Robust Control for Steer-by-Wire Systems in Road Vehicles
Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy
Hai Wang
Faculty of Engineering and Industrial Sciences Swinburne University of Technology
Melbourne, Australia
2013
i
Abstract
N the last two decades, the advances of steering systems for road vehicles have
experienced three stages, that is, mechanical steering systems, hydraulic and electro-
hydraulic-power-assisted steering systems, and electric-power-assisted steering systems.
To further improve the safety and reliability of road vehicles, the automotive industry is
currently working on Steer-by-Wire (SbW) systems that are considered to be the next
generation of steering systems in road vehicles. The most distinctive feature of an SbW
system is that the mechanical shaft used to link the hand-wheel with the front wheels in
the conventional steering system is removed and instead, two electric motors are
introduced to steer the front wheels and provide the driver with a feeling of the steering
effort, respectively. The benefits of using SbW in road vehicles are that the overall
steering performance and cruising comforts can be improved, driving safety can be
enhanced, and power consumption and the long-term cost can be further reduced.
Recently, SbW control systems have been intensively studied to achieve good steering
performance following the fast driver input with minimal lag under different road
conditions. However, the steering performances are not satisfied by using the existing
SbW control designs, when road conditions are varying.
In this thesis, the mathematical modelling of SbW systems is first explored, and a
sliding mode control (SMC) scheme is developed for SbW systems with uncertain
dynamics. Unlike conventional control techniques, the SMC is designed based on the
bound information of uncertain system parameters and uncertain tyre self-aligning
torque as well as motor torque pulsation disturbances. The SMC ensures that the
I
ii
controlled SbW systems behave with a strong robustness against the system
uncertainties and disturbances, and the asymptotic convergence of the steering angle
tracking error is achieved.
In order to further improve the tracking error convergence, a nonsingular terminal
sliding mode (NTSM) control for SbW systems is developed. In addition to the finite
time error convergence, many superior characteristics of NTSM control are
demonstrated, including, better tracking accuracy, stronger robustness and disturbance
rejection against significantly varying road conditions.
In practice, an SbW system is actually a partially known system. A new robust
control scheme is developed, with a feedback controller for stabilizing the nominal
steering system and a sliding mode compensator for eliminating the effects of the
uncertain dynamics in the SbW system. The steering performances of all proposed
steering control algorithms are verified with real-time experiments that are carried out
on the SbW platform in the Robotics Laboratory at Swinburne University of
Technology.
iii
Declaration
This is to certify that:
1. This thesis contains no material which has been accepted for the award to the
candidate of any other degree or diploma, except where due reference is made in
the text of the examinable outcome.
2. To the best of the candidate’s knowledge, this thesis contains no material
previously published or written by another person except where due reference is
made in the text of the examinable outcome.
3. The work is based on the joint research and publications; the relative
contributions of the respective authors are disclosed.
________________________
Hai Wang, 2013
iv
v
Acknowledgements
The research work presented in this thesis has been carried out in the Robotics &
Mechatronics Lab at Swinburne University of Technology, Australia. There are a great
number of people without whose support and encouragement I would not have come
this far. First of all, I would like to thank my supervisors, Professor Zhihong Man and
Dr. Weixiang Shen, for their profound knowledge, insightful advice, thoughtful
guidance, and great support throughout the past 4 years of my PhD study. I am deeply
grateful to Professor Zhihong Man who has given me so much in my time as his
research student. He has devoted a lot to continuously guiding me to be a good
researcher, sharing the unceasing passion for research with me, and helping me cultivate
the spirit of perseverance. Without his strong encouragement throughout my PhD work,
I would have never completed this truly amazing intellectual journey.
I would also like to express my deep appreciation to other lecturers of Robotics &
Mechatronics, Dr. Zhenwei Cao, Dr. Jinchuan Zheng, and Dr. Jiong Jin, who have
offered me tremendous support in my research work. During the past few years, their
insightful advice has helped me to significantly improve the quality of this thesis. My
sincere gratitude extends to the finance staff, Adriana and Sam for their support and
patience in equipment purchase. Thanks must also go to the technical staff, Walter
Chetcuti, Krys Stachowicz, Mikhail Mayorov, David Vass, and Meredith Jewson, for
the countless hours they spent with me in resolving the issues with the SbW research
platform. Great thanks also go to Garry Strachan for proofreading my thesis and provide
me with many useful comments to promote my thesis to a higher level.
vi
It is difficult to imagine the past 4 years without the support and companionship of
my colleagues and friends in the Robotics & Mechatronics Lab and AD 222. Many
thanks go to Fei Siang, Matt, Kevin, Aiji, Sui Sin, Edi, Ehsan, Hossein, Mehdi, and
Ruwan. Thanks, too, to many of my friends in Melbourne, including Xiaopeng, Wenjie,
Zhenghua, Zhe, Chaohong, Xiudong, and Jiayan, for their friendship, help, and support
throughout my time in Melbourne. They all certainly made my time more enjoyable,
and for that, I am grateful.
I owe big thanks to the Faculty of Engineering and Industrial Sciences, Swinburne
University of Technology, for giving me this wonderful opportunity to do my PhD at
Swinburne, awarding me the SUPRA scholarship, and providing me with a comfortable
and conducive working environment.
Finally, I would like to offer my most heartfelt thanks to my host family, providing
me with continuous support and encouragement. Thanks to my parents for their endless
love and positive nudges, and my sisters and brothers-in-law who are always the ones I
turn to for advice, sending caring emails to encourage me. Last but not least, I owe a lot
to my lovely wife, Tao, who has been very supportive, patient, and understanding of my
PhD career. Thank you for your love, your encouragement, and for leaving your stable
life in China to join my PhD journey in Melbourne. This thesis would not have been
possible without your support.
vii
Contents
1 Introduction 1
1.1. Steer-by-Wire Systems and Control 2
1.2. Sliding Mode Control Systems 4
1.3. Motivations 5
1.4. Objectives and Major Contributions of the Thesis 5
1.5. Organizations of the Thesis 7
2 Background and Literature Review 9
2.1. History of Steering Systems 10
2.2. Conventional Steering Systems 11
2.3. Power Assisted Steering Systems 17
2.3.1. Hydraulic Power Assisted Steering 17
2.3.2. Electric Power Assisted Steering 20
2.4. Basics of Steer-by-Wire Systems 23
2.4.1. Hand-wheel Subsystem of Steer-by-Wire Systems 25
2.4.2. Front Wheel Subsystem of Steer-by-Wire Systems 29
2.5. Nonlinearities and Disturbances in Steer-by-Wire Systems 32
2.5.1. Coulomb Friction 32
2.5.2. Torque Pulsation Disturbances 33
viii
2.5.3. Tyre Aligning Moment 37
2.6. Existing Modelling for Steer-by-Wire Systems 43
2.7. Basis Control Methodologies for Steer-by-Wire Systems 46
2.7.1. Conventional Linear Feedback Control for Steer-by-Wire
Systems 46
2.7.2. Adaptive Control for Steer-by-Wire Systems 48
2.8. Lyapunov Stability Theory 49
2.8.1. Stability-related Definitions 50
2.8.2. Direct Method of Lyapunov Stability 51
2.9. Sliding Mode Control Theory 54
2.9.1. Sliding Mode Control Design 55
2.9.2. The Chattering Problem 60
2.9.3. Reaching Law Method for SMC Design 63
2.10. Finite Time Sliding Mode Control 66
2.10.1. Terminal Sliding Mode Control 66
2.10.2. Nonsingular Terminal Sliding Mode Control 69
2.11. Summary 70
3 Sliding Mode Control for Steer-by-Wire Systems with AC
Motors in Road Vehicles 71
3.1. Introduction 71
3.2. Problem Formulation 75
3.2.1. Mathematical Modelling 75
3.2.2. Bounds of System Parameters and Disturbances 77
3.3. Design of A Robust Sliding Mode Controller 80
3.4. Numerical Simulation 84
ix
3.4.1. Parameters of SbW System and Vehicle Dynamics 84
3.4.2. Control Law 87
3.4.3. Simulation Environment 88
3.4.4. Simulation Results 88
3.5. Experimental Studies 99
3.5.1. Experimental System Setup 99
3.5.2. Experimental Results 101
3.6. Conclusion 105
4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire
Systems with Uncertain Dynamics 107
4.1. Introduction 107
4.2. Problem Formulation 110
4.3. Design of An NTSM Steering Controller 111
4.4. Experimental Results 118
4.5. Conclusion 126
5 Robust Control for Steer-by-Wire Systems with Partially
Known Dynamics 129
5.1. Introduction 129
5.2. Problem Formulation 133
5.2.1. Modelling 133
5.2.2. Disturbances 137
5.2.3. Bounded Property of System Lumped Uncertainty 139
5.2.4. Bounded Property of Steering-Wheel Angular Acceleration
141
x
5.3. Design of A Robust Control Scheme 143
5.3.1. Controller Design for System with Uncertainty 145
5.3.2. A Robust Exact Differentiator 150
5.4. Experimental Studies 154
5.4.1. Experimental System Identification 154
5.4.2. Experimental Results 156
5.5. Conclusion 164
6 Conclusions and Future Work 165
6.1. Summary of Contributions 165
6.2. Future Research 167
6.2.1. Sliding Mode-based Adaptive Control for SbW Systems 167
6.2.2. Sampled Data Systems 168
6.2.3. Observer Design for SMC-based SbW Systems 168
6.2.4. Vehicle Stability Control for SbW-equipped Vehicles 169
Appendix A Proof of Bounded Property of Lumped Uncertainty
in (5.32) and (5.33) 171
Author’s Publications 175
Bibliography 177
xi
List of Figures
2.1 Drive-by-Wire system 11
2.2 Conventional steering system 12
2.3 Block diagram of conventional steering system 15
2.4 The Hydraulic power assisted steering system as a part of the vehicle’s closed
loop 16
2.5 Simplified representation of the HPAS system 17
2.6 Block diagram of the HPAS system 19
2.7 EPAS system model 20
2.8 Steer-by-Wire system 24
2.9 Steer-by-Wire hand-wheel subsystem 25
2.10 Steering wheel torque versus lateral acceleration 27
2.11 Steer-by-Wire front wheel subsystem 29
2.12 Side view of steering actuator assembly together with universal joint 31
2.13 Block diagram of Steer-by-Wire system 31
2.14 Coulomb friction 33
2.15 Linear bicycle model 38
xii
2.16 Tyre force at front central wheel (a)Tyre forces and (b)Self-aligning torque 39
2.17 Tyre lateral force versus slip angle 40
2.18 Rack-actuating type of front wheel system 45
2.19 Tie-rod-actuating type of front wheel system 45
2.20 The chattering phenomenon 60
2.21 Saturation function ( ) 61
2.22 Continuation approximation method for ( ) 62
3.1 Control performance of SM controller (a) Tracking performance
(b) Tracking error (c) Control torque (d) Self-aligning torque and upper
bound (e) Torque pulsation disturbances and upper bound (f) Tracking in
the 25th second 90
3.2 Control performance of BL-SM controller (a) Tracking performance
(b) Tracking error. (c) Control torque (d) Self-aligning torque and upper
bound (e) Torque pulsation disturbances and upper bound. (f) Tracking in
the 25th second 91
3.3 Control performance of PD controller (a) Tracking performance
(b) Tracking error (c) Control torque (d) Self-aligning torque
(e) Torque pulsation disturbances (f) Tracking in the 25th second 93
3.4 Control performance of controller (a) Tracking performance
(b) Tracking error (c) Control torque (d) Self-aligning torque
(e) Torque pulsation disturbances (f) Tracking in the 25th second 94
3.5 Control performance of BL-SM controller (a) Tracking performance.
xiii
(b) Tracking error (c) Control torque 97
3.6 Control performance of PD controller (a) Tracking performance.
(b) Tracking error (c) Control torque 98
3.7 Control performance of controller (a) Tracking performance
(b) Tracking error (c) Control torque 99
3.8 The SbW Experimental Platform 100
3.9 Control performance of BL-SM controller (experiments) (a) Tracking
performance (b) Tracking error (c) Control torque 102
3.10 Control performance of PD controller (experiments) (a) Tracking
performance (b) Tracking error (c) Control torque 103
3.11 Control performance of controller (experiments) (a) Tracking
performance (b) Tracking error (c) Control torque. 104
4.1 Control performance of PD control at case 1 (a) Tracking performance
(b) Tracking error (c) Control torque 120
4.2 Control performance of BL-SM control at case 1 (a) Tracking performance
(b) Tracking error (c) Control torque 121
4.3 Control performance of BL-NTSM control at case 1 (a) Tracking
performance (b) Tracking error (c) Control torque 122
4.4 Control performance of PD control at case 2 (a) Tracking performance
(b) Tracking error (c) Control torque 123
4.5 Control performance of BL-SM control at case 2 (a) Tracking performance
(b) Tracking error (c) Control torque 124
xiv
4.6 Control performance of BL-NTSM control at case 2 (a) Tracking performance
(b) Tracking error (c) Control torque 125
5.1 The full SbW system control diagram 153
5.2 The SbW experimental platform (a) Steering-wheel subsystem.
(b) Front-wheel subsystem 153
5.3 Frequency responses of the SbW system model 155
5.4 Control performance of proposed controller (a) Tracking performance.
(b) Tracking error (c) Control torque 158
5.5 Control performance of BL-SM controller (a) Tracking performance.
(b) Tracking error (c) Control torque 159
5.6 Control performance of controller (a) Tracking performance
(b) Tracking error (c) Control torque 161
5.7 Control performance of NFC (a) Tracking performance (b) Tracking error
(c) Control torque 162
xv
List of Tables
2.1 EPAS system parameters 22
2.2 Parameters of tyre dynamics 40
3.1 Nominal parameter values of the SbW system in equation (3.5) 85
3.2 Nominal parameters of PMAC motor 85
3.3 Parameters of vehicle dynamics and motor harmonic torque for simulation 86
3.4 Values of control parameters 101
4.1 Parameters of the SbW system 111
4.2 Performance comparisons of controllers in Chapter 4 126
5.1 Nominal parameters of the SbW system model in equation (5.12) 156
5.2 Performance comparisons of controllers in Chapter 5 164
xvi
xvii
List of Abbreviations and Acronyms
ABS – antilock braking system
BbW –-Brake-by-Wire
BLC – boundary layer compensator
BL-NTSM – boundary layer NTSM
BL-SM – boundary layer sliding mode
CG – centre of gravity
EPAS – electric power assisted steering
ESC – electronic stability control
GPS – global positioning system
HPAS – hydraulic power assisted steering
INS – inertia navigation system
LTI – linear time-invariant
NFC – nominal feedback controller
NTSM – nonsingular terminal sliding mode
PAS – power assisted steering
PD – proportional derivative
xviii
PMAC – permanent magnet asynchronous current
RCS – robust control scheme
REDs – robust exact differentiators
RMS – root mean square
SbW – Steer-by-Wire
SM – sliding mode
SMC – sliding mode control
TbW – Throttle-by-Wire
TSM – terminal sliding mode
VGRS – variable gear ratio steering
VSC – vehicle stability control
VSSs – variable structure systems
Chapter 1 Introduction
1
Chapter 1
Introduction
HE automotive industry has experienced rapid development and growth over the
last century. Engineers and scientists from many fields, such as information
technology, advanced materials, defence systems, and aerospace, have collaborated with
the automotive industry in order to boost vehicle performance and enhance passenger
safety. Recently, X-by-Wire technology has become more and more popular in
automotive applications, where the input device used by the driver is connected to the
actuation power subsystem by electrical wires, which was formerly connected by
mechanical or hydraulic means. Examples such as Throttle-by-Wire (TbW), Brake-by-
Wire (BbW), and Steer-by-Wire (SbW) are common [1-4]. Among these, due to the fact
that the steering systems play an essential role for the driver to interact with the vehicle,
SbW systems and their control are receiving great attention from the automotive
industry for the purpose of precisely regulating the operations of vehicles [5-12].
Though SbW control systems have achieved a great success by improving the
stability of road vehicles and the comfort of drivers, it is still not competent to ensure
good steering performance, particularly when the vehicle frequently experiences
unexpected varying road conditions. Therefore, robust control for SbW systems should
to be designed to ensure a robust steering performance under varying road environments.
T
Chapter 1 Introduction
2
1.1 Steer-by-Wire Systems and Control
Over the last two decades, with the rapid development and growth of electronic control
systems, like the antilock braking system (ABS) [3, 4] and electronic stability control
(ESC) [12, 13], electronic technology has significantly influenced automotive
engineering and made tremendous achievements. In terms of automotive steering
systems, hydraulic power assisted steering (HPAS) systems [14-16] have been replaced
by electric power assisted steering (EPAS) systems [17-20] in production vehicles in
order to provide more efficient power steering and easily tune the assist level based on
the vehicle type, road speed, and driver preference. SbW systems representing the next
generation of steering systems have been introduced and received great attention among
automotive engineers and researchers.
In SbW systems, no mechanical linkage is used between the hand-wheel and the
steered front wheels. The front wheels are steered by an electric motor via the rack and
pinion gear box by following the hand-wheel steering commands. Meanwhile, a small-
power electric motor is mounted on the hand-wheel side to provide drivers with a
feeling of the steering effort. The introduction of an SbW system in road vehicles brings
many advantages over the existing ones in terms of improving steering performance,
enhancing vehicles’ manoeuvrability and driving safety. Specifically, with no
mechanical shaft between the hand-wheel and front wheels, noise vibration and
harshness from the road surface changes do not transfer to the drivers’ hands through
the hand-wheel. Furthermore, the removal of the mechanical shaft alleviates a potential
physical risk in case of an accident. In addition, the fixed steering ratio and steering
effort can now be flexibly adjusted in SbW systems to optimize steering response and
feel.
On one hand, the mathematical modelling of SbW systems becomes of vital
importance in SbW control systems. Although many studies on the modelling of SbW
systems have been carried out, either the steering motor dynamics are ignored or the
Chapter 1 Introduction
3
tyre disturbances are not included in the system modelling [10, 11, 21-25]. The detailed
modelling of SbW systems, from the steering motor to the steered front wheels, has not
been fully studied yet. On the other hand, various controllers have been developed for
SbW systems over the last few years for improving the stability of vehicles and the
comfort of drivers. The most widely used control methods are the linear feedback
control method, with the conventional proportional and derivative (PD) signals of the
tracking error, and linear quadratic control method that can be easily implemented in the
real applications [10, 11, 22-25]. However, good steering performance cannot be
achieved when the road surface conditions are varying, such as from an icy or snowy
road to a wet asphalt road. The reason is that the control parameters are only locally
optimized and may not be applicable when large system uncertainties and disturbances
occur. Although some other auxiliary methods for estimating the tyre sideslip angle and
tyre cornering stiffness are used for the compensation of uncertainties and road
disturbances (tyre self-aligning torque), the controller structure accordingly becomes
complicated due to the use of many extra sensors. For instance, the following methods
are commonly used: the adaptive online parameter estimation in [26] and adaptive
control method for virtual steering characteristics using the identified cornering stiffness
in [27], and observer designs for observing the tyre self-aligning torque in [28, 29].
However, how the uncertain parameters and unknown disturbances under varying road
environment can be accurately estimated online to guarantee a robust steering
performance is still an open issue.
It is seen from the above discussion that the difficulties in the design of high quality
SbW control systems are twofold: (i) A complete mathematical model of a SbW system
should be obtained, involving the main steering system dynamics and uncertainties as
well as disturbances; (ii) How the effects of system uncertainties and highly nonlinear
road disturbance torque variations, due to different road conditions on the steering
performance, can be eliminated. Therefore, it is essential to derive the integrated
mathematical model of SbW systems and design robust control algorithms that are
Chapter 1 Introduction
4
expected to eliminate the effects of uncertain steering system dynamics and
disturbances.
1.2 Sliding Mode Control Systems
A closed-loop sliding mode control (SMC) system is a variable structure system (VSSs)
[30, 31]. SMC has been extensively studied for more than half a century and used in
many practical applications [32, 33]. The essence of SMC is that a sliding mode
controller is designed to drive a sliding variable (vector) to reach a prescribed switching
manifold (sliding surface) in a finite time, the controller can then play the role of
maintaining closed-loop dynamics on the sliding surface in the sense that the closed-
loop dynamics can asymptotically converge to the equilibrium point. Such a
convergence of the closed-loop dynamics constrained on the sliding mode surface is not
affected by the system uncertainties and external disturbances.
In general, an SMC design is divided into two phases. The first phase is to design a
sliding surface such that the system state trajectory that is restricted to the surface
possesses the desired dynamics. The second phase is to design a switching control for
driving the closed-loop dynamics to reach the sliding surface and then be maintained
within a neighbourhood of the switching manifold for all subsequent time.
An important feature of SMC is that a sign function is used in the control signal to
ensure that, after reaching the sliding mode surface, the closed-loop dynamics can be
constrained on the sliding mode surface. However, nearly all closed-loop SMC systems
suffer from chattering because of the sign function involved in the sliding mode control,
which is the major disadvantage of SMC in practical applications [32, 35]. Therefore,
two well-known methods have been proposed to eliminate chattering, i.e., boundary
layer technique [33-35], and the continuous approximation method [35, 38-40].
Chapter 1 Introduction
5
For most SMC systems, a linear sliding mode surface is often adopted to describe the
desired system dynamics. A robust controller is designed to drive the sliding variable to
reach the sliding mode surface and the asymptotic convergence of system state variables
can then be obtained on the sliding mode surface. Although the parameters of the linear
sliding mode can be adjusted in order to obtain an arbitrarily fast convergence rate, the
system states on the sliding mode surface cannot possess the finite-time convergence
characteristic. Therefore, to solve this problem, terminal sliding mode (TSM) control
and nonsingular terminal sliding mode (NTSM) control techniques were developed by
Man and Yu [44], and Feng [45], for the purpose of achieving finite time convergence
of the system dynamics on the terminal sliding mode surface. Meanwhile, the gain of
the TSM controller can be significantly reduced in comparison with the high gain of
linear sliding mode controllers.
1.3 Motivations
Steering systems, as the human-vehicle interface, have been intensively studied.
Nowadays, power assisted steering (PAS) technology has improved the steering ease
and comfort in road vehicles. To further improve the safety and reliability of road
vehicles, from an engineering point of view, the development of SbW systems has the
priority in automotive industry in the next ten years. Although many control approaches
have been developed for SbW systems, good steering performance cannot be fully
achieved because of varying road conditions and variations of steering system
parameters and disturbances. Therefore, there is an urgent need to develop robust
steering controllers to eliminate the effects of uncertain steering system parameters and
varying road conditions. This accordingly has led the research work of this thesis.
1.4 Objectives and Major Contributions of the Thesis
In this thesis, mathematical modelling and robust control for SbW systems with
uncertain dynamics are studied. A few sliding mode control and terminal sliding mode
Chapter 1 Introduction
6
control algorithms using the bound information of uncertain system parameters and
disturbances are developed. The focus of controller designs is on eliminating the effects
of uncertainties and disturbances on the steering performance and significantly
improving the safety and reliability of road vehicles under different road conditions.
The following outlines the major contributions of the thesis.
1. Develop a complete mathematical model for SbW systems and design a robust
SMC scheme, using the bound information of uncertain system dynamics and
disturbances, to achieve high-quality steering performance.
2. Design a robust NTSM control scheme to ensure the finite time error
convergence of the closed-loop SbW system and strong robustness against
uncertain dynamics.
3. Develop a novel robust control scheme for the SbW systems with partially
known dynamics, in the sense that the nominal system can be stabilized by a
nominal feedback controller and the uncertain dynamics in the closed-loop SbW
system can be compensated by a sliding mode compensator.
4. Implement the proposed control algorithms in real-time experiments on the SbW
platform to verify the excellent performance and efficacy of the proposed
control methods in comparison with the conventional control methodologies.
In summary, the work of the thesis will significantly enhance the research on SbW
control systems for achieving the desired steering performance and especially offering
strong robustness against varying road conditions.
Chapter 1 Introduction
7
1.5 Organizations of the Thesis
This thesis presents a robust control designs for SbW systems in order to improve
steering accuracy and precision under various system parameter variations and road
disturbances.
The organization of this thesis is as follows.
Chapter 2 provides a survey of the basics of steering systems as well as SMC systems.
The existing modelling and control methodologies of steering systems are presented,
and the total disturbances in the SbW systems are discussed in detail. The designs of
SMC systems, chattering phenomenon with solutions, and reaching law method are
reviewed, and the recently developed terminal sliding mode control schemes are also
discussed.
Chapter 3 explores the mathematical modelling of SbW systems. A sliding mode
controller using the bound information of the uncertain system parameters and
disturbances is then designed to ensure the front wheel steering angle to asymptotically
track the hand-wheel commands. Both the simulation and experimental results are
presented to confirm the steering performance of the closed-loop SbW system.
Chapter 4 considers an NTSM control scheme for SbW systems with uncertain
dynamics. The design concentrates on the convergence performance and robustness of
the NTSM-based SbW control system. It is shown that due to the finite time
convergence and high accurate tracking capabilities, the NTSM control behaves with a
better level of tracking performance and robustness compared with the SMC scheme
proposed in Chapter 3. Experimental results are presented to verify the analysis.
Chapter 5 presents a robust control scheme for SbW systems with partially known
dynamics. In practice, an SbW system is a partially known system with an unknown
Chapter 1 Introduction
8
portion. After that, the chapter moves on to design a nominal feedback controller for
stabilizing the nominal system and introduce a sliding mode compensator for
eliminating the effects of the unknown parts. The upper bound of the unknown system
uncertainty is described in detail. Experimental results are also given to show the
validity of the proposed robust control scheme.
Finally, this thesis is summarized and concluded in Chapter 6 where topics for future
work are suggested.
Chapter 2 Background and Literature Review
9
Chapter 2
Background and Literature Review
This chapter will provide a broad review of the various generations of steering systems
followed by a discussion on the existing modelling and control methodologies of SbW
systems. Following that, is a review of the fundamental theory of sliding mode control
and consideration of how the sliding mode control can be used to design robust control
schemes for systems with uncertain dynamics. In Section 2.1 ~ Section 2.3, how SbW
systems came into being in road vehicles is briefly reviewed, and the main components
in SbW systems are described in detail. In Section 2.4, the major nonlinearities and
disturbances in SbW systems are presented. In Section 2.5 and Section 2.6, the existing
mathematical modelling for SbW systems is reviewed and the control methodologies of
SbW systems are summarized. In Section 2.7, systems with partially known dynamics
and control are briefly revisited for facilitating and simplifying the controller designs in
the following chapters. In Section 2.8 ~ Section 2.10, focus is given on reviewing the
basic concepts as well as definitions of the sliding mode control (SMC) theory.
Furthermore, an outline is given of several typical SMC schemes and finite time SMC
methodologies that will be adopted in the controller designs for the SbW systems in the
subsequent chapters.
Chapter 2 Background and Literature Review
10
2.1 History of Steering Systems
HROUGHOUT the history of steering systems in road vehicles, the steering
systems are generally classified into three generations: (i) mechanical steering
systems [14]; (ii) hydraulic- and electro-hydraulic-power-assisted steering (HPAS and
EHPAS) systems [15, 16, 46-51]; and (iii) electric-power-assisted steering (EPAS)
systems [17-20, 52-57]. Obviously, the common point of these three assisted steering
systems is that the intermediate mechanical link used to connect the hand-wheel to the
steered front wheels, through the rack and pinion gearbox is still retained on the hand-
wheel side. Unfortunately, with the use of the mechanical shaft with a fixed gear ratio, it
is not easy to assist in adjusting the driver’s steering command in accordance with the
different driving conditions and environments as well as the vehicle’s status. Also, the
steering shaft is a potentially risky component for drivers in case of an accident.
In order to address the problems, by-wire technology that has already been
completely applied to fly-by-wire flight control systems in modern aircraft, is
consequently introduced to replace mechanical or hydraulic systems by electronic ones.
Automakers have also employed by-wire technology for throttle and brakes in
production vehicles with the benefits of lowering power consumption and improving
vehicle control, such as TbW and BbW systems. The next challenging target is to apply
the new technology to the next generation of steering systems, which are termed SbW
systems. Figure 2.1 shows the concept design of the Drive-by-Wire system that includes
TbW, BbW, and SbW systems [58]. As introduced in Chapter 1, several potential
benefits of SbW systems compared with the conventional steering systems are the
improvement of the overall steering performance and vehicle stability control, reduction
in the automotive power consumption, and enhancement of the safety and comfort for
the drivers as well as passengers during normal driving and emergencies.
Although there are no production vehicles equipped with SbW systems today, the
SbW systems have drawn a great deal of attention from the automotive industry due to
T
Chapter 2 Background and Literature Review
11
Figure 2.1: Drive-by-Wire system Credit: Motorola.
their potential benefits and have been demonstrated in several concept vehicles such as
ThyssenKrupp Presta Steering’s Mercedes-Benz Unimog, General Motors’ Hy-wire and
Sequel, and the Mazda Ryuga. More recently, Nissan has announced a project to install
a new SbW system in its Infinity line of automobiles by the end of 2013, which will be
the first time in the world that an SbW system will be available in a mass produced
automobile.
