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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

Robust control for steer-by-wire systems in road vehicles · Robust Control for Steer-by-Wire Systems in Road Vehicles . Submitted in total fulfilment of the requirements of the degree

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Page 1: Robust control for steer-by-wire systems in road vehicles · Robust Control for Steer-by-Wire Systems in Road Vehicles . Submitted in total fulfilment of the requirements of the degree

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

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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

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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.

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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

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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.

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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.

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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

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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

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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

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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

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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

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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.

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(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

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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

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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

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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

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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

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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

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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

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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

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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].

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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

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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.

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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

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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.

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Chapter 2 Background and Literature Review

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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.

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Chapter 2 Background and Literature Review

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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

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Chapter 2 Background and Literature Review

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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

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Chapter 2 Background and Literature Review

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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.

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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

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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-

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Chapter 2 Background and Literature Review

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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

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Chapter 2 Background and Literature Review

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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.

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Chapter 2 Background and Literature Review

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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

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Chapter 2 Background and Literature Review

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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:

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Chapter 2 Background and Literature Review

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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-

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Chapter 2 Background and Literature Review

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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

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Chapter 2 Background and Literature Review

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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)

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Chapter 2 Background and Literature Review

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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

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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-

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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

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Chapter 2 Background and Literature Review

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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

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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.

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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

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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

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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-

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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

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Figure 2.12: Side view of steering actuator assembly together with universal joint.

Figure 2.13: Block diagram of Steer-by-Wire system.

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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

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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

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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)

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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

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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:

( )

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Chapter 2 Background and Literature Review

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( )(

( ) ( ))

( ) (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

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Chapter 2 Background and Literature Review

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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

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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.

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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 .

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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

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( ) (

) (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

[ ]

[

]

[

]

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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.

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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

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Figure 2.18: Rack-actuating type of front-wheel system.

Figure 2.19: Tie-rod-actuating type of front-wheel system.

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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

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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-

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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

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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].

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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)

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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

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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 ( ) .

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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)

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( ) 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.

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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 :

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( ) ( ) (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)

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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

[ ( ) ( ) ]

{ ( ) ( )[ (‖ ‖ ( ) ) ( )]}

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( ) | | [ ( )

] ‖ ‖ ( ) | | [

( )

]

| |‖ ‖‖ ( )‖ | | [ ( )

] ‖ ‖ ( ) | | [

( )

] (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.

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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.

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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

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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)

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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]:

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( )

| | (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:

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( ) ( ) (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)

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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

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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)

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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:

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( ) ( ( )

⁄ ( ) ( )) (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

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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).

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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.

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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

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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

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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.

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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.

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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.

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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)

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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)

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(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

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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

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( )

| |

(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

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| | (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

[ ]

( )

{[

(

( ))]}

( )

[

( )

] ( )

( )

| |

[ ( | |) | |

]

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( )

(| |

) (| |

| |

)

(| |

) (| |

)

(| |

( )

) (

| || | )

| | (

) | |

(

) | |

| |

(

| |) | |

(

| |)

| | (

| |) | | (

) | |

| |

(

| |) 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)

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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(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

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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:

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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)

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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(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

.

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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

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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.

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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.

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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.

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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)

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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.

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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.

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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

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[

] (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 :

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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.

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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.

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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).

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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

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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)

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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-

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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

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Chapter 3 Sliding Mode Control for Steer-by-Wire Systems with AC Motors in Road Vehicles

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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.

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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

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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

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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

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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.

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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

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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

(

( )

(

) )

[

( )

(

)( )]

{

( )

(

)[

(

( )) ]}

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

(

) ( )

( )

(

)

| |

( )

(

)

[ ( ) | |

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

114

( )

( ⁄ ) ( )]

( )

(

)

( )

(

)

( )

(

)

( )

(

) ( )

( )

(

)

( )

(

)(| |

)

( )

(

)(| |

| |

)

( )

(

)(| |

)

( )

(

)(| |

( )

)

(

| || | )

( )

(

)(| |

)

( )

(

)| | (

| |)

( )

(

)| | (

| |

| |)

( )

(

)| | (

| |

)

( )

(

)| | (

| |

)

( )

(

)| | (

| |

) (

| || | | || |)

( )

(

)| | (

)

( )

(

)| |

(

) | |

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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

( )

| |

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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(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

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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)

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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.

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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.

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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.

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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.

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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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

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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

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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-

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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.

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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

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Chapter 4 Nonsingular Terminal Sliding Mode Control for Steer-by-Wire Systems with Uncertain Dynamics

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effectiveness of the proposed NTSM control scheme. Further work is to design a novel

sliding mode observer-based NTSM control scheme for SbW systems.

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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

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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

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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

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Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics

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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.

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Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics

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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

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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

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( ) (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)

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(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)

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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

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( )(

)

( )(

)

(

) ( ) (

) (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

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(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:

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| | | | (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

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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:

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(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)

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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

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(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

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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.

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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

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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

( )

(

)

(

)

(

)

(

)

( ) ( ) (

| || | |

| | |)

( ) [ ( )]

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(

) | | |

| | |

| || || | | |

| |

| | |

| | | | | |

| | |

| || || | | || || | | | | || |

| |

| | | || |

| |

| |( | |)

| |

| | 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

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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)

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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

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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 :

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(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

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Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics

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Figure 5.1: Full SbW system control diagram.

Figure 5.2: SbW experimental platform. (a) Steering-wheel subsystem. (b) Front-wheel subsystem.

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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

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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.

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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)

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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.

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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.

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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

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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:

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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

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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.

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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.

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Chapter 5 Robust Control for Steer-by-Wire Systems with Partially Unknown Dynamics

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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

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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

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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.

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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].

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Chapter 6 Conclusions and Future Work

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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

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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

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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.

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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

( )

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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

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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.

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Appendix A

174

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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.

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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.

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