On Electrohydraulic Pressure Control for Power Steering Applications Alessandro Dell’Amico

On Electrohydraulic Pressure Control for Power Steering Applications Alessandro Dell’Amico
Linköping Studies in Science and Technology
Dissertations No. 1739
On Electrohydraulic Pressure Control
for Power Steering Applications
Active Steering for Road Vehicles
Alessandro Dell’Amico
Division of Fluid and Mechatronic Systems
Department of Management and Engineering
Linköping University, SE–581 83 Linköping, Sweden
Linköping 2016
Copyright © Alessandro Dell’Amico, 2016
On Electrohydraulic Pressure Control for Power Steering Applications
Active Steering for Road Vehicles
ISBN 978-91-7685-838-7
ISSN 0345-7524
Distributed by:
Division of Fluid and Mechatronic Systems
Department of Management and Engineering
Linköping University
SE-581 83 Linköping, Sweden
Printed in Sweden by LiU-Tryck, Linköping 2016.
Per la mia famiglia, Barbara e Leonardo
Dreams are what happen when you’re
asleep. Achieving is what you do when
you’re at work.
”
Abstract
This thesis deals with the Electrohydraulic Power Steering system for road
vehicles, using electronic pressure control valves. With an ever increasing demand for safer vehicles and fewer traffic accidents, steering-related active safety
functions are becoming more common in modern vehicles. These are functions
that aim at assisting the driver in more or less critical situations and could be
anything from applying a guiding steering torque to taking over the steering
completely. Future road vehicles will also evolve towards autonomous vehicles,
with several safety, environmental and financial benefits. A key component
in realising such solutions is active steering. This refers to the possibility to
control the steering angle or steering torque with an electronic signal.
The power steering system was initially developed to ease the driver’s workload by assisting in turning the wheels. This is traditionally done through an
open-centre hydraulic system and heavy trucks must still rely on fluid power,
due to the heavy work forces. The system provides a robust solution but suffers
from poor energy efficiency and lacks the possibility of active control. Since the
purpose of the original system is to control the assistive pressure, one way would
be to use proportional pressure control valves. Since these are electronically
controlled, active steering is possible and with closed-centre, energy efficiency
can be significantly improved on.
In this work, such a system is analysed in detail with the purpose of investigating the possible use of the system for Boost curve control, the most common
control strategy in power steering systems, and position control for autonomous
driving. Commercially available valves are investigated since they provide an
attractive solution. A model-based approach is adopted, where simulation of
the system is an important tool. Detailed models at both a component level
and a system level are developed and form the basis of the analysis and control
design. Another important tool is hardware-in-the-loop simulation. A test rig
of the system, that also includes the original system, is developed. The rig
supports the modelling and validation process, as it also makes it possible to
verify control concepts on a real system.
This work has shown how proportional pressure control valves can be used
for Boost curve control and position control and what implications this has
i
on a system level. As it turns out, the valves add a great deal of phase shift
and with the high gain from the Boost curve, this creates a control challenge.
The problem can be handled by tuning the Boost gain, pressure response and
damping and has been effectively shown through simulation and experiments.
For position control, there is greater freedom to design the controller to fit the
system. The pressure response can be made fast enough for this case and the
phase shift is much less critical.
Keywords: Electrohydraulic power steering, pressure control, closed-centre
steering, Boost curve control, autonomous driving, hardware-in-the-loop
simulation, state feedback control, nonlinear control
ii
Populärvetenskaplig
sammanfattning
I denna avhandling studeras elektrohydraulisk servostyrning för fordon, där
eletroniska tryckstyrande ventiler används. Då kravet på säkrare fordon och
färre olyckor ständigt ökar blir aktiva säkerhetsfunktioner som använder sig
av servostyrningen allt vanligare. Dessa funktioner syftar till att assistera
föraren i mer eller mindre farliga situationer. Detta kan göras genom att ett
styrande moment guidar föraren i rätt riktning, till att styrningen tas över helt.
Framtida fordon kommer även att utvecklas mot autonoma fordon, vilket ger
flera säkerhetsrelaterade-, miljörelaterade- och ekonomiska fördelar. Nyckeln
ligger i att realisera aktiv styrning. Med detta menas att hjulvinkeln, eller det
styrande momentet, går att kontrollera via en elektronisk signal.
Servostyrning i fordon utvecklades från början med syfte att underlätta
förarens arbete, genom att assistera när föraren vrider på ratten. Traditionellt
har detta gjorts genom ett hydrauliskt, s.k. öppet-centrum, system och lastbilar måste än idag förlita sig på denna teknik då lasterna är väldigt stora. Denna
lösning är robust men har dålig verkningsgrad och kan inte styras aktivt. Då
syftet med original systemet är att styra tryck kan detta istället göras med proportionella tryckstyrande ventiler. Genom att de är elektroniskt styrda och har
stängt centrum kan aktiv styrning realiseras, samtidigt som bränslebesparingar
är möjliga.
I detta arbete analyseras detta system i detalj. Undersökningarna tittar på
reglering via Boost kurva, vilket är det mest förekommande sättet i dagens
system, samt positionsreglering för autonom körning. Möjligheten att använda
kommersiellt tillgängliga ventiler studeras då detta skulle vara en attraktiv lösning. Arbetet med att undersöka systemet är modellbaserat där simulering är
ett viktigt verktyg. Flera matematiska modeller har utvecklats på både komponentnivå och systemnivå och ligger till grund för reglerdesign och de analyser
som gjorts. Ett annat viktigt verktyg är ”hardware-in-the-loop” simulering,
som används som en del i valideringsprocessen och verifieringsprocessen av
koncept.
Detta arbete visar på hur man kan använda tryckstyrande ventiler som ser-
iii
vostyrning och hur dessa påverkar systemet. Ventilernas beteende skiljer sig
från ursprungssystemet och ger en del utmaningar när Boost kurvan används
som reglerstrategi. Dessa kan hanteras genom rätt inställning av prestanda,
dämpning och förstärkning i reglerloopen. Detta har visats genom simulering
och experiment. För positionsreglering finns större frihet att utveckla regulatorn för att uppnå dom krav som finns och ventilens prestanda är inte lika
kritisk.
iv
Acknowledgements
The work presented in this thesis has been carried out at the Division of Fluid
and Mechatronic Systems (Flumes) at Linköping University. It was partially
financed by Scania AB.
I am very greatful for having the opportunity to be part of a research project
and to be part of Flumes. I am very greatful to my supervisor and Head of
the Division, Professor Petter Krus. Thank you for all support and for always
having time for a discussion, no matter the subject or issue. A big thank you
goes to Dr. Jochen Pohl. Your input has always been very valuable and I
am very greatful for all interesting conversations we had during these years. I
also wish to thank our former Head of Division, Professor Jan-Ove Palmberg.
Thank you for letting me be a part of this Division and for always showing an
interest in my work.
I want to thank all my present and former colleagues at the division. Thank
you for making this a great place to work at. A special thank goes to Mikael
Axin and Karl Petterson. We have many beatiful memories from all over the
world and I am very greatful for that. I also want to thank the technical staff at
the faculty, Ulf Bengtsson, Per Johansson and Peter Karlsson, for their support
with all the testing and for letting me escape to the workshop when I needed to.
A special thank you goes to Ian Hutchinson. Thank you for always delivering
on a very tight schedule.
My greatest gratitude goes to my wife Barbara and my son Leonardo. Barbara, thank you for all your support and patience during these years. Leonardo,
thank you for giving me the greatest joy in life.
v
vi
Notation
αf
Front slip angle
[rad]
αr
Rear slip angle
[rad]
β
Bulkmodulus
[Pa]
δ
Jet stream angle
[rad]
δa
Desired closed loop damping
[-]
δm
Mechanical damping
[-]
λ
Flux linkage
µ
Permeability of the material
[H/m]
ωa
Desired closed loop response
[rad/s]
ωc
Resonance of control chamber
[rad/s]
ωh
Hydraulic resonance
[rad/s]
ωm
Mechanical resonance
[rad/s]
ωs
System volume resonance
[rad/s]
ωv
Valve break frequency
[rad/s]
Ωz
Yaw rate
[rad/s]
Φ
Damping function
φ
Magnetic flux
φp
Damping function derivative
[sPa/m3 ]
φq
Damping function derivative
[-]
θsw
Steering wheel angle
[rad]
A1
Opening area of open-centre valve
[m2 ]
[Wb-turns]
[Pa]
[Wb]
vii
A2
Opening area of open-centre valve
[m2 ]
Ac
Pressure-sensing area
[m2 ]
Ap
Piston area
[m2 ]
As
Valve opening to pump
[m2 ]
At
Valve opening to tank
[m2 ]
Acc
Large pressure-sensing area
[m2 ]
Acr
Area of damping orifice
[m2 ]
Agap
Gap area
[m2 ]
Ain
Meter-in area
[m2 ]
Aout
Meter-out area
[m2 ]
B
Magnetic field
[T]
B
Viscous friction coefficient
[Ns/m]
bv
Viscous friction coefficient
[Ns/m]
brw
Damping coefficient of the rack
[Ns/m]
bsw
Damping coefficient of steering wheel and column
[Ns/m]
Cq
Discharge coefficient
Crw
Equivalent tyre stiffness
Cαf
Front tyre cornering stiffness
[N/rad]
Cαr
Rear tyre cornering stiffness
[N/rad]
f0
Spring pretension
f1
Resistive current function coefficient
[A/W3b ]
f2
Resistive current function coefficient
[A/W2b ]
f3
Resistive current function coefficient
[A/Wb ]
Ff
Flow force
[N]
Fs
Solenoid force
[N]
Ffin
Flow force, meter-in
[N]
Ffout
Flow force, meter-out
[N]
Ffrw
Friction at the rack
[N]
Ffstat
Static friction level
[N]
viii
[-]
[N/m]
[N]
Fload
Load force
[N]
Fyf
Front lateral force
[N]
Fyr
Rear lateral force
[N]
H
Magnetic field intensity
i
current
[A]
id
Dissipative current
[A]
ir
Resistive current
[A]
Iencl
Enclosed current by magnetic path
[A]
Jrw
Total road wheel inertia
[kgm2 ]
Jsw
Inertia of steering wheel and column
[kgm2 ]
K
Spring stiffness
[N/m]
K1
Pressure-flow coefficient of damping orifice
[m3 /sPa]
K1
Pressure-flow coefficient of damping orifice
[m3 /sPa]
Kb
Boost gain
[Pa/Nm]
Kc
Flow-pressure coefficient
[m3 /sPa]
Ke
Equivalent spring stiffness
[N/m]
Kh
Hydraulic stiffness
[N/m]
Ki
Integral gain
Kp
Proportional gain
[V/Pa]
Kq
Flow gain
[m2 /s]
KT
Spring stiffness of torsion bar and column
[Nm/rad]
Kv
Static loop gain
[m2 Pa/N]
L
Inductance
[H]
lf
Front axle distance
[m]
lr
Rear axle distance
[m]
m
Mass
[kq]
mv
Mass of spool
[kg]
Mfsw
Friction at the column
Mrw
Total mass of rack
[A/m]
[V/sPa]
[Nm]
[kg]
ix
N
Number of turns
p
Pressure
[Pa]
pc
Pressure of damping chamber
[Pa]
pL
Load pressure
[Pa]
ps
System pressure
[Pa]
pBoost
Boost pressure
[Pa]
pd1
Dissipative current function coefficient
[A/V3 ]
pd2
Dissipative current function coefficient
[A/V2 ]
pd3
Dissipative current function coefficient
[A/V]
pd4
Dissipative current function coefficient
[A]
pfs1
Force function coefficient
[N/V5 ]
pfs2
Force function coefficient
[N/V4 ]
pfs3
Force function coefficient
[N/V3 ]
pfs4
Force function coefficient
[N/V2 ]
pref
Reference pressure
q
Volumetric flow
[m3 /s]
qc
Flow through damping orifice
[m3 /s]
ql
Load flow
[m3 /s]
qs
Pump flow
[m3 /s]
qs
System flow
[m3 /s]
qs
System flow
[m3 /s]
qv
Valve flow
[m3 /s]
qin
Meter-in flow
[m3 /s]
qout
Meter-out flow
[m3 /s]
R
Coil resistance
[Ω]
Re
Reynolds number
RT
Steering system gear ratio
[rad/m]
Rv
Static characteristic of the valve
[Ns/m5 ]
t
Time
x
[-]
[Pa]
[-]
[s]
Td
Driver’s torque
[Nm]
Ttb
Torsion bar torque
[Nm]
U
Control signal
[V]
V
Voltage
[V]
V1
Cylinder volume
[m3 ]
V2
Cylinder volume
[m3 ]
Vc
Damping volume
[m3 ]
VL
Inductive voltage
[V]
Vl
Load volume
[m3 ]
Vs
System volume
[m3 ]
Vs
Total voltage
vv
Spool velocity
[m/s]
vx
Longitudinal velocity
[m/s]
vy
Lateral velocity
[m/s]
w
Area gradient
[m]
xv
Spool position
[m]
xrw
Rack position
[m]
[V]
xi
xii
Papers
This thesis is presented as a monograph. Parts of the work are based on the
following publications:
Journal articles
Dell’Amico, Alessandro and Petter Krus (2015a). “Modeling, simulation and
experimental investigation of an electrohydraulic closed-centre power steering system”. In: IEEE/ASME Transactions on Mechatronics 20.5, pp. 2452–
2462.
Dell’Amico, Alessandro and Petter Krus (2015b). “Modelling and experimental
validation of a nonlinear proportional solenoid pressure control valve”. In:
accepted for publication in the International Journal of Fluid Power.
Conference articles
Dell’Amico, Alessandro and Petter Krus (2013). “A Test rig for Hydraulic
Power Steering Concept Evaluation using Hardware-in-the-loop Simulation”. In: The 8th International Conference on Fluid Power Transmission
and Control (ICFP13). Hangzhou, China.
Dell’Amico, Alessandro and Petter Krus (2014). “Closed-centre Hydraulic
power steering by direct pressure control”. In: the 9th JFPS International
Symposium on Fluid Power. Matsue, Japan.
Dell’Amico, Alessandro, Jochen Pohl, and Petter Krus (2010). “Modeling and
Simulation for Requirement Generation of Heavy Vehicles Steering Gears”.
In: Fluid Power and Motion Control (FPMC 2010). Bath, UK.
Dell’Amico, Alessandro, Jochen Pohl, and Petter Krus (2012). “Conceptual
evaluation of closed-centre steering gears in road vehicles”. In: The 7th FPNI
PhD Symposium on Fluid Power.
Licentiate thesis
Dell’Amico, Alessandro (2013). Pressure Control in Hydraulic Power Steering
Systems.
xiii
Additional articles
Following articles do not directly relate to the work in this thesis but constitute
an important part of the process.
Dell’Amico, Alessandro, Marcus Carlsson, et al. (2013). “Investigation of a
digital actuation system on an excavator arm”. In: 13th Scandinavian International Conference on Fluid Power. Linköping, Sweden.
Dell’Amico, Alessandro, Liselott Ericson, et al. (2015). “Modelling and experimental verification of a secondary controlled six-wheel pendulum arm
forwarder”. In: the 13th European Conference of ISTVS. Rome, Italy.
Dell’Amico, Alessandro, Magnus Sethson, and Jan-Ove Palmberg (2009).
“Modeling, Simulation and Experimental Verification of a Solenoid Pressure Control Valve”. In: The 11th Scandinavian International Conference
on Fluid Power (SICFP09). Linköping, Sweden.
Ericson, Liselott, Alessandro Dell’Amico, and Petter Krus (2015). “Modelling
of a secondary controlled six-wheel pendulum arm forwarder”. In: the 14th
Scandinavian International Conference on Fluid Power. Tampere, Finland.
xiv
Contents
1 Introduction
1.1 Aims and research questions
1.2 Method . . . . . . . . . . .
1.3 Delimitations . . . . . . . .
1.4 Contributions . . . . . . . .
1.5 Outline . . . . . . . . . . .
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3 Modelling of the hydraulic power steering system
3.1 Nonlinear model . . . . . . . . . . . . . . . . . . . .
3.1.1 Mechanical submodel . . . . . . . . . . . . .
3.1.2 Hydraulic submodel . . . . . . . . . . . . . .
3.1.3 Load model . . . . . . . . . . . . . . . . . . .
3.2 Measurements and parameter identification . . . . .
3.3 Validation of nonlinear model . . . . . . . . . . . . .
3.4 Linear modelling and analysis . . . . . . . . . . . . .
3.4.1 Hydraulic system analysis . . . . . . . . . . .
3.4.2 HPAS system analysis . . . . . . . . . . . . .
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
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4 Active steering
4.1 Active safety and increased comfort . . .
4.2 Realising active steering . . . . . . . . .
4.2.1 Superposition of steering angle .
4.2.2 Superposition of steering torque
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2 Description of the steering system
2.1 Design and functionality . . . . . . . . . .
2.2 Assistance . . . . . . . . . . . . . . . . . .
2.3 Hydraulic system . . . . . . . . . . . . . .
2.4 Steering wheel torque . . . . . . . . . . .
2.5 Energy aspects and alternative systems .
2.6 Steering feel and steering system influence
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on vehicle
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4.3
Selected technology for active steering . . . . . . . . . . . . . .
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5 Pressure control
5.1 General modelling and analysis . . . . . . . . . . . . . . . . . .
5.1.1 Nonlinear model of the hydro-mechanical subsystem . .
5.1.2 Static characteristic . . . . . . . . . . . . . . . . . . . .
5.1.3 Linear model and analysis . . . . . . . . . . . . . . . . .
5.2 Modelling and analysis of a commercial proportional solenoid
pressure control valve . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Measurements and modelling of the solenoid . . . . . .
5.2.2 Measurements and modelling of the valve . . . . . . . .
5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Electrohydraulic closed-centre steering by pressure control
6.1 System description . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Stability analysis of the electrohydraulic CC-system by using
servo valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 System modelling . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 System analysis . . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Controlling the pressure . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Linear state feedback controller . . . . . . . . . . . . . .
6.3.2 Nonlinear state feedback controller . . . . . . . . . . . .
6.3.3 Discussion on controller strategy . . . . . . . . . . . . .
6.4 Boost curve control with pressure control valves . . . . . . . . .
6.4.1 System equations . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Updating the controller . . . . . . . . . . . . . . . . . .
6.4.3 Simulation model and results . . . . . . . . . . . . . . .
6.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Position control for autonomous steering . . . . . . . . . . . . .
6.5.1 System equations . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Results from simulation . . . . . . . . . . . . . . . . . .
6.5.4 Analysis and discussion . . . . . . . . . . . . . . . . . .
6.6 Heavy truck steering system application . . . . . . . . . . . . .
6.6.1 System modelling . . . . . . . . . . . . . . . . . . . . . .
6.6.2 Response of open-centre system . . . . . . . . . . . . . .
6.6.3 Controller design . . . . . . . . . . . . . . . . . . . . . .
6.6.4 Simulation of complete system . . . . . . . . . . . . . .
6.6.5 Linear analysis . . . . . . . . . . . . . . . . . . . . . . .
6.6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.7
Implementation and verification . . . . . . . . . . . . . . . . . .
6.7.1 Pressure controller . . . . . . . . . . . . . . . . . . . . .
7 Hardware-in-the-loop simulation
7.1 Servo valves . . . . . . . . . . . .
7.1.1 Hardware description . .
7.2 Pressure control valves . . . . . .
7.2.1 Hardware description . .
7.3 Force control . . . . . . . . . . .
7.3.1 Mathematical description
7.3.2 Controller design . . . . .
7.3.3 Results . . . . . . . . . .
7.3.4 Discussion . . . . . . . . .
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8 Discussion
169
9 Conclusions
173
A Parameter values
A.1 Parameters for HPAS system simulation . . . . . . . . . . .
A.2 Parameter values for commercial pressure control valve . . .
A.3 Parameters for steering system with servo valves simulation
A.4 Parameters for truck system simulation . . . . . . . . . . .
A.5 Parameters for force controller simulation model . . . . . .
Bibliography
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xvii
1
Introduction
With an ever increasing demand for safer vehicles and fewer traffic accidents,
there is a clear trend within the vehicle industry where steering-related active
safety functions are becoming more common. Active safety systems provide
assistance to the driver in more or less critical situations. The most common
ones on the market today are the Anti-lock Brake System, the Electronic Stability Program and Traction Control systems. Including the steering system in
increasing safety has several benefits (J. Ackermann, Bünte, and D. Odenthal
(1999)) and the system can intervene to avoid a collision with an oncoming
vehicle or keep the car on the road. Typical functions are Emergency Lane Assist (Eidehall et al. (2007)), collision avoidance systems (Schorn et al. (2006)),
roll-over prevention and vehicle stabilisation (Koehn and Eckrich (2004)). The
future will also see road vehicles evolve into autonomous vehicles, with several
safety, environmental and financial benefits. The key to realising steering related active safety or autonomous driving is active steering, referring to the
possibility to control the steer angle or torque with an electronic signal.
The power steering system is the driver’s mean to change the direction of the
vehicle. Power steering was developed to ease the driver’s workload in steering
by adding an assistive force to the driver’s applied force. Traditionally, this has
been done with a hydraulic system, Hydraulic Power Assisted Steering (HPAS),
consisting mainly of a fixed pump, an open-centre valve and a cylinder (Marcus
Rösth (2007)). The valve is actuated by the driver’s applied torque on the
steering wheel. The actuation of the valve results in a controlled pressure in
either side of the cylinder, generating the required force to assist the driver. The
charaterisation of the assistive pressure to the input of the driver is called the
Boost curve. This is a simple and robust solution. The drawback is the inherent
compromise between steering effort and road feel, the low energy efficiency and
the lack of the possibility for active steering.
The solution to the energy problem in the passenger car industry was the
introduction of the Electric Power Assisted Steering (EPAS) system. This has
become a very common solution, though HPAS systems are still found in the
1
On Electrohydraulic Pressure Control for Power Steering Applications
premium car and sport car segments. In addition to the energy savings, the
electric motor also allows the assistive torque to be actively controlled and
thereby opens up for implementation of active safety functions. Due to the
high axle load on heavy vehicles, EPAS systems are not possible with today’s
limitations in the electric system of the vehicle. A hydraulic power steering
system is therefore necessary in this segment. The most common solution to
active steering for trucks is a combination of an electric motor in parallel to the
hydraulic system. There are, however, several implications. The electric motor
introduces inertia, friction and backlash. The open-centre system remains with
its poor energy efficiency. A large electric motor can do the majority of the
work but increases cost, could be difficult to pack and the large inertia needs
to be handled. A smaller electric motor could be packed more easily but the
open-centre hydraulic system needs to do much of the work. The electric motor
can only work through the Boost curve of the hydraulic system, which could
be a limitation.
Since the purpose of the hydraulic system is to control the assistive pressure,
a possible solution would be to use proportional pressure control valves. These
are actuated by a solenoid, thereby providing an electronic interface and they
have closed-centre, which opens up for a reduction in energy consumption.
In this work, this system is referred to as Electrohydraulic Power Assisted
Steering (EHPAS) and should not be confused with the electric motor driven
pump system, which is also often referred to as EHPAS (Forbes, Baird, and
Weisgerber (1987)). An electrohydraulic solution takes advantage of the high
power density from fluid power systems, which gives a very compact solution.
The valves could completely replace the original system, giving only one Boost
curve, or work as an add-on system. The latter solution could be beneficial
with respect to safety issues.
Since the electrohydraulic pressure control valve with closed-centre behaves
in a completely different manner to the hydromechanical open-centre valve,
there is a need to understand how this affects the steering system performance
and how it can be controlled. The purpose is to keep the Boost curve control
strategy in order to maintain, as far as possible, the behaviour of the system.
It is also interesting to investigate the possible use of a commercially available
valve, since this would reduce development cost.
1.1
Aims and research questions
The main aim of this thesis is to investigate the possibility to use proportional
solenoid pressure control valves in power steering applications for realisation
of active steering. The valve behaves fundamentally differently to the original
valve and an understanding of the impact on the steering system performance is
therefore required. The intended application is Boost curve control and position
control. The latter is aimed at autonomous driving of vehicles, while the former
is the most common control strategy of today’s power steering systems. The
2
Introduction
process includes a deep understanding and analysis of the original system,
which works as a reference point. A deep understanding of the control valves
is also necessary, which leads to detail modelling and control design. Finally,
the closed-centre system is analysed for performance and stability to give an
idea of the usefulness of the valves in this application.
The aim of the thesis can be formulated into the following research questions,
which relate to the different chapters and sections in the thesis.
1) How can the valve be modelled and what are the important phenomena
that need to be considered in order to capture the essential behaviour of
the valve?
2) How can the pressure control valve be controlled to increase and tune the
performance and what control srategies are possible?
3) What are the main challenges with Boost curve control when using electronically controlled closed-centre valves and how can they be handled?
4) Under what circumstances does the pressure control valve represent a
feasible solution to Boost curve control of the power steering system and
how do they relate to the original system?
5) In what way can the pressure control valves be used for autonomous
driving?
6) Why should a truck steering system be considered as a potential candidate
for the solution with pressure control valves?
Question 1) is treated in chapter 5, while the remaining questions are treated
in chapter 6.
1.2
Method
The working approach is model-based, resulting in a framework where the original and new systems can be analysed in order to draw conclusions. The main
idea of the work flow can be illustrated by figure 1.1. Initial experiments are
conducted on the original system in order to derive model parameters and validation data. The original system is modelled in order to provide a reference
system and is validated against measurements. This results in an analysis in
the time domain and the frequency domain. Both are valuable tools. Parts
of the model are then reused for the modelling of the electrohydraulic system.
Derived models are used for control design, which is a requirement of the system architecture. The system is simulated and the models are also used for
analysis of the system, from which conclusions can be drawn regarding its behaviour. Final experiments are conducted to verify conclusions from simulation
and analysis. The process is an iterative process where several steps could be
undertaken several times.
3
On Electrohydraulic Pressure Control for Power Steering Applications
Experiments
Modelling
Original system
Analysis:
Validation
time domain
frequency domain
Modelling
EHPAS
system
Control
Design
Simulation
Time domain
Frequency domain
Figure 1.1
1.3
Analysis
Conclusions
Experiments
Verification
Validation
Work flow illustrating the methodological approach.
Delimitations
The work in this thesis is bounded to study the steering system alone and
the impact of the closed-centre system on power steering system performance.
The approach is rather a comparison between the closed-centre system and
the original system and the impact of vehicle dynamics is therefore excluded.
Since the real reference system is related to the test rig, no vehicle is therefore
present. A load is, however, necessary and an arbitrary load model, such as a
vehicle model or a tyre hysteresis model, can be used. The same load model is
used for both systems in order to equate the results from both systems.
Although the technology studied in this work is most interesting and perhaps suitable for heavy vehicles, the application has been a steering system
for a passenger car. As will be discussed, there is no fundamental difference
between how the steering systems work, especially with regard to the hydraulic
system. The main difference lies in the pressure level, the flow rate level and the
mechanical linkage. The reason why a passenger car steering system has been
4
Introduction
studied as the application is related only to the available hardware during the
project. The test rig, which is explained in this work, is from a passenger car.
In order to facilitate the comparison of simulation results to measurements,
the test rig consitutes the application in this work. However, the results are
applicable to any steering system with the same working principle, whether it
is for a passenger car or a truck.
Energy aspects of the steering system are not considered. It is well known
that the traditional hydraulic power steering system is not primarly designed
to be energy efficient. It is part of the fundamental technology and several
researchers have investigated this, as well as alternative layouts of the system
in order to improve energy efficiency. Whether or not it is an issue depends on
the application, but in the end it must be answered by the vehicle manufacturer.
It is therefore assumed in this work that the chosen technology offers significant
improvements in fuel consumption but will be treated as a very positive side
effect. The primary target thus remains realisation of active steering.
Safety aspects are not considered in this work. Of course, the new system
brings additional safety-critical aspects compared to the original system. For
instance, the system relies on active control of the assistive pressure, based on
feedback control loops. At any point the system must not apply unexpected
values and must handle erronuous sensor data and power loss. This is left for
future work and discussion. However, the derived simulation and hardware-inthe-loop simulation environments provide a good platform for such an investigation.
The most common control strategy in power steering systems is Boost curve
control. It is an inherent solution of the original system’s hardware layout.
Even though other control concepts are possible, the approach in this work
is to keep this strategy in order to minimise the modification of the original
system.
1.4
Contributions
The contributions in this work are the following:
− A model-based framework that analyses the closed-centre power steering
system in both the time domain and the frequency domain.
− A model and analysis of a commercial solenoid pressure control valve
is derived, with the level of detail sufficient to describe its particular
behaviour. The model includes the behaviour of the electromagnetic
characteristic of the solenoid, the detailed behaviour of the pressure-flow
characteristic and varying flow induced forces.
− A control structure of the pressure control valve that increases performance and handles disturbances.
5
On Electrohydraulic Pressure Control for Power Steering Applications
− An analysis of the control challenges for Boost curve controlled power
steering with electrohydraulic valves and possible solutions. Since the
work is focused on Boost curve control, the control strategy is already
defined and constitutes a limitation.
− Position control of the steering rack with pressure control valves for realisation of autonomous driving.
− An analysis of the difference in performance and control conditions between a passenger car steering system and a truck steering system.
− A hardware-in-the-loop test rig for simulation and verification of the
electrohydraulic closed-centre power steering system.
1.5
Outline
Chapter 2 gives a description of the original steering system and its purpose
and functionality, both on a system level and a component level. This is to
give an understanding of the application. This is followed by mathematical
modelling and analysis of the original system in chapter 3. The analysis is
important since it gives an understanding of the system’s behaviour. The
models are used for comparison with the closed-centre system and parts of
the model are reused when modelling the closed-centre system. Chapter 4
gives an overview of existing solutions in realising active steering and discusses
the selected solution in this work. Chapter 5 covers the detailed modelling
and analysis of the commercial pressure control valve and contains a general
discussion about pressure control. In chapter 6 the analysis of the pressure
control valve with the application is outlined. This covers control design, Boost
curve control analysis and position control. The test rig that was developed and
used in this work is described in section 7. A brief discussion of the work can be
found in chapter 8 while detailed discussions on specific topics are given in the
relevant chapters. Finally, the conclusions, related to the research questions,
are outlined in chapter 9.
6
2
Description of the
steering system
The steering system of a vehicle serves two main purposes: to provide a means
to control the direction of the vehicle and to provide a feedback information
channel to the driver about the road conditions. Driving a vehicle can be seen
as a closed loop system, which is well described by Marcus Rösth (2007). The
intention of the driver is to follow a predefined route. This is done by controlling
the steering wheel angle. Feedback channels are a visual channel, the driver sees
where the vehicle is headed, the lateral acceleration from the vehicle and the
steering wheel torque from the steering system. There are several requirements
the steering system should fulfill, as is explained by Pfeffer (2006). These
are related to the response of the vehicle to steering inputs, returnability of
the steering wheel, torque build-up at on-centre, handling limits should be
perceptible and during parking a low steering wheel torque and small steering
angles are desired. With high axle loads, large tyres and front wheel drive, an
assistance system is required to meet the demands for low steering wheel torque
and comfort. Traditionally, this is done with hydraulic power, but in recent
years the most common technology is electric power steering for passenger cars.
For heavy vehicles and trucks, the axle loads are significantly higher and with
today’s electric system limitations, hydraulic power is still in use.
This chapter gives an overview of the power steering system, both the mechanical system and the hydraulic system. A comparison between the passenger
car and heavy vehicles is presented. Energy aspects are discussed, as well as
steering feel. This chapter gives the background to the system modelled in
chapter 3.
7
On Electrohydraulic Pressure Control for Power Steering Applications
(a) Rack and pinion system.
Figure 2.1
2.1
(b) Steering box system.
Illustrations of steering system arrangements.
Design and functionality
There are basically two types of steering system arrangement: the rack and
pinion and the worm gear steering box. Both arrangements are illustrated
in figure 2.1. Even though both types can be found in both passenger cars
and heavy vehicles, it is most common for the rack and pinion to be used in
passenger cars and the worm gear steering box in heavy vehicles. The main
benefits with the rack and pinion is the reduced complexity: it is much lighter,
easier to accomodate in front wheel drive vehicles and has a better efficiency,
which gives the driver a more accurate feeling of the tyre-ground interaction, see
e.g. Gillespie (1992) and Genta and Morello (2009). The worm gear steering
box allows for a greater gear ratio, which is needed with the heavy loads in
trucks. The basic working principle of the two arrangements is the same. The
difference lies in the mechanical linkage, which is more complicated for the
worm gear steering box, with more moving parts that generate friction.
Figure 2.2 shows a photograph that compares the steering gears of two arrangements. Note the difference in size of the assistance piston. The rack and
pinion system converts a rotational motion into a translational motion of the
rack. The steering wheel is attached to the column, which connects to the
pinion via a torsion bar. The torsion bar is the main control element of the
assistance system. The rack is supported by a sliding block, called a yoke. The
yoke is pushed by a preloaded spring in order to reduce backlash between the
pinion and the rack. It also adds friction, which is controlled by the preload.
The most common worm gear is the recirculating ball gear, see e.g. Stone
and Ball (2004). A nut is meshed onto a threaded worm. Inside the nut there
are helical channels where balls rotate and move. The channels with balls are
closed, which makes the balls recirculate. The worm rotates with the rotation
8
Description of the steering system
Pressure area
Recirculating ball screw
Rack and pinion
Figure 2.2
Photograph comparing the rack and pinion system and the recirculating ball screw steering box.
of the steering column, which makes the nut move up and down. The nut is
meshed with a sector that rotates with the movement of the nut. The pitman
arm is in turn attached to the sector at one end and to the mechanical linkage
at the other. The purpose of the channels with balls is to reduce friction and
backlash.
2.2
Assistance
In order to ease the driver’s workload during steering manoeuvres the steering
system is equipped with an assistance system. The assistance system is in
parallel to the mechanical system. This ensures the haptic feedback to the
driver. There is a compromise, however, since with the assistance system there
is also a loss of information to the driver. The assistance system is charatierised
by the assistance curve, also known as the Boost curve. Two examples are
shown in figure 2.3, where figure 2.3a could represent a passenger car and
figure 2.3b a heavy vehicle. The Boost curve shows how much assistance is
applied depending on the torque applied by the driver. The shape of the Boost
curve is a consequence of the requirements of the steering system, that is, low
effort to steer during parking and good feel during high speed driving. Two
regions are marked in figure 2.3a: on-centre driving and parking. On-centre
driving refers to for example to highway driving with small steering inputs.
Here the need for good feedback to the driver is high. The steering torque is
9
On Electrohydraulic Pressure Control for Power Steering Applications
150
100
80
Parking
100
40
Assistance pressure [bar]
Assistance pressure [bar]
60
20
0
−20
On-centre
−40
−60
50
0
−50
−100
−80
−100
−4
−3
−2
−1
0
Torque [Nm]
1
2
3
(a) Passenger car.
4
−150
−10
−8
−6
−4
−2
0
2
Torque [Nm]
4
6
8
10
(b) Heavy vehicle.
Figure 2.3
Typical assistance curves of a passenger car to (left) and a heavy
vehicle (right). Regions for on-centre driving and parking are also marked in the
figure.
low, as well as the assistance. It is during parking that the loads are highest.
The need for information feedback to the driver is less critical and the steering
should be comfortable. Therefore, the assistance is high in this scenario. From
here onwards, the Boost curve will be shown as the absolute value, which is
common.
The traditional assistance system is a hydraulic system. In this case, the
Boost curve represents the pressure applied by the hydraulic system. In case