2.2 Conventional Steering Systems
This subsection will review the working principle of the conventional steering systems
in road vehicles and the corresponding drawbacks in detail.
It should be noted that, for the HPAS systems and the EPAS systems, the essence is
to provide steering assistance to the driver through a mechanical shaft using a hydraulic
Chapter 2 Background and Literature Review
12
Figure 2.2: Conventional steering system.
and an electronically-controlled electric motor, respectively. However, from the
mechanical shaft point of view, because the total steering torque is directly transferred
to the steered wheels from the mechanical shaft, consequently a more general model for
conventional steering systems has been used as shown in Figure 2.2 [11]. Basic
information regarding the model and the performance of HPAS and EPAS systems will
be briefly discussed in the next subchapter and the details can be found in literature [15-
20, 46-57].
The conventional steering system that can be modelled as a multi-mass system is
composed of several basic components: the hand-wheel, the steering column, the
mechanical shaft, the steering rack, the pinion, the tie rods connecting the steering rack
to the steered wheels, the steering arms, and the two steered wheels. Note that no
backlash exists in the rack and pinion gear teeth and the spring effect in the tie rods is
also negligible. The dynamic equations of the conventional steering system in Figure
2.2 are given below.
Chapter 2 Background and Literature Review
13
First, the dynamic equation of the hand-wheel side is described by the following
second-order differential equation:
(2.1)
where and are the moment of inertia and the viscous friction coefficient of the
hand-wheel side elements, is the input torque exerted by the driver on the hand-
wheel, and is the reaction torque applied on the hand-wheel side elements by the
shaft which can be modelled as
(
)
(
)
(
)
(
) (2.2)
where and are the damping coefficient and the torsional stiffness of the shaft,
respectively, is the front wheel steering angle, is the torque ratio between the
steering torque to be defined later and , and is the angle ratio between and
. It can be seen that the reaction torque is proportional to both the hand-wheel
rotational angle and its angular velocity.
The dynamics of the steered front wheels about their vertical axes crossing the wheel
centres can be described by
(2.3)
where and are the moment of inertia and the viscous friction coefficient of the
steered front wheels, is the self-aligning torque reflecting the interaction between the
road surface and the steered front wheels while the vehicle is turning, and is the
Chapter 2 Background and Literature Review
14
steering torque exerted on the steering arm by the shaft through the rack and pinion
gearbox which is given by
(
) (
)
(
) (
) (2.4)
Thus, similar to the reaction torque , the steering torque is also proportional to
the hand-wheel rotational angle and its angular velocity. Due the rack and pinion
gearbox mechanism, the steering torque and the reaction torque satisfy the
following relationship:
(2.5)
It is seen from (2.5) that, the two ratios and have the same values in the
conventional steering systems that are determined by the rack and pinion gearbox
mechanism. However, in a practical situation, it is because of the inherent
characteristics that these two ratios are highly nonlinear and related to the front wheel
steering angle, that the hand-wheel position should be adjusted frequently by the driver
to provide the desired trajectory.
For a further understanding of the fundamental principle of the conventional steering
system, the conventional steering system is described in a block diagram as shown in
Figure 2.3, which is based on the equations given above (2.1)-(2.4). In Figure 2.3, it can
be seen that the components on the hand-wheel part at the bottom are represented by the
total moment of inertia and the total viscous friction coefficient of the hand-wheel
side elements, respectively. Similarly, the components on the steered wheel part at the
top are denoted by the total moment of inertia and the total viscous friction coeffic-
Chapter 2 Background and Literature Review
15
Figure 2.3: Block diagram of conventional steering system.
ient of the shaft, respectively. In particular, the shaft in the middle part of the
diagram is described by the damping coefficient and the torsional stiffness of the
shaft, respectively.
Furthermore, when it comes to analyzing the characteristics of the tyre self-aligning
torque , two different situations are thought to affect the steering performance. Firstly,
when the vehicle keeps still and the driver starts to rotate the hand-wheel, the stiction
torque is in opposition to the steering torque which is generated by the tyre contact
patches when turning. In this case, the tyre self-aligning torque which has the same
value as the stiction torque is transferred to the steered front wheel side by means of the
elastic twisting of the tyres. When the driver keeps on rotating the hand-wheel to enable
the steering torque to exceed the stiction torque, the steered front wheels start to turn
Chapter 2 Background and Literature Review
16
Figure 2.4: Hydraulic power assisted steering system as a part of a vehicle’s closed loop.
whereby becomes the dry friction torque of the road surface. After that, when the
vehicle is moving, the significant portion of appears, which is a function of the
steering geometry, especially the caster and kingpin angles and is strongly associated
with vehicle longitudinal velocity and the steering angle.
With the help of the block diagram of a conventional steering system given in Figure
2.3, it has been found in [11] that, the conventional mechanical shaft performs as a
proportional derivative (PD) regulator, that is, the torque exerted on the steered wheels
by the mechanical shaft is proportional to both the driver’s rotational torque on the
hand-wheel wheel and its changing rate. However, it is because the gains of the PD
regulator are represented by the inherent mechanical parameters of the shaft that the
responsiveness of the conventional steering systems cannot be adjusted. Thus, it is very
difficult for a driver to quickly adjust the steering angle to a desired position in case of
accidents and critical situations, which leads to the unguaranteed safety and reliability
of the vehicles in practice.
Chapter 2 Background and Literature Review
17
Figure 2.5: Simplified representation of an HPAS system.
Source: Marcus 2007.
2.3 Power Assisted Steering Systems
The key task of a power steering system in road vehicles is to decrease the steering
effort of the driver by applying additional torque to the mechanical steering column in
certain driving situation such as lower speed manoeuvring and parking cases. Generally,
HPAS and EPAS control systems are the two most typical representatives in power
assisted steering systems for improving the steering feel as perceived by the driver and
ensuring adequate assistance levels based on the measured steering column torque using
the torque sensor.
2.3.1 Hydraulic Power Assisted Steering
The HPAS system layout is generally the same from car to car, as shown in Figure 2.4,
where the main components of the HPAS system are the hydraulic pump, the rotary
spool valve, and the rack piston. This figure describes the power steering unit with a
Chapter 2 Background and Literature Review
18
more detailed view, where the steering wheel is connected to the steering rack via the
valve considered to be the controlling element in the steering unit. This valve
displacement together with the hydraulic system regulates the pressure in the cylinder
for the purpose of adding appropriate assistance to the steering rack. Due to the fact that
the complexity and usability of the model is not a linear relationship, in this work, we
use a simplified representation of the applied HPAS system, as shown Figure 2.5. In
order to the study the modelling of the HPAS system and analyse its stability, we
provide the following suitable linear model, in which all motions are linear and related
to the motion of the steering rack.
The force of the steering wheel (hand-wheel) is described by the following equation,
where the torque applied by the driver, , is transferred to a linear force by the
radius of the pinion gear, :
∑ ( )
(2.6)
where is the displacement of the steering wheel, is the displacement of the rack,
is the mass of the steering wheel, is the viscous damping in the steering wheel,
and is the equivalent spring coefficient in the torsion bar.
The rack dynamics including the tyres are expressed as
∑ ( ) ( ) (2.7)
where is the mass of the rack, and are the viscous damping of the wheel and
the rack, respectively, is the maximal external load, and is the lateral spring
coefficients in the tyre.
The hydraulic system can be described by the following equation:
Chapter 2 Background and Literature Review
19
Figure 2.6: Block diagram of HPAS system.
(2.8)
where is the load pressure, is the hydraulic capacitance, is the linearized flow-
pressure coefficient, is flow gain, is cylinder area, and is the linear
displacement of the valve.
The actuation of the valve is generated by the difference between the steering wheel
position and the rack position, which is of the form:
(2.9)
It should be noted that a power steering system is a closed-loop system in which the
rack position is the controlled variable. The steering wheel angle is the input and
reference angle. Thus, it is useful to adopt a linear system for understanding the funda-
Chapter 2 Background and Literature Review
20
Figure 2.7: EPAS system model.
mental principle of the HPAS system. Based on the equations (2.6)-(2.9), we can obtain
the block diagram of the HPAS system, as shown in Figure 2.6, where the driver’s
torque is the input and the steering rack position is the output. It is easily seen that the
motion of the system is completely transferred to the linear motion associated with the
steering rack dynamics. Readers of great interests may refer to [46-52] for the
specifications of the modelling and control of HPAS systems.
2.3.2 Electric Power Assisted Steering
Due to the fact that the efficiency of conventional HPAS systems especially for high-
way driving is quite low, EPAS systems as an alternative have recently entered the
market for smaller and medium sized vehicles. The advantages of using EPAS systems
are clear: engine independence and fuel economy, tenability of steering sense,
modularity and quick assembly, and environmental compatibility. The operation of an
EPAS system equipped with a brushed DC motor is shown in Figure 2.7. When a driver
Chapter 2 Background and Literature Review
21
rotates the hand-wheel, the steering torque is detected by a torque sensor installed
between the hand-wheel and the motor. The measured driver torque is used to determine
the amount of assisted torque provided by the electric motor. The assisted torque which
is generated by the tunable torque boost based on the vehicle speed and the applied
driver’s torque is combined with the driver’s torque to provide the total steering torque
for the front wheels. The dynamic models of the EPAS system including the hand-wheel
rotation and the motor dynamics are described as follows:
( ) (2.10)
(
)
( )
(2.11)
(2.12)
where ( ⁄ ) , ( ⁄ ) , ⁄ , is the
driver torque, is the road reaction force, and are the rotational angles of the
hand-wheel and motor shaft, respectively, is the motor voltage, and is the rack
position. The EPAS system model parameters are given in Table 2. 1.
In addition, the steering torque applied to the steering column , the road reaction
torque , and the assisted torque are given by
(
) (2.13)
(2.14)
Chapter 2 Background and Literature Review
22
Table 2.1: EPAS system parameters.
Parameter Definition
and Moments of inertia of steering
column and motor
and
Viscous damping coefficients of steering column and motor shaft
Steering column stiffness
and
Frictions of steering column and motor
Rack mass
Rack viscous damping
Steering column pinion radius Tyre spring rate Motor torque, voltage constant Motor inductance
Motor resistance
Motor current N Motor gear ratio
(2.15)
Thus, using the equations (2.10)-(2.15), we can derive the linear model of the EPAS
system in the state-space form:
{
[ ] [ ]
(2.16)
where [ ] is the state vector of the EPAS system, is the
Chapter 2 Background and Literature Review
23
DC motor voltage, [ ] denotes the vector of system unknown inputs
consisting of the driver’s torque and the road reaction torque, respectively, is the
applied steering torque on the steering column reflecting the steering feel on the driver’s
hands, [ ] is the measured hand-wheel and motor shaft angle, the system
matrices are of the following forms:
[
(
)
]
[
]
,
[
]
[
], [
]
The objectives of the EPAS control system are to ensure appropriate assisted torque,
system stability, vibration attenuations, improved hand-wheel returnability and freely
controlled performance. In addition, the control system must be robust against system
modelling errors and parameter uncertainties as well as road disturbances. Readers can
refer to [20, 54-57] for a detailed description in terms of the modelling and control
issues.
2.4 Basics of Steer-by-Wire Systems
In order to overcome the difficulties caused by the mechanical shaft in conventional
steering systems, a new technique called SbW systems has been developed and has rec-
Chapter 2 Background and Literature Review
24
Figure 2.8: Steer-by-Wire system.
eived a great deal of attention for researchers and engineers in the automotive industry
[10, 11, 21-27, 60-63]. It is expected that SbW systems will completely replace the
mechanical shaft in modern vehicles to provide significantly improved safety and
reliability in the near future.
The modern SbW systems shown in Figure 2.8 have the following distinct
characteristics: Firstly, the intermediate mechanical link in conventional road vehicles
used to connect the hand-wheel to the steered front wheels, through the rack and pinion
gearbox is eliminated. Secondly, the front wheel steering motor flexibly coupled to the
rack and pinion gearbox is adopted to steer the front wheels for tracking the hand-wheel
reference angle that is provided by the installed hand-wheel angle sensor. Thirdly, the
purpose of the hand-wheel feedback motor installed on the hand-wheel side is to
provide drivers with the true feelings of the effects of self-aligning torque acting
between the front tyres and the road surface. Control of the hand-wheel feedback motor
is based on the error information between the hand-wheel reference angle sensor and the
Chapter 2 Background and Literature Review
25
Figure 2.9: Steer-by Wire hand-wheel subsystem.
actual front wheel steering angle, measured by the hand-wheel angle sensor and the
pinion angle sensor, respectively.
Thus, the SbW system as shown in Figure 2.8 is separated into two subsystems: the
hand-wheel and the front wheel subsystems, respectively. The hand-wheel subsystem
consists of a hand-wheel, a hand-wheel angle sensor, and a hand-wheel feedback motor
to provide the driver with force feedback. The front-wheel subsystem is composed of a
pinion angle sensor, a rack and pinion gearbox, and a steering motor that provides the
necessary torque to steer the front wheels through the rack and pinion gearbox.
2.4.1 Hand-wheel Subsystem of Steer-by-Wire Systems
When the mechanical connection between the hand-wheel and the front wheels is
removed in the SbW systems, the natural source of force feedback does not exist any
longer. The forces in the hand-wheel part are considered to be one of the major sources
Chapter 2 Background and Literature Review
26
of information that the driver uses to manipulate the vehicle, and thus artificial force
feedback is an essential part for assisting the driver to obtain the feelings of the road
surface.
Figure 2.9 shows the physical description of the hand-wheel subsystem as
implemented in the experimental platform, which is to be introduced in the next chapter.
It is the combination of the three components that achieves the characteristics of the
torque feedback in the SbW systems. The hand-wheel angle sensor is attached to the
hand-wheel to provide the reference information for the front wheels to follow.
Meanwhile, the data obtained from the hand-wheel angle sensor is transmitted to the
hand-wheel feedback control unit for generating the corresponding torque input signal
for the feedback motor. The main purpose of the hand-wheel feedback motor is to
provide the driver with the feeling of the effects of self-aligning torque between the
front wheels and the road surface, based on the error information between the reference
angle and the actual steering angle. In this thesis, the hand-wheel dynamics as illustrated
in Figure 2.9 can be described by the following second-order differential equation:
[ ] [
] [ ] [
] ( ) (2.17)
where , , and are the moment of inertia, the viscous friction coefficient and the
torsional stiffness of the hand-wheel shaft, respectively, is the hand-wheel rotational
angle, is the input torque provided by the driver, and is the feedback torque
generated by the hand-wheel feedback motor controlled by a control unit based on the
tracking error between the reference angle from the hand-wheel, and the steering angle,
providing the driver with the true feeling of the steering effort. Moreover, in this thesis,
the control unit is chosen as a PD regulator as in literature [11, 64] and the control
parameters of the regulator must be chosen in the sense that the closed-loop in the hand-
wheel side is stable.
Chapter 2 Background and Literature Review
27
Figure 2.10: Steering wheel torque versus lateral acceleration.
The desired reference angle for the front wheel to follow can be expressed as:
(2.18)
where is the ratio between the hand-wheel rotational angle and the front wheel
steering angle. It is interesting to note that the ratio in SbW systems can be
adaptively adjusted based on different driving conditions. For instance, during low
speed driving situations like parking, can be decreased to achieve an easy level of
steering effort for the driver while the value of can be increased at high speeds for
maneuvering the vehicle steadily and avoiding steering sensitivity.
It is important to note that, steering force feedback may take into account the
following components: inertia and damping whereby the driver can experience similar
feelings as the mechanical steering system; tyre self-aligning torque generated by the
Chapter 2 Background and Literature Review
28
tyre lateral force; jacking effect resulting from the vertical tyre force and suspension
travel which is also a function of the steering angle. Thus, the steering force feedback
varies non-linearly with respective to the variables mentioned above in a variety of road
conditions. It should be emphasized that among the above factors, the tyre self-aligning
torque is considered to be the most essential factor in the force feedback. As shown in
Figure 2.10, a typical illustration of the relationship between the steering wheel torque
and the lateral acceleration is presented. In particular, an ideal level of torque feedback
to the driver should be guaranteed through the steep gradient in the hand-wheel torque
at small lateral acceleration. According to researches into steering effort in conventional
light vehicles, normal driving requires the steering wheel torque to range from 0 to 2
Nm, but it goes up to 15 Nm in an emergency situation [65, 66]. Thus, this important
characteristic in steering effort should be taken into consideration in relation to both the
selection of an appropriate feedback motor and the control of the feedback motor to
assist the driver with manoeuvring the vehicle.
Recently, a number of researches on force feedback have been intensively carried out
to recreate smooth reaction torque so that the driver can feel the interaction between the
tyres and the road. In [67, 68], on-centre force feedback (zero steering angle) was found
to contribute more force to the driver than other types and a detailed model was
demonstrated in a practical vehicle. In [69], a single controller was designed not only to
control the front wheels but also to provide the driver with force feedback, such that
both the tracking between the hand-wheel angle and the front wheel steering angle and
accurate force feedback can be guaranteed. This work was continued by Oh in 2004,
Park in 2005, and Kim in 2008, respectively [22, 24, 70], where a PID control method
was used to control the hand-wheel feedback motor to enable the hand-wheel to easily
rotate at parking and tight at high speeds. In [26, 27], the impedance control method was
adopted in the hand-wheel feedback motor to adjust the dynamic behaviour of the
steering system owing to its robustness.
As pointed out explicitly in the above literatures, steering force feedback is indeed an
Chapter 2 Background and Literature Review
29
Figure 2.11: Steer-by-Wire front wheel subsystem.
essential way to recreate the feelings of a conventional vehicle in SbW systems, which
is considered to be an challenging issue in SbW systems. However, in this thesis, the
focus is on formulating the mathematical modelling and robust control designs for the
front wheel subsystem in SbW systems and thus the topic of steering force feedback
will not be discussed further.
2.4.2 Front Wheel Subsystem of Steer-by-Wire Systems
This subsection will review the structure of the front wheel subsystem of SbW systems.
Although the mechanical design of SbW system basically alters a bit from car to car
which will be illustrated in detail by the modelling part in the later subsection, the
fundamental principle of the front wheel subsystem is still the same, as shown in Figure
2.11. The two front wheels are steered by the actual torque generated by the
steering actuator assembly via the rack and pinion gearbox and the steering arm. The
steering actuator assembly together with its servo driver is controlled by a control unit
based on the tracking error between the front wheel steering angle and the hand-
Chapter 2 Background and Literature Review
30
wheel reference angle . In order to steer the two front wheels sufficiently in different
driving situations, a steering actuator assembly including a steering motor and a
gearhead is attached to the auxiliary shaft connected to one side of the universal joint
while the other side of universal joint is flexibly coupled to the pinion side, as seen in
Figure 2.12 for details. The reason why the universal joint should be used is that the
steering actuator assembly cannot be share the same axis with the pinion side owing to
the limited space of the engineroom in a road vehicle. By introducing the universal joint,
not only can the space issue be effectively avoided, but the steering torque can be
transmitted with slight variations to meet the steering requirement [58, 71].
From the perspective of design, it should be noted that the dynamics of the steered
front wheels are similar to those in conventional steering systems in (2.1)-(2.4), where
the only difference is that the steering torque on the front wheel side components is
generated by the corresponding steering actuator instead of the mechanical shaft torque
being transferred from the hand-wheel due to a twisting motion. Similarly, it has been
shown in [11] that, the block diagram of the SbW system consisting of the hand-wheel
and front wheel subsystems is given in Figure 2.13. The steering actuator assembly and
the feedback motor with the corresponding drivers are denoted by and , which
are controlled by two control units denoted by and , respectively. The steering
characteristic of the front wheel subsystem, thus, can be obtained from the generated
motor torque via the rack and pinion gearbox with the steering ratio .
Furthermore, when it comes to choosing a suitable motor and gearhead for the front
wheel subsystem of SbW systems in practical application, two points need to be
considered. First, as mentioned in the above, normal driving requires steering torque
from 0 to 2 Nm while a maximum level of 15 Nm torque is needed in the emergency
driving conditions. Because the steering motor normally works in the nominal operating
range, a gearhead with a sufficient ratio should be chosen to ensure the maximum
steering torque. Second, in terms of the steering rate, studies suggest that maximum
steering rate with 540 degrees per second occurs during emergency manoeuvres while
Chapter 2 Background and Literature Review
31
Figure 2.12: Side view of steering actuator assembly together with universal joint.
Figure 2.13: Block diagram of Steer-by-Wire system.
Chapter 2 Background and Literature Review
32
the normal driving requires an average steering rate of 500 degrees per second. Thus,
considering the case of using the gearhead, the determination of the steering motor with
high nominal speed should be well performed in order to satisfy the steering rate
requirement in both normal and emergency driving situations.
Due to the fact that the front wheel subsystem plays an essential role in realizing the
steering purpose, in order to capture an excellent level of steering performance, the
nonlinearities and disturbances existing in the front wheel subsystem need to be
considered and how to deal with the effects of nonlinearities and disturbances becomes
of vital importance in the subsequent controller design. The following section will list
the typical nonlinearities and disturbances in SbW systems with the given modellings in
detail.
2.5 Nonlinearities and Disturbances in Steer-by-Wire Systems
As seen from the front wheel subsystem in the above literature [10, 11, 21-27], only the
tyre aligning moment is considered as a disturbance in the steering system. However,
in complex SbW systems, there exist large quantities of nonlinearities and disturbances
coming internally and externally, such as SbW system parameter variations, Coulomb
friction, motor torque pulsation disturbances, and most significantly varying tyre
aligning moment due to frequent road surface changes. All these uncertainties and
disturbance will affect the steering performance of SbW systems.
2.5.1 Coulomb Friction
In fact, many types of friction are involved in the steering motion of SbW systems such
as Coulomb, viscous and static frictions, etc. However, as pointed out explicitly in [10],
the Coulomb friction as shown in Figure 2.14 is considered to be the main friction
present in the steering actuator assembly and the steering system and is defined as
Chapter 2 Background and Literature Review
33
Figure 2.14: Coulomb friction.
( ) (2.19)
with as the Coulomb friction constant, and ( ) as the sign function with
( ) {
(2.20)
2.5.2 Torque Pulsation Disturbances
As for SbW systems, in the final analysis, it is to control the front wheel steering motor
that plays an essential role to handle the uncertain dynamics and disturbances. It is
known that brushless DC motors have been used as actuators for SbW applications due
to their higher torque/weight ratio, maintenance freedom of commutators, lower rotor
moment of inertia, and better heat dissipation [209-211]. The effects of back-EMF and
Chapter 2 Background and Literature Review
34
motor parameter variations on the winding current tracking responses are considered to
be the main disturbances in the DC motor torque generation. However, in this thesis, we
have chosen that the front wheel steering motor is a three phase permanent magnet
asynchronous current (PMAC) motor. Compared with the control of DC motor-based
SbW systems, the control of an AC motor-based SbW system is by nature more
complicated due to its features in terms of designing control architecture and handling
system nonlinearities. The derivation of the mathematical model of the PMAC motor
and the sources of torque pulsation disturbances has led to a vast amount of research
activities over the past two decades [72-80].
When considering the rotor rotating coordinates (d-q axes) of the motor as the
reference coordinates, the d-q axes stator voltages of the three phase PMAC motor can
be modeled as follows [72-77]:
(2.21)
( ) (2.22)
where and are the d-q axes stator voltages, respectively, and are the d-q axes
stator currents, respectively, and are the d-q axes stator synchronous inductances,
respectively, is the d-axis flux linkage due to the permanent magnet, is the
stator resistance, and is the rotor electrical speed, which is related to the rotor
mechanical speed as
with as the number of poles (even number).
Generally, the current and can be calculated from and (obtained from current
measurements) by using Clarke and Park transformations [75, 79].
The corresponding electromagnetic torque is expressed as follows:
[ ( ) ] (2.23)
Chapter 2 Background and Literature Review
35
In industrial applications, a field-oriented control principle is widely adopted to
control a PMAC motor. Thus, in order to simplify the model and reduce the costs, the
desired current component in the direct axis is set to zero, that is, . Usually,
with the current loop controller, can be easily regulated to zero. In this case, the
torque expression in (2.23) can be rewritten as
(2.24)
In the ideal case, the q-axis reference current can be directly achieved from (2.24)
due to the constant d-axis flux linkage . However, the torque disturbance
always exists in motor torque generation. Consequently, the actual flux linkage
is described as follows [76]
∑ ( )
( ) ( )
(2.25)
where , ,and are the known constant average dc, the 6th, and the 12th
harmonics amplitude of the d-axis flux linkage, respectively, and are the
6th, and the 12th harmonic terms, respectively, and is the electrical angle of the rotor.
For the purpose of simplicity, only the 6th and 12th harmonics are considered as the
principal sources of torque pulsations in this thesis.
Furthermore, as shown in the literatures [75, 78-80], the dc current offsets always
exist in the motor terminals in virtue of the digital-to-analog converter offsets of the
motion controller as well as the current offset error of the current tracking amplifier.
Thus, sinusoidal torque disturbances and the corresponding velocity ripples at the
Chapter 2 Background and Literature Review
36
system output are inevitably generated, which will severely affect the high-precision
tracking performance in practice.
Let ( ) ( ) ( ) be the desired currents at the motor terminals and the dc
current offsets in the measured currents of phases a and b be , respectively.
Then, the third phase current offset is calculated as ( ), and the actual
currents are of the forms , , and , respectively.
The currents can be calculated by using Clarke and Park transformations
based on the three phase currents with the offsets (abc frame to dq0 frame). Therefore,
the actual current can be expressed as
( ) ( ) (2.26)
where the desired d-axis current is given by
( )
[ ( ) (
)
( ) (
)
( ) (
)] (2.27)
and the current disturbance is of the form:
{ [ (
) (
)] [ (
) (
)]}
√ ( )√(
) ( ) (2.28)
where is the phase related to and .
Then, based on the above analysis, we re-write (2.24) as follows:
( )
Chapter 2 Background and Literature Review
37
( )(
( ) ( ))
( ) (2.29)
where is the desired torque signal for the motor, and represents the total
pulsation disturbances in the generation of motor torque, which satisfies as
[( )(
( ) ( )) ( )]
[( )]
( )
( ) ( )
( ) (2.30)
where and are the 6th and the 12th harmonic torque amplitudes,
respectively.
In the above equation, it can be easily seen that the amount of the torque disturbance
mainly depends on the 6th and the 12th harmonic torque amplitudes, and
, and the current disturbance . Thus, in designing the torque control signal
for the steering motor, these perturbations need to be taken into account and
compensated effectively in the meantime.
2.5.3 Tyre Aligning Moment
As mentioned previously, the tyre alignment moment which is treated as the most
significant disturbance is generated by the tyre lateral force which reflects the
interactions between the front wheels and road surface. As shown in the literature [10,
28], Paul and Yung-Hsiang have presented the tyre dynamics and described the effects
Chapter 2 Background and Literature Review
38
Figure 2.15: Linear Bicycle Model.
of the tyre aligning moment on the steering performance by experimental studies. For
further analysis of vehicle dynamics, a road vehicle equipped with the SbW system in
Figure 2.8 is represented by the two-wheel planar bicycle model with states of vehicle
body slip angle at the center of gravity (CG) and yaw rate as shown in Figure 2.15.
The two front wheels and the two rear wheels of the vehicle are represented by a central
front wheel and a central rear wheel, respectively. is the vehicle velocity at the CG,
is the steering angle of the central front wheel, and are the longitudinal and
lateral components of the CG velocity, and are the velocities of the central front
wheel and central rear wheel, and are the distances of the central front wheel and
central rear wheel from the CG of the vehicle, and
are the lateral forces of the
central front wheel and central rear wheel, respectively, and are the tyre slip
angles of the central front wheel and central rear wheel.
Figure 2.16(a) and (b) show the tyre forces and the self-aligning torque for the steered
central front wheel of the bicycle model, respectively, in which is the front wheel
Chapter 2 Background and Literature Review
39
Figure 2.16: Tyre force at front central wheel (a) Tyre forces. (b) Self-aligning torque.
longitudinal driving force, is the front wheel cornering force, is the front wheel
cornering stiffness coefficient, and are the front wheel mechanical and pneumatic
trails, respectively. In addition, the parameters of the tyre dynamics are listed in Table
2.2. As shown in Figure 2.16 (b), the tyre aligning torque generated by the tyre lateral
force is given by [83, 84]:
( ) (2.31)
where is the mechanical trail, describing the distance between the tyre centre and the
point on the ground about the tyre pivots as a result of the wheel caster angle, and is
the pneumatic trail, the distance from the centre of tyre to the application of the lateral
force .
