
to ensure that the lateral force is not reduced too much because of combinedslip. Future research should investigate the benefits that tyre force measurement
could bring into ESP. Furthermore, when untripped roll-over avoidance is considered [110], the inverse principle could be used. There, one objective is to
reduce the tyre lateral force using heavy braking. Detection of roll-over can
be done based on the vertical force measurement. Future research could therefore also investigate the benefits that tyre force measurement could bring to
untripped roll-over mitigation. Finally, future research should strive at integrating the different subsystems for addressing, with a force-based approach,
the longitudinal, yaw and roll instabilities.
143
144
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SUMMARY
154
Summary
Global Chassis Control and Braking Control
using Tyre Forces Measurement
Mobility and traffic safety is a major concern in the society today. Many accidents
take place because the vehicle is not following the trajectory that the driver desires.
To avoid such accidents, an increasing number of active safety systems are introduced
in modern vehicles. Still, most of the time, those systems are designed independently,
which can lead to non-optimal performances and unexpected interactions. To properly
consider the coupling between the different actuators like the brakes and steering,
coupling coming for example from the nonlinear tyre dynamics, a Global Chassis
Control strategy is necessary. This thesis defines a new 2-level framework for Global
Chassis Control.
At the chassis level, the distribution of the total control action to all the actuators
is formulated as a constrained convex optimization problem, independent from uncertain parameters. As the driver is constantly changing his inputs to the vehicle, the
solution of the optimization should be constantly updated. For that purpose, a new
continuous optimization method, the Hybrid Descent Method, has been developed.
This method defines a dynamical system such that its trajectory converges, first to
a solution allowed by the constraints, and then to the optimum of the related optimization problem. In order to introduce the right number of degrees of freedom in
the optimization, the actuators are grouped into building blocks depending on their
coupling with the different tyre forces. Tyre constraints are handled in a realistic way,
without the common assumption that the whole tyre characteristic in known long in
advance.
At the wheel level, the control of the tyre forces is made simple and robust to
uncertainties in the road conditions thanks to the use of tyre force measurement.
Such tyre forces measurement is at the moment too expensive to be implemented.
In current production vehicles, only sensors measuring motion are available, like accelerometers or gyroscopes. At the moment, SKF is developing a technology enabling
the measurement of the load in their bearings. This technology allows the introduction of cheap force sensors in all vehicles, which will allow the implementation of new
control algorithms like those presented in this thesis.
A particularly critical problem in vehicle dynamics control is to get the largest
possible braking force out of the tyres while maintaining vehicle stability. This the155
SUMMARY
sis contributes to the development and experimental validation of Anti-lock Braking
Systems (ABS). As a result of a collaboration with CNRS/Supelec, Paris, France,
a theoretical hybrid ABS based on wheel deceleration has been improved to handle
actuation delay; and a new wheel slip regulator based on a cascaded approach has
been implemented. Furthermore, new algorithms using force measurement are shown
to be simpler to tune, better performing and more robust to road conditions than
acceleration-based alternatives. The first ABS algorithm directly considering the loss
of potential to generate lateral tyre forces because of braking is an achievement of
this thesis. All algorithms for straight-line braking have been validated on the tyrein-the-loop experimental facility, which was made suitable for ABS testing during this
research.
156
Samenvatting
Global Chassis Control en Braking Control
met gebruik van bandkracht meting
Mobiliteit en verkeersveiligheid zijn van essentieel belang in de hedendaagse samenleving. Veel ongelukken vinden plaats omdat het voertuig niet het door de bestuurder
gewenste traject volgt. Om dit soort ongelukken te voorkomen, is er een toenemend
aantal actieve veiligheid systemen in moderne voertuigen geı̈ntroduceerd. Toch zijn
deze systemen in het algemeen onafhankelijk ontwikkeld, wat kan leiden tot nietoptimaal prestatie en onverwachte interactie. Om de koppeling tussen de verschillende actuatoren, bijvoorbeeld veroorzaakt door niet-lineaire banddynamica, goed
mee te nemen in het ontwerp, is een Global Chassis Control strategie noodzakelijk.