of an electric power steering system, the Boost curve represents the torque
applied by the electric motor.
2.3
Hydraulic system
The hydraulic system can be treated equal for both the rack and pinion system
and the recirculating ball gear system. Both systems are illustrated in figures
2.4 and 2.5, respectively. The main components of the system are a constant
flow pump, rotational valve and cylinder. The pump is usually a fixed displacement vane pump driven by the combustion engine. It has an integrated
pressure relief valve and a flow control valve. The rotational valve is controlled
by the torsion bar. The spool is attached to the upper end of the torsion bar.
The lower part of the torsion bar is attached to the pinion, where the valve
body is also attached. The valve is of open-centre type with several control
edges. At centre position, all control edges are equally opened and all ports
are connected to tank. When the driver applies a torque on the steering wheel,
the torsion bar is twisted. This causes an angular displacement between the
spool and the valve body. This in turn will close the control edges on one side
and open it on the other side. On the closing side, the pressure will increase
10
Description of the steering system
Driver
input
Steering column
Valve spool
Valve body
Tank
Pump
Torsion bar
Pinion
Figure 2.4
Schematic of the valve assembly of the rack and pinion steering
system.
Tank
Pump
Recirculating
balls
Driver
input
Torsion
bar
Piston
Rotating
nut
Figure 2.5
Schematic of the valve assembly of the recirculating ball screw
steering system.
and generate the assistance force needed. Thus, the purpose of the hydraulic
system is to control pressure. The torsion bar is linear, which means that the
control of the valve is proportional to the torque applied by the driver.
11
On Electrohydraulic Pressure Control for Power Steering Applications
2.4
Steering wheel torque
The steering wheel torque is the fastest information channel to the driver about
the road conditions. The torque translated through the steering system comes
from the tyre-road contact, steering geometry and suspension effects and friction in the steering system itself. In order to accurately describe the steering
wheel torque related to vehicle characteristic and handling, a precise model of
the vehicle and tyres is required. This is studied by Pfeffer (2006), where an
enhanced bicycle model is suggested. The forces from the tyre-ground contact
act in the x-, y- and z-planes and generate a torque around the kingpin axis.
These are analysed by Gillespie (1992). The sum of the left and right wheel
makes part of the steering wheel torque. The contribution from the vertical
force is mainly a result of the lateral inclination of the kingpin axis and produces a centering torque while the front of the vehicle is lifted. The lateral
force exists to counteract the centrifugal force of the vehicle during cornering.
It produces a torque through the caster angle. The tractive forces produce a
torque through the scrub radius in the opposite direction and is therefore only
present when there is a force imbalance. The aligning torque is a result of the
lateral force acting at a distance behind the centre of the tyre-ground contact,
called pneumatic trail. The torque opposes the steering action.
2.5
Energy aspects and alternative systems
One of the major drawbacks with the hydraulic power steering system is its
poor energy efficiency. Due to the open-centre of the valve and a fixed pump
that always delivers flow, some part of the oil will always go to tank. The
exception would be extreme cases where the valve is fully displaced and the
open-centre path is closed, but this is hardly the case. The flow through the
open-centre path creates a pressure drop even when no steering occurs and
causes a power loss. The level of assistive pressure from the Boost curve is
flow-dependent, which is shown in chapter 3. Since the pump is driven from
the combustion engine, the pump must therefore be designed to provide enough
oil even at idling speed. It is also here where the highest assistive pressure is
needed. As soon as the engine speed increases, so also does the pump speed,
resulting in an increase in oil flow since the pump is fixed. The surplus oil is fed
back to the tank by the internal flow control valve of the pump. This property
of the system is one of the major reasons for the poor energy consumption, see
Marcus Rösth (2007). Additional losses also come from the pressure drop over
the load ports.
In the passenger car industry, the Electric Power Assisted Steering (EPAS)
system has become the most common system. The main advantage initially
was low energy consumption. Here, the hydraulic system is replaced with an
electric motor that provides the level of assistance necessary either through
the steering column or directly through the rack, see Badawy, Zuraski, et al.
12
Description of the steering system
(1999). The low energy consumption comes from the torque being applied only
when necessary and this dramatically reduces energy consumption compared
to the HPAS system, see Wellenzohn (2008). Challenges with the EPAS system
have been the steering feel, which is compromised due to the high inertia of the
electric motor, friction, backlash, and torque ripples. Safety aspects are also a
concern. A pure electric power steering system, however, is not a possibility
for heavy vehicles due to the greater axle loads and a hydraulic system is
still necessary. The exception would be hybrid vehicles, where a high voltage
battery allows the use of an EPAS system, as was studied by Morton, Pickert,
and Armstrong (2014).
Extensive research can be found that is aimed at improving the energy consumption of hydraulic power steering systems. A common approach is to drive
the pump from an electric motor instead of from the combustion engine, which
is discussed by Pfeffer, D. N. Johnston, et al. (2005), Forbes, Baird, and Weisgerber (1987), Suzuki et al. (1995), Inaguma, Suzuki, and Haga (1996), Gessat
(2007) and Yu and Szabela (2007). The main purpose is to actively control
the oil flow depending on the driving scenario and which can be turned down
to a very low level during high speed driving. Other advantages of the system
include improved packaging and the possibility to vary the level of assistance
for different scenarios and in that way to some extent tune the steering feel.
The solution can also be used with a hydromechanical closed-centre rotational
valve, as presented by Suzuki et al. (1995) and Inaguma, Suzuki, and Haga
(1996). With a small overlap of the valve the pump flow can be turned off
when no steering action is applied. The characteristic of the steering system is
also affected, providing a region of manual steer around zero position. An electronically controlled 4/3-valve with closed-centre is used by Bootz, Brander,
and Stoffel (2003), in order to reduce energy consumption. The system uses
a constant pressure, provided from fixed pump driven from an electric motor
or the combustion engine via a clutch. Two accumulators are used to vary
the system pressure depending on driving scenario. With this system, energy
consumption was significantly reduced compared to the open-centre system.
Several solutions to control the output flow of a combustion engine driven
pump for open-centre systems are also presented by Bootz (2006). For a fixed
pump, a by-pass valve and an electro-hydraulic proportional valve can be introduced to control the oil flow. Instead of a fixed pump, a hydromechanical or an
electrohydraulic variable pump can be used. If an accumulator is used in the
supply system, initial flow demands can be taken care of by the accumulator
and the pump can be run at very low speed. The use of an accumulator has
the highest potential to reduce the energy consumption. The tests by Bootz
(2006) showed that only 28% of the power, compared to the original system,
was needed for the given driving cycle. In the hybrid power steering system for
heavy vehicles, presented by Wiesel et al. (2009), an electric motor is combined
with the open-centre power steering system. The use of energy can be optimised for any specific driving scenario. On the hydraulic supply side, different
13
On Electrohydraulic Pressure Control for Power Steering Applications
solutions were investigated: a by-pass pump, a variable pump and a pump with
clutch. The variable pump showed the highest potential with a reduction of up
to 80% for the given driving cycle.
2.6
Steering feel and steering system influence on
vehicle handling
A good steering feel is important when driving to allow the driver to accurately
place the vehicle, Adams (1983). Steering feel can be defined as "vehicle reaction and steering wheel torque feedback within the driving task of directional
control", Pfeffer (2006). As mentioned by Adams (1983), there is a different
need of feel depending on driving situation. During low speed manoeuvre the
driving can be undertaken with less feel, while at high speed cornering a higher
feel is needed to give the driver information about the vehicle’s response.
Several researchers have investigated steering system properties on on-centre
characteristics. There are several measures that are possible to quantify that
have an influence on the on-centre characteristics. These are derived by, for
example, Norman (1984). Several vehicles were tested with sinusoidal input
and parameters were defined from cross plots, such as steering wheel angle vs.
lateral acceleration, which gives steering sensitivity, steering hysteresis and yaw
rate lag time; steering wheel torque vs. lateral acceleration, which gives the
returnability, indication of friction, steering torque gradient that indicates road
feel and steering effort; steering wheel torque vs. steering wheel angle, which
gives stiffness. Other quantitative measures are defined analytically by Baxter
(1988). The stiffness is defined as the gradient of the rack force over steering
angle and increases with the Boost gain. The steering gear feel is defined as
the gradient of the steering wheel torque over rack force and decreases with
the Boost gain. There is thus a contradiction between the two. An example is
for on-centre driving where the Boost gain is low to give a good feel, but also
generates a low stiffness.
An early paper by Segel (1964) investigated the influence steering force (or
steering torque) versus lateral acceleration, that is torque gradient, has on
driver rating of vehicle controllability. It was shown that there is a strong
relation between steering torque gradient and driver ratings, e.g. too light
a torque gradient results in difficulty in precise positioning of the vehicle as
too high a gradient was experienced as the response was slowed down. The
importance of tuning the torque gradient for a good steering feel is also stressed
by Harrer, Pfeffer, and N. D. Johnston (2006). A typical relation between
the steering wheel torque and the lateral acceleration is shown in figure 2.6.
Due to the Boost curve the gradient decreases with lateral acceleration. At
low lateral acceleration the gradient is steep to provide a good on-centre feel.
An analytical expression for the steering wheel torque characteristic versus
lateral acceleration at steady state was derived by Pfeffer and Harrer (2008),
14
Description of the steering system
Steering Wheel Torque
by defining the steering assistance ratio. A linear ratio is desirable, which
gives two parameters to tune. The Boost curve can then be defined from the
assistance ratio.
Lateral Acceleration
Figure 2.6
Steering wheel torque as a function of lateral acceleration.
Several researchers have also investigated the relationship between the
driver’s subjective rating regarding the controllability of the vehicle and vehicle characteristic. Jaksch (1979) found that there is a strong relationship
between subjective ratings and yaw velocity response time, steering wheel angle
gradient and steering wheel torque gradient. Rothhämel (2010) investigated
the correlation between steering feel and vehicle handling properties for heavy
vehicles. The steering wheel returnability is correlated to the torque gradient
while torque magnitude and friction levels are correlated to the driver’s opinion
of the heaviness of driving and stability.
It can be concluded that the steering wheel torque is very important for
steering feel. In this work it is therefore assumed to be sufficient to study how
the steering wheel torque is affected by the closed-centre system. Since no references are available, the original system will fill this role. Vehicle performance
is also a contributor to the steering feel. By using the same arbitrary model
for both systems when comparing, the effects of the vehicle can be excluded.
It is important to have a model that predicts the steering wheel torque.
15
On Electrohydraulic Pressure Control for Power Steering Applications
16
3
Modelling of the
hydraulic power
steering system
The modelling of the original hydraulic power steering system serves several
purposes in this work. Since the alternative electro-hydraulic concept will work
only to replace the existing hydraulic system, a model of the original system
will therefore be useful for evaluation and development of the electro-hydraulic
system. A simulation environment facilitates much of the development work,
where control strategies and concepts can be evaulated without the need for
hardware. Since the original system also works as a frame of reference, a good
understanding of the system is essential. A model will help in this, where the
effect of different parameters can be understood better. A frequency analysis
of the system can further explain certain behaviour.
This chapter describes the modelling procedure for the hydraulic power steering system. To better compare results, the system studied is the test rig explained in detail in chapter 7, where the steering system has been isolated from
the rest of the vehicle. A nonlinear model of the system is first derived and
validated with measurements from the test rig. The model is then linearised
and analysed in the frequency domain, where stability can be examined.
3.1
Nonlinear model
For modelling purposes, the hydraulic power steering system can schematically
be described as shown in figure 3.1. The system is only studied from a steering
wheel angle θsw to a rack displacement xrw . It is convenient to divide the model
into several submodels, which is illustrated in figure 3.2. A driver model serves
the purpose of applying the right amount of torque at steering wheel in order
17
On Electrohydraulic Pressure Control for Power Steering Applications
to follow given steering wheel angle driving cycle. This means that different
steering systems can be evaluated for the same cycle. The main idea is to be
able to compare different steering systems and the difference in behaviour is
seen in the driver’s torque. The development of a driver model is not within
the scope of this work and a PID controller, implemented as a lead-lag filter, is
choosen and manually tuned. This will not be explained further in this work.
The driver submodel communicates with the mechanical submodel by sending
the torque. The mechanical submodel will in turn calculate a steering wheel
angle based on this input. The mechanical submodel also communicates with
the hydraulic submodel and the load model. The mechanical submodel sends
the torque at the torsion bar to the hydraulic submodel, which in turn calculates
the pressure difference in the assistance cylinder. This is sent back to the
mechanical submodel. The load model could be any model that generates the
right amount of load felt by the steering rack during different driving scenarios.
Detailed models of the load are not within the scope of this work and simple,
but sufficient, models are used. For highway driving, the classic bicycle model
is used. The highest load case represents the vehicle at stand-still. For this
scenario, the load, or tyre, is simply regarded as a spring. The load takes
the rack position from the mechanical submodel as input and calculates the
force, which is sent back. The mechanical, hydraulic and load submodels are
described in detail below. The equations are defined in continuous form, but all
implementation has been done with the Transmission Line Method and Bilinear
transform, see e.g. Auslander (1968), since it ensures a robust numerical solver
and the models can be re-used for implementation in a real-time computer.
θsw
F
xrw
Figure 3.1
18
Schematic of the hydraulic power steering system.
Modelling of the hydraulic power steering system
Load
xrw
θswref
Driver
Tsw
θsw
F
Mechanical
submodel
Ttb
∆p
Hydraulic
submodel
Figure 3.2 The structure of the simulation model of the hydraulic power
steering system. The interaction between the submodels is also illustrated.
3.1.1
Mechanical submodel
The mechanical subsystem can be modelled with different levels of complexity
depending on the purpose and a great deal of research has been done on the
subject. Preferably, the model should not be unnecessarily complex since it will
increase computational time and be more difficult to analyse. Starting from a
2 DoF system, models up to a 5 DoF system have been considered where every
inertia of the system is taken into account. Pfeffer (2006), see also Pfeffer,
Harrer, et al. (2006), derives a 5 DoF model with the purpose of studying the
steering wheel torque. However, a 2 DoF model is also derived for the same
purpose and also shows that the two models perform in a similar fashion. A 2
DoF model is also used to study the power consumption of an HPAS system by
Pfeffer, D. N. Johnston, et al. (2005). Baxter (1988) also uses a simple model
and ignores friction to derive expressions for steering gear feel and stiffness.
Post and Law (1996) measure the friction and stiffnesses of a rack and pinion
steering gear and a recirculating ball screw steering gear and model the systems
with both a high degree of freedom model and a 2 DoF model implemented with
a vehicle model. The importance of including nonlinear effects, such as friction,
stiffness and boost curve, in order to predict the behaviour of the system, was
shown. Neureder (2002) studied steering wheel nibble, i.e. vibrations in the
steering wheel, and derived a simple model of the steering system. Marcus
Rösth (2007) derived both a 2 DoF model of a passenger car steering system
and a 3 DoF model of a truck steering system, both models with pressureand speed-dependent friction. Linear models were derived to study chattering
and a nonlinear model to study catch-up. The same approach to modelling
the mechanical subsystem is also used in EPAS systems, as shown by Badawy,
Zuraski, et al. (1999). A column-assisted EPAS was modelled and a higher
and a lower degree model were compared to show similar results. Ueda et al.
(2002) investigated how changes in friction in different parts of the steering
19
On Electrohydraulic Pressure Control for Power Steering Applications
system affect the system characteristics. A 4 DoF model was used for this. In
the early paper by Segel (1966), the author derives a simplified model in order
to predict the behaviour of the vehicle. The steering system is modelled with 2
DoF with Coulomb friction together with a 3 DoF linear vehicle model. In the
work done by Zaremba and Davis, 1995, the choice was a 2 DoF model for the
mechanical structure, with the purpose of analysing stability in the frequency
domain of an EPAS system. Since the system studied is the test rig without
wheel assembly and vehicle and since the operating range where the steering
system is studied in this work is similar to that in, for example, Pfeffer (2006),
it should be sufficient to use a 2 DoF model for the purpose of describing the
behaviour of the system and to predict the steering wheel torque under different
conditions.
Friction is an important aspect to properly describe the system and a proper
friction model is therefore needed. There is friction in the column and in the
rack. The rack friction is strongly pressure-dependent, but friction also comes
from the yoke, mesh and steering valve sealings. One possible way would be to
implement a Coulomb friction. As long ago as 1966, Segel (1966) stressed the
importance of including friction, and found that without a certain amount of
Coulomb friction in the steering system the vehicle would be unstable, in studies
of a pure mechanical steering system. However, investigations by Neureder
(2002) suggest that the friction in the steering system behaves differently than
a Coulomb friction. It was found that cars exhibit nibble even at low-level
forces, which contradicts the behaviour of Coulomb friction. Ueda et al. (2002)
also investigated friction and suggested a spring/friction model to best describe
its behaviour. This means that the friction force behaves like a spring up to
a certain level where the force remains constant. Pfeffer (2006) used the same
approach and further developed the model. This is also adopted in this work.
The equations of motion describing the steering system are described by
equation 3.1 and equation 3.2. The steering wheel and column constitute the
upper inertia, Jsw , and the rack with mass constitutes a translational mass,
Mrw . Steering wheel torque is denoted Tsw , stiffness Kt , steering gear ratio
Rt , viscous friction coefficient b, load force Fload , cylinder area Ap , column
friction Mfsw and rack friction Ffrw (pL ). The steering wheel is denoted index
sw and the rack index rw. The load pressure is denoted pL and is defined as
the difference between the right and left cylinder chambers, pL = p1 − p2 . The
rack friction is dependent on the pressure in the assistance cylinder.
Jsw θ̈sw = Td − KT (θsw − xrw RT ) − bsw θ̇sw − Mfsw
(3.1)
Mrw ẍrw = pL Ap + KT (θsw − xrw RT ) Rt − brw ẋrw − Fload − Ffrw (pL ) (3.2)
The friction model is illustrated in figure 3.3. It is modelled as an exponential
spring with limited force, Flim , and hysteresis, and is well documented by
Pfeffer (2006). The column friction uses this model without modifications.
The rack friction uses the same model as well but with an extra term for the
pressure dependancy, which is assumed to be linear. It follows the relation in
20
Modelling of the hydraulic power steering system
equation 3.3 where cpF f is a constant that describes the pressure dependancy.
Ffrw (pL ) = Ffstat + cpF f
p1 + p2
2
(3.3)
Equivalent spring stiffness of the rack
Ff
Flim
x
−Flim
Figure 3.3
3.1.2
Representation of the friction model according to Pfeffer, 2006.
Hydraulic submodel
The open-centre valve consists of several control edges but is preferably
modelled as a lumped Wheatstone bridge, as shown in figure 3.4. Opposite orifices are assumed to be equal. This has proven to be useful by M.
Rösth and J-O. Palmberg (2007). A more general model and analysis of an
open-centre valve can be found in Merritt (1967), which is also applicable here.
For the modelling, the tank pressure is assumed to be zero and the pump is
assumed to deliver a constant flow qp . The model consists of the equations
describing the flow through the four orifices, q1 to q4 , equations 3.4 to 3.7, and
the continuity equation to describe the pressure change in the system volume
Vs and the two cylinder volumes V1 and V2 , equations 3.8 to 3.10. Variables
are defined in figure 3.4, where Ap is the piston area, Cq is the discharge
coefficient, ρ is the oil density and β is the bulkmodulus. The volume are
allowed to change their size with piston stroke, i.e., dV = Ap xrw .
Flow equations
21
On Electrohydraulic Pressure Control for Power Steering Applications
qp
Vs
qs
ps
q1
q2
A1
A2
p1
p2
ql
V1
V2
xrw
A2
Figure 3.4
q4
q3
A1
The open-centre valve represented as a Wheatstone bridge.
r
2
(ps − p1 )
ρ
r
2
q2 = Cq A2
(ps − p2 )
ρ
r
2
q3 = Cq A1
p2
ρ
r
2
q4 = Cq A2
p1
ρ
q1 = Cq A1
(3.4)
(3.5)
(3.6)
(3.7)
Continuity equations
qp − qs =
Vs
ṗs
β
V1
ṗ1
β
V2
q2 − q3 = Ap ẋrw + ṗ2
β
q1 − q4 = Ap ẋrw +
(3.8)
(3.9)
(3.10)
Equations 3.4 to 3.10 represent how the system is implemented in the simulation environment. For further analysis of the system, a few assumptions can be
22
Modelling of the hydraulic power steering system
made and some variables defined to simplify the model. If it is assumed that
the orifices in diagonally opposite arms are symmetric and matched and that
the flow is also equal, as described by equations 3.11 and 3.12, it follows that
the system pressure ps can be expressed as shown in equation 3.13.
q1 = q3
(3.11)
q2 = q4
(3.12)
ps = p1 + p2
(3.13)
The load pressure pL is defined by equation 3.14.
pL = p1 − p2
(3.14)
Combining equations 3.13 and 3.14, the cylinder volume pressures p1 and p2
can be defined as shown in equation 3.15 and 3.16.
ps + pL
(3.15)
2
ps − pL
p2 =
(3.16)
2
The load flow ql is defined by equation 3.17 or equation 3.18. The system flow
qs is defined by equation 3.19 or equation 3.20.
p1 =
ql = q1 − q4
(3.17)
ql = q3 − q2
(3.18)
qs = q1 + q2
(3.19)
qs = q3 + q4
(3.20)
At steady-state, the load flow corresponds to the piston velocity. Combining
the definitions of the cylinder volume pressures, the system flow and load flow,
the hydraulic system can be described by equations 3.21 to equation 3.24, where
the system is expressed in the variables system pressure ps , load pressure pL ,
system flow qs and load flow ql .
r
r
ps − pL
ps + pL
q l = C q A1
− C q A2
(3.21)
ρ
ρ
r
r
ps − pL
ps + pL
qs = Cq A1
+ C q A2
(3.22)
ρ
ρ
Vs
qp − qs =
ṗs
(3.23)
β
V0
ql = Ap ẋrw + ṗL
(3.24)
β
V1 V2
V0 =
V1 + V2
23
On Electrohydraulic Pressure Control for Power Steering Applications
At steady-state, the load pressure and system pressure can be derived from
equations 3.21 and 3.22 and expressed by equations 3.25 and 3.26,
2 2 !
ρqp2
1+q
1−q
−
(3.25)
pL (Ttb , q) =
8Cq2
A (Ttb )
A (−Ttb )
2 2 !
ρqp2
1−q
1+q
ps (Ttb , q) =
+
(3.26)
8Cq2
A (Ttb )
A (−Ttb )
where q = qqpl . The opening areas depend on the displacement of the valve,
that is, the twisting of the torsion bar. The opening areas can therefore be
expressed as functions of the torsion bar torque, as shown in equations 3.27
and 3.28. The load pressure in equation 3.25 is therefore also a function of the
torsion bar and represents the Boost curve in analytical form. As can be seen,
it depends on the opening areas, i.e. the geometry of the valve and the pump
flow. It also varies with piston speed.
A1 = A(Ttb )
(3.27)
A2 = A(−Ttb )
(3.28)
The opening areas can be derived in several ways. One way is to measure the
static Boost curve together with the pump pressure for a given pump flow. The
opening areas can then be calculated according to equations 3.29 and 3.30. The
Boost curve for the test rig is measured and the opening areas calculated in
section 3.2.
r
qp
ρ
A1 (Ttb ) =
(3.29)
2Cq ps (Ttb ) − pL (Ttb )
r
ρ
qp
A2 (Ttb ) =
(3.30)
2Cq ps (Ttb ) + pL (Ttb )
3.1.3
Load model
The steering rack is subjected to forces due to the interaction between the
tyre and ground. An exact model of the behaviour of the load is not within
the scope of this work and arbitrary, but reasonable, load models have been
used. Different forces and behaviour of the load arise during different driving
scenarios. For example, the highest loads arise when the vehicle is at stand
still. The tyre then behaves like a spring until it begins to slip and the force
becomes more or less constant. The simplest way is to only study small steering
wheel angle inputs and see the tyre as a pure spring. This has been done when
analysing the closed-centre system for stability, which is described in chapter 6.
For most study cases, however, the bicycle model has been used as load, which
can be found in the book by Wong (2001). Small steering wheel and slip angles
are assumed. The model only considers lateral acceleration and any forces that
24
Modelling of the hydraulic power steering system
could arise from longitudinal accerelation are ignored. The longitudinal velocity
vx is therefore constant. The lateral acceleration v̇y and the yaw acceleration
Ω̇z about the z-axis of the vehicle are described by equations 3.31 and 3.32,
respectively, where m is the mass of the vehicle and J is the inertia about the
z-axis. The equations are defined in the vehicle’s fixed coordinate system, with
parameters defined by figure 3.5.
Y
lf
lr
vr
Ωz
αr
vx
y
Fy r
Figure 3.5
x
vf
vy
Fy f
αf
X
Bicycle model used as load model.
mv̇y = Fyf + Fyr − mvx Ω
(3.31)
J Ω̇z = Fyf lf − Fyr lr
(3.32)
The front and rear lateral forces, Fyf and Fyr , depend on the tyre model. For
moderate lateral accelerations it is sufficient to consider a linear behaviour of
the tyre and the lateral forces are given by equations 3.33 and 3.34, where Cαf
and Cαr are the tyre cornering stiffnesses and α is the slip angle at the front
and rear. The bicycle model lumps the inner and outer wheel and the cornering
stiffness is therefore considered for both wheels.
Fyf = Cαf αf
Fyr = Cαr αr
(3.33)
(3.34)
The slip angles are defined by equation 3.35 and 3.36. The road wheel angle
is here denoted θrw and is input to the model. Since the steering system is
isolated about the rack, the rack position is tranformed to a corresponding
road wheel angle.
l f Ωz + v y
vx
lr Ωz − vy
αr =
vx
αf = θrw −
(3.35)
(3.36)
With the parameters according to table 3.1, the yaw response shown in figure
3.6 can be produced with a step input in road wheel angle of 0.01 rad at 70
kph.
The force on the steering rack is given by the front lateral force, which
generates a torque through the pneumatic trail and caster angle, as explained
25
On Electrohydraulic Pressure Control for Power Steering Applications
0.045
0.04
Yaw velocity [rad/s]
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
Figure 3.6 Yaw response of the vehicle used as load model to a step input
in road wheel angle of 0.01 rad at 70 kph.
Parameter
m
J
Cαf
Cαr
lf
lr
Table 3.1
Value
1.66 · 103
4.88 · 103
2.06 · 105
10.65 · 105
0.7025
2.0125
Unit
kg
kgm2
N/rad
N/rad
m
m
Vehicle parameter values
in chapter 2, and is transformed to a force on the rack through the suspension
geometry. Here it works only through the total pneumatic trail, pt . Whichever
load model, it is dependent on the rack position. The load model could therefore
be treated as part of the control loop or as a disturbance. Hereafter, when the
load is referred to the vehicle model it is the vehicle model presented.
3.2
Measurements and parameter identification
Several parameters of the steering system model can be identified by measurements of the test rig. The test rig provides a very convenient way to study
specific parts of the system, which could be difficult when installed in the vehicle. More on the test rig can be found in chapter 7. Parameters identified
are the Boost curve, opening areas of the valve, rack friction, column friction,
column and torsion bar stiffness, and steering wheel and column inertia.
26
Modelling of the hydraulic power steering system
Boost curve
The Boost curve is measured by clamping the rack. A steering wheel torque
is applied slowly in both directions. Pump and cylinder pressures are recorded
together with the torsion bar torque. The measured boost curve is shown in
figure 3.7.
100
90
Assistance pressure [bar]
80
70
60
50
40
30
20
10
0
−4
−3
−2
−1
0
1
Torsionbar torque [Nm]
2
3
4
Figure 3.7 The measured Boost curve of the test rig shows differential cylinder pressure vs. torsion bar torque.
From the boost curve the opening areas of the valve can be calculated using
equations 3.30 and 3.29. Both areas are shown in figure 3.8.
Rack friction
According to Pfeffer (2006), it is the pressure-dependent friction that is the
most dominant friction in the rack. The friction is measured by disconnecting
the column from the input shaft of the valve. The load cylinder is run as a
position controller with a sinusoidal input with an amplitude of 0.004 m and a
frequency of 0.5 rad/s. Both the force required to drive the rack and the rack
position are registered. The servo valves are controlled to maintain a constant
pressure in both chambers. Each measurement is made for 5, 10, 20, 40, 60
and 80 bar in each chamber. The results of the measurements can be seen in
figure 3.9 and the increased amplitude of the hysteresis curve with increased
pressure is clearly visible. A linear relation between the friction and the sum of
the cylinder pressures is assumed according to equation 3.3. The static friction
27
On Electrohydraulic Pressure Control for Power Steering Applications
−6
4
x 10
A1
3.5
Area opening [m2]
3
2.5
2
1.5
A2
1
0.5
−4
−3
Figure 3.8
−2
−1
0
1
Torsion bar torque [Nm]
2
3
4
The calculated opening areas of the open-centre valve.
level Ffstat is set to 357 N and the pressure coefficient cpF f to 2.94 · 10−5 N/Pa.
Column friction
The column friction is measured by again using the load cylinder as a position
servo with a sinusoidal input with an amplitude of 0.004 m and a frequency of
0.5 rad/s. The steering column is attached to the valve and the steering wheel
is free. The torsion bar torque then gives an indication of the friction in the
upper inertia. The chamber pressures are held at 10 bar. It turned out that at
zero pressure the result is very oscillative, probably due to the friction in the
rack. Figure 3.10 shows the torsion bar torque vs. the steering wheel angle.
The result is still quite oscillative but the amplitude of the hysteresis curve is
still clearly around 0.3 Nm.
Column and torsion bar stiffness
The column stiffness is determined by measuring the torsion bar torque and
steering wheel angle with the rack clamped. This gives the total stiffness of
the upper inertia, i.e. the torsion bar and steering wheel column. The result is
shown in figure 3.11. The inclination of the curve gives the stiffness and is set
to 1.06 Nm/deg.
28
Modelling of the hydraulic power steering system
800
5 bar
10 bar
20 bar
40 bar
60 bar
80 bar
600
RackForce [N]
400
200
0
−200
−400
−600
−800
−5
Figure 3.9
−4
−3
−2
−1
0
1
RackPos [m]
2
3
4
5
−3
x 10
Rack force vs. rack position for different chamber pressures.
0.6
0.4
Torsion bar torque [N]
0.2
0
−0.2
−0.4
−0.6
−0.8
−30
−20
−10
0
10
20
Steering wheel angle [deg]
30
40
Figure 3.10 Torsion bar torque vs. the steering wheel angle with a freely
rotating steering wheel. The mean friction level at ±0.3 Nm is also indicated.
Steering wheel and column inertia
The inertia of the upper system, i.e. the steering wheel and column, is measured
by clamping the rack and applying a torque at the steering wheel. When
29
On Electrohydraulic Pressure Control for Power Steering Applications
10
8
Steering wheel torque [Nm]
6
4
2
0
−2
−4
−6
−8
−8
−6
−4
−2
0
2
Steering wheel angle [deg]
4
6
8
Figure 3.11
The figure shows the torsion bar torque vs. the steering wheel
angle with the rack clamped.
the torque is suddenly released, the steering wheel will excite free oscillations
according to equation 3.37
r
KT
f=
(3.37)
Jsw
The result is shown in figure 3.12. The frequency is 5.15 Hz. Since the stiffness
KT is known, the inertia can be calculated to 0.058 kgm2 . The damping
coefficient is tuned to get a satisfactory result from simulations of the same
test case compared to the measurements. A damping of 0.06 was achieved in
this way.
30
Modelling of the hydraulic power steering system
12
10
Steering wheel torque [Nm]
8
6
4
2
0
−2
−4
−6
−8
0
0.5
1
Figure 3.12
3.3
1.5
2
2.5
Time [s]
3
3.5
4
4.5
5
The free oscillations of the steering wheel.
Validation of nonlinear model
The model of the hydraulic power steering system is validated by comparing
the outputs of the model to measurements for the same input cycle. The input
is the steering wheel angle, which is measured and fed as a reference value to
the driver model. The outputs are the torsion bar torque, the rack position, the
pump pressure and the assistance pressure in the cylinder. The steering wheel
agle was manually applied with increased velocity. The vehicle model was used
and set to run at 70 kph. The steering wheel angle reached an amplitude of
±30° and a velocity of 400°/s. The pump pressure reached ∼40 bar. The oil
temperature was maintained between 40 and 45° C.
Figure 3.13 shows a comparison between the simulated and measured steering wheel angle, which shows the performance of the driver controller. The
driver model is able to follow the reference with very good result. Figure 3.14
shows a comparison between the simulated and measured torsion bar torque,
which is the most important indication that the model is accurate enough to be
used in the evaluation process of the electrohydraulic power steering system.
Figure 3.15 shows a comparison between the simulated and the measured rack
position. Figure 3.16 shows a comparison between the simulated and the measured pump pressure. Figure 3.17 shows a comparison between the simulated
and the measured pressure differences, i.e. the assistance pressure. All figures
indicate that the model performs sufficiently well for the given driving cycle.
31
On Electrohydraulic Pressure Control for Power Steering Applications
30
Steering wheel angle [deg]
20
10
0
30
−10
−20
−30
35
−30
36
5
10
15
20
25
Time [s]
30
35
40
Figure 3.13
Torsionbar torque [Nm]
Comparison between simulated (dashed) and measured (solid)
steering wheel angle.
3
2
1
0
−1
−2
Torsionbar torque [Nm]
4
6
8
10
12
Time [s]
14
16
18
20
4
2
0
−2
35
36
37
38
39
40
Time [s]
Figure 3.14
Comparison between simulated (dashed) and measured (solid)
torsion bar torque.
32
Modelling of the hydraulic power steering system
−3
Rack position [m]
4
x 10
2
0
−2
−4
4
6
8
10
12
Time [s]
14
16
18
20
−3
x 10
Rack position [m]
4
2
0
−2
−4
35
36
37
38
39
40
Time [s]
Figure 3.15
Comparison between simulated (dashed) and measured (solid)
Pump pressure [bar]
rack position.
30
20
10
Pump pressure [bar]
4
6
8
10
12
Time [s]
14
16
18
20
60
50
40
30
20
10
35
36
37
38
39
40
Time [s]
Figure 3.16
Comparison between simulated (dashed) and measured (solid)
pump pressure.
33
Pressure difference [bar]
Pressure difference [bar]
On Electrohydraulic Pressure Control for Power Steering Applications
20
0
−20
4
6
8
10
12
Time [s]
14
16
18
20
50
0
−50
35
36
37
38
39
40
Time [s]
Figure 3.17
Comparison between simulated (dashed) and measured (solid)
pressure differences between left and right chamber of the assistance cylinder.
34
Modelling of the hydraulic power steering system
3.4
Linear modelling and analysis
The complete hydraulic power steering system is linearised and analysed in the
frequency domain. Such an analysis is very useful. It helps to better understand
the system, stability can be examined and any particular behaviour will be
visible. The analysis is divided into first study the hydraulic system and then
the complete system.
3.4.1
Hydraulic system analysis
The hydraulic system described by equations 3.21 to 3.24 is analysed. The first
step is to linearise the flow equations, which gives the results in equations 3.38
and 3.39. The ∆-symbol indicates a change in the state.
∂ql
∂ql
∂ql
∆Ttb +
∆ps +
∆pl
∂Ttb
∂ps
∂pl
∂qs
∂qs
∂qs
∆qs =
∆Ttb +
∆ps +
∆pl
∂Ttb
∂ps
∂pl
∆ql =
(3.38)
(3.39)
The partial derivatives are defined by equations 3.40 to 3.45, where index 0
indicates the working point for which the equation is linearised.
r
r
∂qs
ps0 − pl0
ps0 + pl0
+ Cq w2
= Kq2
= Cq w1
∂Ttb
ρ
ρ
Cq A1
∂qs
C q A2
= q
= Kc1
+ q
ps0 −pl0
∂ps
2
ρ 2 ps0 +pl0 ρ
(3.40)
(3.41)
ρ
ρ
∂qs
C q A2
Cq A1
+ q
= Kc2
=− q
ps0 +pl0
ps0 −pl0
∂pl
2
ρ
2
ρ
ρ
ρ
r
r
∂ql
ps0 − pl0
ps0 + pl0
= Cq w1
− Cq w2
= Kq1
∂Ttb
ρ
ρ
∂ql
Cq A1
C q A2
= q
− q
= −Kc2
∂ps
2 ps0 −pl0 ρ 2 ps0 +pl0 ρ
(3.42)
(3.43)
(3.44)
ρ
ρ
C q A1
C q A2
∂ql
=− q
− q
= −Kc1
p
−p
∂pl
s0
l0
l0
ρ
2
ρ 2 ps0 +p
ρ
ρ
(3.45)
w is the area gradient and is defined according to equation 3.46.
w=
∂A(Ttb )
∂Ttb
(3.46)
The linearised system equations are transformed into the Laplace domain
35
On Electrohydraulic Pressure Control for Power Steering Applications
and are defined by equations 3.47 to 3.50, where s is the Laplace operator.
∆ql = Kq1 ∆Ttb − Kc2 ∆ps − Kc1 ∆pl
(3.47)
∆qs = Kq2 ∆Ttb + Kc1 ∆ps + Kc2 ∆pl
Vs
∆qp − ∆qs =
∆ps s
β
V0
∆ql = Ap ∆xrw s + ∆pl s
β
(3.48)
(3.49)
(3.50)
From the linearised system equations the transfer function of load pressure pL
can be derived from an input in torsion bar torque. The transfer function is
defined in equation 3.51.
1 + KVc1s β s
Kc1
·
∆pL = 2
2 s2
0
Kc1 − Kc2
+ 2δ
ω0 s + 1
ω02