Further, Figure 2.17 shows the typical nonlinear relationship between the lateral force
and the tyre slip angle . It can be easily seen that when the tyre slip angle is
less than about 4 degrees, the lateral force increases in a straight line. Its increment
becomes less and less after this value and eventually saturates at around 8-10 degrees.
Chapter 2 Background and Literature Review
40
Table 2.2: Parameters of tyre dynamics.
Parameter Definition
Front wheel steering angle Front tyre slip angle Vf Front wheel velocity Vehicle inertia around CG M Vehicle mass
Front wheel longitudinal driving
force Front wheel lateral force
Front wheel cornering force
Front wheel cornering stiffness
Front wheel self-aligning torque
Front wheel mechanical trail
Front wheel pneumatic trail
Figure 2.17: Tyre lateral force
versus slip angle .
Chapter 2 Background and Literature Review
41
However, for a normal passenger car, its lateral motion usually occurs within the linear
region of the tyre operation. Thus, the slope of the line is the tyre cornering stiffness and
the tyre lateral force can be modelled as follows:
(2.32)
where is the front tyre cornering stiffness coefficient which is defined as the gradient
of the lateral force curve in Figure 2.17 at and also serves as an important value
in the evaluation of tyre cornering characteristics.
According to the bicycle model of the road vehicle in Figure 2.15, we can obtain the
relationship between the front tyre slip angle and the steering angle of the front
wheel as follows [27], [62]:
( ( )
( ))
(2.33)
where the tyre operation is in the linear region with
(2.34)
Thus, the front tyre lateral force in (2.32) can be rewritten as:
(
) (2.35)
and the self-aligning torque can be expressed as
Chapter 2 Background and Literature Review
42
( ) (
) (2.36)
Furthermore, it is worth noting that in (2.36), the steering angle of the steered front
wheels and the yaw rate of the vehicle are all measured by using the steering angle
sensor and the yaw rate sensor, respectively, in practical automotive applications. The
slip angle of the CG of the vehicle can be calculated from the bicycle model of the
road vehicle in Figure 2.15 as follows:
(
) (2.37)
However, in the following chapters of the thesis, a yaw rate sensor may not be
applied in the simulations. Thus, alternatively, the dynamics of the yaw motion of the
road vehicle with an SbW system based on the bicycle model in Figure 2.15 can be used
to calculate the yaw rate , which is described by the following state-space equation [10,
27]:
(2.38)
where
[ ]
[
]
[
]
Chapter 2 Background and Literature Review
43
where is the vehicle mass and is the vehicle inertia around CG.
It should be noted that in the thesis, for calculating the self-aligning torque in
(2.36), the vehicle body slip angle and the yaw rate can be roughly obtained by
equation (2.37) and the yaw rate sensor (or equation (2.38)), respectively. However,
there are some applications for estimating the body slip angle by means of observer
designs instead of using equation (2.37). Such applications can be found in literature [27,
81]. For these applications, the proposed observers are either using vehicle yaw rate or
combined with lateral acceleration at CG of vehicle as measurable state variables to
estimate . Meanwhile, there exists another class of real-time applications to directly
approximate the tyre aligning moment from the steering system model using different
observer techniques, such as a linear Luenberger observer, a nonlinear observer, and a
sliding mode observer [28, 57, 82].
2.6 Existing Modelling for Steer-by-Wire Systems
So far, we have learnt the fundamental principle and the main nonlinearities and
disturbances of SbW systems. In this sub-chapter, we will present the existing different
modellings of SbW systems and also address the corresponding drawbacks of the
modelling techniques.
Since Amberkar et al. [85] propose a system-safety process for a by-wire system,
different actuating types of SbW systems have been developed recently in several
versions, namely, pinion-actuating type, rack-actuating type, and tie-rod-actuating type
of SbW systems in the literature.
The pinion-actuating type is the most popular modelling technique for SbW systems
in which the conventional rack and pinion steering mechanism is retained and the
steering actuator assembly is flexibly coupled to the pinion via the universal joint. This
modelling type that is also adopted in this thesis can be clearly seen from Figure 2.11.
Chapter 2 Background and Literature Review
44
Meanwhile, among the literature using this modelling technique, the front wheel
dynamics of a vehicle’s SbW system is described with a second-order model [10, 11, 21,
22, 26]. In [10], a simple second-order model is used to describe the steering system
dynamics based on the observation of the experimental results while the motor
dynamics have not been considered yet. Similarly, Bertoluzzo et al. [11] and Baviskar
[21] use two second-order differential equations to represent the dynamics of the front
wheel side and hand-wheel side, respectively, considering the effect of tyre dynamics.
Although the steering actuator assembly is installed on the pinion side, the front wheel
steering dynamics have been represented by the rack motion using second-order model
where the motor dynamics are considered and the tyre dynamics are ignored [22]. Cetin
et al. [26] describe the front wheel directional unit as a second-order model, however,
neither the tyre dynamics nor the motor dynamics are considered in the dynamical
equation.
Inspired by the use of the pinion-actuating type, some researchers further proposed a
scheme of using rack-actuating steering mechanism in SbW systems, as shown in
Figure 2.18. The rack-actuating type possesses the advantage of satisfying Ackermann
geometry where a six-bar mechanical linkage and a motor together with a ball screw
gear are adopted in a rack system. Park’s paper [24] is the most representative one to
illustrate this modelling methodology. It demonstrates that the modelling of the front
wheel system can be described by a second-order differential equation in which the rack
displacement serves as the state variable measured by the rack position sensor. A similar
idea is used by Kim et al. [23] to achieve the steering characteristic and improve the
stability and manoeuvrability of vehicle.
Meanwhile, Kim et al. [70] have also developed a tie-rod actuating type of front-
wheel system in SbW systems, in which two independently actuating electric motors are
adopted in the tie-rod sides. As can be in Figure 2.19, the front wheel steering
mechanism in the tie-rod-actuating SbW system does not include a rack linkage and
thus, in particular, it is of importance for the two motors to closely emulate the steering
Chapter 2 Background and Literature Review
45
Figure 2.18: Rack-actuating type of front-wheel system.
Figure 2.19: Tie-rod-actuating type of front-wheel system.
Chapter 2 Background and Literature Review
46
performance of a conventional steering system, following the Ackerman geometry
derived from a six-bar linkage in the steering mechanism. However, using the tie-rod
actuating type, the synchronization of the operations of two motors should be taken into
account to ensure the two front wheels to have identical steering angles.
2.7 Basic Control Methodologies for Steer-by-Wire Systems
Heretofore, we have figured out three different actuating and modelling types of a front
wheel system in SbW systems. Now in this subsection, based on mathematical
modelling, we will review the existing representative control methodologies and their
corresponding performance when taking into account the effects of uncertain vehicle
dynamics and highly nonlinear disturbances due to varying road conditions.
2.7.1 Conventional Linear Feedback Control for Steer-by-Wire Systems
Conventional PD control is an important and preferred linear feedback control method
in steering controller design among researchers and automotive engineers due to the
easy design and implementation in SbW systems of road vehicles. The reason why the
integral term is not included in the controller design is that it may introduce integrator
windup especially in extreme steering situations when a rapid change of direction is
required to avoid obstacles. Early in the paper [10], it was reported that Paul and Gerdes
used the PD control technique with feedforward compensation in SbW control design.
In terms of calculating the feedforward term of the tyre self-aligning moment, the
vehicle sideslip angle is estimated using global positioning system (GPS) and inertia
navigation system (INS) sensor measurements, which do not depend on any parameter
uncertainties and variations. Because many intelligent sensors are needed to ensure a
good steering performance, it may restrict the use of this control system in real
application and thus a simple but effective control scheme for SbW systems is still open.
Berrtoluzzo et al. [11] combine two schemes including torque and speed schemes using
a PD controller such that good overall steering performance both in steering the front
Chapter 2 Background and Literature Review
47
wheels and returning the steering effort to the driver can be obtained. Unfortunately,
because the effect of the tyre self-aligning moment has not presented and the different
road conditions in the experiments have not been considered yet, the steering
performance is not verified sufficiently. Similarly, in the applications among the
literature [22-24, 26], PD control for either the rack-actuating type or the tie-rod-
actuating type of an SbW system is adopted with the problem that the effect of the tyre
alignment moment caused by the varying road conditions is not fully considered in
evaluating the performance of the steering controller. In [29], Ohara et al. design a
steering angle control scheme which is composed of a PD controller and a linear
disturbance observer to minimize the estimated error of the side slip angle. Since there
are no considerations of the variations of road conditions, consequently if the road
surfaces are frequently changing, the linear disturbance observer may not perform well,
which could result in a deterioration in the steering control performance. This problem
is solved by Yamaguchi et al. in [27] who present an integrated control scheme using a
PD control and a complementary nonlinear disturbance observer based on real-time
identified tyre cornering stiffness and tyre slip angle using weighted recursive least-
square (RLS) algorithm. Due to the use of a recursive RLS algorithm, all of the data in
the past needs to be collected and processed giving rise to the large time consumptions
of the calculations, which is not practical in the applications.
As discussed above, when an SbW system experiences large parameter uncertainties
and some unexpected external disturbances, such as sudden changes in the road surface,
the pre-well-tuned PD control cannot assign the closed-loop poles to the desired
position and thus, the steering performance and robustness of SbW control systems will
degenerate. Although a high-gain PD control may achieve an acceptable level of
tracking performance with high accuracy and strong robustness, the gains of the PD
control must be frequently adjusted based on the detection and estimation of both the
uncertain dynamics and varying road conditions.
All these limitations of PD control mentioned above have created new potential rese-
Chapter 2 Background and Literature Review
48
arch opportunities in the optimal control of SbW systems to improve the handling and
stability of the steering system in varying driving and road conditions. In [59], Marumo
and Nagai propose a state feedback control scheme such as a linear quadratic control so
that the actual rolling angle of the SbW motorcycle is ensured to track the desired
rolling angle provided by the driver. A better tracking performance is obtained in
comparison with conventional PD control since the rolling motion can be decoupled
from other motions that could influence the rolling motion in the original coupled
motorcycle system. Very recently, Marumo and Katagiri in [60] show that, by applying
the optimal control theory to the controller design together with the lane-keeping system,
the SbW system behaves with both a good level of tracking performance and a strong
robustness under not only the steering torque disturbance but the lateral force
disturbance.
2.7.2 Adaptive Control for Steer-by-Wire Systems
It can be recognized that for the above conventional PD and optimal control methods,
good steering performance can only be achieved if it is based on the proper selections of
the control parameters and accurate estimates of the road surface conditions and the
classis sideslip angle. However, how the optimal control parameters and the estimates of
the tyre lateral forces can be accurately online obtained under varying road environment
to ensure a robust steering performance is still an open issue. Therefore, in order to
obtain the robustness of the SbW control systems, some adaptive control methodologies
are proposed in [21, 86, 26, 62]. For these applications, the steering system parameters
and the disturbances are adaptively adjusted online such that the need to use accurate
parameter settings and proper estimates of disturbances can be eliminated. Baviskar et
al. [21, 86] develop an adaptive SbW tracking controller using steering angle and
angular velocity information to ensure an asymptotic error convergence. To avoid the
use of the measurements of the driver input torque and the reaction torque between the
front wheels and the road surface, two torque observers are proposed to eliminate the
needs of torque measurements. In [26], using the adaptive online parameter estimation
Chapter 2 Background and Literature Review
49
method, the directional control unit parameters of the SbW system are first estimated
and then used in an adaptive-pole-placement-model-based controller in order to closely
follow the reference position provided by the original hydraulic rack. Since there is no
consideration for the tyre aligning moment in the SbW system, whether this adaptive
control algorithm is still able to ensure good tracking performance under large
variations of road surface conditions has not been confirmed yet. To overcome this
problem, Kazemi’s paper [62] gives a solution in which a nonlinear adaptive sliding
mode controller is proposed to improve the vehicle handling characteristics with no
requirement for the information of system parameters and uncertainty bound. Only the
simulation results are presented to verify the control performance and the relevant expe-
riments are still in question.
2.8 Lyapunov Stability Theory
Stability theory plays an essential role in system theory and engineering. For linear
time-invariant (LTI) systems, system stability can be determined by the use of Routh’s
stability criterion or Nyquist stability criterion, by checking whether all the system poles
are located in the left-half complex plane. However, the linear stability criteria are not
applicable for nonlinear systems with nonlinearities and possible time-varying
parameters.
For nonlinear systems, the most typical technique for analysing their stability is
Lyapunov stability theory which is named after Aleksandr Lyapunov, a Russian
mathematician and engineer who laid the foundation of the theory. In this section, we
will present some stability-related concepts and then discuss the direct method of
Lyapunov stability in detail, which is indeed necessary in understanding stability
analysis of sliding mode control systems [106, 107].
Chapter 2 Background and Literature Review
50
2.8.1 Stability-related Definitions
Consider a nonlinear system as follows:
( ) (2.39)
where is the state variable vector with as the order of the system, and
( ) is the nonlinear function of ( ) and time .
D 2.1: For the system in (2.39), the state is an equilibrium point provided
that satisfies the following equation:
( ) for all (2.40)
D 2.2: The state equilibrium point of the system in (2.39) is said to be stable in
the Lyapunov sense provided that, for each spherical region ( ) of radius about the
equilibrium point , i.e.,
‖ ‖ (2.41)
where ( ) is the bounded initial value of the system trajectory in the state-space
at , there must be a spherical region ( ) of radius about , i.e.,
‖ ‖ (2.42)
where ‖ ‖ is the Euclidian norm defined as:
‖ ‖ √( ) ( ) (2.43)
Chapter 2 Background and Literature Review
51
and thus, the trajectory starting in ( ) does not leave ( ) as goes to infinity.
D 2.3: The state equilibrium point of the system in (2.39) is said to be
asymptotically stable provided that, it is stable in the sense of Lyapunov and for each
spherical region ( ), i.e.:
‖ ‖ (2.44)
where ( ) is the bounded initial value at ,
as (2.45)
Given the above local stability definitions of the nonlinear system in (2.39) around
the equilibrium point, we can easily obtain the global stability definition for the system
in (2.39).
D 2.4: The state equilibrium point of the system in (2.39) is said to be globally
asymptotically stable provided that, the equilibrium point is globally asymptotically
stable for any initial state ( ).
2.8.2 Direct Method of Lyapunov Stability
Generally, the Lyapunov stability theory consists of two methods which are the
linearization method (the first method) and the direct method (the second method),
respectively. The linearization method identifies the local stability of nonlinear systems
based on the premise that we can obtain a linearized system with regard to the system
equilibrium point at the system origin. In accordance with the locations of all the
eigenvalues of the linearized system in the complex plane, the local stability
characteristics of a nonlinear system can then be determined. However, the direct
method provides a different way to determine the stability of nonlinear systems by
Chapter 2 Background and Literature Review
52
selecting an “energy-like” function and examining its changing rate. This indicates that,
using the direct method, we can still determine the system stability without the
requirement to obtain the explicit eigenvalues of the linearized system in the linearized
method [40, 108-111].
In control engineering, the “energy-like function” is called the candidate of Lyapunov
function with two important features: (i) it must be a positive function of the system
state variables; (ii) it can only be zero at the system origin. From the mathematical point
of view, an “energy-like” function is a positive definite function as defined below:
D 2.5: A scalar function ( ) is said to be positive definite in a region including the
system origin provided that ( ) for all nonzero states in the region and
( ) when is at the origin.
In D 2.5, the scalar function ( ) is simply the function of the state variable vector ,
and it does not depend explicitly on time . If a scalar function ( ) depends on both
the state variable vector and time , we give the following definition:
D 2.6: A scalar function ( ) is said to be positive definite in a region including the
system origin provided that there exists a positive function ( ) such at
( ) ( ) for all (2.46)
( ) for all (2.47)
D 2.7: A scalar function ( ) is said to be positive semi-definite in a region including
the system origin provided that ( ) for all states except at the system origin and
some other points in the region where ( ) .
Chapter 2 Background and Literature Review
53
D 2.8: A scalar function ( ) is said to be positive semi-definite in a region
including the system origin provided that there exists a positive function ( ) such
that ( ) ( ) for all , except at the system origin and some other points in
the region where ( ) .
Consider a system described by the following state-space equation:
( ) (2.48)
where is the state variable vector, ( ) is a linear or nonlinear
function of and time , and ( ) for all .
Theorem 2.1: (Local Stability) If there exists a scalar function ( ) with continuous
first-order partial derivatives in a ball , with its radius and centred at the system
origin, such that
( ) (2.49)
( ) (2.50)
then ( ) is a Lyapunov function and the equilibrium point at the system origin is
asymptotically stable.
Theorem 2.2: (Global Stability) If there exists a scalar function ( ) with continuous
first-order partial derivatives, such that
( ) (2.51)
( ) (2.52)
Chapter 2 Background and Literature Review
54
( ) as ‖ ‖ (2.53)
then ( ) is a Lyapunov function and the equilibrium point at the system origin is
globally asymptotically stable.
It is worth noting from the above theorems that, the total energy of a dynamical
system is continuously dissipated when time goes to infinity, and ultimately the
dynamic system will reach a stable equilibrium point. Furthermore, for the system under
consideration, more than one Lyapunov function may exist and thus the Lyapunov
function of a system is not unique. In addition, the conditions of the two theorems of the
Lyapunov stability are only sufficient, so the failure of a Lyapunov function candidate
to satisfy the conditions for stability does not mean that the system is unstable, but it
means that such a stability property cannot be established by using this Lyapunov
function candidate.
2.9 Sliding Mode Control Theory
Sliding mode control is a very powerful technique that has been widely used for the
tracking control and stabilization of both linear and nonlinear systems with parameter
uncertainties and bounded input disturbances [32, 34, 146]. Sliding mode control
systems, as a special class of variable-structure systems [30], were first proposed by
Russian control scientists, Emel’yanov and Barbashin, in the early 1960s [112, 113],
and were not investigated outside of Russia until the middle 1970s when a book by Itkis
and a survey paper by Utkin were published in English [31, 114, 115]. The essence of
SMC is that in a vicinity of a prescribed switching manifold, the velocity vector of the
controlled state trajectories always points toward the switching manifold. Such motion
is induced by imposing discontinuous control actions, commonly in the form of
switching control strategies. An ideal sliding mode exists only when the system state
satisfies the dynamic equation governing the sliding mode surface for all time, which
generally requires an infinite switching to assure the sliding motion.
Chapter 2 Background and Literature Review
55
The basic idea of SMC is that [108]
The desired system dynamics are first defined on a sliding surface in the state
space.
A controller is then designed, using the output measurements and system
uncertainties bounds, to drive the closed-loop system to reach the sliding surface.
The desired dynamics of the closed-loop system is then obtained on the sliding
surface.
The main purpose of this sub-chapter is to introduce the most basic and elementary
concepts related to SMC, such as equivalent control, robustness property, reaching law
method and dynamics in the sliding mode. All these concepts will play an essential role
in designing the robust control methodologies for SbW systems in the following
chapters.
2.9.1 Sliding Mode Control Design
Consider the following controllable nth-order system:
( ) ( ) (2.54)
where [ ] is the system variable vector, both ( ) and
( ) are linear or nonlinear functions of the stable variable vector , and
is the control input.
For the design of a sliding mode controller, we need to first define a so-called sliding
variable :
Chapter 2 Background and Literature Review
56
( ) ( ) (2.55)
where [ ] is the sliding mode parameter vector, and the
parameters ( ) should be properly selected such that the characteristic
polynomial of the following equation is strictly Hurwitz:
( ) ( ) (2.56)
The system dynamics in the sliding mode are determined by selecting the parameters
( ) to obtain a desired asymptotic convergence characteristic.
According to the linear control theory, in order to guarantee the solution of the
differential equation in (2.56) to be asymptotically stable, the sliding mode parameters
( ) should be chosen, such that all the eigenvalues of the differential
equation in (2.56) have negative real parts. Therefore, the task of the sliding mode
controller is to drive the sliding variable to converge to zero, and then the desired
system dynamics prescribed in (2.56) will be obtained.
In SMC, expression (2.56) is called sliding mode surface in the state space. The
controller designed to drive the sliding variable to converge to zero is called sliding
mode controller.
As a matter of fact, the most important point is how to derive the condition to ensure
the system state to reach and retain in the sliding mode. The Lyapunov’s direct method
can be used in the SMC design for the system (2.54) with the desired asymptotically
stable dynamics (2.56). In general, the following Lyapunov function is often adopted for
the SMC design:
(2.57)
Chapter 2 Background and Literature Review
57
Differentiating ( )with respect to time , we have
( )
[ ( ( ) ( ) )]
[ ( ) ( ) ] (2.58)
In practice, we do not know ( ) and ( ) exactly. However, we assume that the
following upper and lower bounds are known:
‖ ( )‖ ( ) (2.59)
( ) (2.60)
with a positive constant .
Then, we may choose the SMC signal in the following form:
(‖ ‖ ( ) ) ( ) (2.61)
with a constant and the sign function ( ) has the same definition as in (2.20).
Using (2.61) in (2.58), we obtain
[ ( ) ( ) ]
{ ( ) ( )[ (‖ ‖ ( ) ) ( )]}
Chapter 2 Background and Literature Review
58
( ) | | [ ( )
] ‖ ‖ ( ) | | [
( )
]
| |‖ ‖‖ ( )‖ | | [ ( )
] ‖ ‖ ( ) | | [
( )
] (2.62)
Considering (2.59) and (2.60), (2.62) becomes
| |‖ ‖‖ ( )‖ | |‖ ‖ ( ) | |
| |‖ ‖[ ( ) ‖ ( )‖] | |
| | (2.63)
Then, according to the direct method of Lyapunov stability, the sliding variable will
asymptotically converge to zero. After , the desired closed-loop system dynamics
defined on the sliding mode surface in (2.56) are obtained. Thus, the state variable
vector will asymptotically converge to zero on the sliding mode surface .
As seen from the above discussion, the following characteristics of SMC have been
noted:
Due to the signum function used in the sliding mode controller, the controller
has a variable structure, which is the reason why the SMC sometimes is called
variable structure control.
The convergence of a closed-loop SMC system is divided into two parts: First,
the sliding variable is driven by the sliding mode controller to converge to
zero. Second, in the sliding mode, , the state variable vector
asymptotically converges to zero.
Chapter 2 Background and Literature Review
59
After the sliding variable is driven to zero, the closed-loop system dynamics
are only determined by the desired dynamics in (2.56) and thus, the closed-loop
system is insensitive to system uncertainties on the sliding mode surface. It is
because SMC systems possess the property of robustness with respect to system
uncertainties, that SMC becomes a powerful tool in the control of uncertain
systems and significantly motivates the subsequent researchers in the area.
However, it should be noted that the system remains affected by perturbations
during the reaching phase (that is to say before the sliding surface has been
reached).
If the system in (2.54) has a bounded input disturbance ( ), satisfying the
following bounded condition:
| ( )| (2.64)
where is the upper bound of ( ).
The system in (2.54) can be rewritten as follows:
( ) ( )( ( )) (2.65)
It can be proved that, if the constant in the control signal in (2.61) satisfies the
following equality:
(2.66)
the sliding variable in (2.55) can be driven to zero, and the desired system
dynamics in the sliding mode in (2.56) can then be obtained. Thus, the SMC
behaves with a strong robustness property with regard to the bounded input
disturbances.
Chapter 2 Background and Literature Review
60
Figure 2.20: The chattering phenomenon.
In addition, it should be addressed that for the system in (2.54), in this thesis, we only
consider the SMC design for the case that the system parameters and external
disturbances are not exactly known but their bounded information is known in prior.
However, there are other typical SMC strategies such as equivalent control-based SMC
that consists of an equivalent control component and a switching control component. In
this method, the system parameters are assumed to be known and will not be discussed
in this thesis. Readers who are interested in this part may refer to [31, 32, 35, 40, 44] for
further detail.
2.9.2 The Chattering Problem
It is known that in order to account for the presence of modelling imprecision and
disturbances, SMC laws have to be discontinuous across the sliding surface ( ). Since
the implementation of the associated control switchings is necessarily imperfect, this
leads to control chattering as shown in Figure 2.20 [35, 39, 40, 116-118]. Chattering is
in general highly undesirable in practice, since it involves extremely high control
Chapter 2 Background and Literature Review
61
Figure 2.21: Saturation function ( ).
activity, and furthermore, may excite high-frequency dynamics neglected in the course
of modelling (such as unmodelled structural modes, neglected time-delays, and so on).
Chattering also results in a high wear of the moving mechanical parts and high heat
losses in electrical power circuits. That is the reason why various methods have been
proposed to reduce or eliminate this chattering [34, 40, 119-123].
The first approach is called the boundary layer control method [34], where the
chattering is remedied by smoothing out the control discontinuity in a thin boundary
layer neighbouring the switching surface. Accordingly, the SMC in (2.61) becomes
(‖ ‖ ( ) ) ( ) (2.67)
where ( ) is the saturation function as shown in Figure 2.21 and defined as
( ) {
| |
( ) | | (2.68)
Chapter 2 Background and Literature Review
62
Figure 2.22: Continuation approximation method for ( ).
where the positive constant is called the boundary layer thickness.
It is worth noting that the boundary layer controller offers a continuous
approximation to the chattering control input inside the boundary layer, and guarantees
attractiveness to the boundary layer and ultimate boundedness of the output tracking
error within a neighbourhood of the origin depending on the value of . However, the
drawback is that the output tracking error cannot converge to zero and the robustness of
SMC is also compromised. Therefore, in the selection of the parameter , a tradeoff
between the control chattering and the tracking error as well as robustness should be
carefully taken.
Another solution to handle the chattering problem is based on the continuous
approximation method in which the sign function ( ) in (2.61) is substituted for the
following continuous approximation ( ) as shown in Figure 2.22 [38, 124-127]:
Chapter 2 Background and Literature Review
63
( )
| | (2.69)
where is a small positive number. This method results in a high-gain control when the
states are in the close neighbourhood of the switching manifold.
In conclusion, continuation approaches eliminate the high-frequency chattering at the
price of losing zero-convergence of the out tracking error and invariance property. A
high degree of robustness can still be maintained within a small boundary layer width.
In addition, the resulting physical system often exhibits low-frequency oscillation due to
unmodelled dynamics.
Since chattering is also caused by the nonideal reaching at the end of the reaching
phase [41, 128, 129], the third approach in the next sub-section deals directly with the
reaching process to reduce the control chattering. In addition, high-order SMC has also
attracted an increasing attention owing to its effectiveness of alleviating the chattering
magnitude and maintaining the main characteristics of invariance and accuracy. Readers
of great interest may refer to the standard work [165-171] for further detail.
2.9.3 Reaching Law Method for SMC Design
In addition to the aforementioned two continuation approaches to soften the chattering,
tuning the reaching law approach has also been extensively studied and has proved to be
efficient in chattering reduction [41]. The reaching law is a differential equation
specifying the dynamics of a switching manifold ( ). The differential equation of an
asymptotically stable ( ) is itself a reaching condition, i.e., ( ) ( ) .
Additionally, by the proper selection of the parameters in the differential equation, the
dynamic quality of an SMC system in the reaching mode can be controlled. Generally, a
practical reaching law is of the form:
Chapter 2 Background and Literature Review
64
( ) ( ) (2.70)
where
[ ],
( ) [ ( ) ( )]
[ ],
( ) [ ( ) ( )]
( ) , ( )
Three practical special cases are given as follows:
Constant rate reaching
( ) (2.71)
It is apparently seen that this law forces the sliding variable ( ) to reach the sliding
mode surface ( ) at a constant rate . The superiority of this reaching law
is its simplicity. Nevertheless, if the selection of is too small, the reaching time will
be too long. Moreover, a large value of will lead to significant chattering.
Constant plus proportional rate reaching
( ) (2.72)
Chapter 2 Background and Literature Review
65
Clearly, by adding the proportional rate term , the state is forced to approach the
switching manifolds faster when is large. It can be shown that the system state will
be driven from an initial state to the switching manifold in a finite time [130], and
the time is given by
| |
(2.73)
Power rate reaching
| | ( ) (2.74)
This power rate reaching law increases the reaching speed when the state is far from
the sliding mode surface ( ) , while reduces the rate when the state is near the
manifold. The result is a fast reaching and low chattering reaching mode. Integrating
(2.74) from to yields
( ) ( ) (2.75)
It is easily seen that the reaching time is finite. Thus, the power rate reaching law
indeed provides a finite reaching time. Additionally, because the discontinuous term
( ) has been removed on the right side of (2.72), this reaching law successfully
alleviates the chattering effects. Some other exponential reaching law methods based on
the power rate reaching have been proposed and the details can be found in [42, 43,
132].
It is worth nothing that the reaching law method simultaneously takes care of
ensuring the reaching condition, influencing the dynamic quality of the system during
the reaching phase, and providing the means for controlling the chattering level. Thus,
the reaching law method can be applied to both linear and nonlinear SMC systems with
Chapter 2 Background and Literature Review
66
the system perturbations and internal and external disturbances, for the purpose of not
only improving the performance of the reaching mode but also reducing the amplitude
of chattering.
