Modelling and Control of Vehicle Suspension Control Systems

Modelling and Control of Vehicle

Suspension Control Systems

Jiangtao Cao

Institute of Industrial Research

University of Portsmouth

A thesis submitted for the degree of

Doctor of Philosophy

Yet to be decided

UoPlogo.eps

Acknowledgements

I wish to express my gratitude to my supervisor, Dr. Honghai Liu, for

his constant support, guidance and encouragement. I feel especially

blessed since he is not only my supervisor but also my friend.

Of equal importance are my thanks to Prof. Ping Li and Dr. David

J. Brown. This research was supported in part by PML Flightlink

Ltd., British Council and University of Portsmouth. While everyone

at QED workgroup was supportive of my work, I am particularly

pleased to thank Mr. Martin Boughtwood and Mr. Chris Hilton for

their generous support.

Many thanks to my colleagues at Institute of Industrial Research.

Special thanks go to Dr. Ian Morgan, Mr. Edward Smart, Mr. Zhao-

jie Ju and Mr. Medhi Khoury for sharing their knowledge and exper-

tise with me.

Throughout the past years as a graduate student, there have been

many joyful moments in life, the moments which have been made

more enjoyable by my friends. Of equal importance are my happy

memories of Piyush Goel, Josh Fahimi, Chee Seng Chan, Xin Wen,

Yang Wang, Jian Ma, Jie Ma, Yuhui Shao.

I wish to express thanks to all my family members for their constant

support, encouragement, understanding. I will be short on words to

express my gratitude to my parents, what they have done for me

over the years. They have never stopped loving, even in the hardest

moment. They have always been my biggest inspiration. This work

would not be possible without their love support.

Abstract

In this thesis, firstly, a state of the art on computational intelligence

approaches in active vehicle suspension control systems is surveyed.

With a focus on the problems raised in practical implementations by

their non-linear and uncertain properties, it explores existing methods

in fuzzy inference systems, neural networks, genetic algorithms and

their combination for suspension control issues. Exactly due to these

conclusions of literature review, a new half-vehicle suspension model

is built. A novel framework of type-2 fuzzy control system for vehicle

active suspension is proposed and its closed-loop stability with the

sufficient conditions is carried out.

From a comprehensive consideration of a real car, an improved half-

vehicle suspension model is built in this thesis. The proposed model

can not only describe the real coupling between front and rear vehi-

cle body, but also be convenient to integrate suspension system with

brake control and anti-roll control systems. Hence this model is of

benefit to design the following active suspension control system.

A novel adaptive fuzzy logic controller is designed for vehicle ac-

tive suspension system. The proposed method utilizes interval type-2

fuzzy membership functions to deal with not only non-linearity and

uncertainty caused by irregular road inputs and complex suspension

dynamics, but also the potential uncertainty of experts knowledge and

experience which occur in typical fuzzy logic methods. An adaptive

strategy with closed-loop feedback regulation is proposed to improve

the existed type-reduction methods of type-2 fuzzy system. Simula-

tions on quarter-vehicle and half-vehicle active suspension models are

studied to evaluate the proposed control system. In comparison with

passive and typical fuzzy methods, the proposed method can obtain

better control performance.

For the closed-loop stability analysis of proposed control system, with

the Takagi-Sugeno (T-S) fuzzy consequents, the Lyapunov stabiliza-

tion method is implemented to verify the closed-loop stability. The

sufficient stability conditions for proposed vehicle active suspension

control system are deduced.

To review above all, with the improved vehicle suspension model, the

adaptive fuzzy controller and its stability analysis, this thesis builds

a completely intelligent control system for vehicle active suspension

system. The simulation results demonstrate its efficiency and prac-

ticability. And it is convenient to be implemented for the industrial

applications.

Contents

List of Figures vi

List of Tables ix

1 Introduction 1

1.1 A Brief Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problems and Challenges . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Overview of Approaches and Contributions . . . . . . . . . . . . . 4

1.4 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature Review 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Active Suspension System Linear Models and Control . . . 12

2.2.1.1 Quarter Vehicle Active Suspension Model . . . . 12

2.2.1.2 Half Vehicle Active Suspension Model . . . . . . 16

2.2.2 Non-linear Model of Active Suspension Model . . . . . . . 21

2.3 Adaptive Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Adaptive Fuzzy Sliding Mode Control . . . . . . . . . . . . . . . . 25

2.4.1 Conventional Sliding Mode Control . . . . . . . . . . . . . 26

2.4.2 Fuzzy Sliding Mode Control System . . . . . . . . . . . . . 29

2.4.2.1 Alleviating SMC Chattering . . . . . . . . . . . . 29

2.4.2.2 Fuzzy Logic Complementary to SMC . . . . . . . 33

2.5 Adaptive Neural Network Control . . . . . . . . . . . . . . . . . . 35

2.6 Genetic Algorithms Based Adaptive Optimization and Control . . 37

iv

CONTENTS

2.7 Integrated Adaptive Control Methods . . . . . . . . . . . . . . . . 41

2.7.1 Adaptive Neuro-fuzzy Control . . . . . . . . . . . . . . . . 41

2.7.2 Adaptive Genetic-based Optimal Fuzzy Control . . . . . . 43

2.7.3 GA-ANNs Combined Control . . . . . . . . . . . . . . . . 45

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Improved Vehicle Active Suspension Model 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 A Rigid Tyre Model . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 The Improved Half-vehicle Active Suspension Model . . . . . . . . 52

3.3.1 The Linear Half-vehicle Model . . . . . . . . . . . . . . . . 54

3.3.2 Linear Model Performance Analysis . . . . . . . . . . . . . 56

3.3.3 The Non-linear Half-vehicle Model . . . . . . . . . . . . . 61

3.3.4 Non-linear Model Analysis . . . . . . . . . . . . . . . . . . 62

3.4 The Improved LQG Design . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1 The Improved LQG . . . . . . . . . . . . . . . . . . . . . . 66

3.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 67

3.4.2.1 Step Road Inputs . . . . . . . . . . . . . . . . . . 67

3.4.2.2 Random Road Inputs . . . . . . . . . . . . . . . 69

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Interval Type-2 Fuzzy Control System 74

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Interval Type-2 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . 77

4.2.1 The Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . 77

4.2.2 The Interval Type-2 Fuzzy System . . . . . . . . . . . . . 79

4.2.3 Type-reduction and Defuzzification Methods . . . . . . . . 81

4.3 The Adaptive Interval Type-2 FLC . . . . . . . . . . . . . . . . . 84

4.3.1 The Framework of Adaptive IT2 FLC . . . . . . . . . . . . 85

4.3.2 The LMS method . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.3 The PSO method . . . . . . . . . . . . . . . . . . . . . . . 88

4.4 Simulations on the Quarter-vehicle Model . . . . . . . . . . . . . 90

4.4.1 Adaptive IT2 FLC with the LMS method . . . . . . . . . 91

4.4.2 Adaptive IT2 FLC with the PSO method . . . . . . . . . . 100

v

CONTENTS

4.5 Simulations on the Half-vehicle Model . . . . . . . . . . . . . . . 105

4.5.1 The adaptive IT2 FLC with the LMS method . . . . . . . 107

4.5.2 The IT2 FLC with the PSO method . . . . . . . . . . . . 112

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Stability Analysis of Closed-loop Systems 119

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Typical T-S Fuzzy Control Systems and the Stability Conditions . 120

5.2.1 T-S Fuzzy Model and Control System . . . . . . . . . . . . 120

5.2.2 The Stability Conditions with Lyapunov Stability Theory . 122

5.3 The General Interval Type-2 T-S Fuzzy System . . . . . . . . . . 124

5.3.1 The Interval Type-2 T-S Fuzzy System . . . . . . . . . . . 124

5.3.2 The Interval Type-2 T-S Fuzzy Control System . . . . . . 126

5.4 Stability Analysis of the IT2 T-S Fuzzy Control System . . . . . . 127

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Conclusions 133

6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.1 The Improved Models . . . . . . . . . . . . . . . . . . . . 134

6.2.2 The Interval Type-2 Fuzzy Controller with LMS . . . . . . 134

6.2.3 The Interval Type-2 Fuzzy Controller with PSO . . . . . . 135

6.2.4 Closed-loop Stability Analysis . . . . . . . . . . . . . . . . 135

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3.1 Expansion of Type-2 Fuzzy Inference Engine . . . . . . . . 136

6.3.2 Relaxation on Stability Conditions . . . . . . . . . . . . . 136

6.3.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

References 153

A Publications 154

B Fuzzy Rules Table 156

vi

List of Figures

2.1 Six degree-of-freedom vehicle model (Nagai, 1993) . . . . . . . . 13

2.2 Two degree-of-freedom quarter-vehicle model . . . . . . . . . . . . 14

2.3 Half-vehicle suspension model . . . . . . . . . . . . . . . . . . . . 17

2.4 Non-linear properties of suspension system [wheel stroke(m) versus

suspension force(N)](Kim & Ro, 1998) . . . . . . . . . . . . . . . 22

2.5 The adaptive FLC scheme in Huang & Chao (2000) . . . . . . . . 25

2.6 Effects of parameters G and K (Kaynak, 1998) . . . . . . . . . . 30

2.7 Fuzzy adaptive sliding mode control scheme for Active Suspension

Control System (ASCS) in Chen et al. (1995) . . . . . . . . . . . 30

2.8 The fuzzy adaptive controller scheme in Zhang et al. (2007) . . . 32

2.9 The adaptive fuzzy sliding mode controller scheme in Huang & Lin

(2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.10 Scheme of the hydraulic active suspension system in Kucukdemiral

et al. (2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.11 The scheme of indirect adaptive control based on ANNs in Guo

et al. (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.12 The 5 degree-of-freedom half vehicle model employed in Baumal

et al. (1998) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.13 The force control scheme with skyhook damper, virtual damper

and road-following spring in Tsao & Chen (2001) . . . . . . . . . 40

2.14 The adaptive neural network fuzzy control system with time-delay

compensator in Dong et al. (2006) . . . . . . . . . . . . . . . . . . 42

2.15 The FNNC scheme in Dong et al. (2006) . . . . . . . . . . . . . . 42

2.16 The PBGA fuzzy control system in Nawa et al. (1999) . . . . . . 44

vii

LIST OF FIGURES

2.17 An example of the fuzzy system encoded in a chromosome in Nawa

et al. (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 The rigid tyre model . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 The half vehicle model . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 The pitch angles of a vehicle body with different Vcr . . . . . . . . 57

3.4 The accelerations of front vehicle body with different Vcr . . . . . 57

3.5 The accelerations of rear vehicle body with different Vcr . . . . . . 58

3.6 The front suspension travel with different Vcr . . . . . . . . . . . 59

3.7 The rear suspension travel with different Vcr . . . . . . . . . . . . 59

3.8 The front tyre dynamic loading with different Vcr . . . . . . . . . 60

3.9 The rear tyre dynamic loading with different Vcr . . . . . . . . . 60

3.10 The vertical accelerations of front vehicle body with non-linear model 63

3.11 The vertical accelerations of rear vehicle body with non-linear model 63

3.12 The pitch angles of vehicle body with non-linear model . . . . . . 64

3.13 The front suspension travel with non-linear model . . . . . . . . 64

3.14 The rear suspension travel with non-linear model . . . . . . . . . 65

3.15 The front tyre dynamic loading with non-linear model . . . . . . 65

3.16 The rear tyre dynamic loading with non-linear model . . . . . . . 66

3.17 Pitch angle comparison with typical LQG and different rolling ve-

locities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.18 Pitch angle comparison with different controller at Vcr= 35 m/s

(upper three lines) and 25 m/s (lower three lines) . . . . . . . . . 69

3.19 Pitch angle comparison with different controller and random road

input ( Vcr=35m/s) . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.20 Pitch angle comparison with different controller and random road

input (Vcr=25m/s) . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 The interval type-2 fuzzy membership functions . . . . . . . . . . 78

4.2 The interval type-2 fuzzy logic system . . . . . . . . . . . . . . . . 80

4.3 The framework of proposed IT2 fuzzy controller . . . . . . . . . . 85

4.4 The structure of adaptive IT2 FLC with LMS method . . . . . . 87

4.5 The structure of adaptive IT2 FLC with PSO method . . . . . . . 89

4.6 The interval type-2 fuzzy membership functions of three inputs . . 91

viii

LIST OF FIGURES

4.7 The membership functions of actuator force fa . . . . . . . . . . . 92

4.8 The frequency response of vehicle body acceleration zb . . . . . . 95

4.9 The frequency response of tyre dynamic load . . . . . . . . . . . . 95

4.10 The frequency response of vehicle body acceleration on B class

road surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.11 The frequency response of vehicle body acceleration on C class

road surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.12 The frequency response of vehicle body acceleration on D class

road surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.13 The frequency response of vehicle body acceleration (1: sprung

mass +50%, 2: sprung mass -50%) . . . . . . . . . . . . . . . . . 98

4.14 The frequency response of vehicle body acceleration (1: Ks1 +10%,

2: Ks1 -10%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.15 The frequency response of vehicle body acceleration zb . . . . . . 101

4.16 The frequency response of tyre dynamic load . . . . . . . . . . . . 102

4.17 The interval type-2 fuzzy membership functions of five inputs . . 105

4.18 The frequency response of vehicle front body acceleration . . . . . 108

4.19 The frequency response of vehicle rear body acceleration . . . . . 108

4.20 The frequency response of front tyre dynamic load . . . . . . . . . 109

4.21 The frequency response of rear tyre dynamic load . . . . . . . . . 109

4.22 The frequency response of pitch angle acceleration . . . . . . . . . 110

4.23 The different pitch dynamics with different vehicle speed . . . . . 111

4.24 The frequency response of vehicle front body acceleration . . . . . 113

4.25 The frequency response of vehicle rear body acceleration . . . . . 113

4.26 The frequency response of front tyre dynamic load . . . . . . . . . 114

4.27 The frequency response of rear tyre dynamic load . . . . . . . . . 114

4.28 The frequency response of pitch angle acceleration . . . . . . . . . 115

4.29 The different pitch dynamics with changing vehicle speed . . . . . 116

ix

List of Tables

2.1 Comparison of capabilities of different adaptive methodologies(Fukuda

& Kubota, 1999) . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Road roughness values classified by ISO (Degree of roughness S(Ω)×10−6) 15

3.1 Nominal parameters of half-vehicle active suspension model . . . . 56

3.2 The coefficients of non-linear forces . . . . . . . . . . . . . . . . . 63

3.3 Random road input parameters and the weighting parameters . . 67

3.4 Vehicle performance comparison with Vcr = 30m/s . . . . . . . . 71

3.5 Vehicle performance comparison with Vcr = 35m/s . . . . . . . . 71

3.6 Vehicle performance comparison with Vcr=25 m/s . . . . . . . . . 72

4.1 The parameters of quarter vehicle active suspension . . . . . . . . 91

4.2 The rules of fuzzy controller . . . . . . . . . . . . . . . . . . . . . 93

4.3 The RMS values comparison in time domain . . . . . . . . . . . . 95

4.4 The RMS values comparison on B class road surface . . . . . . . . 97

4.5 The RMS values comparison on C class road surface . . . . . . . . 98

4.6 The RMS values comparison on D class road surface . . . . . . . . 98

4.7 The RMS comparison of body acceleration in time domain . . . . 99

4.8 The RMS comparison of tyre dynamic loads in time domain . . . 99

4.9 The RMS values comparison in time domain . . . . . . . . . . . . 102

4.10 The comparison of RMS values of body acceleration . . . . . . . . 103

4.11 The comparison of RMS values of tyre dynamic load . . . . . . . 103

4.12 The comparison of RMS values of suspension travel . . . . . . . . 104

4.13 The comparison of RMS values of control force . . . . . . . . . . . 104

4.14 The RMS values comparison with constant vehicle speed . . . . . 111

x

LIST OF TABLES

4.15 The RMS values comparison with changing vehicle speed . . . . . 112

4.16 The RMS values comparison with constant vehicle speed . . . . . 116

4.17 The RMS values comparison with changing Vcr . . . . . . . . . . . 117

B.1 The fuzzy rules for half-vehicle active suspension (part 1) . . . . . 156






xi

Chapter 1

Introduction

It is well known that the suspension system performs multiple tasks such as

maintaining contact between vehicle tyres and the road, addressing the stability

of the vehicle, and isolating the frame of the vehicle from road-induced vibration

and shocks. With the development of mechanical and electronics technology, the

requirements of ride comfort and driving performance have been major devel-

opment objectives of modern vehicles to satisfy the expectations of customers.

Hence, the design of an appropriate suspension system is always an important

research topic for achieving the desired vehicle quality.

1.1 A Brief Background

There are many performance parameters which need to be optimized in vehicle

suspension system. Among them there are four main parameters which should be

considered carefully in designing a suspension system, i.e., ride comfort (directly

related to acceleration sensed by passengers), body motion (bounce, pitch and

roll of sprung mass are created by cornering and acceleration or deceleration),

road handling (associated with the contact forces of tyres and road surface),

suspension travel (referring to the displacement between the sprung mass and

unsprung mass) (Hrovat, 1997). It is a challenging issue for one suspension system

to simultaneously minimize all four parameters. Hence, how to obtain a proper

trade-off between these performances is the main task for successfully designing

a vehicle suspension system.

1

1.1 A Brief Background

To improve the vehicle performance, many kinds of suspension systems have

been designed for different types of vehicles. In general, there are three main

branches of suspension system, i.e., a passive, semi-active and active suspension

system. Passive suspension systems consist of conventional springs and dampers

which are used in most cars. The springs and dampers are assumed to have al-

most linear characteristics. In passive suspension systems, these elements have

fixed characteristics, and so, have no mechanism for feedback control(Naude &

Snyman, 2003a,b; Tamboli & Joshi, 1999). Semi-active suspensions provide con-

trolled real-time dissipation of energy(Choi et al., 2001; Margolis, 1982; Oueslati

& Sankar, 1994; Yao et al., 2002). This is achieved through a mechanical device

called an active damper which is used in parallel to a conventional spring. The

main feature of this system is the ability to adjust the damping of the suspension,

without any use of actuators. Active suspension systems employ pneumatic or

hydraulic actuators which in turn creates the desired force in the suspension sys-

tem (Crolla & Abdel, 1991; Hac, 1992; Hrovat, 1997; Thompson & Davis, 1988).

The actuator works in parallel to a spring and damper. An active suspension

system requires sensors to be located at different points of the vehicle to measure

its dynamic information of part of a vehicle. This information is used in the

real-time controller to drive the actuator in order to provide the exact amount of

force required.