Vs
s
1+
Kq2 Kc2
Kc1 + K
β
Kq2 Kc2


q1
Kq1 1 +
∆Ttb − Ap ∆xrw s

Vs
Kq1 Kc1
1 + Kc1 β s
≈
GKc (s)
GKq (s)Kq ∆Ttb − Ap ∆xrw s
Kc
(3.51)
where
s
2 − K2
Kc1
c2
β
V0 Vs
Kc1
V0 + Vs
1
√
δ0 = p 2
2
2 Kc1 − Kc2
V0 Vs
ω0 =
2
2
− Kc2
Kc1
Kc1
Kq2 Kc2
Kq = Kq1 1 +
Kq1 Kc1
Kc =
At steady-state, the transfer function describes the static Boost curve, which
in linear form is defined by equation 3.52.
∆pL =
Kq
Ap
Ap
∆Ttb −
∆xrw s = Kboost ∆Ttb −
∆xrw s
Kc
Kc
Kc
(3.52)
The frequency of the transfer function is plotted in figure 3.18. The frequency
response depends on the working point and the figure shows the response at
zero load flow for torque levels 0 Nm, 1 Nm, 2 Nm, 3 Nm and 3.5 Nm. The
response is very similar to a first order system with a maximum phase shift
of 90°. What is interesting to note is how the response changes with higher
36
Modelling of the hydraulic power steering system
torque, starting at around 150 Hz at 0 Nm and becoming as slow as approx.
5 Hz at 3.5 Nm. This will have an impact on the closed loop system of the
complete steering system, as will be seen in the analsis in the next chapter, and
can also be useful information when designing new control loops.
Amplitude [dB]
140
Increased torque
120
100
80
60
−1
10
0
1
10
10
2
10
3
10
0
Phase [deg]
−20
−40
−60
−80
Increased torque
−100
−1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
Figure 3.18 Frequency response from a torsion bar torque input to a load
pressure at zero load flow for different torque levels. The upper plot shows the
amplitude curve and the lower plot the phase curve.
3.4.2
HPAS system analysis
The hydraulic system makes up part of the complete steering system. By
exluding static and pressure-dependent friction, the mechanical subsystem is
already linear. By transforming equations 3.1 and 3.2 into the Laplace domain
and with the hydraulic subsystem transfer function, the complete system in
linear form is described by equations 3.53 to 3.56.
Jsw ∆θsw s2 = ∆Td − KT (∆θsw − RT ∆xrw ) − brw ∆θsw s
2
Mrw ∆xrw s = ∆pL Ap + KT (∆θsw − RT ∆xrw ) RT − brw ∆xrw s
GKc (s)
GKq (s)Kq ∆Ttb − Ap ∆xrw s
∆pL =
Kc
∆Ttb = Kt (∆θsw − Rt ∆xrw )
(3.53)
(3.54)
(3.55)
(3.56)
From the linearised system equations a block diagram can be derived, shown
in figure 3.19.
37
On Electrohydraulic Pressure Control for Power Steering Applications
∆Td
∆θsw
1
Jsw s2 +bsw s+KT
KT RT
+
+
KT RT
+
KT
Kq GK
q
+
GK
c
Kc
+
+
Ap
1
Mrw s2 +brw s+KT R2
T
∆xrw
−
−
Ap s
RT
Figure 3.19
Block diagram of the hydraulic power steering system.
The block diagram clearly shows that the steering system can be treated as
a position control loop. The driver applies a torque which through the inertia
of the steering wheel and column generates an angle at the steering wheel.
There is a torque from the rack through the steering gear. The steering wheel
angle serves as input, or reference, to the control loop. The output is the rack
position. The corresponding rack position is compared to the steering wheel
angle and the error constitutes the twisting of the torsion bar. The twisting of
the torsion bar is the input to the valve and activates it. The valve controls the
pressure that pushes on the piston of the assistance cylinder and together with
the manual torque applied by the driver through the steering gear accelerates
the rack. Hence, there are two control loops: the outer position control loop
and the inner pressure control loop. The pressure control loop for the hydraulic
power assisted steering system is an open control loop without any feedback.
From the block diagram an expression for the closed loop transfer function
from a steering wheel angle to a rack position can be derived according to
equation 3.57. The valve controls the gain and increases the damping. The
spring stiffness is also increased.
KT RT2 + GKc GKq ApKT Kq RT /Kc
Rt ∆xrw
=
(3.57)
GKc GKq Ap Kq KT RT
A2
∆θsw
Mrw s2 + brw + Kpc GKc s + KT RT2 +
Kc
In order to study stability the loop gain transfer function Ag (s) can also be
derived from the block diagram according to equation 3.58.
Rt Kt Kq GKq GKc Ap Grw
Kc + A2p GKc Grw s
1
=
2
Mrw s + brw s + KT RT2
Ag (s) =
Grw
38
(3.58)
Modelling of the hydraulic power steering system
The frequency response of the loop gain at zero load flow is shown in figure
3.20 and the closed loop frequency response in figure 3.21.
50
Amplitude [dB]
Increased torque
0
−50
−100
−150
0
1
10
2
10
3
10
4
10
10
Phase [deg]
0
−100
Increased torque
−200
−300
−1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
4
10
Figure 3.20 Frequency response of the loop gain Ag (s). The upper plot
shows the amplitude curve and the lower plot the phase curve.
39
On Electrohydraulic Pressure Control for Power Steering Applications
Increased torque
Amplitude [dB]
0
−20
−40
−60
−80
1
10
2
3
10
0
10
4
10
Increased torque
Phase [deg]
−50
−100
−150
−200
−250
1
10
2
10
Frequency [Hz]
3
10
4
10
Figure 3.21
Frequency response of the closed loop, from a steering wheel
angle input to a corresponding rack displacement. The upper plot shows the
amplitude curve and the lower plot the phase curve.
3.5
Discussion
The simulation model of the hydraulic power steeering system has been shown
to be accurate within the working region of interest. With the help of the test
rig, the different components of the system could be evaluated independently
of each other. This has been valuable for the simulation model.
From the analysis it is clear seen that the purpose of the hydraulic system
is to control and generate a certain amount of pressure in order to assist the
displacement of the steering rack. As explained previously, this is done by
changing the control edges of the valve and the characteristic of the hydraulic
system is therefore set by the geometry of the valve. The complete steering
system is a positioning follower. This is easily seen from the block diagram in
figure 3.19, where the reference input is the steering wheel angle applied by
the driver and the output signal is the rack displacement. The steering system
therefore constitutes a closed control loop that must be designed for stability.
The gain of this control loop is set by the hydraulic system, which in practice
is the Boost curve. This is a critical point of the steering system. While the
torque level increases, so does the boost pressure to assist the driver. While
the boost pressure increases the gain of the control loop decreases, together
with the stability margins. This is most critical during exiting a corner when
the steering rack is pushed back by the external forces on the rack. The boost
40
Modelling of the hydraulic power steering system
pressure varies with load flow and is at its highest during this scenario. When
the rack is pushed back, it will compress the chamber of the assistance cylinder
with higher pressure and momentarily increase the pressure, and thereby the
gain. If the stability becomes too small ocillations can occur in the system,
causing small vibrations called chattering. This is analysed in detail by M.
Rösth and J-O. Palmberg (2007). The positioning control can also be seen as
a proportional controller with the gain set by the Boost curve.
The hydraulic system also contributes a dynamic, affecting the outer control
loop. The bode plot of the pressure dynamic in figure 3.18 illustrates this. What
is interesting is that for higher torque level, i.e. for a higher boost gain, the
response of the pressure dramatically decreases. This has the effect of "pushing
down" the amplitude curve of the loop gain as the torque increases, giving back
some of the stability margin lost with the higher gain. This is best explained by
studying figure 3.22, which shows the same loop gain frequency response as in
figure 3.20 but only for the highest torque level. The dashed curve represents
the loop gain with a fixed response, the same as for the lowest torque level
where the response is at its fastest. The solid curve is the normal curve. It can
be seen that the amplitude margin is higher with a slower response. However,
this comes at the cost of a lower phase margin. There is a compromise between
the two. How this information can be utilised is studied in detail in chapter 6,
section 6.2.
Amplitude [dB]
50
0
−50
−100
−150
0
1
10
2
10
3
10
4
10
10
Phase [deg]
0
−100
−200
−300
−1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
4
10
Figure 3.22 Frequency response of the loop gain at highest torque level and
zero load flow. The dashed curve is for a response with fixed dynamic. The solid
curve is with a variable dynamic.
In addition to controlling the gain of the positioning control loop, the valve
41
On Electrohydraulic Pressure Control for Power Steering Applications
also contributes a great deal of damping. The damping can be seen as part of
the Boost curve, i.e. there is a gain and a damping coefficient, or be seen as a
part of the plant, which is the case here. The damping is important in terms
of steering feel but also affects the control from a stability point of view. Like
the gain the damping is not controllable but is set by the valve’s geometry.
Next chapter will look into how the system can be modified in order to realise
active steering.
42
4
Active steering
Active steering is referred to as the possibility to control the output torque
required to turn the wheels or manipulate the steering angle introduced by the
driver with an electronic signal. The need for active steering comes from an
increased attention to safer vehicles. Steering-related active safety is becoming more common and several solutions can already be found on the market.
Another aspect is the possibility to increase the comfort and experience of the
driver through the steering system and enhance the steering feel. A third aspect
is autonomous driving. There are several safety, economic and environmental
benefits that can be realised with autonomous driving and it is only possible if active steering is in place, see e.g. Tai, Hingwe, and Tomizuka (2004).
This chapter gives a brief overview of different solutions to increase safety and
existing solutions to how active steering is realised on the hardware side.
4.1
Active safety and increased comfort
There is active safety and passive safety. Typical passive safety systems are
the seat belt and airbag, while active safety systems that rely on some kind
of control and usually requires sensors and actuators. Typical active safety
functions are the Anti-lock Brake System, Electronic Stability Program, ESP,
and Traction Control. These systems use the brakes and engine torque control
to increase safety. An additional way is to incorporate the steering system as
well.
Several studies on yaw disturbance attenuation and roll-over avoidance using
active steering have been made by J. Ackermann, Bünte, and D. Odenthal
(1999), J. Ackermann and Bünte (1997), Jürgen Ackermann (1998), Jürgen
Ackermann and Dirk Odenthal (1999). Yaw attenuation refers to when the
yaw rate is no longer observable from the lateral acceleration of the front axle.
The driver applies a desired yaw rate and the system measures the actual yaw
rate and rejects any disturbances. A disturbance could come from a cross-
43
On Electrohydraulic Pressure Control for Power Steering Applications
wind or µ-split, i.e. different grip on different tyres. The authors mention that
active steering is more efficient than using the braking system for the same
purpose but requires an additional actuator. These studies make use of an
angle overlay system, which is described in the next section. Experiments were
also conducted to show the benefit of the system. A roll-over prevention system
for vehicles with an elevated centre of gravity was also developed, where the
authors claim that active steering is more efficient than active damping. A
PD-controller of roll rate and roll acceleration has been developed, where the
controller gains are scheduled with velocity and height of centre of gravity.
The active steering system developed by BMW and ZF, presented by Koehn
and Eckrich (2004), is also described in the next section. However, several
safety and comfort functions can be applied with the system, such as variable
steering ratio. Here the steering angle is increased compared to the driver’s
angle at low speed, e.g. during parking, in order to reduce the steering action.
At high speed, the driver’s steering angle is instead decreased to help stabilize
the vehicle. The acutator can also be used to stabilize the vehicle during
oversteering, like ESP. Using the steering system is more efficient since a faster
reaction of the vehicle is achieved compared to using the braking system.
The research project by Eidehall et al. (2007) resulted in a method to evaluate active safety functions as well as a new function, Emergency Lane Assist,
ELA. The problem mentioned with lane keeping assist functions is to separate
intended lane changes and dangerous situations. The ELA function therefore
only intervenes when there is a dangerous lane change and prevents the lane
change by adding a torque to the steering wheel angle, i.e. a torque overlay
system as described in the next section. Information to the system comes from
a sensor and vision system that provides the distance to the lane markers, the
angle of the host vehicle in relation to the lane, the road curvature and the
relative position in relation to observed vehicles.
A study of an emergency manoeuvre of an articulated vehicle was conducted
by McCann and Nguyen (2004). An angle overlay system is assumed to be
available to the truck, referred to as Active Front Steering. The benefit of such
a system is studied for an evasive manoeuvre where both braking and steering are applied. The AFS is used to stabilize the truck and trailer and avoid
jack-knifing. Different metrics were developed to quantify the performance of
the AFS interventions. The results show the trade-off between stabilizing the
vehicle and the driver’s perception of vehicle control and that there exists an
optimal range. McCann and Le (2008) also investigated the use of gain scheduled assist torque in trucks with an Electrohydraulic Power Steering system,
i.e. the hydraulic power steering system with electric motor driven pump. The
assist torque is varied by controlling the pump speed and is linearly proportional to the rate of change of the articulation angle. The results show that the
gain scheduling opposes the driver’s tendency to overshoot the steering angle
and helps stabilize the vehicle without interfering with the driver’s perception
of the vehicle’s handling characteristics, since the lateral position change and
44
Active steering
wheel steering angle are similar to the case without gain scheduling.
A collision avoidance system is developed and evaluated by Schorn et al.
(2006). A passenger car is equipped with active front steering. Obstacles are
detected using a vision system and a desired trajectory is calculated to avoid
a collision. Two methods for trajectory control were evaluated, both with successful steering of the vehicle. A steering assistance system is developed by
Enache et al. (2009) with the intention to assist the driver in steering the vehicle to the centre of the lane during situations where the driver lacks attention.
An Electric Power Steering System is used, where the steering torque is controllable. A switching strategy is used to switch to the assistance steering during
the driver’s inattentions.
Active steering can also be combined with active braking for yaw stabilisation, as studied by Tjønnås and Johansen (2010). Five actuators are considered:
braking at each wheel and steering angle correction. Several modules at different levels handle the vehicle motion control objective, braking control, and
steering angle corrections. The design is such that the actual torque about the
yaw axis tends to the desired calculated value by satisfying actuator constraints
and minimal control effort. An active pedestrian safety system developed by
Keller et al. (2011) also uses the brakes and steering to avoid collisions. The
system combines sensing, situations analysis, decision making and vehicle control. The decision-making prioritises collision avoidance by first warning the
driver. If the driver does not do what is necessary, full braking will be applied.
If collision is not to be avoided, evasive steering will take place. This is done
by applying a torque at the steering wheel with an additional actuator. During
evasive steering, the motion controller considers the nonlinear dynamics of the
vehicle during such a scenario.
With active steering it is also possible to tune the steering feel. One way of
doing this is described by Birk (2010). The purpose of the described system
is to suppress disturbances, such as side-wind or uneven road, and provide a
good steering feel based on what the driver’s intentions are. The benefit is also
that the steering feel remains the same regardless of driving situation, vehicle
class, vehicle velocity, front axle load, tyre choice, etc.
4.2
Realising active steering
As mentioned earlier, active steering is realised either by alternating the steer
angle applied by the driver or in some way alternating the torque applied by
the driver, see illustration in figure 4.1. The two approaches do not exclude
each other. In fact, a combination can provide for full steer-by-wire functionality, which is explained by S. Müller, A. Kugi, and W. Kemmetmüller (2005).
Steer-by-wire systems decouple the steering wheel and the road wheels by removing the mechanical connection and the actuation of the road wheels can
be performed using elecrical or hydraulic power. This kind of system is not
considered in this work. A superposition of the steering angle is mainly used
45
On Electrohydraulic Pressure Control for Power Steering Applications
Td
θsw
Gear
box
DC motor
θDC
TDC
Tpinion
θpinion
Figure 4.1 Schematic principle of active steering. For superpositioning of the
steering angle, the DC motor adds or subtracts an angle to the steering wheel
angle θsw through the gear box, i.e. θpinion = θsw + θDC . For torque overlay,
the driver’s torque Td can be altered by the DC motor, i.e. Tpinion = Td + TDC .
to alter the steering ratio or for stabilisation of the vehicle, as described above.
In order to change the level of assist and the steering feel in that sense, a
superposition of the torque is necessary.
4.2.1
Superposition of steering angle
Together with BMW AG, ZF Lenksysteme GmbH has developed a superposition steering system which was first installed on the BMW 5 series. The
system, as described by Koehn and Eckrich (2004), uses a planetary gear inserted between the steering wheel and steering gear. The planetary gear has
external toothing and can be rotated by an additional actuator, in this case a
brushless DC motor, through a worm gear. When the DC motor is deactivated,
the steering system works like a conventional system. When the DC motor is
activated, the steering angle from the driver and the steering angle from the
DC motor are superimposed through the planetary gear. This system could
be installed together with any power amplification system, such as hydraulic
or electric. If a hydraulic power steering system was used, the superposition
can generate a higher rack velocity compared to a conventional system and the
maximum required flow is therefore higher. The system is illustrated in figure
4.2.
The solution by Audi Dynamic Steering (2011) is based on the same principle
but instead uses a so-called harmonic drive with an electric motor to change
the gear ratio between the steering wheel and the road wheels. The harmonic
drive is very stiff, free of play, and very efficient. Audi Dynamic Steering (2011)
46
Active steering
θsw
DC motor
Planetary
gear box
θpinion
Figure 4.2 Schematic principle of active steering by BMW/ZF. The DC motors adds to or subtracts from the steering wheel angle through a planetary
gearbox.
have integrated their gear in the steering column rather than at the steering
gear, which was the case for Koehn and Eckrich (2004). A similar solution to
that of Audi Dynamic Steering (2011) is also adopted by Lexus Variable Gear
Ratio Steering, VGRS (n.d.) for their Variable Gear Ratio Steering, VGRS.
Similar solutions to that of Koehn and Eckrich (2004) can by found, e.g.
by Bohm (2007) or by Rothhämel (2011), where the latter is adopted to heavy
vehicles. The main difference between heavy vehicles and passenger cars is that
the former can hardly be steered, or for some configurations not at all, without
an assistance system. The solution today is to use an extra pump that will
provide with hydraulic power if the primary unit fails. This is expensive and
the solution by Rothhämel (2011) is to use superposition of the steering angle.
The mechanical configuration can be designed with a higher gear ratio in order
to facilitate manual steering in case of hydraulic loss. During normal driving,
the superposition angle system corrects the gear ratio to the right one.
4.2.2
Superposition of steering torque
In the passenger car industry the most common approach to power assisted
steering is a pure elecrical system, the Electric Power Assisted Steering, EPAS,
which is described by for instance Badawy, Zuraski, et al. (1999). An electric
motor is attached either to the column, the steering gear, or the rack and assists
the driver by adding a torque to the driver’s torque. Since the electric motor
is controlled from an ECU, active steering is already realised. For heavier
applications, such as commercial vehicles, the load experienced by the steering
system is too high and an EPAS system would stress the elecrical system of
the vehicle too much. A study by Roos (2005) shows this and a pure electrical
47
On Electrohydraulic Pressure Control for Power Steering Applications
system is not interesting until a 600 V supply system is a reality, which can
be found in for example hybrid buses. Morton, Pickert, and Armstrong (2014)
studied such a system where the self-aligning torque is recovered through the
steering system
This work assumes that a hydraulic system is required to deliver the amount
of assistive force required to turn the wheels. There are different principles for
how to change the torque experienced by the driver. One is to focus solely on
the relation between the driver’s torque and the level of assist. Another would
be to actively regulate the pressure difference across the assistance cylinder
piston.
From the model of the open-centre valve in section 3.1.2, it was seen that
the assistance pressure depends on the flow from the pump and the opening
areas of the valve. If one or the other can be altered by an electric signal,
the pressure could be actively controlled. The Electrohydraulic Power Assisted
Steering system, EHPAS, which is presented by for example Forbes, Baird,
and Weisgerber (1987), Gessat (2007) and Badawy, Fehlings, et al. (2004),
was primarly designed to overcome the poor energy efficiency of the HPAS
system. Intead of driving the pump from the combustion engine directly, an
electric motor is used to drive the pump. In this way the flow can be regulated
and during no steering action the flow can be shut off or heavily reduced.
Compared to the conventional system, fuel savings of up to 0.2 L per 100 km
have been reported. With this possibility the level of assist can also be changed
during different scenarios. If a fixed pump driven from the combustion engine is
used instead, the flow to the valve can be controlled by using an electronically
controlled valve prior to the open-centre valve that by-passes some of the oil
to tank. Such a system is described by Duffy (1988).
The variation in pump flow is restricted to only affecting the level of assist
without affecting the internal difference between the two cylinder chambers.
Autonomous driving is not possible with such a system. One way to control
the pressure difference across the piston is to actuate the open-centre valve
with an electronic actuator. An example of this is the Active Pinion presented
by M. Rösth, J. Pohl, and J-O. Palmberg (2003), illustrated in figure 4.3. With
this concept an offset in valve displacement can be created that generates an
offset in pressure difference across the assistance cylinder; a torque can thus
be added to or subtracted from the driver’s torque. The displacement of the
valve is realised by a DC motor attached to an excenter, generating a linear
motion. This linear motion is transferred to a rotational motion by an inclined
groove in the valve body. The working envelope of the system is restricted to
the offset angle of the valve body that can be created. A benefit with such a
system is the low force required to actuate the valve and a relatively small DC
motor can be used.
Several other similar solutions exists that aim to create an offset displacement in the valve body. One is presented by Leutner (2005), where a stepper
motor is placed coaxially with the valve. The stator is connected to the pin-
48
Active steering
Steering wheel torque
θsw
DC motor
θAP
θpinion
Figure 4.3 Schematic of the Active Pinion concept developed by M. Rösth,
J. Pohl, and J-O. Palmberg (2003). The DC motor generates a displacement of
the valve, θAP , which shifts the boost curve.
ion and the rotor to the valve body. Another variant is the iHSA, intelligent
Hydraulic Steering Assist, developed by tedrive steering, see iHSA, intelligent
Hydraulic Steering Assist (n.d.). This solution is available for both rack and
pinion systems and recirculating ball screw and displaces the valve with an
electric motor and a planetary gear set.
The steering torque can also be modified by adding or subtracting a torque directly with an electromagnetic actuator. The Magnasteer presented by Pawlak,
Graber, and Eckhardt (1994) uses an electromagnetic actuator consisting of a
homopolar rotary actuator with a multipole ring permanent magnet that acts
directly on the torsion bar.
For heavy vehicles an electromechanical actuator can be utilised together
with the hydraulic system. The torque from the driver and from the electric
motor are added at the steering column, before the steering gear. Since the hydraulic system is still in place, the electric motor does not have to provide the
full assistance torque. A solution by ZF Lenksysteme is presented by Wiesel et
al. (2009) and is referred to as hybrid steering, see the illustration in figure 4.4.
The electric motor can add or subtract torque on the driver’s torque. There is
also a possible fuel saving by additionally controlling the pump-delivered flow.
When low torque levels are required, the assistance from the electric motor
alone is sufficient to turn the wheels and the pump flow delivered to the valve
can be reduced or even shut off. When the load on the steering system is higher,
49
On Electrohydraulic Pressure Control for Power Steering Applications
Steering wheel torque
DC
motor
DC motor torque
Pitman arm
Figure 4.4 Schematic of the Hybrid steering, see Wiesel et al. (2009). The
electric motor works in parallel to the hydraulic system and can add or subtract
a torque to the driver’s torque.
for example during parking manuouevers, the pump flow can be increased and
the hydraulic system will also provide the level of assistance torque required
to turn the wheels. TRW also has a similar solution, the ColumnDrive, presented by Williams (2009). The electric motor on the steering column works
to generate a synthetic torque feedback experienced by the driver while the
hydraulic system generates the necessary amount of assistance torque. In this
way, the steering feel can be improved on, such as a variable stiffness depending
on the driving scenario, no hysteresis and better returnability. The ColumnDrive was also investigated together with a closed-centre hydraulic system by
Yu and Szabela (2007). The valve is a rotational valve like the conventional
system but in neutral position the flow paths to tank are closed. The valve
is designed with a small overlap to reduce leakage but also to ensure a stable
system. With the closed-centre valve the system can be supplied by an electric
motor driven pump connected to an accumulator and the energy consumption
can be reduced. The overlap of the valve has a negative effect on the steering
feel generating what is called a "bump" steer. This is improved on by tuning the
steering feel with the ColumnDrive. Volvo Trucks also uses a similar system
to generate a superimposed steering torque through the hydraulic system, see
Volvo Dynamic Steering (2013). Here the electric motor is attached directly on
the hydraulic steering gear and the system is used to improve both the steering
feel and the handling of the vehicle.
50
Active steering
A completely different approach to varying the level of assist is to replace the
conventional hydraulic system with a system that uses electronically controlled
valves. Any pressure can then be set in the assistance cylinder based on any
input of choice. If the valve also has a closed centre the supply system can be of
a flow-regulated pump of some kind connected to an accumulator. One possible
solution is studied by S. Müller, A. Kugi, and W. Kemmetmüller (2005), where
a directional 4/3-valve is used. The energy saving potential is high with such
a system where the closed-centre system only used 18% of that of the opencentre system for the given driving cycle. Since a directional valve is intended
to control the flow, the system requires an active closed loop control of the
pressure. With regard to the steering feel, there is a trade-off between a low
leakage and a good feel. To control the pressure fast enough, an offset voltage
to the valve is required but this increases the leakage flow and reduces the
energy efficiency.
Another possibility is to use valves intended to control the pressure instead,
such as pressure-reducing or pressure-relief valves, or a combination of both,
that independently control the pressure in each cylinder chamber. Several similar solutions can be found. Wenzel, Stobbs, and Ward (1996) proposed a
system with two 3-way pressure reducing valves, one for each cylinder chamber. A working pressure is controlled by a pressure control valve depending
on the driving scenario. The 3-way valves then reduces the pressure to each
chamber to generate the right amount of assist. Another approach presented by
Pluschke and Stehli (2001) is to use similar 3-way valves to control the pressure
in the assistance cylinder but in combination with a supply system consisting
of pump with accumulator. The same system is also studied by Wolfgang Kemmetmüller, Steffen Müller, and Andreas Kugi (2007). A nonlinear controller
is developed to control the pressure in the assistance cylinder with the desired
performance and impedance matching is used to generate the right amount of
assistance. From a failsafe perspective such valve will open to tank in case of
loss of electricity. The vehicle will therefore alway be steerable through the
mechanical connection. Since the pressure is independently controlled in each
chamber, the same pressure could also be set in both chambers, creating a
hydraulic lock, for safety reasons. A variant of the same system is proposed
by Mesiti and Avenatti (1995). One electronically controlled pressure control
valve is used together with a 4/3-directional valve. The pressure control valve
controls the required pressure to generate the assistance torque while the directional valve, that is mechanicallly connected to the steering column, connects
the pressure control valve to one or the other chamber of the assistance cylinder
depending on the direction of the steering movement.
51
On Electrohydraulic Pressure Control for Power Steering Applications
θsw
Pressure control
valve
ECU
xrw
Figure 4.5
Schematic principle of closed-centre power steering with pressure
control valves.
4.3
Selected technology for active steering
The choice of technology in this work is the use of independently controlled
pressure control valves as depicted in figure 4.5. There are several reasons why
such a system is beneficial:
− The purpose of the hydraulic system in the power steering application
is to control the pressure. It is therefore logical to use valves that are
designed to control the pressure and the system can keep its fundemental
structure.
− The valves are of closed-centre type. It has already been explained that a
closed-centre system has great potential in reducing energy consumption.
However, it requires a different structure of the supply system.
− Hydraulic systems are known for a high power density compared to electric systems. The valves are compact and provide a space-efficient solution.
− The actuation power of the valve is low. If a constant torque is to be
applied by the steering system, e.g. during steady-state cornering, little
energy is used to achieve this compared to an electric motor.
− Hydraulic power is today still necessary in order to overcome the high axle
loads present in heavy vehicles. Since the hydraulic system is already in
place, by using pressure control valves there is no need for additional
technology.
52
Active steering
− Pressure control valves already exist on the market for several different
applications. If it were possible to use an off-the-shelf valve, the solution
could also be cost efficient.
The next chapter will give a deeper analysis of the pressure control valve.
53
On Electrohydraulic Pressure Control for Power Steering Applications
54
5
Pressure control
Based on the analysis of the HPAS system in chapter 3 the main purpose of
the hydraulic system is to provide a pressure to assist the driver when turning
the wheels. The open-centre valve controls the pressure through its metering
edges. The valve is acutated by the driver when the steering wheel is turned,
which results in a twisting of the torsion bar and a distorsion of the metering
edges of the valve. Pressure can be controlled in different ways by letting either
a pump with variable displacement or a fixed pump with variable speed control
the pressure or by using a valve. Generally, a valve is both faster and cheaper
compared to a pump. One way is to use a kind of directional valve, e.g. a
proportional valve or a servo valve. These mainly control flow by displacing the
spool and to control the pressure they need to be actively controlled. Another
way is to let the valve sense the pressure to control it directly. This can be
solved hydro-mechanically and such a valve is called a pressure control valve
and is considered in this work. Commonly of closed-centre type, it removes any
additional flow paths to tank and the required flow can be greatly reduced. A
valve of such a type that is also actuated by an electro-magnetic mechanism, a
solenoid, can also be controlled from a computer.
A pressure control valve can be either a pressure relief valve or a pressure
reducing valve. The relief valve is one of the most common components in
every hydraulic systems, where it secures the maximum pressure. It can also
be used to set a certain pressure level in a system. The valve is typically a seat
valve with the schematics shown in figure 5.1. The controlled pressure acts on
the seat whereas at the other end a spring is exerting a force, keeping the seat
in balance. If the pressure is larger than the corresponding spring force, the
valve will open to tank and relieve the pressure. The pretension of the spring
corresponds to the opening pressure, or reference pressure, of the valve. In
order to control the pressure, the pressure must be higher than the reference
value and a flow will pass the seat.
A pressure-reducing valve reduces the system pressure from a higher to a
lower value and is designed to maintain a certain pressure regardless of varia-
55
On Electrohydraulic Pressure Control for Power Steering Applications
Tank
System
Figure 5.1
Schematic of a pressure relief valve.
tions in the supply pressure as long as the supply pressure is higher than the
intended system pressure. This valve is usually of spool type, shown schematically in figure 5.2. Here again a spring exerts a force on the valve, keeping
the valve in a normally opened position. For both types of valve, the static
characteristics are constant in the ideal case, i.e. the pressure is the same for
a given reference value regardless of the flow through the valve. In reality the
pressure can change significantly with the flow. This has to do with both the
mechanical spring and the flow forces acting on the seat or the spool.
Up stream
Down stream
Figure 5.2
Schematic of a pressure reducing valve.
A pressure valve can be of either a single-stage or a two-stage type. In
the former case, the spool is controlled directly by the spring force. In the
latter case, the pilot flow is needed to create a pilot pressure that acts on the
main stage. The two-stage valve has much better static characteristics and can
handle much higher flow rates than the single-stage valve. However, it is more
complicated in its design. A pressure valve that is actuated by a solenoid is
said to be proportional. For the single-stage valve the solenoid acts directly
on the main spool. A current through the solenoid generates a force that is
proportional to the current. The force corresponds to the reference pressure of
the valve, hence the name proportional pressure control valve. A proportional
solenoid can only provide moderate forces and becomes a limiting factor in the
design of the valve. In the two-stage case, the solenoid acts on the pilot-stage
56
Pressure control
instead.
In the context of this work, a pressure control valve is a valve with both
relieving and reducing capabilities. It can also be referred to as a reducing
valve with relieving capability. Since the single-stage valve is simpler in its