2.10 Finite Time Sliding Mode Control
As described in Section 2.9, SMC design with the linear sliding mode surface has been
adopted for describing the desired performance of the closed-loop systems in detail, that
is, the system state variables reach the system origin asymptotically on the linear sliding
mode surface. When in the sliding mode, the closed-loop response becomes totally
insensitive to both internal parameter uncertainties and external disturbances. Despite
that the parameters of the linear sliding mode can be adjusted in order to obtain an
arbitrarily fast convergence rate, the system states on the sliding mode surface cannot
converge to zero in a finite time.
Recently, a new technique called terminal sliding mode (TSM) control has been
intensively studied for achieving finite time convergence of the system dynamics in the
terminal sliding mode [44, 99, 133-137]. In comparison with the linear sliding mode
based SMC design, TSM possesses the superior characteristics of fast and finite time
convergence, which particularly improves the high precision control performance by
accelerating the convergence rate near an equilibrium point.
2.10.1 Terminal Sliding Mode Control
Consider the following second-order uncertain nonlinear system:
( ) ( ) ( ) (2.76)
Chapter 2 Background and Literature Review
67
where [ ] is the system state vector, ( ) and ( ) are smooth nonlinear
functions of , and ( ) represents the system uncertainties and disturbances satisfying
‖ ( )‖ where , and is the scalar control input.
In order to obtain the terminal convergence of the system state variables, the first-
order terminal sliding variable is defined as follows:
⁄ (2.77)
where is a designed positive constant, are the two positive odd integers
satisfying the following condition:
(2.78)
The sufficient condition for the existence of TSM is
| | (2.79)
where is a constant. According to [44], for the case of ( ) , the time for
the system states to reach the sliding mode is finite and satisfies the following
inequality:
| ( )|
(2.80)
In order to ensure the terminal sliding variable to reach the terminal sliding mode
surface , we adopt the following sliding mode controller:
Chapter 2 Background and Literature Review
68
( ) ( ( )
⁄ ( ) ( )) (2.81)
In the terminal sliding mode, the system dynamics are determined by the following
nonlinear differential equation:
⁄ (2.82)
It has been shown in [44, 133, 135] that is the terminal attractor of the system
(2.82). The finite time that is taken to travel from ( ) to ( ) is
then given by
∫
⁄
( )
( )| ( )|
⁄ (2.83)
Expression (2.83) means that, in the terminal sliding mode (2.82), both the system states
and converge to zero in finite time.
However, it can be observed from TSM control in (2.81) that, the term ⁄
may
cause singularity if when . It should be noted that this situation does not
occur in the ideal terminal sliding mode because when , ⁄ and thus
as long as , i.e. ⁄ , the term ⁄
( ) ⁄ is
nonsingular. It indicates that the singularity problem may occur in the reaching phase
when there is insufficient control to guarantee while .
It can be concluded that the TSM controller in (2.81) is not capable of ensuring a
bounded control signal for the case of when before the system states
reach the terminal sliding mode surface . In addition, the singularity problem may
also occur even after the sliding mode is reached. The reason is that owing to the
computation errors and uncertain effects, the system states cannot be assured to always
Chapter 2 Background and Literature Review
69
retain in the terminal sliding mode particularly near the system equilibrium point
( ) and the case of while may occur from time to time.
2.10.2 Nonsingular Terminal Sliding Mode Control
For the sake of solving the singularity problem in conventional TSM control systems, a
new singular terminal sliding mode (NTSM) control was proposed in [45] for the
purpose of completely avoiding the singularity problem and successfully applied in
many practical systems [138-145]. The new nonsingular terminal sliding mode variable
is defined as
⁄ (2.84)
where , , and have been defined in (2.77). It should be noted that when , the
NTSM surface (2.84) is equivalent to the terminal sliding mode surface (2.77) and thus,
the time taken to reach the equilibrium point in the terminal sliding mode is the
same as in (2.83).
For the system in (2.76) with an NTSM variable in (2.84), if we use the following
NTSM control
( ) ( ( )
⁄ ( ) ( )) (2.85)
where ⁄ , , the NTSM surface will be reached in finite time.
Then, the system states and will converge to zero in finite time. It is worth noting
that the NTSM control in (2.85) will not lead to any singularity problem due to the
condition ⁄ , which is the main advantage compared with conventional TSM
control in (2.85).
Chapter 2 Background and Literature Review
70
2.11 Summary
A review of steering systems and SMC systems has been presented in this chapter. First,
the conventional and power assisted steering systems have been briefly reviewed. Then,
the existing mathematical modelling, nonlinearities and disturbances, and basic control
methodologies of SbW systems have been revisited. Finally, the theory of Lyapunov
stability and SMC systems has been surveyed in detail and a few key issues have been
highlighted.
It has been seen that the practical SbW systems are highly nonlinear systems where
there exist large levels of disturbances and uncertainties such as system parameter
variations, motor torque pulsation disturbances, and the most significant tyre-self
aligning torque due to road surface changes. Also, it has been pointed out that due to the
superior characteristics of simplicity and power capability in dealing with parameter
variations and disturbances, SMC becomes one of the most effective nonlinear control
techniques. Thus, of particular interest are that it is possible to apply SMC technique to
SbW systems for the purpose of improving steering performance under different road
conditions. This topic has not been fully explored and will be thoroughly studied in the
following chapters.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
71
Chapter 3
Sliding Mode Control for Steer-by-Wire
Systems with AC Motors in Road
Vehicles
In this Chapter, the modelling of Steer-by-Wire (SbW) systems is further studied and a
sliding mode control scheme for the SbW systems with uncertain dynamics is developed.
It is shown that an SbW system, from the steering motor to the steered front wheels, is
equivalent to a second-order system. A sliding mode controller can then be designed
based on the bound information of uncertain system parameters, uncertain self-aligning
torque and uncertain torque pulsation disturbances, in the sense that not only the strong
robustness with respect to large and nonlinear system uncertainties can be obtained, but
also the front wheel steering angle can converge to the hand-wheel reference angle
asymptotically. Both the simulation and experimental results are presented in support of
the excellent performance and effectiveness of the proposed scheme.
3.1 Introduction
ANY researchers and engineers in automotive industry are currently working
on the Steer-by-Wire (SbW) systems that are known as the next generation of
steering systems. The advantages of using SbW systems in the road vehicles are to
M
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
72
improve the overall steering performance, lower the power consumption, and enhance
the safety and comfort of the passengers. The modern SbW systems have the following
distinct characteristics: (i) The mechanical link in conventional road vehicles used to
connect the hand-wheel to the steered front wheels, through the rack and pinion gearbox,
is removed; (ii) The hand-wheel angle sensor is installed on the steering column to
provide the reference signal for the front wheel steering angle to follow; (iii) The
steering motor, coupled to the rack and pinion gearbox, is adopted to steer the front
wheels based on the reference information provided by the hand-wheel angle sensor. In
addition, a feedback motor is employed on the hand-wheel side to provide drivers with
the feeling of the effects of self-aligning torque between the front wheels and the road
surface, based on the error information between the reference angle and the actual
steering angle measured indirectly by the pinion angle sensor.
Recently, many studies on the mathematical modelling of SbW systems have been
carried out. In [10], the dynamics of the test vehicle’s SbW system was described with a
simple second-order model based on the observation of the experimental results by
ignoring tyre forces and considering tyre-to-road contact. In [11, 21], two second-order
models considering the effect of tyre forces and vehicle dynamics were utilized in both
the steered-wheel side and the hand-wheel side, respectively. However, the dynamics of
the two motors are not included in the SbW system modelling. In [22, 23], the hand-
wheel, the front wheels and two motors are all represented by the second-order
differential equations, two independent closed loops are then designed for the hand-
wheel with the feedback motor and the front wheels with the stee ring motor,
respectively. In [24, 25], the hand-wheel with the feedback motor and the front wheel
directional assembly described by the rack motion were represented by two second-
order models. The whole closed-loop SbW system is developed based on the
relationship between the rack displacement and the hand-wheel rotational angle.
However, the disadvantages of this SbW modelling structure are that the tyre dynamics,
especially the tyre self-aligning torque, were not considered, and the effect of the self-
aligning torque on the steering performance cannot be compensated effectively in the
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
73
SbW controller design.
Although the mathematical models of SbW systems have been extensively explored
as briefly discussed in the above, the detailed modelling, viewing from the front wheel
steering motor to the front wheels, has not been fully studied yet. Considering the fact
that a complete SbW system consists of three main components: the front wheel
steering motor, the rack and pinion gearbox, and the steered front wheels, it is essential
to develop a full mathematical model for SbW systems in order to understand the
interactions of these components in practical operation, and design robust controllers for
achieving excellent steering performance against uncertain system parameters and the
tyre self-aligning torque that is treated as the most significant disturbance on the SbW
systems.
In most existing SbW control systems, several control methods have been used to
realize perfect steering characteristics. In [10, 11, 22-25], the conventional proportional-
derivative (PD) control technique was popularly used with the aim of enabling front
wheels to closely follow the driver’s command. In [59, 60], a state feedback controller
using the linear quadratic control technique was developed, aiming at driving the rolling
angle of the SbW motorcycle to track the reference angle. In [27], in order to realize the
virtual steering characteristics, an adaptive control method was applied for controlling
the front-wheel actuators through the estimation of the front tyre cornering stiffness. In
[26], the adaptive online estimation method was used to identify the uncertain
parameters of the vehicle directional-control and driver-interaction units.
However, because the controllers are designed based on the partial mathematical
model in these schemes, good steering performance may not be guaranteed when the
road conditions are varying. Particularly, the poles of the closed-loop SbW systems with
the PD control may change their locations on the complex plane with the variable road
conditions. Such a change may result in the instability of SbW systems.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
74
In this chapter, we will further study the modelling of SbW systems by taking into
account the dynamics from the steering motor to the front wheels. It is worth noting that,
in addition to the inertia, the damping, and the friction, the tyre self-aligning moment
and torque pulsation disturbances are also considered in the system modelling. We then
propose a sliding mode control scheme for SbW systems in order to achieve good
steering performance.
It is well known that for both linear and nonlinear systems, sliding mode control
technique is widely used for tracking control and stabilization with bounded uncertainty
information [135, 151, 172-180].With the proper choice of sliding mode surface, the
stability of the closed-loop system can be obtained asymptotically. It will be shown that
the sliding mode (SM) controller be designed in this chapter is capable of driving the
steering angle to closely follow the hand-wheel command with a strong robustness
against uncertainties. The merit of this control scheme, from the viewpoint of design, is
that only the bound information of the unknown system parameters, self-aligning torque
and torque pulsation disturbances is required for designing SM controller. It will be
confirmed from the simulation and experiment sections that, the designed SM controller
will drive the sliding variable to reach the sliding surface, and then the tracking error
between the steering angle and its reference signal can asymptotically converge to zero
on the sliding mode surface.
The rest of the chapter is organized as follows. In Section 3.2, the mathematical
modelling of SbW systems is formulated, and the tyre self-aligning torque as well as the
total motor torque pulsation disturbances are briefly analysed. In Section 3.3, an SM
steering controller is proposed and the convergence of tracking error dynamics and
robustness with respect to system uncertainties are discussed in detail. In Section 3.4
and Section 3.5, the numerical simulations as well as the experimental studies are
carried out, respectively, to show the good performance of the proposed SMC. Section
3.6 gives conclusions.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
75
3.2 Problem Formulation
3.2.1 Mathematical Modelling
The basic principle of an SbW system in road vehicles is shown in Figure 2.8 [10, 11].
It is seen that the SbW system can be divided into two parts: the upper part includes the
hand-wheel, the hand-wheel angle sensor, and the feedback motor, respectively; and the
lower part is composed of the steering motor, the pinion angle sensor, the rack and
pinion gearbox, and the steered front wheels.
The hand-wheel feedback motor is controlled in the sense that it can provide the
driver to feel the interactions between the front wheels and road surfaces during driving.
The front wheel steering motor generates the actual torque for steering the two front
wheels through the rack and pinion gearbox and the steering arm. The control of the
steering motor aims at driving the front wheel steering angle to closely follow the hand-
wheel reference command.
In this chapter, we model the steering system, from the steering actuator to the steered
front wheels, as a motor driving a load (the steered wheels) through the rack and pinion
gearbox. First, the dynamic equation of the front wheel steering motor is described by
the following second-order differential equation [63, 181]:
(3.1)
where and are the moments of the inertia and the viscous friction of the steering
motor, respectively, is the shaft angle of the steering motor, is the torque
exerted on the motor shaft by the two steered wheels through the rack and pinion
gearbox, represents the motor torque pulsation disturbances that will be described
below, and is the torque control input for the steering motor.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
76
For the further analysis, the road vehicle equipped with the SbW system is
represented by the linear bicycle model (refer to Figure 2.15 for details) [10]. The
steered central front wheel can be treated as the load of the steering motor and rotates
about the vertical axis crossing the wheel centre. Therefore, the rotation of the central
front wheel satisfies the following dynamic equation [28, 182, 181]:
( ) (3.2)
where and are the moment of the inertia and the viscous friction of the front
wheels, respectively, is the torque applied on the steering arm by the steering motor
through the rack and pinion gearbox, is the self-aligning torque which reflects the
interaction between the road surface and the front wheels while the vehicle is turning,
( ) is the Coulomb friction in the steering system which has been defined in
(2.19).
Assuming that there is no backlash between the rack and gear teeth, we have the
following relationships held [181]:
(3.3)
where and are the tooth numbers of rack and pinion gearbox, respectively, r is a
scale factor to account for the conversion from the linear motion of the rack to the
rotation at the steering arm or the steering angle of the steered front wheels.
Further, differentiating (3.3) twice, we obtain the following relationships about the
motor shaft angle , the steering angle , and their derivatives [181]:
(3.4)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
77
Then using (3.4) in (3.2), we have
( ) (3.5)
where
(
)
(3.6)
(
)
(3.7)
and the equivalent drive torque signal
(3.8)
Remark 3.1: It is seen from (3.5) that the SbW system, from the steering motor to the
steered front wheels, is equivalent to a second-order direct drive system. Although many
researchers in [10, 11, 21-25] have extensively considered the modelling issue of SbW
systems, this is the first time to systematically derive a complete mathematical model
for the SbW system in Figure 2.8.
3.2.2 Bounds of System Parameters and Disturbances
Although the moments of the inertias and , the effective viscous friction
coefficients and , the conversion parameter r in (3.6)-(3.7), and the Coulomb
friction constant are all unknown in practice, the following bounded conditions can be
assumed:
(3.9)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
78
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
where , , , , , , , , , and are positive constants. Thus,
considering (3.6) and (3.7), we can express the upper and lower bounds of and
as follows:
(
)
(3.15)
(
)
(3.16)
(
)
(3.17)
Then, and satisfy the following bounded conditions:
(3.18)
(3.19)
In addition to the uncertainties in , , , , r and , another significant
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
79
uncertainty exists in the self-aligning torque that is the total aligning moment of the
tyre. Please refer to Figure 2.16 (a) and (b) for the details of the tyre forces and the self-
aligning torque at the steered central front wheel of the bicycle model, respectively, and
the parameters of the tyre dynamics are listed in Table 2.2.
Given by the expression of the self-aligning torque in (2.36), for the design of the
sliding mode controller in the next section, the upper bound of the self-aligning torque
is estimated as follows:
| | (3.20)
where
( ) |(
)| 3.21)
where , are the upper bounds of the mechanical trail and the pneumatic trail,
respectively, and is the upper bound of the front tyre cornering stiffness coefficient.
In terms of the total pulsation disturbances in the motor torque generation given in
(2.30), we obtain the following bound information for the controller design. For the first
part, we have
( ) ( ) | | | | (3.22)
where and are the upper bounds of the 6th and the 12th harmonic torque
amplitudes, respectively.
For the second part in (2.30), the bound information is given as
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
80
( )
| |
√
(3.23)
Therefore, the upper bound of can be estimated as follows:
(3.24)
where
√
(3.25)
3.3 Design of A Robust Sliding Mode Controller
In this section, we will develop a robust SM controller for the SbW system in (3.5) with
uncertain dynamics, under the condition that the information of the upper and the lower
bounds of the unknown system parameters, the self-aligning torque, and the total torque
pulsation disturbances in (3.9)-(3.14), (3.21), and (3.25) are known, respectively. The
controller design and stability analysis are presented in the following theorem:
Theorem 3.1: Consider the SbW system in (3.5) with the uncertainty bounds in (3.9)-
(3.14), (3.21), and (3.25), respectively. The tracking error asymptotically converges
to zero if the motor control torque is designed as:
( ) [ ( | |) | |
] (3.26)
where is the tracking error between the front wheel steering angle and
the hand-wheel reference angle , is the upper bound of the second-order
derivative of , which satisfies as
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
81
| | (3.27)
and the sliding variable s is defined as:
(3.28)
with the designed positive parameter.
Proof: Choosing a Lyapunov function candidate
and differentiating V with
respect to time, we have
[ ]
( )
{[
(
( ))]}
( )
[
( )
] ( )
( )
| |
[ ( | |) | |
]
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
82
( )
(| |
) (| |
| |
)
(| |
) (| |
)
(| |
( )
) (
| || | )
| | (
) | |
(
) | |
| |
(
| |) | |
(
| |)
| | (
| |) | | (
) | |
| |
(
| |) for | | (3.29)
(3.29) indicates that the sliding variable s is asymptotically stable. However, if , the
upper bound of , is chosen such that
{| |} (3.30)
where is a positive constant number. (49) can then be written as:
| | (3.31)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
83
(3.31) ensures that the sliding variable s converges to the sliding mode surface in a finite
time [40]. The sliding mode controller in (3.26) can constrain the closed-loop error
dynamics on the sliding mode surface, and the tracking error between the steering angle
and the reference signal can then exponentially converge to zero.
Remark 3.2: As the sign function sign(s) is involved in the sliding mode control signal
in (3.26), the chattering may occur in the control input. Using the boundary layer
control technique in [35, 40, 183], we can modify the control law in (3.26) as follows:
( ) [ ( | |) | |
] (3.32)
where
( ) {
| |
( ) | | (3.33)
and the constant .
(3.32) is called the boundary layer sliding mode (BL-SM) controller. As shown in [35,
40, 183], the output tracking error cannot converge to zero as the sign function is
replaced by the sigmoid function. However, by properly choosing the value of the
positive constant in (3.33), the tracking error can be small enough to satisfy the
tracking precision requirement in practice.
Remark 3.3: In terms of the dynamics of the hand-wheel, please refer to Subchapter
2.4.1 for the further details.
Remark 3.4: It is well known that the variable gear ratio steering (VGRS) has been
widely used in advanced road vehicles recently [22, 184-186]. Here we would like to
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
84
address that the proposed sliding mode control scheme in this chapter is also applicable
for the SbW systems with VGRS. The variable gear ratio
is actually embedded in the
steering ratio
, which has been involved in all of the parameters of the equivalent
second-order model in (3.5). By properly choosing the upper and lower bounds of all
parameters of the SbW system model in (3.5) with VGRS, the proposed SMC can
ensure the good steering performance.
3.4 Numerical Simulation
In order to show the good performance of the proposed SM controller, a simulation is
carried out in comparison with the PD controller with feed-forward torque and the
controller for SbW systems.
3.4.1 Parameters of SbW System and Vehicle Dynamics
In this simulation, both the front wheel steering motor (PMAC motor) and the feedback
motor are chosen as the same in order to agree with the SbW platform used in the
experimental section, where the steering motor is connected to a gearhead with ratio .
The nominal parameters of the SbW system are listed in Table 3.1. The nominal
parameters of the three-phase PMAC motors are given in Table 3.2. Then we assume
that the dc current offsets in phase a and phase b are and ,
respectively. In addition, the parameters of the vehicle dynamics and motor harmonic
torque used in the simulation are given in Table 3.3 [61, 187].
In this simulation, we assume that the central front wheel parameters and ,
the steering motor parameters and , the conversion parameter , the actual gear
ratio , the tyre parameters , , and , the 6th and 12th harmonic torque
amplitudes and , the dc current offsets and are all unknown, but the
following uncertainty bounds are known:
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
85
Table 3.1: Nominal parameters of the SbW system in equation (3.5).
Parameter Value (
) 2.6 ( ) 12 (
) 0.02129 ( ) 0.038 (
) 0.0791 ( ) ) 0.15
( ) 0.2
3
6 12 8.5
( ) 2.68
Table 3.2: Nominal parameters of PMAC motor.
Parameter Value
Rated speed (rpm) 2000
Rated torque (Nm) 4.77
Rated power (kW) 1.0
Rated voltage (V) 200
Rated current (A) 5.3
Peak instantaneous torque
(Nm)
14.3
Number of poles 6
Constant magnet flux (Wb) 0.2
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
86
Table 3.3: Parameters of vehicle dynamics and motor harmonic torque for simulation.
Parameter Value
, ( ) 0.016, 0.023
, ( ) 1.2, 1.05
( ) 1300
( ) 2000
( ) for
wet asphalt road
45000, 45000
( ) for
snowy road
12000,12000
( ) for
dry asphalt road
80000,80000
( ) for
wet asphalt road
0.022, 0.005
( ) for
snowy road
0.01,0.003
( ) for
dry asphalt road
0.038, 0.007
(3.34)
(3.35)
(3.36)
(3.37)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
87
(3.38)
(3.39)
(3.40)
and (3.41)
{
(3.42)
, (3.43)
√( ) (
) (3.44)
Furthermore, for the yaw motion of the road vehicle based on the bicycle model in
Fig. 2.15, please refer to (2.38) in chapter 2 for details.
3.4.2 Control Law
Due to the gearhead connected to the front wheel steering motor, the model equation of
the SbW system derived in (3.5) is re-written as the following state-space equation:
[
] [
] [
] [
( )
] [
] (3.45)
where is defined as
.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
88
The corresponding SM controller is of the form:
( ) [ ( | |) | | ] (3.46)
where and are the lower and the upper bounds of , and defined as
, and
. is selected as and the sliding mode
parameter is chosen as .
In addition, the control gains of the PD regulator for controlling the feedback motor
are chosen as and , respectively.
3.4.3 Simulation Environment
In order to demonstrate the effectiveness and robustness of the proposed SM controller,
the simulation environment is set up as follows:
Driver’s input torque is a periodic sinusoidal signal as ( ) .
Three different road conditions (wet asphalt, snowy, and dry asphalt roads) are
set for 0-15s, 15-25s, and 25-35s, respectively.
The vehicle velocity is set as .
The Euler method with the sampling interval is adopted to solve the
closed-loop differential equations in this simulation.
3.4.4 Simulation Results
The steering performance of the SM controller is shown in Figure 3.1(a) and (b), while
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
89
the associated control input is depicted in Figure 3.1(c). It can be seen that the front
wheel steering angle is driven to closely follow the hand-wheel reference angle in the
whole period. Although the road conditions are suddenly changed at 15s and 25s,
respectively, the good steering performance can still be achieved. Such an excellent
steering performance indicates that the SM controller is capable of eliminating the
effects of uncertain road conditions on the steering performance. Figure 3.1(d) and (e)
show the upper bounds of the disturbances needed for the SM controller design no
matter how they change. Particularly, Figure 3.1(f) shows that the steering performance
at time 25s is not affected much owing to the robustness of SM controller. Additionally,
due to the discontinuous control input when crossing the SM surface, the chattering
occurs unavoidably, which can be solved by adopting the BL-SM controller in (3.32).
In order to further show the control performance, the root mean square (RMS) for the
tracking error as a performance evaluation index is utilized for the sake of clear
comparison [188]:
√(∑
( ))
(3.47)
where n is the number of the iterations. The RMS for the proposed SM controller during
the simulation period (35s) is .
BL-SM controller is also used for removing the chattering in the simulation. Figure
3.2(a)-(f) show the steering performance with the BL-SM controller, where the constant
is chosen as and the RMS during the simulation period (35s) is .
It is s from Figure 3.2(c) that with the proper choice of , not only the undesired
chattering in the control signal is effectively removed, but also the amplitude of the
control torque is greatly reduced. Moreover, as shown in Figure 3.2(f), the road
condition variations do not affect the BL-SM control performance due to its robustness.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
90
Figure 3.1: Control performance of SM controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque. (d) Self-aligning torque and upper bound. (e) Torque
pulsation disturbances and upper bound. (f) Tracking in the 25th second.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
91
Figure 3.2: Control performance of BL-SM controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque. (d) Self-aligning torque and upper bound. (e) Torque
pulsation disturbances and upper bound. (f) Tracking in the 25th second.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
92
For further comparison, Figure 3.3(a)-Figure 3.3(f) show the steering performance of
the steering system using a PD controller with feed-forward torque control method [10,
26, 189]:
( ) (3.48)
where and are the proportional and derivative control gains, respectively. The
last two terms on the right-side of (3.48) are adopted for the purpose of reducing the
effects of disturbances on the steering performance.
Based on the system model and motor characteristics described in section 3.2, two
most suitable control gains in (3.48) are determined as follows:
, (3.49)
It is clearly seen from Figure 3.3(a) to Figure 3.3(f) that the steering performance
with the PD control is not as good as the ones as shown in Figures 3.1 and 3.2 with the
proposed SM control schemes in this chapter. The reason is that the PD controller with
fixed gains is unable to deal with the time-varying road conditions. This point can be
easily seen, from Figure 3.3(f) that, at 25s, the front wheels sideslip seriously due to the
varying road condition and after that the tracking performance has greatly deteriorated.
In addition, the RMS for the steering performance with the PD control in (3.48) is
that is much larger than the ones in the SbW systems with the SM and BL-
SM controllers presented in Figures 3.1 and 3.2.
Figure 3.4(a)-Figure 3.4(f) show the performance of the SbW system with the
following controller [190]:
(
) (3.50)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
93
Figure 3.3: Control performance of PD controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque. (d) Self-aligning torque. (e) Torque pulsation
disturbances. (f) Tracking in the 25th second.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
94
Figure 3.4: Control performance of controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque. (d) Self-aligning torque. (e) Torque pulsation
disturbances. (f) Tracking in the 25th second.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
95
where and are the gains of the nominal feedback control, the error state vector
[ ] , and (
)
that is the optimal control gain of the control for
minimizing the effects of the following total disturbance on the steering performance:
( ) (3.51)
where represents the modelling uncertainties.
It is noted from (3.51) that the hand-wheel angular acceleration is required in the
controller design. In fact, it is difficult to measure in practice. In this simulation
as well in the following experiments, is obtained by using the filtering method that
has been widely used in engineering practice [10].
The performance index for the control is given as
∫ ‖ ( )‖
‖ ( )‖
∫ ‖ ( )‖
(3.52)
where Q and P are the weighting matrices, is a prescribed attenuation level as
. P can be found by solving the following Riccati matrix equality:
(3.53)
where [
], [ ] , and is a designed positive constant.
The parameters are set as 0.1, 0.1, 1, and 150, respectively. In terms of the
controller design, the matrix Q is selected to be . Using Matlab, the matrix P is
found as
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
96
[
] (3.54)
and the control gain is
[ ] (3.55)
It has been seen from Figure 3.4(a)-Figure 3.4(f) that, the steering performance of the
SbW system with control has been improved compared with the one of using the
PD control, but still not as good as the ones with the SM and BL-SM controllers. Figure
3.4(f) shows that the control cannot eliminate the effects of the variations of the
road conditions on the steering performance. Further, the RMS for the control based
SbW system is that is larger than the ones of the SM and BL-SM
controllers and lower than the one of the PD controller.
Remark 3.5: It is seen from the above simulation results that the influence of the self-
aligning torque on the steering performance is much greater than that of the torque
pulsation disturbances. For instance, the maximum values of the self-aligning torque
and torque pulsation disturbances are about 150 Nm and 6 Nm for wet asphalt road, 40
Nm and 4 Nm for snowy road, and 220 Nm and 9 Nm for dry asphalt road, respectively.
The advantage of the proposed sliding mode control methodology can eliminate not
only the effect of the torque pulsation disturbances, but also the one of the self-aligning
torque on the steering performance. This point has been clearly seen from both the
stability analysis and the simulation studies in Section 3.3 and Section 3.4, respectively.
Remark 3.6: It is noted from the above simulation studies and [21] that the self-
aligning torque under three different road conditions behaves like three hyperbolic
tangent disturbance signals of the steering angle with different amplitudes. We may thus
use the following hyperbolic tangent signal to model the self-aligning torque :
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
97
Figure 3.5: Control performance of BL-SM controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
{
( )
( )
( )
(3.56)
where , , and are chosen as , , and , to ensure that
the amplitudes of in three different road conditions are about 150 Nm, 40 Nm and
220 Nm, respectively. The corresponding bounds of , , and are selected as 550,
180, and 980, respectively.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
98
1
Figure 3.6: Control performance of PD controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
Figure 3.5-Figure 3.7 show the steering performances of the SbW system equipped
with the BL-SM controller, the PD controller with feed-forward torque method, and
controller, respectively. The corresponding RMS for these three controllers are
, , and , respectively. It can be clearly seen that, given by
the hyperbolic tangent disturbance signals for the tyre self-aligning torque in (3.56),
the simulation results are similar to the ones presented in Figure 3.1-Figure 3.4, which
has verified the efficacy of the expression (3.56) for the tyre self-aligning torque.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
99
Figure 3.7: Control performance of controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
3.5 Experimental Studies
In this section, we will verify the effectiveness and the advantages of the proposed SM
controller on an SbW experimental platform in Figure 3.8 in Robotics and
Mechatronics Lab at Swinburne University of Technology.