Due to fewer mechanical constraints, more degrees of freedom and a stronger

capability to deal with unknown road surfaces, there is an increasing interest in

design and control of active suspension systems for automotive engineers and re-

searchers during the past three decades. Research has shown that a linear optimal

control scheme provides an efficient way to design an active suspension system

which can improve the vehicle ride and handling performance together (Hrovat,

1997; Nagai, 1993). However, it is based on the assumption that there exists a

perfect (broad bandwidth) actuator, which can generate the required force fast

enough and the system can be linearized within some operation regions. For

a real vehicle suspension system, its dynamics are inherently non-linear, even

with some uncertainties. Therefore, adaptive control schemes and robust control

strategies have been proposed to undertake the role of providing self-tuning feed-

back gains and to take the aforementioned four sets of parameters into account

2

1.2 Problems and Challenges

to ensure optimal operation of the system in different driving conditions and road

surfaces(Gordon et al., 1991; Hac, 1987; Sunwoo & Cheok, 1990; Sunwoo et al.,

1991). Much research considered non-linear, uncertain and unmodelled parts of a

real suspension system by using non-linear models and non-linear control meth-

ods(Alleyne & Hedrick, 1995; Alleyne et al., 1993; Gordon et al., 1991). With

the significant development of computational intelligence in the past two decades,

intelligent control methods provide an extensive freedom for control engineers to

deal with practical problems associated with active suspension control systems

(Chen et al., 1995; Cherry & Jones, 1995; Fernando & Viassolo, 2000; Huang &

Chao, 2000; Huang & Lin, 2003; Lian et al., 2005; Rao & Prahlad, 1997; Ting

et al., 1995; Yeh & Tsao, 1994).

1.2 Problems and Challenges

As mentioned in early research about active suspension systems, the dynamics

of suspension and the actuator were assumed to be linear or piecewise linear

and the majority of control laws were built on linear suspension mathematical

models. However, in real applications, there are some issues to bring out the non-

linear and uncertain dynamics of suspension systems, e.g., mechanical non-linear

properties, the coupling with other vehicle control systems and the disturbance by

random road inputs. Furthermore, more and more information from each part of

vehicle components are integrated into one central control unit in modern vehicles,

the problem of obtaining the proper description of suspension information from

other related subsystems becomes more and more complex. So the existed theory

analysis results with assumed models were proved to suffer from severe limitations

(Gao et al., 2006; Li et al., 2006).

For active suspension control systems, the key role is to optimize suspension

performances in real-time with multiple constraints. With the development of

vehicle technology, some devices for special vehicles, such as military vehicles,

can be transferred to normal cars. Meanwhile, some new requirements of the

suspension system have been demanded by customers and vehicle companies,

such as energy saving (Cao et al., 2008). Consequently, a new control framework

needs to be designed to satisfy these new requirements. Alternatively, a more

3

1.3 Overview of Approaches and Contributions

adaptive capability of controller is required to keep satisfied riding and handling

performance on different circumstance.

Almost all of the suspension control systems are closed-loop control systems,

so the closed-loop stability is very important to be guaranteed when they are

employed in real systems. Based on Lyapunov stability methods, some stability

analysis of suspension control systems with linear control strategies have been

studied. However, it is still a challenge for suspension control system to build a

proper surface or systematic analysis method, especially for an intelligent con-

troller (Feng, 2006).

In recent years, research work on improving the active suspension control

systems has been challenged in four major directions: Comprehensive studies on coupling information between suspension systems

and other vehicle control systems, as well as developing the decoupling

methods for suspension control. Achieving more adaptability and appropriate performance active suspension

control methods or control framework while retaining simplicity and real-

time computing efficiency. Development of estimating a platform for active suspension control sys-

tems and stability analysis methods of closed-loop control systems which

considers the effect of uncertainty and non-linearity in applications. Improving the manufacture of high performance actuators and micro control

units, as well as reforming the sensory structures used by information fusion

technology.

1.3 Overview of Approaches and Contributions

To take into account the previous section, this thesis makes contributions to the

first three of the four problem areas described in Section 1.2. The contributions

are driven by three new ideas which are described below:

4

1.4 Outline of Thesis

1. An improved half-vehicle model for active suspension control system is pro-

posed to make up the existing models by considering the coupling of longi-

tudinal motion between the front and the back of vehicle body. This model

can more precisely describe the real dynamics of vehicle suspension system.

2. Based on the improved model, a novel architecture of the interval type-2

fuzzy controller is proposed to control the vehicle non-linear active suspen-

sion system. Under the proposed control framework, the adaptive strategy

with two optimization methods (i.e., Least Means Squares (LMS) and Par-

ticle Swarm Optimization (PSO)) is designed to derive the expected active

control forces which bring better ride comfort and handling performance.

Furthermore, the proposed method has inherent capability to deal with the

potential uncertain information of fuzzy knowledge base which is deduced

from expert experience.

3. With the proposed control framework, the stability of vehicle active sus-

pension closed-loop control system is analysed by the quadratic Lyapunov

stability method. The sufficient conditions are derived for guaranteeing its

global stability.


To fulfil the proposed approaches, the thesis has been structured as follows.

Chapter 2 provides an overview of computational intelligence approaches in

active vehicle suspension control systems with a focus on the problems raised in

practical implementations by their non-linear and uncertain properties. After a

brief introduction on active suspension models, it explores state of the art in fuzzy

inference systems, neural networks, genetic algorithms and their combination for

suspension control issues. Discussion and comments are provided based on the

reviewed simulation and experimental results. The future research directions and

challenges for active suspension control are discussed.

Chapter 3 proposes an improved half-vehicle active suspension model to

explore the nature longitudinal coupling phenomenon between the front and back

part of vehicle body. It addresses the first idea in Chapter 1. To achieve this

5


improved model, firstly, a rigid tyre model is introduced to study the vehicle

speed effect on vertical vibration. Secondly, the improved model is built by

integrating the tyre model into the linear and non-linear half-vehicle suspension

models. Finally, the open-loop and closed-loop performances of the improved

model are analysed with step road inputs and random road inputs.

Chapter 4 builds a novel framework of adaptive interval type-2 fuzzy con-

troller for vehicle non-linear active suspension systems. It undertakes the second

approach in Chapter 1. To design the new control architecture, the first is a

brief review of interval type-2 fuzzy system. Subsequently, the proposed adaptive

control framework is built. With the proposed control structure, the adaptive

method is designed to self-tune lower bounds and upper bounds of interval type-

2 fuzzy reasoning results by two optimization methods (i.e., LMS and PSO). The

control aims are not only to improve the ride comfort and handling performance

of vehicle suspension system, but also to reduce the suspension travel and the lon-

gitudinal vibration. Finally, under several different test conditions, case studies

based on quarter-vehicle and half-vehicle models are demonstrated.

Chapter 5 presents the closed-loop stability analysis for proposed control

system. It addresses the third idea in Chapter 1. To address the stability analy-

sis, initially, the typical T-S fuzzy control system and its stability conditions by

the Lyapunov stability theory are revisited. By formalising the proposed control

system into a general interval type-2 fuzzy control system, the closed-loop sta-

bility is analysed with the quadratic Lyapunov method. Finally, the sufficient

conditions for global stability are derived.

Chapter 6 summaries the thesis with a discussion of the methods employed

and future work.

Appendices includes a list of publications and followed by the fuzzy control

rules for half-vehicle active suspension fuzzy controllers.

6

Chapter 2

Literature Review

2.1 Introduction

A suspension system is one of the important components of a vehicle, which plays

a crucial role in handling performance and the ride comfort characteristics of a

vehicle. It has two main functionalities, one is to isolate the vehicle body with its

passengers from external disturbance inputs which mainly come from irregular

road surfaces. It always relates to riding quality. The other is to maintain a firm

contact between the road and the tyres to provide guidance along the track. It

is called handling performance.

In a conventional passive suspension system which comprises of springs and

dampers, a trade-off is needed to resolve the conflicted requirements of ride com-

fort and good handling performance. The reason is that stiff suspension is re-

quired to support the weight of vehicle and to follow the track; on the other

hand, soft suspension is needed to isolate the disturbance from the road. Hence

there exists a significantly growing interest in the design and control of active

suspension systems from automotive engineers and researchers in the past three

decades. An active suspension system is characterized by employing some kind

of suspension force generator using pneumatic, magneto-rheological or hydraulic

actuators. Practical applications of active suspension systems have been facil-

itated by the development of microprocessors and electronics from the middle

of 1980s(Esmailzadeh & Bateni, 1992; Hrovat, 1987; Thompson & Davis, 1988,

1991). Related surveys on theories and applications of active suspension control

7

2.1 Introduction

systems were provided by Hrovat (1997); Nagai (1993), with fast-growing com-

putational intelligence methodologies significantly driving recent advances in this

research area in the past decade.

The design of a vehicle active suspension control system is a long-standing con-

trol engineering problem, which is rooted in multi-parameter optimization with

a real-time requirement. It includes ride comfort, body motion, road handling

and suspension travel (Taghirad, 1997; Taghirad & Esmailzadeh, 1998; Williams,

1997). Ride comfort directly relates to the acceleration sensed by passengers.

Body motion means bounce, pitch and roll of sprung mass which are created

by cornering, acceleration or deceleration. Road handling is associated with the

contact forces of tyres and the road surface. Suspension travel refers to the dis-

placement between a sprung mass and an unsprung mass. It is really a challenging

issue for one active suspension system to simultaneously optimise all four sets of

parameters. Hence, the ability to handle related trade-offs is crucial for success-

fully designing an active suspension control system. Research in the past three

decades has shown that a linear optimal control scheme provides an efficient way

to design an active suspension system which can improve the vehicle ride and

handling performance together (Hrovat, 1997; Nagai, 1993). This is based on the

assumption that there exists a perfect (broad bandwidth) actuator, which can

generate the required force fast enough and the system can be linearized within

some operation regions. However, a real vehicle suspension system is inherently

non-linear, even with some uncertainties. Therefore, adaptive control schemes

have to undertake the role of providing self-tuning feedback gains and to take

the aforementioned four sets of parameters into account to ensure optimal opera-

tion of the system in different driving conditions and road surfaces(Gordon et al.,

1991; Hac, 1987; Sunwoo & Cheok, 1990).

A classical form of adaptive scheme for a vehicle active suspension system was

introduced in the late 1980s by Hac (1987). This is the starting point of the adap-

tive control scheme, in which a set of feedback gains are varied by the change of

power spectral density of terrain roughness obtained by processing the measure-

ment data. Another comparison of adaptive Linear-Quadratic-Gaussian (LQG)

and non-linear controllers for active suspensions was presented by Gordon et al.

(1991). A model reference adaptive control scheme was proposed by Alleyne et al.

8

2.1 Introduction

(1993) which resulted in better performance than the active suspension system

with a non-adaptive controller and passive suspension system. Also in this thesis,

10% to 30% variances of sprung mass and stiffness coefficients were examined to

check the adaptation capability based on a single degree-of-freedom (DOF) quar-

ter vehicle model. Sunwoo & Cheok (1990) proposed an explicit adaptive control

for an active suspension system which is based on a self-tuning controller design.

It consisted of on-line low-order recursive parameters estimation, a closed-form

algebraic gain computation and manipulation of the control parameters. Some

other works on adaptive control of active suspension systems can be found in

Alleyne (1997); Alleyne & Hedrick (1995); Kim (1996); Kim & Ro (1998); Lu &

DePoyster (2002). Up to this point, most researchers have dealt with a linear

model in developing control laws or using adaptive control scheme to conquer

the limited non-linear properties of suspension systems. However, if the system

is highly non-linear over the range of operation, its adaptive schemes may show

severe limitations. For instance, if a wheel stroke is so strong that the stiffness

of a suspension is beyond the linear range, it might be practically impossible to

identify parameters through ordinary identification(Kim & Ro, 1998; Palkovics &

Venhovens, 1992; Sunwoo et al., 1991). In the early 1990s many studies began to

consider the non-linearities, uncertainties and unmodelled parts of a real suspen-

sion system, which requires the use of a non-linear model and some non-linear

forms of control scheme(Alleyne et al., 1993; Slotine & Li, 1991). In practice,

these non-linear models made active suspension control systems so complex and

too challenging to employ.

In industrial applications, control engineers often have to deal with complex

systems, which have multiple variable and multi-parameter models with perhaps

non-linear coupling. The conventional approaches for understanding and predict-

ing the behaviour of such systems based on analytical techniques can be proved

to be inadequate, even at the initial stages of establishing an appropriate math-

ematical model(Kaynak et al., 2001). The computational environment used in

such an analytical approach is perhaps too categorical and inflexible in order to

cope with the intricacy and the complexity of real world industrial systems. It

turns out that, in dealing with such systems, one has to face a high degree of un-

certainty and tolerate imprecision. Trying to increase precision can be very costly.

9

2.1 Introduction

Thanks to the significant development of soft computing or computational intel-

ligence in the past decades, it has provided alternative ways to non-linear system

modelling and control. Generally speaking, the principal constituents of intelli-

gent computing include Fuzzy Logic (FL), artificial neural networks (ANNs) and

evolutionary computing (EC). FL is mainly concerned with imprecision and ap-

proximate reasoning, ANNs are mainly concerned with learning and curve fitting,

and EC is mainly concerned with global optimization based on the natural se-

lection and genetics. These intelligent computing methodologies have resulted in

the development of the “intelligent control” field, which consists of novel control

approaches based on FL, ANNs, EC, and other techniques induced from artificial

intelligence and their combination. These methods provide an extensive freedom

for control engineers to deal with practical problems of vagueness, uncertainty,

or imprecision. These intelligent methods are good candidates for alleviating the

problems associated with active suspension control systems (Zhang et al., 2007).

In comparison with hard computing, soft computing provides the tolerance

for imprecision and uncertainty which is exploited to achieve a practically ac-

ceptable solution at a reasonable cost, tractability, and high machine intelligence

quotient (MIQ). Zadeh argues that soft computing, rather than hard computing,

should be viewed as the foundation of machine intelligence. A full comparison

of their capabilities in different application fields was constructed by Fukuda and

Shimojima in Table 2.1, together with those of control theory and artificial intel-

ligence(Fukuda & Kubota, 1999).

10

2.1

Intro

ductio

n

Table 2.1: Comparison of capabilities of different adaptive methodologies(Fukuda & Kubota, 1999)

Mathematical

Model

Learning

Data

Operator

Knowl-

edge

Real

Time

Knowledge

Repre-

sentation

Non-

linearity

Optimisation

Control

Theory

Good or

Suitable

Unsuitable Needs

other

methods

Good or

Suitable

Unsuitable Unsuitable Unsuitable

Neural

Network

Unsuitable Good or

Suitable

Unsuitable Good or

Suitable

Unsuitable Good or

Suitable

Fair

Fuzzy

Logic

Fair Unsuitable Good or

Suitable

Good or

Suitable

Needs

other

methods

Good or

Suitable

Unsuitable

other

Artificial

Intelli-

gence

Needs other

methods

Unsuitable Good or

Suitable

Unsuitable Good or

Suitable

Needs

other

methods

Unsuitable

Genetic

Algo-

rithms

Unsuitable Good or

Suitable

Unsuitable Needs

other

methods

Unsuitable Good or

Suitable

Good or Suitable

11

2.2 Background

This chapter reviews the necessary background for active suspension control

systems which provides a foundation for the methods proposed in later chap-

ters. It is organized as follows. Section 2.2 revises the modelling of an active

suspension system. Section 2.3 reviews adaptive fuzzy control methods; Section

2.4 presents adaptive fuzzy sliding mode control approaches; Section 2.5 revises

neural networks based control systems, and Section 2.6 presents adaptive genetic

algorithm control methods. Section 2.7 describes combination methods based on

neural networks, fuzzy inference and generic algorithms. Finally it is concluded

in Section 2.8 with discussions and future work.

2.2 Background

A vehicle body is generally a rigid body with six DOF motions shown in Fig.

2.1. It consists of longitudinal, lateral, heave, roll, pitch and yaw motions. These

motions are restricted by suspension geometries in vehicles and are more or less

coupled with one another. Moreover, as the suspensions have a mechanical struc-

ture with unsprung mass, coupling also occurs between the sprung and unsprung

masses. Regardless of such coupling problems, the reduced-order mathematical

model is useful for designing an active suspension control system. Therefore a

quarter-vehicle model or a half-vehicle model is often used for theoretical analysis

and design of active suspension systems (Hrovat, 1997; Nagai, 1993).

In this section, a linear quarter-vehicle model and a linear half-vehicle model

of an active suspension system are introduced. Based on the linear models, their

LQG controllers are designed. Practical active suspension system models are also

analysed in terms of non-linear properties and uncertain dynamic disturbances.

2.2.1 Active Suspension System Linear Models and Con-

trol

2.2.1.1 Quarter Vehicle Active Suspension Model

The quarter-vehicle model was initially developed to explore active suspension

capabilities. It gave birth to the concepts of skyhook damping and fast load

12

2.2 Background

Figure 2.1: Six degree-of-freedom vehicle model (Nagai, 1993)

leveling which are now being developed toward actual, large-scale production ap-

plications. In this section, we define,

mb: quarter body mass (or sprung mass) (Kg);

mw: wheel mass (or unsprung mass) (Kg);

Ks: suspension spring stiffness (N/m)

Kt: tyre stiffness (N/m);

c: damping coefficient (Ns/m);

G0: road roughness coefficient (m3/cycle);

U0: vehicle original forward velocity (m/s);

f0: low cut-off frequency (Hz);

z0: road displacement (m);

zw: wheel displacement (m);

zb: body displacement (m);

fa: control force (N);

The quarter vehicle model is shown in Fig. 2.2. The dynamic differential equa-

tions of this suspension system can be represented as

mbzb = fa + c(zw − zb) + Ks(zw − zb) (2.1)

mwzw = −fa − c(zw − zb) − Ks(zw − zb) − Kt(zw − z0) (2.2)

For better ride comfort, a perfect road surface model is necessary to design vehicle

active suspension control system. There are many possible ways to analytically

13

Chapter1/Chapter1Figs/EPS/carmodel.eps

2.2 Background

Figure 2.2: Two degree-of-freedom quarter-vehicle model

describe the road inputs, which can be classified as shock or vibration (Hrovat,

1997). Shocks are discrete events of relatively short duration and high intensity,

e.g., a pronounced bump or pothole on an otherwise smooth road. Vibrations,

on the other hand, are characterised by prolonged and consistent excitations that

are called “rough” roads. In this section, the rough road is considered. The

International Organization for Standardization (ISO) has proposed a series of

standards of road roughness classification using the Power Spectral Density (PSD)

values (ISO 2631), as shown in Table 2.2. Due to the ISO, the road displacement

PSD can be described as

G(n) = G(n0)(n

n0)−w (2.3)

Here, n is the space frequency (m−1) and time frequency f is f = nv (v is the

vehicle speed), n0 is the reference space frequency, G(n) is the road displacement

PSD, G(n0) is road roughness coefficient shown in Table 2.2, w is the linear fitting

coefficient, always w = 2. Based on the standard road surface description, the

road surface input model is built through an forming filter by Gaussian white

noise and successfully used in many presented works (Taghirad, 1997; Yu et al.,

2000). The equation of the road surface input is:

z0 = −2πf0z0 + 2π√

G0U0w0 (2.4)

14

Chapter1/Chapter1Figs/EPS/quarter-vehicle.eps

2.2 Background

where f0 is low cut-off frequency, G0 is road roughness coefficient , w0 is a Gaus-

sian white noise, and U0 is the vehicle speed.