design, and therefore cheaper, it is the type of valve considered in the following
analysis.
This chapter provides the modelling and an in-depth analysis of a singlestage pressure control valve. The chapter is divided into two parts. The first
part gives a general analysis of the hydro-mechanical part of the valve, both
from a static and a dynamic viewpoint. As it turns out, the hydro-mechanical
subsystem constitutes a closed-loop control system. Both the nonlinear model
and a linear model are presented. The second part of the chapter applies
the theory on a commercially available valve. This valve is a proportional
electronically controlled valve. The modelling and analysis must therefore also
consider the solenoid. The theory is also extended in order to apply to the
valve under consideration.
5.1
General modelling and analysis
xv
ps
pl
pt
Damping chamber
Solenoid
force
Damping orifice
Figure 5.3
Schematic of a pressure control valve.
The pressure control valve, illustrated in the simplified schematic in figure
5.3, works both as a reducing and a relieving valve. Since a single-stage valve
is considered, the valve is actuated by a solenoid force acting directly on the
spool. A spring force with a certain pretension works in the opposite direction.
The load pressure pl is also sensed by the valve against a defined area Ac which
acts in the same direction as the spring. The solenoid force corresponds to the
reference pressure. If the load flow ql is zero, the valve will be in equilibrium
when the corresponding force from the load pressure is equal to the solenoid
force. If the load pressure is lower than the corresponding solenoid force, the
valve will open up to the pump to fill the load volume with oil. If the pressure is
higher than the force, the valve will open up to tank to relieve the pressure. The
valve is then said to be self-regulating. This will also be clear from the linear
57
On Electrohydraulic Pressure Control for Power Steering Applications
analysis presented below. If a load flow is present in either direction, the valve
will find an equilibrium in an offset position to the closed one. The spring force
will be different in this case due to the compression of the spring and since the
valve is also subjected to flow-dependent flow forces the load pressure will have
to change as well in order to maintain equilibrium. This change in pressure
with flow is defined by the static characteristic of the valve. In many cases a
small orifice is placed between the load volume and the small volume in which
the load pressure is sensed, in this case called a damping chamber. This is to
increase damping, which is further explained below. The modelling is divided
into two parts. First, the nonlinear equations describing the valve’s behaviour
outlined. From these the static characteristics of the valve is defined. The
equations are then linearised for an analysis in the frequency domain, which
reveals some important behaviour of the valve.
The modelling of pressure control valves has been treated by several researchers. Merritt (1967) provides modelling and an analysis of single-stage
pressure relief and reducing valves, with a focus on the dynamic response. A
similar analysis of a pressure relief valve in both the frequency and time domain
was made by Jan-Ove Palmberg (1983). A pilot-controlled solenoid pressure relief valve is modelled and analysed using the bondgraph technique by Dasgupta
and Watton (2005). With a simplified model of the solenoid, the paper presents
a good model of the valve. A direct-acting solenoid valve for clutch control is
modelled and analysed by Walker, Zhu, and Zhang (2014). A nonlinear detailed model, where the solenoid characteristic is mapped, is presented. The
model agrees well with both step inputs and steady state. A general linearised
analysis of a pressure relief valve is presented in Jan-Ove Palmberg, Andersson,
and Malmros (1983). It is shown how most valves can be characterised by a
dominating break frequency and compliance and the system being connected to
the valve by the hydraulic capacitance and the flow-pressure coefficient. This
was also verified by experiments. A design process for a proportional pressure
relief valve is proposed by Erhard, Weber, and Schoppel (2013). An inverse
simulation technique is used to derive a design for both the hydraulic subsystem and the solenoid characteristics. The valve is simulated using both CFD
for the hydraulic subsystem and the Finite Element Method for the solenoid.
5.1.1
Nonlinear model of the hydro-mechanical subsystem
Equation of motion
The movement of the spool is dictated by the equation of motion in equation
5.1. The forces that act on the spool are the solenoid force Fs , the pressure in
the damping chamber pc , which is the same as the static pressure in the static
case, the spring force, the viscous friction, the pretension f0 and the flow force
Ff . The PWM controlled solenoid usually has superimposed dither frequency,
which results in micromovements of the spool. This eliminates the Coulomb
58
Pressure control
friction to such an extent that it can be excluded in the model.
mv ẍv = Fs − pc Ac − f0 − Kxv − bv vv − Ff (xv , ps , pl )
(5.1)
The reference pressure is set by the solenoid force and has the relation shown
in equation 5.2.
Fs = pref Ac + f0
(5.2)
Flow equations
The valve has two main flow paths: from pump to load volume, qvs , and from
load volume to tank, qvt . A small part of the main flow will go to the damping
chamber. The flows are described by equations 5.3 and 5.4.
r
2
(ps − pl )
(5.3)
qvs = Cq As (xv )
ρ
r
2
qvt = Cq At (−xv )
pl
(5.4)
ρ
Load volume
The pressure change in the load volume depends on the net flow into the volume
and is decribed by the continuity equation in equation 5.5. ql is the load flow.
qvs − qvt − qc = ql +
Vl
ṗl
β
(5.5)
Control chamber
The pressure in the control chamber depends on both the flow through the
damping restrictor, equation 5.6, and the continuity equation for the volume,
equation 5.7.
r
2
(pl − pc )
(5.6)
qc = Cq Acr
ρ
Vc
qc = −Ac vv + ṗc
(5.7)
β
Flow forces
The valve is subjected to flow forces that tends to close the valve. These are
flow-dependent and exist for both main flows through the valve. The total flow
force is expressed in equation 5.8.
Ff (xv , ps , pl ) = |2Cq As (xv )(ps − pl )cos(δ)| − |2Cq At (−xv )pl cos(δ)|
5.1.2
(5.8)
Static characteristic
The static characteristic of the valve defines how the pressure will change for
a given flow and, as said before, is an effect of the spring compression and
59
On Electrohydraulic Pressure Control for Power Steering Applications
flow forces acting on the spool. It can also be interpreted as how accurately
the valve is able to control the pressure. However, as will be seen from the
dynamic analysis later on, there is a compromise between control accuracy and
the stability of the valve. The valve is first analysed when it is open to pump
and later when it is open to tank. The equations describing the static behaviour
of the valve are equations 5.9 and 5.10. In order to be able to solve for the
pressure, the opening area is considered to be linear, i.e. A = wxv , where w is
the area gradient. In reality the opening area is most likely a nonlinear function
of the spool displacement and should therefore be linearised in order to solve
the equations.
r
2
(ps − pl )
(5.9)
qv = Cq wxv
ρ
pref Ac − pl Ac − Kxv − Kf (ps − pl )xv = 0
(5.10)
The static equations can be solved for the spool displacement, equation 5.11,
and the flow, equation 5.12.
Ac (pref − pl )
K + Kf (ps − pl )
K0 Ac (pref − pl ) √
qv =
ps − pl
Ke
xv =
(5.11)
(5.12)
where
r
2
ρ
Kf = 2Cq wcos(δ)
K0 = Cq w
Ke = K + Kf (ps − pl )
The term Ke can be regarded as an equivalent spring constant. For a given
pressure drop a larger spool displacement will result in a larger flow and thus
a larger flow force. The static characteristic can be solved by differentiating
equation 5.12.
K 0 Ac √
pref − pl
Kf (pref − pl )
∂qv
=
ps − pl −1 −
+
∂pl
Ke
2(ps − pl )
Ke
√
K0 Ac ps − pl
Ac Kq
1
=−
=−
(5.13)
≈−
Ke
Ke
Rv
Ke
⇒ Rv = −
(5.14)
Ac K q
q
where Kq = Cq w ρ2 (ps − pl ). The static characteristic is denoted Rv , equation
5.14, and corresponds to the incline of the pressure-flow curve. It is very
60
Pressure control
dependent on the operating point, such as spool displacement and pressure. In
the case of a positive spool displacement, the static characteristic is negative
meaning that a larger flow will reduce the load pressure, like a reducing valve.
The static characteristic of a negative spool displacement can be derived in
the same way. The static equations for this case are expressed in 5.15 and 5.16.
r
2
pl
ρ
pref Ac − pl Ac + Kxv + Kf pl xv = 0
qv = Cq wxv
(5.15)
(5.16)
These can again be solved for the spool position, equation 5.17, and the flow,
equation 5.18.
Ac (pref − pl )
K + Kf pl
K0 Ac (pref − pl ) √
pl
qv = −
Ke
xv = −
(5.17)
(5.18)
where Ke = K + Kf pl . The static characteristic is then derived by differentiating equation 5.18 and the result is expression 5.19.
Rv =
Ke
Ac Kq
(5.19)
q
where Kq = Cq w ρ2 (pl ). In this case, for a negative spool displacement, the
static characteristic is positive and a larger flow will increase the pressure, like
a relief valve.
5.1.3
Linear model and analysis
A linear analysis in the frequency domain is very useful since the valve can
be studied with regard to stability. Many properties are revealed and a better
understanding of the valve is possible. The first step is to linearise the system
equations, which is done around a specific operating point. Only the pressure
reducing case is considered for this analysis. The pressure relief case is identical.
The linearised system equations are presented in equations 5.20 to 5.29. The
∆-symbol indicates a change in the state variable. Index 0 indicates the state
variable’s value at a given operating point.
Equation of motion
mv ∆ẍv = ∆pref Ac − ∆pc Ac − K∆xv − bv ∆ẋv
− Kf x ∆xv − Kf p (∆ps − ∆pl )
(5.20)
61
On Electrohydraulic Pressure Control for Power Steering Applications
The derivatives are defined according to equations 5.21 and 5.22.
∂Ff (xv0 , ps0 , pl0 )
(ps0 − pl0 ) = 2Cq wcos(δ)(ps0 − pl0 )
∂xv
∂Ff (xv0 , ps0 , pl0 )
=
xv0 = 2Cq wcos(δ)xv0
∂pl
Kf x =
(5.21)
Kf p
(5.22)
The derivative Kf p is usually small in the context and will be ignored from
here onwards.
Flow equation
∆qv = Kq ∆xv + Kc (∆ps − ∆pl ) − ∆qc
(5.23)
The derivatives are defined according to equations 5.24 and 5.25.
∂qv (xv0 , ps0 , pl0 )
= Cq w
Kq =
∂xv
r
2
(ps0 − pl0 )
ρ
(5.24)
q
Kc =
Cq wxv0 ρ2
∂qv (xv0 , ps0 , pl0 )
∂qv (xv0 , ps0 , pl0 )
=−
= √
∂ps
∂pl
2 ps0 − pl0
(5.25)
Load volume
The linearised continuity equation of the load volume is expressed in equation
5.26.
Vl
∆qv − ∆ql = ∆ṗl
(5.26)
β
Control chamber
The linearised flow and continuity equations for the control chamber are defined
in equations 5.27 and 5.28.
∆qc = K1 (∆pl − ∆pc )
Vc
∆qc = −Ac ∆vv + ∆ṗc
β
(5.27)
(5.28)
The derivative is defined according to equation 5.29.
q
2
C
A
q
cr
ρ
∂qc
∂qc
√
K1 =
=−
=
∂pl
∂pc
2 pl0 − pc0
(5.29)
Laplace transformation
In order to perform an analysis in the frequency domain the linearised system equations are transformed into the Laplace domain. These equations are
62
Pressure control
defined in equations 5.30 to 5.34, where s is the Laplace operator.
mv ∆xv s2 = ∆pref Ac − ∆pc Ac − K∆xv − Bv ∆xv s
− Kf x ∆xv
Ke = K + Kf x
⇒ mv ∆xv s2 = ∆pref Ac − ∆pc Ac − Ke ∆xv − Bv ∆xv s
(5.30)
∆qv = Kq ∆xv + Kc (∆ps − ∆pl ) − ∆qc
Vl
∆qv = ∆ql + ∆pl s
β
Vc
∆qc = −Ac ∆xv s + ∆pc s
β
∆qc = K1 (∆pl − ∆pc )
(5.31)
(5.32)
(5.33)
(5.34)
Equations 5.30 to 5.34 can be rearranged into equations 5.35 to 5.37 and used
to form a block diagram
∆xv =
Ac ∆pref − ∆pc
Kq mv s2 + Bv s + Ke
(5.35)
∆pc =
Ac
K1 ∆xv s + ∆pl
1 + KV1cβ s
(5.36)
Vc
Ac
Vc
s +Kq 1 +
+
(Kc ∆ps − ∆ql ) 1 +
s ∆xv
K1 β
K1 β
Kq
Vc
Vl
s
1+
s
(5.37)
= Kc ∆pl 1 +
Kc β
K1 β
The block diagram describing the valve’s dynamic behaviour is shown in figure
5.4, where the following definitions are used.
K1 β
V
rc
Ke
ωm =
mv
r
Bv
1
δm =
2
Ke mv
Kc β
ωs =
Vl
ωc =
The behaviour of the valve depends very much on the damping restrictor.
If no restrictor is used, the term K1 would be inifite and the valve would be
dominated by a lowly damped mechanical resonance. Such a system would be
difficult to control and the valve would have to be designed for each specific
63
On Electrohydraulic Pressure Control for Power Steering Applications
∆ql − Kc ∆ps
1
1+ s
ωc
∆pref
∆p1
+
Ac
Ke
−
1
s2 + 2δm s+1
2
ωm
ωm
xv
Kq
1+
1 + Ac
ωc
Kq
+
−
s
Kc
1
1+ s
ωc
1+ s
ωs
−
Ac
s
K1
1+ s
ωc
1
1+ s
ωc
Figure 5.4
A general block diagram representation of the hydro-mechanical
subsystem of the pressure control valve.
application. This was concluded by both Merritt (1967) and Jan-Ove Palmberg
(1983). To analyse the effect of the damping restrictor the inner control loop
around the mechanical resonance can be studied, as shown in equation 5.38.
Ac
s
1
+
Ke
ωc
∆xv
(5.38)
=
2δm
s
s2
∆p1
1+
s
+
1
+ s
+
2
ωc
ωm
ωm
ωv
e
where ωv = KA1 K
2 . There are basically two approaches depending on the size
c
of the damping restrictor. If the restrictor fulfils the criteria in equation 5.39,
the result is that of a valve without a damping restrictor but with an increased
mechanical damping. The resulting block diagram representation of this system
is shown in figure 5.5. The resonance ωc is very large and has been ignored.
K
There is also a pumping effect from the spool with resonance frequency Acq ,
which is usually very high and is therefore also ignored as well in the following
analyses.
A2
(5.39)
K1 ≥ √ c
2 Ke mv
In this system, stability must be checked for each application since it depends
∆ql − Kc ∆ps
∆pref
+
−
Figure 5.5
Ac
Ke
1
s2 + 2δm s+1
2
ωm
ωm
xv
+
Kq
−
Kc
1
1+ s
ωs
∆pl
A block diagram representation of a valve with a large damping
restrictor.
on the system parameters. The other solution is to have a damping restrictor
64
∆pl
Pressure control
that fulfils the criteria in equation 5.40.
s
K1 ≤ 2Ac
Vc
βmv
(5.40)
The result of this is that the expression in equation 5.38 can be factored into
the expression in equation 5.41.
∆xv
=
∆p1
1+
Ac
Ke
s
ωv
1+
s2
2
ωh
s
ωc
+
2δh
ωh s
+1
(5.41)
q
A2c β
Kh
ωc
where ωh =
mv , Kh = Vc , δh = 2ωh . With the restrictor small enough,
the oil inside the damping chamber will be entrapped and the result is that
the mechanical resonance has been replaced by a hydraulic resonance. This
resonance is much higher and will not dominate the system any more. It can
therefore be ignored in the analysis, together with ωc . The system will instead
be dominated by a low frequency lag, ωv , and the lag from the system volume.
The resulting block diagram representation is shown in figure 5.6. The stability
must be designed with respect to the phase margin, rather than the amplitude
margin, as is the case with a large damping restrictor. Nevertheless, the design
is very insensitive to variations in system parameters.
∆ql − Kc ∆ps
∆pref
+
−
Figure 5.6
Ac
Ke
1
1+ s
ωv
xv
+
−
Kq
Kc
1
1+ s
ωs
∆pl
A block diagram representation of a valve with a large damping
restrictor.
What can be seen from both cases is that the static loop gain Kv is defined
by equation 5.42 and is also defined by the static characteristic Rv of the valve.
This means that a desirable low static characteristic will yield a high loop gain
and a valve that is difficult to control unless the damping is high enough.
Kv =
Ac Kq
1
=
Ke Kc
Rv Kc
(5.42)
In the subsequent section, the above theory will be applied to a commercial
valve. A model is derived and the valve is analysed using the techniques above.
65
On Electrohydraulic Pressure Control for Power Steering Applications
5.2
Modelling and analysis of a commercial proportional solenoid pressure control valve
In this section a commercial valve is analysed as a potential candidate for
the electro-hydraulic power steering system and the work relates to research
question 1). Using a commercially available valve is an attractive solution
since development costs can be reduced. The valve is the Hydac PDR08-11
proportional solenoid pressure control valve, which is primarily designed as a
pilot stage for a spool valve. A simplified schematics of the valve is shown in
figure 5.7. The valve is a single-stage valve. It has three ports: one to pump,
Core
Coil
Spool
Tank
Pump
Load
Figure 5.7
Simplified schematics of the Hydac pressure control valve.
one to tank and one to load system. The valve is actuated by a solenoid that
pushes to open the valve to the pump directly on the spool. In the opposite
direction a preloaded spring pushes on the spool. The spring is very weak in the
context and serves only to keep the spool in place when no pressure or solenoid
force is present. The load pressure also acts on the spool on a predefined surface
in the opposite direction to the solenoid force. The solenoid is driven by a 200
Hz PWM-modulated voltage with an amplitude of 12 V. The low frequency
ensures a dither effect in the signal, which will give micromovements of the
spool and reduce stick-slip. The force from the solenoid is a function of the
current through the coil. The reference pressure is therefore set by the solenoid
current. Since the coil’s resistance can vary with temperature the current will
also vary for a given voltage. The common solution is to use some kind of
66
Pressure control
current controlller to keep it at a desired value. This, however, is not used
here. As described in the section above, this type of valve is self-regulating.
A small restrictor is placed between the load volume and a much smaller
volume above the spool. This solution serves to increase the damping, as explained in the previous section. A further analysis of the restrictor is presented
below, which reveals its true impact on the valve. The opening areas of the
valve consist at the pump side of 12 circular holes around the valve body, each
with a diameter of 1.5 mm. On the tank side, there are also 12 holes in the
same manner. However, a single small hole of 0.8 mm diameter is placed before
the larger holes with an offset of 0.3 mm. This generates a much smoother area
gradient on the tank side compared to the pump side.
With the centre position of the spool at 0 mm, the spool’s stroke is from -1
mm to 0.85 mm. The core’s stroke is approximately from -1.6 mm to 0.85 mm.
The modelling of the valve serves to explain both the valve’s particular
behaviour and help the control design process. Measurements of the valve,
shown in section 5.2.2, show non-typical behaviour, where the pressure-reducing
curves have negative static characteristics. As it turns out, the design of the
valve makes it less suitable for the intended application. However, the valve
shows some intriguing behaviour and there are several lessons to be learnt during the modelling procedure. The behaviour of the valve suggests that it is
important to consider in detail the flow-pressure relationship and flow forces of
the valve, as well as the solenoid’s characteristics when modelling the valve.
Merritt (1967) gives a good overview of different types of fluid flow. The
flow is usually turbulent and the discharge coefficient can be assumed to be a
constant value. The flow is then proportional to the square root of the pressure
drop. In cases where the flow is more laminar, the discharge coefficient is
instead modelled as a function of the square root of the Reynolds number.
In a strict laminar case, this yields that the flow is directly proportional to
the pressure drop. Wu, Burton, and Schoenau (2002) propose an exponential
model of the discharge coefficient as a function of the Reynolds number. Since
the Reynolds number itself is dependent on the flow, this model requires an
iterative process or a pre-defined look-up table to solve. Ellman and Piché
(1996) propose a two-region flow model for numerical reasons. This is also done
by Åman, Handroos, and Eskola (2008), who use a cubic spline curve for the
laminar and transition regions. This method has proven to be computationally
effecient compared to the traditional orifice flow model. Borutzky, Barnard,
and Thoma (2002) propose a single formula for the discharge coefficient for
better numerical effeciency. This results in a linear and a quadratic term in
the pressure flow relation.
The flow forces of the open-centre directional valve were measured and analysed by Amirante, Vescovo, and Lippolis (2006). Depending on the position of
the spool, the flow forces initially act to open the valve, whereas they change
direction to close the valve after a certain opening. Stationary flow forces were
also analysed on compensated spool valves by Borghi, Milani, and Paoluzzi
67
On Electrohydraulic Pressure Control for Power Steering Applications
(2000). Computional Fluid Dynamics (CFD) was used to analyse the jet flow
angle and the flow forces in general for different designs.
A detailed model of a solenoid is presented by N. Vaughan and J. Gamble
(1996). By measuring the current and voltage of the solenoid, the magnetization curves can be derived, which also include hysteresis effects. A PWM-driven
solenoid for ABS valves is modelled by Branciforte et al. (2011). The fast dynamics are modelled using a fractional order system, while the hysteresis is
covered by using neural networks.
5.2.1
Measurements and modelling of the solenoid
In this section the measurements and modelling procedure of the solenoid is
presented. The test rig is first explained, followed by the mathematical description. A validation of the model is presented last. Magnetic properties are
difficult to measure and no material properties of the coil are available. The
measurements and modelling procedures are therefore based on the work done
by N. Vaughan and J. Gamble (1996), which requires the voltage, current and
force of the solenoid to be measured.
Test rig for solenoid and measurements
A schematic of this test rig is shown in figure 5.8a and a photo in figure 5.8b.
A low-speed electro-mechanical actuator with a load cell attached to its end
Solenoid
Core
Load
cell
Low speed actuator
(a) A schematic of the measurement set-up
of the solenoid.
Figure 5.8
(b) A photo of the measurement
set-up of the solenoid.
Solenoid test rig.
was used to control the position of the magnet core. The solenoid was clamped
such that the core was resting against the load cell in the horizontal position.
The solenoid was supplied with 12 V. Two types of test were conducted. The
first tests measured the static relationship between current and force. Since
the core was only resting against the load cell and no spring was used, no
PWM-modulation was used. Instead, the supply voltage was slowly increased
while current and force were registered. This was done for 25 positions over
the entire stroke. The other tests applied step input signals, also at 25 different
positions over the entire stroke. The voltage, current and force were measured.
68
Pressure control
Before each test the solenoid was heated by applying maximum current to avoid
fluctations in the coil resistance. All tests were sampled at a rate of 0.1 ms.
The results from the first tests are shown in figures 5.9 and 5.10. As can be
seen in figure 5.9, the force is quite nonlinear for small currents and there is
a variation with coil stroke which tends to be less influencial within a certain
region. This is confirmed by figure 5.10 which shows the solenoid force against
stroke at different current levels. The static behaviour is strongly nonlinear but
have a region where the behaviour tends to be more linear. This is where the
valve has its working region. In these measurements, zero position corresponds
to the coil’s end position, furthest away from the spool. The fluctations in
current have been considered in the figure 5.10. The results from the second
60
50
Increased position
Force [N]
40
30
20
10
0
0
Figure 5.9
0.2
0.4
0.6
0.8
1
1.2
Current [A]
1.4
1.6
1.8
2
Static measurements of solenoid force against current.
type of test, are shown in figures 5.11, 5.12 and 5.13 for coil positions -1.6mm,
0.01mm and 0.81mm. The current shows the expected first order behaviour
and the force follows this.
69
On Electrohydraulic Pressure Control for Power Steering Applications
Current = 0.5 [A]
Current = 1 [A]
20
6
4
2
15
10
0
1
2
x [mm]
Current = 1.5 [A]
5
3
50
70
40
60
Force [N]
Force [N]
25
Force [N]
Force [N]
8
30
20
10
0
1
2
x [mm]
Current = 2.1 [A]
3
50
40
0
1
2
30
3
0
1
x [mm]
2
3
x [mm]
Figure 5.10
Static characteristic of the solenoid represented as force against
coil position at different current levels.
14
12
10
Voltage [V]
8
6
4
2
0
−2
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
Figure 5.11 Dynamic response of solenoid voltage for coil positions -1.6mm,
0.01mm and 0.81mm.
70
Pressure control
2
Current [A]
1.5
1
0.5
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
Figure 5.12 Dynamic response of solenoid current for coil positions -1.6mm,
0.01mm and 0.81mm.
60
50
Force [N]
40
30
20
10
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
Figure 5.13
Dynamic response of solenoid force for coil positions -1.6mm,
0.01mm and 0.81mm.
71
On Electrohydraulic Pressure Control for Power Steering Applications
Mathematical description
φ
VL L
i
Vs
Vs
VR R
(a) A simplified schematic of a linear
solenoid.
Figure 5.14
Air
gap
(b) The magnetic path of the solenoid.
Representation of a solenoid.
A solenoid is basically a coil wired around an armature and in its simplest
form it can be seen as a resistor in series with an inductor as shown in figure
5.14. By applying a current through the coil a magnetic field is produced where
the magnetic field intensity H is described by Ampere’s law as in equation 5.43,
see e.g. Chapman (1998)).
I
Hdl = Iencl
(5.43)
where Iencl is the current enclosed by the magnetic path. If the magnetic path
of the solenoid is seen as a closed core with an air gap and the coil wrapped
around one leg, as illustrated in figure 5.14, the magnetic field intensity is
described by equation 5.44.
Ni
H=
(5.44)
l
How large the magnetic flux φ becomes depends on the material in the core,
and in this case also the air gap. The relation between the magnetic field’s
intensity and flux density is given by equation 5.45.
B = µH
(5.45)
where µ is the permeability of the material. The total magnetic flux is given by
φ = BA. The permeability, however, is not constant for different fields. The
easiest way to describe the relation between field intensity and field density
is with a magnetization curve. A typical curve is shown in figure 5.15. The
magnetization curve shows two typical phenomena: hysteresis, i.e. losses in
the core, and saturation in the magnetic flux. Faraday’s law, equation 5.46,
states that if a magnetic flux passes through a turn of a coil, a voltage will be
induced that is proportional to the rate of change of the flux in time.
V =−
72
dφ
dt
(5.46)
Pressure control
λ
id
i
ir i
Figure 5.15 A typical magnetization curve represented as flux linkage
against current.
The total flux passing through all turns is given by λ =
that Faraday’s law can be expressed as in equation 5.47.
VL =
dλ
dt
PN
i=1
φi . This yields
(5.47)
where VL is the induced voltage over the solenoid. If it is assumed that the
same flux passes through all windings, the flux linkage becomes λ = N φ. This
in turn gives an expression for the flux density as B = NλA . Since both B and
H are difficult to measure, the magnetization curve can instead be expressed
with flux linkage λ and current i, which are easy to measure. The current
is directly measurable. Since the inductive voltage is the difference between
the total voltage and the resistive voltage according to equation 5.48, the flux
linkage λ can be derived by integrating the inductive voltage as in equation
5.49.
VL = Vs − iR
(5.48)
Z
λ=
Z
(Vs − VR )dt =
(Vs − iR)dt
(5.49)
The inductive voltage of the solenoid, calculated from the measured total
voltage and current, is shown in figure 5.16a. The coil resistance is calculated
from the voltage and current at steady-state condition. The corresponding
magnetization curves are then calculated from the inductive voltage with the
result shown in figure 5.16b. The slope of the curves varies for different coil
positions.
The idea presented by N. Vaughan and J. Gamble (1996) is to model the
solenoid as a resistor in series with a nonlinear inductor, which also considers
the hysteresis seen from the magnetization curves. The total current through
the solenoid consists of an energy-restoring part and an energy-dissipating part,
as described in equation 5.50. The restoring function depends on both the flux
73
On Electrohydraulic Pressure Control for Power Steering Applications
0.18
8
0.16
6
0.14
4
0.12
Flux linkage [Wb]
Inductive voltage [V]
10
2
0
−2
Increased position
0.1
0.08
0.06
−4
0.04
−6
0.02
−8
0
0.1
0.2
0.3
Time [s]
0.4
0.5
(a) Inductive voltage calculated from the
measured voltage and current for coil positions -1.6mm, 0.01mm and 0.81mm.
Figure 5.16
0
0.6
0
0.5
1
Time [s]
1.5
2
(b) Measured magnetization curves for
coil positions -1.6mm, 0.01mm and
0.81mm.
Inductive voltage and magnetization curves.
linkage and the core position, while the dissipating function depends only on
the inductive voltage.
i = ir + id = f (λ, x) + d(VL )
(5.50)
The restoring function is found by calculating the mean current from the magnetization curves at each core position, as illustrated in figure 5.15, and fitting
a polynomial function. The best fit for this solenoid is described by equations
5.51 to 5.54. Figure 5.17 shows the comparison of the fitted function coefficients
to measured ones against the core position.
ir = f1 λ3 + f2 λ2 + f3 λ
(5.51)
5
4
3
2
(5.52)
5
4
3
2
(5.53)
f1 = pf11 x + pf12 x + pf13 x + pf14 x + pf15 x + pf16
f2 = pf21 x + pf22 x + pf23 x + pf24 x + pf25 x + pf26
2
f3 = pf31 x + pf32 x + pf33
(5.54)
The dissipating function is also fitted with a polynomial from the calculated
dissipating current, with a best fit as in equation 5.55. The result is shown in
figure 5.18. The dissipating current also shows a variation with coil position but
it is assumed to be of less importance and is therefore ignored. The validation
will confirm this assumption later.
id = pd1 VL3 + pd2 VL2 + pd3 VL + pd4
(5.55)
The existence of an air gap within the solenoid is the cause of a magnetic
potential difference across the air gap that produces an attractive force between
74
Pressure control
400
380
360
f1 [A/wb3]
340
320
300
280
260
240
220
−1.5
−1
−0.5
Core position [mm]
0
0.5
0
0.5
0
0.5
(a) f1
0
−10
f2 [A/wb2]
−20
−30
−40
−50
−60
−1.5
−1
−0.5
Core position [mm]
(b) f2
20
18
14
3
f [A/wb]
16
12
10
8
−1.5
−1
−0.5
Core position [mm]
(c) f3
Figure 5.17 Coefficients for the restoring current function. Dashed curves
are the fit.
75
On Electrohydraulic Pressure Control for Power Steering Applications
0.25
0.2
0.15
d
Dissipating current i [A]
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
−8
−6
−4
−2
0
2
Inductive voltage [V]
4
6
8
10
Figure 5.18 Fitted dissipating current (solid) compared to calculated ones
for coil positions -1.6mm, 0.01mm and 0.81mm.
the armature and the opposite stator. This force acts directly on the spool. N.
Vaughan and J. Gamble (1996) proposed here again a nonlinear model of the
force as a function of the flux linkage, since the flux linkage is easily obtained.
However, since the force against current was convenient to measure, the force
is modelled as a function of current and position. Again a polynomial model
is fitted, described in equations 5.56 to 5.60. Due to the strongly nonlinear
behaviour of the solenoid, the fit is limited to the valve’s working region where
the behaviour is more linear. The results from the fit are shown in figure 5.19.
Fs (i, x) = pfs1 i5 + pfs2 i4 + pfs3 i3 + pfs4 i2
(5.56)
3
2
(5.57)
3
2
(5.58)
3
2
(5.59)
3
2
(5.60)
pfs1 = pfs11 x + pfs12 x + pfs13 x + pfs14
pfs2 = pfs21 x + pfs22 x + pfs23 x + pfs24
pfs3 = pfs31 x + pfs32 x + pfs33 x + pfs34
pfs4 = pfs41 x + pfs42 x + pfs43 x + pfs44
A resulting block diagram representation of the solenoid model is shown in
figure 5.20. A time lag is added to the dissipating current to prevent it from
growing too fast.
76
Pressure control
10
3.5
3
5
2.5
2
pf 2 [N/A4]
s
s
pf 1 [N/A5]
0
1.5
1
−5
0.5
0
−10
−0.5
−1
0
0.5
1
1.5
Position [mm]
2
−15
2.5
0
0.5
(a) pfs1
1.5
Position [mm]
2
2.5
2
2.5
(b) pfs2
20
45
15
40
10
35
5
pf 4 [N/A2]
30
25
s
0
s
pf 3 [N/A3]
1
−5
20
−10
15
−15
10
−20
−25
0
0.5
1
1.5
Position [mm]
2
2.5
5
0
0.5
(c) pfs3
Figure 5.19
1
1.5
Position [mm]
(d) pfs4
Coefficients for the force function. Dashed curves are the fit.
Vs +
VL
1
s
λ
−
x
f (λ, x)
ir
1
1+τd s
d(VL )
VR
Figure 5.20
id
+
+
F
R
Fs (i, x)
A block diagram representation of the solenoid model.
Validation
The validation of the solenoid model is done by applying step voltages of 12
V and 6 V at the fixed coil positions -0.6 mm, 0.095 mm and 0.81 mm, which
is within the working range of the valve. The voltage reaches steady-state
before a step to 0 V is applied. The steady-state characteristic is therefore
77
On Electrohydraulic Pressure Control for Power Steering Applications
also validated from the same measurements. The coil resistance is calculated
at steady-state current and used in the simulation model. Figure 5.21 shows
the solenoid current for a 12 V step input. Figure 5.22 shows the solenoid force
for a 12 V step input. Figure 5.23 shows the calculated inductive voltage for
a 12 V step input. Figure 5.24 shows the calculated magnetization curves for
a 12 V step input. Figure 5.25 shows the solenoid force for a 6 V step input.
Figure 5.26 shows the solenoid force for a 6 V step input. Figure 5.27 shows
the calculated inductive voltage for a 6 V step input. Figure 5.28 shows the
magnetization curves for a 6 V step input.
78
Pressure control
60
2
50
1.5
Force [N]
Current [A]
40
1
30
20
0.5
10
0
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.5
0.6
0.5
0.6
(a) x = -0.6 mm
(a) x = -0.6 mm
60
2
50
1.5
Force [N]
Current [A]
40
1
30
20
0.5
10
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0
0.6
0
0.1
(b) x = 0.095 mm
0.2
0.3
Time [s]
0.4
(b) x = 0.095 mm
60
2
50
40
Force [N]
Current [A]
1.5
1
30
20
0.5
10
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
(c) x = 0.81 mm
Figure 5.21 Validation of the
solenoid current for a 12 V step
input at positions -0.6 mm, 0.095
mm and 0.81 mm. Dashed line is
simulation results.
0
0
0.1
0.2
0.3
Time [s]
0.4
(c) x = 0.81 mm
Figure 5.22
Validation of the
force for a 12 V step input at positions -0.6 mm, 0.095 mm and 0.81
mm. Dashed line is simulation results.
79
On Electrohydraulic Pressure Control for Power Steering Applications
0.18
10
0.16
0.14
5
Flux linkage [Wb]
VL [V]
0.12
0
−5
0.1
0.08
0.06
0.04
−10
0.02
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0
(a) x = -0.6 mm
0.5
1
Current [A]
1.5
2
(a) x = -0.6 mm
0.18
10
0.16
0.14
5
Flux linkage [Wb]
VL [V]
0.12
0
−5
0.1
0.08
0.06
0.04
−10
0.02
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0
(b) x = 0.095 mm
0.5
1
Current [A]
1.5
2
(b) x = 0.095 mm
0.18
10
0.16
0.14
5
Flux linkage [Wb]
VL [V]
0.12
0
−5
0.1
0.08
0.06
0.04
−10
0.02
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0
0.5
(c) x = 0.81 mm
1
Current [A]
1.5
2
(c) x = 0.81 mm
Figure
5.23
Validation of
the inductive voltage over the
solenoid for a 12 V step input at
positions -0.6 mm, 0.095 mm and
0.81 mm. Dashed line is simulation results.
80
Figure 5.24
Validation of the
magnetization curves for a 12 V
step input at positions -0.6 mm,
0.095 mm and 0.81 mm. Dashed
line is simulation results.
Pressure control
30
1
25
0.8
Force [N]
Current [A]
20
0.6
15
0.4
10
0.2
5
0
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.5
0.6
0.5
0.6
(a) x = -0.6 mm
(a) x = -0.6 mm
30
1
25
0.8
Force [N]
Current [A]
20
0.6
15
0.4
10
0.2
0
5
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0
0.6
0
0.1
(b) x = 0.095 mm
0.2
0.3
Time [s]
0.4
(b) x = 0.095 mm
30
1
25
0.8
Force [N]
Current [A]
20
0.6
15
0.4
10
0.2
5
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
(c) x = 0.81 mm
Figure 5.25 Validation of the
solenoid current for a 6 V step input at positions -0.6 mm, 0.095
mm and 0.81 mm. Dashed line is
simulation results.
0
0
0.1
0.2
0.3
Time [s]
0.4
(c) x = 0.81 mm
Figure 5.26
Validation of the
force current for a 6 V step input at positions -0.6 mm, 0.095
mm and 0.81 mm. Dashed line is
simulation results.
81
On Electrohydraulic Pressure Control for Power Steering Applications
6
0.12
4
0.1
Flux linkage [Wb]
VL [V]
2
0
−2
0.08
0.06
0.04
−4
0.02
−6
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0
0.2
(a) x = -0.6 mm
0.4
0.6
Current [A]
0.8
1
0.8
1
0.8
1
(a) x = -0.6 mm
6
0.12
4
0.1
Flux linkage [Wb]
VL [V]
2
0
−2
0.08
0.06
0.04
−4
0.02
−6
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0
0
(b) x = 0.095 mm
0.2
0.4
0.6
Current [A]
(b) x = 0.095 mm
6
0.12
4
0.1
Flux linkage [Wb]
VL [V]
2
0
−2
0.08
0.06
0.04
−4
0.02
−6
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
(c) x = 0.81 mm
Figure
5.27 Validation of
the inductive voltage over the
solenoid for a 6 V step input at
positions -0.6 mm, 0.095 mm and
0.81 mm. Dashed line is simulation results.
82
0
0
0.2
0.4
0.6
Current [A]
(c) x = 0.81 mm
Figure 5.28
Validation of the
magnetization curves for a 6 V
step input at positions -0.6 mm,
0.095 mm and 0.81 mm. Dashed
line is simulation results.
Pressure control
5.2.2
Measurements and modelling of the valve
This section presents the measurements, modelling of the hydro-mechanical
subsystem and validation of the complete valve. The test rig for measuring
the static and dynamic characteristics of the complete valve is explained first,
followed by a mathematical description of the hydro-mechanical subsystem.