Deze strategie zoekt een globaal optimum voor het regelen van het hele voertuig. Dit
proefschrift definieert een nieuw 2-laags kader voor Global Chassis Control.
Op het niveau van het voertuig, wordt de distributie van de totale regelactie over
alle actuatoren geformuleerd als een convex optimalisatie probleem onder beperkende
voorwaarden, onafhankelijk van onzekere parameters. Omdat de bestuurder voortdurend zijn input naar het voertuig verandert, moet de oplossing van de optimalisatie
doorlopend bijgewerkt worden. Daarvoor is, een nieuwe continue optimalisatie methode, de Hybrid Descent Methode, ontworpen. Die methode definieert een dynamisch
systeem zodat zijn traject eerst convergeert naar een, binnen beperkende voorwaarden
toegestane oplossing en vervolgens naar het optimum van het gerelateerde optimalisatie probleem. Om het juiste aantal vrijheidsgraden te introduceren in de optimalisatie, zijn de actuatoren gegroepeerd tot bouwstenen, afhankelijk van hun koppeling
met de bandkrachten. Beperkende voorwaarden ontleend aan het band-wegdek contact, zijn behandeld op een meer realistische manier: zonder de gebruikelijke aanname
dat de hele bandkarakteristiek vooraf bekend is.
Op het niveau van de wiel is het regelen van de bandkrachten robuust gemaakt
tegen onzekerheden in het band-wegdek contact dankzij het gebruik van bandkracht
meting. In huidige productievoertuigen zijn alleen sensoren die de beweging meten,
zoals versnellingmeter of gyroscoop beschikbaar. Het bepalen van de bandkrachten is
echter waardevol. Op dit moment is SKF een technologie aan het ontwikkelen waardoor het meten van de belasting van de wiellagers mogelijk wordt. Deze technologie
maakt meten van de bandkrachten in productievoertuigen wellicht ook economisch
haalbaar.
157
SAMENVATTING
Een bijzonder uitdagend probleem in het regelen van voertuigdynamica is het
halen van de hoogst mogelijke remkracht uit de banden terwijl voertuigstabiliteit
wordt behouden. Dit proefschrift draagt bij aan de ontwikkeling en experimentele
validatie van een Anti- blokkeer Rem Systeem (ABS). Als een resultaat van een
samenwerking met CNRS/Supelec, Parijs, werd een theoretisch hybride ABS algoritme, gebaseerd op wielversnelling verbeterd; en een nieuwe slip regelaar gebaseerd
op de cascade benadering werd geı̈mplementeerd. Verder wordt getoond dat nieuwe
algoritmes die gebruik maken van bandkrachtmeting simpeler, beter en robuuster
tegen onzekere wegdek conditie kunnen zijn, in vergelijking van versnellinggebaseerde
alternatieven. Het eerste ABS algoritme dat rechtstreeks het verlies van potentieel
om laterale krachten te genereren meeweegt in de remregelaar is een resultaat van dit
proefschrift. Alle algoritmes voor rechtdoor remmen zijn gevalideerd op de “tyre-inthe-loop” experimentele faciliteit, die geschikt gemaakt is voor ABS testen tijden dit
promotieonderzoek.
158
Résumé
Contrôle Global de Châssis et Contrôle du Freinage
utilisant la mesure des forces du pneu
La mobilité et la sécurité routière sont des enjeux majeurs dans notre société
d’aujourd’hui. Beaucoup d’accidents ont lieu parce que le véhicule ne suit pas la
trajectoire désirée par le conducteur. Pour éviter ce genre d’accidents, un nombre
croissant de systèmes de sécurité active sont introduits dans les véhicules modernes.
Cependant, la plupart du temps, ces systèmes sont conçus indépendamment les uns
des autres, ce qui conduit à des performances sub-optimales et des interactions inattendues. Pour prendre en compte le couplage entre les différents actionneurs comme
les freins et la direction, couplage introduit par exemple par la dynamique non-linéaire
des pneus, une stratégie de Contrôle Global de Châssis est nécessaire. Cette thèse
définit un nouveau cadre à deux niveaux pour le Contrôle Global de Châssis.