Table 2.2: Road roughness values classified by ISO (Degree of roughness

S(Ω)×10−6)

Road Class Range Geometric mean

A(very good) <8 4

B(good) 8-32 16

C(Average) 32-128 64

D(Poor) 128-512 256

E(very poor) 512-2048 1024

F 2048-8192 4096

G 8192-32768 16384

H > 32768

Equations 2.1, 2.2 and 2.4 are combined to give the state space representation

of the quarter-vehicle model:

X = AX + BU + FW (2.5)

Y = CX + DU (2.6)

where

X =[

zb zw zb zw z0

](2.7)

Y =[

zb zw zw − zb zw − z0

](2.8)

U = [fa] , W = [w0] (2.9)

A =

− cmb

cmb

−Ks

mb

Ks

mb0

cmw

− cmw

Ks

mw−Ks+Kt

mw

Kt

mw

1 0 0 0 00 1 0 0 00 0 0 0 −2πf0

(2.10)

15

2.2 Background

B =

1mb

− 1mw

000

, F =

0000

2π√

G0U0

, D =

1mb

− 1mw

00

(2.11)

C =

− cmb

cmb

−Ks

mb

Ks

mb0

cmw

− cmw

Ks

mw−Ks+Kt

mw− Kt

mw

0 0 −1 1 00 0 0 1 −1

(2.12)

Based on the proposed model, linear optimal control theory is used to design

the active suspension controller. To obtain the better handling performance and

ride comfort, the performance index can be written as a weighted sum of mean

square values of output performance variables including body acceleration, wheel

to body displacement and dynamic tyre deflection.

J = limT→∞

1

T

∫ T

0

q1 (zw − zb)

2 + q2(zw − z0)2 + q3z

2b

dt (2.13)

Changing equation 2.13 into a general matrix format, it follows that

J = limT→∞

1

T

∫ T

0

[XTQX + UTRU + 2XTNU ]dt (2.14)

where Q, R, N can be solved from equation 2.1, 2.2, 2.4. Assuming that an opti-

mal state observer, i.e. Kalman filter, is available to get a satisfactory estimation

of state vector X, based on the separation theorem, an optimal control force is

U = −R−1BTPX = −KX (2.15)

where K represents the gain matrix; and P is the solution of the following classical

algebraic Riccati equation

PA + ATP − (PB + N)R−1(BTP + NT) = −Q (2.16)

2.2.1.2 Half Vehicle Active Suspension Model

The half-vehicle model including pitch and heave modes was invented to simulate

the ride characteristics of a simplified whole vehicle, which led to significant

16

2.2 Background

Figure 2.3: Half-vehicle suspension model

improvement in ride and handling. Let f and r denote the front and rear, x and

z be the longitudinal forward direction and vertical up direction in this thesis,

we define,

df : distance from the front axle to the center of gravity (m);

dr: distance from the rear axle to the center of gravity (m);

Ib: pitch inertia (Kgm2);

U0: vehicle forward speed(m/s);

zf0: road displacement at the front wheel (m);

zr0: road displacement at the rear wheel (m);

zwf : front wheel displacement (m);

zbf : front body displacement (m);

zwr: rear wheel displacement (m);

zbr: rear body displacement (m);

The half-vehicle model is shown in Fig. 2.3.

With the assumption of a small pitch angle, the following are obtained,

zbf = zb − df · θ, zbr = zb + dr · θ (2.17)

From equation 2.17, the pitch angle can be written as:

θ =zbr − zbf

df + dr

(2.18)

17

Chapter1/Chapter1Figs/EPS/halfcar.eps

2.2 Background

and hence the model equations of motion can be written as follows:

zwfmwf = −Ktf (zwf − zf0) − [faf + cf (zwf − zbf )+ Ksf(zwf − zbf )]

(2.19a)

zwrmwr = −Ktr(zwr − zr0) − [far + cr(zwr − zbr)+ Ksr(zwr − zbr)]

(2.19b)

zbmb = faf + cf(zwf − zbf ) + Ksf(zwf − zbf ) + far

+ cr(zwr − zbr) + Ksr(zwr − zbr)(2.19c)

θIb = −df [faf + cf(zwf − zbf ) + Ksf(zwf − zbf )]+ dr[far + cr(zwr − zbr) + Ksr(zwr − zbr)]

(2.19d)

Substituting equation 2.17 into 2.19c and 2.19d, we have the following,

zbf =(

1mb

+d2

f

Ib

)[faf + cf(zwf − zbf ) + Ksf(zwf − zbf )]

+(

1mb

− dfdr

Ib

)[far + cr(zwr − zbr) + Ksr(zwr − zbr)]

(2.20a)

zbr =(

1mb

− dfdr

Ib

)[faf + cf (zwf − zbf ) + Ksf(zwf − zbf )]

+(

1mb

+ d2r

Ib

)[far + cr(zwr − zbr) + Ksr(zwr − zbr)]

(2.20b)

Using filtered white noise w1, w2 as the road inputs, the road input equations for

the front and rear wheels respectively are

zf0 = −2πf0zf0 + 2π√

G0U0w1 (2.21a)

zr0 = −2πf0zr0 + 2π√

G0U0w2 (2.21b)

So far we have a state vector as

Xhalf = [ zbr zwr zbf zwf zbr

zwr zbf zwf zr0 zf0 ]T(2.22)

Combining vehicle model equations 2.18, 2.19a, 2.19b,2.19c, 2.19d 2.20a, 2.20b,

and road input equations 2.21a and 2.21b, the system representation in state

space form is given by,

Xhalf = AXhalf + BUhalf + Fwhalf (2.23a)

18

2.2 Background

Yhalf = CXhalf + DUhalf + vhalf (2.23b)

where A, B, C, D, F are differential equation coefficient matrices, Xhalf is the

state vector, Yhalf is the output vector, here Yhalf is defined in equation 2.24,

Uhalf is control input matrix, whalf is road inputs, vhalf is measurement noise.

Yhalf = [ zbf zbf − zwf zwf − zf0

zbr zbr − zwr zwr − zr0 ]T(2.24)

The matrices A, B, C, D, F , Uhalf , whalf are shown in the following equations 2.25-

2.31,

D =

[α3 0 0 α2 0 0α2 0 0 α1 0 0

]T

(2.25)

F =

[0 0 0 0 0 0 0 0 2π

√G0U0 0

0 0 0 0 0 0 0 0 0 2π√

G0U0

]T

(2.26)

Uhalf =

[faf

far

], whalf =

[w2

w1

](2.27)

where α1 denotes ( 1mb

+ d2r

Ib), α2 denotes ( 1

mb− df dr

Ib) and α3 denotes ( 1

mb+

d2f

Ib).

Based on the proposed model, linear optimal control theory is used here to design

the active suspension controller. For obtaining the better handling and ride

comfort, the performance index can be written as a weighted sum of mean square

values of output performance variables including body acceleration, wheel to body

displacement and dynamic tyres deflection, as shown in equation 2.28.

J = limT→∞

1T

∫ T

0[q1(zwf − zf0)

2+q2(zbf − zwf )2

+ρ1zbf + q3 (zwr − zr0)2 + q4(zbr − zwr)

2 + ρ2zbr]dt(2.28)

The optimal LQ control gain can be found by solving from Riccati equation,

similar to the quarter vehicle model.

19

2.2

Back

gro

und

A =

−α1cr α1cr −α2cf α2cf −α1Ksr α1Ksr −α2Ksf α2Ksf 0 0cr

mwr− cr

mwr0 0 Ksr

mwr−Ksr+Ktr

mwr0 0 Ktr

mwr0

−α2cr α2cr −α3cf α3cf −α2Ksr α2Ksr −α3Ksf α3Ksf 0 0

0 0cf

mwf− cf

mwf0 0

Ksf

mwf−Ksf +Ktf

mwf0

Ktf

mwf

1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 −2πf0 0

0 0 0 0 0 0 0 0 0 −2πf0

(2.29)

B =

[α2 0 α3 − 1

mwf0 0 0 0 0 0

α1 − 1mwr

α2 0 0 0 0 0 0 0

]T

(2.30)

C =

−α2cr 0 0 −α1cr 0 0α2cr 0 0 α1cr 0 0−α3cf 0 0 −α2cf 0 0α3cf 0 0 α2cf 0 0

−α2Ksr 0 0 −α1Ksr 1 0α2Ksr 0 0 α1Ksr −1 1−α3Ksf 1 0 −α2Ksf 0 0α3Ksf −1 1 α2Ksf 0 0

0 0 0 0 0 −10 0 −1 0 0 0

T

(2.31)

20

2.2 Background

2.2.2 Non-linear Model of Active Suspension Model

Many researchers have dealt with a linear model in developing control laws. How-

ever, considering the inherent non-linearities and uncertainties, it is not sufficient

to represent the real system with a linear model as in Section 2.2.1. In the early

1990s many studies began to consider non-linearities, uncertainties and unmod-

elled parts of a real suspension system, which required the use of a non-linear

model and some adaptive or robust form of control scheme (Alleyne & Hedrick,

1995; Alleyne et al., 1993; Gordon et al., 1991; Hrovat, 1997; Kim & Ro, 1998;

Sunwoo & Cheok, 1990). In this section, the non-linear properties are introduced

and the general non-linear models of suspension systems are illustrated.

As Hrovat (1997) remarked, for many operations, the linear system approxi-

mation was appropriate. However, there were some situations which amplify the

non-linear effects. One was created by discrete-event disturbances, such as single

bumps or potholes, which can cause a highly non-linear phenomenon. Another

was dry friction. Based on the quarter-vehicle model shown in Section 2.2.1.1,

Kim & Ro (1998) modelled the connecting forces (e.g., spring force, damping

force) as non-linear functions using measured data. In the linear model, these

connecting forces were described as linear functions of the system states. Fig. 2.4

showed major non-linearities in a real suspension system. In Kim’s paper, the

non-linear spring properties were mainly due to two parts. One was the bump

stop which restricted the wheel travel within a given range and prevents the tyre

from contacting the vehicle body. The other was the strut bushing which con-

nected the strut with the body structure and reduced vibrations from the road

input. These two non-linear effects can be included in the spring force fs with

non-linear characteristics versus suspension rattle space (zw-zb). Based on the

measured data in Kim & Ro (1998), Kim modelled the spring force fs and the

damping force by high-order polynomial functions. The spring force was described

as a third-order polynomial function shown as equation 2.32,

fs = fsl + fsn = k1∆x + (k0 + k2∆x2 + k3∆x3) (2.32)

where fsl is the linear part of the spring force and fsn is the non-linear part of

the spring force. The coefficients can be obtained from fitting the experimental

data.

21

2.2 Background

Figure 2.4: Non-linear properties of suspension system [wheel stroke(m) versus

suspension force(N)](Kim & Ro, 1998)

Also the damping force fd was modelled as a second-order polynomial function

by fitting the measured data, shown as below

fd = fdl + fdn = c1∆x + c2∆x2 (2.33)

where the fdl is the linear part and the fdn is the non-linear part of damper force,

the coefficients can be obtained from fitting the experimental data.

Except for the non-linear properties presented by the spring force and damp-

ing force, the vertical tyre force is highly non-linear, especially when the load

condition changed by a significant amount. Even the vertical tyre force became

zero when the tyre lost contact with the road. Kim et al modeled the tyre force

as:ftl = kt(z0 − zw) when(z0 − zw) > 0ftn = 0 when(z0 − zw) ≤ 0

(2.34)

where ftl denotes the linear tyre force, and ftn denotes the non-linear tyre force.

In order to show the effect of the asymmetric tyre stiffness on the response

of the quarter-car model, some simulation results were shown to investigate the

22

Chapter1/Chapter1Figs/EPS/nonlinear.eps

2.3 Adaptive Fuzzy Control

effect of non-linear tyre force under the different amplitudes of road input(Kim

& Ro, 1998). From the results, it was clear that vehicle non-linearities should

be considered in developing a more accurate system model, from which a more

reliable control algorithm can be developed.

In this thesis, two kinds of non-linear suspension system models are provided

for the controller design and performance analysis. Considering the non-linear

parts shown by equations 2.32 and 2.33, the active suspension system can be

written as a multiple-input multiple-output (MIMO) non-linear model:

X = F (X) + BU + d (2.35)

where F (X) is a non-linear function including the non-linear forces fs, ft and fd, U

is the input of the suspension system and d is the unknown external disturbance.

The other non-linear model can be described as a hybrid model with linear part

and non-linear part:

X = AX + BU + d (2.36)

where AX+BU is the linear model of the suspension system based on fsl, fdl and

ftl, d represents the non-linearity and uncertain parts of the suspension system.


The control performance of a traditional controller greatly depends on the accu-

racy of the known system dynamic model according to Section 2.2.1. In order

to meet the practical requirements of an active suspension system, it is crucial

to derive or identify an appropriate model for the traditional controller design.

Estimating uncertain effects is even more challenging due to random noise occur-

ring in road inputs. Hence some model-free intelligent controllers were introduced

to solve these problems, e.g., Fuzzy Logic Control (FLC)(Huang & Chao, 2000;

Huang & Lin, 2004; Rao & Prahlad, 1997; Yeh & Tsao, 1994). The FLC is cred-

ited with being an adequate methodology for designing robust controllers that

are capable of delivering a satisfactory performance in the face of uncertainty and

imprecision. As a result, the FLC has become a popular approach to non-linear

and uncertain system control in recent years.

23


There are different ways to construct FLC for vehicle suspension control. The

most common method to construct the FLC is by eliciting the fuzzy rules and

its membership functions based on expert knowledge or experience. The most

common problem which occurs is that they cannot fully handle or accommo-

date the linguistic and numerical uncertainties associated with dynamic natural

changing road inputs as they use precise fuzzy sets. In order to overcome this

weakness, adaptive FLC was designed to self-tune the fuzzy rules or member-

ship functions(Huang & Chao, 2000; Huang & Lin, 2004; Lian et al., 2005; Rao &

Prahlad, 1997; Yang et al., 2006; Yeh & Tsao, 1994). In this section, the adaptive

FLC designs and applications on active suspension systems are reviewed.

The key components of a FLC are a set of linguistic fuzzy control rules and

an inference engine to comprehend these rules. These fuzzy rules offer a transfor-

mation between the linguistic control knowledge of an expert and the automatic

control strategies of an actuator. Every fuzzy control rule is composed of an

antecedent and a consequent. A general form of the rules, Ri, can be expressed

as,

Ri : IF x1 is Di1 · · · and xn is Di

n , THEN u is Ei

where Ri stands for the ith rule, i = 1 · · ·n, Dij stands for the linguistic value

of the premise variable xj and Ei denotes the linguistic value of the consequence

output u. A mapping from the universe of discourse Dij to the universe of dis-

course Ei is performed by the inference mechanism.

The structures and parameters of control rules dominate the performance of

fuzzy control. From the control point of view, it is crucial that related parameters

or structures are modified automatically by evaluating the results of fuzzy control.

For instance, Huang & Chao (2000) proposed an adaptive FLC for an active

suspension system. This adaptive FLC scheme is shown in Fig. 2.5

The inputs of FLC were the vertical position error and error change of the

vehicle sprung mass. Its output was the control voltage increment. The an-

tecedents membership functions consisted of 11 equal triangular type functions.

The voltage increment membership function was a set of 15 equal triangular type

functions. Its self-tuning property was implemented by tuning the scaling factors

S1, S2, S3. Then the membership functions were adapted to improve the FLC

24

2.4 Adaptive Fuzzy Sliding Mode Control

Figure 2.5: The adaptive FLC scheme in Huang & Chao (2000)

performance. Its 121 fuzzy rules were employed to suppress the sprung mass

vibration amplitude due to road inputs.

In order to evaluate the fuzzy control system, a two DOF quarter-vehicle

suspension model was established. The suspension mechanism included a spring,

mass and a hydraulic control loop. A hydraulic servo system was used to generate

various road surfaces and an optical linear scale and a linear potentiometer were

employed to measure the sprung mass and road surface vertical displacements

respectively. Based on this realistic suspension model, the dynamic response of

active suspension system was provided for vehicle ride performance on a rough

concave-convex road with 25mm obstacles. The maximum displacement of the

vehicle body was less than 5mm and it converged within 0.5s. The control signal

was very smooth and easy to employ in the practical vehicle. However, its ad-

justed scaling factors were chosen by experiments and many simulations, which

limited the flexible and adaptive abilities of the adaptive FLC. In order to over-

come this problem, researchers have compensated for this type of adaptive FLC

by employing non-linear optimal algorithms, such as Genetic Algorithms (GA)

and/or ANNs to self-tune the parameters of their membership functions and fuzzy

rules. These kinds of adaptive FLC will be covered in Section 2.7.


Sliding Mode Control (SMC) nowadays enjoys a wide variety of application areas,

such as general motion control applications, robotics, process control, aerospace

25

Chapter1/Chapter1Figs/EPS/fig13.eps


applications and vehicle active suspension systems. The main reason for this

popularity is its attractive properties including good control performance for

non-linear systems, applicability to MIMO systems, and well-established design

criteria for discrete-time systems. Robustness is its most significant property.

Loosely speaking, when a system is in a sliding mode, it is insensitive to pa-

rameter changes or external disturbances. However, the SMC also suffer from

the following disadvantages in practical application. Firstly, there is the problem

of chattering, which is the high-frequency oscillations of the controller output

which is brought by the high speed switching for the establishment of a sliding

mode. Chattering is very undesirable and dangerous in practice because it may

excite unmodelled high-frequency dynamics resulting in unforeseen instabilities.