The complete valve model is then validated. The analysis of the complete
system is performed according to section 5.1.
Test rig for valve and measurements
Supply unit
Solenoid
pressure valve
Load
volume
Variable orifice
(a) A schematic of the measurement
set-up of the valve.
Figure 5.29
(b) A photo of the measurement
set-up of the valve.
Solenoid pressure control valve test rig.
The test rig for measurements of the complete valve is shown in Fig. 5.29.
The valve is connected to a constant pressure source at 200 bar. A small
accumulator of 1 L is installed to minimise variations in the supply pressure.
The load consists of a fixed volume of 0.1 L. A flow-meter is installed on the
supply line, as well as a temperature sensor. The temperature varied between
37° C and 42° C for all tests. Two variable orifices are installed, one between
the load volume and tank and the other between the pump and load volume.
These are used to apply a positive or negative load flow. Load, supply and tank
pressures are measured with transducers. The magnet core housing is fitted
with a small screw at its end. The screw is used to bleed the core housing, since
it is filled with oil. When the screw is removed, the hole gives access to the
magnet core. A position transducer was installed to measure the core position.
As long as the core is in contact with the spool, the core position corresponds
to the spool position. The position transducer is spring-loaded to ensure it is
always in contact with the core, at the same time as the spring force is kept
low enough to have no impact on the valve.
Two types of test were conducted in order to capture the valve’s static and
dynamic characteristics. To measure the static characteristic of the valve, a
constant duty cycle was applied. Either of the two variable orifices was slowly
opened to increase the load flow while pressure, flow and spool position were
83
On Electrohydraulic Pressure Control for Power Steering Applications
registered. This was done from 30% - 100% duty cycle with steps of 10%,
both for positive and negative load flow, i.e. the valve is working either as a
reducing valve or as a relief valve. Figure 5.30a shows the pressure flow curves
and figure 5.30b shows the spool position against flow. At a certain negative
flow the magnet and core separate and the core is pushed to its end position.
When measuring the dynamic characteristic, the load flow was set at zero.
Steps were applied in the input signal for different levels and amplitudes while
pressure, current and spool position were registered. If too sudden an input
signal was applied when relieving the pressure, the magnet core and spool could
separate and cause great undershots in the pressure. This was clearly seen
from the measurements. Since this is an unwanted behaviour and is outside
the scope of this paper, a rate limiter was implemented for a decreasing input
signal. Different rates were evaluated. Figure 5.31 shows the response for a
step input from 40% to 90% duty cycle and a negative rate of 5000% 1/s, which
results in a large undershot in pressure. The rate limiter was tuned so that
the core and spool stayed in contact. A negative rate of 2000% 1/s turned out
to be sufficient. Figure 5.32 shows the response for a step input from 46% to
100% duty cycle.
200
0.8
180
0.6
160
0.4
120
Position [mm]
Pressure [bar]
140
100
80
0.2
0
−0.2
60
−0.4
40
−0.6
20
0
−5
0
5
10
Flow [l/min]
15
(a) Pressure vs. flow curves.
Figure 5.30
20
−5
0
5
10
Flow [l/min]
15
20
(b) Position vs. flow curves.
Static measurements of the solenoid pressure control valve.
Mathematical description
This section describes the mathematical model of the hydro-mechanical subsystem. When studying the static characteristic of the valve in figure 5.30, a
few things can be concluded that will affect the model of the valve. First, it
can be seen that the spool works with an offset from the centre position towards the tank side, i.e. a negative stroke. This is due to the difference in area
gradient and pressure drop over the meter-in and meter-out orifices. To cover
the leakage from the pump side, the valve needs to open somewhat to tank.
84
Pressure control
180
0.2
160
0
140
−0.2
−0.4
Spool position [mm]
Pressure [bar]
120
100
80
60
−0.6
−0.8
−1
−1.2
40
−1.4
20
0
−1.6
0
0.1
0.2
0.3
0.4
0.5
0.6
−1.8
0.7
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.7
Time [s]
(a) Pressure response.
(b) Spool position.
Figure 5.31
Dynamic response of valve for a step input from 40% - 90% with
negative rate of 5000% 1/s.
200
0.2
180
0.1
0
Spool position [mm]
Pressure [bar]
160
140
120
100
−0.1
−0.2
−0.3
80
−0.4
60
−0.5
40
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
(a) Pressure response.
0.7
−0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time [s]
(b) Spool position.
Figure 5.32 Dynamic response of valve for a step input from 46% - 100%
with negative rate of 2000% 1/s.
It is therefore important to consider this when modelling the valve. It is done
by regarding both the gap leakage and area opening on the pump side as two
orifices, each with different characteristics. The gap leakage on the tank side
is less important due to the smoother area gradient.
The second thing concluded is that generally the valve is working with low
flows. It is far from certain that the flow reaches fully turbulent flow and
the common assumption that the discharge coefficient is constant might be far
from the truth. This is checked by calculating the discharge coefficient from
measurements. The discharge coefficient is not purely a physical property and
any assumption and modelling mismatch will affect its behaviour.
Third, the static characteristic of the valve is very specific when working as
a pressure-reducing valve. As can be seen in figure 5.30a, the pressure begins
85
On Electrohydraulic Pressure Control for Power Steering Applications
to increase for larger flows, which is not the expected behaviour of a pressurereducing valve. The typical behaviour of a solenoid is an increased force with
stroke, i.e. for a smaller air gap, which is also seen in Figure 5.9. The increase
in force, however, is too small to fully explain the increase in pressure. The
explanation lies in the flow forces, as will be described below.
The model considers both the meter-in and the meter-out port of the valve
and is outlined in equations 5.61 to 5.82.
Spool motion
mv̇ = Fs (i, x)−Ac p+Acc pc −Kx−Bv −Ffin (x, ps , p)+Ffout (x, p)−f0 (5.61)
The spool is subjected to flow forces Ff in both directions, a spring force with
preload f0 and viscous friction with coefficient B. The total mass is denoted m,
Ac is the pressure sensing area, K is the spring stiffness and p is the pressure.
Static friction and stick-slip phenomena are assumed to be negligible due to
the dither frequency in the input signal. The spool’s stroke is limited between
-0.55 mm and 0.85 mm, which according to measurements is the working range
of interest.
Continuity equation
The pressure change depends on both the meter-in and meter-out flow as well
as a load flow as described by equation 5.62. The load flow could be a cylinder
piston moving.
V
(5.62)
qin − qout − qc − ql = ṗ
β
Orifice flow
The meter-in, qin , and meter-out, qout , flows are given by equation 5.63 and
5.64, where Cq is the discharge coefficient and ρ is the oil density.
r
2
(ps − p)
(5.63)
qin = Cq (x, ps , p)Ain (x)
ρ
r
2
qout = Cq (x, p)Aout (x)
p
(5.64)
ρ
On the pump side, the valve is modelled with two orifices, one for the gap
leakage and the other when the spool passes the holes. Equation 5.63, which
assumes turbulent flow, is used in both cases. The opening area is modelled
according to equation 5.66 for the meter-in orifice and equation 5.67 for the
meter-out orifice. The meter-in area considers the 12 circular holes and a plane
leakage gap at each hole. On the tank side, the opening area is modelled in
the same manner but also considering the small initial hole. The gap leakage
is modelled with the opening area in equation 5.65, which is a plane gap. The
reason to also inlcude the gap in the hole opening areas is to get a smooth
86
Pressure control
transition from the gap to the holes. When implementing the opening areas,
the range of the spool position x is considered with respect to the trigonometric
functions.
Agap = 2rhgap
r − x · 10−3
1 2
r 2arccos
Ain (x) = 12
2
r
−3
r − x · 10
−sin 2arccos
+ 2rhgap
r
1 2
rs − |x| · 10−3
2
Aout (x) = max πrs ,
r 2arccos
2 s
rs
−3
rs − |x| · 10
−sin 2arccos
rs
r − (|x| − 0.3) · 10−3
1 2
r 2arccos
+ 12
2
r
r − (|x| − 0.3) · 10−3
−sin 2arccos
+ 12· 2rhgap
r
(5.65)
(5.66)
(5.67)
Since the flow can be anything from laminar to turbulent, the discharge coefficient Cq is modelled to take this into account. The discharge coefficient
can be calculated from equation 5.63 or equation 5.64 and using the data from
figure 5.30. The discharge coefficient is shown in figure 5.33a for the pump
side and in figure 5.33b for the tank side. The discharge coefficient is split into
1.2
0.45
0.4
1
Discharge coefficient [−]
Discharge coefficient [−]
0.35
0.8
0.6
0.4
0.3
0.25
0.2
0.15
0.1
0.2
0.05
0
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Spool position [mm]
(a)
0.6
0.7
0.8
0
−0.5
−0.4
−0.3
−0.2
Spool position [mm]
−0.1
0
(b)
Figure 5.33 Measured (solid) and modelled (dashed) discharge coefficient
for the pump side a) and tank side b) at corresponding 30%, 60% and 90% duty
cycle.
two parts. For gap leakage, the flow is assumed proportional to the pressure
drop. According to the literature, Merritt (1967), the discharge coefficient is
87
On Electrohydraulic Pressure Control for Power Steering Applications
proportional to the square root of the Reynolds number Re , which is calculated
according to equation 5.68.
qdh
Aν
4q
⇒ Re =
Oν
Re =
dh =
4A
O
(5.68)
The discharge coefficient for the gap leakage is then modelled as in equation
5.69.
p
(5.69)
Cqgap = gg (x) Re
The function gg (x) is a geometrical term that will both compensate for any
errors in the area opening function and take into account the effective cross
area of the fluid stream. Studying the discharge coefficient, it can be seen that
the value tends to approach a constant value when the spool crosses the holes
at 0 mm. The flow goes from laminar to a mix of laminar and turbulent flow
as the spool opens more. The discharge coefficient is no longer proportional to
the square root of the Reynolds number and a more complicated relation would
require an iterative process to solve. This is not practical when implementing
the model in simulation software. In this case, the discharge coefficient is instead modelled as in equation 5.70, where it only depends on the spool position
and pressure.
Cqhole (x, p) = g(x)pγ(x)
(5.70)
The function g(x) is again a geometrical term, while γ(x) handles the transition
from laminar to turbulent flow and varies between 0 and 0.5. An arctan function
turned out to be suitable and the three functions γ(x), gg (x) and g(x) are
described in equation 5.71 to 5.73.
gg (x) = 0.02477e1.314x + 0.0642e24.41x
(5.71)
g(x) = 0.3973sin(3.493x − 0.6221)+
3.789sin(6.942x + 1.088)+
3.626sin(7.047x + 4.173)+
0.0009295sin(22.45x + 2.653)
(5.72)
γ(x) = ((−arctan(5(x − 0.3))) + π/2)0.3/π
(5.73)
The gap leakage on the tank side is assumed to be very small due to the
smoother area gradient and no separate measurements of it were recorded. It
is therefore lumped together with the main flow and the pressure-flow characteristic is modelled with the same method as on the tank side, resulting in the
functions γr (x) and gr (x), shown in equations 5.74 and 5.75.
gr (x) = 0.1267e−
|x|−0.5556 2
0.1016
+ 10880e−
|x|−3.269 2
0.7823
γr (x) = ((−arctan(10(|x| − 0.4))) + π/2)0.3/π
88
(5.74)
(5.75)
Pressure control
The resulting functions are also plotted in figures 5.33a and 5.33b. To obtain a
smooth, continuous transition from the gap to the holes at the pump side the
so called logistic function, equations 5.76 and 5.77, was used.
Cq = Cqgap · (1 − L(x)) + Cqhole · L(x)
1
L(x) =
1 − e−1000x
(5.76)
(5.77)
Damping orifice and chamber
The flow into the damping chamber, equation 5.78, depends on the size of the
damping orifice and the leakage flow over the spool land. This flow is assumed
to be laminar and is modelled with the constant Kd , which is tuned to give
a good match with measurements. The pressure change due to this flow is
modelled with the continuity equation as in equation 5.79.
qc = Kd (p − pc )
Vc
qc + Acc v =
ṗc
β
(5.78)
(5.79)
Flow forces
Since the mechanical spring is very weak, the static characteristic of the valve
is mainly defined by the flow forces. The flow force is defined in equation 5.80,
where only stationary flow forces are considered. ∆p is the pressure drop over
the orifice and is either the difference between pump and load pressure or the
difference between load and tank pressure.
Ff = 2Cq (x, ∆p)A(x)∆pcos(δ)
(5.80)
Since the discharge coefficient is known, the flow angle can be calculated from
the spool equilibrium in equation 5.61. The resulting models are defined in
equations 5.81 and 5.82. When implementing the models, the working range
of the spool is considered. For convenience, the cosine of the flow angle is
modelled rather than the flow angle itself.
Pump side
x+3.129 2
cos(δ) = max 0.7, 6.151 · 1013 e− 0.5478
x−0.154 2
+0.1908e− 0.106
(5.81)
Tank side
|x|+1.741 2
cos(δ) = max 1, 932e− 0.7617
|x|−0.5226 2
+0.3796e− 0.2083
(5.82)
89
On Electrohydraulic Pressure Control for Power Steering Applications
Validation
The validation is performed by comparing measurements to simulation results.
This is done for both static and dynamic measurements. Figure 5.34 shows
the comparison of the measured and simulated pressure against flow and spool
position against flow in the static case. Figures 5.35 to 5.37 compare the
current, pressure and spool position of the measured and simulated results.
The different cases are respectively 22%-62% duty cycle, 46%-100% duty cycle
and 65%-85% duty cycle. Parameter data for the simulation model is found in
the appendix.
0.8
200
180
0.6
160
Spool position [mm]
Pressure [bar]
140
120
100
80
0.4
0.2
0
60
−0.2
40
−0.4
20
0
−10
−5
0
5
10
Flow [l/min]
15
20
25
−10
−5
(a) Pressure vs. flow
0
5
10
Flow [l/min]
15
20
25
(b) Spool position vs. flow
Figure 5.34 Validation of the valve for static measurements. Measurements
are solid lines and simulation results are dashed lines.
1.6
120
0.1
0
1.4
100
−0.1
1.2
−0.2
0.8
Spool position [mm]
Pressure [bar]
Current [A]
80
1
60
40
−0.3
−0.4
−0.5
−0.6
0.6
−0.7
20
0.4
−0.8
0.2
0
0.1
0.2
0.3
Time [s]
0.4
(a) Current
0.5
0.6
0.7
0
0
0.1
0.2
0.3
Time [s]
0.4
0.5
(b) Pressure
0.6
0.7
−0.9
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.7
(c) Spool position
Figure 5.35
Validation of the dynamic response of the valve for a step input
from 22% - 62% with a negative rate of 2000% 1/s. Measurements are solid lines
and simulation results are dashed lines.
Simplified model
As explained in the general case in section 5.1, the purpose of the damping
restrictor, is to increase the damping. Depending on the size of the restrictor
90
Pressure control
2.2
200
0.2
2
180
0.1
160
1.6
1.4
0
Spool position [mm]
Pressure [bar]
Current [A]
1.8
140
120
100
−0.1
−0.2
−0.3
1.2
80
1
0.8
−0.4
60
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
40
0.7
−0.5
0
(a) Current
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
−0.6
0.7
(b) Pressure
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.7
(c) Spool position
Figure 5.36 Validation of the dynamic response of the valve for a step input
from 46% - 100% with a negative rate of 2000% 1/s. Measurements are solid
lines and simulation results are dashed lines.
1.75
160
1.7
150
0.1
0.05
1.65
0
140
1.5
1.45
Spool position [mm]
Pressure [bar]
Current [A]
1.6
1.55
130
120
110
1.4
−0.05
−0.1
−0.15
−0.2
100
−0.25
1.35
90
1.3
1.25
0
0.1
0.2
0.3
Time [s]
0.4
(a) Current
0.5
0.6
0.7
80
−0.3
0
0.1
0.2
0.3
Time [s]
0.4
0.5
(b) Pressure
0.6
0.7
−0.35
0
0.1
0.2
0.3
Time [s]
0.4
0.5
0.6
0.7
(c) Spool position
Figure 5.37 Validation of the dynamic response of the valve for a step input
from 65% - 85% with a negative rate of 2000% 1/s. Measurements are solid lines
and simulation results are dashed lines.
the results is either a very high hydraulic frequency with a low first order lag
or an increased mechanical damping. With the parameters in table A.2 in the
appendix, the valve fulfils criterion 5.39 and the restrictor only increases on the
mechanical damping. This means that the nonlinear model can be simplified by
removing the damping orifice and chamber and thus one state. What changes
in the model is the equation of motion, equation 5.83, where the pressure acting
on the spool is now the same as the load pressure and a new viscous friction,
and equation 5.84 where the damping flow no longer takes place.
mv̇ = Fs (i, x) − Ac p − Kx − Bsimple v
− Ffin (x, ps , p) + Ffout (x, p) − f0
V
qin − qout − ql = ṗ
β
(5.83)
(5.84)
The new total mechanical damping, which is represented by the viscous friction coefficient, is tuned to obtain an equal behaviour between the full model
and the simplified model. The simplified model is advantageous when it comes
to simulation. The very small damping chamber of the full model generates
91
On Electrohydraulic Pressure Control for Power Steering Applications
a very stiff differential equation system which requires a small time step during simulation. The simplified model can be run with a larger time step and
simulation is faster.
Linear modelling and analysis
A linear model of the system is very useful in both understanding the system
and for control design, where linear control theory can be applied. The linear
model is derived from the simplified nonlinear model. Since the characteristic of
the valve is mainly defined by the flow forces, the solenoid is simplified to consist
of a resistor and a linear inductor L. By applying the same method described
in section 5.1, the linearised and Laplace transformed system is defined by
equation 5.85 to equation 5.87, where s is the Laplace operator.
1
(∆Vs − R∆i)
L
1
∆xs =
(Kf i ∆i − Ac ∆p − Ke ∆x − B∆xs)
m
V
∆ps =
Kq ∆x − Kc ∆p + Kcps ∆ps + Ac ∆xs − ∆ql
β
∆is =
(5.85)
(5.86)
(5.87)
The derivatives are defined according to equations 5.88 to 5.95, where index 0
indicates the working point for which the derivative is calculated.
∂Fs (i0 , x0 )
∂i
∂Fs (i0 , x0 )
Kf x =
∂x
∂Ffin (x0 , ps0 , p0 )
Kf fin =
∂x
∂Ffout (x0 , p0 )
Kf fout =
∂x
Ke = K − Kf x + Kf fin − Kf fout
Kf i =
∂qin (x0 , ps0 , p0 ) ∂qout (x0 , p0 )
−
∂x
∂x
∂qin (x0 , ps0 , p0 ) ∂qout (x0 , p0 )
Kc = −
+
∂p
∂p
∂qv (x0 , ps0 , p0 )
Kcps =
∂ps
Kq =
(5.88)
(5.89)
(5.90)
(5.91)
(5.92)
(5.93)
(5.94)
(5.95)
The linearised equations are represented by the block diagram in figure 5.38.
From the block diagram the loop gain transfer function Ag (s) can be defined
according to equation 5.96, which is used to study the stability margins of the
valve.
Ac (Kq + Ac s)
Ag (s) =
(5.96)
(ms2 + Bs + Ke )(Kc + Vβ s)
92
Pressure control
∆ql − Kcps ∆ps
∆Vs
Kf i
R+Ls
+
1
ms2 +Bs+Ke
x
−
+
Kq + Ac s
1
Kc + V s
β
∆p
−
Ac
Figure 5.38
A block diagram representation of the valve.
The loop gain is evaluated for two working points. A working point should
correspond to a point where the system is in steady-state. Any point on the
pressure-flow chart in figure 5.34a and the corresponding pressure, current and
spool position can be chosen. It is of interest to study what happens when the
pressure starts to increase with a higher flow in pressure-reducing mode. The
two points are listed in table 5.1 and are two points along the curve at 70% duty
cycle. The first working point is where the pressure is declining with the flow.
The second point is where the pressure is increasing with the flow, yielding
negative characteristics. The loop gain frequency response is shown in figure
5.39. As can be seen, the system is stable in both cases. In the second case,
however, the phase margin begins at -180° since the linearised spring constant is
negative. Being quite low, the spring constant only affects the system at very
low frequencies, allowing the system to maintain a positive, but low, phase
margin at the crossover frequency. The resonance from the resulting pumping
motion of the spool is very high and can be ignored.
point 1
point 2
Table 5.1
i
1.5 [A]
1.49 [A]
xv
-0.0326 [mm]
0.2482 [mm]
p
120.4 [bar]
109.6 [bar]
The working points for loop gain frequency analysis.
For controller design in chapter 6, it is of interest to study the closed loop
transfer function of the valve, defined in equation 5.97.
Go (s) =
Gc (s) =
(ms2
Kq + Ac s
+ Bs + Ke )(Kc +
Kf i
Go (s)
R + Ls 1 + Ac Go (s)
V
β
s)
(5.97)
Six points covering the working region of interest are studied and defined in
table 5.2. Figure 5.40 shows the resulting frequency plots. As seen, both
the gain and resonance frequency varies with working point. The frequency
resonance is in general quite low.
93
On Electrohydraulic Pressure Control for Power Steering Applications
50
Magnitude (dB)
0
−50
−100
−150
−200
Phase (deg)
−250
0
−90
−180
−270
0
10
2
4
10
10
Frequency (Hz)
6
10
Figure 5.39
Loop gain frequency response of the valve. Dots indicate the
amplitude and phase margins.
point
point
point
point
point
point
Table 5.2
5.2.3
1
2
3
4
5
6
i
0.57 [A]
0.6 [A]
0.6 [A]
1.27 [A]
1.29 [A]
1.29 [A]
xv
-0.46 [mm]
-0.2 [mm]
0.01 [mm]
-0.4 [mm]
-0.1 [mm]
0.046 [mm]
p
27.5 [bar]
22.3 [bar]
10.2 [bar]
108 [bar]
98.4 [bar]
88.8 [bar]
The working points for closed loop frequency analysis.
Discussion
The valve is constructed in a way that limits its performance. As can be seen
from static characteristic curves in figure 5.30, in pressure-relieving mode the
flow capacity is far from the capacity in pressure-reducing mode. The reason to
this is probably that the magnet core connects with the spool in the damping
chamber where the pressure is high. When the pressure is higher than the
corresponding solenoid force, the magnet core is separated from the spool and
pushed to its end position. This is also seen from the measurements. The
solenoid loses its ability to control the pressure and what happens to the spool
is not known since the spool position was not measured. This restriction makes
the valve unsuitable for the intended application, where the flow requirement
is equal in both directions.
94
Pressure control
Bode Diagram
140
Magnitude (dB)
120
100
80
60
40
Point 1
Point 2
Point 3
Point 4
Point 5
Point 6
Phase (deg)
20
0
−90
−180
−270
−360
−1
10
Figure 5.40
0
1
10
10
Frequency (Hz)
2
10
Loop gain frequency response of the valve.
The static characteristic of the valve is mostly defined by the flow forces.
The variation in force from the solenoid is comparatively small for the small
stroke of the spool. The reason for a change in direction of the pressure in
reducing mode is explained by studying the flow forces. Figure 5.41 shows
the calculated flow forces at 70% duty cycle compared to the pressure. The
normal behaviour is an increasing flow force with a higher flow rate, which
initially occurs. However, the jet stream angle is continuously reduced with
an increased area opening. At a certain opening, the jet stream angle reduces
more than the flow is increased resulting in a reduced flow force. Equilibrium
states that the pressure must therefore increase.
Since the static characteristic is defined by the flow forces, it also very likely
to vary depending on circumstances. For instance, the smallest disturbance
might change the jet stream angle and as a result a different behaviour is
attained. The very flat characteristic yields low stability margins, as shown
in figure 5.39. This could lead to difficulties when controlling the valve with
an additional control loop. An alternative way to accomplish a good static
characteristic and still maintain good stability margins and a more predictible
behaviour would be to increase the size of the valve. However, a large solenoid
force would be required and a large solenoid would not be practical and would
be very slow. If the intention is to control the valve with an additional control
loop, the static characteristic of the valve would be of less importance. The
valve could instead be designed to use a smaller and faster solenoid.
The discharge coefficient compensates for the fact that the actual crosssectional area of the stream is different to the area opening of the valve. The
95
125
5
120
4
115
3
110
2
105
1
Force [N]
Pressure [bar]
On Electrohydraulic Pressure Control for Power Steering Applications
100
0
5
10
15
20
0
25
Figure 5.41
Measured pressure (solid) and calculated flow force (dashed)
against the flow at 70% duty cycle.
model of the discharge coefficient will also contain the uncertainties regarding the modelling of the area opening. There are uncertainties regarding the
possible underlap or overlap of the spool and the gap between the spool and
the spool body. The flow path of the leakage is also unknown but assumed to
follow a certain direction and this will also affect the shape of the discharge
coefficient. Nevertheless, the discharge coefficient clearly shows how the flow
transfers from laminar in the gap region towards turbulent flow. The shape of
the curve is also similar to the literature, Merritt (1967).
The effect of the solenoid on the loop gain response has been ignored in the
analysis. There are two effects that probably have the most impact. One is the
induced electromotive force, emf, that will change the current as the magnet
core moves. A change in current is also a change in force. A change in force
with the movement results in a damping. The other effect comes from the
magnet core being surrounded by oil, which will increase the viscous friction
as it moves in the magnet housing and as a result the damping is increased.
The total viscous friction of the simplified model, which is tuned to obtain a
good match with the actual valve, contains all effects that contribute to the
damping.
96
6
Electrohydraulic
closed-centre
steering by pressure
control
In this chapter the electrohydraulic closed-centre power steering system is analysed for the intended applications. These are Boost curve control and position
control. Boost curve control is the inherent solution of the open-centre power
steering system and is also widely used for e.g. electric power steering systems.
It is therefore the intention to keep this strategy for the electrohydraulic power
steering system in this work in order to, as far as possible, maintain the fundamental behaviour of the original system. Position control refers to the rack
motion and is related to autonomous driving.
A description of the system is presented in section 6.1. A generalised analysis of the closed-centre steering system, with emphasis on stability for Boost
curve control, is performed in section 6.2. This is done with high-response
servo valves. The analysis shows the implications of a Boost curve controlled
electrohydraulic power steering system with electronic closed-centre valves. A
possible way of how this can be addressed is also presented. The ideas from
that analysis are then extended to include the system with pressure control
valves in chapter 6.4. The results from chapter 5 and chapter 3 are combined
to derive a complete model of the closed-centre steering system. This is done by
replacing the model of the open-centre hydraulic system in the steering system
model in chapter 3, with the model of the pressure control valve in chapter 5.
An external controller for controlling the performance of the valves is derived in
section 6.3 and how the system can be utilised for position control is presented
in section 6.5.
97
On Electrohydraulic Pressure Control for Power Steering Applications
6.1
System description
There are several ways in which the system can be realised depending on the
requirements of and the intentions with the system. This is not analysed in
detail in this work but only discussed briefly. One way to realise the system is
to replace the open-centre system with the pressure control valves from chapter
5. Schematically, the system could look as shown figure 6.1. The supply system
needs to be modified to supply a constant pressure rather than a constant flow.
This can be done in several ways but preferably by charging an accumulator to
the required pressure by controlling the flow output of the pump. An electric
motor driven pump or a pump with a shunt-valve can be used. In the remainder
it is assumed that the pressure can be delivered.
There is no connection between the torsion bar and the valves as in the
original system. The torque applied by the driver needs to be measured instead.
This requires an additional sensor to be installed on the steering column. The
existing torsion bar could also be used by attaching strain gauges. This was
done to the test rig in chapter 7. Additional pressure sensors are also required,
one for each chamber of the assistance cylinder.
θsw
Constant pressure
Pressure control
valve
Controller
xrw
Figure 6.1
Schematic principle of closed-centre power steering with pressure
control valves.
It could also be possible to realise the system as an add-on to the existing
power steering system. Under normal circumstances the open-centre valve is
deactivated by shutting off the supply and tank lines and activating the closedcentre system. In a case of a failure the chambers of the assistance cylinder
could be connected to the open-centre valve instead, providing a solution with
fallback.
98
Electrohydraulic closed-centre steering by pressure control
6.2
Stability analysis of the electrohydraulic CCsystem by using servo valves
θsw
Constant pressure
Servo
valve
Controller
xrw
Figure 6.2
Illustration of the steering system with servo valves.
This section adresses the stability properties of the electrohydraulic power
steering system with closed-centre valves. The purpose is to keep the analysis
generalised and the results are applicable to any configuration of such a system,
like the system with pressure control valves presented at the beginning of this
chapter. In the system studied in this section, the hydraulic system of the
original system is replaced by fast-acting servo valves from Moog. Four 4/3servo valves are used. They are connected so that they individually control the
meter-in and meter-out flow of each chamber of the assistance cylinder. The
system can schematically be illustrated as in figure 6.2. Since the servo valves
have a very high bandwidth, the system becomes more flexible which facilitates
the study.
In chapter 3 the conventional power steering system was analysed with respect to stability. It was seen how the gain of the position control loop is
defined by the Boost curve and how this affected stability. The pressure loop
was also analysed, where it was seen how the response changed with working
point, from being fast at low load pressure to become slow at high load pressure.
The pressure response also affects the steering system stability. This section
presents how the knowledge about the original system can be utilised in the
control of the closed-centre system and how this affects stability and performance. The closed-centre system is analysed both in the frequency domain and
the time domain and compared to the original system. Both a nonlinear and a
99
On Electrohydraulic Pressure Control for Power Steering Applications
linear model of the system are derived for this purpose. The test rig, which is
explained in chapter 7, was equipped with the servo valves and used to verify
the results seen from simulation. As in previous analyses, the supply system
is not within the scope of this work. It is assumed that a constant pressure is
available.
6.2.1
System modelling
The nonlinear and linear models are presented in this section. Since the mechanical system is the same as the original system, the modelling proceduce for
the mechanical subsystem is referred to in chapter 3, section 3.1.1, equations
3.1 and 3.2. The idea is to study the system over the entire pressure range.
This means that a heavy load is required, e.g. a parking scenario. The load is
set to be a linear spring with the displacement of the rack, with stiffness Crw .
It can be seen as a tyre deflecting during stand-still, before it starts slipping.
An accurate load model is not necessary for the scope of this study and the
spring stiffness is set to an arbitrary number in order to ensure that the steering
system works in the entire pressure region.
Nonlinear model
xv1
xv2
V , β, p
ps
qin
qout
Figure 6.3 Representation of the pressure control system with servo valves
for one chamber of the assistance cylinder.
Since both chambers of the assistance cylinder are controlled independently
and are equal, it is sufficient to study one chamber. The system is illustrated
in figure 6.3. The flows through the meter-in and the meter-out orifices are
described by equations 6.1 and 6.2, respectively. The pressure derivative of the
volume is described by equation 6.3 and is also affected by the motion of the
100
Electrohydraulic closed-centre steering by pressure control
assistance cylinder xrw .
r
2
(ps − p)
ρ
r
2
qout = Cq A2
p
ρ
V
qin − qout = Ap ẋrw + ṗ
β
qin = Cq A1
(6.1)
(6.2)
(6.3)
The opening area of each orifice is assumed to be linear to the spool displacement xv with an underlap according to equations 6.4 and 6.5, where w is the
area gradient and u is the underlap.
A1 = wxv1 + u
(6.4)
A2 = wxv2 + u
(6.5)
The valve spool motion is modelled with a second order dynamic as in equation
6.6, where U is the input signal, ωv is the response and δv is the damping
coefficient.
ẍv = U ωv2 − 2δv ωv ẋv − ωv ẋv
(6.6)
A servo valve has an internal position control of the spool. In order to capture
this behaviour and better resemble the real system a PI-controller is used to
control the spool position according to equation 6.7, where the error e(t) is the
difference between the desired spool position and the actual position according
to equation 6.8. The controller gains Kp and Ki are tuned for each valve to fit
the measurements.
Z
U (t) = Kp e(t) + Ki e(t)dt
(6.7)
e = xvref − xv
(6.8)
Linear model
The system equations are linearised and transformed into the Laplace domain.
The flow equations and continuity equation are shown in equations 6.9 to 6.11,
where s is the Laplace operator and ∆ indicates a change in state variable.
∆qin = Kq1 ∆xv1 + Kc1 (∆ps − ∆p)
∆qout = Kq2 ∆xv2 + Kc2 (∆p)
V
∆qin − ∆qout = Ap ∆xrw s + ∆s
β
(6.9)
(6.10)
(6.11)
101
On Electrohydraulic Pressure Control for Power Steering Applications
The meter-in and meter-out valves are controlled simultaneously as shown in
equation 6.12. This results in equations 6.9 to 6.11 being able to be formed
into equation 6.13, with ∆ql = Ap ∆xrw and hydraulic capacitance as Cs = Vβ .
∆xv2 = −∆xv1 = −∆xv
(6.12)
(Kq1 + Kq2 ) ∆xv + Kc1 ∆ps − ∆ql =
V
(Kc1 + Kc2 ) ∆p + ∆ps
β
⇒ Kq ∆xv + ∆ql = (Kc + Cs s) ∆p
(6.13)
The derivatives are defined according to equations 6.14 to 6.19, where
indicates the working point around which the system is linearised.
r
2
Kq1 = Cq w
(ps0 − p0 )
ρ
r
2
p0
Kq2 = Cq w
ρ
q
Cq wxv1 0 ρ2
Kc1 = p
2 (ps0 − p0 )
q
Cq wxv2 0 ρ2
Kc2 =
√
2 p0
index 0
(6.14)
(6.15)
(6.16)
(6.17)
Kq = Kq1 + Kq2
(6.18)
Kc = Kc1 + Kc2
(6.19)
The spool motion is assumed to behave like a second order system for the
linearised analysis according to equation 6.20.
∆xv =
6.2.2
∆xvref
s2
ωv2
+ 2 ωδvv s + 1
(6.20)
Control
In order to control the static characteristic, the controller needs to be of integrator type. The valve dynamic also introduces some phase shift and in order
to compensate for this a derivative part is also introduced in the controller. The
system is strongly non-linear and to improve the performance, the controller
is adaptive in that it compensates for the varying system dynamics. This can
be seen from equation 6.13. The controller is implemented as a lead-lag filter
as shown in equation 6.21, taking the difference between the reference pressure and measured pressure as input. A similar approach was used by Ramdén
102
Electrohydraulic closed-centre steering by pressure control
(1999).
Gcntrl (s) =
Kp0
K̂q
!
V
s+1
β K̂c
V
s+γ
β K̂c
1
ωv s
α
ωv s
+1
!
(6.21)
+1
The estimated system variables K̂q and K̂c are calculated according to the
expression equations 6.14 to 6.19, and γ and α are tuned for good performance.
In the subsequent analyses, it is assumed that K̂q and K̂c are perfectly estimated. With equations 6.13, 6.20 and 6.21, the pressure control loop for the
closed-centre system with servo valves can be derived in the block diagram
shown in figure 6.4. This loop constitutes the inner control loop of the power
steering system, as will be shown.
∆ql
∆pref
Gcntrl (s)
+
1
s2 + 2δv s+1
2
ωv
ωv
−
Kq
+
1
Kc +Cs s
∆p
−
Figure 6.4
Block diagram representation of the pressure control loop with
servo valves.
The advantage of incorporating the system variables in the controller is seen
from studying the loop gain transfer function of the pressure control loop. If
γ and α are set to zero the loop gain transfer function is defined by equation
6.22. The working point dependent variables are cancelled and a pure integrator
remains, ensuring the convergence to zero of the error.
Ag (s) = Kp0
s2
ωv2
1
ωv s + 1
v
+ 2δ
ωv s +
1
1 Cs s
(6.22)
The closed loop transfer function can also be derived from the block diagram.
The effective force from the pressure that acts on the cylinder piston is the
difference between the two chambers. By defining the load pressure pL =
p1 − p2 , the transfer function can be defined as in equation 6.23. The reference
pressure is changed for one chamber at a time. However, the pressure is affected
by the motion of the piston, which occurs for both chambers simultaneously.
∆pL = ∆p1 − ∆p2 =
Ag (s)
∆pref
1 + Ag (s)
| {z }
Gc (s)