Au niveau du châssis, la distribution de la commande totale vers les actionneurs est
formulée comme un problème d’optimisation convexe sous contraintes, indépendant
des paramètres incertains. Comme le conducteur change constamment sa commande
au véhicule, la solution de l’optimisation doit être adaptée en permanence. A cette fin,
une nouvelle méthode d’optimisation en continu, la Méthode de Descente Hybride, a
été développée. Cette méthode définit un système dynamique tel que sa trajectoire
converge d’abord vers une solution acceptable au regard des contraintes, et ensuite
vers l’optimum du problème d’optimisation connexe. Dans le but d’introduire le
bon nombre de degrés de liberté dans l’optimisation, les actionneurs sont groupés en
blocs en fonction de leur couplage avec les différentes forces du pneu. Les contraintes
liées au pneu sont gérées de manière réaliste, sans faire l’hypothèse classique que la
caractéristique complète du pneu est connue longtemps à l’avance.
Au niveau des roues, le contrôle des forces générées par le pneu est rendu plus
simple et plus robuste par rapport aux conditions de route grâce à l’utilisation de la
mesure des forces du pneu. Cette mesure des forces n’est pas encore possible en raison des coûts trop importants. Dans les véhicules actuellement en production, seuls
des capteurs mesurant le mouvement sont disponibles, comme des accéléromètres et
des gyroscopes. En ce moment, SKF développe une technologie rendant possible la
mesure de la charge appliquée sur les roulements à billes. Cette technologie permettra
l’introduction de capteurs de forces à faible coût dans tous les véhicules ce qui permettra la mise en oeuvre de nouveaux algorithmes comme ceux présentés dans cette
159
SUMMARY
thèse.
Un problème particulièrement important en contrôle de la dynamique du véhicule
est de retirer la plus grande force de freinage possible du pneu tout en maintenant la
stabilité latérale. Cette thèse contribue au développement et à la validation expérimentale
de systèmes de freinage anti-blocage (Anti-lock Braking System - ABS). A la suite
d’une collaboration avec le FNRS/Supelec, Paris, France, un ABS hybride théorique
basé sur la décélération des roues a été amélioré pour supporter un retard dans
l’actionnement du frein et un nouveau régulateur de taux de glissement des roues
basé sur une approche en cascade a été mis en oeuvre. De plus, il est montré que de
nouveaux algorithmes utilisant la mesure des forces du pneu peuvent être plus simples
à régler, meilleurs et plus robustes aux conditions de route par rapport aux alternatives basées sur l’accélération. Le premier algorithme ABS qui considère directement
la réduction du potentiel du pneu à produire des forces latérales à cause du freinage
est un accomplissement de cette thèse. Tous les algorithmes ayant trait au freinage
en ligne droite ont été validés sur le banc d’essais “tyre-in-the-loop” qui a été adapté
pendant cette thèse pour permettre des tests ABS.
160
Curriculum Vitae
Mathieu Gerard was born in Gosselies, Belgium, in 1983. He studied electrical engineering at the University of Liege, Belgium, and received his master degree Summa
Cum Laude in 2006. During his studies, he spent one year in the department of
Automatic Control of Lund University, Sweden, in the framework of the exchange
program Erasmus. His master thesis topic was tyre-road friction estimation and he
was advised by Prof.dr. Anders Rantzer and dr. Brad Schofield.
Between 2006 and 2010, Mathieu Gerard worked on his PhD project at Delft
Center for Systems and Control, Delft University of Technology, The Netherlands.
His PhD research was about global chassis control and braking control using tyre
forces measurement, and was performed under the supervision of Prof.dr.ir. Michel
Verhaegen. During his PhD project, Mathieu Gerard obtained the DISC certificate
for fulfilling the course program requirements of the Dutch Institute for Systems and
Control, and supervised a number of M.Sc. and B.Sc. students.
In 2009, Mathieu Gerard started developing mobile applications to bring conference programs into mobile phones, and co-founded Conference Compass together with
dr. Jelmer van Ast. He is now Managing Director at Conference Compass B.V.
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