3.5.1 Experimental System Setup
It is seen from Figure 3.8 that Mitsubishi HF-SP102 (A) AC motors are used as the
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
100
Figure 3.8: The SbW Experimental Platform.
steering motor and the feedback motor, respectively. The steering motor is connected
with the gearhead, and the corresponding servo-driver is selected as MR-J3100A
manufactured by Mitsubishi Inc. An angle sensor (59006-10 turn, MoTeC) is installed
on the pinion to measure the front wheel steering angle indirectly. In the hand-wheel
side, the feedback motor is mounted on the steering column to provide the feeling of the
interactions between the steered wheels and road surface.
The nominal parameters of the SbW platform and AC servo motor in experiments are
the same as the ones in the simulation section. Both the servo drivers are operated in
torque control mode, driven by a +/- 8V reference signal. The servo motor is provided
with a current by the servo driver, which is linearly proportional to the reference input
voltage. Then, the torque generated by the servo motor is proportional to the input
current. Thus, the torque generated by the servo motor is linear with the analogue torque
command (input voltage).
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
101
Table 3.4: Values of control parameters.
Parameter Value
12
1.35
-3.6
-1.2
The proposed control algorithm is implemented on a HP personal computer using
Matlab/Simulink/Real-time Workshop. The Advantech PCI 1716 Multifunction Card is
installed in the PC for real-time control applications. The sampling period is chosen as
, and the Euler method is adopted for this real-time experiment.
In order to reduce the cost of the SbW system in real applications, the velocity of the
front wheel steering angle is computed by differentiating and low-pass filtering the
position signal measured by the position sensor [191].
3.5.2 Experimental Results
The bound information of all the SbW system parameters is same as the one used in the
simulation. To avoid the chattering in the control signal, the BL-SM controller is
utilized in the following experiment. The values of BL-SM controller parameters
( ) and PD controller parameters ( and ) are determined in Table 3.4,
while the control parameters of controller are kept the same as the ones in
simulations. On the other hand, the control gains of the PD regulator for the hand-wheel
feedback motor are also set as same as the ones in the simulation section.
After the system is set up, the current offsets of phase a and b are measured with the
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
102
Figure 3.9: Control performance of BL-SM controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
values of 0.1 A and 0.05 A, respectively. It should be emphasized that, although the
current offsets are varying with time and temperature, the design of the proposed
controller is not affected because only the bound information of the current offsets is
required, as shown in (3.44). For confirming the robustness of the proposed scheme, the
following voltage signal that models the self-aligning torque in (3.56) is added to the
system:
{
( )
( )
( )
(3.57)
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
103
Figure 3.10: Control performance of PD controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
where ( ) is the combined ratio including the steering ratio and the
gearhead ratio defined in (65), , , and are the same as those determined in (76),
and ( ) is the nominal ratio between the motor torque and the input voltage of the
servo driver given in the manual.
The experimental results with the BL-SM controller, the PD controller, and the
controller are shown in Figure 3.9-Figure 3.11, and the corresponding RMS are
, , and , respectively. It is observed that the
steering performances of the three controllers are nearly the same on the snowy road at
time period of 15-25s. This is because the equivalent disturbance is very mall and the
three controllers are working closely at the idea condition. However, in the periods of 0-
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
104
Figure 3.11: Control performance of controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
15s and 25-35s, the steering performances of both the PD controller and the
controller are deteriorated seriously due to the large variation of disturbances. Only the
SbW system equipped with the proposed SM controller performs very well and behaves
with a strong robustness against the large changes of the external disturbance.
Remark 3.7: It has been noted that the tracking performances in the simulations are
better than the ones in the experimental results. The reasons are as follows: (i) All the
mechanical parts, such as rack and pinion gearbox, are assumed to match perfectly, that
is, no backlash exists in the mathematical model; however, the backlash indeed exists in
the rack and pinion gearbox in practice, which is actually the main factor of affecting
the steering accuracy; (ii) In the experiments, we have observed that the small structural
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
105
resonances of mechanical parts occur sometimes during the operations, which have also
affected the tracking precisions; (iii) It is noticeable that the low sampling rate of the
microcontroller and the low resolution of the angle sensors are the other factors of
degenerating the steering performance of the SbW system. Thus, the use of the high
quality rack and pinion gearbox and the proper adjustment of the structure of the SbW
system to avoid the structural resonances can play an essential role of further improving
the tracking performance. In addition, an advanced microcontroller with a fast sampling
rate and the sensors with higher resolutions are required to achieve more accurate
tracking precision in SbW systems.
3.6 Conclusion
In this chapter, the mathematical modelling of SbW systems has been further
explored and a robust sliding mode steering controller has been proposed. It has been
seen that the proposed sliding mode controller can efficiently alleviate the effects of
uncertain system parameters and the variations of the road conditions as well as torque
pulsation disturbances. Both the simulation and experimental results have verified the
excellent steering performance of the proposed scheme. The further work on designing
a sliding mode-based adaptive controller and the sliding mode observer-based diagnosis
system are under the authors’ investigation.
Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles
106
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
107
Chapter 4
Nonsingular Terminal Sliding Mode
Control for Steer-by-Wire Systems with
Uncertain Dynamics
In this chapter, a robust nonsingular terminal sliding mode (NTSM) control scheme for
Steer-by-Wire (SbW) systems with uncertain dynamics is developed. Based on the
equivalent second-order model of SbW systems derived in Chapter 3, we further
propose an NTSM steering controller in order to achieve a faster, finite time
convergence rate and higher tracking precision using the bound information of uncertain
system parameters and disturbances. It is shown that, by the use of the proposed NTSM
control scheme, both the finite time convergence of the steering angle tracking error and
the strong robustness against large system uncertainties can be ensured. Experimental
results are provided to verify the superior steering performance of the proposed control
scheme in terms of good tracking performance and robustness property.
4.1 Introduction
UTOMOTIVE steering systems have evolved from mechanical steering systems
to hydraulic power assisted and electric power assisted steering systems during
the past century. The common characteristic of these steering systems is that a
A
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
108
mechanical shaft is used to connect the hand-wheel with the front wheels. The
mechanical shaft performs as a proportional-derivative (PD) regulator, that is, the torque
exerted on the front wheels by the mechanical shaft is proportional to the driver’s torque
and its changing rate [11]. Since the parameters of the PD regulator cannot be adjusted,
it is very difficult for a driver to quickly adjust the front wheel steering angle to the
desired position in case of accidents and critical situations, which results in a risk and
unreliability of road vehicles. Therefore, a new form of technology that is receiving
great attention in the automotive industry is the Steer-by-Wire (SbW) system, where the
conventional mechanical linkage between the hand-wheel and front wheel is removed
and substituted by two electrical motors. The modern SbW systems possess several
advantages: improved overall steering performance, reduced power consumption and
maintenance costs, and enhanced driving safety as well as comfort for the passengers.
Over the last few years, various kinds of control schemes have been developed for
SbW systems. It is well known that linear control schemes, e.g., the proportional
derivative (PD) control have been widely applied in the SbW systems owing to its
simplicity of implementation [10, 11, 22-25]. Nevertheless, in the practical SbW
systems, there exist unavoidable uncertain vehicle and steering system dynamics, and
highly nonlinear self-aligning torque, due to frequent road surface changes. It is because
the conventional PD control cannot assign the closed-loop poles to the desired location
when dealing with large road variations, it is difficult to ensure a satisfactory level of
steering performance in the entire operating range.
Due to the incompetency of PD control in dealing with varying driving conditions
and road environments, many authors devote themselves to the control approaches with
the strategies that the controllers are designed based on the feedforward compensation
of the tyre self-aligning torque using the estimations of the road surface condition and
vehicle body slip angle [10, 27-29]. However, good steering performance can only be
guaranteed when accurate estimations are obtained, which imposes an additional
restriction on the controller design in practical situations. In the meanwhile, several
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
109
adaptive control techniques for SbW systems have been developed in the direction of
achieving good steering performance [26, 21]. However, how to obtain accurate online
estimations of system parameters and unknown road disturbances is still challenging. In
addition, it is difficult to select the proper convergence rate of the adaptive controller
when the vehicle experiences varying road conditions.
Recently, the sliding mode (SM) control technique has been introduced to SbW
control systems [62, 193, 194, 206] due to the advantages of simplicity and robustness
against parameter variations and disturbances [31-36]. It is because the most commonly
used linear sliding surface is adopted in these SM-based SbW control systems, that the
asymptotic stability of the closed-loop system is guaranteed in the sliding mode, but the
steering angle tracking error cannot converge to zero in finite time. It should be noted
that the finite-time control is of particular interest because systems with finite-time
convergence possess some nice features including better tracking performance,
robustness and disturbance rejection properties. Thus, in order to improve the dynamic
response of the closed-loop SbW system, an alternative way is to introduce a nonlinear
sliding surface known as a terminal sliding surface [44] in the terminal sliding mode
(TSM) controller, which has the capability of ensuring the finite-time convergence of
the tracking error. Despite the fact that the finite time convergence characteristic of the
output tracking error is successfully achieved, there exists a singularity problem due to a
negative fractional power in the TSM controller. Hence, a nonsingular terminal sliding
surface has been accordingly proposed in the nonsingular terminal sliding mode (NTSM)
controller as a remedy to the singularity problem [45].
In this chapter, we will develop an NTSM steering control scheme for the SbW
systems for the purpose of achieving better tracking performance and robustness against
system parameter variations and unknown road environments. It will be shown from the
experimental results that the proposed NTSM steering controller in this chapter can
drive the nonsingular terminal sliding variable to reach the nonsingular terminal sliding
surface and the closed-loop error dynamics can then enjoy a finite time convergence
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
110
characteristic.
This rest of the chapter is organized as follows. In Section 4.2, a model of SbW
systems is given. Section 4.3 designs an NTSM control scheme and the finite time
stability of tracking error dynamics and robustness with respect to uncertain dynamics
are discussed in detail. In Section 4.4, experimental studies are carried out to validate
the effectiveness and feasibility of the proposed control scheme. Finally, the
conclusions are addressed in Section 4.5.
4.2 Problem Formulation
Consider the following second-order equivalent model of the SbW systems:
( )
(4.1)
where is the steering motor control torque, and are given as
(
)
(4.2)
(
)
(4.3)
where the corresponding system parameters are listed in Table 4.1.
Remark 4.1: It is worth noting that, only the simple model of SbW systems is given
here while the details on how to obtain this model can be found in Chapter 3. Following
Chapter 3, we also consider the case that we do not exactly know the values of the SbW
systems and disturbances, but the bound information of the uncertain parameters and
disturbances are assumed to be known. Readers may refer to Chapter 3 for determining
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
111
Table 4.1: Parameters of the SbW system.
Symbols Description Rotational angles of hand-wheel, front-
wheels, and steering motor shaft Torques generated by driver, feedback
motor, and steering motor Self-aligning torque Coulomb friction in the SbW system Coulomb friction constant Motor torque pulsation disturbance Equivalent drive torque
Moments of inertia of front-wheels, steering motor and equivalent system
Viscous frictions of front wheels, steering motor and equivalent system
A scale factor accounting for the conversion from the linear motion of the rack to the rotation of the front-wheels.
Rack & pinion system’s gear ratio
Tooth number of rack & pinion gearbox
the upper and lower bounds of the system parameters and the upper bound of the
disturbances in detail.
4.3 Design of an NTSM Steering Controller
Assume that the upper and lower bounds of the steering system parameters in (3.9)-
(3.14) and the upper bounds of the self-aligning torque in (3.21) and the motor torque
pulsation disturbance in (3.24) are all known. In this section, an NTSM steering
controller is proposed in order to achieve good tracking performance, such as faster
convergence and better tracking precision. It demonstrates how the controller drives the
front wheel steering angle to track the desired hand-wheel rotational angle in finite time.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
112
Given the hand-wheel reference command (please refer to Section 2.4.1 for
further details of hand-wheel dynamics), we define the tracking error between the front
wheel steering angle and the hand-wheel reference angle as follows:
(4.4)
Generally, in order to use the NTSM technique, an nonsingular terminal sliding
surface is designed as
( ) (4.5)
where , are the positive odd integers and satisfy .
The NTSM steering controller is given by:
( ) [ ( ) | |
( )
( ⁄ ) ( )] (4.6)
where ( ) is the sign function defined in (2.9), is the upper bound of the
second-order derivative of , which satisfies
| | (4.7)
Theorem 4.1: Consider the SBW system model in (4.1) under the assumption that all
the upper and lower bounds of the system parameters in (3.9)-(3.14) and the upper
bounds of the self-aligning torque in (3.21) and the motor torque pulsation in (3.25) are
known. If the motor control torque is designed as in (4.6), the tracking error can
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
113
then converge to zero in finite time.
Proof: Consider a Lyapunov function candidate
and differentiating V with
respect to time, we have
(
( )
(
) )
[
( )
(
)( )]
{
( )
(
)[
(
( )) ]}
( )
(
)
( )
(
)
( )
(
)
( )
(
)
( )
(
) ( )
( )
(
)
| |
( )
(
)
[ ( ) | |
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
114
( )
( ⁄ ) ( )]
( )
(
)
( )
(
)
( )
(
)
( )
(
) ( )
( )
(
)
( )
(
)(| |
)
( )
(
)(| |
| |
)
( )
(
)(| |
)
( )
(
)(| |
( )
)
(
| || | )
( )
(
)(| |
)
( )
(
)| | (
| |)
( )
(
)| | (
| |
| |)
( )
(
)| | (
| |
)
( )
(
)| | (
| |
)
( )
(
)| | (
| |
) (
| || | | || |)
( )
(
)| | (
)
( )
(
)| |
(
) | |
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
115
( )
(
)| |
(
| |)
( )
(
)| |
(
| |)
( )
(
)| |
(
| |) | | (
) | | (4.8)
Consider the bounded information in (3.9)-(3.14), (3.21) and (3.25), and are two
positive odd integers with , then (4.8) can be expressed as
( )
(
)| |
(
| |) for | | | | (4.9)
Expression (4.9) is the sufficient condition for the nonsingular terminal sliding variable
to converge to zero in finite time according to the Lyapunov stability theory [35, 40].
Then, for | | , the NTSM control signal can be rewritten as:
( ) [ ( ) | |
] (4.10)
Using (4.10) in (4.1), we have
( )
( )[ ( ) | |
]
(4.11)
For , (4.11) can be expressed as
( )
| |
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
116
(4.12)
Thus,
| |
( )
(
) | |
(
| |)
(
| |)
(
| |) (
)
(
| |)
where is a positive constant.
Similarly, for , we obtain .
Hence, there exists to be a vicinity of so that for an arbitrarily small ,
we can obtain
( ) { | }
and for , for . Therefore, by LaSalle’s theorem [40], every
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
117
trajectory starting in ( ) approaches in finite time.
Therefore, it is easily seen that the NTSM can be reached from anywhere in
finite time,
( ) (4.13)
such that the output tracking error can converge to zero in finite time [35, 40], which
means that the front wheel steering angle will closely track the reference hand-wheel
angle in finite time.
Remark 4.2: As the sign function sign(s) is involved in the control signal in (4.6),
chattering may occur in the control input. Based on the principle of the boundary layer
control technique in [35, 40, 183], the following boundary layer NTSM (BL-NTSM)
control input can be derived:
( ) [ ( ) | |
( )
( ⁄ ) ( )] (4.14)
where
( ) { | | ⁄
( ) (4.15)
( ) { | | ⁄
( )
(4.16)
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
118
with the boundary layer constants .
It should be noted that, although the output tracking error cannot converge to zero as
the sign function is replaced by the sigmoid function in NTSM control systems, by the
proper choice of the value of the positive constants and , the steering angle
tracking error can be small enough to satisfy the tracking precision requirement in
practice [35, 40, 183].
4.4 Experimental Results
In order to demonstrate the performance of the proposed NTSM control approach, the
SbW system platform equipped with the BL-NTSM control with no chattering has been
tested in real-time experiments, compared with the conventional boundary layer sliding
mode (BL-SM) control and the PD control, respectively.
In terms of the experimental setup, we use the same SbW platform as in Chapter 3 as
shown in Figure 3.8. The nominal parameters of the SbW system and the PM AC motor
can be found in Table 3.1 and Table 3.2, respectively. The sampling interval is chosen
as .
In consistent with the obtained voltage model for the tyre self-aligning torque in
Chapter 3, we give the following voltage signal which is the input to the steering motor
for representing three different road conditions:
{
( )
( )
( )
(4.17)
where , , and to ensure that the amplitudes of the
disturbance for three road conditions are different.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
119
In order to test the performance of the proposed NTSM controller in different driving
operation, the following two cases are taken into account:
Case 1) Road condition variation: With the periodic sinusoidal steering signal (standard
slalom maneuver), the wet asphalt, icy, and dry asphalt road conditions are set for 0-15 s,
15-25 s, 25-35 s, respectively.
Case 2) Variable step-like steering commands: On the fixed wet asphalt road condition,
the step-like steering commands are changing from 0 to 0.3 rad at 5 s, from 0.3 to -0.3
rad at 15 s, and from -0.3 to 0.3 rad at 25 s, respectively.
The bounded values of the uncertain system parameters and disturbances are kept the
same as in Chapter 3. The control parameters of the proposed control system are given
as follows while the ones of the comparative PD and BL-SM control are kept the same
as in Chapter 3.
PD control: , ,
BL-SM control: , ,
BL-NTSM control: , , . (4.18)
Please note all the control parameters of these three controllers are determined for the
purpose of achieving both good transient and steady-state control performances with
consideration of the requirement for stability and the possible operating conditions.
First, the steering performances of three SbW control systems with a periodic
sinusoidal steering signal under three different road conditions are shown in Figure 4.1-
Figure 4.3.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
120
Figure 4.1: Control performance of PD control in case 1. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
Figure 4.1(a)-(c) shows the steering performance, tracking error, and control torque
input, respectively, with the conventional PD control scheme in case 1. It is clearly seen
that, after the road surface changes from icy to dry asphalt road conditions ( ),
the steering performance has significantly deteriorated and the PD control is not capable
of driving the front wheels to properly track the hand-wheel command. This is because
under a large variation of road conditions, the pre-well-tuned PD control cannot assign
the poles of the closed-loop SbW system to the desired location and thus, the steering
performance accordingly degenerates.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
121
Figure 4.2: Control performance of BL-SM control in case 1. (a) Tracking performance.
(b) Tracking error. (c) Control torque.
Figure 4.2(a)-(c) shows the performance of the SbW system with the BL-SM
controller in case 1. It is seen that the BL-SM equipped SbW control system behaves
much better than PD control in terms of steering performance and robustness, especially
on the dry asphalt road during the last 15 seconds. This is because the BL-SM control
using the bound information of the system uncertainties and disturbances is able to
eliminate the effects of the uncertain system dynamics. However, due to the asymptotic
convergence characteristic of the tracking error in the linear sliding surface of the BL-
SM control, superior tracking performance cannot be achieved.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
122
Figure 4.3: Control performance of BL-NTSM control in case 1. (a) Tracking
performance. (b) Tracking error. (c) Control torque.
Figure 4.3(a)-(c) shows the experimental results of the steering performance, tracking
error, and control torque for the steering motor, respectively, with the proposed BL-
NTSM control in case 1. It is shown that the steering performance has been improved a
lot with a smaller tracking error, as compared with the one using the PD control in
Figure 4.1(a)-(c) and the BL-SM control in Figure 4.2(a)-(c), respectively. Such an
excellent steering performance of the proposed BL-NTSM control is largely due to the
fact that nonsingular terminal sliding surface in (5.5) possesses faster and higher
precision tracking characteristics compared with previous comparative control systems.
It is worth noting that the ideal finite-time error convergence of the BL-NTSM control
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
123
Figure 4.4: Control performance of PD control in case 2. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
scheme cannot be obtained in real applications due to the large variation of the
disturbances, the steering tracking error can still converge to an acceptable region,
which is one of the major superiorities over the conventional PD and BL-SM control
methods.
Also, in order to show the steering performance in an emergency steering situation, a
series of experiments are performed for the three SbW control systems under the
condition of variable step-like steering commands in case 2. Figures 4.4-4.6 show the
steering performance of the SbW system with the PD control, the BL-SM control, and
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
124
Figure 4.5: Control performance of BL-SM control in case 2. (a) Tracking performance.
(b) Tracking error. (c) Control torque.
proposed BL-NTSM control, respectively. After evaluating the performance of three
controllers, the following facts have been noted: (i) It can be clearly seen that the BL-
NTSM control gives a faster response than the PD control and the BL-SM control in
tracking the step-like steering commands, particularly at 25s. (ii) It is also seen that a
steady state error occurs in all control methods when the step-like steering command
takes place; however, the proposed BL-NTSM controller leads to smaller steady state
error than the PD and BL-SM controllers, which further confirms the high precision
tracking performance of the proposed control. (iii) Under the wet asphalt road condition,
when the step-like steering command suddenly changes, both the PD and proposed BL-
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
125
Figure 4.6: Control performance of BL-NTSM control in case 2. (a) Tracking
performance. (b) Tracking error. (c) Control torque.
NTSM control result in an undershoot while the BL-SM control causes an overshoot,
but the proposed BL-NTSM control performs with the smallest magnitude.
To quantify the improvement of the proposed BL-NTSM control over the PD and
BL-SM control, the root mean square (RMS) of the tracking error in (3.47) is used. The
performance comparisons of three control schemes during the period (5 s – 35 s) in two
cases are summarized in Table 4.2. It can be seen that the proposed BL-NTSM control
outperforms the PD and BL-SM control in terms of the RMS values in two cases, owing
to the finite time convergence and stronger robustness features of the proposed control.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
126
Table 4.2: Performance comparisons of controllers in Chapter 4.
RMS error
(rad)
PD control
Case 1 0.0931
Case 2 0.1135
BL-SM control
Case 1 0.0541
Case 2 0.0604
Proposed BL-NTSM control
Case 1 0.0351 Case 2 0.0367
Remark 4.2: It is worth noting that the steady state error for the step-like steering input
that exists in all three controllers is mainly due to the inserted voltage disturbance
representing the wet asphalt road and the actual friction in the steering system as well as
the ground contact surface [21]. It is because of the excellent performance of the
proposed BL-NTSM control scheme that these disturbances have been effectively
compensated resulting in the smallest steady state error, which enables the proposed
control to be one of the most favorable control methods in SbW control systems.
4.5 Conclusion
In this chapter, an NTSM control method has been proposed for SbW systems with
uncertain dynamics. It has been that the proposed control scheme exhibits excellent
robust steering performance against the system parameter uncertainties and varying road
conditions, owing to the characteristics of the finite-time convergence and strong
robustness of the NTSM control methodology. Different driving conditions including
the standard slalom maneuver for three road conditions and the step-like steering
command have been carried out in real-time experiments to demonstrate the
Performance
Controller
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
127
effectiveness of the proposed NTSM control scheme. Further work is to design a novel
sliding mode observer-based NTSM control scheme for SbW systems.
Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics
128
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
129
Chapter 5
Robust Control for Steer-by-Wire
Systems with Partially Known Dynamics
This chapter proposes a robust control scheme (RCS) for Steer-by-Wire (SbW) systems
with partially known dynamics. The dynamic model of an SbW system consists of a
partially known system and an unknown portion. Then, a nominal feedback controller is
used to stabilize the nominal model. A sliding mode compensator is introduced to
remove the impacts of the unknown parts of the SbW system based on the system
uncertainty bound. In addition, robust exact differentiator technique is utilized to
estimate the required derivatives of the measured position signals. It is shown that, not
only the designed RCS is greatly simplified with the aid of the partially known
knowledge of the SbW system, but also ensures the robust steering performance with
respect to large system uncertainties. As a result, the tracking error between the actual
front steer angle and the steering-wheel angle can enjoy the asymptotic zero-
convergence characteristic. Some experimental studies are given to verify the excellent
performance and the efficacy of the proposed RCS.
5.1 Introduction
TEER-BY-WIRE (SbW) systems have drawn considerable attention among
engineers in automotive industry and researchers over the last two decades. S
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
130
Compared with the conventional steering systems in road vehicles, the SbW systems
aim at bettering the overall steering performance, reducing the energy consumption, and
boosting the passengers’ safety and comfort. The SbW systems are distinct from the
conventional steering systems in the following aspects. First, the intermediate
mechanical shaft between the steering-wheel and the steered front wheels is eliminated.
Second, the front wheels are steered in the direction of closely following the steering-
wheel angle by means of a front-wheel motor, coupled to the steering rack though a
pinion gear. Third, a steering-wheel motor is employed on the steering-wheel side to
assist drivers in perceiving the influences of tyre self-aligning moment acting between
the front tyres and the road.
In the SbW systems, there exist large quantities of disturbances and uncertainties
internally or externally, such as parameter variations, Coulomb friction, motor torque
pulsation disturbances, and the significant variations of tyre self-aligning torque due to
the road surface changes. Because the conventional proportional-derivative control is
incapable of ensuring the good steering performance especially when the road
conditions are frequently varying [10, 11, 22-25], the robust control of the SbW systems
is still challenging.
Recently, a number of researchers have used advanced control techniques as
alternatives on the SbW systems in the purpose of realizing good steering characteristics.
In [59, 60], a state feedback control scheme using the linear quadratic control technique
was proposed to drive the rolling angle of the motorcycle equipped with the SbW
system to track the reference signal. In [27, 192], an adaptive control methodology was
adopted in order to realize the virtual steering characteristics for controlling the front-
wheel actuators based on the real-time estimation of the cornering stiffness and tyre slip
angle. In [26], the adaptive pole placement controller based on the adaptively estimated
parameters of the vehicle direction-control unit was used to reduce the tracking error. In
[182, 62, 193, 194, 207], the sliding mode control was successfully employed in the
SbW systems using the bound information of the uncertain parameters and perturbations
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
131
due to its powerful capability in dealing with parameter variations and perturbations for
highly nonlinear systems [99, 140, 101, 149, 178, 179, 196].
However, from the perspective of engineering applications, automotive engineers are
increasingly concerned with the acquisition of the bound information of the uncertain
SbW parameters and perturbations. Not only may the sliding mode controller design
become complicated, but also the large control gain is required owing to the pre-
determined large bounds of system parameters and disturbances. To tackle these issues,
in this paper, we treat an SbW system as a partially known system and an unknown part
called lumped uncertainty. This design method has been successfully employed in many
practical systems such as general nonlinear uncertain systems [87-92, 94-96], practical
robot manipulator systems [93, 97-101], active suspension system of vehicles [102,
103], and mechanical system with partially known nonlinear dynamic friction models
[104, 105]. In terms of the SbW systems, the nominal parameters of the mechanical
components including the front-wheel motor, front tyres as well as the rack and pinion
gearbox, and the identified disturbances in normal operating conditions are included in
the partially known system; and the unknown parts result from the parameter variations
and the unpredicted changes of the disturbances especially the tyre self-aligning torque
when the vehicle frequently experiences varying road conditions and environments,
which will highly affect the steering performance of the SbW systems. Thus, the
partially known knowledge of the SbW system can be employed in reducing the
complexity of the controller design in which a small uncertainty bound is needed to
eliminate the effects of the unknown lumped uncertainty. This in turn not only improves
the steering performance, but decreases the amplitude of the control signal, which is
practical and desirable in real applications.
As inspired from the control of robotic manipulators [101, 99], we prove that only
three parameters of the lumped uncertainty bound are required in prior for the controller
design using the position and speed signals. A nominal feedback controller (NFC) is
then used to stabilize the nominal model, followed by a sliding mode (SM) compensator
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
132
which is introduced to remove the influences of the lumped uncertainty. As a result, the
proposed robust control scheme (RCS) consists of the NFC and SM compensator. In
addition, both the angular speeds of the steering-wheel and the front wheels are
unmeasurable and need to be approximated in the proposed RCS. Thus, the so-called
robust exact differentiators (REDs) using super-twisting algorithm [197] are adopted to
achieve the required derivatives of the position measurements owing to the superior
finite-time reachability of estimated derivative values to the exact ones as well as the
strong robustness with respect to the measurement noise.