1
1


−
+
 ∆ql
 (Kc + Cs s) (1 + Go ) (Kc + Cs s) (1 + Go ) 
|
{z
} |
{z
}
chamber 1, = Gq1
(6.23)
chamber 2, = Gq2
103
On Electrohydraulic Pressure Control for Power Steering Applications
For a comparison between the original system and the closed-centre system
to be valid, it is necessary to control the closed-centre system in a similar
fashion to the original system. The original system is controlled by the Boost
curve. The reference pressure of the servo valve closed-centre system on a linear
form is therefore expressed as shown in equation 6.24, with KqOC and KcOC
being the flow gain and flow-pressure coefficient for the open-centre system,
respectively. The derivation of the linear Boost curve was shown in chapter 3.
Two different Boost curves are used: the curve as in the original system and
a flatter curve. Both are shown in figure 6.5. A pre-filter is also applied to
shape the response. Two cases are evaluated. One is a pre-filter generating
open-centre dynamic, that is the nonlinear dynamic model of the open-centre
valve from chapter 3. The other is a low-pass filter with 50 Hz break frequency.
For good performance in the test rig, the mean pressure was raised by 50 bar
in each cylinder and an offset in spool displacement of the valve was applied.
This does not affect the assistance pressure, but might generate more friction.
∆pref = Gf ilter (s)
1
(Kqoc (∆θsw − RT ∆xrw ) KT − Ap ∆xrw s)
Kcoc
(6.24)
90
80
Assistance pressure [bar]
70
60
50
40
30
20
10
0
−8
−6
−4
−2
0
2
Torsion bar torque [Nm]
4
6
8
Figure 6.5 Boost curves used for control of the closed-centre system with
servo valves. The solid line shows the curve as in the original system. The
dashed curve is an alternative used for comparison.
6.2.3
System analysis
With the linearised equations 6.23, 6.24 and the mechanical system with equations 3.53, 3.54 and 3.56 from chapter 3, the block diagram of the system can
104
Electrohydraulic closed-centre steering by pressure control
be derived, shown in figure 6.6. The result resembles the original system, with
the position control loop. The difference is the contribution from the servo
valves. The pressure control loop with transfer function Gc constitutes an inner control loop, where the position control loop is the outer loop. There is also
a feedback from the rack movement, which affects the pressure loop. The gain
Kqoc
Kcoc depends on which of the two Boost curves is used. The filters G1 and G2
depend on which dynamic pre-filter is applied. For the case where the dynamic
is defined as in the original system, then G1 (s) = GKq (s) and G2 (s) = GKc (s),
see equation 3.51. In the case where the pre-filter with fixed break frequency
is used, then G1 (s) = 1 and G2 (s) = 1+1 s , where ωb is set to 50 Hz. The
ωb
mechanical resonance from the steering wheel and column, Gsw , and the rack,
Grw , are defined by equations 6.25 and 6.26, respectively. It should be noted
that the spring load Crw is a part of the mechanical transfer function and is
not treated as a disturbance.
1
Jsw + bsw s + Kt
1
Grw (s) =
2
Mrw s + brw s + Crw + Kt Rt2
Gsw (s) =
(6.25)
s2
(6.26)
∆Td
+
∆θsw
Gsw
+
KT RT
+
Gq1
KT
Kqoc
G1
Kcoc
+
+
G2
Ap s
−
+
+
Ap
Gc
K T RT
∆xrw
Grw
−
−
Gq2
−
Ap s
Ap s
Kcoc
RT
Figure 6.6
Block diagram of the hydraulic power steering system with servo
valves.
From the block diagram the closed loop transfer function from an applied
steering wheel angle applied by the driver to a corresponding rack position
is defined by equation 6.27. This function can be compared to the original
system, equation 3.57 in chapter 3. The dynamic is affected by the inner
pressure control loop. The damping can here be controlled but depends on how
well the pressure control loop can follow the desired pressure. The disturbance
sensitivity of the pressure loop, defined by transfer functions Gq1 and Gq2 , also
105
On Electrohydraulic Pressure Control for Power Steering Applications
affects the damping.
Rt Xrw
=
θsw
KT R2 +Gc ApKT Kqoc RT /Kcoc
T
Gc Ap Kqoc KT RT
Gc
2
Mrw s2 + brw + Gq1 +Gq2 +
A2
p s+Crw +KT RT +
Kcoc
Kcoc
(6.27)
In order to study stability for the different cases the loop gain transfer function Ag (s) is defined by equation 6.28.
Ag (s) =
RT KT Kqoc Ap Gc Grw
Kcoc (1 + Grw A2p (Gq1 + Gq2 )s) + A2p Gc Grw s
(6.28)
The frequency response of the loop gain is shown in Fig. 6.7 for zero load flow
and with the steering wheel in the centre position. The two cases are shown
for a very high torque level at 3.9 Nm, which corresponds to the highest gain
level. In the first case the pre-filter of the controller is set to have the same first
order dynamic as the open-centre valve. This is the dashed line. This results
in a positive amplitude margin and a stable system. In the second case, the
pre-filter is set at 50 Hz. This is much higher than the first case, where the
response was about 5 Hz. This results in a negative amplitude margin and an
unstable system.
Amplitude [dB]
30
20
10
0
−10
−20
−30
−1
10
0
10
1
10
2
10
Phase [deg]
0
−100
−200
−300
−400
−1
10
Figure 6.7
0
10
Frequency [Hz]
1
10
2
10
Frequency plot of the open loop of the closed-centre power steering system. Solid curve represents with a pre-filter at 50 Hz, and dashed curve
represents with a pre-filter with open-centre dynamic.
106
Electrohydraulic closed-centre steering by pressure control
6.2.4
Simulation
The system is simulated in order to study its behaviour in the time domain.
The basic model is the same as used when simulating the original system.
The hydraulic system is, however, replaced with the model of the servo valves.
To obtain a better resemblance to the actual system, i.e. the test rig, the
load cylinder with force controller is also simulated. That model is detailed in
chapter 7. Parameter values of the simulation model are found in table A.3 in
the appendix.
6.2.5
Results
The results in this section present first the results from the pressure controller.
The second part presents the results from the simulation and measurements of
the different test cases. All results from the test rig have been sampled at a
rate of 10 kHz.
Pressure control response
Figure 6.8 shows the responses of the pressure steps for the two different prefilters used as well as without pre-filter. The top plot is without pre-filter. A
high gain was chosen, which generated a fast response but some oscillations and
long settling time. The time to reach 63% of the final value is about 2.2 ms.
The settling time within 2% is about 16.5 ms. The oscillations can be adjusted
for with the pre-filter, which is shown in the bottom plot. The faster curves are
with pre-filter at 50 Hz and the slower curves with the pre-filter set to opencentre dynamic. With 50 Hz pre-filter the time to reach 63% of the final value is
about 7 ms. The model, shown as dashed lines, is able to predict the behaviour
of the pressure loop sufficiently well compared to measured results, shown as
solid lines. The figure also shows the reference curves for each case, shown as
dashed-dotted lines. There is a clear difference in bandwidth between the two
cases with pre-filter. The time lag between the reference and actual pressure is
not of importance here, but merely illustrates the time lag introduced by the
valves.
Steering system behaviour
Figure 6.9 shows a comparison between the test rig and simulation for the opencentre system and closed-centre system with three cases. Figure 6.9a shows the
assistive pressure, with a solid curve for measurement and a dashed curve for
simulated results. Figure 6.9b shows the measured and simulated torsion bar
torque. The top plots are for the open-centre system. The second top plots are
for the closed-centre system with the pre-filter set to open-centre dynamic. The
system is stable. The third top plots are for the system with 50 Hz pre-filter.
As the pressure increases, instability finally occurs and the system returns to
107
On Electrohydraulic Pressure Control for Power Steering Applications
Pressure [bar]
120
100
80
60
0
0.005
0.01
0.015
0.02
0.025 0.03
Time [s]
0.035
0.04
0.045
0.05
160
Pressure [bar]
140
120
100
80
60
0
0.05
0.1
0.15
Time [s]
Figure 6.8
Comparison between measured (solid) and simulated (dashed)
pressure responses. The dashed-dotted line is the reference pressure for the
respective case. In the top plot, the pressure response without pre-filter is shown.
The lower plot shows the pressure response with pre-filter, where the faster curves
represent the 50 Hz pre-filter and the slower curves the dynamic open-centre
model as pre-filter.
a stable state as the pressure decreases again. The lower plots are the closedcentre system with 50 Hz pre-filter and the flatter Boost curve, shown as an
alternative Boost curve in figure 6.5. The steering wheel angle was manually
applied with an amplitude of 22-25° and no faster than 150°/s. In all cases,
the simulation results also show the same behaviour.
6.2.6
Discussion
The results clearly show the behaviour and design aspects of the power steering
system with electonically controlled closed-centre valves. While there is a fixed
relation between the Boost curve and the response of the pressure control for
the original system, there is no such relation in this case. Since the pressure
controller can not be made fast enough to be negligible for the outer position
control loop, its response must be designed and adapted to the rest of the
system in order to preserve stability. The three bottom plots in figure 6.9,
which are results from the closed-centre system, illustrate this. Comparing the
second and third plots, where the system had the same static Boost curve but
different pre-filter, it is seen that the system becomes unstable with the faster
response. This might not be intuitively since a fast response might be necessary
108
Electrohydraulic closed-centre steering by pressure control
6
100
Torque [Nm]
Pressure [bar]
4
50
0
−50
2
0
−2
−4
−100
8.4
8.5
8.6
8.7
8.8
8.9
Time [s]
9
9.1
9.2
−6
8.4
9.3
8.5
8.6
8.7
8.8
8.9
Time [s]
9
9.1
9.2
9.3
6
100
Torque [Nm]
Pressure [bar]
4
50
0
−50
2
0
−2
−4
−100
−6
6
6.1
6.2
6.3
6.4
Time [s]
6.5
6.6
6.7
6
6.1
6.2
6.3
6.4
Time [s]
6.5
3.9
4
Time [s]
4.1
6.6
6.7
6
100
Torque [Nm]
Pressure [bar]
4
50
0
−50
2
0
−2
−4
−100
−6
3.5
3.6
3.7
3.8
3.9
4
Time [s]
4.1
4.2
4.3
4.4
3.5
3.6
3.7
3.8
4.2
4.3
4.4
5
50
Torque [Nm]
Pressure [bar]
100
0
−50
0
−5
−100
4.6
4.7
4.8
4.9
5
5.1
Time [s]
(a)
5.2
5.3
5.4
5.5
5.6
4.6
4.7
4.8
4.9
5
5.1
Time [s]
5.2
5.3
5.4
5.5
5.6
(b)
Figure 6.9 Comparison between measured (solid) and simulated (dashed)
pressure (a) and torsion bar torque (b). The top plots are the open-centre
system while the other plots are the closed-centre system. The second top plots
are with open-centre dynamic as pre-filter. The third top plots are with 50 Hz
pre-filter. The bottom plots are with 50 Hz pre-filter and flat boost curve.
for a good steering feel. However, as touched on also in chapter 3, the response
from the original valve is also slow at high load pressure and it is a way to
ensure stability for Boost curve control. There is a compromise though. The
response can not be too slow. Too slow response might generate poor steering
feel and reduce the phase margin to such an extent that stability becomes an
issue from that point of view. In the bottom plot, the Boost curve is much
flatter. Also less damping was provided compared to the previous cases. Even
though the response is faster here, stability is not an issue. Of course, the
steering wheel torque is higher, but whether or not this is feasible or how the
steering feel is affected is another discussion and not whithin the scope of this
work. The result shows that everthing is linked together. How the response of
109
On Electrohydraulic Pressure Control for Power Steering Applications
the pressure control loop should be shaped depends on the system parameters,
what Boost curve is used and the valve itself.
The two top plots should show similar results. A small discrepancy is, however, observed. The closed-centre system has a small tendency to oscillate here.
There are probably several reasons for this. One might be the additional phase
shift of the servo valves and the pressure control loop. This is shown in figure
6.8. Phase shift is also introduced from filtering of the rack position. The rack
position is used to calculate the load but also to derive the velocity for calculation of the reference pressure. The derivation of the velocity requires quite
a low break frequency of the filter. The closed-centre system also has slightly
more friction in the assistance cylinder since the sum of the pressures is higher,
as explained previously.
The method and control strategy for this system will be applied to the system
with pressure control valves in the following sections.
6.3
Controlling the pressure
This section deals with an external software controller for the commercial pressure control valve modelled and analysed in chapter 5, section 5.2, and relates
to research question 2). The reason for designing a controller is to increase
and shape the performance, ensure steady-state error convergence to zero and
to handle the effect of varying system properties. The control of fluid power
systems is decribed by Eriksson (2010) as challenging due to several nonlinearities being present. These are nonlinear flow-pressure characteristic, hysteresis,
dead-band and saturation. Classical PID control could still be effective if the
performance specification is not too demanding (Burrows (2000)). An indirect
adaptive controller for hydraulic servosystems was developed by Plummer and
N. Vaughan (1996) in order to handle the nonlinearities. The controller responded well to changes in the plant parameters. In a review article by Edge
(1997), several different control strategies are outlined. These include, among
others, PID control, state-feedback control, adaptive control, nonlinear correction and robust control. A comparison of sliding mode control with state
feedback control and PID control of a proportional valve was conducted by
J. B. Gamble and N. D. Vaughan (1996). The sliding mode controller showed
best performance and was derived with least design effort. It was also more
robust against disturbances compared to the other approaches.
The results from chapter 5 are used to formulate and design the controller.
The frequency response of the valve, shown in figure 5.40, revealed both a
varying dynamic behaviour with working point and a low resonance frequency.
In order to increase performance, the controller needs to address this. A pole
placement approach is used by state feedback. Two controllers are designed
and evaluated in simulation: a linear state feedback controller and a nonlinear
state feedback controller. It is interesting to see how severe the variable plant
dynamic is to the linear controller and what performance benefits there are
110
Electrohydraulic closed-centre steering by pressure control
with a nonlinear controller. In simulation it is assumed that all states are measurable. In the test rig only two states are measurable: current and pressure.
The other two states, spool velocity and position, are estimated in a simplified
manner, where the simulation model runs in real time. It is then assumed that
the calculated states are close enough to the actual states. This assumption is
based on the accuracy of the model and is verified in the test rig.
6.3.1
Linear state feedback controller
There are several examples where linear control design has been successful for
fluid power systems. Gunnarsson and Krus (1993) used LQG control design
for position control of a hydraulic crane. They showed that with a simple
linearised model it was possible to derive a controller that results in a well
damped behaviour of the crane. The LQG method was also used for the velocity control of a hydrostatic transmission by Lennevi and Jan-Ove Palmberg
(1995), where the effect of a variable system dynamic was also disussed. Eriksson, Marcus Rösth, and Jan-Ove Palmberg (2009) used a LQ-design approach
for MIMO-control (multiple-input multiple-output) of an individual metering
system. Parts of the nonlinear behaviour were cancelled out by hardware by
using compensators. It was shown that a simplified model was sufficient for
the control design. Berg and Ivantysynova (1999) derived a linear cascaded
controller with the LQG-LTR method for a secondary controlled hydrostatic
drive. A linear inversion model of the servo valve was part of the controller to
reach a high bandwidth of the swash plate control.
The idea here with this controller is to design it for the most critical working
point and then verify the performance over the entire working region through
simulation of the nonlinear model with controller. The design approach requires
the model to be expressed in state-space form according to equation 6.29, see
e.g. Glad and Ljung (2006), where u is the input signal, v is the disturbance
and y is the output signal. The input signal is the voltage to the solenoid and
the disturbance is the load flow.
ẋ = Ax + Bu + F v
y = Cx
(6.29)
With the linearised system equations of the valve in chapter 5, section 5.2, the
state-space vector is defined by equation 6.30. The matrices A, B, C and F
are defined by equations 6.31 to 6.34.


∆i
∆v 

x=
∆x
∆p
(6.30)
111
On Electrohydraulic Pressure Control for Power Steering Applications


0
0
−Ke /m −Ac /m 


0
0
βKq /V −βKc /V


0
 0 

F =
 0 
β/V
−R/L
0
Kf i /m −B/m
A=
 0
1
0
βAc /V


1/L
 0 

(6.32)
B=
 0 
0
C= 0
0
0
1
(6.31)
(6.33)
(6.34)
In order to make the error convergence to zero, an integrator must be introduced. This is done by adding an extra state to the state-space vector, which
results in the system described by equation 6.35 or 6.36, where r is the reference
signal.
ẋ
A 0 x
B
F
0
=
+
u+
v+
r
ẇ
C 0 w
0
0
1
(6.35)
x
y= C 0
w
⇒
ẋi = Ai xi + Bi u + Fi v + Br r
y = Ci xi
(6.36)
The control scheme can now be formulated as shown in figure 6.10, where L
is a four dimensional row vector and Ki is a constant. A pre-filter is also
added to the reference value in order to further shape the response, according
to equation 6.37, where ω is the break frequency. It is implemented to avoid
unnecessary phase shift for higher frequencies.
Gpf (s) =
1 + 0.01s
ω
1 + ωs
(6.37)
The idea is to shape the response with the pre-filter so as to obtain the same
behaviour as the open-centre valve, that is, fast at low load pressure and slow
at high load pressure. As disussed in section 6.2, the response must be tuned
with respect to the circumstances. In the evaluation of the controller the prefilter is set with a break frequency of 100 Hz at low pressure with a decline to 5
Hz at 90 bar. As seen from the analysis of the original system in chapter 3, the
valve adds damping to the motion of the rack. The disturbance flow ql in figure
6.10 comes from the movement of the rack. If the controller is set to cancel the
disturbance, the contribution to damping from the valve is also removed. This
is undesirable. In addition to affecting the steering feel, damping is needed
to increase the stability margins of the steering system position control loop.
The reference value is therefore affected by the disturbance flow. For a positive
flow the reference pressure is set to increase and for a negative flow it is set to
112
Electrohydraulic closed-centre steering by pressure control
∆ql
∆pref
Gpf (s)
+ −
1
s
Ki
+ u
−
Valve
+ −
∆p
Volume
x
L
Figure 6.10
Linear state-space feedback control scheme.
decrease. The change is linear to the flow with an arbitrary constant for the
evaluation of the controller.
The controller is designed for the working point with highest gain and slowest
resonance, that is, point 5 in figure 5.40 in chapter 5. The L-vector and gain
Ki are calculated with the help of Matlab Control Systems Toolbox by
placing the poles of the system at a desired location. Manually selecting the
poles gives the freedom to shape the response. One way of understanding where
to place the poles is to look at the poles of the plant. They naturally vary with
working point. The poles for the selected design point are defined in table 6.1.
The pole location at zero corresponds to the integrator. The system is then
dominated by a second order dynamic with very low damping and a first order
dynamic that is somewhat faster. The idea is to keep the dynamic structure of
0
-8247.5
-15.5 +77i
-15.5 -77i
-54.8
Table 6.1
Eigenvalues for the selected design point of the linearised plant.
the plant but move the poles into a faster and more damped region. The poles
of the closed loop system are placed according to table 6.2.
-600
-10000
-250 +10i
-250 -10i
-500
Table 6.2
Pole locations of the closed loop system.
When using the pressure control valves in an application it can be useful to
113
On Electrohydraulic Pressure Control for Power Steering Applications
have the transfer function of the closed loop response. For Boost curve control
of the power steering or position control, which are described later in sections
6.4 and 6.5, respectively, a change in reference pressure as well as a change in
load flow can be present simultaneously. A change in load flow would come
from the movement of the steering rack. The pressure response can be defined
as in equation 6.38, with the transfer function from a reference pressure to the
pressure, Gc (s), as in equation 6.39 and from a disturbance flow to the pressure,
Gf (s), as in equation 6.40. Matrix I is the unity matrix. The dynamic of the
closed loop is defined by the poles of the matrix Aci in equation 6.41.
∆p = Gc (s)∆pref + Gf (s)∆ql
Gc (s) = Ci (sI − Aci )
−1
Br
−1
Gf (s) = Ci (sI − Aci )
Aci = Ai − Bi L
Fi
(6.38)
(6.39)
(6.40)
(6.41)
Evaluation through simulation
The controller is evaluated with the help of simulating the nonlinear model for
different working points. All simulations were conducted with a step time of
1 · 10−4 s. Figure 6.11 shows the response with zero load flow for both high
and low pressure. A step in reference pressure is applied from 20 bar to 30 bar,
from 30 bar to 50 bar and back to 30 bar. From 20 bar to 30 bar the response
has a small overshoot of 2.4%, 63% of the final value is reached within 16 ms
and within 2% of the final value within 45 ms. From 30 bar to 50 bar the
overshoot is 0.4%, 63% of the final value is reached within 19 ms and within
2% of the final value is reached within 40 ms. From 50 bar to 30 bar the
undershot is 1.2%, 63% of the final value is reached within 13 ms and within
2% of the final value is reached within 47 ms. For the last step the control
signal is saturated. The response is slower at higher load pressure due to the
setting of the pre-filter.
Figure 6.12 shows the result when a load flow is applied. The load flow is
filtered with break frequency 2 Hz to be more realistic. At the lower pressure
level the reference pressure change with flow gain is set to 1 · 1010 Pa·s/m3 and
for the higher pressure level to 5 · 1010 Pa·s/m3 .
114
Electrohydraulic closed-centre steering by pressure control
55
120
50
110
45
Pressure [bar]
Pressure [bar]
100
40
35
30
80
70
25
60
20
15
90
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
50
2
1.2
1.3
1.4
(a)
1.5
1.6
Time [s]
1.7
1.8
1.9
2
1.7
1.8
1.9
2
1.7
1.8
1.9
2
(b)
12
8.5
8
10
7.5
7
Control signal [V]
Control signal [V]
8
6
4
6.5
6
5.5
5
2
4.5
0
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
4
2
1.2
1.3
1.4
1.1
1.5
1
1.4
0.9
1.3
0.8
0.7
1.2
1.1
0.6
1
0.5
0.9
0.4
1.2
1.3
1.4
1.5
1.6
Time [s]
(e)
1.6
Time [s]
(d)
Current [A]
Current [A]
(c)
1.5
1.7
1.8
1.9
2
0.8
1.2
1.3
1.4
1.5
1.6
Time [s]
(f)
Figure 6.11 Simulation results of pressure response with linear controller.
Load flow is zero. a) and b) show the actual pressure (solid) and reference
pressure (dashed) for low and high pressure levels, respectively. c) and d) show
the corresponding control signals to the valve. e) and f) show the corresponding
current through the solenoid.
115
On Electrohydraulic Pressure Control for Power Steering Applications
28
140
26
130
24
120
Pressure [bar]
Pressure [bar]
22
20
18
16
100
90
80
14
70
12
10
110
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
60
2
1.2
1.3
1.4
(a)
1.7
1.8
1.9
2
1.7
1.8
1.9
2
1.7
1.8
1.9
2
8.5
6
8
5
7.5
Control signal [V]
Control signal [V]
1.6
Time [s]
(b)
7
4
3
7
6.5
2
6
1
5.5
0
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
5
2
1.2
1.3
1.4
(c)
1.5
1.6
Time [s]
(d)
1
1.5
0.9
1.45
1.4
0.8
1.35
Current [A]
0.7
Current [A]
1.5
0.6
1.3
1.25
0.5
1.2
0.4
1.15
0.3
0.2
1.1
1.2
1.3
1.4
1.5
1.6
Time [s]
(e)
1.7
1.8
1.9
2
1.05
1.2
1.3
1.4
1.5
1.6
Time [s]
(f)
Figure 6.12 Simulation results of pressure response with linear controller.
A 2 Hz filtered load flow of 4 L/min is applied at 1.2 s and of -8 L/min at 1.6
s. a) and b) show the actual pressure (solid) and reference pressure (dashed)
for low and high pressure levels, respectively. c) and d) show the corresponding
control signals to the valve. e) and f) show the corresponding current through
the solenoid.
116
Electrohydraulic closed-centre steering by pressure control
6.3.2
Nonlinear state feedback controller
In the previous section, section 6.3.1, a linear controller was designed for the
valve. Obviously, it is a compromise since the system dynamic varies with
the working point and the same performance is not guaranteed over the entire
working envelope. One way to handle the nonlinearities of the system is to use
a nonlinear controller that transforms the system into a linear and controllable
system using feedback. The method Exact linearisation via feedback, see e.g.
Isidori (1995) or Glad and Ljung (2003), is investigated for the valve. When the
linear system is derived, linear control design can be applied. Pole placement,
as in section 6.3.1, is also used here to define the performance of the closed
loop system.
Feedback linearisation for hydraulic systems has been investigated by several researchers. Hahn, Piepenbrink, and Leimbach (1994) derived a nonlinear
controller for an electrohydraulic servo system. The efficiency and robustness
against parameter uncertainties were showed through simulation. The same
approach was also used by Chiriboga, Thein, and Misawa (1995) to control a
load-sensing system. However, none of these authors have implemented the
controller on a real system. This was done by Seo, Venugopal, and Kenné
(2007). A feedback linearisation controller was derived and implemented for
a rotational hydraulic drive. The controller was compared to a classic PID
controller and it was shown that nonlinearities in the plant were effectively
handled by the nonlinear controller, where the PID controller suffered from
these variations.
The system is defined in the form shown in equation 6.42. The solenoid model
is simplified to take only the nonlinear current-force relationship into account,
while a linear inductor is considered. The hydro-mechanical subsystem is the
simplified model from chapter 5. This gives the model for control design defined
by equation 6.43.
ẋ = f (x) + g(x)u
y = h(x)
iR
di
=−
+ g(x)u
dt
L
dv
= (Fs (i, x) − pAc − Kx − Bv − f0
dt
− Ffin (x, ps , p) + Ffout (x, p))/m
dx
=v
dt
dp
β
= (qin (ps , p, x) − qout (p, x) + Ac v + ql )
dt
V
g(x) = 1/L
(6.42)
(6.43)
y=p
117
On Electrohydraulic Pressure Control for Power Steering Applications
The system in equation 6.42 is transformed into the system in equation 6.44.
ż1 = z2
ż2 = z3
·
·
·
żr−1 = zr
żr = b(z) + a(z)u
(6.44)
żr+1 = ψr+1 (z, u)
·
·
·
żn = ψn (z, u)
y = z1
Here, r is the relative degree and n the number of states. According to Isidori
(1995), a system is said to have relative degree r at a point x0 if the relations
in 6.45 are fulfilled.
Lg Lkf h(x) = 0 for all x in a neighborhood of x0 and all k < r − 1
Lg Lr−1
h(x0 ) 6= 0
f
(6.45)
L is called the Lie derivative and is defined according to equations 6.46 and
6.47.
∂
∂
+ · · · + fn
∂x1
∂xn
∂
∂
Lg = g1
+ · · · + gn
∂x1
∂xn
Lf = f1
(6.46)
(6.47)
The new coordinates are defined as in equation 6.48.
zi = φi (x)
118
1≤i≤n
(6.48)
Electrohydraulic closed-centre steering by pressure control
The coordinates are calculated with the help of the Lie derivatives as in 6.49.

 

h(x)
φ1 (x)
 φ2 (x)   Lf h(x) 

 


 

·
·

 





·
·

 





·

  r−1·

 φr (x)  =  L (x) 
(6.49)


  f
φr+1 (x) χr+1 (x, u)

 


 

·
·

 





·
·

 





·
·
φn (x)
χn (x, u)
In this case, the relative degree is equal to the number of states and no extra
functions except the Lie derivatives are needed. By setting the control signal
u as in equation 6.50, with input signal v, the transformed system becomes as
in 6.51. This system is linear and controllable.
u=
1
(−b(z) + v)
a(z)
(6.50)
ż1 = z2
ż2 = z3
·
·
(6.51)
·
żn−1 = zn
żn = r
y = z1
The Lie derivatives for the system in 6.43 are calculated using Mathematica, resulting in some cumbersome expressions that are not listed here. The
first step is to control the relative degree of the system. The system in 6.43 has
a relative degree 3, that is less than the number of states. The transformation
into linear coordinates is therefore in the form in 6.44. The system is only
partially linear. There are different approaches to overcome this. One is to
find another output, c(x), so that the relative degree of the system becomes n.
That output should fulfill 6.52.
Lg Ljf c(x) = 0,
j = 0, ..., n − 2
Lg Ln−1
c(x) 6= 0
f
(6.52)
Another way, which is chosen here, is simply to keep the structure in 6.44.
As mentioned earlier, the system is partially linear with a linear subsystem of
dimension r, in this case 3, which is responsible for the input-output behaviour
119
On Electrohydraulic Pressure Control for Power Steering Applications
and a possibly nonlinear subsystem of dimension one. The controller is then
evaluated with the nonlinear model of the valve derived in chapter 5. The
feedback control is designed to place the poles of the linear system at desired
locations. The control signal u remains as in equation 6.50. By using the
original state vector x and the control law to place the poles, the control signal
can be expressed as in equation 6.53, where K is the feedback gain vector.
u=
2
−L3f h(x) + Σi=0
Ki Lif h(x)
Lg L2f h(x)
(6.53)
The new transformed system to be controlled can also be expressed in state
space form, as in 6.54, with matrices Az , Bz and Cz defined in equations 6.55,
6.56 and 6.57.
ż = Az z + Bz v
(6.54)
y = Cz z

 