Secondly, a SMC is extremely vulnerable to measured noise since its input de-

pends on the sign of a measured variable which is very close to zero. Thirdly, the

SMC may employ unnecessarily large control signals to overcome the paramet-

ric uncertainties. Lastly, there exists difficulty in the calculation of the so-called

equivalent control. The integration of a FL system in a SMC has been witnessed

in many successful applications where an attempt to relieve the implementation

difficulties of the SMC are made by the addition of the FL system (Efe et al.,

2000; Yoo & Ham, 1998). On the other hand, some significant research works

have originated due to different difficulties, i.e., the difficulties in carrying out a

rigorous stability analysis of FLCs.

2.4.1 Conventional Sliding Mode Control

Let us consider the following nth order MIMO non-linear system,

X = F (X) + B(X)U + d (2.37)

where X ∈ Rn denotes the state vector of a system and is assumed to be avail-

able for measurement, U ∈ Rq denote the inputs of the plant, and d represent

the unknown bounded external disturbances, F (X) and B(X) are non-linear,

uncertain, continuous and bounded functions.

26


Suppose that the functions F (X) and B(X) can be written as the sum of a

well-characterized nominal function and a bounded uncertainty:

F (X) = F0(X) + ∆F (X), ‖∆F (X)‖ < MF

B(X) = B0(X) + ∆B(X), ‖∆B(X)‖ < MB(2.38)

where MF and MB are positive constants, and ‖.‖ denotes the Euclidian norm.

System equation 2.37 can be rewritten in the following form:

X = F0(X) + B0(X)U + D. (2.39)

where D = ∆F (X) + ∆B(X)U + d, and D ≤ αd; αd is a positive constant.

The design of a SMC involves two steps. The first step is to select switching

hyperplane called sliding surface to prescribe the desired dynamic characteristics

of a controlled system; The second step is to design discontinuous control such

that the system enters a sliding surface and remains in it. Regarding to the

system given by equation 2.39, the sliding surface S is generally selected as,

S(X) = GX = 0 (2.40)

where S(X) denotes a set of switching hyperplanes, and G is a constant q × n

matrix to be determined.

The main object in a SMC is to force the system states to the sliding surface.

Once the states are on the sliding surface, the system errors converge to zero

with an error dynamics dictated by the matrix G. The solution S(X) = 0 is

rigorous but in practise difficult to use for a controller design. A better approach

for a controller design is to introduce the equivalent control methods for defin-

ing the system behaviour on its sliding surface. If the dynamic of a system is

exactly known and no disturbance affects the system, the equivalent control can

be defined by equation 2.40 and 2.41:

S(X) = GX = GF0(X) + GB0(X)U + GD = 0. (2.41)

The condition in equation 2.40 is such that the system is on its sliding surface

and the condition in equation 2.41 shows that the system does not leave the

surface. Let us assume GB0 is non-singular, then the equivalent control can be

obtained by

Ueq = −(GB0)−1[GF0(X) + GD]. (2.42)

27


In order to satisfy the sliding conditions despite uncertainty on the dynamic

F0(X), a term which is discontinuous across the surface S = 0 can be added to

Ueq as in equation 2.43 below:

U = Ueq + Usw (2.43)

where Usw is the switch control and is defined as:

Usw = −(GB0)−1Ksgn(S) (2.44)

where K denotes the switching gain and sgn(S) denotes the sign function and is

defined as,

sgn(S) =

+1, S > 00, S = 0

−1, S < 0.(2.45)

The ability to maintain the stability of a designed control system is determined

by the selection of a Lyapunov function. The control must be chosen such that its

candidate Lyapunov function satisfies Lyapunov stability criteria. For instance,

herein a SMC control’s Lyapunov function is selected as

V =1

2ST (X)S(X). (2.46)

This function is positive definite because V (S = 0) = 0 and V (S) > 0 ∀S 6= 0. It

is such that the derivative of the Lyapunov function is negative definite. That is:

dV

dt=

1

2

d

dt(S2

i ) = SiSi < −ηi |Si| . (2.47)

where Si is a component of vector S, and ηi is positive. Then the derivative of

the sliding parameter is described as:

Si ≤ −ηisgn(Si). (2.48)

Substituting U in equation 2.43, the stable switching condition is reached.

28


2.4.2 Fuzzy Sliding Mode Control System

Considering the SMC designed by equations 2.40-2.45, its implementation needs

two necessary conditions, one of which is the exact system model or the sys-

tem dynamics and the other is a high frequency switching control. In practical

systems, these conditions will be constrained by the non-linear dynamics or un-

certain disturbance and physical actuators. In the last two decades, fuzzy logic

has been employed to improve SMC in terms of efficiency and practical issues.

Two types of fuzzy sliding mode control are introduced in this section. They are

employed to solve two SMC weaknesses, i.e., alleviation of SMC chattering and

modelling the non-linear or uncertain characteristics of practical systems.

2.4.2.1 Alleviating SMC Chattering

Fuzzy logic is employed to self-tune the discontinuous switching control law in

order to overcome the chattering phenomenon in SMC. Considering the switching

control law in terms of equation 2.44, there are two parameters (G and K) to

be optimised. Their effects on the system performance are shown in Fig. 2.6.

Parameter G determines the slope of the sliding line, which means the larger G

is, the faster the system response. Due to the fact that an over-large value of G

can cause overshoot or instability, it would be advantageous to adaptively vary its

slope in such a way that the slope is increased as the magnitude of its error gets

smaller. Curve labelled “1” corresponds to the case when K is large. The system

states reach the sliding line in a short time, but overshoot it by a considerable

amount. Curve labelled “2” reflects the case with a small K parameter. Neither

curve “1” nor “2” is desired. Curve “3” can be obtained via fuzzy adaptive

algorithms in which parameter “K” is increased only when the states are close

to its sliding line.

For instance, Chen et al. (1995) proposed a fuzzy adaptive sliding mode con-

troller for an active suspension system. The proposed quarter car active suspen-

sion model was as follows,

X = AX + BU + EW + D (2.49)

29


Figure 2.6: Effects of parameters G and K (Kaynak, 1998)

Its sliding surface was defined as:

S(X) = GX = x2 + λx1 = 0, λ > 0 (2.50)

Likewise, the SMC control Ueq and UN were chosen as below:

Ueq = b−1[−a1x1 − (a2 + λ)x2], UN = b−1Ksgn(S). (2.51)

The proposed fuzzy adaptive SMC scheme is shown in Fig. 2.7. Note that the

actual inputs of this fuzzy adaptive SMC controller are S and its derivative S.

(a) Fuzzy control scheme for active

suspension control system

(b) Self-tuning fuzzy logic controller

Figure 2.7: Fuzzy adaptive sliding mode control scheme for ASCS in Chen et al.

(1995)

The output is the hitting control. Fuzzification and defuzzification stood for

an interface between the crisp values of reality and the linguistic values of in-

30





ference. A map from the universe of input discourse to the universe of output

discourse was carried out by the inference mechanism. The controller was orga-

nized at two levels. At the basic level, the conventional fuzzy control rule sets

and inference mechanism were constructed to generate a fuzzy control scheme. At

the supervising level, the control performance was evaluated to modify system

parameters, especially for adaptively tuning its scaling factors. The proposed

fuzzy control rules were outlined in Chen et al. (1995). Here, for instance, if Ss

is NB and Sδs is NB, then uf is PB. It represented the fact that “ Ss is NB ”

meant s was far from the sliding surface in the negative and “Sδs is NB” meant

divergent speed was very large, therefore a large amount of positive uf should be

provided to force it backward. On the other hand, if Ss was NB and Sδs was PB,

S currently converged to the sliding surface, hence there was no need to give a

very large output because a small positive uf was able to drive S gently to the

surface and to prevent overshoot.

In order to investigate an active suspension performance based on the above-

mentioned fuzzy SMC, a pseudo-random disturbance road input was employed

to test robustness of the controller under the condition that spring mass distur-

bance was increased by 30% and damping coefficient and spring stiffness were

decreased by 30% from the nominal values. The simulation results demonstrated

that the controlled suspension deflection was smaller than its counterpart of a

LQG optimal control but larger than that of a conventional SMC. Regarding

the riding quality, the fuzzy SMC achieved the best performance of sprung mass

acceleration. The simulation results also illustrated that the road handling abil-

ity maintained by the fuzzy SMC outperformed that of a LQG controller and a

conventional SMC. Similar conclusions were also drawn for the perturbed condi-

tions.

Additionally, Zhang et al. (2007) also proposed a fuzzy adaptive SMC for an

active suspension system. The main difference from Chen’s research was the way

in which a sliding surface was constructed. In Yun’s paper, the sliding surface

was built on the basis of conventional sliding surface s and its derivative s as

below:

σ = s + λs (2.52)

31


where λ was a positive value, and its Lyapunov stability condition must be sat-

isfied:

V = σσ < 0 (2.53)

The equivalent control can be obtained:

Ueq = −(GB)−1[(GA + λG)AZ + (GA + λG)BU ] (2.54)

UN = −(GB)−1εsgn(σ). (2.55)

Then the SMC control output was achieved:

U = Ueq + UN (2.56)

Finally it led to the controller output:

U(n) = U(n − 1) + U(n). (2.57)

Figure 2.8: The fuzzy adaptive controller scheme in Zhang et al. (2007)

The scheme of a fuzzy adaptive tuning controller is cited in Fig. 2.8. The

simulations in the time domain and the frequency domain were carried out on a

quarter car active suspension system. In the time domain analysis, the compar-

ison between a LQG controller and the fuzzy adaptive SMC controller showed

that the proposed controller can significantly decrease its sprung mass accelera-

tion from peak value to zero. However, the proposed fuzzy adaptive SMC simul-

taneously needed higher active forces than the LQG controller. In the frequency

32

Chapter1/Chapter1Figs/EPS/adfc.eps


domain analysis, the fuzzy adaptive SMC improved the frequency response from

the road input to the sprung mass acceleration, especially in the frequency range

4 − 8 Hz. Based on ISO2631, the human body is very sensitive to vertical vibra-

tion in the frequency range 4− 8 Hz. That is to say that the proposed controller

can significantly improve the ride quality. Considering the existence of uncertain

parameters, the sprung mass was assumed to change in a bounded range of ±50%. The simulation results demonstrated that the maximum acceleration of the

active suspension using the proposed controller was on average 54% smaller than

a passive suspension system.

2.4.2.2 Fuzzy Logic Complementary to SMC

Referring to a traditional SMC design, the equivalent control law always depends

on its system model, due to the fact that an exact system model is very diffi-

cult to achieve for a more complex non-linear system. A practical method for a

non-linear problem is linearized around given operation points such that the well-

developed linear control theory can be applied into the local region with apparent

ease. However, it leads to a new problem of how to aggregate each locally lin-

earized model into a global model which represents the corresponding non-linear

system. Fuzzy logic offers a solution to the problem without the need of a mathe-

matical model and constant gain limitation(Chen & Chang, 1998). Huang & Lin

(2003) proposed an adaptive fuzzy sliding mode controller for an active vehicle

suspension system. Its system control block diagram is given in Fig. 2.9. FLC

was employed to approximate the non-linear function of the equivalent control

law, Ueq. The voltage output of an actuator in each sampling step was derived

from fuzzy inference, instead of from the nominal model at the sliding surface.

It significantly diminished the chattering phenomenon of the traditional sliding

mode control.

The input signal of this type of fuzzy logic control was the sliding surface

variable, S in equation 2.58, in terms of its sprung mass position and velocity

deviations. Its output signal was control voltage, U , which was the output of the

hydraulic servo actuator.

S = x2 + λx1. (2.58)

33


Figure 2.9: The adaptive fuzzy sliding mode controller scheme in Huang & Lin

(2003)

The fuzzy input variable S consisted of eleven equal-span triangular mem-

bership functions, which were employed for the fuzzy output variable U through

eleven fuzzy inference rules. The tunable consequent parameters of those peaks of

the triangular membership functions were initialized to zero by default. A novel

on-line parameters tuning algorithm was proposed to adjust the consequent pa-

rameters and monitor the system control performance. A quarter car 2 DOF

active suspension system was designed and built for investigating its dynamic

performance and control effect. The suspension system was tested under three

different conditions. One was a rough road with 40mm amplitude sinusoidal wave,

the other two were a rough concave-convex road with a randomly dynamic 40mm

height and a rough road with a random amplitude. The experimental results

showed that the proposed method had significantly suppressed the sprung mass

position oscillation amplitude. In addition, the control voltage was smooth and

the speed converged quickly.

Additionally, Kucukdemiral et al. (2005) proposed a fuzzy logic method to

handle the non-linear system model and uncertain disturbance for an active sus-

pension system. The adaptive SMC is shown in Fig. 2.10, where the control

was given by u = ufz + uvs. Here, uvs denoted the switching control which im-

proved by a boundary layer and alleviated the chattering; ufz was obtained from

FLC with the input S. To evaluate the proposed controller, the simulation en-

vironment was controlled as follows: vehicle speed was 72 km/h and two types

of road surfaces were employed for controller performance evaluation including a

34


2.5 Adaptive Neural Network Control

Figure 2.10: Scheme of the hydraulic active suspension system in Kucukdemiral

et al. (2005)

standard bump-type surface profile with 10cm length × 10cm height and a ran-

dom road profile generated to simulate stabilized road with 1cm × 1cm pebbles.

Four types of controllers were employed on the active suspension system. When

the standard bump-type surface profile was used, the proposed controller clearly

produced the shortest response time of 0.85s and the lowest peak value of 0.4

cm. Under the condition of random road input, the adaptive fuzzy SMC had

overwhelming success over other controllers. Besides, since it has a single input

FLC as the main controller, the rule base of FLC drastically decreased when it

was compared with traditional FLCs.


Due to their non-linear mapping and learning ability, ANNs have been one of

dominant methods for designing robust, adaptive and intelligent control systems

(Feng et al., 1997b). For further information on ANNs control systems please

refer to Agarwal (1997); Vemuri (1993).

35

Chapter1/Chapter1Figs/EPS/hass.eps


An adaptive non-linear controller is required for the non-linear and uncertainty

during operation in an active suspension system. For instance, Guo et al. (2004)

designed an adaptive controller with an ANNs-based identifier to control a semi-

active suspension with a magneto-rheological damper based on a quarter vehicle

model. The ANNs control system scheme is shown in Fig. 2.11. In principle, the

direct neural network control takes the error between the ideal reference signal

and the system response as the error of back propagation. However this error does

not offer good information for updating the weights of neural networks because

of potential uncertainty on the non-linear model. In Guo’s paper, an indirect

adaptive ANNs control strategy was proposed to approximate the input error.

The structure of the neural network controller is shown in Fig. 2.11. The NNC

was the neural network controller, the NNI was the neural network identifier,

and the TDL was tapped delay. Due to the unavailability of the inverse model of

the non-linear dynamic system, not only did the NNI trace the system response,

but it also calculated the back propagation error for the NNC. The topological

structure of the NNC consisted of three layers with 4 × 9 × 1 nodes, including

one hidden layer. The NNI structure was the same as the NNC. The sigmoid

function served as the activation function for both the hidden and output layers;

the back propagation algorithm was used to update the weights.

Figure 2.11: The scheme of indirect adaptive control based on ANNs in Guo et al.

(2004)

For evaluating the adaptive NN-based control system, numerical simulations

36


2.6 Genetic Algorithms Based Adaptive Optimization and Control

and experiments were carried out for the quarter vehicle equipped with a magneto-

rheological damper. The fundamental natural frequency of the quarter vehicle

model was chosen as 1.8 Hz and the road profile was given based on the road

classification of the ISO database. The numerical simulation and experiment

results convincingly showed the vertical acceleration of vehicle body to be con-

siderably reduced with the indirect ANNs controller than the traditional ANNs

controller. For example, the root-mean-square (RMS) of acceleration of the vehi-

cle body subject to the random road disturbance of C grade was reduced by 38%

when the direct ANNs controller was used and by 55% when the indirect adap-

tive ANNs controller was implemented in numerical simulation. In comparison

with passive suspension, the semi-active suspension with indirect adaptive ANNs

controller reduced the acceleration of the vehicle body under the sinusoidal road

excitation of C grade by 41% in the experiment. On the other hand, the indirect

adaptive NNC worked very fast since the neural networks included only a single

hidden layer and the neural network identifier received good training before the

experiments took place.

2.6 Genetic Algorithms Based Adaptive Opti-

mization and Control

Genetic algorithms, one kind of stochastic global optimization techniques, has

been successfully applied in a variety of research and industrial fields, especially

in optimisation and control(Baumal et al., 1998; Davis, 1991; Goldberg, 1989;

Holland, 1975; Moran & Nagai, 1993; Tsao & Chen, 2001, 1997, 1998). For

instance, GAs have demonstrated their effectiveness in multiple peak problems

with local optimum solutions with approval in robust search around complex

spaces. The main difference between GAs and conventional optimisation and

search procedures are: 1) working with a coding set of the parameters, not the

parameters themselves; 2) searching from a population of points, not a single

point, they are capable of handling large search spaces; 3) using probabilistic

transition rules, rather than deterministic ones(Kaynak, 1998). However, it also

37


needs to be pointed out that the main disadvantage of GAs is that their optimal

speed is too slow to use in real-time applications.

Considering the control strategy in active suspension systems, Baumal et al.

(1998) utilised the GA in a five DOF half vehicle model shown in Fig. 2.12. In

their research, all the involved parameters were comprised into one constraint

optimal description with eight unknown parameters and seven constraints. That

means the active control and passive mechanical parameters were the designed

variables to be optimised. Two active elements provided forces proportional

to the absolute vertical velocity of the points on the car body directly above

the rear and front wheels. These devices, characterised by proportionality con-

stants cf and cr, were known as skyhook dampers. The design variables were

the set:x=k1, c1, k3, c3, cr, k4, c4, cf. The constraints were obtained from the

three performances of vehicle suspension systems: 1) ride comfort; 2) road-holding

ability; 3) the suspension working space. Two constraints were for the body ac-

celeration and the seat acceleration. The other five constraints were for the seat,

suspension and tyre deflections. Given the optimised initial set, there were three

steps to implement the genetic algorithm. Firstly, the reproduction, which was

performed by copying a current generation string into a new population (par-

ent pool) according to the chosen fitness. The fitness depended on the objective

function value and constrained violations. Secondly, the crossover, which was

the exchange of design characteristics among randomly selected pairs from the

parent pool. Finally, the mutation was achieved by switching a 0 with a 1, or

vice-versa, at a binary site. The algorithm stopped when the maximum fitness

design comprised at least 30% of a newly created generation. The reproduction

stage itself was a simulation of the survival of the fittest designs. Moreover, in

order to improve the efficiency of the GA, the binary strings and fitness values

for each unique design of the current generation were stored in a linear search

look-up table. If a design string in the next generation matched one in the ta-

ble, then the fitness did not have to be re-calculated. This significantly avoided

the GAs weakness by improving computing time, especially for expensive fitness

evaluations. With five independent runs of the GA, the optimal values were ob-

tained and compared with the local optimisation search technique and the passive

suspension design. The results showed that the proposed GA can carry out the

38


best parameters with the least computing time among the three methods. The

active and passive suspension system seat acceleration responses were compared

to evaluate its dynamics performance. The response of the active system showed

that the road disturbance had little effect on the seat acceleration, and indicated

that GAs had strong potential to incorporate global optimisation methods for

suspension system design.