It will be seen from the following sections that the proposed RCS has several
remarkable advantages: Firstly, the closed-loop SbW system using the SMC as well as
REDs not only increases the robustness against parameter variations and disturbances,
but is capable of driving the front wheels to accurately track the steering-wheel
command. Secondly, because the partially known information of the SbW system is
used, the lumped uncertainty bound in the SM compensator is determined with only
three parameters, which significantly simplifies the implementation for practical
application. Thirdly, it is because of the proposed controller consisting of an NFC and a
SM compensator to eliminate the influences of the small lumped uncertainty, the control
gain is greatly reduced resulting in the smaller control amplitude in comparison with the
conventional sliding mode control schemes [182, 62, 193, 194].
In the rest of this chapter, the dynamic model of SbW systems is presented, the main
disturbances in the SbW systems are briefly analysed, and the property of lumped
uncertainty bound is explored in Section 5.2. In Section 5.3, an RCS using the SM
compensator and REDs is proposed and the asymptotic error convergence as well as the
stability analysis is detailedly discussed. Some experimental studies are performed for
demonstrating the good steering performance of the proposed controller in comparison
with other three comparative controllers in Section 5.4. Section 5.5 gives the conclusion
and some further work.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
133
5.2 Problem Formulation
A simplified model of a steering mechanism equipped with an SbW system can be
described in Figure 2.8 (please refer to Chapter 2). As seen from Figure 2.8, the SbW
system consists of two subsystems: the steering-wheel subsystem including the steering-
wheel, the steering-wheel motor coupled to the steering-wheel column, and the steering-
wheel angle sensor to measure the steering-wheel commanded angle, and the front-
wheel subsystem composed of a steering rack, the front-wheel motor coupled to the
steering rack via the pinion gear, the pinion angle sensor to indirectly measure the front
steer angle, and the front wheels.
It is recognized that the steering-wheel motor provides the driver with the feeling of
the reaction torque between the front tyres and the road as the vehicle is turning.
Meanwhile, the front-wheel motor is to steer the front wheels in the sense that the actual
front steer angle is able to accurately track the steering-wheel command.
5.2.1 Modelling
In this chapter, we model the steering system between the front-wheel actuator and the
two front wheels as a motor driving a load (the two front wheels) via the steering arms
as well as the rack and pinion steering mechanism.
First, the mechanical dynamic equation of the front-wheel motor is given by [182,
193]:
(5.1)
where is the front-wheel motor moment of inertia, is the front-wheel motor
viscous friction, is the angular position of the motor shaft, represents the torque
applied to the shaft of the front-wheel steering motor by the front wheels via the
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
134
steering arms and the rack and pinion gearbox, is the motor torque pulsation
disturbances that will be described below, and is the torque control input for the front-
wheel motor.
The rotation of the two steered front wheels can be expressed as:
(5.2)
where , and are the moment of inertia and the viscous friction of the front
wheels, respectively, is the front steer angle, represents the torque transmitted to
the steering arm of the front wheels by the coupled front-wheel motor via the rack and
pinion gearbox, is the self-aligning moment generated by the tyre cornering forces
during turning, and is the Coulomb friction in the motor assembly and the steering
system which has defined in (2.19) (please refer to Sub-chapter 2.5).
Considering that no backlash exists between the rack and pinion gear teeth, we obtain
the following relationships about , , and their derivatives [182]:
(5.3)
where is the steering ratio.
Then, using (5.3) in (5.2) and eliminating , we have
(5.4)
where and are the total inertia and damping coefficient of the SbW system
model in (5.4), respectively, which are defined as
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
135
( ) (5.5)
( ) (5.6)
(5.4) can be rewritten as:
(5.7)
Let
(5.8)
(5.9)
( )
(5.10)
(5.11)
We have
(5.12)
With the presence of the system uncertainties, (5.8)-(5.11) can be expressed as
(5.13)
(5.14)
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
136
(5.15)
(5.16)
and
(5.17)
where
,
, , , and are the nominal values of the system
parameters, ( )
, , , and
are the nominal values of the
system disturbances that can be identified by the preliminary experiments, , ,
( ), , , and denote the unknown bounded uncertainties
introduced by the system parameters and disturbances. In particular, is defined as
the known external tyre self-aligning torque on the wet asphalt road. Thus, is the
difference between the actual tyre self-aligning torque and the pre-determined self-
aligning torque for the wet asphalt road condition.
Now, under the above analysis, (5.12) can be modified as:
(5.18)
where represents the lumped uncertainty as follows:
(5.19)
The following system with no lumped uncertainty is defined as the nominal system:
(5.20)
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
137
Remark 5.1: It is worth noting that the lumped uncertainty in (5.19) consists of the
parameter variations in the mechanical components including the front-wheel motor,
rack and pinion gearbox as well as front wheels, and also the estimation errors of the
disturbances such as Coulomb friction, tyre self-aligning torque and motor torque
pulsations. The analysis of the disturbances will be described below for further properly
determining the bound of the lumped uncertainty in (5.19).
5.2.2 Disturbances
1) Self-aligning Torque: On the SbW systems, the tyre self-aligning torque is treated
as the most significant disturbance torque and its descriptions can be found in Sub-
chapter 2.4.3 [10, 27].
Given by the expression of self-aligning torque in (2.25), we obtain the following
forms of and in (5.16):
( )(
)
(5.21)
( )(
)
(5.22)
where is the pre-determined cornering stiffness coefficient at front tyre for the wet
asphalt road condition.
Thus, the modelled error in the self-aligning torque part in (5.16) is obtained as
(5.23a)
Further, using (5.21) and (5.22), we have
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
138
( )(
)
( )(
)
(
) ( ) (
) (5.23b)
Remark 5.2: Please note that those parameters , , , and will also affect the
determination of the self-aligning torque in practice. However, the corresponding
effects are sufficiently smaller than those caused by the front tyre cornering stiffness
coefficient and the steering ratio . In this paper, therefore, we only consider the
effects of the variable parameters and on the steering performance of the SbW
system.
2) Motor Torque Pulsation Disturbances: In this study, the front-wheel motor is a
PMAC motor. Flux harmonics and DC current offsets are considered to be the two main
sources of the torque pulsation disturbances . The detailed descriptions can be found
Sub-chapter 2.5.2.
Given by the expression of in (2.28) and (2.30), we obtain the nominal value of
the torque pulsation disturbance as follows
( ) ( )
√ ( )√(
) ( ) (5.24)
where and are nominal values of and , and are
nominal values of and , respectively.
In addition, the modelling uncertainty in the torque pulsation disturbances in
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
139
(5.17) can be described as follows:
( ) ( )
√ ( ) (5.25)
where and represent the uncertainties in the 6th and 12th harmonic torque
amplitudes, respectively, and denotes the uncertainties in the dc current offsets.
5.2.3 Bounded Property of System Lumped Uncertainty
As seen from (5.19), the lumped uncertainty contains the acceleration signal of the
front steer angle, which leads to the difficulty in determining the upper bound of the
lumped uncertainty. Thus, it is essential that the upper bound of the lumped uncertainty
can be represented using only the position and velocity signals. The bound property of
lumped uncertainty will be discussed in the next subsection.
For the further analysis, three assumptions are made as follows [101].
A.5.1: The moment of inertia and viscous damping of the SbW system model and in
(5.12) are upper bounded by the two positive constants and , respectively, and are
given by:
| | (5.26)
| | (5.27)
A.5.2: Based on the above analysis of the disturbances, the disturbances are also upper
bounded by the following:
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
140
| | | | (5.28)
| | | | (5.29)
| | | | (5.30)
where ( ) are positive constants.
A.5.3: By considering the SbW system model in (5.12), the proposed control system
adopts the following polynomial-type of controller that is bounded by the following
function:
| | | | | | (5.31)
where , , and are positive constants.
Under the assumptions (A.5.1)-(A.5.3), the lumped uncertainty in (5.19) is assumed
to be bounded as follows:
| | (5.32)
where
| | | | (5.33)
where , , and are designed positive constants. The corresponding proof is given
in Appendix A.
Remark 5.4: As shown in the above discussion, the bounded property of the lumped
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
141
uncertainty is significantly related to the control input, i.e., if the acceleration signal is
not included in the control input, the system lumped uncertainty can then be bounded by
a positive function using only the angular position and speed information. Only the
structures of the controller are required to be known in advance for determining the
form of the uncertainty bound. It should be noted that, only the three parameters , ,
and in the controller design should be properly chosen based on the detailed analysis
of the disturbances through trial and error method. However, the values of (
), , , and are not explicitly needed in the controller design and only play an
important role in the stability analysis.
Remark 5.5: It is worth noting that the steering-wheel angular acceleration is required
in the acquisition of the error dynamics of the closed-loop SbW system. However, it is
because high-frequency noise is always introduced in the derivatives of the measured
position signal by using the numerical differentiation method that the steering-wheel
angular acceleration should be avoided and substituted by its upper bound in the
controller design in practice. The upper bound of the steering-wheel angular
acceleration will be determined from the steering-wheel dynamical equation as below.
5.2.4 Bounded Property of Steering-Wheel Angular Acceleration
In this paper, the steering-wheel dynamics can be expressed as [193]:
(5.34)
where is the steering-wheel column moment of inertia, is the steering-wheel
column viscous friction coefficient, is the steering-wheel column stiffness coefficient,
is the steering-wheel angular position, represents the provided feedback torque
using the steering-wheel motor, and is the driver input torque.
The desired reference signal of the SbW system is given by:
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
142
(5.35)
where denotes the scale factor between the actual steering-wheel angle and the
actual front steer angle .
Then, the output tracking error is defined as
(5.36)
Re-arranging (5.34) with the help of (5.35), we obtain
(5.37)
Thus, in (5.37) can be bounded by the following positive function:
| | | | | | (5.38)
where are positive constants.
Further, using (5.36) in (5.38), we have
| | (| |) (| |)
(| | | |) (| | | |) (5.39)
where
| | | | | | | | (5.40)
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
143
Remark 5.6: It is seen from (5.40) that, the values of and are significantly related
to the mechanical parameters , , and , which can be determined through the
preliminary system identification of the steering-wheel. More importantly, the value of
largely depends on the maximum steering torque that can be found in the studies
of steering effort for a typical automobile in emergency manoeuvres. Compared with
the method that is upper bounded within a positive constant [193], the bounded
condition in (5.40) is more accurate and appropriate in the controller design.
This chapter aims at designing an RCS for the closed-loop SbW systems, such that
the front wheels are able to exactly track the steering-wheel reference angle with a
strong robustness against uncertainties.
5.3 Design of A Robust Control Scheme
In this section, three steps are considered in the SbW control system design. First, an
NFC is used to ensure the asymptotic convergence of the output tracking error for the
nominal model. Second, an SM compensator is introduced for eliminating the effects of
the lumped uncertainty such that the tracking error of the SbW system with large
uncertainties can have the asymptotic zero-convergence feature. Third, the REDs are
utilized to get the required time derivatives of the measured position signals.
Using the system model in (5.18), we obtain the error dynamics of the SbW model as
follows
(5.41)
where is the new bounded lumped uncertainty in the closed-loop error dynamics and
defined as
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
144
(5.42)
Then, without considering the new bounded lumped uncertainty in (5.42), we get the
following error dynamics for the nominal system in (5.20):
(5.43)
The nominal control is chosen of the following form:
( ) (5.44)
where and are the feedback gains designed later.
Then using (5.44) in (5.43), we obtain the following error dynamic equation for the
nominal system:
(
) (5.45)
It is worth noting that, the control gains and need to be properly selected such
that the characteristic polynomial in equation (5.45) is strictly Hurwitz, that is, the
polynomial roots lie in the open left-half of the complex plane. Then, the tracking error
will exponentially converge to zero.
Remark 5.7: It is observed from (5.42) that, the steering-wheel angular acceleration
is included in the new lumped uncertainty and thus, the use of the acceleration signal
in the nominal controller can be effectively avoided. Then, the effect of the new
lumped uncertainty will be eliminated by the following SM compensator design
using the bound information of the new lumped uncertainty, which is more practical in
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
145
engineering applications.
5.3.1 Controller Design for System with Uncertainty
Here, we consider the SM comensator design for the SbW system in (5.18). The control
input in system (5.18) is modified as:
(5.46)
where is the NFC in (5.44) and is the compensator for dealing with the influences
of the system lumped uncertainty.
By substituting (5.46) and (5.44) into (5.41), the closed-loop error dynamics can be
obtained as:
(5.47)
For the SM compensator design, the upper bound of is estimated as follows:
| | (5.48)
where
(5.49)
where and are given in (5.33) and (5.40), respectively.
Generally, for using the sliding mode technique in the compensator design, two steps
are required to be considered.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
146
First, a linear sliding surface is defined as follows:
(5.50)
where is a positive parameter. The asymptotic error convergence is ensured with the
sliding surface and the convergence rate is mainly determined by the value of .
Second, the corresponding control input should be designed such that the output
tracking error can be driven into the sliding manifold with finite-time convergence,
indicating that the following reaching condition is satisfied:
(5.51)
In this study, therefore, we consider the following reaching law [41, 130]:
( ) (5.52)
where and are two positive constants, and ( ) is the sign function defined in
(2.20).
Then, in terms of the compensator design and stability analysis of the proposed
control scheme in (5.46), we give the following theorem.
Theorem 5.1: Considering the closed-loop SbW system (5.18) with the error dynamics
(5.47) and the new lumped uncertainty bound (5.49), the tracking error will approach
zero asymptotically if the control law is designed such that
(5.53)
where is the NFC given by expression (5.44) and is the sliding mode compensator
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
147
given as follows:
( ) ( | || | |
| | |)
( ) (5.54)
where is the upper bound of given in (5.49), and are two designed
parameters as in (5.45).
Proof: Considering a Lyapunov function
and taking the time derivative of V,
we obtain
( )
(
)
(
)
(
)
(
)
( ) ( ) (
| || | |
| | |)
( ) [ ( )]
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
148
(
) | | |
| | |
| || || | | |
| |
| | |
| | | | | |
| | |
| || || | | || || | | | | || |
| |
| | | || |
| |
| |( | |)
| |
| | for | | (5.55)
Expression (5.55) ensures that the sliding variable s reaches the sliding manifold in
finite time [40]. Thus, the designed controller in (5.53) can constrain the error dynamics
of the closed-loop SbW system on the sliding mode surface in finite time, and the output
tracking error can then exponentially approach zero.
Remark 5.8: Here we would like to address that, and are the two essential
parameters affecting the convergence of the sliding surface. First, a larger results in a
faster reaching time; however, if is increased significantly, a high control input will
be required, which is always limited due to the steering actuator constraint [41]. Second,
for a given value of , not only a higher value of leads to a faster reaching time as
seen from (5.55), but also can be treated as one good strategy to offset the effect of
selecting a small value for the upper bound of lumped uncertainty ; however, too
large values of will result in more serious chattering. Thus, a compromise between
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
149
the reaching speed and the control input as well as chattering should be taken. Third, the
three parameters of the uncertainty bound , , and in determining in (5.33) and
the ones , , and in determining the steering-wheel angular acceleration bound in
(5.40) also affect the control performance. Thus, all these parameters should be properly
selected to achieve good tracking performance under actuator constraint in practical
applications.
Remark 5.9: It has been observed from (5.33) that the upper bound of significantly
depends on the value of that is largely determined by the modelled error in the tyre
self-aligning torque part | | . However, the bound information of | | is usually
adopted in the conventional sliding mode controller [182, 193], rather than the modelled
error of the tyre self-aligning torque part | | in this paper. Because the upper bound
of the modelled error in the self-aligning torque part | | is much smaller than the one
of the actual self-aligning torque, both the control amplitude and chattering can be
reduced. This is one of the significant merits for the proposed RCS in comparison with
the conventional sliding mode controller.
Remark 5.10: Due to the signum function sign(s) included in the SM compensator of
the proposed controller in (5.54), there exists the chattering in the control signal. This
chattering issue can be tackled by substituting the following boundary layer
compensator (BLC) for the SM compensator in (5.54):
( ) ( | || | |
| | |)
( ) (5.56)
where ( ) is the saturation function and defined as
( ) {
| |
( ) | | (5.57)
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
150
with the boundary layer thickness .
With the use of the BLC in the proposed RCS, the output tracking error cannot
converge to zero since the signum function is substituted for the saturation function
[183, 35, 40]. However, through the proper choice of the positive constant , the
tracking error can still be small enough to satisfy the tracking precision requirement in
practice.
5.3.2 A Robust Exact Differentiator
It is observed from the designed controller in (5.44) and (5.54) that the angular speeds
of the steering-wheel and the front wheels, that is, and , are required in the
practical implementation. However, in this paper, only the angular positions and
can be measured by the two angle sensors. Hence, the two angular speed signals must
be estimated based on the position measurements. Traditionally, the following
approximations to estimate and are adopted:
( ) ( ) ( )
(5.58)
( ) ( ) ( )
(5.59)
where is the sampling period.
However, due to the existence of the high-frequency noise, the outputs of the above
approximation algorithm cannot be accurate and inevitably lead to the chatters in the
control signal. Therefore, in this application, the so-called REDs are employed for
estimating the time derivatives of the measured signals due to the finite-time
reachability of the estimated derivatives to the exact ones and strong robustness
property with respect to the measurement noise [191, 197-201]. The basic idea of the
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
151
differentiator can be illustrated by the following simple example:
First, let the measured front steer angle be differentiated, and the second derivative
of have a known upper bound as follows:
| | (5.60)
To differentiate the measured front steer angle , we consider the following
auxiliary equation:
(5.61)
Consider the following sliding variable for the differentiator, which is the
difference between x and :
(5.62)
Thus, differentiating (5.62), we obtain:
(5.63)
Then, the control law is given by the so-called super-twisting algorithm:
( ) (5.64)
√ ( ) (5.65)
where is the output of the differentiator, and are two positive constants
satisfying the following sufficient conditions for the convergence of to :
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
152
(5.66)
√
(5.67)
As shown in [197], the sliding variable of the differentiator in finite time, such
that
(5.68)
(5.69)
In terms of the steering-wheel, another differentiator is also introduced to
approximate the steering-wheel angular speed, where the two positive constants are
and , and the associated bounded condition of the second derivative of steering-wheel
angle can be always found based on (5.40).
Remark 5.11: It is worth noting that, the design of the proposed controller and the
differentiators can be separated satisfying the separation principle owing to the finite-
time reachability of the estimated derivative values to the exact ones in the differentiator
[197-199]. Therefore, as long as the dynamics of the differentiator are sufficiently fast
for achieving the exact values of the derivatives of the measured signals, the dynamics
of the closed-loop SbW system can be stabilized based on Theorem 1. It should be
addressed that, and ( ) are strictly positive constants that determine the
differentiation accuracy and required to be properly chosen for ensuring the finite-time
convergence. In addition, because the inequalities in (5.66) and (5.67) are only
sufficient conditions, the parameters and ( ) in this paper are first initialized
by simulations and then adjusted experimentally through trial-and-error steps.
The full SbW system control diagram is summarized in Figure 5.1 and the steering
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
153
Figure 5.1: Full SbW system control diagram.
Figure 5.2: SbW experimental platform. (a) Steering-wheel subsystem. (b) Front-wheel subsystem.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
154
performance of the proposed RCS will be validated by experimental studies in the
following section.
5.4 Experimental Studies
To verify the efficacy and advantages of the proposed RCS, a series of experiments are
carried out to compare the proposed controller with other three controllers: the boundary
layer sliding mode (BL-SM) controller, the controller, and the NFC.
5.4.1 Experimental System Identification
The SbW experimental platform in Robotics and Mechatronics Lab at Swinburne
University of Technology is used the same as the Chapter 3 and 4, as shown in Figure
5.2. The nominal parameters of the PMAC are given in Table 3.2. The sampling period
is chosen as .
After setting up the whole system, the nominal values of the current offsets of phase a
and b as well as the motor harmonic torque at the certain time are measured or partly
collected from the manual [193]. In terms of the Coulomb friction present in the
steering system, it is easy to determine the value of Coulomb friction by recording the
input voltage when the two front wheels start changing.
For the sake of determining the nominal parameters of the SbW system, the closed-
loop system transfer function is identified from experimental frequency response data,
where the feedback gain is 1 [10]. Due to the fact that the Coulomb friction and motor
torque harmonics included in the system model, these nonlinearities need to be
compensated when the plant model is identified. In addition, the tyre forces are ignored
in the identification process. The frequency responses of the actual and identified model
are shown in Figure 5.3. It is shown that the corner frequency of the system is around
2.5 rad/s, and the identified model matches the measured model well in the frequency
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
155
Figure 5.3: Frequency responses of the SbW system model.
below 20 rad/s. The identified closed-loop transfer function of the SbW system is given
by:
( ) ( )
( )
(5.70)
Based on the preliminary experiments and the above transfer function, the nominal
values for the identified SbW system model in (5.12) are easily calculated and listed in
Table 5.1.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
156
Table 5.1: Nominal parameters of the SbW system model in equation (5.12).
Parameter Value
0.064
0.16
18
( ) 3.04
( ) 0.03
( ) 0.005
( ) 0.1
( ) -0.06
( ) 0.0791
( ) 0.15
( ) 0.2
12
5.4.2 Experimental Results
It has been seen from (2.36) that, the tyre dynamic model is approximately represented
by the linear region of the nonlinear self-aligning torque , where the tyre slip angle
is smaller than 4 degrees. However, to test the robustness of the proposed RCS, the
following voltage signal is added onto the front-wheel motor control input, which
models the nonlinear self-aligning torque including the tyre slip angle greater than 4
degrees for three different road conditions, that is, wet asphalt, snowy, and dry asphalt
roads, respectively [21]:
{
( )
( )
( )
(5.71)
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
157
where ( ) represents the tyre-dependent parameters in three different road
conditions, and chosen as , , and , respectively, and
is the ratio between the actual self-aligning torque and the input voltage of
the servo driver. Then, according to expression (5.22), the pre-determined nominal
parameter on the wet asphalt road is set as .
In order to eliminate the chattering in the closed-loop SbW system, the proposed
controller using the BLC in (5.56) is utilized in the experiments. The parameters of the
designed controller and the differentiators, and the uncertain bound parameters in (5.33)
and (5.40) are determined as follows:
(5.72a)
(5.72b)
, , (5.73a)
, , (5.73b)
For the sake of clear comparison, both the maximum and root mean square (RMS)
values for the tracking error are utilized as a performance evaluation index and defined
as [188, 158]:
(| |) (5.74a)
√(∑
( ))
(5.74b)
where n is the number of the iterations.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
158
Figure 5.4: Control performance of proposed controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
Figure 5.4(a) and (b) show the steering performance of the proposed RCS, while the
control input is depicted in Figure 5.4(c). It can be seen that the front wheels are able to
closely track the steering-wheel command during the whole period (35s). Although the
disturbance torque is suddenly changed at 15s and 25s, the satisfactory steering
performance can always be obtained, which implies that with the aid of the partially
known information, the proposed controller has the strong capability of dealing with the
effects of large system uncertainties and disturbances. Particularly, the closed-loop
system behaves very well during the last 10 seconds under the largest self-aligning
torque. Moreover, the proposed control signal is somewhat noisy but bounded without
any obvious chattering shown in Figure 5.4(c), which is applicable in real applications.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
159
Figure 5.5: Control performance of BL-SM controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
For comparison purpose, Figure 5.5(a)-Figure 5.5(c) show the steering performance
using the following BL-SM controller [193]:
( ) [ ( | |) | |
] (5.75)
where the sliding variable , the upper bounds of , and are
chosen as , , and , respectively, the
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
160
lower bound of is chosen as , the upper bound of is selected as
, the upper bounds of , , and in determining are selected as 550, 180,
and 980, respectively, the parameters for the upper bound of in (2.19) are
√( ) (
) , , and , and
the saturation function has the same form as (5.57) with the boundary layer thickness
.
It is observed that, although the steering performance with the BL-SM controller is
acceptable, it is not as superior as the one with the proposed controller. The main
reasons are: (i) In order to remove the significant chattering caused by the large pre-
determined bound values of the uncertain parameters as well as disturbances, the
saturation function with a large boundary layer thickness is used resulting in the
significantly degenerated tracking performance; (ii) Because the pre-determined
boundary layer thickness is too large and not updated for the wet asphalt and snowy
road conditions at first 15s and middle 15s, respectively, the steady state errors during
these two periods are accordingly increased. However, the partial system knowledge is
employed in the proposed control and the BLC with a small boundary layer thickness is
to only compensate the small lumped uncertainty of the SbW system. Thus, good
tracking performance can be better retained compared with the BL-SM controller. This
is one of the most noticeable superiorities over BL-SM controller. In addition, as
described in Remark 5.3, the control amplitude of the proposed RCS is indeed reduced
compared with that of the BL-SM controller. It further indicates that the strategy of
adopting the lumped uncertainty bound can be effectively applied for the practical SbW
systems in modern vehicles.
For further comparison, the steering performances of using the following
controller [190] and the NFC in (5.44) are shown in Figure 5.6 and Figure 5.7,
respectively:
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
161
Figure 5.6: Control performance of controller. (a) Tracking performance. (b)
Tracking error. (c) Control torque.
( )
(5.76)
where the error vector [ ] , are the NFC parameters given in (54),
and ( )
is the optimal control gain of the control for minimizing the
effects of the bounded lumped uncertainty given in (5.42).
The performance index for the control is given by
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
162
Figure 5.7: Control performance of NFC. (a) Tracking performance. (b) Tracking error.
(c) Control torque.
∫ ‖ ( )‖
‖ ( )‖
∫ ‖ ‖
(5.77)
where Q and P are the weighting matrices, is a prescribed attenuation level as
. P can be found by solving the following Riccati matrix equality:
(5.78)
where [
], [ ] , and is a designed positive constant.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
163
Both the parameters are set as 0.1. The matrix Q is selected to be , and the
matrix P is found as
[
] (5.79)
and the control gain is
[ ] (5.80)
It is seen that the steering performance of the control in Figure 5.6 is better than
one of the NFC in Figure 5.7, while both are not good as the proposed control and the
BL-SM controller. Even though the control behaves certain robustness due to the
locally optimized control parameters, the strong robustness property cannot be
maintained under the large uncertainties and varying road conditions and thus, the
tracking performance has greatly deteriorated. In terms of the NFC, it is observed that
good tracking performance is obtained on wet asphalt road during the first 15s due to
the appropriate pre-determined tyre-dependent parameter. However, it is because the
pre-determined tyre-dependent parameter is no longer valid when the road surface
changes to snowy and dry asphalt roads at middle and last 15s that the tracking
performance and robustness have degenerated significantly. Thus, both controllers
cannot eliminate the effects of the varying road conditions on the steering performances.
In addition, the performance comparisons of the proposed controller, BL-SM and
controllers, and NFC during the period (5s-35s) are listed in Table 5.2. It is seen that, by
using the partial system knowledge, the proposed controller not only achieves best
steering performance with the results that the both the maximum and RMS values for
the tracking error are much smaller than those of other three types of controllers, but
also exhibits a strong robustness property against large system uncertainties and varying
road conditions.
Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics
164
Table 5.2: Performance comparisons of controllers in Chapter 5.
Max. error (rad)
RMS error (rad)
Proposed controller 0.0409 0.0139 BL-SM controller 0.1134 0.0541 controller 0.1334 0.0676
NFC 0.2209 0.0802
5.5 Conclusion
This chapter has proposed an RCS for SbW systems with partially known dynamics
against uncertainties and varying road conditions. The major contributions of this study
are that the SbW system has been treated as a partially known system with an unknown
lumped uncertainty and the bounded property of the lumped uncertainty has been
investigated in detail, which can greatly simplify the controller design of the SbW
system in practice. An NFC has been used to stabilize the nominal model and an SM
compensator using the small lumped uncertainty bound has been introduced for
eliminating the effects of the lumped uncertainty. It has been seen that the SbW system
with the developed RCS exhibits a strong robust steering performance and ensures that
the front steer angle can closely track the steering-wheel angle. In addition, the robust
exact differentiators have been utilized effectively to eliminate the need of
measurements for angular speeds. The experimental results have verified the excellent
robustness and steering performance of the developed RCS. The future research work to
design the robust terminal sliding mode control with a neural network uncertainty
estimator for SbW systems is under the authors’ investigation.
Performance
Controller
Chapter 6 Conclusions and Future Work
165
Chapter 6
Conclusions and Future Work
In this chapter, the major contributions of the analysis and evaluation of the proposed
robust SbW control systems are summarized as well as the key conclusions from this
thesis. In addition, a few open topics and possible future research directions based on
this thesis are discussed.
6.1 Summary of Contributions
HIS thesis investigates robust control methodologies for SbW systems with
uncertain dynamics. In particular, we have studied the mathematical modelling of
SbW systems which involves the dynamics from the steering motor to the steered front
wheels. Given the obtained system model, we have developed three different robust
steering control approaches based on the sliding mode technique for SbW systems
which are able to not only achieve excellent and robust steering performance against
system parameter variations and varying road conditions, but also simplify the control
structure and reduce implementation costs. To reiterate, the key contributions of the
thesis are summarized as follows.