0
0 1 0
(6.56)
(6.55)
Bz = 0
Az = 0 0 1
1
0 0 0
Cz = 1 0 0
(6.57)
Just as for the linear controller, an integrator is needed in order to handle the
steady-state error. This is done in the same way by adding an extra state, w.
The complete system becomes as in 6.58 or 6.59, with the reference value r.
ż
Az 0
z
Bz
0
=
+
v+
r
ẇ
Cz 0
w
0
1
(6.58)
z
y = Cz 0
w
⇒
zi = Azi zi + Bzi v + Br r
y = Czi zi
(6.59)
The system can be described by the block diagram in figure 6.13. Since the
system is linearised, it can be presented in the Laplace domain. The same prefilter, Gpf (s), as in the linear case, is used. The reference value is ∆pref and the
output is the pressure ∆p. The intergrator gain Ci is added to the closed loop.
The feedback gains Ki and Ci are calculated using the Matlab Control
Systems Toolbox, based on the system in 6.59 and placing the poles as in
table 6.3. The locations of the poles are chosen to match the response of the
linear controller in order to compare the two controllers.
Implementation of simulation model
When implementing the controller, it is required to limit the allowed range
of the spool stroke. Since the controller involves the inverse of the model, a
120
Electrohydraulic closed-centre steering by pressure control
Linearised plant
∆pref
Gpf (s)
+
Ci
1
s
−
+
v
−
−L3 h(x)+v
f
Lg L2 h(x)
f
u
ẋ = f (x) + g(x)u
∆p
y = h(x)
x
z
Ki
Figure 6.13
Block diagram representation of the feedback linearisation system. The dashed rectangle marks the linearised plant.
-220
-300
-350
-500
Table 6.3
Pole locations of the linearised closed loop system.
spool position of zero could lead to numerical issues. The controller therefore
only accepts positive spool displacements and the minimum value was limited to
above zero. The controller was also divided into a part for positive displacement
and negative displacement, since the model differs depending on the spool
position. Switches were used to change between the two controllers.
Evaluation through simulation
The controller is evaluated with the help of simulating the nonlinear model
at the same working points as for the linear controller. All simulations were
conducted with a step time of 1 · 10−6 s. Figure 6.14 shows the response with
zero load flow for both high and low pressure. A step in reference pressure
is applied from 20 bar to 30 bar, from 30 bar to 50 bar and back to 30 bar.
No overshoot is seen this time. From 20 bar to 30 bar the response reaches
63% of the final value within 16 ms and within 2% of the final value within
32 ms. From 30 bar to 50 bar, 63% of the final value is reached within 19 ms
and within 2% of the final value is reached within 40 ms. From 50 bar to 30
bar, 63% of the final value is reached within 16 ms and within 2% of the final
value is reached within 30 ms. As for the linear controller the control signal
is saturated for the last step. The same pre-filter is also used, with slower
dynamic at higher load pressure.
121
On Electrohydraulic Pressure Control for Power Steering Applications
55
120
50
110
45
Pressure [bar]
Pressure [bar]
100
40
35
30
80
70
25
60
20
15
90
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
50
2
1.2
1.3
1.4
(a)
1.5
1.6
Time [s]
1.7
1.8
1.9
2
1.7
1.8
1.9
2
1.7
1.8
1.9
2
(b)
12
8.5
8
10
7.5
Control signal [V]
Control signal [V]
8
6
7
6.5
6
4
5.5
2
5
0
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
4.5
2
1.2
1.3
1.4
(c)
1.5
1.6
Time [s]
(d)
1.5
0.95
0.9
1.4
0.85
1.3
Current [A]
Current [A]
0.8
0.75
0.7
1.2
1.1
0.65
0.6
1
0.55
0.9
0.5
0.45
1.2
1.3
1.4
1.5
1.6
Time [s]
(e)
1.7
1.8
1.9
2
0.8
1.2
1.3
1.4
1.5
1.6
Time [s]
(f)
Figure 6.14 Simulation results of pressure response with the nonlinear controller. Load flow is zero. a) and b) show the actual pressure (solid) and reference
pressure (dashed) for low and high pressure levels, respectively. c) and d) show
the corresponding control signals to the valve. e) and f) show the corresponding
current through the solenoid.
122
38
140
36
130
34
120
32
110
Pressure [bar]
Pressure [bar]
Electrohydraulic closed-centre steering by pressure control
30
28
100
90
26
80
24
70
22
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
60
2
1.2
1.3
1.4
12
12
10
10
8
8
6
4
2
2
1.2
1.3
1.4
1.5
1.6
Time [s]
1.7
1.8
1.9
2
1.7
1.8
1.9
2
1.7
1.8
1.9
2
6
4
0
1.6
Time [s]
(b)
Control signal [V]
Control signal [V]
(a)
1.5
1.7
1.8
1.9
0
2
1.2
1.3
1.4
(c)
1.5
1.6
Time [s]
(d)
1
1.5
0.9
1.45
1.4
0.8
Current [A]
Current [A]
1.35
0.7
0.6
1.3
1.25
0.5
1.2
0.4
1.15
1.2
1.3
1.4
1.5
1.6
Time [s]
(e)
1.7
1.8
1.9
2
1.1
1.2
1.3
1.4
1.5
1.6
Time [s]
(f)
Figure 6.15 Simulation results of pressure response with the nonlinear controller. A 2 Hz filtered load flow of 4 L/min is applied at 1.2 s and of -8 L/min at
1.6 s. a) and b) show the actual pressure (solid) and reference pressure (dashed)
for low and high pressure levels, respectively. c) and d) show the corresponding
control signals to the valve. e) and f) show the corresponding current through
the solenoid.
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On Electrohydraulic Pressure Control for Power Steering Applications
6.3.3
Discussion on controller strategy
Both the linear and the nonlinear controller provide for a good response of the
pressure. The pre-filter was set to generate a faster reponse at low pressure
and a slower response at high pressure. This is in accordance with the results
from section 6.2. This is also advantageous since at low pressure there is more
control energy available to boost the performance of the valve.
There is a compromise when designing the linear controller. It must be stable for all working points and is therefore designed for a point near the worst
point from a stability view. The performance is therefore also as designed for
at this point. At other points the performance might deteriorate. This is seen
in figure 6.11 where there are small overshoots at the lower pressure levels.
These are, however, very small and should not cause any issues. This will be
seen in sections 6.4 and 6.5, where the controller is used in the applications.
The nonlinear controller, on the other hand, generates a linear behaviour over
the entire working range. This is at the cost of slightly more control energy.
The nonlinear controller is also far more complicated and the expressions of
the required derivatives are very large. This could be problematic when implementing the controller in a real-time computer. A much smaller time step
is also necessary for the simulation and would probably also be the case in
real time. The nonlinear controller also requires full knowledge of the plant
and could therefore be more sensitive to parameter variations. In reality there
is always some discrepancy between the model and the real plant. The linear controller should be more robust against parameter variations. However,
this should be further investigated by alternating system parameters in the
simulation. It could also be worth investigating possible simplifications of the
nonlinear controller in order to facilitate implementation and to increase the
step time.
The reason the linear controller is sufficient might be because the plant dynamic does not vary enough to cause any problems. The reason for this might
be the valve’s internal hydromechanical feedback loop. One way of improving
the performance of the linear controller could be to use gain scheduling in order to meet the variable plant dynamic. With the derived model, the plant
dynamics are known and can be cancelled out to some extent. The nonlinear
controller also incorporates the load flow and handles disturbances somewhat
better than the linear controller. One way to improve disturbance rejection of
the linear controller is to make the reference pressure react to the movements
of the load cylinder through a high-pass filter, but is anyway limited to the
mechanical response of the valve.
Both controllers run a risk of saturating the control signal. This was not
an issue in the test cases but could cause problems in other cases since none
of the controllers is designed to handle saturation. The approach is instead to
avoid, as far as possible, saturation of the control signal. One way to handle
saturation would be to implement an anti-windup algorithm to prevent the
integrative part of the controller increasing the control signal when the valve
124
Electrohydraulic closed-centre steering by pressure control
is saturated.
Both controllers are sensitive to disturbances. This is because the disturbance acts directly on the output derivative. The nonlinear controller handles
this better and also has to work less. The disturbance is part of the controller
and can be estimated from the rack velocity. How the effect of the disturbance
affects the application is investigated in sections 6.4 and 6.5. It was seen already in the generalised analysis of the close-centre power steering system that
the disturbance affects the damping and should therefore not cause any control
issues. It remains, however, to be seen how it affects the steering feel. The
spikes seen in the control signal of the nonlinear controller comes from the fact
that the controller is in practice an inverted model and when the spool position
goes towards zero the control signal will increase.
The chosen control strategy is the linear one. The performance gain from
the nonlinear controller, at this point, is not worth the effort of implementing
it. The linear controller is much simpler and easier to implement and its performance is good over the entire working region. In the subsequent sections
the controller is referred to as the linear controller.
6.4
Boost curve control with pressure control
valves
Boost curve control of the power steering system is the most common form
of control. It is the fundamental heritage of the system layout of the original
system and is also widely used for electric power steering systems. In order to
keep the structure and behaviour from the original system as far as possible,
it is interesting to use the same control strategy for the closed-centre power
steering system.
In this section the pressure control valves with the developed controller is
investigated for use with Boost curve control of the closed-centre power steering
system and the work relates to research question 4). The knowledge gained
from the generalised analysis in section 6.2 will also be adapted in this case.
The main idea is to keep the commercial valve from chapter 5 and limit the
operating range of the power steering system to the allowed ranged of the valve
for the sake of the analysis. The results from this section could then easily be
extended to involve a valve with a symmetric flow range. The main analysis
is aimed at investigating under what circumstances the valves can be used for
Boost curve control. This includes how the stability of the power steering loop
is affected, as well as closed loop performance. The work in this section relates
to research question 4) in chapter 1.
The control scheme can be illustrated as shown in figure 6.16. The steering
wheel torque is measured. This can be done in different ways, e.g. by adding
a torque sensor or making use of the existing torsion bar. The latter is chosen
here where the torsion bar torque is available. In the test rig (chapter 7) it
125
On Electrohydraulic Pressure Control for Power Steering Applications
has been realised by attaching strain gauges to the torsion bar. The measured
torque is fed to the Boost curve, which calculates the required assistive pressure. Two pressure control valves are needed that are controlled individually.
The linear controller from section 6.3.1 is applied to each valve. From a control
perspective, the pressure control constitutes an inner control loop and the position control of the steering system constitutes an outer control loop. This was
seen from the generalised analysis in section 6.2 and will also be studied later
in this section. Because of the Boost curve, the outer loop is already defined
and limits what can be done. The degrees of freedom are in the inner loop,
which must be tuned to reach the desired performance. The implications of
this will be shown in the analysis.
Boost curve
Torque
+
pref
Ctrl
+
pref
u1
−
Ctrl
u2
−
p1
Figure 6.16
p2
Illustration of the control scheme with Boost curve.
The investigation is divided into several steps. The first step is to see if
the original Boost curve, measured from the test rig, can be used and under
what conditions. If the valves are fast enough, tuning with a pre-filter should
be sufficient to ensure stability, as was shown in section 6.2. However, the
servo valves used in section 6.2 are much faster than the pressure control valves
(compare this with the results from section 6.3.1). If the valves do not meet the
required response, it is also possible to increase damping to ensure stability, but
126
Electrohydraulic closed-centre steering by pressure control
this will also alter the steering feel. The effect of this will be analysed. Hence,
there are two parameters to tune, pre-filter break frequency and damping. The
next step is to tune the Boost curve instead. The aim is to keep the damping
as close as possible to the original system and the Boost curve is altered to give
a stable system. The pre-filter break frequency will also be tuned for this case.
Nonlinear simulation and linear model analysis will be applied to arrive at the
conclusions.
6.4.1
System equations
The system equations of the electrohydraulic power steering system consists of
already derived models. The mechanical system is the same as for the original
system, which was defined in chapter 3, and the pressure control was defined
in section 6.3.1. For convenience, the equations are rewritten here but directly
in the Laplace domain, equations 6.60 and 6.61.
Jsw ∆θsw s2 = ∆Td − KT (∆θsw − RT ∆xrw ) − brw ∆θsw s
(6.60)
2
Mrw ∆xrw s = ∆pL Ap + KT (∆θsw − RT ∆xrw ) RT
− brw ∆xrw s − Crw xrw
(6.61)
An external load is considered in the analysis. Both a vehicle model or a
tyre model generate road wheel angle dependent forces. In a linear sense, it
can be regarded as a spring with constant Crw . This spring stiffness has a
large impact on the behaviour of the system, where a stiffer spring reduces the
gain. If a load was not present, the steering wheel torque would be zero. It is
therefore justified to assume some kind of stiffness in the analysis. The load
pressure ∆pL is defined as the difference between the two cylinder chambers
as shown in equation 6.62. The response for one chamber is already defined
in equation 6.38. It is assumed that the reference pressure is only changed for
one chamber at a time but the movement of the rack affects both chambers,
where chamber one is expanded and chamber two is compressed for a positive
rack displacement.
∆pL = ∆p1 − ∆p2 = Gc (s)∆pref − 2Gf (s)∆ql
− Gc (s)Φ(∆pref , ∆ql )
∆Ttb = Kt (∆θsw − Rt ∆xrw )
(6.62)
(6.63)
The disturbance transfer function Gf (s) will be different for the left and right
cylinder chamber if the working points are different. The reference pressure,
∆pref , is defined by the Boost curve, which takes the torsion bar torque as
input. An exponential function can be used to represent the static Boost curve
as in equation 6.64, a and b are chosen to obtain the desired curve. The torsion
bar is defined in equation 6.63, as previously. The function Φ(pref , ql ) is an
additional term to control the static characteristic of the valve and adds to
127
On Electrohydraulic Pressure Control for Power Steering Applications
the damping of the rack movement. Different functions are possible, such as a
linear function to the flow or one that also involves the pressure.
pref = Kboost (Ttb ) = aebTtb − ae−bTtb
6.4.2
(6.64)
Updating the controller
The pressure control valves have significantly more phase shift than the original
open-centre valve. Since the pressure control loop constitutes an inner loop
of the power steering position control loop, the additional phase might have
a negative impact on stability. The closed loop of the pressure controller is
dominated by two poles at around 40 Hz, which is seen from the closed loop pole
location of the linear state feedback controller. One approach to improve this
is to add a compensation filter outside the pressure loop. This filter basically
moves the two dominating poles to give faster response. The compensation
filter is on the form shown in equation 6.65.
Gcomp (s) =
a0 s2 + a1 s + a2
b0 s2 + b1 s + b2
(6.65)
The numerator is set to cancel the poles at 40 Hz, while the denominator is set
to correspond to poles at 300 Hz. In order to avoid too large input signal, a
low-pass filter is placed before the compensator filter. The break frequency of
the low-pass filter can then be tuned according to section 6.2. An exponential
function can be used to make the break frequency decay with the reference
pressure, according to equation 6.66.
ωpre = 15e−0.01pref + 5 [Hz]
(6.66)
where pref is in unit Pa. The filter is limited to a lowest level at 5 Hz.
6.4.3
Simulation model and results
As in the previous implementations of the model, the Transmission Line
Method is also used here. The layout of the simulation model is illustrated
in figure 6.17, which shows the interaction between model components. The
model is divided into submodels, such as Boost curve, valves, volumes, mechanical part of the steering system and load model. With the Transmission
Line Method the interchange between TLM components is the characteristic
impedance and wave variable and not pressure. However, for simplicity the
pressure is used in the figure.
Four test cases are used to compare the electrohydraulic steering system to
the original system and are as follows:
1) The vehicle model used as load, set with a constant speed of 70 kph.
The steering wheel angle is a chirp signal up to 2 Hz and with 50° in
amplitude. This resulted in a later acceleration of up to 3 m/s2 and a
flow rate of up to 3.6 L/min.
128
Electrohydraulic closed-centre steering by pressure control
p
Boost
curve
vp
qv
Valve
Volume
Wheel
angle
p
pref
pref
p
Valve
p
qv
Steering
system
Load
model
Force
Volume v
p
Torque, ql
Figure 6.17 Layout of the simulation model implementation of the electrohydraulic power steering system.
2) A hysteresis model with maximum force at 2490 N is used as load. The
steering wheel angle is a chirp signal up to 1 Hz and with 120° in amplitude. The flow rate reached 4 L/min.
3) The same hysteresis model is used with maximum force set at 7000 N.
The steering wheel angle is a chirp signal up to 0.1 Hz and with 200° in
amplitude which gave a flow rate of up to 0.8 L/min.
4) Weave test at 80 kph. The vehicle model used as load and the steering
input is a sine wave at 0.25 Hz and ±15◦ angle.
Both the pre-filter break frequency and damping was tuned to ensure stability.
First, the Boost curve from the original system is used and a comparison is made
for the three test cases. A flatter Boost curve is then derived and implemented
for both systems. It can be expressed by equation 6.67. By simply assuming
a relation between load pressure and pump pressure, similar to what has been
meausured, a valve design can be derived by calculating the opening areas of
the open-centre valve with the alternative Boost curve, according to equations
3.29 and 3.30. The two Boost curves are shown in figure 6.18. The first three
tests are used to compare the two systems over time and for different load cases,
with both the original Boost curve and the flatter one. The fourth test is to
derive cross-plot for steering feel, where the results from the two Boost curves
are plotted in the same figure.
pBoost = 3.5e0.5Ttb − 3.5e−0.5Ttb
(6.67)
With the original Boost curve, the first three test cases are plotted in figures
6.19 to 6.21. Four subplots are shown: the steering wheel angle in a), the
steering wheel torque in b), the torsion bar torque in c) and the load pressure
in d). The damping function is tuned in and is defined by equation 6.68. The
129
On Electrohydraulic Pressure Control for Power Steering Applications
200
180
Alternative
curve
Assistive pressure [bar]
160
140
Original
curve
120
100
80
60
40
20
0
−8
−6
−4
−2
0
2
Torsion bar torque [Nm]
4
6
8
Figure 6.18 Original Boost curve shown as dashed and alternative Boost
curve shown as solid.
pre-filter was tuned and the function in the previous section was used.
Φ(p, ql ) = 2 · 104 pref ql
max(Φ) = 5 · 1010
min(Φ) = 1 · 10
(6.68)
9
The three cases with the alternative Boost curve are shown in figures 6.22
to 6.24. Since the Boost curve is much flatter, hence a lower gain, a different
damping function could be used. This is shown in equation 6.69.
Φ(p, ql ) = 6 · 103 pref ql
min(Φ) = 5 · 109
(6.69)
Case 4) is shown in figure 6.25. It shows both the original Boost curve and
the alternative Boost curve and compares the open-centre system with the
electrohydraulic closed-centre system.
130
Electrohydraulic closed-centre steering by pressure control
50
10
40
Steering wheel torque [Nm]
Steering wheel angle [deg]
30
20
10
0
−10
−20
5
0
−5
−30
−40
−10
−50
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
5
0
0.5
1
1.5
2
(a)
2.5
Time [s]
3
3.5
4
4.5
5
3
3.5
4
4.5
5
(b)
5
50
4
40
3
20
1
Pressure [bar]
Torsion bar torque [Nm]
30
2
0
−1
10
0
−10
−20
−2
−30
−3
−40
−4
−50
−5
0
0.5
1
1.5
2
2.5
Time [s]
(c)
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.19 Simulation results for case 1) with original Boost curve. Solid
lines represents the EHPAS system and dashed lines the HPAS system.
131
On Electrohydraulic Pressure Control for Power Steering Applications
10
8
100
Steering wheel torque [Nm]
Steering wheel angle [deg]
6
50
0
−50
4
2
0
−2
−4
−6
−8
−100
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
−10
5
0
0.5
1
1.5
2
(a)
2.5
Time [s]
3
3.5
4
4.5
5
3
3.5
4
4.5
5
(b)
5
50
4
40
3
20
1
Pressure [bar]
Torsion bar torque [Nm]
30
2
0
−1
10
0
−10
−20
−2
−30
−3
−40
−4
−50
−5
0
0.5
1
1.5
2
2.5
Time [s]
(c)
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.20 Simulation results for case 2) with original Boost curve. Solid
lines represents the EHPAS system and dashed lines the HPAS system.
132
Electrohydraulic closed-centre steering by pressure control
5
200
4
150
3
Steering wheel torque [Nm]
Steering wheel angle [deg]
100
50
0
−50
2
1
0
−1
−2
−100
−3
−150
−4
−200
0
2
4
6
Time [s]
8
10
−5
12
0
2
4
6
Time [s]
(a)
10
12
10
12
(b)
4
100
3
80
60
2
40
1
Pressure [bar]
Torsion bar torque [Nm]
8
0
−1
20
0
−20
−40
−2
−60
−3
−4
−80
0
2
4
6
Time [s]
(c)
8
10
12
−100
0
2
4
6
Time [s]
8
(d)
Figure 6.21 Simulation results for case 3) with original Boost curve. Solid
lines represents the EHPAS system and dashed lines the HPAS system.
133
On Electrohydraulic Pressure Control for Power Steering Applications
10
50
8
40
6
Steering wheel torque [Nm]
Steering wheel angle [deg]
30
20
10
0
−10
−20
−30
4
2
0
−2
−4
−6
−40
−8
−50
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
−10
5
0
0.5
1
1.5
2
(a)
2.5
Time [s]
3
3.5
4
4.5
5
3
3.5
4
4.5
5
(b)
50
6
40
30
20
2
Pressure [bar]
Torsion bar torque [Nm]
4
0
10
0
−10
−2
−20
−4
−30
−40
−6
0
0.5
1
1.5
2
2.5
Time [s]
(c)
3
3.5
4
4.5
5
−50
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.22 Simulation results for case 1) with alternative Boost curve. Solid
lines represents the EHPAS system and dashed lines the HPAS system.
134
Electrohydraulic closed-centre steering by pressure control
10
Steering wheel torque [Nm]
Steering wheel angle [deg]
100
50
0
−50
−100
5
0
−5
−10
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
0
5
0.5
1
1.5
2
8
40
6
30
4
20
2
10
0
−2
−20
−30
0.5
1
1.5
2
2.5
Time [s]
(c)
4
4.5
5
3
3.5
4
4.5
5
−10
−6
0
3.5
0
−4
−8
3
(b)
Pressure [bar]
Torsion bar torque [Nm]
(a)
2.5
Time [s]
3
3.5
4
4.5
5
−40
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.23 Simulation results for case 2) with alternative Boost curve. Solid
lines represents the EHPAS system and dashed lines the HPAS system.
135
On Electrohydraulic Pressure Control for Power Steering Applications
200
6
150
4
Steering wheel torque [Nm]
Steering wheel angle [deg]
100
50
0
−50
−100
2
0
−2
−4
−150
−6
−200
0
2
4
6
Time [s]
8
10
0
12
1
2
3
4
(a)
5
Time [s]
6
7
8
9
10
(b)
80
6
60
4
Pressure [bar]
Torsion bar torque [Nm]
40
2
0
−2
20
0
−20
−40
−4
−60
−6
−80
0
2
4
6
Time [s]
(c)
8
10
12
0
2
4
6
Time [s]
8
10
(d)
Figure 6.24 Simulation results for case 3) with alternative Boost curve. Solid
lines represents the EHPAS system and dashed line the HPAS system.
136
12
Electrohydraulic closed-centre steering by pressure control
3
2.5
2
2
Steering wheel torque [Nm]
Steering wheel torque [Nm]
1.5
1
0.5
0
−0.5
−1
−1.5
1
0
−1
−2
−2
−2.5
−15
−10
−5
0
5
Steering wheel angle [deg]
10
15
−3
−15
−10
(a)
−5
0
5
Steering wheel angle [deg]
10
15
(b)
Figure 6.25 Steering wheel torque against steering wheel angle for case 4). a)
is with original Boost curve and b) is with alternative Boost curve. Dashed lines
represent the original system with open-centre valve and solid lines represent the
electrohydraulic closed-centre system.
6.4.4
Analysis
In order to proceed with the analysis, the Boost curve and damping need to
be linearised. The Boost curve, represented by an exponential function in
equation 6.64, depends only on the torsion bar torque and becomes as shown
in equation 6.70 when linearised, with the derivative defined in equation 6.71.
Index 0 indicates the working point.
dpref (Ttb,0 )
∆Ttb = Kb ∆Ttb
dTtb
Kb = abebTtb,0 + abe−bTtb,0
∆pref =
(6.70)
(6.71)
The linearisation of the damping funcion is derived in equation 6.72, with
derivatives in equation 6.73.
∆Φ = φq ∆pref + φp ∆ql
φq = φql,0
φp = φpref,0
(6.72)
(6.73)
The expression for the load pressure, defined in equation 6.62, can now be
expressed as shown in equation 6.74 by introducing the linearised Boost curve
and damping.
∆pL = Gc (s)(1 − φq )∆pref − 2Gf (s)Ap ∆xrw s − Gc (s)φp Ap ∆xrw s
(6.74)
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On Electrohydraulic Pressure Control for Power Steering Applications
With the linearised equations 6.60, 6.61, 6.63 and 6.74, a block diagram
reprenting the system can be derived as shown in figure 6.26. Here is
G∗c (s) = Gpre (s)Gcomp (s)Gc (s) and Grw (s) = Mrw s2 +b1rw s+Crw .
∆Td
+
∆θsw
Gsw (s)
+
KT RT
+
+
KT
G∗
c (s)
Kb (1 − φq )
+
+
KT RT
+
Ap
∆xrw
Grw (s)
−
−
−
2Gf (s)
Ap s
φp A p s
RT
Figure 6.26
Block diagram of the electrohydraulic power steering system
with pressure control valves.
From the block diagram, expressions for the loop gain transfer function
Ag (s), used for stability analysis, and closed loop transfer funtions can be
derived. These are shown in equations 6.75 and 6.76, respectively.
Ag (s) =
Rt ∆xrw
=
∆θsw
RT KT Kb (1 − φq )Ap G∗c (s)Grw (s)
(1 + 2A2p Grw (s)Gf (s)s) + G∗c (s)A2p Grw (s)φp s
KT R2 +G∗
c (s)ApKT Kb (1−φq )RT
T
2
∗
2
∗
Mrw s2 + brw +2Gf (s)A2
p +Gc (s)φp Ap s+Crw +KT RT +Gc (s)Ap KT Kb (1−φq )RT
(
)
(6.75)
(6.76)
From the closed loop transfer function it is seen that both the transfer function
from a disturbance to a pressure change of the valve controller, Gf (s), and
the parameter φp add to the damping of the rack motion. The stability is
first examined for the original Boost curve at 3.9 Nm in figure 6.27. Figure
6.27a shows the frequency response for the two systems and it is seen that
stability is maintained. However, the stability margin is significantly lower for
the electrohydraulic system. Figures 6.27b shows the frequency response of the
closed loop system. The two systems follow each other well up to about 10 Hz.
The case with the flatter alternative Boost curve is examined for a torque
level at 8 Nm. The frequency response of the loop gain is shown in figure 6.28a,
comparing the open-centre system to the electrohydraulic system. Stability is
maintain but is lower for the electrohydraulic system, like in the previous case.
The closed loop response is shown in figure 6.28b, where the two systems follow each other up to about 10 Hz. It is also interesting to study the effect
138
Electrohydraulic closed-centre steering by pressure control
Bode Diagram
Bode Diagram
20
0
Magnitude (dB)
0
−100
−200
−20
−40
−60
−300
−80
−400
0
−100
180
90
−180
Phase (deg)
Phase (deg)
Magnitude (dB)
100
−360
−540
−720
−1
10
0
−90
−180
0
10
1
10
2
10
Frequency (Hz)
(a)
3
10
4
10
5
10
−270
0
10
1
10
2
10
Frequency (Hz)
3
4
10
10
(b)
Figure 6.27 Stability plots for the case with the original Boost curve. Dashed
lines represent the open-centre system and solid lines the electrohydraulic closedcentre system. Plot a) shows the loop gain frequency response and plots b) the
closed loop response.
of the flow disturbance. Figure 6.28c shows the comparison of the closed loop
responses, but with the dynamic flow disturbance removed from the electrohydraulic system. The response is much smoother and the two systems now
follow each other up to about 30 Hz.
6.4.5
Discussion
The results show a clear trend that it is very challenging to achieve the desired
performance under the circumstances in this analysis. Boost curve control
is a restrictive strategy since it limits what can be done in terms of control.
The main challenge is the additional phase shift of the pressure control valve.
Compared to the open-centre valve, the pressure control valve has a phase shift
of 360° and additional 90° with the integrator, while the latter only has 90°
phase shift in total. While it was sufficient to tune the pressure response for
the system with high performance servo valves in section 6.2, it was not for
the system with the pressure control valves. The damping was also tuned to
ensure stability.
For both cases with different Boost curves, the electrohydraulic steering system was stable. Seen was seen from both simulation and linear analysis. However, with the original Boost curve, the damping was set higher than the original
system. By using a flatter Boost curve, the gain decreases and it was possible
to use a lower damping level. It was also possible to tune the damping to have
a similar characteristic to the open-centre system, which is also seen from the
cross-plot steering wheel torque over steering wheel angle in figure 6.25. An
issue here was a slightly jerky behaviour in the steerning wheel torque. This is
139
On Electrohydraulic Pressure Control for Power Steering Applications
Bode Diagram
Bode Diagram
20
0
Magnitude (dB)
0
−100
−200
−20
−40
−60
−300
−80
−400
0
−100
180
90
−180
Phase (deg)
Phase (deg)
Magnitude (dB)
100
−360
−540
−720
−1
10
0
−90
−180
0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
5
10
−270
0
10
1
10
(a)
2
10
Frequency (Hz)
3
10
4
10
(b)
Bode Diagram
20
Magnitude (dB)
0
−20
−40
−60
−80
Phase (deg)
−100
90
0
−90
−180
−270
0
10
1
10
2
10
Frequency (Hz)
3
10
4
10
(c)
Figure 6.28
Stability plots for the case with the alternative Boost curve.
Dashed lines represent the open-centre system and solid lines the electrohydraulic
closed-centre system. Plot a) shows the loop gain frequency response and plots
b) the closed loop response. The effect of the dynamic flow disturbance is seen
in plot c), where it is removed from the electrohydraulic system.
a result of the dynamic effect of the flow disturbance on pressure, which was
difficult to handle. One way is to increase the slope of the static characteristic of the valve to such an extent that it masks the dynamic response of the
disturbance, although this will also increase damping. The effect it has on the
steering feel, however, must be investigated and related to how much damping
is desired. The effect of the dynamic disturbance is also seen in figures 6.28b
and 6.28c, where the latter figure shows a better closed loop response.
It should be noted that the results very much depend on the circumstances.
Both the vechicle model and hysteresis model are simplified and their credibility
is not verified. Research has shown that the whole system has an impact on the
140
Electrohydraulic closed-centre steering by pressure control
steering feel, see e.g. Pfeffer (2006), and excluded dynamic and effects could
have a positive or a negative influence on the results. This is not within the
scope of this work and since the same load is used for both steering systems,
it is possible to make a relative comparison and judgement.
6.5
Position control for autonomous steering
In this section position control of the steering rack with the help of the pressure
control valves is analysed. As has been discussed and shown, the power steering
system is a positioning follower where the driver sets the reference position. In
this section, however, the driver is out of the picture and the control is undertaken solely by a computer. This is interesting when it comes to autonomous
driving or active safety functions, such as collision avoidance, as disussed in
chapter 4. For low speed manoeuvering it can be sufficient to only control the
position of the rack, M. Rösth, J. Pohl, and J-O. Palmberg (2006). At high
speed, however, it is the lateral position of the vehicle that is to be controlled.
One way to do this is to use a nested controller, as is done by Tai, Hingwe, and
Tomizuka (2004), where the inner controller is for the steering system. The
linear state feedback controller for the valves, section 6.3.1, is considered here.
The system equations are presented first, followed by the design of the controller, results, and finally a brief discussion. The controller is designed in
the linear frequency domain and later evaluated with the nonlinear simulation
model of the complete system.
6.5.1
System equations
The system is the same steering system as in previous cases, with the difference
that the driver’s torque is non-existent. Since the pressure is being controlled
by software, is can be assumed that the response is linear. The steering system
is also linear and the system equations can be expressed directly in the Laplace
domain by equations 6.77 and 6.78, where s is the Laplace operator.
Jsw s2 = −KT (∆θsw − RT ∆xrw ) − bsw θsw s
2
Mrw s = ∆pL Ap + KT (∆θsw − RT ∆xrw )RT − brw ∆xrw s
(6.77)
(6.78)
The load pressure ∆pL is defined as the difference between the two cylinder
chambers, as shown in equation 6.79. The response for one chamber is already
defined in equation 6.38. It is assumed that the reference pressure is only
changed for one chamber at a time but the movement of the rack affects both
chambers, where chamber one is expanded and chamber two is compressed for
a positive rack displacement.
∆pL = ∆p1 − ∆p2 = Gc (s)∆pref − 2Gf (s)∆ql =
Gc (s)∆pref − 2Gf (s)Ap ∆xrw s
(6.79)
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On Electrohydraulic Pressure Control for Power Steering Applications
Equation 6.78 can now be updated according to equation 6.80. Damping can
also be added from the valves. It is set to be the dominant contributor and
varies with load pressure. For the calculation of the frequency response it is
set to brw = 4 · 104 Ns/m.
Mrw = Ap Gc (s)∆pref + KT (∆θsw − RT ∆xrw )RT
− (brw + 2A2p Gf (s))∆xrw s
(6.80)
Equations 6.77 and 6.80 can be combined to solve the transfer function from a
change in load pressure to a change in rack displacement, as shown in equation
6.81. The frequency response is shown in figure 6.29. At low frequencies the
plant can be approximated with a pure integrator. At around 5 Hz is the
anti-resonance from the steering wheel and column is visible.
Ap
∆xrw =
Mrw + (brw +
2A2p Gf (s))s
+ KT RT2 −
∆pL
2 R2
KT
T
Jsw s2 +brw s+KT
(6.81)
Bode Diagram
Magnitude (dB)
−150
−200
−250
Phase (deg)
−300
0
−45
−90
−135
−180
−1
10
0
10
1
2
10
10
Frequency (Hz)
3
10
4
10
Figure 6.29 Frequency response of the plant, from a change in load pressure
pL to a change in rack displacement xrw .
6.5.2
Control
The purpose of the controller is to follow a reference position of the steering
rack. In order to avoid steady-state error, an integrator needs to be introduced
into the controller. One way of doing so is presented by Krus and Gunnarsson
142
Electrohydraulic closed-centre steering by pressure control
(1993) and is suitable for systems that can be approximated with a pure integrator. It is a PI-controller with a pre-filter that handles the overshoot from
the integrative part and the system can be represented by the block diagram
shown in figure 6.30, where b∗rw = brw + 2A2p Gf (s) is the effective damping
coefficient. The pre-filter for the pressure controller in section 6.3 is not used
here. The shaping of the response is handled only by the outer position control
loop.
1
Jsw s2 +bsw s+KT
K T RT
∆pL
ref
∆xref
+
Gpf (s)
+
GP I (s)
+
Ap Gc (s)
−
Figure 6.30
KT RT
1
2
Mrw s2 +b∗
rw s+KT RT
xrw
Block diagram of the position controller.
The plant is approximated to equation 6.82, with Kx according to equation
6.83.
Kx
s
Ap K T
Kx =
brw KT + bsw KT RT2
Gplant (s) =
(6.82)
(6.83)
The PI-controller is defined as shown in equation 6.84, with proportional gain
K.
ω
I
GP I (s) =
+1 K
(6.84)
s
By introducing the parameters in equations 6.85, 6.86, 6.87 and with the prefilter defined as in equation 6.88, the closed loop Gcl (s) becomes as shown in
equation 6.91. The response of the system is then defined by ωa and δa . The
control parameters of the PI-controller are calculated according to equations
6.89 and 6.90.
ωc = KKx
√
ωa = ωc ωI
ωa
δa =
2ωI
s/ωa + 1
Gpf (s) =
s/ωI + 1
ωa
ωI =
2δa
2δa ωa
K=
Kx
(6.85)
(6.86)
(6.87)
(6.88)
(6.89)
(6.90)
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On Electrohydraulic Pressure Control for Power Steering Applications
Gcl (s) =
s/ωa + 1
s2 2 2δa
ωa ωa s
(6.91)
+1
The reference value to the pressure controller, calculated by the position controller, is the load pressure, that is, the difference between the left and right
cylinder chamber. However, since each valve has its own controller, the reference load pressure is calculated to an individual reference pressure for each
pressure controller. This is done according to equations 6.92 and 6.93. They
are based on the desired load pressure and the sum of the cylinder pressures,
which is calculated according to equation 6.94. In this way one chamber pressure is kept at 20 bar while the other chamber pressure is increased. Since the
minimum pressure is larger than zero, the performance is increased somewhat
and it gives room to control the static characteristic. The static characteristic
of the pressure controller is defined by 6.95, and is linear to the pressure in
the respective chamber. It is, however, limited to a minimum and a maximum
value. This allows the damping of the rack to be controlled.
psum + pL
− Dq ql
2
psum − pL
+ Dq ql
=
2
= |pL | + 40 · 105
p1ref =
(6.92)
p2ref
(6.93)
psum
(6.94)
10
10
Dq = 7500p, max = 7 · 10 , min = 1 · 10
6.5.3
(6.95)
Results from simulation
Simulation results of the position controller are shown in figures 6.31 and 6.32.
In figure 6.31, the vehicle model, set at a constant speed of 70 kph, is used as
load. The reference position to the controller is a sine sweep with a frequency
up to 2 Hz. In figure 6.32 the load is a hysteresis model to represent the tyreground force during parking, that is, at very low speed. The reference position
to the controller is a step input with a rate limiter set at ±60 mm/s. The
controller gains were set to ω = 8 Hz and δa = 1.5.
For both test cases the performance of the position controller is good. A
small lag and a small overshoot are visible. The performance of the pressure
control valves is also sufficient, with an even smaller lag. What is seen, though,
are oscillations in rack velocity and torsion bar torque.
6.5.4
Analysis and discussion
The biggest challenge in achieving good performance is the effects from the twomass system created from the rack and column attached through the torsion
bar. When the rack starts to move, oscillations are initiated at the steering
144
Electrohydraulic closed-centre steering by pressure control
6
100
80
4
40
Load pressure [bar]
Rack position [mm]
60
2
0
−2
20
0
−20
−40
−60
−4
−80
−6
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
−100
5
0
0.5
1
1.5
80
20
60
15
40
10
20
0
−20
−15
1
1.5
2
2.5
Time [s]
(c)
4
4.5
5
3
3.5
4
4.5
5
0
−10
0.5
3.5
−5
−60
0
3
5
−40
−80
2.5
Time [s]
(b)
Torsion bar torque [Nm]
Rack velocity [mm/s]
(a)
2
3
3.5
4
4.5
5
−20
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.31 Simulation results of position controller with vehicle model as
load. Reference values are shown as dashed lines. Plot a) shows the rack position,
plot b) shows the load pressure, plot c) shows the rack velocity and plot d) shows
the torsion bar torque.
wheel. This causes the torsion bar to twist and generate a disturbing torque
on the rack. The oscillating steering wheel tries to resist the movement of the
rack, seen as an anti-resonance in the bode plot in figure 6.29. A higher gain
would be necessary but that would also reduce the stability margins. However,
as long as the input is slower than the anti-resonance, as is the case from the
results, the anti-resonance should not cause any issues.
The results are also affected by the load, as seen from the results where the
rack velocity oscillates more during the parking manoeuvre. The tyre load in
that is set arbitrarily and how well it reflects a real tyre is not verified. For
instance, more damping provided from the load would reduce the oscillations.
It also depends on how much damping is provided from the valve and the
controller gain. With a high gain the controller forces the rack to follow the
reference better, which might cause the pressure and rack velocity to oscillate.
A smaller gain makes the movement smoother but introduces more lag.
145
On Electrohydraulic Pressure Control for Power Steering Applications
40
100
30
80
60
20
Load pressure [bar]
Rack position [mm]
40
10
0
−10
20
0
−20
−40
−20
−60
−30
−40
−80
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
−100
5
0
0.5
1
1.5
80
20
60
15
40
10
20
0
−20
−15
1
1.5
2
2.5
Time [s]
(c)
4
4.5
5
3
3.5
4
4.5
5
0
−10
0.5
3.5
−5
−60
0
3
5
−40
−80
2.5
Time [s]
(b)
Torsion bar torque [Nm]
Rack velocity [mm/s]
(a)
2
3
3.5
4
4.5
5
−20
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.32
Simulation results of position controller with hysteresis model
as load, which represent a parking manoeuvre. Reference values are shown as
dashed lines. Plot a) shows the rack position, plot b) shows the load pressure,
plot c) shows the rack velocity and plot d) shows the torsion bar torque.
6.6
Heavy truck steering system application
In this section the pressure control valves for use in a heavy truck power steering
system are investigated. This relates to research question 6). With a few
assumptions and using the framework derived for the previous analyses, it is
possible to get an idea of how the approach with pressure control valves suit
this application.
There are several differences between the steering system in a heavy truck
and a passenger car that relate to the hydraulic system. The vehicle is much
larger and heavier, which generates much higher loads on the steering system.
The higher mass means that the inertia is larger, which obviously increases the
phase shift from the mechanical system. Since the mechanical system is a part
of the loop gain, it will have consequencies for the stability margins. With the
heavy loads, a high force from the hydraulic cylinder is required so the piston
146
Electrohydraulic closed-centre steering by pressure control
area is much larger. The pump flow is slightly less than twice as high compared
to a passenger car and the cylinder volumes are at least ten times larger. The
torsion bar could also be stiffer. Other differences are more difficult to specify
but the mechanical linkage is far more complex for a steering box than a rack
and pinion system. This should generate more friction and damping. Naturally,
the Boost curve is different and is set to be flatter than previous. The steering
wheel is larger and consequently a higher torque is applied for the same for as
in the passenger car. This leads to a flatter Boost curve with lower gain.
Relating the differences to the requirements of the valve, the most striking
one is the much larger volumes while the pump flow is only less than twice as
high. This should lead to a much slower response of the original open-centre
valve. By assuming the open-centre valve to be a reference for performance the
requirement on the pressure control valve is much lower than in the passenger
car, but this also depends on the rest of the system dynamic. With the larger
volumes, the pressure control valve should also be less prone to react to flow
disturbances, which, was seen from the previous analysis, could have a negative
effect on performance and steering feel.
The model is explained first, followed by an analysis of the open-centre valve.
The controller for the pressure control valve needs to be tuned for the new
system. The analysis of the complete system is then similar to previous cases,
section 6.4, with simulation of the nonlinear system equations and a linear
analysis in the frequency domain.
6.6.1
System modelling
Since all models are already derived at this point, only minor alterations and
different parameters are needed. In reality the mechanical linkage is quite
flexible and a higher order model than two degrees of freedom is needed, see
M. Rösth, J. Pohl, and J-O. Palmberg (2006). However, for the sake of the
analysis and see the trends for the application, the system is simplified into
a system with two degrees of freedom, which makes it possible to re-use the
models already derived. The model is defined from a steering wheel angle to a
road wheel angle. In between is the cylinder piston moving in a linear motion.
It is treated only as a gear, where the rotational motion of the column gives
a linear motion of the piston, that turns in a rotational motion of the wheels
through the steering box and linkage. The system equations are similar to as
previous and are defined by equations 6.96 and 6.97.
Jsw θ̈sw = Td − KT (θsw − RT θrw ) − bsw θ̇sw − Mfsw
(6.96)
Jrw θ̈rw = Ap pL /Rl − KT (θsw − RT θrw )RT − brw θ̇rw
− ML − Ffrw (pL )/Rl
(6.97)
The total inertia of the road wheels is denoted Jrw and Rl transforms the linear
motion of the piston to a corresponding rotational motion at the wheels. RT
is the total gear ratio of the steering system, while ML is the load torque from
147
On Electrohydraulic Pressure Control for Power Steering Applications
the tyre-ground interaction. Both the pressure dependent friction and column
friction are set to be equal as in the previous case. The pressure is defined by
the hydraulic system, which is the same model as derived in chapter 3. The
static Boost curve is defined by equation 6.98 and is shown in figure 6.33.
pBoost = 1.813e0.408Ttb − 1.813e−0.408Ttb
(6.98)
For an open-centre system the pump pressure is related to the load pressure,
140
120
Pressure [bar]
100
80
60
40
20
0
−10
−8
Figure 6.33
−6
−4
−2
0
2
4
Torsion bar torque [Nm]
6
8
10
Boost curve for heavy truck steering system.
that is, as the load pressure increases so will also the pump pressure. The
pump pressure is not known here but by assuming a relation between the pump
pressure and the load pressure, the opening areas of the open-centre valve can
be calculated. The static pressure drop of a truck steering system valve is lower
than in the passenger car. The pump pressure is set to be 2.5 bar higher than
the load pressure, equation 6.99. The load pressure is already known from the
assumed Boost curve.
ppump = |pboost | + 2.5 bar
(6.99)
The opening areas can now be calculated according to equations 3.29 and 3.30
and this completes the model of the open-centre system. The remaining parameters are found in the appendix.
When it comes to the electrohydraulic pressure control valves, it is justified to
set a higher possible flow rate at the tank port. Recall that the examined valve
has an asymmetric flow rate, which makes it not suitable for the application.
However, it can be modified in order to have a symmetric flow rate and such
148
Electrohydraulic closed-centre steering by pressure control
valves exists on the market. The allowed range is therefore extended to allow
a higher flow rate at the tank port.
6.6.2
Response of open-centre system
It is interesting to study the pressure response of the open-centre valve since
it is a benchmark for the pressure control valves. There are a few factors that
should tend to make the response faster and other to make it slower compared
to the passenger car system. A larger flow rate and a flatter Boost curve tends
to increase the response while larger volumes decreases the response. The
response is shown in figure 6.34 at zero load flow for torque levels 1, 6 and 10
Nm. The bandwidth, defined at -3 dB from the static level, is 80, 9 and 1.7 Hz
correspondingly. The response is thus slower than in the passenger car case,