Figure 2.12: The 5 degree-of-freedom half vehicle model employed in Baumal

et al. (1998)

Tsao & Chen (2001) also proposed an active suspension force controller using

GAs with maximum stroke constraints based on their former research (Tsao &

Chen, 1997, 1998). In contrast to the traditional approach, the maximum absolute

values of suspension strokes were employed in the objective function to achieve

better ride comfort within the stroke limitation. A GA was employed to search

for the parameters of damping ratios and spring constants to achieve an optimum

trade-off among ride comfort, handling quality and suspension stroke limitation

simultaneously. The force control scheme of a half-vehicle model is shown in Fig.

2.13, where bdf and bdr denote the skyhook damping ratios, krf and krr denote

39

Chapter1/Chapter1Figs/EPS/5half.eps


the road-following spring stiffness coefficients, ktf and ktr denote the known tyre

stiffness coefficients, btf and btr denote the virtual tyre damping ratios, bcf and kcf

denote the damping ratio and spring stiffness for compensating front end dynam-

ics. Utilising this control scheme, the active suspension control was converged into

an optimal problem where eight parameters(bdf , bdr, krf , krr, btf , btr, bcf , kcf) were

optimised simultaneously by GA. Two driving conditions were tested on the ac-

tive force controller. One was a steep ramp road with forward speed V = 10m/s

and the other was a sinusoidal bump road with V = 40m/s. The simulations

were carried out for the three cases in each driving condition. Each case was

ended after 500 generation runs. The comparisons of the performance among

these cases showed that the proposed force controller using GA achieved great

ride and handling quality while the suspension stroke was restricted to be less

than or equal to the passive system. In terms of the dynamic performance, the

heave and pitch angle motions of the suspension system were shown and com-

pared with the passive suspension system. Especially considering the comparison

of the suspension displacement, the summation of the quadratic values of the

suspension displacements in the active system was larger than that of the passive

suspension. But the maximum displacement was smaller and the vibration had

been absorbed during the transient period. These results can explain why the

maximum absolute value instead of the summation quadratic form of suspension

displacement can achieve better performance.

Figure 2.13: The force control scheme with skyhook damper, virtual damper and

road-following spring in Tsao & Chen (2001)

40

Chapter1/Chapter1Figs/EPS/skyhook.eps

2.7 Integrated Adaptive Control Methods


Control strategies are reviewed in this section based on the combination of pre-

sented methodologies in previous sections.

2.7.1 Adaptive Neuro-fuzzy Control

Much attention has been paid on the combination of ANNs and fuzzy systems

(Nurnberger et al., 1999). The advantage is that the fuzzy systems can compen-

sate the tuning ability of their rules by using the learning algorithms of ANNs;

on the other hand, the ANNs system can also improve the transparency and

interpretability by rule-based fuzzy reasoning construction. Generally speaking,

an ANNs-fuzzy system can be viewed as a special 3-layer feed forward neural

network, and the fuzzy rules are trained by an ANNs algorithm. With both

advantages of neural network and fuzzy logic, the neuro-fuzzy system had been

successfully employed to solve a wide range of industry problems, especially on

non-linear and uncertain systems.

For instance, Dong et al. (2006) employed an adaptive ANNs-fuzzy controller

for a quarter-vehicle magneto-rheological suspension system. The control system

is shown in Fig. 2.14. This controller consisted of a Fuzzy Neural Network Con-

troller(FNNC) and a Time Delay Compensator(TDC). The FNNC calculated the

control force according to error and the change of the error, the TDC was a neu-

ral network model which predicted for compensating the suspension’s time-delay.

For the quarter-vehicle model, the input was the damper force, the output was

the sprung mass vertical acceleration, and road input was treated as a distur-

bance. The FNNC scheme shown in Fig. 2.15, where two linguistic variables

were the inputs of the network and seven fuzzy sets were defined for each input

as NB, NM, ZE, PS, PM, PB in the first layer; the second layer included 14

neurons to correspond to all the fuzzy sets; the third layer contained 49 neurons

to do the fuzzy reasoning based on the defined fuzzy rules. The FNNC output

41


was defined as:

u =

Rn∑i=1

µiwi

Rn∑i=1

µi

=∑

µiwi (2.59)

where Rn=49, and µ was the chosen Gaussian distribution. Weights wi of the

last layer were learned by back propagation algorithm. The simulation and ex-

perimental results showed that the proposed FNNC with TDC can significantly

reduce the acceleration peak value with 42.3% decrease in comparison of a passive

suspension system.

Figure 2.14: The adaptive neural network fuzzy control system with time-delay

compensator in Dong et al. (2006)

Figure 2.15: The FNNC scheme in Dong et al. (2006)

42

Chapter1/Chapter1Figs/EPS/annf.eps

Chapter1/Chapter1Figs/EPS/fnn.eps


Additionally, Wu et al. (2005) proposed a fuzzy controller based on the neuro-

fuzzy model for a half-vehicle active suspension system. The half-vehicle active

suspension was modelled as a non-linear system including heave, pitch and motion

of the front and rear wheels. The proposed neuro-fuzzy network was a self-

organizing inference network with 6 layers to derive the corresponding Takagi-

Sugeno (T-S) fuzzy model. The learning structure included both precondition

and consequence identification of fuzzy IF-THEN rules. Based on supervised

learning algorithms, the parameters of linear equations in the consequent parts

were adjusted by recursive least squares(RLS) algorithms, and the parameters in

the precondition part were adjusted by back propagation algorithm to minimize

a given cost function. Based on the T-S fuzzy model, a fuzzy controller was

designed to get the optimal active force. The simulation results showed the

proposed optimal fuzzy controller can improve the ride comfort by minimizing

both the displacements and accelerations of the vehicle centre and the pitch angle

simultaneously.

2.7.2 Adaptive Genetic-based Optimal Fuzzy Control

Due to the fact that IF-THEN rules in a fuzzy inference system are not always

available, automatic design methods and rule acquisition procedures for fuzzy

systems are required and have been proposed mostly based on GA and/or ANNs

in the past four decades. The key advantage of the hybrid system combining GA

and FL is that almost all the tasks of fuzzy system design can be accomplished

automatically. Because the GA can converge to the global optimal solution, FL

parameters of inference rules and membership functions are able to be determined

by a hybrid system itself. For GA-fuzzy control systems please refer to Herrera

et al. (1995, 1998); Homaifar & McCormick (1995); Magdalena et al. (2004).

Nawa et al studied a GA-fuzzy control system with the aid of pseudo bacte-

rial GAs (PBGA) and employed this controller to an active suspension system

shown in Fig. 2.16. Its encoding method is demonstrated in Fig. 2.17. Differ-

ing from the traditional canonical binary encoding, the parameters were put into

the chromosome, each of which encoded the rules of fuzzy system. Since every

43


Figure 2.16: The PBGA fuzzy control system in Nawa et al. (1999)

rule contained the information of antecedent and consequent variables, each chro-

mosome encoded the parameters of the membership functions. Triangular type

membership functions were employed so that the parameters of the membership

function were in pairs of centre and width as shown in Fig. 2.17. This encod-

ing method gave a high degree of freedom for the GA, which can optimise the

variables to be employed in the rules, the rules themselves, and the parameters

of membership functions. Therefore, this encoding was desirable to simultane-

ously evolve the rules and the membership functions, minimize the probability of

arriving at a local optimal point.

The GA algorithm can be briefly described as follows, 1) generation of the ini-

tial population; 2) genetic operations: mutation-evaluation-selection-replacement;

3) crossover and produce the new generation. An adaptive method was used

in a crossover operation instead of randomly deciding the chromosomes cutting

points. The adaptive crossover operator took into account the moving average

of the degrees of truth values of the fuzzy rules when deciding where to cut the

chromosome. The moving average was defined as the average of the accumulated

truth values of the rules. The accumulated truth value of a fuzzy rule was the

sum of the truth values for each one of the entries in the training data. It was

a measure of quality. If a rule possessed a high value of accumulated truth, it

meant that the rule was intensively and frequently triggered during the evalu-

ation process. Consequently, this was an indication of the utility and possible

44

Chapter1/Chapter1Figs/EPS/pbga.eps


effectiveness of that rule. On the other hand, if a rule possessed a low value of

accumulated truth, this was an indication that the rule did not play an important

role in the system. Four approaches were employed on the semi-active suspension

control system. The first method was a GA with fixed membership functions as

defined in Goldberg (1989). The second method is a GA with the possibility of

defining the membership functions and rules of a fuzzy controller simultaneously.

The third approach used the PBGA with a traditional crossover operator and

the fourth approach was the PBGA with adaptive crossover operation. The sim-

ulation results showed the proposed adaptive PBGA fuzzy controller worked well

to find out better rules and obtained the best performance of these four control

strategies. The results also indicated that this PBGA fuzzy controller focused

more on the actuation but the encoding methods increased the total number of

membership functions in the system.

Figure 2.17: An example of the fuzzy system encoded in a chromosome in Nawa

et al. (1999)

2.7.3 GA-ANNs Combined Control

A combination of a genetic algorithm and a neural network was employed to

design an active suspension controller by Tang & Zhang (2005). The GA searched

for the optimal acceleration of the vehicle body, which served as the objective

output of the ANNs control system. The neural network had two hidden layers

and the input, hidden and output neurons were 1, 10, 3 and 1, respectively. An

adaptive leaning rate was applied to decrease the training by keeping the learning

reasonably high, while insuring stable learning. The input of the ANNs was the

time response of the acceleration of the sprung mass, the objective output was the

optimised suspension control force. The proposed GA-ANNs combined controller

45

Chapter1/Chapter1Figs/EPS/ga.eps

2.8 Summary

and a LQG controller were employed to evaluate the control performance. The

simulation results demonstrated that the NN controller with optimal acceleration

parameters computed by GA-based optimisation provided better ride comfort in

the time domain.

2.8 Summary

Computational intelligence based adaptive control approaches are required due

to the real-time, non-linear and uncertain nature properties of active suspension

systems. This chapter provided an account of state of the art of adaptive ac-

tive suspension control systems with intelligent methodologies. Their advantages

and disadvantages are concluded based on theoretical analysis, analysing simu-

lations and the experimental results of the reviewed systems. In what follows,

we enumerate some open questions and scientific problems that suggest following

chapters and future research.

1. Employing intelligent control based on a more practical suspension model or

hybrid model with other vehicle control subsystems. Most of the reported

research on active suspension intelligent control has studied the suspension

performance under the linear quarter-vehicle or half-vehicle model. A com-

prehensive consideration of a real car can lead to a more practical model

which will bring further distinct functional and safety-related benefits. Also

a hybrid model will be convenient to integrate the other control subsystems,

such as brake control, steering control and anti roll control, etc., to a hy-

brid intelligent system and it will be benefit for analysis of a unit vehicle

performance.

2. Integration of multi-objective optimisation methods and fuzzy logic rea-

soning. Considering the trade-off between ride comfort and road handling

quality, the optimal objective will be changed with the requirements of

different road surfaces. Then the adaptive multi-objective optimisation

methods with high real-time computing efficiency need to be researched,

especially for the application of modern vehicle active suspension systems.

46

2.8 Summary

Some new intelligent methods, such as the type-2 fuzzy logic system, have

provided an alternative way for vehicle active suspension control.

3. Pursuing the balance of accuracy and interpretability in an active suspen-

sion intelligent control system such as FLC, ANNs or their integrating meth-

ods. Though intelligent systems have been widely investigated in many do-

mains, their future will lie in the careful integration of the best constituent

technologies beyond simply combining individual methods.

4. Evaluating intelligent control methodologies from the perspective of prac-

tical applications. It is necessary to build an evaluating system to compare

the different intelligent systems according to application requirements such

as computing cost, number of tuning parameters, closed-loop stability con-

ditions and the interface to faulty diagnosis, etc. The comparison results

will be beneficial to intelligent control system applications and to the future

research.

47

Chapter 3

Improved Vehicle Active

Suspension Model

3.1 Introduction

With the development of electronics technology, the requirements of ride comfort

and driving performance are major objectives in designing modern vehicles to

satisfy the expectations of customers. It is well known that the suspension system

is important when considering the ride comfort and handling capability. Hence,

the design of an appropriate suspension system is always an important research

topic for achieving the desired vehicle ride quality.

Three types of mathematical vehicle models consisting of a quarter-vehicle

model, half-vehicle model and full-vehicle model have been developed for analyz-

ing and designing active suspension control systems. These models have inves-

tigated the trade-off between the ride comfort and safety. Firstly, the quarter-

vehicle model was initially developed to explore active suspension capabilities

and gave birth to the concepts of skyhook damping and fast load leveling which

are now being developed toward actual, large-scale production applications(Gao

et al., 2006; Hrovat, 1997). Furthermore, the half-vehicle model including pitch

and heave modes was invented to simulate ride characteristics of a simplified

whole vehicle, which led to significant improvement in ride and handling (EI-

Demerdash et al., 1999; Mansour & Crolla, 2000; Wang et al., 2006; Yu et al.,

2000; Yu & Ma, 2005). The full-vehicle model had been proposed with the aim

48

3.2 A Rigid Tyre Model

of closely studying coupled states of four quarter-vehicle models and handling

trade-offs between ride comfort and safety (Hrovat, 1997; Yoshimura & Emoto,

2003).

Although the above three mathematical suspension models have been imple-

mented to design different active suspension control systems, few models con-

sidered the effect of vehicle speed change, such as acceleration or deceleration.

In fact, when the vehicle speed is significantly changed by braking or acceler-

ating, the variant rolling torque on wheels will break down the existed torque

balance of suspension system and affect the vehicle ride comfort and handling

performance. The key reason is the coupling phenomenon between the suspen-

sion system and the braking or driving system (Pacejka, 2006; Trachtler, 2004;

Yoshimura & Watanabe, 2003). Some research has been done on the effect of

latitude speed change for heavy vehicles, especially when the vehicles turned a

sharp corner (Lin et al., 1996; Sampson & Cebon, 2003; Wang & Shen, 2008).

Here, from a longitudinal speed change point, a rigid tyre model is built and in-

tegrated into a half vehicle model to present a more real and practical dynamics

for later research in Chapter 4.

This chapter is organized as follows: Section 3.2 analyses a rigid tyre model

and its non-linear dynamics by braking or accelerating are presented; by integrat-

ing the tyre model into the existing half-vehicle suspension model, an improved

half-vehicle active suspension model is proposed in Section 3.3; Section 3.4 de-

signs a linear optimal controller for evaluating the improved model; finally we

conclude this chapter with discussions in Section 3.5.


The kinematics of a pneumatic tyre are important for the vehicle performance

of ride comfort and handling. Generally speaking, there are three major sources

of tyre vibrations. They are brake torque fluctuations, road unevennesses, and

horizontal and vertical oscillations of the axle (Zegelaar & Pacejka, 1996). In this

section, the first two sources are considered to analysis the dynamics of the tyre

and build a proper model for later research on active suspension system.

49


The diagram of a simplified tyre model is shown in Fig. 3.1, which has been

used in research reported by Zegelaar & Pacejka (1996). This tyre model can

generate the typical tyre vibrations in the frequency range 0-100 Hz which ensures

the tyre tread-band behaves as a rigid body. The whole wheel is separated into

three parts, one is the rim and axle with index a, the second part is the tyre belt

with index b and the third part is the contact patch with index c. For consistency

with the coordinates used in Chapter 2, the vertical direction is noted as Z, the

latitudinal direction is noted as Y and the longitudinal direction is noted as X.

The rotational speed is defined as Ω. The motion equations of this tyre model

are described as follows.

Figure 3.1: The rigid tyre model

mwAc = Ffx (3.1)

mwzw = Fz (3.2)

Iayθa = Tby − Tay (3.3)

Iby θb = −rFcx − Tby. (3.4)

where mw is the mass of whole wheel, Ac is the longitudinal acceleration, Ffx is

the longitudinal resultant force, zw is the vertical displacement of the wheel, and

50

Chapter2/Chapter2Figs/EPS/tyre.eps


Fz is the vertical resultant force seen in Chapter 2. Iay and Iby are the inertia

about the y-axis of the rim and tyre belt, respectively. The torques of rim and

tyre belt are Tay and Tby. Fcx is the longitudinal force in the contact patch.

The main aim is to build the relation between the longitudinal force Fcx and

vehicle forward speed V or wheel rotational velocity Ω. Since the longitudinal

force is related with vertical force and the tyre slip conditions, here, the tyre slip

characteristics are presented by a brush type model which describes the realistic

slip performance using three parameters: half the contact length la, the tread

element stiffness per unit of length cpx and the friction coefficient lµ. Then the

longitudinal force Fcx is presented as:

Fcx =

lµFz

3 |θxζcx| − 3 |θxζcx|2 + |θxζcx|3

sgn(ζcx), |ζcx| ≤ 1

θx

lµFzsgn(ζcx), |ζcx| > 1θx

(3.5)

where, the tyre parameters θx is:

θx =2cpxl

2a

3lµFz

(3.6)

and the slip function ζcx can be solved by the first order slip model:

σcζcx + Vcrζcx = −Vcsx (3.7)

Here, the rolling velocity Vcr=rΩ, r is defined as effective rolling radius, and

the slip velocity Vcsx can described as:

Vcsx = V − Vcr = V − rΩ (3.8)

Only considering the case of full adhesion, the relaxation length of contact patch

σc is given by

σc =1

2la. (3.9)

From the equations 3.5 - 3.9, based on the tyre model, the relationship is

built between the longitudinal force of the tyre and the vehicle forward speed.

The longitudinal force will affect the vehicle body pitch motion induced by the

suspension system. Then a more faithful vehicle suspension model is proposed in

Section 3.3 to integrate the pitch motion with the vertical dynamic.