In Chapter 3, we have further explored the mathematical modelling for an SbW
system and systematically derived a complete model for the SbW system which
includes the dynamics of the steering motor, the front wheel, and the tyre dynamics. It
T
Chapter 6 Conclusions and Future Work
166
has been shown that an SbW system, from the steering motor to the steered front wheels,
is equivalent to a second-order model. Therefore, given the obtained model, we have
designed a sliding mode steering controller using the bound information of the uncertain
system parameters and disturbances. It has been shown that the proposed sliding mode
controller is capable of not only efficiently alleviating the effects of uncertain system
parameters and varying road conditions as well as motor torque pulsation disturbances,
but also providing excellent steering performance.
In Chapter 4, a nonsingular terminal sliding mode steering control scheme has been
developed in order to achieve finite time convergence characteristic and stronger
robustness with regard to system parameter uncertainties and varying road conditions.
The proposed controller is able to ensure that the steering angle tracking error not only
reaches the terminal sliding mode in finite time, but also converges to zero in finite time.
Different driving conditions are performed in the experiments to show a faster and
higher precision tracking performance and stronger robustness feature of the proposed
control scheme than several comparative control systems.
In Chapter 5, a novel robust control scheme for SbW systems with partially known
dynamics has been proposed. It has been shown that an SbW is treated as a partially
known system with an unknown part. Based on these two parts, a nominal feedback
controller has been designed to stabilize the SbW nominal model and a sliding mode
compensator has been introduced to cancel the effects of the unknown parts of the SbW
system with the aid of the obtained lumped uncertainty bound. In addition, inspired by
the control of robotic manipulators, the upper bound of the lumped uncertainty has been
derived from only three predetermined parameters. The proposed control system
behaves with a strong robustness against large system uncertainties and ensures the
front wheel steering angle to closely track the hand-wheel command. The superior
steering performance of the proposed robust control is supported by the comparative
experimental results.
Chapter 6 Conclusions and Future Work
167
6.2 Future Research
In addition to the contributions, the work of this thesis also interestingly sparks several
ideas that could constitute the basis of future research.
6.2.1 Sliding Mode-based Adaptive Control for SbW Systems
As can be seen from our proposed robust control schemes for SbW systems, the bound
information of the unknown system parameters and disturbances is required in the
controller designs to deal with the effects of the system uncertainties. However, the
bounds are difficult to obtain in real applications. If the bounds are selected too large,
serious chattering will occur, which will result in wear and tear of the mechanical
system structure and also excite undesired system dynamics. If the bounds are too small,
the closed-loop system stability may not be satisfied. Thus, we need a simple bound
estimator for estimating the bounds of system parameters and disturbances. One way of
achieving this is to formulate a sliding mode-based adaptive controller in which a set of
parameter and disturbance adaptive laws are used to adaptively adjust the controller
parameters in the sense that the output tracking error can asymptotically converge to
zero and the closed-loop system behaves with a strong robustness with respect to
parameter uncertainties and disturbances. This control method has played a very
important role in the control of electric motor and robotic manipulators in recent years
[152-156].
In addition, it could also be extended to neural-network-based control systems where
different kinds of neural networks (RBF neural network, recurrent Hermite neural
network and so on) can be used to adaptively learn the bounds of uncertain dynamics in
a compact set for the purpose of facilitating adaptive control gain adjustment. Readers
with great interest in this research topic can refer to some practical applications in the
literature [157-164].
Chapter 6 Conclusions and Future Work
168
6.2.2 Sampled Data Systems
All the proposed control algorithms in this thesis are designed for continuous-time SbW
systems. However, in practical situations, the controllers are normally implemented in
digital electronics owing to the increasing affordability of microprocessor hardware.
Therefore, it is necessary to carry out research investigations into the discrete-times
sliding mode control and terminal sliding mode control for SbW systems with uncertain
dynamics.
6.2.3 Observer design for SMC based-SbW systems
As can be seen from the disturbance analysis in Chapter 2, the most significant
disturbance torque in the SbW systems is the tyre self-aligning torque. In the
conventional control systems, disturbance observer designs have been used to
compensate for the effect of the total aligning torque, such as the linear Luenberger
disturbance observer [26, 29], nonlinear observer[28, 82]. However, the main drawback
of the above methods is that they require additional instrumentation. In terms of the
SMC systems, the control gain has to be chosen as a high value when meeting large
road disturbances, which results in serious chattering. Although the chattering can be
reduced by the use of the boundary layer technique, there is always a compromise
between the chattering and the tracking performance as well as robustness. Therefore, it
is encouraging to introduce a disturbance observer for SMC to not only alleviate the
chattering problem but also retain excellent control performance. The idea is to
construct the control law by combining the SMC feedback with the disturbance
estimation based-feedforward compensation directly. With the aid of the feedforward
compensation for disturbances, control gain can be accordingly selected smaller
resulting in the reduction of the chattering. In addition, compared with the conventional
observer techniques like the Luenberger observer, it is suggested that the sliding mode
observer and TSM observer can be better applied to SMC systems due to the superior
Chapter 6 Conclusions and Future Work
169
advantages of only using bound information and strong robustness property against
system uncertainties.
On the other hand, it should be pointed out that when the vehicle experiences varying
road conditions, parameter variations and unpredicted changes of the tyre self-aligning
torque often occur, which will affect the performance of the mechanical components of
SbW systems in terms of achieving high accuracy, a high response speed, and high
efficiency. Hence, it is necessary to make use of an observer technique particularly the
sliding mode and TSM observers for estimating the mechanical parameters including
the moment of inertia, damping coefficients of the steering motor and steered front
wheels. The real-time estimations can then be used in the corresponding fault diagnosis
of SbW systems. For instance, when a tyre is leaking causing variations of tyre pressure
and the mechanical parameters of the SbW systems, it is an innovative approach to
develop an integrated diagnosis mechanism based on the real-time estimation of all
these mechanical parameters such that the potential fault of an SbW system can be
effectively predicted in advance.
6.2.4 Vehicle Stability Control for SbW equipped vehicles
It is well known that vehicle stability control (VSC) systems, also referred to as yaw
stability control systems, enable vehicles to avoid spinning and drifting out. For
enhancing safety in critical driving conditions, vehicles equipped with SbW systems as
one of the typical VSC systems are receiving considerable attention from the
automotive industry and researchers in terms of improving handling performance and
stability [10, 27, 29, 192, 202-205]. It is because the front wheel steering angle serves as
an input in the vehicle yaw motion dynamical equations that steering control plays an
essential role in the vehicle stability control. Once the steering system is controlled
properly through the use of the proposed robust controllers mentioned in the previous
chapters, it can be further extended to consider the whole vehicle stability control.
Therefore, it is exciting to combine steering control with vehicle stability control
Chapter 6 Conclusions and Future Work
170
together in the SbW equipped vehicles. By using this approach, not only can the
steering angle be controlled well to follow the reference command, but also good yaw
rate tracking performance and a small sideslip angle are obtained. Similarly, by using
sliding mode control technique, it is not necessary to know the real cornering stiffness
of the front and rear tyres and only their upper bounds are needed to achieve robust
control performance. It will be greatly helpful to achieve superior VSC performance in
different road conditions and driving environment.
Appendix A
171
Appendix A
Proof of Bounded Property of Lumped Uncertainty in (5.32) and (5.33)
Based on the equation in (5.18), the acceleration term is obtained as:
( )
(A.1)
Using the expression (A.1) in (5.19), we obtain
[ ( )
]
( )
(A.2)
Then, collecting in the left side, we get
( )
Appendix A
172
( )
( ) (A.3)
Thus
| | |
| | | |
| | | |
| (| |
| | | |) |
| (| | | | | |) (A.4)
Considering assumptions A.5.1 and A.5.2, we have the following inequalities:
|
| (A.5)
|
| (A.6)
|
| (A.7)
| | | | | | | | | | (A.8)
| | | | | | | | | | (A.9)
where ( ) are positive constants.
Using expressions (A.5)-(A.9) in (A.4), we have
Appendix A
173
| | | | | | ( | | | |)
( | | | |)
| | ( ) ( )| |
( )| | (A.10)
It is apparently seen from (A.10) that, the upper bound of the lumped uncertainty is
related to the form of the designed controller. If the control input satisfies A.5.3 in (5.31)
without containing the acceleration signal, then the expression in (A.10) becomes
| | ( | | | |) ( )
( )| | ( )| |
( )| |
( )| | (A.11)
Then, we directly obtain expressions (5.32) and (5.33) from (A.11), where
(A.12)
(A.13)
(A.14)
This completes the proof.
Appendix A
174
Author’s Publications
175
Author’s Publications
Peer Reviewed Journal Papers
[1] H. Wang, H. Kong, Z. Man, D. M. Tuan, Z. Cao, and W. Shen, “Sliding mode
control for Steer-by-Wire systems with AC motors in road vehicles,” IEEE
Transactions on Industrial Electronics, vol. 61, no. 3, pp.1596-1611, March 2014.
[2] D. M. Tuan, Z. Man, C. Zhang, H. Wang, and F. Tay, “Robust sliding mode
learning control for Steer-by-Wire systems,” accepted by IEEE Transactions on
Vehicular Technology, 2013.
[3] H. Wang, Z. Man, W. Shen, and D. M. Tuan, “Robust control for Steer-by-Wire
systems with partially known dynamics,” IEEE Transactions on Industrial
Informatics, 2013, under review.
[4] D. M. Tuan, Z. Man, C. Zhang, J. Zheng, and H. Wang, “Robust sliding mode
learning control for causal nonminimum phase nonlinear systems,” Asian Journal
of Control, 2013, under review.
[5] D. M. Tuan, Z. Man, C. Zhang, J. Jin, and H. Wang, “Robust sliding mode
learning control for uncertain discrete-time MIMO systems,” IET Control Theory
and Applications, 2013, under review.
Author’s Publications
176
[6] H. Wang, Z. Man, W. Shen, J. Zheng, J. Jin, and D. M. Tuan, “Novel nonsingular
terminal sliding mode control for Steer-by-Wire systems with uncertain dynamics,”
IEEE Transactions on Mechatronics, 2013, to be submitted for publication.
[7] H. Kong, X. Zhang, W. Bao, and H. Wang, “The application of granular
computing in electric vehicle fault diagnosis,” Australian Journal of Electrical &
Electronics Engineering, 2013, under review.
Conference Publications
[8] H. Wang, Z. Man, H. Kong, and W. Shen, “Terminal sliding mode control for
steer-by-wire system in electric vehicles,” in Proceedings of the 7th IEEE
Conference on Industrial Electronics and Applications (ICIEA 2012), Singapore,
Jul 2012, pp. 919-924.
[9] H. Wang, Z. Man, W. Shen, and J. Zheng, “Robust sliding mode control for steer-
by-wire systems with AC motors in road vehicles,” in Proceedings of the 8th
IEEE Conference on Industrial Electronics and Applications (ICIEA 2013),
Melbourne, Australia, Jun 2013, pp. 674-679.
[10] F. Tay, Z. Man. J. Jin, S. Khoo, J. Zheng, and H. Wang, “Sliding mode based
learning control for interconnected systems,” in Proceedings of the 8th IEEE
Conference on Industrial Electronics and Applications (ICIEA 2013), Melbourne,
Australia, Jun 2013, pp. 816-821.
Bibliography
177
Bibliography
[1] H. Inagaki, K. Akuzawa, and M. Sato, “Yaw rate feedback braking force
distribution control with control-by-wire braking system,” in Proceedings
of the International symposium on Advanced Vehicle Control (AVEC),
Yokohama, Japan, 1992.
[2] E. A. Bretz, “By-wire cars turn the corner,” IEEE Spectrum, vol. 38, no. 4,
pp. 68-73, 2001.
[3] S. Anwar and B. Zheng, “An antilock-braking algorithm for an eddy-
current-based brake-by-wire system,” IEEE Transactions on Vehicular
Technology, vol. 56, no. 3, pp. 1100-1107, 2007.
[4] W. Xiang, P. C. Richardson, C. Zhao, and S. Mohammad, “Automobile
brake-by-wire control system design and analysis,” IEEE Transactions on
Vehicular Technology, vol. 57, no. 1, pp. 138-145, 2008.
[5] J. Tajima and N. Yuhara, “Effects of steering system characteristics on
control performance from the viewpoint of steer-by-wire system design,”
SAE Technical Paper Series 1999-01-0821, 1999.
[6] S. Horiuchi and N. Yuhara, “An analytical approach to the prediction of
handling qualities of vehicles with advanced steering control system using
Bibliography
178
multi-input driver model,” ASME Journal of Dynamics Systems,
Measurement, and Control, vol. 122, no. 3, pp.490-497, 2000.
[7] S. Mammar and D. Koenig, “Vehicle handling improvement by active
steering ,” Vehicle System Dynamics, vol. 38, no. 3, pp. 211-242, 2002.
[8] N. Yuhara and J. Tajima, “Advanced steering system adaptable to lateral
control task and driver’s intention,” Vehicle System Dynamics, vol. 36, no.
2-3, pp. 119-158, 2000.
[9] N. Yuhara, J. Tajima, S. Sano, and S. Takimoto, “Steer-by-wire-oriented
steering system design: Concept and examination,” Vehicle System
Dynamics, vol. 33, pp. 692-703, 2000.
[10] P. Yih and J. C. Gerdes, “Modification of vehicle handling characteristics
via Steer-by-Wire,” IEEE Transactions on Control Systems Technology,
vol. 13, no. 6, pp. 965-976, 2005.
[11] M. Bertoluzzo, G. Buja, and R. Menis, “Control schemes for Steer-by-
Wire systems,” IEEE Industrial Electronics Magazine, vol. 1, no. 1, pp.
20-27, 2007.
[12] M. Aga and A. Okada, “Analysis of vehicle stability control (VSC)’s
effectiveness from accident area,” in Proceedings of the 18th International
Technical Conference on the Enhanced Safety of Vehicles, Nagoya, Japan,
2003.
[13] A. Zanten, “Evolution of electronic control systems for improving the
vehicle dynamic behavior,” in Proceedings of the International
Symposium on Advanced Vehicle Control (AVEC), Hiroshima, Japan, pp.
7-15, 2002.
Bibliography
179
[14] J. W. Post, “Modelling, simulation and testing of automobile power
steering system for the evaluation of on-centre handling,” PhD.
dissertation, Graduate School Clemson Univ., Clemson, SC, 1995.
[15] Y. Xue and J. Watton, “Modelling of a hydraulic powering steering
system,” International Journal of Vehicle Design, vol. 38, no. 2, 2005.
[16] D. Ammon, M. Borner, and J. Rauh, “Simulation of the perceptible
feed-forward and feed-back properties of hydraulic power-steering
systems on the vehicle's handling behavior using simple physical
models,” Vehicle System Dynamics, vol. 44, Supplement, pp. 158-170,
2006.
[17] D. Peter and R. Gerhard, “Electric power steering-the first step on the
way to steer-by-wire,” in SAE International Congress and Exposition,
no. 1999-01-0401, Detroit, Michigan, USA, March 1999.
[18] J. H. Kim and J. B. Song, “Control logic for an electric power steering
system using assist motor,” Mechatronics, vol. 12, no. 3, pp. 447-459,
2000.
[19] A. T. Zaremba, M. K. Liubakka, and R. M. Stuntz, “Control and steering
feel issues in the design of an electric power steering system,” in
Proceedings of American Control Conference, Philadelphia, PA, pp.36-
40, 1998.
[20] M. Parmar and J. Y. Yung, “A sensorless optimal control system for an
automotive electric power assist steering system,” IEEE Transactions on
Industrial Electronics, vol. 51, no. 2, pp. 290-298, 2004.
Bibliography
180
[21] A. Baviskar, J. R. Wagner, and D. M. Dawson, “An adjustable steer-by-
wire haptic-interface tracking controller for ground vehicles,” IEEE
Transactions on Vehicular Technology, vol. 58, no. 2, pp. 546-554, 2009.
[22] S. W. OH, H. C. Chae, S. C. Yun, and C.S. Han, “The design of a
controller for the steer-by-wire system,” JSME International Journal,
vol. 47, no. 3, pp. 896-907, 2004.
[23] C. J. Kim, J. H. Jang, and S. K. Oh, “Development of a control
algorithm for a rack-actuating steer-by-wire system using road
information feedback,” Proc. International Mechanical Engineering,
Part D: Journal of Automobile Engineering, vol. 222 pp. 1559-1571,
2008.
[24] T. J. Park, C. S. Han, and S. H. Lee, “Development of the electronic
control unit for the rack-actuating steer-by-wire using the hardware-in-
the-loop simulation system,” Mechatronics, vol. 15, no. 8, pp. 899-918,
2005.
[25] P. Setlur, J. R. Wagner, D. M. Dawson, and D. Braganza, “A trajectory
tracking steer-by-wire control system for ground vehicles,” IEEE
Transactions on Vehicular Technology, vol. 55, no. 1, pp. 76-85, 2006.
[26] A. E. Cetin, M. A. Adli, and D. E. Barkana, “Implementation and
development of an adaptive steering-control system,” IEEE
Transactions on Vehicular Technology, vol. 59, no. 1, pp. 75-83, 2010.
[27] Y. Yamaguchi and T. Murakami, “Adaptive control for virtual steering
characteristics on electric vehicle using Steer-by-Wire system,” IEEE
Trans. Ind. Electron., vol. 56, no. 5, pp. 1585-1594, 2009.
[28] Y. H. J. Hsu, S. M. Laws, and J. C. Gerdes, “Estimation of Tire slip
Bibliography
181
angle and friction limits using steering torque,” IEEE Transactions on
Control Systems Technology, vol. 18, no. 4, pp. 896-907, 2010.
[29] H. Ohara and T. Murakami, “A stability control by active angle control
of front-wheel in a vehicle system,” IEEE Transactions on Industrial
Electronics, vol. 55, no. 3, pp. 1277-1285, 2008.
[30] S. V. Emelyanov, “Control of first order delay systems by means of an
astatic controller and nonlinear corrections,” Automation and Remote
Control, no. 8, pp. 983-991, 1959.
[31] V. I. Utkin, “Variable structure systems with sliding mode,” IEEE
Transactions on Automatic Control, vol. 22, no. 2, pp. 212-222, 1977.
[32] V. I. Utkin, Sliding Modes in Control and Optimization, Berlin,
Germany: Springer-Verlag, 1992.
[33] X. Yu and J.-X. Xu, Variable Structure Systems: Towards the 21st
Century, vol. 274, Lecture Notes in Control and Information Sciences.
Berlin, Germany: Springer-Verlag, 2002.
[34] J.-J. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall,
Englewood Cliffs, NJ, 1991.
[35] C. Edwards and S. Spurgeon, Sliding mode control: Theory and
applications. London: Taylor and Francis, 1998.
[36] K. D. Young, “A variable structure model following control design for
robotic manipulators,” IEEE Transactions on Automatic Control, vol. 33,
no. 5, pp. 556-561, 1988.
Bibliography
182
[37] C. Abdallah, D. Dawson, p. Dorato, and M. Jamshidi, “Survey of robust
control for rigid robots, ” IEEE Control Systems Magazine, vol. 11, no.
2, pp. 24-30, 1991.
[38] J. A. Burton and A. S. I. Zinober, “Continuous approximation of
variable structure control,” International Journal of System Science, vol.
17, no. 6, pp. 875-885, 1986.
[39] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer's guide
to sliding mode control,” IEEE Transactions on Control Systems
Technology, vol. 7, no. 3, pp. 328-342, 1999.
[40] H. K. Khalil and J. W. Grizzle, Nonlinear systems. Upper Saddle River:
Prentice hall, 2002.
[41] W. B. Gao and J. C. Hung, “Variable structure control of nonlinear
systems: a new approach,” IEEE Transactions on Industrial Electronics,
vol. 40, no. 1, pp.45-55, 1993.
[42] C. J. Fallaha, M. Saad, H. Y. Kanaan, and K. A. Haddad, “Sliding-mode
robot control with exponential reaching law,” IEEE Transactions on
Industrial Electronics, vol. 58, no. 2, pp. 600-610, 2011.
[43] X. Zhang, L. Sun, K. Zhao, and L. Sun, “Nonlinear speed control for
PMSM system using sliding-mode control and disturbance
compensation techniques,” IEEE Transactions on Industrial Electronics,
vol. 28, no. 3, pp. 1358-1365, 2013.
[44] Z. Man and X. Yu, “Terminal sliding mode control of MIMO linear
systems,” IEEE Transactions on Circuits and Systems I: Fundamental
Bibliography
183
Theory and Applications, vol. 44, no. 11, pp. 1065-1070, 1997.
[45] Y. Feng, X. Yu, and Z. Man, "Non-singular terminal sliding mode
control of rigid manipulators," Automatica, vol. 38, no. 12, pp. 2159-
2167, 2002.
[46] Y. Inaguma, K. Suzuki, and K. Haga, “An energy saving technique in an
electro- hydraulic power steering ehps system,” in SAE International
Congress and Exposition, no. 960934, Detroit, Michigan, USA, March
1996.
[47] T. Wong, “Hydraulic power steering system design and optimization
simulation,” in SAE 2001 World Congress, no. 2001-01-0479, Detroit,
Michigan, USA, March 2001.
[48] A. B. Proca and A. Keyhani, “Identification of power steering system
dynamic models,” Mechatronics, vol. 8, no. 3, pp. 255-270, 1998.
[49] V. Kokotovic, J. Grabowski, V. Amin, and J. Lee, “Electro hydraulic
power steering system,” in SAE International Congress and Exposition,
no. 1999-01-0404, Detroit, Michigan, USA, March 1999.
[50] J. Hur, “Characteristics analysis of interior permanent-magnet
synchronous motor in electrohydraulic power steering systems,” IEEE
Transactions on Industrial Electronics vol. 55, no. 6, pp. 2316-2323,
2008.
[51] M. Rosth, “Hydraulic power steering system design in road vehicles,”
PhD. dissertation, Department of Mechanical Engineering, Linkoping
Uni., Linkoping, Sweden, 2007.
Bibliography
184
[52] A. W. Burton, “Innovation drivers for electric power-assisted steering,”
IEEE Control Systems Magazine, vol. 23, no. 6, pp. 30-39, 2003.
[53] X. Chen, T. Yang, X. Chen, and K. Zhou, “A generic model-based
advanced control of electric power-assisted steering systems,” IEEE
Transactions on Control Systems Technology, vol. 16, no. 6, pp. 1289-
1300, 2008.
[54] Y. G. Liao and H. I. Du, “Modelling and analysis of electric power
steering system and its effect on vehicle dynamic behaviour,”
International Journal of Vehicle Autonomous. Systems, vol. 1, no. 2, pp.
153–166, 2003.
[55] P. Zhao, X. Lin, J. Chen, and J. Men, “Parametric design and application
of steering characteristic curve in control for electric power steering,”
Mechatronics, vol. 19, no. 6, pp. 905-911, 2009.
[56] A. Marouf, M. Djemai, C. Sentouh, and P. Pudlo, “Sensorless control of
electric power assisted steering system,” in Proceedings of 20th
Mediterranean Conference on Control & Automation, Barcelona, pp.
909-914, 2012.
[57] A. Marouf, M. Djemai, C. Sentouh, and P. Pudlo, “A new control
strategy of an electric-power-assisted steering system,” IEEE
Transactions on Vehicular Technology, vol. 61, no. 8, pp. 3574-3589,
2012.
[58] P. Yih, “Steer-by-Wire: implications for vehicle handling and safety,”
PhD dissertation, Graduate School Stanford University, CA, 2005.
Bibliography
185
[59] Y. Marumo and M. Nagai, “Steering control of motorcycles using steer-
by-wire system,” Vehicle System Dynamics, vol. 45, no. 9, pp. 445-458,
2007.
[60] Y. Marumo and N. Katagiri, “Control effects of steer-by-wire system for
motorcycles on lane-keeping performance,” Vehicle System Dynamics,
vol. 49, no. 8, pp. 1283-1298, 2011.
[61] C. D. Gadda, S. M. Laws, and J. C. Gerdes, “Generating diagnostic
residuals for Steer-by-Wire vehicle,” IEEE Transactions Control
Systems Technology, vol. 15, no. 3, pp. 529-540, 2007.
[62] R. Kazemi and A. A. Janbakhsh, “Nonlinear adaptive sliding mode
control for vehicle handling improvement via steer-by-wire,”
International Journal of Automotive Technology, vol. 11, no. 3, pp. 345-
354, 2010.
[63] Y. Fujimoto, “Robust servo-system based on two-degree-of-freedom
control with sliding mode,” IEEE Transactions on Industrial Electronics,
vol. 42, no. 3, pp. 272-280, 1995.
[64] M. Bertoluzzo, G. Buja, R. Menis, and G. Sulligoi, “An approach to
steer-by-wire system design,” in Proceedings of IEEE Conference on
Industrial Technology (ICIT) 2005, Hong Kong, pp. 443-447, 2005.
[65] A. Liu and S. Chang, “Force feedback in a stationary driving simulator,”
in Proceedings of the IEEE International Conference on Systems, Man,
and Cybernetics, vol. 2, pp. 1711-1716, 1995.
[66] R. S. Sharp and R. Granger, “On car steering torque at parking speeds,”
Bibliography
186
International Mechanical Engineering, Journal of Automobile
Engineering, vol. 217, pp. 87-86, 2003.
[67] M. Segawa, S. Kimura, T. Kada, and S. Nakano, “A study of reactive
torque control for steer by wire system,” in Proceedings of the
International Symposium on Advanced Vehicle Control (AVEC),
Hiroshima, pp. 653-657, 2002.
[68] E. Ueda, E. Inoue, Y. Saki, M. Hasegawa, H. Takai, and S. Kimoto,
“The development of detailed steering model for on-centre handling
simulation,” in Proceedings of the International Symposium on
Advanced Vehicle Control (AVEC), Hiroshima, pp. 658-662, 2002.
[69] P. Setlur, D. Dawson, J. Chen, and J. Wagner, “A nonlinear tracking
controller for a haptic interface steer-by-wire systems,” in proceedings
of the 41st IEEE Conference on Decision and Control, Las Vegas,
Nevada, pp. 3112-3117, 2002.
[70] C. Kim, J. Jang, S. Yu, S. Lee, C. Han, and J. K. Hedrick, “Development
of a control algorithm for a tie-rod-actuating steer-by-wire system,” in
Proceedings of International Mechanical Engineering Part D; Journal
of Automobile Engineering, vol. 222, no. 9, pp. 1534-1557, 2008.
[71] Self-study programme 225, “The electro-mechanical power steering
system,” VOLKSWAGEN AG.
[72] K. K. Shyu, C. K. Lai, Y. W. Tsai, and D. I. Yang, “ A newly robust
controller design for the position control of permanent-magnet
synchronous motor,” IEEE Transactions on Industrial Electronics,
vol.49, no. 3, pp. 558-565, 2002.
Bibliography
187
[73] Y. X. Su, C. H. Zheng, and B. Y. Duan, “Automatic disturbances
rejection controller for precise control of permanent-magnet
synchronous motors,” IEEE Transactions on Industrial Electronics, vol.
52, no. 3, pp. 814-823, 2005.
[74] I. C. Baik, K. H. Kim, and M. J. Youn, “Robust nonlinear speed control
of PM synchronous motor using boundary layer integral sliding mode
control technique,” IEEE Transactions on Control Systems Technology,
vol. 8, no. 1, pp. 47-54, 2000.
[75] W. C. Gan and L. Qiu, “Torque and velocity ripple elimination of AC
permanent magnet motor control systems using the internal model
principle,” IEEE Transactions on Mechatronics, vol. 9, no. 2, pp. 436-
447, 2004.
[76] J. X. Xu, S. K. Panda, Y. J. Pan, T. H. Lee, and B. H. Lam, “A modular
control scheme for PMSM speed control with pulsating torque
minimization,” IEEE Transactions on Industrial Electronics, vol. 51, no.
3, pp. 526-536, 2004.
[77] G. J. Wang, C. T. Fong, and K. J. Chang, “Neural-network-based self-
tuning PI controller for precise motion control of PMAC motors,” IEEE
Transactions on Industrial Electronics, vol. 48, no. 2, pp. 408-415, 2001.
[78] G. Ferretti, G. Magnani, and P. Rocco, “ Modeling, identification, and
compensation of pulsating torque in permanent magnet AC motors,”
IEEE Transactions on Industrial Electronics, vol. 45, no. 6, pp. 912-920,
1998.
[79] W. Qian, S. Panda, and J. Xu, “Torque ripple minimization in PM
Bibliography
188
synchronous motors using iterative learning control,” IEEE Transactions
on Power Electronics, vol. 19, no. 2, pp. 272-279, 2004.
[80] H. Liu and S. Li, “Speed control for PMSM servo system using
predictive functional control and extended state observer,” IEEE
Transactions on Industrial Electronics, vol. 59, no. 2, pp. 1171-1183,
2012.
[81] H. Fujimoto, N. Takahashi, A. Tsumasaka, and T. Noguchi, “Motion
control of electric vehicle based on cornering stiffness estimation with
yaw-moment observer,” in Proceedings of 9th IEEE International
Workshop on Advanced Motion Control, pp. 206-211, 2006.