mainly due to the larger volumes. It is therefore possible that the response of
the pressure control valve can be lower than previous but must of course handle
a larger flow rate.
Bode Diagram
Magnitude (dB)
140
120
10 Nm
6 Nm
1 Nm
100
80
60
Phase (deg)
40
0
1 Nm
6 Nm
−45
10 Nm
−90
−1
10
Figure 6.34
0
10
1
2
10
10
Frequency (Hz)
3
10
4
10
Pressure response of the open-centre valve at zero load flow with
centred piston.
6.6.3
Controller design
Next step is to re-design the controller for the valves. The same linear controller
approach will be used. The same linear model from section 6.3.1 is still valid
149
On Electrohydraulic Pressure Control for Power Steering Applications
here. Only the size of the volume changes. The same placement of the poles
of the closed loop system can not be kept since it would require a much larger
control energy, which is limited. The poles are instead placed as shown in table
6.4. The performance of the controller is seen in figure 6.35. The response
time, defined as time to reach 63% of final value, is 30 ms. When the flow
disturbance enters the controller has to do a lot of work, but the flow rate is
high and the influence on the pressure is minor. A linear static characteristic
was set with rate 1 · 1010 Pa·s/m3 .
-500
-9000
-80
-81
-400
Table 6.4
Pole location of the closed loop system.
60
10
9
55
8
50
Voltage [V]
Pressure [bar]
7
45
40
35
6
5
4
3
30
2
25
20
0.8
1
1
1.2
1.4
Time [s]
1.6
1.8
2
0
0.8
1
(a)
1.2
1.4
Time [s]
1.6
1.8
2
(b)
Figure 6.35 Simulation result of the pressure controller. Plot a) shows the
reference pressure (dashed) and actual pressure (solid). Plot b) shows the control
signal. A disturbance flow of 10 L/min is entering the volume at 1.2 s and leaving
the volume 1.6 s. The flow disturbance is low-pass filtered with a break frequency
of 2 Hz.
6.6.4
Simulation of complete system
The electrohydraulic was tested by simulation for a steering wheel angle as a
chirp signal up to 1 Hz and 50° in amplitude. The load flow reach 6.5 L/min.
A vehicle model was used as load and all parameters are found in the appendix.
The result of the simulation is seen in figure 6.36. The damping function was
150
Electrohydraulic closed-centre steering by pressure control
on the same form as previous and is shown in equation 6.100. A low-pass filter
with the break frequency at 1 Hz was used in conjunction with the damping
function in order to break up the dependancy on the pressure. The break
frequency of the pre-filter was used as shown in equation 6.101 and was limited
to a lowest level at 2 Hz.
Φ(p, ql ) = 1 · 104 pref ql
min(Φ) = 1 · 1011
min(Φ) = 1 · 10
(6.100)
9
ωpre = 15e−0.05pref + 2 [Hz]
60
(6.101)
8
6
40
Steering wheel torque [Nm]
Steering wheel angle [deg]
4
20
0
−20
2
0
−2
−4
−6
−40
−8
−60
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
−10
5
0
0.5
1
1.5
2
(a)
20
6
15
4
3.5
4
4.5
5
3
3.5
4
4.5
5
10
2
Load pressure [bar]
Torsion bar torque [Nm]
3
(b)
8
0
−2
−4
5
0
−5
−10
−6
−15
−8
−10
2.5
Time [s]
0
0.5
1
1.5
2
2.5
Time [s]
(c)
3
3.5
4
4.5
5
−20
0
0.5
1
1.5
2
2.5
Time [s]
(d)
Figure 6.36 Simulation results for a truck steering system and vehicle model.
Solid lines represents the EHPAS system and dashed lines the HPAS system.
151
On Electrohydraulic Pressure Control for Power Steering Applications
6.6.5
Linear analysis
The linear analysis becomes identical to the passenger car case in section 6.4.
The loop gain and closed loop transfer functions are only updated to fit the
new system parameters in equations 6.102 and 6.103. The resulting loop gain
frequency response and closed loop frequency response at 10 Nm are shown in
figure 6.37. Both systems are stable and the electrohydraulic system follows
the open-centre system up to about 20 Hz.
Ag (s) =
Rt ∆xrw
=
∆θsw
RT KT Kb (1 − φq )Ap /Rl G∗c (s)Grw (s)
(1 + 2A2p /Rl2 Grw (s)Gf (s)s) + G∗c (s)A2p /Rl2 Grw (s)φp s
(6.102)
KT R2 +G∗
c (s)Ap/Rl KT Kb (1−φq )RT
T
2
2
∗
2
∗
2
Jrw s2 + brw +2Gf (s)A2
p /Rl +Gc (s)φp Ap /Rl s+Crw +KT RT +Gc (s)Ap /Rl KT Kb (1−φq )RT
)
(
(6.103)
Bode Diagram
Bode Diagram
0
0
Magnitude (dB)
Magnitude (dB)
100
−100
−200
−100
−400
0
−150
360
−180
180
Phase (deg)
Phase (deg)
−300
−50
−360
−540
−720
−2
10
0
10
2
10
Frequency (Hz)
(a)
4
10
0
−180
−360
−1
10
0
10
1
2
10
10
Frequency (Hz)
3
10
4
10
(b)
Figure 6.37
Stability plots for the truck steering system. Dashed lines represent the open-centre system and solid lines the electrohydraulic closed-centre
system. Plot a) shows the loop gain frequency response and plots b) the closed
loop response.
6.6.6
Discussion
At this point the model of the truck steering system is very simplified, but
it is possible to see some trends in the behaviour. The larger volumes of the
assistance cylinder means that the pressure control valve can not reach the same
performance as previous. A larger valve could, on the other hand, increase the
performance. The larger volumes also mean that the open-centre valve has
a slow response in pressure. It should therefore be possible to set the break
frequency of the pre-filter at a lower value than in the passenger car system.
The larger volumes also makes the valve less prone to react to disturbances in
152
Electrohydraulic closed-centre steering by pressure control
the flow. The steering torque is much smoother in the simulation than what
was seen before. The larger inertia of the steering system should also help
to smoothen any disturbance. At the same time, the larger inertia introduces
more phase shift that reduces the stability margins.
6.7
Implementation and verification
This section aims at implementing the developed controllers and discuss some
issues that were encountered and how they were addressed. The aim is not to
illustrate a fully functional solution but to verify the ideas. As will be seen,
one of the major concerns was the restrictions in the hardware. The system is
described in section 7.2, chapter 7.
6.7.1
Pressure controller
The linear state feedback controller was chosen for the evaluation of the applications. A state feedback controller relies on the existent measurements of the
state, which is not the case here. The solenoid current and pressure are easily
obtained, but this is not the case for the spool position and velocity. During
the measurements of the valve characteristics, a position transducer was indeed
installed on the valve. This caused a high leakage flow rate, which was not an
issue during the specific test but would not be very practical in the steering
system test rig. A state observer is therefore needed to estimate the spool
position and velocity.
The initial approach was to derive a model with very good accuracy, which
could also be use as state observer. With a feedback loop the estimated state
would convergence towards the actual state. Since the valve is nonlinear, however, a nonlinear state observer is necessary, for example, the Extended Kalman
filter. This requires that the Jacobian is calculated at each time step.
It turned to be crucial to keep the time step as low as possible in order to
minimise the time lag in the system. The model, as it was initially implemented
for the simulations, was to complex to run at the desired time step in real time.
Some alterations were needed with the objective to try and reduce the computational time. The jet stream angle was set to fixed values, that is, on the
pump side cos(δ) = 0.7 and on the tank side cos(δ) = 1. The jet stream angle
changes more dramatically for large spool displacements and since the valves
would only work in a small region in the test rig, they can be kept constant.
The nonlinear characteristic of the discharge coefficient and current-force relationship of the solenoid were kept but implemented with look-up tables to avoid
the many trigonometric functions. Since the current was measurable, it was
directly fed to the corresponding look-up table to obtain the force. The lowest
possible time step with this configuration and two valve models implemented
was 0.1 ms, which was not possible with the initial model. With additional controllers implemented in the real-time computer, the lowest possible time step
153
On Electrohydraulic Pressure Control for Power Steering Applications
was instead 0.2 ms. A state observer was never implemented. The required Jacobian would be complex and would require several additional look-up tables,
that would increase the time step further. It was therefore decided to only use
the model to estimate the states and that the increased time step would have a
greater negative impact on the result than the estimation error. A linear state
estimator is also a possibility, but the linearised states are only valid for an
infinitely small change in input signal. It was also believed that the nonlinear
model would do a better job alone.
Another contributor to the time lag is the conversion to a PWM signal. The
valve requires a PWM frequency in the range 100 - 200 Hz. This is required
in order to obtain the dither effect, which minimises the stick-slip friction and
improves the performance. However, this also means that the control signal
only updates every 5 ms, which adds a great deal of time lag in the system.
The result from this can be verified from simulation by adding a time lag to
the model. The resuls is shown in figure 6.38, with a clearly unstable system.
The figure shows also the result with a 0.2 ms time lag. In this case, the
control signal is still oscillatory but the pressure is still controlled. A possible
solution is to increase the PWM frequency above the time step of the realtime computer. This requires that the dither frequency is superimposed on the
control signal to the valve. However, the intended PWM frequency has been
purposely chosen and the valve has undergone extensive testing in the factory
to obtain best possible performance. The time required to find a new set of
parameters with equally good performance, was just not simply possible within
the time frame of this work. The PWM frequency was set to 100 kHz with a
dither frequency at 200 Hz and 10% duty cycle.
A concern with the valve is the fact that high pressure acts on the magnet
core where it is in contact with the spool, see chapter 5. It was seen from
obervations how the magnet core and spool could separate and the valve is no
longer controllable. The valve was never designed for this application but it is
likely that this effects is easily altered. However, it constitutes a restriction for
the tests in this work and possible effects of this will be seen.
Measurement results and discussion
Slightly slower pole placements, compared to simulation, were set and the locations are defined in table 6.5. The resulting pressure response for a filtered
step from 30 bar to 50 bar and back is shown in figure 6.39a, and from 50
to 70 bar and back in figure 6.39b. The corresponding solenoid currents are
shown figures 6.39c and 6.39d, respectively. The step increase and step decline
are measured for two different data sets but are plotted in the same figure for
convenience. The time step was 0.1 ms and a FIR filter at 200 Hz was used for
the currents. The supply pressure was set to 150 bar. This was used to filter
the effects of the dither frequency.
The response for a pressure increase is quite good and not far from what was
seen when simulating the model with controller. The desired pole locations
154
Electrohydraulic closed-centre steering by pressure control
12
1.8
180
1.6
160
1.4
140
1.2
120
Current [A]
Control signal [V]
8
6
4
Pressure [bar]
10
1
0.8
100
80
0.6
60
0.4
40
2
0.2
0
0.5
0.55
0.6
0.65
Time [s]
0.7
0.75
20
0
0.5
0.8
0.55
0.6
(a)
0.65
Time [s]
0.7
0.75
0
0.5
0.8
0.55
0.6
(b)
12
0.7
0.75
0.8
0.7
0.75
0.8
42
1
10
0.65
Time [s]
(c)
0.95
40
0.9
38
6
Pressure [bar]
0.85
Current [A]
Control signal [V]
8
0.8
36
34
0.75
4
32
0.7
2
30
0.65
0
0.5
0.55
0.6
0.65
Time [s]
0.7
(d)
0.75
0.8
0.5
0.55
0.6
0.65
Time [s]
0.7
(e)
0.75
0.8
28
0.5
0.55
0.6
0.65
Time [s]
(f)
Figure 6.38 Simulation of linear state space controller with an introduced
time lag. Plots a), b) and c) show the control signal, solenoid current and
pressure with a 5 ms time lag. Plots d), e) and f) show the same signals with a
0.2 ms time lag.
-400
-10000
-200+10i
-200-10i
-350
Table 6.5
Pole locations of the linearised closed loop system.
are set to obtain a somewhat slower response. A faster response would give an
unstable system. The response for a pressure decrease, however, is poor. This
is most likely caused by a separation of the magnet core and spool. The current
is very oscillatory and behaves similar as shown if figure 6.38. This indicates
that the time lag is somewhat too long and might also be a reason to why
the pole locations would have to be set to obtain a slower response. Another
concern was the lack or repeatability. The performance of the valve with the
higher PWM frequency has not been optimised and there is likely that some
friction still remains. If this could be improved, it is possible that also the performance could be improved. Another possible improvement could come from
the implementation of a more accurate state observer. The indication from
155
55
75
50
70
45
65
Pressure [bar]
Pressure [bar]
On Electrohydraulic Pressure Control for Power Steering Applications
40
60
35
55
30
50
25
−0.1
−0.05
0
0.05
0.1
0.15
Time [s]
0.2
0.25
0.3
45
−0.1
0.35
−0.05
0
0.05
(a)
0.1
0.15
Time [s]
0.2
0.25
0.3
0.35
0.2
0.25
0.3
0.35
(b)
1.4
1
0.95
1.3
0.9
1.2
0.8
Current [A]
Current [bar]
0.85
0.75
1.1
1
0.7
0.65
0.9
0.6
0.8
0.55
0.5
−0.1
−0.05
0
0.05
0.1
0.15
Time [s]
(c)
0.2
0.25
0.3
0.35
0.7
−0.1
−0.05
0
0.05
0.1
0.15
Time [s]
(d)
Figure 6.39
Results from the pressure controller. Plots a) and b) show the
pressure for both chambers (solid and dashed lines). The faster curves represents
the reference pressure (also solid). Plots c) and d) show the corresponding
solenoid currents.
the results, however, are positive and verifies that a linear state feedback controller is a possible solution. Future work would involve an Extended Kalman
filter, or possibly a linear filter, to better estimate the states, reduce the time
lag, and reduce friction by optimising PWM and dither frequencies. These are
beside the alternation done to the valve. Due to this behaviour, other control
strategies could not be confirmed at this stage.
156
7
Hardware-in-theloop
simulation
Hardware-in-the-loop, HWIL, is the simulation technique where a part of the
software implemented simulation model is replaced by actual hardware. This
has the advantage of being able to test parts of the system without going for the
full prototype, especially for large systems. As described by Sannelius (1999),
there are basically two approaches. One is where new components are tested
as hardware and the rest of the system is simulated. The other approach is to
simulate the new component while the system is tested as hardware. The first
approach is adopted in this work, which can be illustrated by figure 7.1.
Hardware
Software
Steering
system
Vehicle
model
Figure 7.1 General approach of hardware-in-the-loop simulation where the
vehicle is implemented as a simulation model and the hardware consists of the
steering system.
Another classification of HWIL simulation is related to where in the system
the transformation from hardware to simulation takes place. For hydraulic
systems it can take place in the hydraulic line where pressure and flow is transformed. An example of this is explained in the work by Ramdén (1999). Here
is a valve tested and the load simulated by controlling pressure and flow. Sannelius (1999) also presents how HWIL simulation is used for testing of hydro-
157
On Electrohydraulic Pressure Control for Power Steering Applications
static transmissions. In this work it is the force and velocity that is transformed
between the hardware and simulation.
The simulation is divided so the steering system constitutes the hardware
and the vehicle is represented by the simulation model. There are two main
purposes with the test rig in this work. First is to develop a simulation model,
described in chapter 3. With the test rig at hands the system under study can
be isolated and different parts can easily be tested, compared to a system that
is in the vehicle. The other purpose is the have a more realistic environment for
the investigation and evaluation of the closed-centre steering system. A pure
simulation model is always just a reflection of the reality and possible issues
that might have been overseen in the simulation may arise in the actual system.
A human can also be integrated in the loop with HWIL simulation. One of
the most important aspects of the steering system is the steering feel. This is
a subjective measure and a qualitative judgement of the closed-centre system
is possible with the test rig.
The work consists of two parts. For the first part the test rig was equipped
with fast acting servo valves for a general study of a closed-centre power steering
system. For the second part the servo valves were replaced by the pressure
control valves described in chapter 5, section 5.2. Several controllers were
developed and implemented. A force controller takes care of providing the right
amount of load on the steering system. The servo valves require a controller
to control the pressure. This is also the case for the pressure control valves.
This chapter gives an overview of the hardware involved in the test rig. The
force controller is also derived and analysed. The analyses of the controllers
for the servo valves and pressure control valves have already been outlined in
chapter 6.
7.1
7.1.1
Servo valves
Hardware description
A schematic of the test rig is shown in figure 7.2 and a photograph is figure
7.3. The test rig is built up around a rack-and-pinion steering system. A
fixed displacement bent-axis pump with size 5 cc/rev driven by an electric
motor is used to supply the open-centre valve with a constant flow. It is not
an original pump used in vehicles but since the pump is excluded from the
analyses, it is only of interest to deliver a constant flow. Several sensors are
used to measure the pump pressure as well as both chamber pressures. To
measure the torque applied by the driver, strain gauges are attached directly
to the torsion bar. This means that the actual torsion bar torque is measured
instead of the steering wheel torque and an additional torsion bar is not needed.
Since the torsion bar torque is linear to the twist of the torsion bar, hence the
twist of the open-centre valve, the measurement of the torque gives a direct
measure of the valve actuation. The steering wheel angle and rack position are
158
Hardware-in-the-loop simulation
Original
system
Load cylinder
Shut-off
valves
Force
sensor
Servo
valve
Controller
Vehicle-tyre
model
Figure 7.2
Servo valve
pack
Software
Schematic description of the test rig.
measured with potentiometers. A weight of 20 kg is attached to one end of the
rack to give the system the right inertia. At the other end of the rack, a load
cylinder is attached to generate the load experienced by the steering system
due to the tyre-ground interaction. Between the load cylinder and the rack,
a force sensor measures the force applied by the load cylinder and is used in
a closed loop controller. The load cylinder is an identical rack from another
steering system. One concern about using such a cylinder is the amount of
friction generated from the internal sealings. This is discussed below in section
7.3. A Moog servo valve is used to control the load cylinder.
Parallel to the original system, four high performance 4/3 servo valves from
Moog control the meter-in and meter-out flow to each chamber of the assistance
cylinder. The approximate bandwidth is 200 Hz at 10% stroke. These valves are
over-dimensioned for the application and the result is that they work with very
small movements, which should be beneficial from a performance point of view.
They are also arranged in such a way that each valve deliver double the flow,
i.e. the two load ports are connected to a single load port. All servo valves use
an external supply system of 200 bar. Argus valves are used to switch between
the original system and the servo valve system. The Argus valves are placed in
159
On Electrohydraulic Pressure Control for Power Steering Applications
the line to each cylinder chamber. The reason for keeping the original system
is to always have the reference system available for a qualitative comparison
between the open-centre and closed-centre systems. A potograph of the Moog
servo valves is shown in figure 7.4.
Open-center
valve
Tank
Pump
Rack
Force
sensor
Load
valve
Load cylinder
Figure 7.3
Photograph of the test rig.
Servo valves
Load
hose
Load
hoses
Figure 7.4
160
Argus
valve
Photograph of the Moog servo valve pack installed in the test rig.
Hardware-in-the-loop simulation
7.2
Pressure control valves
7.2.1
Hardware description
Hydac
valves
Rack
Load
hoses
Figure 7.5
Photograph of the Hydac valves installed in the test rig.
In this part of the work the servo valves have been replaced with two pressure
control valves, one for each chamber of the assistance cylinder, as shown in
figure 7.5. They use a constant pressure supply of 200 bar. A small accumulator
of 1 L is placed in the supply line right before the valves. This will help to
keep the supply at a more even level when sudden flow demands are required
by the valves. The valves are controlled with a 200 Hz PWM duty cycle.
7.3
Force control
The force controller’s task is to generate the right load on the steering rack
depending on the driving scenario. The reference load is calculated from a
software implemented model, which typically is a vehicle model. For the design
and analysis of the force controller, a mathematical model is derived and is
described below. The model considers the load cylinder with hydraulic system
and controller. It is attached to a moving mass through a very stiff spring. The
moving mass is positioned controlled and represents the motion of the steering
rack, which in the test rig is attached to the load cylinder. The analysis is
based on the schematic description of the system in figure 7.6.
161
On Electrohydraulic Pressure Control for Power Steering Applications
xp
Load cylinder
Rack
FL
Force
sensor
Fref
Servo
valve
Controller
ps
Constant
pressure
Figure 7.6 Layout of the force controller of the load cylinder. The actual
force between rack and load cylinder is measured and fed to the controller. The
force is controlled with a servo valve.
7.3.1
Mathematical description
The model consists of three parts: the servo valve, the cylinder volumes and
the moving mass of the cylinder. The model has been somewhat simplified, but
as will be shown, it is accurate enough to predict the behaviour of the system.
Servo valve
The servo valve is modelled as four variable orifices and a second order dynamics
for the movement of the spool. The four orifices connects the supply port p
and the tank port t to the respective load port a and b. The orifice flows are
describe by equation 7.1 to 7.6, where Cq is the flow coefficient and ρ is the oil
density.
r
2
qpa = Cq A(xv )
(ps − pa )
ρ
r
2
(pb − pt )
qbt = Cq A(xv )
ρ
r
2
(ps − pb )
qpb = Cq A(−xv )
ρ
r
2
(pa − pt )
qat = Cq A(−xv )
ρ
qa = qpa − qat
qb = −qbt + qpb
162
(7.1)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
Hardware-in-the-loop simulation
The opening area A is assumed to linear with spool displacement and is
modelled according to equation 7.7 and 7.8, where w is the area gradient and
u is the underlap of the areaopening.
A(xv ) = wxv + u
(7.7)
A(−xv ) = |wxv − u|
(7.8)
(7.9)
The displacement xv of the spool is decribed by the differential equation 7.10
where U is the control signal to the valve, ωv is the response of the valve and
δv is the damping coefficient.
ẍv = U ωv2 − 2ωv δv ẋv − ωv2 xv
(7.10)
Cylinder volumes
The cylinder volumes are modelled with the continuity equation according to
equation 7.11 and 7.12, where Ap is the piston area, xp is the piston position,
V is the cylinder volume and β is the bulkmodulus.
V
ṗa
β
V
qb + Ap ẋp = ṗb
β
qa − Ap ẋp =
(7.11)
(7.12)
Cylinder motion
The motion of the cylinder is modelled with the help of Newton’s second law
according to equation 7.13, where m is the mass of the piston and rod, b is
the viscous friction coefficient, Ff (psum ) is the static friction as described in
chapter 3 and FL is the external load acting on the cylinder which is to be
controlled.
mẍp = pa Ap − pb Ap − bẋp − Ff (psum ) − FL
(7.13)
7.3.2
Controller design
For the design of the controller the system can be somewhat simplified and
analysed in the frequency domain. It is assumed that there is no underlap
of the spool and since the cylinder is symmetrical the flow is equal on both
chambers defined by equation 7.14.
qa = qb = q
(7.14)
From the flow equations the relation in equation 7.15 is valid and the load
pressure pL is defined according to equation 7.16.
ps = pa + pb
(7.15)
pL = pa − pb
(7.16)
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On Electrohydraulic Pressure Control for Power Steering Applications
The flow equation for the servo valve can now be rewritten in terms of the load
pressure as in equation 7.17.
r
1
q = Cq A(xv )
(ps − pL )
(7.17)
ρ
The same is also true for the continuity equation, which results in equation
7.18.
V
ṗL
(7.18)
q − Ap ẋp =
2β
In order to analyse the system in the frequency domain the system equations
need to be linearised and transformed into the Laplace domain. This results
in equation 7.19 to 7.23, where s is the Laplace operator and ∆-sign indicates
a small change in the system state variable. The control signal U is calculated
by multiplying the error in force to a controller Greg (s).
∆q = Kq ∆xv − Kc ∆pL
V
∆q − Ap ∆xp s =
∆pL s
2β
∆U
∆xv = s2
2δv
+
ω2
ωv s + 1
(7.19)
(7.20)
(7.21)
v
m∆xp s2 = ∆pL Ap − ∆FL − b∆xp s
(7.22)
∆U = Greg (s)(∆Fref − ∆FL )
(7.23)
The derivatives are defined according to equation 7.24 and 7.25, where index 0
represents the working point for which the system is linearised.
r
1
∂q
Kq =
= Cq w
(ps0 − pL0 )
(7.24)
∂xv
ρ
q
Cq wxv0 ρ1
∂q
Kc = −
= √
(7.25)
∂pL
2 ps0 − pL0
From the linearised system equations a block diagram can be derived that represents the control loop. This block diagram is shown in figure 7.7.
From
Ap ∆xp s
∆Fref
Greg (s)
+
1
s2 + 2δv s+1
2
ωv
ωv
−
Kq
+
(ms2 + bs)∆xp
1
Kc + V s
2β
−
Ap
+
−
Figure 7.7
164
Block diagram representation of the force control loop.
∆FL
Hardware-in-the-loop simulation
the block diagram the transfer function of the loop gain can be defined according to equation 7.26.
Ag (s) = Greg (s) s2
ωv2
+
2δv
ωv s
Kq Ap
+ 1 Kc +
V
2β s
(7.26)
The controller Greg (s) is designed as a lead-lag filter according to equation
7.27.
1 + τi s 1 + τd s
(7.27)
Greg (s) = Kp
γ + τi s 1 + ατd s
The controller parameters are defined in table 7.1. The complete system with
Parameter
Kp
τi
γ
τd
α
Table 7.1
Value
2.5 · 10−5
0.24
0.001
0.003
0.2
Controller parameter values.
controller can now be studied for stability by plotting the frequency response
of the loop gain Ag (s). The result will vary depending on the working point.
In this case the system is studied for the worst point, which is for a very small
spool displacement and a large pressure drop. Figure 7.8 shows the frequency
response with system parameter values defined in table A.5 in the appendix.
The amplitude margin is at this point 8.11 dB at 303 Hz and the phase margin
is 49.1° at 178 Hz. The controller is then evaluated by simulating the nonlinear
model. The results are shown in the next section.
7.3.3
Results
Figure 7.9 shows the results of the simulation of the force controller where the
rack is moving with an increased frequency up to 2 Hz and an amplitude of 3
mm. The controller is able to follow the reference value with satisfying result.
At the turning points there is a "bump" in the actual force. Figure 7.10 shows
the results from the test rig, where figure 7.10a shows the rack position applied
and figure 7.10b shows the corresponding reference and measured force. A
proportional gain Kp of 0.7 · 10−5 was used in the figure. The other control
parameters were the same as in the simulations. The result is similar to the
simulation and the measured force shows the same behaviour at the turning
points. This is believed to be caused by the friction in the load cylinder and is
analysed in detail below.
165
On Electrohydraulic Pressure Control for Power Steering Applications
Bode Diagram
150
Magnitude (dB)
100
50
0
−50
Phase (deg)
−100
0
−90
−180
−270
−4
10
Figure 7.8
−2
10
0
2
10
Frequency (Hz)
4
10
10
Frequency plot of loop gain of the force controller system.
2000
1500
1000
Force [N]
500
0
−500
−1000
−1500
−2000
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 7.9 Simulation results of the force controller model. Dashed line is
reference force and solid line is actual force.
7.3.4
Discussion
The force controller worked satisfyingly. What was noticed, however, is a small
"bump" in force at the turning points. This have an affect on the steering wheel
torque but since it is always present the situation is equal regardless of what
hydraulic system is evaluated. Interesting is that the simulation also shows the
same behaviour, which indicates that the reason to this effect is caught by the
model. The hypothesis is that it is the static friction in the load cylinder that
166
Hardware-in-the-loop simulation
3
2000
1500
2
Rack force [N]
Rack position [mm]
1000
1
0
−1
500
0
1000
−500
−1000
−2
−1500 −1000
8
−3
0
2
4
6
8
10
Time [s]
12
14
(a) Rack position.
Figure 7.10
16
18
20
−2000
0
2
4
10
6
8
10
Time [s]
12
14
16
18
20
(b) Rack force. Dashed curve is reference
force and solid curve is the actual force.
Comparison of reference and actual force of the force controller.
causes this. This can be easily testet by simply removing the static friction from
the model of the load cylinder. The viscous friction is still present. The result is
shown in figure 7.11, where the resulting force of the controller is compared for
the case with and without friction. It is seen that without friction the behaviour
is much better, which confirms the hypothesis. It is difficult to overcome this
in reality though. One way is to increase the controller gains but they are
limited by the stability margins of the system. Another way would simply be
to remove the sealings. A leakage over the cylinder piston is not an issue in a
laboratory environment.
167
On Electrohydraulic Pressure Control for Power Steering Applications
1800
1600
Force [N]
1400
1200
1000
800
600
1.6
1.65
1.7
1.75
1.8
1.85
Time [s]
1.9
1.95
2
2.05
2.1
Figure 7.11 Simulation results of the force controller model. Solid line represents the results when a cylinder with friction is used. Dashed line represents
the results when a cylinder without static friction is used.
168
8
Discussion
The biggest difference between the open-centre valve from the original system
and the electrohydraulic pressure control valves with closed-centre, used in this
work, is the much higher phase shift. The open-centre valve is attached to and
controlled by the twisting of the torsion bar, which in turn is a result of the
torque applied by the driver. This means that the controlling of the valve is
done directly by the driver and the only lag that occurs comes from the oil
compressibility in the volumes. This is not the case for the electrohydraulic
valve. There are several parts that all add to the phase shift. The valve
is activated based on measurements of the torsion bar torque. The signal is
processed and sent to the solenoid. The force is increased with a certain lag
and then acts on the inertia of the moving mass. When the mass has been
moved, a flow can enter the volume and the pressure starts to change.
Since the pressure control loop constitutes the inner control of the power
steering system, the additional phase shift has a negative impact on stability
and must be handled. This is most severe for Boost curve control, where the
outer control loop is already defined with a high gain at high pressure levels.
Preferably, a fast valve gives little additional phase shift in the critical region
and the performance of the pressure control loop can be controlled with a
pre-filter and set to be the dominating dynamic. In this way stability can be
maintained. This was seen and also verified in the test rig, from the analysis
in section 6.2, chapter 6. In this analysis high-response servo valves were used
to control the pressure.
If the valves are too slow, ensuring stability with a pre-filter will not be
sufficient. The pre-filter also adds to the phase-shift. This was the case for
the pressure control valves, which were analysed with Boost curve control in
section 6.4, chapter 6. The performance and analysis of the system very much
depend on the circumstances given by the system properties and driving scenarios. Another approach to ensure stability is also to tweak the damping.
Since the pressure is controlled from a computer, any desired pressure can be
set. Allowing the pressure change with flow will affect the damping. An in-
169
On Electrohydraulic Pressure Control for Power Steering Applications
creased damping will increase stability margins but also affect the steering feel.
However, the critical region is high pressure, which occurs during parking or
low-speed manouevering, and steering feel is less important in this region. It
is, though, fair to conclude that the additional phase shift gives a limitation
in possible performance variations. The best solution is simply to use a flatter
Boost curve that has a lower gain. The performance of the valve then only
affects the steering feel. One way to analyse this is to look at the hysteresis curve, steering wheel torque over steering wheel angle, where it was seen
that the electrohydraulic closed-centre system can be tuned to have similar
performance to the open-centre system if the Boost gain is kept low enough.
The challenge with Boost curve control leads to the question of if this is
the proper control strategy at all. Boost curve control in an inherent solution
of the original open-centre system but has clear restrictions. It can be seen
as a pure proportional controller with very high gain when the load increases.
Other approaches exist where the Boost curve is more or less moved out from
the position control loop of the power steering system, see e.g. Birk (2010) or
Wolfgang Kemmetmüller, Steffen Müller, and Andreas Kugi (2007). This, however, is a different topic and not analysed in this work. Which control strategy
is used must ultimatiley be chosen by the vehicle manufacturer. The approach
in this work has been that the manufacturer wants to keep the fundamental
behaviour from the original system.
Position control with electrohydraulic pressure control valves, analysed in
section 6.5, chapter 6, gives a greater degree of freedom to design the outer
position control loop. It seems that the reqiurements on the valve are less
strict and the results show that the controller is able to follow the reference
signal with little time lag. This could also be seen as an indication of whether
other control strategies other than Boost curve control could be used. Such a
control strategy could be based on a position control loop, where the reference
position is set by the driver’s steering wheel angle. Position control with the
power steering system is an important subject and the near future will see
autonomous vehicles on the roads. The results show that it is possible to
achieve with pressure control valves. Another important feature is that steering
feel is not an issue if the driver is only a passenger.
An accurate model of the different parts of the system has been very useful
during the work. Both the model of the power steering system and the pressure
control valve model were validated against measurements and the models show
a very good agreement with the measurements. The models have played an
important role in control design, in analysing the system, and in verifying the
concept. Since the models have a high accuracy, when combining the models to
generate the concept with closed-centre pressure control valves, high confidence
in the results can be maintained.
The modelling procedure for the valve stresses the importance of studying
and including the nonlinear behaviour of the pressure-flow characteristic and
flow forces. It turned out that assumptions like a constant discharge coefficent
170
Discussion
and jet stream angle were not valid. The reason is most likely the low flow level,
while the stroke was quite large, which means the flow is in the mixed region
of laminar and turbulent flow. The weak mechanical spring of the valve also
means that the induced flow forces dominate the characteristic’s behaviour,
which in turn is defined by, among other things, the discharge coefficent and
jet stream angle.
An external controller for the pressure control valve was designed. This was
required in order to increase the performance and to ensure that the reference
value was followed. A state feedback approach was investigated. The reason
was the ability to place the closed loop poles at a location where the desired
performance can be obtained. This requires knowledge of the states and two
of the states, namely current and pressure, were measured. Both a linear and
a nonlinear controller were developed and analysed. The linear controller is a
compromise since it does not consider the variations in system dynamics and
characteristic and was designed for a working point near the worst case. The
nonlinear controller, on the other hand, cancels the nonlinearities of the plant
and transforms the system into a linear, controllable system. The requirement
is a good knowledge of the plant. Basically, it involves the inverse of the plant
and is something that needs to be considered during the design process in order
to avoid numerical issues.
Both controllers performed well. This was under the assumption that all
states were obtainable. The nonlinear controller showed best performance and
it was also better at handling flow disturbances. The linear controller had
to do more work when a flow disturbance appeared. However, the nonlinear
controller is far more complex to implement and it is left for future work to
investigate possible simplications and robustness against parameter variations.
The linear controller was therefore chosen to be implemented with Boost curve
control and position control. It was also possible to run the model with a larger
time step, which is beneficial since it reduces the computational time. It would,
however, be interesting to see if any performance gains can be obtained for the
applications with the nonlinear controller. With higher flow rates, the valve
shows a less linear characteristic. Higher flow rates occur, for example, in the
truck steering system. This is more difficult to handle for the linear controller
and the nonlinear controller would most likely do a better job here. When it
comes to the flow disturbances, it was clear that they had a negative impact on
the steering wheel torque. One way to handle this was, as mentioned previously,
to make the static characteristic flatter, which increases the damping. How
much the driver would actually notice the torque disturbance or how much
damping can be applied is a question related to steering feel and not treated
here. Improving the performance of the controller was also possible by adding a
compensation filter at the reference pressure. With the state feedback controller
the valve can be seen as linear and since the closed loop pole locations are
known, a compensation filter can compensate for the phase shift.
The valve has never been intended for the applications studied in this work,
171
On Electrohydraulic Pressure Control for Power Steering Applications
while the analysis and results show how the valve can be used. It is possible
that small alterations to the valve could improve performance. The valve has a
very flat static characteristic. In order to achieve that with a single-stage valve,
the gain of the valve’s hydromechanical control loop needs to be high. This
is achieved with a relatively large pressure sensing area and a weak spring.
A large pressure area requires a large solenoid, which tends to become slow
when its size increases. A weak spring means that the static characteristic is
dominated by the flow forces, which could result in a nonlinear and somewhat
unpredictible characteristic, as was seen in this case. Since an external controller is used, the valve itself does not require a flat characteristic, but rather
a linear characteristic would be more useful and easier to control. This could
be achieved by using a stiffer spring that would instead dominate over the flow
forces. A smaller pressure sensing area could also be used and then a smaller
and faster solenoid could be used. The damping restrictor of the valve could
also be designed to actually replace the low mechanical resonance with a very
high hydraulic resonance. The valve would then have three dominating poles
instead of four, as is the case now. It is possible that such a valve already exists
on the market.
For the truck steering system case, the biggest difference from the valve’s
perspective is the larger cylinder volumes and flow rate. With the larger cylinder volumes the response of the original open-centre valve becomes slower. This
gives an indication that the requirement on the response of the pressure control
valve is lower than in the passenger car case. Larger volumes also means that
the valve is less prone to react to flow disturbances, which was seen from the
analysis in section 6.6, chapter 6. The performance could be further increased
if a pilot-operated pressure control valve was used. The performance of the
investigated valve is limited by the available control energy and relatively low
flow rate. Larger volumes evidently require a higher flow rate. However, a
pilot-operated valve is more complex and expensive, and a single-stage valve is
therefore to be preferred. Single-stage valves with a higher flow rate already
exists on the market. The flatter Boost curve of the truck steering system
reduces the gain of the power steering control loop, which increases the stability margins. There is also more friction in the steering system compared to a
passenger car, if a steering box is assumed and compared to a rack and pinion
system. Friction will decrease the impact of the flow disturbance on steering
wheel torque.
Variations in temperature have been ignored during this work. For each test,
the temperature was maintained at a constant working level. However, road
vehicles operate in a wide range of temperatures and the control loop must
handle this. A robust controller will be able to do this and it is left for future
work to investigate this further.
172
9
Conclusions
Relating to the research questions in chapter 1, the following conclusions can
be drawn:
1) The electrohydraulic pressure control valve showed a nonlinear static
characteristic. In order to capture this behaviour, it was necessary to
consider the flow-pressure characteristic and the transition from laminar
to turbulent flow. It was also necessary to consider the behaviour of the
flow forces, especially the variation of the jet stream angle with spool
stroke.
2) A state feedback controller is a feasible solution to control and increase
the performance of the valve. A linear controller can be designed with
respect to the worst operating condition. A nonlinear feedback linearisation controller is also a feasible solution that eliminates the nonlinear
characteristic but is more complex to implement.
3) The main challenge with Boost curve control is the high gain at high
pressure levels. Since the pressure control loop constitutes the inner loop
of the power steering control loop, the additional phase shift from the
valve has a great impact on stability. This can be addressed by tuning
the response of the valve, alone or in combination with increasing the
damping. This depends on the initial performance of the valve.
4) The electrohydraulic pressure control valves studied in this work adds a
greater limitation in possible Boost curves than the original system. If a
steep Boost curve is desired, higher damping must be added to the power
steering control loop in order to ensure stability. This affects steering feel
but only at high torque levels, where the steering feel is less important.
A flatter Boost curve gives a greater possibility to tune the system to the
desired performance. For example, a lower damping can be set. However,
a lower damping will make the dynamic response to a flow disturbance
of the valve more apparent and there exists a compromise.
173
On Electrohydraulic Pressure Control for Power Steering Applications
5) Using the pressure control valves for position control of the steering rack
offers a promising solution in realising autonomous driving. The pressure
control loop constitutes the inner loop and can be treated as linear for the
design of the outer position control loop. A PI-controller with a pre-filter
resulted in good performance.
6) A truck steering system has several properties that could reduce the dynamic requirements of the electrohydraulic pressure control valve. The
cylinder volumes are much larger than in a passenger car, while the maximum flow rate is also larger but not to the same extent. The larger
volumes mean that the original pressure loop is slower, which gives an
indication that the dynamic response of the pressure control valve can
also be lower. Larger volumes also make the valve less prone to react to
flow disturbances and the larger mass of the vehicle restricts the steering
demand to a lower level than in a passenger car.
174
A
Parameter values
A.1
Parameters for HPAS system simulation
Parameter
Jsw
Mrw
Kt
Rt
bsw
brw
Crw
Ap
V
Vs
Cq
ρ
β
qp
Ff
mveh
Jveh
Cαf
Cαr
lf
lr
pt
Table A.1
Value
0.053
30
60.73
154
0.23
1.05 · 104
3.86 · 106
8.26 · 10−4
1.02 · 10−4
2 · 10−4
0.67
850
8.7 · 108
9
0.3
1657
4875
2.05 · 105
1.06 · 106
0.7
2
0.055
Unit
kgm2
kg
Nm/rad
rad/m
Nm/s
Ns/m
N/m
m2
m3
m3
kg/m3
Pa
L/min
Nm
kg
kgm2
N/rad
N/rad
m
m
m
Table of parameter values
175
On Electrohydraulic Pressure Control for Power Steering Applications
A.2
Parameter values for commercial pressure control valve
Parameter
pf11
pf12
pf13
pf14
pf15
pf16
pf21
pf22
pf23
pf24
pf25
pf26
pf31
pf32
pf33
pd 1
pd 2
pd 3
pd 4
pfs11
pfs12
pfs13
pfs14
pfs21
pfs22
Value
-69.18
335.12
-449.62
99.99
36.65
324.00
15.23
-72.02
87.96
-7.55
-5,17
-31.69
2.13
-9.48
19.14
-0.0004
-0.0003
0.0472
0.0059
-1.06
4.30
-2.30
-1.55
6.07
-23.45
Unit
A/(W3b m5 )
A/(W3b m4 )
A/(W3b m3 )
A/(W3b m2 )
A/(W3b m)
A/W3b
A/(W2b m5 )
A/(W2b m4 )
A/(W2b m3 )
A/(W2b m2 )
A/(W2b m)
A/W2b
A/(Wb m2 )
A/(Wb m)
A/Wb
A/V3
A/V2
A/V
A
N/(A5 m3 )
N/(A5 m2 )
N/(A5 m)
N/A5
N/(A4 m3 )
N/(A4 m2 )
Parameter
pfs23
pfs24
pfs31
pfs32
pfs33
pfs34
pfs41
pfs42
pfs43
pfs44
m
Ac
Acc
r
hgap
K
f0
B
Bsimplied
V
Vc
ρ
β
ν
Kd
Value
9.16
14.69
-12.83
47.77
-17.55
-41.77
10.76
-40.61
23.38
47.33
0.04
6.362·10−5
6.048·10−5
1.5·10−3
1.5·10−5
0.5
2
4.47
223.6
1·10−4
4.7·10−11
900
1·109
46·10−6
1.24·10−11
Unit
N/(A4 x)
N/A4
N/(A3 x3 )
N/(A3 x2 )
N/(A3 x)
N/A3 )
N/(A2 x3 )
N/(A2 x2 )
N/(A2 x)
N/A2 )
kg
m2
m2
m
m
N/mm
N
Ns/m
Ns/m
m3
m3
kg/m3
Pa
m2 /s
m5 /Ns
Table A.2 Parameter values for commercial pressure control valve simulation
model.
176
Parameter values
A.3
Parameters for steering system with servo
valves simulation
Parameter
Jsw
Mrw
Kt
Rt
bsw
brw
Crw
Ap
Vp
Cq
ρ
β
qp
Ff
ωv
δv
w
Table A.3
Value
0.053
30
60.73
154
0.23
1.05 · 104
3.86 · 106
8.26 · 10−4
1.02 · 10−4
0.67
850
8.7 · 108
9
0.3
150
0.7
6 · 10−5
Unit
kgm2
kg
Nm/rad
rad/m
Nm/s
Ns/m
N/m
m2
m3
kg/m3
Pa
l/min
Nm
Hz
m2 /U
Table of parameter values
177
On Electrohydraulic Pressure Control for Power Steering Applications
A.4
Parameters for truck system simulation
Parameter
Jsw
Jrw
Kt
Rt
Rl
bsw
brw
Crw
Ap
V
Vs
Cq
ρ
β
qp
Ff
mveh
Jveh
Cαf
Cαr
lf
lr
pt
Table A.4
178
Value
0.09
16
126
21
17.7
1.8
2500
2·106
0.0092
1 · 10−3
0.5 · 10−3
0.67
900
1· 109
16
0.3
8000
2.5 · 104
3.3 · 105
2 · 105
0.9
2.8
0.075
Unit
kgm2
kg2
Nm/rad
rad/m
rad/m
Nm/s
Nms
Nm/rad
m2
m3
m3
kg/m3
Pa
L/min
Nm
kg
kgm2
N/rad
N/rad
m
m
m
Table of parameter values
Parameter values
A.5
Parameters for force controller simulation
model
Parameter
Ap
Cq
m
b
ps
w
V
u
β
δv
ωv
ρ
Table A.5
Value
8.26 · 10−4
0.67
10
5000
200
1.1 · 10−5
1.02 · 10−4
1 · 10−7
8.67 · 108
0.7
200
850
Unit
m2
kg
Ns/m
bar
m
m3
m2
Pa
Hz
kg/m3
System parameter values for force controller simulation model.
179
On Electrohydraulic Pressure Control for Power Steering Applications
180
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