51

3.3 The Improved Half-vehicle Active Suspension Model

3.3 The Improved Half-vehicle Active Suspen-

sion Model

Compared with the typical half-vehicle active suspension model which has been

shown in Chapter 2.2.1, an improved model is presented and its structure is shown

in Fig. 3.2. Let f and r denote the front and rear, x and z be the longitudinal

forward direction and vertical up direction, and additional variables are defined

as,

h: body center of gravity height from ground(m);

Ac: vehicle longitudinal forward acceleration(m/s2);

Fx: longitudinal force from tyre(N);

Fz: vertical force from road surface(N);

The vertical and longitudinal dynamics of this half-vehicle active suspension

Figure 3.2: The half vehicle model

system are represented and the motion equations of force balance and torque

balance can be written as follows:

zwfmwf = ftf − Ff (3.10a)

52

Chapter2/Chapter2Figs/EPS/halfcarmodel.eps


zwrmwr = ftr − Fr (3.10b)

zbmb = Ff + Fr (3.10c)

θIb = −dfFf + drFr − dtfFxf − dtrFxr. (3.10d)

Here, ftf and ftr are the front and rear vertical tyre forces, Ff and Fr are the

front and rear resultant forces of the suspension system, and Ff = faf +fsf +fdf ,

Fr = far +fsr +fdr. faf and far are front and rear actuator forces, fsf and fsr are

front and rear spring forces, fdf and fdr are front and rear damping forces. Fxf and

Fxr are the longitudinal forces from the front tyre and rear tyre, dtf = zbf−zwf +h

and dtr = zbr − zwr + h, θ is the pitch angle which is described by equations 3.11

- 3.12, θ is the pitch angle acceleration of the vehicle body.

zbf = zb − df · θ, zbr = zb + dr · θ (3.11)

From equation 3.11, the pitch angle can be written as:

θ =zbr − zbf

df + dr

(3.12)

Based on the half-vehicle active suspension structure in Fig. 3.2, the equations

3.10a - 3.10d are rewritten as follows:

zwfmwf = Ktf (zf0 − zwf ) − [faf + cf (zwf − zbf ) + Ksf(zbf − zwf)] (3.13a)

zwrmwr = Ktr(zr0 − zwr) − [far + cr(zwr − zbr) + Ksr(zbr − zwr)] (3.13b)

zbmb = faf + cf(zwf − zbf ) + Ksf(zwf − zbf ) + far

+ cr(zwr − zbr) + Ksr(zwr − zbr)(3.13c)

θIb = −df [faf + cf(zwf − zbf ) + Ksf(zwf − zbf )] − Fxf(zbf − zwf + h)+ dr[far + cr(zwr − zbr) + Ksr(zwr − zbr)] − Fxr(zbr − zwr + h).

(3.13d)

Different from the typical model in Chapter 2, the torque balance equation

3.13d includes the tyre dynamics which are described in Section 3.2. With the

53


equation 3.5, the relation between the longitudinal force and vehicle speed is non-

linear. In order to evaluate the improved suspension model, the improved linear

half-vehicle model with linearized longitudinal force is described in Section 3.3.1

and its performance is analysed in Section 3.3.2. With the non-linear longitudinal

force, the improved non-linear half-vehicle model is implemented in Section 3.3.3

and its performance is analysed in Section 3.3.4.

3.3.1 The Linear Half-vehicle Model

A linear half-vehicle model with linear longitudinal force is proposed in this sec-

tion. The non-linear longitudinal force with the rigid tyre model is described in

equations 3.5 - 3.9. For simplicity, the longitudinal force is linearized by ignoring

the high order terms and it can be rewritten as:

Fx = lµFz · 3θxζcx (3.14)

With the normal stationary solution of the first order slip model (i.e. ζcx=-

Vcsx/Vcr), substituting the equations 3.6 - 3.9 into the equation 3.14, then the

linear longitudinal force is defined in equation 3.15.

Fx = −2cpxl2a

Vcr

V + 2cpxl2a (3.15)

Assuming the longitudinal force from front tyre (Fxf) is the same as the force

from the rear tyre (Fxr) (i.e., Fxf = Fxr = Fx), substituting equation 3.15 into the

equations 3.13a - 3.13d, the active suspension dynamic equations are rewritten

as follows,

zbf = ( 1mb

+d2

f

Ib)[faf + cf(zwf − zbf ) + Ksf(zbf − zwf)] + ( 1

mb− df dr

Ib)[far

+cr(zwr − zbr) + Ksr(zbr − zwr)] +Fxdf

Ib(zbf − zbr − zwf + zwr)

(3.16a)

zbr = ( 1mb

− df dr

Ib)[faf + cf (zwf − zbf) + Ksf(zbf − zwf)] + ( 1

mb+ d2

r

Ib)[far

+cr(zwr − zbr) + Ksr(zbr − zwr)] − Fxdr

Ib(zbf − zbr − zwf + zwr)

(3.16b)

which incorporate the relationship between vehicle suspension vertical dynamics

and pitch motion into the model.

54


Using the same road inputs as in equations 2.21a - 2.21b, the state vector of

the half-vehicle active suspension system is defined by equation 3.17.

X =[

zbr zwr zbf zwf zbr

zwr zbf zwf zr0 zf0

]T (3.17)

Combining the improved half-vehicle active suspension motion equations 3.16a,

3.16b, and road input equations 2.21a and 2.21b, the system equation and output

equation in state space form can be rewritten in equations 3.18a and 3.18b.

X(t) = AnewX(t) + BU(t) + Fw(t) (3.18a)

Y (t) = CnewX(t) + DU(t) + v(t) (3.18b)

In comparison to the typical state space model in equations 2.23a - 2.23b, only

matrices Anew, Cnew are changed due to the introduction of longitudinal force

changes, given in equations 3.19 and 3.20.

Anew =

0 0 0 0 α1Ksr − α5 −α1Ksr + α5

0 0 0 0 − Ksr

mwr

Ksr−Ktr

mwr

0 0 0 0 α2Ksr + α4 −α2Ksr − α4

0 0 0 0 0 01 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 0 00 0 0 0 0 0

α2Ksf − α5 −α2Ksf + α5 0 00 0 Ktr

mwr0

α3Ksf + α4 −α3Ksf − α4 0 0

− Ksf

mwf

Ksf−Ktf

mwf0

Ktf

mwf

0 0 0 00 0 0 00 0 0 00 0 0 00 0 −2πf0 00 0 0 −2πf0

(3.19)

55


Cnew =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

α2Ksr + α4 0 0 α1Ksr − α4 1 0−α2Ksr + α4 0 0 −α1Ksr − α4 −1 1α3Ksf − α5 1 0 α2Ksf + α5 0 0−α3Ksf − α5 −1 1 −α2Ksf + α5 0 0

0 0 0 0 0 −10 0 −1 0 0 0

T

(3.20)

where α4 denotes (df Fx

Ib) and α5 denotes (drFx

Ib), and B, D, F, U, W, α1, α2, α3, α4

are described in Chapter 2, Section 2.2.1.

3.3.2 Linear Model Performance Analysis

For evaluating this improved linear half-vehicle active suspension model, some

simulations are done to demonstrate the vertical and pitch vibrations which are

not repeated by other existing models. The nominal parameters of half-vehicle

active suspension system are given in Table 3.1.

Table 3.1: Nominal parameters of half-vehicle active suspension model

parameters values parameters values

mb (Kg) 1794.4 Ktf (N/m) 101115

mwf (Kg) 87.15 Ktr (N/m) 101115

mwr (Kg) 140.04 df (m) 1.3

Ib (Kg m2) 3443.05 dr (m) 1.5

Ksf (N/m) 66824.2 cf (N s/m) 1190

Ksr (N/m) 18615 cr (N s/m) 1000

f0 (Hz) 0.1 cpx (N/s2) 1.7 × 106

la (m) 0.053 h (m) 0.5

In the field of suspension control analysis, researchers use various common

indexes to compare the performance of different systems. Differences in perfor-

mance might be due to damper type, suspension category, or even the suspension

56


control method. The most commonly used benchmarking indexes for ride com-

fort, road handling, and pitch motion performances are used in this thesis. That

is, the performance of ride comfort is indexed by the vehicle body vertical acceler-

ations, the performance of handling is embodied by the suspension displacements

(e.g., the suspension travel and tyre dynamic loading), and the pitch angle is

used to compare the pitch motion of suspension system (Crolla & Abdel, 1991;

EI-Demerdash et al., 1999; Liu et al., 2008b).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time(s)

θ (

rad)

Vcr=20

Vcr=17

Vcr=14

Figure 3.3: The pitch angles of a vehicle body with different Vcr

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.5

0

0.5

1

1.5

Time(s)

Ve

hic

le b

od

y a

cce

lera

tio

ns (

m/s

2)

Vcr=20

Vcr=17

Vcr=14

Figure 3.4: The accelerations of front vehicle body with different Vcr

57

Chapter2/Chapter2Figs/EPS/pass-ang.eps

Chapter2/Chapter2Figs/EPS/pass-fa.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time(s)

Ve

hic

le b

od

y a

cce

lera

tio

ns (

m/s

2)

Vcr=20

Vcr=17

Vcr=14

Figure 3.5: The accelerations of rear vehicle body with different Vcr

In order to analyse the natural vertical and pitch dynamics of the improved

half-vehicle suspension model, the active forces are ignored, the average forward

speed V is 20 m/s and the rolling velocity Vcr are chosen as 20 m/s, 17 m/s

and 14 m/s. From the longitudinal tyre force in equation 3.15, when the forward

speed V is the same as the rolling velocity Vcr, the longitudinal effect to pitch

motion disappears and the improved model is reduced to the typical model. With

step road inputs (z0 = 0.1 m ), the pitch angles of a vehicle body with different

rolling velocities are shown in Fig. 3.3. The comparison of accelerations of the

front and rear vehicle body with different rolling velocities are shown in Fig. 3.4

and Fig. 3.5. The front and rear suspension travel and tyre dynamic loadings

are compared in Fig. 3.6 - 3.9. These simulation results demonstrate that the

longitudinal tyre force can significantly affect the pitch angles of a vehicle body

which are not presented by the existing typical model. Also it is evident that

the longitudinal tyre force has a very small effect on the vertical vibration of

the vehicle body because the vertical accelerations, suspension travels and tyre

dynamic loadings of the improved model are similar with the typical model.

58

Chapter2/Chapter2Figs/EPS/pass-ra.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time(s)

The s

uspensio

n tra

vel (m

)

Vcr=20

Vcr=17

Vcr=14

Figure 3.6: The front suspension travel with different Vcr

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time(s)

The s

uspensio

n tra

vel (m

)

Vcr=20

Vcr=17

Vcr=14

Figure 3.7: The rear suspension travel with different Vcr

59

Chapter2/Chapter2Figs/EPS/pass-fst.eps

Chapter2/Chapter2Figs/EPS/pass-rst.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−12

−10

−8

−6

−4

−2

0

2

4x 10

−3

Time(s)

Th

e t

yre

dyn

am

ic lo

ad

ing

(m

)

Vcr=20

Vcr=17

Vcr=14

Figure 3.8: The front tyre dynamic loading with different Vcr

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10

−8

−6

−4

−2

0

2

4x 10

−3

Time(s)

Th

e t

yre

dyn

am

ic lo

ad

ing

(m

)

Vcr=20

Vcr=17

Vcr=14

Figure 3.9: The rear tyre dynamic loading with different Vcr

60

Chapter2/Chapter2Figs/EPS/pass-ftd.eps

Chapter2/Chapter2Figs/EPS/pass-rtd.eps


3.3.3 The Non-linear Half-vehicle Model

It is evident that a linear system approximation is not appropriate for highly non-

linear systems with uncertain input factors. This is because non-linear effects are

usually amplified in such systems.

For a vehicle suspension system, except for the tyre longitudinal non-linear

dynamics mentioned in Section 3.2, the non-linear dynamics could be created

by dry friction and discrete-event disturbances (e.g., single bumps or potholes),

which cause a highly non-linear phenomenon. The non-linear properties mainly

arise from two sources. The first is the bump stop which restricts the wheel travel

within a given range and prevents the tyre from contacting the vehicle body; the

other is the strut bushing which connects the strut with the body structure and

reduces harshness from the road input. Based on the method in Kim & Ro

(1998), the connecting forces (e.g., spring force, damping force, tyre force) can

be modelled as non-linear functions using measured data. Here, the spring force

fs is estimated by high-order polynomial functions as shown in equation 3.21,

fs = fsl + fsn = k1∆zs + (k0 + k2∆z2s + k3∆z3

s ), (3.21)

where fsl is the linear term of the spring force and fsn is the non-linear term

of the spring force, ∆zs is the displacement of suspension. The coefficients were

obtained by fitting the equation to experimental data. The damping force fd were

also modelled as a second-order polynomial function by fitting the measured data,

as given below,

fd = fdl + fdn = c1∆zs + c2∆zs2, (3.22)

where fdl is the linear term and fdn is the non-linear term of the damper force.

Besides the non-linear properties presented by the spring force and damping

force, the vertical tyre force is also highly non-linear, especially when there are

substantial load changes. The vertical tyre force becomes zero when the tyre loses

contact with the road surface. The tyre force was modelled as below,

ft :

ftl = kt∆zt, zt > 0ftn = 0, zt ≤ 0.

(3.23)

61


where ft is the vertical tyre force, ftl denotes the linear tyre force, ftn denotes

the non-linear tyre force and ∆zt is the tyre displacement.

Integrating all the above mentioned non-linear dynamics in an active suspen-

sion system, this non-linear model is provided below for the further research in

this thesis. Also this non-linear model is built as a reference model of real vehicle

active suspension system for all further simulations of this thesis.

With the non-linearity shown in equations 3.5, 3.21, 3.22, 3.23, the active

suspension system can be written as a MIMO non-linear model:

X = F (X, U) (3.24)

where X denotes the state matrices which includes the displacements and

velocity of the vehicle body (i.e., Zb and Zb), suspension (i.e., Zw and Zw) and

road surface input (i.e., Z0 and Z0) and U denotes the actuator force matrices

(i.e., faf and far), F (X, U) is a non-linear function which presents the suspension

non-linear dynamic description which can be obtained by integrating the linear

model and the non-linear forces (i.e., fs ,ft, fd and Fx). The performance analysis

of this non-linear model is presented in Section 3.3.4.

3.3.4 Non-linear Model Analysis

In order to demonstrate the non-linear characteristics of a vehicle suspension

model described in Section 3.3.3, the simulations without active control forces

are done to show the natural vertical and pitch motions. Since most of the non-

linear dynamics occur around the suspension minimum and maximum travel, a

0.2m step road input is used to simulate a very sharp bump. The parameters

of the vehicle suspension can be found in Table 3.1 and the coefficients of the

non-linear forces are given in Table 3.2. Here, subscript f and r denote the front

and rear.

The vertical accelerations of the front and rear vehicle body are shown in Fig.

3.10 - Fig. 3.11. In comparison with the linear model described in Section 3.3.2,

the non-linear dynamics occurred in the first second when the suspension travel

was close to reaching its maximum expansion and contraction. With the passage

of time, when the vertical vibrations are reduced into the linear ranges by the

62


Table 3.2: The coefficients of non-linear forcesparameters values parameters values

k0f (N) -236 k1f (N/m) 66824

k2f (N/m2) -403 k3f (N/m3) 104

k0r (N) -146 k1r (N/m) 18615

k2r (N/m2) -265 k3r (N/m3) 284

c1f (N s/m) 1190 c2f (N s2/m) 426

c1r (N s/m) 1000 c2r (N s2/m) 215

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

Ve

hic

le b

od

y a

cce

lera

tio

ns (

m/s

2)

linear

nonlinear

Figure 3.10: The vertical accelerations of front vehicle body with non-linear model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

Ve

hic

le b

od

y a

cce

lera

tio

ns (

m/s

2)

linear

nonlinear

Figure 3.11: The vertical accelerations of rear vehicle body with non-linear model

63

Chapter2/Chapter2Figs/EPS/pass-fa-non.eps

Chapter2/Chapter2Figs/EPS/pass-ra-non.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time(s)

θ (

rad)

linear

nonlinear

Figure 3.12: The pitch angles of vehicle body with non-linear model

dampers, the vertical dynamics of this non-linear model are similar to the linear

dynamics. The other performance indexes of a non-linear model, that is, the

suspension travel and tyre dynamic loading, shown in Fig. 3.13 - Fig. 3.16, can

also demonstrate similar dynamics from non-linear to linear.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time(s)

The s

uspensio

n tra

vel (m

)

linear

nonlinear

Figure 3.13: The front suspension travel with non-linear model

Fig. 3.12 shows the pitch angles of a non-linear model and a linear model.

From these pitch motion dynamics, the non-linear model can represent the pitch

motion as a linear model.

64

Chapter2/Chapter2Figs/EPS/pass-ang-non.eps

Chapter2/Chapter2Figs/EPS/pass-fst-non.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time(s)

The s

uspensio

n tra

vel (m

)

linear

nonlinear

Figure 3.14: The rear suspension travel with non-linear model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Time(s)

The tyre

dynam

ic loadin

g (

m)

linear

nonlinear

Figure 3.15: The front tyre dynamic loading with non-linear model

65

Chapter2/Chapter2Figs/EPS/pass-rst-non.eps

Chapter2/Chapter2Figs/EPS/pass-ftd-non.eps

3.4 The Improved LQG Design

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Time(s)

The tyre

dynam

ic loadin

g (

m)

linear

nonlinear

Figure 3.16: The rear tyre dynamic loading with non-linear model

Since all the above dynamics of the proposed non-linear suspension model

are closer to the real physical dynamics, further research in this thesis will use

this non-linear model as the reference suspension model to design the intelligent

active suspension control system and evaluate its ride comfort, handling and pitch

motion performance.


In order to further test the proposed suspension model, especially closed-loop

evaluation with a controller, an improved LQG controller is designed to improve

not only the ride comfort and handling performance, but also the pitch motion.

3.4.1 The Improved LQG

Considering the improved model in Section 3.3.1 and the typical LQG design in

Chapter 2, a new term is added to the control performance index, and shown in

equation 3.25

J = limT→∞

1T

∫ T

0[q1(zwf − zf0)

2+q2(zbf − zwf)2

+ρ1zbf + q3 (zwr − zr0)2 + q4(zbr − zwr)

2

+ρ2z2br + q5(zbr − zbf )

2]dt

(3.25)

66

Chapter2/Chapter2Figs/EPS/pass-rtd-non.eps


where q5 is a new weight value to the control performance of pitch angle. With

this new control performance equation 3.25 and the state space model in Section

3.3.1, the improved optimal control force can be described by equation 3.26.

U(t) = −R−1BTPX(t) = −KnewX(t). (3.26)

where Knew represents the control gain matrix; and P is the solution of the

classical algebraic Riccati equation in equation 2.16.