[82] Y. H. J. Hsu, S. M. Laws, C. D. Gadda, and J. C. Gerdes, “A method to
estimate the friction coefficient and tire slip angle using steering torque,”
in Proceedings of ASME International Mechanical Engineering
Congress Exposition (IMECT), Chicago, 2006.
[83] H. B. Pacejka, Tire and vehicle dynamics, SAE, Warrendale, PA, 2002.
[84] R. Rajamani, Vehicle dyamics and control, Springer, 2006.
[85] S. Amberkar, D. Ambrosio, and B. T. T. Murray, “A system-safety
process for by-wire automotive systems,” in Proceedings of the SAE
International Congress, SAE paper 2000-07-1056, 2000.
[86] A. Baviskar, J. R. Wagner, D. M. Dawson, and P. Setlur, “Steer-by-wire
haptic interface tracking controller experimental demonstration,”
Clemson University , Clemson, SC, CRB Technical Report
CU/CRB/2/24/05/#1, February 2005. [Online]. Available:
Bibliography
189
http://www.ces.clemson.edu/ece/crb/publictn/tr.htm.
[87] K. Astrom and B. Wittenmark, “Adaptive Control,” Pearson Education,
1995.
[88] G. C. Goodwin and K. S. Sin, “Adaptive filtering, prediction and
control,” Englewood Cliffs, NJ: Printice-Hall, 1984.
[89] A. J. Koivo and T. H. Guo, “Adaptive linear controller for robotic
manupulators,” IEEE Transactions on Automatic Control, vol. AC-28,
no. 2, pp. 162-171, 1983.
[90] Y. A. Tsypkin, “Adaptation and learning in automatic systems,”
Academic Press, New York, 1971.
[91] A. Guez, J. Eilbert, and M. Kam, “Neuromorphic architectures for fast
adaptive robot control,” in Proceedings of International Conference on
Robotics and Automation, Philadelphia, PA, 1988.
[92] G. Tao, “Adaptive control of partially known systems,” IEEE
Transactions on Automatic Control, vol. 40, no. 10, pp. 1813-1818,
1995.
[93] C. Aissi and M. F. Chouikha, “An iterative learning method for partially
known dynamical systems,” in Proceedings of the 35th Midwest
Symposium on circuits and systems, Washington, DC, 1992.
[94] R. Mohammadi, “Adaptive control of partially known continuous-time
systems,” in Proceedings of the 2009 IEEE International Conference on
Systems, Man, and Cybernetics, San Antonio, TX, 2009.
Bibliography
190
[95] B. Brogliato and A. T. Neto, “Practical stabilization of a class of
nonlinear systems with partially known uncertainties,” Automatica, vol.
31, no. 1, pp.145-150, 1995.
[96] H. Yu and S. Z. Sabatto, “Real-time intelligent controllers for systems
with partially known dynamic models,” in Proceedings of the 35th
Southeastern Symposium on system theory, Morgantown, WV, 2003.
[97] G. N. Maliotis and F. L. Lewis, “Improved robust adaptive controller for
a class of partially known nonlinear systems,” in Proceedings of the
28th Conference on decision and control, Tampa, Florida, 1989.
[98] K. Y. Lim and M. Eslami, “Robust adaptive controller designs for robot
manipulator systems,” IEEE Journal on Robotics and Automation, vol.
RA-3, no. 1, pp. 54-66, 1987.
[99] Z. Man and M. Palaniswami, “Robust tracking control for rigid robotic
manipulators,” IEEE Transactions on Automatic Control, vol. 39, no. 1,
pp. 154-159, 1994.
[100] Z. Man and M. Palaniswami, “A robust tracking control scheme for
rigid robotic manipulators with uncertain dynamics,” Computer
Electrical Engineering, vol. 21, no. 3, pp.211-220, 1995.
[101] Z. Man and X. Yu, “Adaptive terminal sliding mode tracking control for
rigid robot manipulators with uncertain dynamics,” JSME International
Journal, vol. 40, no. 3, pp. 493-502, 1997.
[102] T. Fukao, A. Yamawaki, and N. Adachi, “Adaptive control of partially
known systems using backstepping application to design of active
Bibliography
191
suspension,” in Proceedings of the 37th Conference on decision and
control, Tampa, Florida, 1998.
[103] T. Fukao, A. Yamawaki, and N. Adachi, “Adaptive control of partially
known systems and application to active suspensions,” Asian Journal of
Control, vol. 4, no. 2, pp. 199-205, 2002.
[104] C. C. Wit and P. Lischinsky, “Adaptive friction compensation with
partially known dynamics friction model,” International Journal of
Adaptive Control and Signal Processing, vol. 11, no. 1, pp. 65-80, 1997.
[105] H. S. Yang, M. C. Berg, and B. Hong, “Tracking control of mechanical
systems with partially known friction model,” Transactions on Control,
Automation, and Systems Engineering, vol. 4, no. 4, 2002.
[106] P. C. Parks, “AM Lyapunov's stability theory-100 years on,” IMA
journal of Mathematical Control and Information, vol. 9, no. 4, pp. 275-
303, 1992.
[107] A.M. Lyapunov, The general problem of the stability of motion.
Translated by A. T. Fuller, London: Taylor & Francis, 1992.
[108] Z. Man, Robotics. Prentice–Hall, Pearson Education Asia Pte Ltd, 2005.
[109] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth
systems,” IEEE Transactions on Automatic Control, vol. 39, no. 9,
pp.1910-1914, 1994.
[110] S. Sastry, Nonlinear systems: analysis, stability, and control. Springer,
New York, 1999.
Bibliography
192
[111] A. C. King, J. Billingham, and S. R. Otto, Differential equations-linear,
nonlinear, ordinary, partial. Cambridge University Press, 2003.
[112] S. V. Emelyanov, “Automatic control systems with variable structure,”
Foreign Technology Div. Wright-Patterson AFB OHIO, 1970.
[113] E. A. Barbashin, V. A. Tabueva, and R. M. Eidinov, “On the stability of
variable structure control systems when sliding conditions are violated,”
Automatic Remote Control, no. 7, pp. 810-816, 1963.
[114] U. Itkis, Control systems of variable structure, 1976.
[115] V. Utkin, Sliding modes and their application in variable structure
systems. Imported Publications, Inc., 1978.
[116] D. R. Yoerger, J.-J.E. Slotine, “Robust trajectory control of underwater
vechicles,” IEEE Journal of Oceanic Engineering, vol. 10, no. 4, pp.
462-470, 1985.
[117] J.-J.E. Slotine, “The robust control of robot manipulators,” The
International Journal of Robotics Research, vol. 4, no. 2, pp. 49-64,
1985.
[118] J.-J.E. Slotine and J. A. Coetsee, “Adaptive sliding controller synthesis
for non-linear systems,” International Journal of Control, vol.43, no.6,
pp.1631-1651, 1986.
[119] G. Bartolini, A. Ferrara, E. Usai, and V. I. Utkin, “On multi-input
chattering-free second-order sliding mode control,” IEEE Transactions
on Automatic Control, vol. 45, no. 9, pp.1711-1717, 2000.
Bibliography
193
[120] I. Boiko, L. Fridman, and R. Iriarte, “Analysis of chattering in
continuous sliding mode control,” Proceedings of the 2005 American
Control Conference, vol. 4, pp. 2439-2444, Jun 2005.
[121] H. Lee and V. I. Utkin, “Chattering suppression methods in sliding
mode control systems,” Annual Reviews in Control, vol. 31, no. 2,
pp.179-188, 2007.
[122] M.-H. Park and K.-S. Kim, “Chattering reduction in the position control
of induction motor using the sliding mode,” IEEE Transactions on
Power Electronics, vol. 6, no. 3, pp.317-325, 1991.
[123] R.-J. Wai, “Sliding-Mode Control Using Adaptive Tuning Technique,”
IEEE Transactions on Industrial Electronics, vol. 54, no. 1, pp.586-594,
Feb 2007.
[124] X. Yu and R. B. Potts, “Analysis of discrete variable structure systems
with pseudo-sliding modes,” International journal of systems science,
vol.23, no.4, pp.503-516, 1992.
[125] R. B. Potts and X. Yu, “Discrete variable structure system with pseudo-
sliding mode,” Journal of Australian Mathematical Society, Series B-
Applied Mathematics, vol. 32, pp. 365-376, 1991.
[126] X. Yu, “Digital variable structure control with pseudo-sliding modes,”
In Variable Structure and Lyapunov Control, Springer Berlin
Heidelberg, pp. 133-159, 1994.
[127] J.Y. Hung, W. Gao, and J. C. Hung, “Variable structure control: a
survey,” IEEE Transactions on Industrial Electronics, vol. 40, no. 1, pp.
Bibliography
194
2-22, Feb 1993.
[128] W. B. Gao and M. Cheng, “Quality of variable structure control
systems,” Control of Decision, vol. 40, no. 4, pp. 1-7, 1989 (in Chinese).
[129] M. Cheng and W. B. Gao, “Hierarchical switching mode of variable
structure systems,” Acta Aeronautica Sinica, vol. 10, no. 3, pp. 126-133,
1989 (in Chinese).
[130] Z. Zhu, Y. Xia, and M. Fu, “Adaptive sliding mode control for attitude
stabilization with actuator saturation,” IEEE Transactions on Industrial
Electronics, vol. 58, no. 10, pp. 4898-4907, 2011.
[131] A. Wang, X. Jia, S. Dong, “A new exponential reaching law of sliding
mode control to improve performance of permanent magnet
synchronous motor,” IEEE Transactions on Magnetics, vol. 49, no. 5,
pp. 2409-2412, 2013.
[132] M. Asad, A. Bhatii, S. Iqbal, “A novel reaching law for smooth sliding
mode control using inverse hyperbolic function,” in proceedings of
International Conference on Emerging Technologies (ICET) 2012, pp.
1-6, 2012.
[133] Z. Man, A. P. Paplinski, and H. R. Wu, “A robust MIMO terminal
sliding mode control scheme for rigid robotic manipulators,” IEEE
Transactions on Automatic Control, vol. 39, no. 12, pp. 2464-2469,
1994.
[134] X. Yu and Z. Man, “Model reference adaptive control systems with
Bibliography
195
terminal sliding modes,” International Journal of Control, vol. 64, pp.
1165 - 1176, 1996.
[135] Z. Man and X. Yu, “Terminal sliding mode control of MIMO linear
systems,” in Proceedings of the 35th IEEE Decision and Control, 1996.
[136] Y. Wu, X. Yu, and Z. Man, “Terminal sliding mode control design for
uncertain dynamic systems,” Systems & Control Letters, vol. 34, pp.
281-287, 1998.
[137] X. Yu and Z. Man, “Multi-input uncertain linear systems with terminal
sliding-mode control,” Automatica, vol. 34, pp. 389-392, 1998.
[138] C. K. Lin, “Nonsingular terminal sliding mode control of robot
manipulators using fuzzy wavelet networks,” IEEE Transactions on
Fuzzy Systems, vol. 14, no. 6, 849-859, 2006.
[139] S. Y. Chen and F. J. Lin, “Robust nonsingular terminal sliding-mode
control for nonlinear magnetic bearing system,” IEEE Transactions on
Control System Technology, vol. 19, no. 3, pp. 636-643, 2011.
[140] J. D. Lee, S. Khoo, and Z. B. Wang, “DSP-based sliding-mode control
for electromagnetic-levitation precise-position system,” IEEE
Transactions on Industrial Informatics, vol. 9, no. 2, pp. 817-827, 2013.
[141] H. Komurcugil, “Non-singular terminal sliding-mode control of DC-DC
buck converters,” Control Engineering Practice, vol. 21, pp. 321-332,
2013.
[142] F. J. Lin, S. Y. Lee, and P. H. Chou, “Intelligent nonsingular terminal
Bibliography
196
sliding-mode control using MIMO elman neural network for piezo-
flexural nanopositioning stage, ” IEEE Transactions on Ultrasonics,
Ferroelectrics, and Frequency Control, vol. 59, no. 12, pp. 2716-2730,
2012.
[143] M. Jin, J. Lee, P. H. Chang, and C. Choi, “Practical nonsingular
terminal sliding-mode control of robot manipulators for high-accuracy
tracking control,” IEEE Transactions on Industrial Electronics, vol. 56,
no. 9, pp. 3593-3601, 2009.
[144] F. J. Lin, P. H. Chuou, C. S. Chen, and Y. S. Lin, “Three-degree-of-
freedom dynamic model-based intelligent nonsingular terminal sliding
mode control for a gantry position stage,” IEEE Transactions on Fuzzy
Systems, vol. 20, no. 5, 971-985, 2012.
[145] Y. Wang, X. Zhang, X. Yuan, and G. Liu, “Position-sensorless hybrid
sliding-mode control of electric vehicles with brushless DC motor,”
IEEE Transactions on Vehicular Technology, vol. 60, no. 2, pp. 421-432,
2011.
[146] K. Shyu, C. Lai, Y. Tsai, and D. Yang, “A newly robust controller
design for the position control of permanent-magnet synchronous motor,”
IEEE Transactions on Industrial Electronics, vol. 49, no. 3, pp. 558-565,
2002.
[147] J. Yang, S. Li, and X. Yu, “Sliding-mode control for systems with
mismatched uncertainties via a disturbance observer,” IEEE
Transactions on Industrial Electronics, vol. 60, no. 1, pp. 160-169, 2013.
[148] R. Pupadubsin, N. Chayopitak, D. Taylor, N. Nulek, S. Kachapornkul, P.
Bibliography
197
Jitkreeyarn, P. Somsiri, and K. Tungpimolrut, “Adaptive integral sliding-
mode position control of a coupled-phase linear variable reluctance
motor for high-precision applications,” IEEE Transactions on Industrial
Electronics, vol. 48, no. 4, pp. 1353-1363, 2012.
[149] C. Milosavljevic, B. Drazenovic, B. Veselic, “Discrete-time velocity
servo system design using sliding mode control approach with
disturbance compensation,” IEEE Transactions on Industrial
Informatics, vol. 9, no. 2, pp. 920-927, 2013.
[150] Q. Xu and Y. Li, “Model predictive discrete-time sliding mode control
of nanopositioning piezostage without modelling hysteresis,” IEEE
Transactions on Control Systems Technology, vol. 20, no. 4, pp. 983-
994, 2012.
[151] A. Izadian, L. A. Hornak, and P. Famouri, “Structure rotation and pull-
in voltage control of MENS lateral comb resonators under fault
conditions,” IEEE Transactions on Control Systems Technology, vol. 17,
no. 1, pp. 51-59, 2009.
[152] Y. Li and Q. Xu, “Adaptive sliding mode control with perturbation
estimation and PID sliding surface for motion tracking of a piezo-driven
micromanipulator,” IEEE Transactions on Control Systems Technology,
vol. 18, no. 4, pp. 798-810, 2010.
[153] Q. Hu, “Robust adaptive sliding mode attitude maneuvering and
vibration damping of three-axis-stabilized flexible spacecraft with
actuator saturation limits,” Nonlinear Dynamics, vol. 55, no. 4, pp. 301-
321, 2009.
Bibliography
198
[154] C. Yang, “Adaptive nonsingular terminal sliding mode control for
synchronization of identical oscillators,” Nonlinear Dynamics, vol.
69, no. 1-2, pp. 21-33, 2012.
[155] K. K. Tan, S. N. Huang, and T. H. Lee, “Robust adaptive numerical
compensation for friction and force ripple in permanent-magnet linear
motors,” IEEE Transactions on Magnetics, vol. 38, no. 1, pp. 221-228,
2002.
[156] R. J. Wai, “Adaptive sliding-mode control for induction servomotor
drive,” IEE Proceedings of Electric Power Applications, vol. 147, no. 6,
pp. 553-562, 2000.
[157] H. C. Liaw, B. Shirinzadeh, and J. Smith, “Robust neural network
motion tracking control of piezoelectric actuation systems for
micro/nanomanipulation,” IEEE Transactions on Neural Networks, vol.
20, no. 2, pp. 356-367, 2009.
[158] R. J. Lian, “Enhanced adaptive self-organizing fuzzy sliding-mode
controller for active suspension systems,” IEEE Transactions on
Industrial Electronics, vol. 60, no. 3, pp. 958-968, 2013.
[159] D. Xu, D. Zhao, J. Yi, and X. Tan, “Trajectory tracking control of
omnidirectional wheeled mobile manipulators: robust neural network-
based sliding mode approach,” IEEE Transactions on Systems, Man,
and Cybernetics-Part B: Cybernetics, vol. 39, no. 3, pp. 788-799, 2009.
[160] G. Debbache and N. Golea, “Neural network based adaptive sliding
mode control of uncertain nonlinear systems,” Journal of systems
Engineering and Electronics, vol. 23, no. 1, pp. 119-128, 2012.
Bibliography
199
[161] F. J. Lin, H. J. Shieh, P. K. Huang, and P. H. Shieh, “An adaptive
recurrent radial basis function network tracking controller for a two-
dimensional piezo-positioning stage,” IEEE Transactions on Ultrasonics,
Ferroelectronics, and Frequency Control, vol. 55, no. 1, pp. 183-198,
2008.
[162] S. Kumarawadu and T. T. Lee, “Neuroadaptive combined lateral and
longitudinal control of highway vehicles using RBF networks,” IEEE
Transactions on Intelligent Transportation Systems, vol. 7, no. 4, pp.
500-512, 2006.
[163] C. M. Lin and C. F. Hsu, “Neufal-network hybrid control for antilock
braking systems,” IEEE Transactions on Neural Networks, vol. 14, no.
2, pp. 351-359, 2003.
[164] F. J. Lin, L. T. Teng, C.Y. Chen, and C. K. Chang, “Robust RBFN
control for linear induction motor drive using FPGA,” IEEE
Transactions on Power Electronics, vol. 23, no.4, pp. 2170-2180, 2008.
[165] S. Laghrouche, F. Plestan, and A. Glumineau, “Higher order sliding
mode control based on integral sliding mode,” Automatica, vol. 43, no. 3,
pp. 531-537, 2007.
[166] A. Levant and L. Fridman. Robustness issues of 2-sliding mode control.
In A.Sabanovic, L.M. Fridman, and S. Spurgeon, editors, Variable
Structure Systems: from Principles to Implementation, chapter 6, pages
131–156. IEE, London, 2004.
[167] A. Pisano, A. Davila, L. Fridman, and E. Usai, “Cascade control of PM
DC drives via second-order sliding-mode technique,” IEEE
Bibliography
200
Transactions on Industrial Electronics, vol. 55, no. 11, pp. 3846-3854,
2008.
[168] A. Levant, “Higher-order sliding modes, differentiation and output-
feedback control,” International Journal of Control, vol. 76, no. 9-10,
pp. 924-941, 2010.
[169] M. Tanelli, C. Vecchio. M. Corno, A. Ferrara, and S. M. Savaresi,
“Traction control for ride-by-wire sport motorcycles: a second-order
sliding mode approach,” IEEE Transactions on Industrial Electronics,
vol. 56, no. 9, pp. 3347-3356, 2009.
[170] B. Beltran, T. A. Ali, and M. Benbouzid, “Higher-order sliding-mode
control of variable-speed wind turbines,” IEEE Transactions on
Industrial Electronics, vol. 56, no. 9, pp. 3314-3321, 2009.
[171] I. Boiko, L. Fridman, A. Pisano, and E. Usai, “Analysis of chattering in
systems with second-order sliding modes,” IEEE Transactions on
Industrial Electronics, vol. 52, no. 11, pp. 2085-2102, 2007.
[172] Z. Man, H. R. Wu, and M. Palaniswami, “An adaptive tracking
controller using neural networks for a class of nonlinear systems,” IEEE
Transactions on Neural Networks, vol. 9, no. 5, pp.947-955, 1998.
[173] V. I. Utkin and K. D. Young, “Methods for constructing discontinuity
planes in multidimensional variable-structure systems,” Automation
Remote Control, vol. 39, no. 10, pp. 1466-1470, 1978.
[174] V. I. Utkin, “Sliding mode control design principles and applications to
electric drives,” IEEE Transactions on Industrial Electronics, vol. 40,
Bibliography
201
no. 1, pp. 23-36, 1993.
[175] E. Kayacan and O. Kaynak, “Sliding mode control-based algorithm for
online leanring in type-2 fuzzy neural networks: application to velocity
control of an electro hydraulic servo system,” International Journal on
Adaptive Control and Signal Processing, vol. 26, no. 7, pp. 645-659,
2012.
[176] E. Kayacan, O. Cigdem, and O. Kaynak, “Sliding mode control
approach for online learning as applied to type-2 fuzzy neural networks
and its experimental evaluation,” IEEE Transactions on Industrial
Electronics, vol. 59, no. 9, pp. 3510-3520, 2012.
[177] Y. Lin, Y. Shi, and R. Burton, “Modeling and robust discrete-time
sliding-mode control design for a fluid power electrohydraulic actuator
(EHA) system,” IEEE Transactions on Mechatronics, vol. 18, no. 1, pp.
1-10, 2013.
[178] O. Kaynak, K. Erbatur, and M. Ertugrul, “The fusion of computationa-
lly intelligent methodologies and sliding-mode control: A survey,” IEEE
Transactions on Industrial Electronics, vol. 48, no. 1, pp. 4-17, 2001.
[179] X. Yu and O. Kaynak, “Sliding mode control with soft computing: A
survey,” IEEE Transactions on Industrial Electronics, vol. 56, no. 9, pp.
3275-3285, 2009.
[180] T. R. Oliveira, L. Hsu, and A. J. Peixoto, “Output-feedback global
tracking for unknown control direction plants with application to
extremum-seeking control,” Automatica, vol. 47, no. 9, pp. 2029-2038,
2011.
Bibliography
202
[181] M. Gopal, Control systems: principles and design, McGraw-Hill, New
Delhi, 2002.
[182] H. Wang, Z. Man, H. Kong, and W. Shen, “Terminal sliding mode
control for steer-by-wire system in electric vehicles,” in Proceedings of
ICIEA 2012, pp. 919-924, 2012.
[183] V. Utkin, Sliding mode control in electro-mechanical systems, Taylor &
Francis, 2009.
[184] J. Kim, T. Kim, B. Min, S. Hwang, and H. Kim, "Mode Control Strategy
for a Two-Mode Hybrid Electric Vehicle Using Electrically Variable
Transmission (EVT) and Fixed-Gear Mode," IEEE Transactions on
Vehicular Technology, vol. 60, no. 3, pp. 793-803, 2011.
[185] Y. Morita, K. Torii, N. Tsuchida, M. Iwasaki, H. Ukai, N. Matsui, T.
Hayashi, N. Ido, and H. Ishikawa, "Improvement of steering feel of
Electric Power Steering system with Variable Gear Transmission
System using decoupling control," 10th IEEE International Workshop
on Advanced Motion Control, pp. 417-422, 2008.
[186] Y. Yao, "Vehicle Steer-by-Wire System Control," SAE Technical Paper,
2006.
[187] G. Baffet, A. Charara, and D. Lechner, “Estimation of vehicle sideslip,
tire force and wheel cornering stiffness,” Control Engineering Practice
vol. 17, no. 11, pp. 1255-1264, 2009.
[188] W. F. Xie, “Sliding-mode-observer-based adaptive control for servo
actuator with friction,” IEEE Transactions on Industrial Electronics, vol.
Bibliography
203
54, no. 3, pp. 1517-1527, 2007.
[189] A. H. Brian, D. Pierre, and C. D. W. Carlos, “A survey of models,
analysis tools and compensation methods for the control of machines
with friction,” Automatica, vol. 30, no. 7, pp. 1083-1138, 1994.
[190] N. C. Shieh, “Robust output tracking control of a linear brushless DC
motor with time-varying disturbances,” IEE Proceedings of Electric
Power Applications, vol. 149, no. 1, pp. 39-45, 2002.
[191] M. Smaoui, X. Brun, and D. Thomasset, “High-order sliding mode for
an electropneumatic system: A robust differentiator-controller design,”
International Journal of Robust Nonlinear Control, vol. 18, no. 4-5, pp.
481-501, 2008.
[192] K. Nam, H. Fujimoto, and Y. Hori, “Lateral stability control of in-
wheel-motor-driven electric vehicles based on sideslip angle estimation
using lateral tire force sensors,” IEEE Transactions on Vehicular
Technology, vol. 61, no. 5, pp. 1972-1985, 2012.
[193] H. Wang, H. Kong, Z. Man, M. T. Do, W. Shen, and Z. Cao, “Sliding
mode control for steer-by-wire systems with AC motors in road
vehicles,” IEEE Transactions on Industrial Electronics, vol. 61, no. 3,
pp. 1596-1611, 2014.
[194] F. Zhou, D. Song, Q. Li, and B. Yuan. “A new intelligent technology of
steering-by-wire by variable structure control with sliding mode,” in
Proceedings of International Joint. Conference on Artificial Intelligence,
pp. 857-860, 2009.
Bibliography
204
[195] X. Yu, B. Wang, and X. Li, “Computer-controlled variable structure
systems: The state-of-art,” IEEE Transactions on Industrial Informatics,
vol. 8, no. 2, pp. 1517-1527, 2007.
[196] M. O. Efe, O. Kaynak, and B. M. Wilamowski, “Stable Training of
computationally intelligent systems by using variable structure systems
technique,” IEEE Transactions on Industrial Electronics, vol. 47, no. 2,
pp. 487-496, 2000.
[197] A. Levant, “Robust exact differentiation via sliding mode technique,”
Automatica, vol. 34, no. 3, pp. 379-384, 1998.
[198] A. Damiano, G. L. Gatto, I. Marongiu, and A. Pisano, “Second-order
sliding-mode control of DC drives,” IEEE Transactions on Industrial
Electronics, vol. 51, no. 2, pp. 364-373, 2004.
[199] A. Colbia-Vega, J. de Leon-Morales, L. Fridman, O. Salas-Pena, and M.
Mata-Jimenez, “Robust excitation control design using sliding-mode
technique for multimachine power systems,” Electric Power System
Research, vol. 78, no. 9, pp. 1627-1634, 2008.
[200] M. Reichhartinger and M. Horn, “Application of higher order sliding-
mode concepts to a throttle actuator for gasoline engines,” IEEE
Transactions on Industrial Electronics, vol. 56, no. 9, pp. 3322-3329,
2009.
[201] Q. Ahmed and A. I. Bhatti, “Estimating SI engine efficiencies and
parameters in second-order sliding modes,” IEEE Transactions on
Industrial Electronics, vol. 58, no. 10, pp. 4837-4846, 2011.
Bibliography
205
[202] S. Mammar and D. Koenig, “Vehicle handling improvement by active
steering,” Vehicle System Dynamics, vol. 38, no. 3, pp. 211-242, 2002.
[203] N. Yu, Yaw control enhancement for buses by active front steering, PhD
Dissertation, The Pennsylvania State University, 2007.
[204] J. Zhang, J. Kim, D. Xuan, and Y. Kim, “Design of active front steering
system with QFT control,” International Journal of Computer
Application Technology, vol. 41, no. 3/4, pp. 236-245, 2011.
[205] B. A. Guvenc, L. Guvenc, and S. Karaman, “Robust yaw stability
controller design and hardware-in-the-loop testing for a road vehicle,”
IEEE Transactions on Vehicular Technology, vol. 58, no. 2, pp.555-571,
2009.
[206] H. Wang, Z. Man, W. Shen, and J. Zheng, “Robust sliding mode control
for steer-by-wire systems with AC motors in road vehicles,” in
Proceedings of the 8th IEEE Conference on Industrial Electronics and
Applications (ICIEA 2013), Melbourne, Australia, pp. 674-679, Jun
2013.
[207] H. Wang, Z. Man, W. Shen, J. Zheng, J. Jin, and D. M. Tuan, “Novel
nonsingular terminal sliding mode control for Steer-by-Wire systems
with uncertain dynamics,” IEEE Transactions on Mechatronics, 2013, to
be submitted for publication.
[208] H. Wang, Z. Man, W. Shen, and D. M. Tuan, “Robust control for Steer-
by-Wire systems with partially known dynamics,” IEEE Transactions
on Industrial Informatics, 2013, under review.
Bibliography
206
[209] R. Pastorino, M. A. Naya, J. A. Perez, and J. Cuadrado, “Geared PM
coreless motor modeling for driver’s force feedback in steer-by-wire
systems,” Mechatronics, vol. 21, no. 6, pp. 1043-1054, 2011.
[210] B. H. Nguyen and J. H. Ryu, “Direct current measurement based steer-
by-wire systems for realistic driving feeling,” IEEE International
Symposium on Industrial Electronics, pp. 1023-1028, 2009.
[211] N. Bajcinca, R. Cortesao, and M. Hauschild, “Robust control for steer-
by-wire vehicles,” Autonomous Robots, vol. 19, no. 2, pp. 193-214,
2005.