3.4.2 Simulation Results

Initially, the response of the improved active suspension model to a step road in-

put is investigated with different acceleration values based on an LQG controller,

as this provides an insight into the relation between vehicle body pitch angle and

longitudinal tyre forces. Furthermore, the control performance of three control

strategies (LQG, Improved LQG (ILQ) and Wheelbase preview LQG (WLQ)) are

assessed on the improved model with a step road input and different longitudinal

acceleration values. More detail about the WLQ can be found in paper (Yu et al.,

2000). With the random road input, three control strategies are also used on the

improved model to compare their control performance.

The nominal values of parameters for the half-vehicle suspension model are

given in Table 3.1, and the coefficients of calculating control gains used in the

simulation are given in Table 3.3.

Table 3.3: Random road input parameters and the weighting parameters

parameters G0 U0 ρ1 ρ2 q1 q2 q3 q4

values 5 × 10−6 30 1 1 80000 100 80000 100

3.4.2.1 Step Road Inputs

In order to examine the dynamic effect of speed changes to the vehicle body

pitch angle, the response of the improved linear half-vehicle suspension model

to step road input is carried out under different rolling velocities. The typical

LQG controller is used and the original forward velocity is 30m/s. A step input

67


of 0.1m is applied on the vehicle model to simulate the sharp road vibration.

Fig. 3.17 shows pitch angle comparison with different rolling velocities as 35m/s,

34m/s, 32m/s, 30m/s, 28m/s, 26m/s, 25m/s. The simulation results showed

that different rolling velocities caused different pitch angles. Higher velocity gap

values brought larger additional pitch angle from zero-velocity-gap pitch angle,

and vice versa.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

pitch a

ngle

(ra

d)

time(second)

Vcr

=35

Vcr

=34

Vcr

=32

Vcr

=30

Vcr

=28

Vcr

=26

Vcr

=25

Figure 3.17: Pitch angle comparison with typical LQG and different rolling ve-

locities

For constraining the additional pitch angle to obtain better vehicle pitch mo-

tion performance, the second simulation is carried out under the improved lin-

ear half-vehicle suspension model, the same rolling velocity and different control

strategies. One controller is the typical LQG controller without considering the

pitch motion information in control performance index as in equation 2.28. Its

control results of pitch angle are shown in Fig. 3.18 by a solid line. The second

controller is the ILQ based on equation 3.25, and weight value q5 is 1. Its control

results of pitch angle are shown in Fig. 3.18 by dot line. The third controller is

the WLQ. The control results are shown in Fig. 3.18 by dash dot line.

68

Chapter2/Chapter2Figs/EPS/acc_angles.eps


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

time(second)

pitch a

ngle

(ra

d)

wheelbase preview LQ

basical LQ

improved LQ


improved LQ

basical LQ

Figure 3.18: Pitch angle comparison with different controller at Vcr= 35 m/s

(upper three lines) and 25 m/s (lower three lines)

As shown in Fig. 3.18, regarding vehicle body pitch angle, the proposed ILQ

controller obtained smaller overshoot and stable values than the other two con-

trollers. The WLQ controller obtained nearly the same stable values as the LQG

controller. But it is sensitive to velocity changing and the dynamic performance

became unsatisfactory when the vehicle speed was significantly changed.

3.4.2.2 Random Road Inputs

In order to provide approximate road input conditions, a random road surface

model is used, which is mentioned in Chapter 2, Section 2.2.1. The road surface

with a roughness coefficient of 5 × 10−6 is used. The vehicle forward speed is

30m/s. With different Vcr values and different controllers, the pitch angle vari-

ance were shown in Fig. 3.19 and Fig. 3.20, and the RMS values of Body

Vertical Acceleration (BVA), Suspension Displacement (SD), Tyres Dynamic

Loading (TDL) and Body Pitch Angle (BPA) were shown in Table 3.4, Table

3.5, Table 3.6.

As shown in Fig. 3.19 and Fig. 3.20, the ILQ controller obtained smaller

69



dynamic and stable values than the other two controllers. The wheel-based pre-

view controller was sensitive to velocity changing and the dynamic performance

became unsatisfactory.

As shown in Table 3.4, all the control performance of the ILQ controller were

the same as LQG controller without considering the longitudinal force. The WLQ

controller obtained better control performance, especially in the rear wheel and

suspension.

Table 3.5 and Table 3.6 showed the effects of the tyre longitudinal force chang-

ing to the vehicle performance, especially to vehicle body pitch angle. The ILQ

controller obtained smaller pitch angle than LQG and WLQ with the rolling ve-

locities 35m/s and 25m/s. The other performance of the vehicle (i.e., BVA, SD,

TDL ) changed a little by using three controllers under different rolling velocities.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

time (second)

pitch a

ngle

(ra

d)


basical LQ

improved LQ

Figure 3.19: Pitch angle comparison with different controller and random road

input ( Vcr=35m/s)

70

Chapter2/Chapter2Figs/EPS/lq1.eps

3.5 Summary

Table 3.4: Vehicle performance comparison with Vcr = 30m/s

BV Af SDf TDLf BV Ar

LQ 0.7731 0.0079 0.0030 0.8349

ILQ 0.7731 0.0079 0.0030 0.8349

WLQ 0.7668 0.0079 0.0030 0.7716

SDr TDLr BPA Vcr

LQ 0.0080 0.0032 0.0025 30

ILQ 0.0080 0.0032 0.0025 30

WLQ 0.0052 0.0027 0.0026 30

Table 3.5: Vehicle performance comparison with Vcr = 35m/s


LQ 0.7708 0.0076 0.0030 0.8400

ILQ 0.7707 0.0079 0.0030 0.8401

WLQ 0.7636 0.0077 0.0030 0.7709

SDr TDLr BPA Vcr

LQ 0.0084 0.0031 0.0049 35

ILQ 0.0083 0.0032 0.0044 35

WLQ 0.0072 0.0027 0.0108 35

3.5 Summary

In this chapter, the improved half-vehicle active suspension models which inte-

grate tyre longitudinal dynamics were proposed. Their dynamic open-loop and

closed-loop response to different road surfaces have been demonstrated.

To more precisely describe the suspension dynamics, a tyre dynamic model

was introduced to connect the tyre longitudinal forces with the vehicle suspension

model. The effect of tyre longitudinal dynamics were represented by two improved

half-vehicle active suspension models. One was the linear model to demonstrate

the typical linear response to road vibrations. It was also used to design the linear

controller, such as the LQG controller. The other was the non-linear model which

can show the real spring, damping forces and tyre non-linear dynamics, especially

when the suspension travel reached its physical limitations.

71

3.5 Summary

Table 3.6: Vehicle performance comparison with Vcr=25 m/s


LQ 0.7636 0.0081 0.0031 0.8306

ILQ 0.7664 0.0081 0.0031 0.8307

WLQ 0.7708 0.0082 0.0030 0.7731

SDr TDLr BPA Vcr

LQ 0.0076 0.0032 0.0105 25

ILQ 0.0077 0.0032 0.0096 25

WLQ 0.0062 0.0027 0.0137 25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

time (second)

pitch a

ngle

(ra

d)


basical LQ

improved LQ

Figure 3.20: Pitch angle comparison with different controller and random road

input (Vcr=25m/s)

72

Chapter2/Chapter2Figs/EPS/lq2.eps

3.5 Summary

All the proposed suspension models were evaluated from open-loop point

(without the active forces) to the closed-loop system (with the linear controller

for active suspension). The simulation results have shown that the improved

models provided a more accurate representation of the pitch motion dynamics of

an active suspension system than existing models. Also the non-linear model can

present more accurate vertical dynamics than the linear models. However, it was

hard to obtain satisfied control performances by the LQG controller because it

was designed to work with a linear model.

Then Chapter 4, a novel intelligent control strategy is proposed to deal with

these non-linear dynamics in the active suspension system and to improve the

capability of constraining the vertical and longitudinal vibrations to the vehicle

body.

73

Chapter 4

Interval Type-2 Fuzzy Control

System

4.1 Introduction

In order to improve the capability of active suspension systems, a lot of research

has been carried out on the design and control of active suspension systems in

the past two decades (Hrovat, 1997; Nagai, 1993; Sun, 2003). Simultaneously, the

control strategies of an active suspension system have been developed in a wide

range from initially linear quadratic controllers (e.g., LQG) to intelligent con-

trollers based on recently new findings in the field of computational intelligence.

To control an active suspension, the control algorithms must be able to deal

with mechanical non-linear dynamics and to operate under imprecise and uncer-

tain conditions, mainly caused by road surfaces. The mechanism behind FLCs

are credited with being a feasible methodology for designing robust controllers

that are able to deliver a satisfactory performance in face of non-linearity, un-

certainty and imprecision. The FLC has been successfully employed in some

practical suspension systems. Chen et al. (1995) proposed a FLC for an active

suspension under the sliding mode control (SMC) frame. The FLC was designed

to reduce the chattering of the SMC and to supervise the control performance to

self-tuning parameters of the control system. Zhang et al. (2007) also proposed

an adaptive fuzzy sliding mode controller for an active suspension system but its

sliding surface was different from Chen’s. Rao & Prahlad (1997) also presented a

74

4.1 Introduction

tunable FLC to bring down the suspension acceleration and deflection to a level

that of a hypothetical reference model. The suspension deflection and velocity

were the two input variables to the FLC while the desired force as its output.

Fernando & Viassolo (2000) proposed a controller consisting of two control loops

to minimize the vehicle body acceleration and to avoid hitting the suspension

limit. The outer loop was as same as Rao’s work, but additional inner loop was

designed to deal with the non-linear hydraulic actuator. More recent research

results can be found in (Cao et al., 2007, 2008; Chen & Huang, 2008; Gao et al.,

2006; Li et al., 2006, 2008; Lin & Lian, 2008; Liu et al., 2008b; Wang & Shen,

2008).

Though there are different ways to construct FLCs for a vehicle active sus-

pension, the most common way is to construct a FLC by eliciting the fuzzy rules

and the Membership Function (MF)s based on expert knowledge or experience.

A FLC handles the uncertainty associated with the inputs and outputs by using

precise and crisp MFs. This means that, once the MFs have been chosen, all

the uncertainty disappears (Hagras, 2004; Mendel & John, 2002). Then there is

the common problem that such a FLC limits introducing uncertain factors from

linguistic rules through predefined membership functions. Recently, fuzzy type-2

methods, which were introduced by Zadeh (1975), have been further developed

to improve the FLCs performance for handling the high level uncertainty (Karnik

& Mendel, 2001; Wu & Mendel, 2009).

Fuzzy type-2 method is that its fuzzy set is further defined by a typical fuzzy

membership function, i.e., the membership degree of belonging for each element

of this set is a fuzzy set in [0, 1], not a crisp number (Liang & Mendel, 2000).

Then the Interval Type-2 Fuzzy Membership Function (IT2MF) is three dimen-

sional and include a footprint of uncertainty, which provided additional degrees of

freedom that made it possible to directly model and handle uncertainty (John &

Coupland, 2007; Mendel, 2007a; Wu & Mendel, 2007, 2008). In comparison with

the type-1 Fuzzy Logic System (FLS), a type-2 FLS has the two-fold advantages

as follows. Firstly, it has the capability of directly handling the uncertain factors

of fuzzy rules caused by expert experience or linguistic description. Secondly, it

is efficient to employ a type-2 FLS to cope with scenarios in which it is difficult or

impossible to determine an exact membership function and related measurement

75

4.1 Introduction

of uncertainties. These strengths have made researchers consider type-2 FLS as

the preference for real-world applications (Astudillo et al., 2007; Hagras, 2007;

Liang et al., 2000; Liu, 2008a; Sepulveda et al., 2007).

About the type-2 FLS, some research has been done to study the set theoretic

operations, properties of membership grades and the uncertainty bounds of type-

2 fuzzy sets(John, 1998; Karnik & Liang, 1999; Mendel et al., 2007). From the

viewpoint of real-time application, Interval Type-2 (IT2)FLS has been widely

studied and utilized in many research fields, such as autonomous mobile robots

control, adaptive control of nonlinear system, noise cancellation, quality control

and wireless communications, etc. That is mainly due to the IT2FLS’s simple

computing methods and less computational expense on type reduction which is

still a bottleneck for other type-2 FLSs to be used in real-time applications.

According to the research of type-2 FLC, considering the real-time control

requirements, many researchers used interval type-2 fuzzy sets to solve the com-

putational complexity of general type-2 fuzzy sets and have brought some appli-

cations(Julio & Alberto, 2007; Wu & Mendel, 2002). However, the computational

expense on type reduction of type-2 FLS also is a bottleneck for a type-2 FLC in

real-time control applications(Mendel, 2007b). Some new alternative ways have

been provided to reduce the computational expense of type-2 FLC, such as the

method proposed by Wu & Mendel (2002). In Wu’s approach, they replaced the

type reduction with lower and upper bounds for the end-points of the type re-

duction sets, and those bounds can be computed without having to perform the

type reduction. But its drawback is the type reduction will be performed during

the design step. However, designing an efficient type-2 FLC with less calculation

and strong adaptive ability to overcome uncertainty of industrial applications is

still an open question.

Due to the difficulties of building proper crisp MFs from uncertainty of ex-

pert knowledge or experience for an active suspension system, inspiring from the

idea of type-2 fuzzy methods, an adaptive FLC with IT2MFs is proposed in this

chapter. A general structure to aggregate uncertainty bounds to the defuzzified

outputs through further optimization is built to improve the vehicle performance.

The main control aims are to minimize vehicle body acceleration to improve ride

76

4.2 Interval Type-2 Fuzzy Systems

comfort and reduce the tyre dynamic loadings to enhance the handling perfor-

mance. The further optimal goals are to reduce the suspension displacements

and tyre dynamic loadings with less control force.

In Section 4.2 the general background of interval type-2 FLS is introduced.

The framework of adaptive interval type-2 FLC is proposed in Section 4.3. Simu-

lation results on quarter and half vehicle active suspension systems are provided

in Section 4.4 and Section 4.5. This chapter is concluded in the Section 4.6.


The interval type-2 FLS has been widely studied in recent decade. Compared to

the general type-2 FLS, its primary advantage is its computational tractability

(Mendel, 2007a). In order to clearly describe the later controller design, the

background of IT2 fuzzy sets and fuzzy systems are introduced in this section.

4.2.1 The Interval Type-2 Fuzzy Sets

An interval type-2 fuzzy set is defined as below:

A =

∫

x∈X

∫

u∈Jx⊆[0,1]

1/(x, u) =

∫

x∈X

∫

u∈Jx⊆[0,1]

1/u

/x (4.1)

where x is the primary variable with domain X, u ∈ U is the secondary variable

with domain Jx at each x ∈ X. Jx is the primary membership of x and it is

defined in equation 4.2. The secondary grades of A are all equal 1.

Jx =

(x, u) : u ∈[µ

A(x), µ

A(x)]

(4.2)

As shown in Fig. 4.1, for a crisp input value a, its primary membership

grade µx(a) belongs to an interval:[µx(a), µx(a)] and all the secondary grades

f(µx(a)), f(µx(a)) are all equal to 1.

77


(a) The primary membership functions of crisp in-

put a(describing an interval type-2 fuzzy set)

(b) The secondary membership function of crisp in-

put a(describing an interval type-1 fuzzy set)

Figure 4.1: The interval type-2 fuzzy membership functions

By this interval type-2 fuzzy set, uncertainty about type-1 fuzzy set will be

conveyed by all the primary MFs. The primary MF can be separated into two

type-1 MFs, i.e., the Upper Membership Function (UMF) and the Lower Mem-

bership Function (LMF). The footprint of uncertainty of A can be described as:

FOU(A) =⋃

∀x∈X

Jx =⋃

∀x∈X

[µ

A(x), µA(x)

](4.3)

With two interval type-2 fuzzy sets A and B in equations 4.4 and 4.5, the interval

type-2 fuzzy set theory operations of union, intersection and complement can be

78

Chapter3/Chapter3Figs/EPS/intelty2mf0.eps

Chapter3/Chapter3Figs/EPS/intelty2mf1.eps


written in equations 4.6- 4.8.

A = 1/FOU(A) = 1/⋃

∀x∈X

[µA(x), µA(x)] (4.4)

B = 1/FOU(B) = 1/⋃

∀x∈X

[µB(x), µB(x)] (4.5)

A ∪ B = 1/⋃

∀x∈X

[µA(x) ∨ µ

B(x), µA(x) ∨ µB(x)] (4.6)

A ∩ B = 1/⋃

∀x∈X

[µA(x) ∗ µ

B(x), µA(x) ∗ µB(x)] (4.7)

A = 1/⋃

∀x∈X

[1 − µA(x), 1 − µA(x)] (4.8)

4.2.2 The Interval Type-2 Fuzzy System

With IT2 fuzzy sets, the framework of the IT2 fuzzy logic system is shown in Fig.

4.2. Basically, an IT2 fuzzy system is composed of five functional blocks: the Fuzzification interface which transforms the crisp inputs into fuzzy

values with linguistic knowledge. the Knowledge base which includes the rule base and database; the rule

base contains fuzzy if-then rules and the database defines the IT2 fuzzy

membership functions of IT2 fuzzy sets. the Decision-making unit which performs the inference operations on the

fuzzy rules; that is the fuzzy reasoning process. the Type-reduction which combines the output sets and performs a cen-

troid calculation to obtain the type-reduced sets (Type-1 fuzzy sets). the Defuzzification interface which transforms the fuzzy reasoning re-

sults to crisp outputs.

Based on this framework, the IT2 fuzzy system will work as follows:

79


Figure 4.2: The interval type-2 fuzzy logic system

1. The crisp inputs are fuzzified into IT2 fuzzy sets which are described by

IT2 membership functions. Then each crisp value will be transferred into a

fuzzy value.

2. These fuzzy values can active the fuzzy inference engine with the fuzzy rule

base to produce the fuzzy reasoning results. The fuzzy rules are same as

type-1 fuzzy system, but all fuzzy values are defined by interval membership

grades obtained from their IT2 fuzzy sets. Then the reasoning results are

all IT2 fuzzy values.

3. With a specially designed type-reduction and defuzzification method, the

IT2 fuzzy reasoning results are combined and calculated to the expected

crisp outputs. Then the IT2 reasoning process is implemented.

A simplified IT2 fuzzy system with ν inputs and one output is considered to

show the basic existed research results about the IT2 fuzzy systems. Assume

there are m IF-THEN fuzzy rules with the form:

R(l): IF x1 is F l1 and x2 is F l

2, . . ., and xν is F lν , THEN y is Gl.

where, l ∈ L := 1, 2, . . . , m.

80

Chapter3/Chapter3Figs/EPS/it2sys.eps


The IT2 MFs for all antecedent IT2 fuzzy sets are defined as µF lk(xk) (k =

1, . . . , ν) and the IT2 MFs for consequent is defined as µGl(y). The major result

is concluded by the following theorem (Liang & Mendel, 2000).

Theorem 1: In an interval singleton type-2 fuzzy system, using product or min-

imum t-norm, for input X = X ′:

1. The result of the antecedent operations is an interval type-1 fuzzy set as

follows.

F l(X ′) =[f l(X ′), f

l(X ′)

]≡[f l, f

l]

=[µ

F l1

(x′1) ∗ µ

F l2

(x′2) ∗ · · · ∗ µ

F lν

(x′ν), µF l

1(x′

1) ∗ µF l2(x′

2) ∗ · · · ∗ µF lν(x′

ν)]

(4.9)

2. The rule Rl fired output consequent set, µBl(y), is the IT2 fuzzy set:

µBl =

∫

bl∈[f l∗µGl(y),f

l∗µ

Gl (y)]

1/bl, y ∈ Y (4.10)

where µGi

(y) and µGl(y) are the lower and upper membership grades of

µGi(y).

3. Supposing that n of the m rules in the IT2 fuzzy system are fired, where

n < m, and the combined output fuzzy set, µB(y), is obtained as:

µB =

∫

b∈[(f1∗µ

G1 (y))∨···∨(fn∗µGn (y)),(f

1∗µ

G1 (y))∨···∨(fn∗µGn (y))

] 1/bl, y ∈ Y

(4.11)

4.2.3 Type-reduction and Defuzzification Methods

With the structure of IT2 system and Theorem 1 in Section 4.2.2, the type-

reduction process provides an IT2 fuzzy set which presents an interval of uncer-

tainty for the output of an IT2 FLS. That is, the uncertain information from

inputs and fuzzy rules can be transferred into the fuzzy results by the IT2 fuzzy

reasoning process. Meanwhile, as mentioned by Mendel & John (2002), the more

uncertainties that occur in an IT2 FLS, the larger the type-reduced fuzzy set

will be, vice-versa. The type-reduction method is important to fully present the

81


capability of covering uncertainty and directly relates to the computational cost

of the IT2 FLS. In this section, two kinds of existing type-reduction methods are

represented.

A general IT2 fuzzy output with different type-reduction methods can be

expressed as:

YTR(X ′) = [yLE(X ′), yRI(X′)] ≡ [yLE , yRI ]

=∫

y1∈[y1LE

,y1RI

]· · ·∫

ym∈[ymLE

,ymRI

]

∫f1∈[f1,f

1]· · ·∫

fm∈[fm,fm

]1/

∑ml=1

f lyl

∑ml=1 f l

(4.12)

where the multiple integral signs denote the union operation. In this thesis, the

center-of-set method is used for type-reduction. yiLE and yi

RI are the left and

right end-points of the centroid of the consequent of the ith rule, f i and fiare

the lower and upper firing degrees of the ith rule. They can be solved by the

equation 4.9.

There are no closed-form formula for yLE and yRI . For computing these two

values, two different methods are introduced.

The KM algorithm:

This method was introduced by Karnik and Mendel (Karnik & Mendel, 2001).

The end-point yRI :

1. The pre-computed yiRI are arranged in ascending order, i.e., y1

RI ≤ y2RI ≤

. . . ≤ ymRI .

2. Compute yRI by equation 4.14. The initial f iRI is (f i + f

i)/2.

yRI =∑m

l=1 f lRIy

lRI

/∑ml=1 f l

RI(4.13)

3. Let y′RI = yRI , find R (1 ≤ R ≤ m − 1) which can satisfy the condition:

yRRI ≤ y′

RI ≤ yR+1RI .

4. Compute the yRI by equation 4.14 with f iRI = f i for i ≤ R and f i

RI = fi

for i > R.

5. Let y′′RI = yRI , if y′′

RI 6= y′RI , then set y′

RI equal to y′′RI and turn to step 3.

6. If y′′RI = y′

RI , then let y′′RI = yRI , stop.

82


The end-point yLE:

1. The pre-computed yiLE are arranged in ascending order, i.e., y1

LE ≤ y2LE ≤

. . . ≤ ymLE.

2. Compute yLE by equation 4.14. The initial f iLE is (f i + f

i)/2.

yLE =∑m

l=1 f lLEyl

LE

/∑ml=1 f l

LE(4.14)

3. Let y′LE = yLE, find L (1 ≤ L ≤ m − 1) which can satisfy the condition:

yLLE ≤ y′

LE ≤ yL+1LE .

4. Compute the yLE by equation 4.14 with f iLE = f i for i ≤ L and f i

LE = fi

for i > L.

5. Let y′′LE = yLE, if y′′

LE 6= y′LE, then set y′

LE equal to y′′LE and turn to step 3.

6. If y′′LE = y′

LE, then let y′′LE = yLE , stop.

The WM algorithm:

This method was proposed by Wu and Mendel (Wu & Mendel, 2002).

The end-point yRI(yRI(x′) ≤ yRI(x

′) ≤ yRI(x′)):

yRI(x′) =

yr(x′) + yr(x

′)

2(4.15)

here,

yr(x′) = max

m∑i=1

fiyi

RI

m∑i=1

fi

,

m∑i=1

f iyiRI

m∑i=1

f i

(4.16)

yr(x′) = y

r(x′)+

m∑i=1

(fi − f i)

m∑i=1

fi

m∑i=1

f i

×

m∑i=1

fi(yi

RI − y1RI)

m∑i=1

f i(ymRI − yi

RI)

m∑i=1

fi(yi

RI − y1RI) +

m∑i=1

f i(ymRI − yi

RI)

. (4.17)

The end-point yLE(yLE

(x′) ≤ yLE(x′) ≤ yLE(x′)):

yLE(x′) =y

LE(x′) + yLE(x′)

2(4.18)

83

4.3 The Adaptive Interval Type-2 FLC

here,

yl(x′) = min

m∑i=1

f iyiLE

m∑i=1

f i

,

m∑i=1

fiyi

LE

m∑i=1

fi

(4.19)

yl(x′) = yl(x

′) −

m∑i=1

(fi − f i)

m∑i=1

fi

m∑i=1

f i

×

m∑i=1

f i(yiLE − y1

LE)m∑

i=1

fi(ym

LE − yiLE)

m∑i=1

f i(yiLE − y1

LE) +m∑

i=1

fi(ym

LE − yiLE)

.

(4.20)

With the above two methods, the yLE(x′) and yRI(x′) are solved and the crisp

output of IT2 FLS can be obtained by the equation 4.21.

y(x′) =yLE(x′) + yRI(x

′)

2. (4.21)


By analyzing the general reasoning process of the IT2 FLS and the two main

kinds of type-reduction methods in Section 4.2, the crisp output is calculated

by averaging the end-points which are reasoning results from the interval fuzzy

inference engine. In fact, this interval between two end-points includes all possible

reasoning results. The average value of this interval region is not guaranteed to be

the expected one for control aim. Also considering the fact that all the uncertain

information is finally embodied into this interval domain, the method of deducing

a better output is very important to show the advantage of IT2 fuzzy system in

dealing with uncertain information.

In this section, a framework for adaptive IT2 FLC is designed to deal with

the uncertainty and imprecision in the vehicle active suspension system. With

designed feedback structure, two optimal algorithms are used to adaptively opti-

mise the interval region and to obtain a crisp output which can bring the better

control performance.

Firstly, a framework of proposed method is designed for control the non-

linear and uncertain active suspension system in Section 4.3.1. Then two optimal

algorithms are described in the adaptive control scheme and undertake the role to

84


provide self-tuning feedback gains changing with different suspension performance

in Section 4.3.2 and Section 4.3.3.

4.3.1 The Framework of Adaptive IT2 FLC

The framework of adaptive IT2 FLC is shown in Fig. 4.3. In comparison with

Figure 4.3: The framework of proposed IT2 fuzzy controller

the conventional IT2 FLS in Fig. 4.2, the proposed structure builds a more

general framework to represent the type-reduction and defuzzification process. If

an optimal goal of the proposed IT2 FLS can be described by equation 4.21, the

convergence of the optimization method is guaranteed, the proposed method is

shrunk to the same form as the conventional IT2 FLS.

In comparison of two kinds of type-reduction methods, the first type of meth-

ods calculate the exact solutions monotonically and super-exponentially fast with

simple formula and they can be run in parallel, but the time delay caused by algo-

rithmic iteration is the bottleneck for real-time applications. On the other hand,

the second type of methods replaces the TR by four uncertainty bounds. These

bounds only depend on the lower and upper firing levels of each rule and the cen-

troid of each rule’s consequent set. For the purpose of computational efficiency,

the proposed method in this thesis uses the second type-reduction method to

calculate the end-points of reasoning results.

85

Chapter3/Chapter3Figs/EPS/it2afcfla.eps


Furthermore, under the proposed structure, the crisp output of the IT2 FLS

represents twofold information. One is the fuzzy reasoning result which is based

on fuzzy rules extracted from expert knowledge or industrial experience; the

other is the further optimal goal which is required by practical issues (e.g., saving

energy) or is impossible to be combined into the fuzzy rules.

Optimization algorithms can be selected in terms of domain-dependent goals

and practical requirements. Here, for real-time control, two optimization algo-

rithms are used, one is the LMS method which is a gradient-based method and

the other is the PSO method which is a recently invented high-performance non-

linear optimizer and requires less computational cost in real-time applications.

4.3.2 The LMS method

The Least Mean Square (LMS) algorithm, introduced by Widrow & Hoff (1960) is

an adaptive gradient-based algorithm. It incorporates an iterative procedure that

makes successive corrections to the weight vector in the direction of the negative

of the gradient vector that eventually leads to the minimum mean square error.

Compared to other optimization algorithms, LMS algorithm is relatively simple

and it does not require correlation function calculation or matrix inversions.

For an active suspension system, considering the natural variability of the

road conditions and the human sensation to vibration in different environments,

an adaptive strategy used in a control system will improve the self-tuning and

robustness of an active suspension system. Also the structure of the proposed

IT2 FLS means that it is possible to use an adaptive strategy on the interval

results. That is, the designed adaptive solution can self-tune the weight value of

end-points or boundaries to minimize optimal performance index. The structure

of this adaptive algorithm on quarter-vehicle suspension system is shown in Fig.

4.4.

The crisp output of proposed IT2 FLC is described as weighted sum of two

end-points. It can be written as:

y = αyLE + (1 − α)yRI (4.22)

86


Figure 4.4: The structure of adaptive IT2 FLC with LMS method

So the LMS optimization method is employed to construct the optimal weight

value α of fuzzy reasoning results. According to the linear quarter-vehicle sus-

pension model in Chapter 3, the sprung mass acceleration with the adaptive FLC

can be rewritten as

y(k) = W T · X(k) (4.23)

where y(k) is the measurable system output(i.e., active suspension sprung mass

acceleration), W is a coefficient vector of α, the vector is [ αmb

(1−α)mb

cmb

Ks

mb]T , X(k)

is a vector including the measurable system inputs and fuzzy control force given

by [yo yo zw − zb zw − zb]. Consider:

e(k) = d(k) − y(k) (4.24)

where d=0 is the reference value of y(k) and e is the error of sprung mass accel-

eration.

The LMS algorithm can be used to compute the weight matrix coefficients

W by minimum the squared error criterion. Since the system total mean square

value can not be obtained, the LMS algorithm adaptive tunes the coefficients

according to the negative gradient of single sample mean square. The iterative

optimal formula of weight coefficient is

W (k + 1) = W (k) − µ∇e2(k) (4.25)

87

Chapter3/Chapter3Figs/EPS/sys-lms.eps


where µ is gain coefficient which related to adaptive speed and stability, ∇e2(k)

is the gradient of single error sample square. Generally, for efficient calculation,

∇e2(k) can be approximated as the gradient of theerror mean square E[e2(k)].

That is

∇e2(k) = −2e(k)X(k) (4.26)

The weight coefficient iterative adaptive optimal algorithm comprises of equations

4.23-4.26. When the gradient is zero, the mean square of the error is minimum and

the weight coefficient is optimum at the same time. Additionally, α is only tunable

weight value in W and all the variables in X can be measured or calculated. Then

the adaptive strategy can optimize the weight coefficients by using the feedback

of the vehicle body acceleration.

The half-vehicle suspension system can be divided into front quarter and rear

quarter suspension for using this adaptive strategy. The coupling between front

and rear vehicle suspension will be presented by fuzzy rules and an antecedent

part, not the consequent part (i.e., control outputs). Simulations on non-linear

quarter-vehicle and half-vehicle suspension models are described in Section 4.4

and Section 4.5.

4.3.3 The PSO method

Although the LMS method can optimize the IT2 fuzzy reasoning results with a

simple iterative process, it depends on a linear model of the suspension system. It

can only work on a linear optimization performance because it is a kind of linear

optimizer. In other words, the optimal solutions by LMS method are obtained

with a linear reference model and they can not be guaranteed to have optimal

solutions for the non-linear models. Furthermore, if the optimization performance

is non-linear, the LMS method can not work.

For dealing with the non-linear optimization problems and satisfying the real-

time control requirements, the PSO method is used to optimize the IT2 fuzzy

reasoning results. The PSO is a population-based stochastic evolutionary al-

gorithm, noted for its capability to search for the global optimum of complex

problems. Since its inception in 1995 (Kennedy & Eberhart, 1995), PSO has

attracted a lot of attention from a variety of engineering fields. In this thesis,

88


the PSO method is integrated into the adaptive strategy of the proposed IT2

FLC. The structure of this adaptive algorithm on the quarter-vehicle suspension

system is shown in Fig. 4.5.

Figure 4.5: The structure of adaptive IT2 FLC with PSO method

Initially, assume that the interval results of IT2 fuzzy reasoning are discretized

to D values by equation 4.27-4.29.

y∗ = min

yLE

, yRI

, y∗ = max yLE , yRI , (4.27)

yd = y∗ + d∆y∗, d = 1, 2, . . . , D (4.28)

∆y∗ =y∗ − y∗

D(4.29)

Then each discrete value yd is regarded as the d-th particle of a swarm and

represented by the vector Yd=(yi1, yi2, . . . , yiD), i = 1, 2, . . . , N , where N is the

size of population. The best particle of the swarm, i.e., the particle with the

lowest function value, is denoted by index g. The best previous position (i.e., the

position corresponding to the best function value) of the d-th particle is recorded

and represented as Pd = (pi1, pi2, . . . , piD), and the position change (velocity) of

the d-th particle is Vd = (vi1, vi2, . . . , viD).

The particles are manipulated with the equations below.

V k+1d = λ(ωV k

d + c1rki1(P

kd − Y k

d ) + c2rki2(P

kg − Y k

d )) (4.30)

89

Chapter3/Chapter3Figs/EPS/sys-pso.eps

4.4 Simulations on the Quarter-vehicle Model

Y k+1d = Y k

d + V k+1d (4.31)

λ is a constriction factor which is used to control and constrict velocities, ω is the

inertia weight, c1 and c2 are two positive constants, called the cognitive and social

parameter respectively, ri1 and ri2 are random numbers uniformly distributed

within the range [0,1]. Equation 4.30 is used to determine the i-th particle’s

new velocity at each iteration, while equation 4.31 provides the new position

of the i-th particle. The performance of each particle is measured according

to a problem-dependent optimal objective function. In this thesis, the optimal

objective functions for quarter-vehicle and half-vehicle suspension systems are

defined in Section 4.4 and 4.5.

The inertia weight ω is considered important for the convergence of PSO

(Parsopoulos & Vrahatis, 2002). In fact, the parameter ω regulates the trade-off

between the global (wide-ranging) and the local (nearby) exploration abilities of

the swarm. Experimental results implied that it was preferable to initially set

the inertia to a large value, to promote global exploration of the search space,

and gradually decrease it to obtain refined solutions(Shi & Eberhart, 1998). In

simulation sections, this method is used off-line to choose a proper inertia weight

for proposed IT2 FLC.


For evaluating the control performance of the proposed controller, the numerical

simulations have been carried out on the quarter vehicle active suspension system

shown in Fig. 2.2. The parameters of active suspension are selected from Taghirad

(1997) and presented on the Table 4.1. Compared to the typical quarter-vehicle

suspension system, this model integrates the non-linear dynamics which have

been described in Chapter 3. Then, based on the nominal spring stiffness Ks

and damper coefficient c in Table 4.1, the related coefficients of their high-order

polynomial functions are also given in Table 4.1.

The simulations of the proposed adaptive IT2 FLC with LMS method is pre-

sented in Section 4.4.1. With similar testing conditions, the adaptive IT2 FLC

90


Table 4.1: The parameters of quarter vehicle active suspension

mb(Kg) mw(Kg) Ks1(N/m) c1(Ns/m) Kt (N/m)

897.2 87.15 66824.2 1190 101115

Ks0(N) Ks2(N/m2) Ks3(N/m3) c0(N) -

-236 -403 104 426 -

with PSO method is demonstrated and simulation results are shown in Section

4.4.2.

4.4.1 Adaptive IT2 FLC with the LMS method

For the purposes of minimizing the vehicle body acceleration, reducing the tyre

dynamic loads and avoiding hitting the suspension physical limit, the vehicle body

displacement zb, velocity zb and the deflection between body and wheel zb − zw

are chosen as the input variables. Scaling factors are used for these three inputs

to appropriately map them to their respective universes of discourse, denoted

by S1, S2 and S3. Considering noise and construct uncertainty in these inputs,

their MFs are designed as the interval type-2 fuzzy membership functions which

is shown in Fig. 4.6.

(a) The membership functions of vehicle

body displacement zb and velocity zb

(b) The membership functions of deflection

between body and wheel zb − zw

Figure 4.6: The interval type-2 fuzzy membership functions of three inputs

The output is the actuator force fa and its membership functions are shown

in Fig. 4.7. A rule base developed by heuristics with suspension displacement,

velocity and deflection between body and wheel as input variables is designed.

91

Chapter3/Chapter3Figs/EPS/mf.eps

Chapter3/Chapter3Figs/EPS/mf2.eps


Figure 4.7: The membership functions of actuator force fa

The rules are governed by three main cases. Firstly, when the body displacement

is around big (positive or negative) and its derivative has a large magnitude, the

power of the vibration is strong and the chief task of control is to improve the

dynamic response with fast and strong force. Secondly, when the body displace-

ment and its derivative is small or zero, stability will be the key aim to control

and the output force will be small or zero. Thirdly, considering the physical real-

ity of the suspension system, the input of the suspension deflection will be used

in the rules only when the suspension displacement is positive big or negative

big. Here, define zb − zw as ∆z, the rules are shown in Table 4.2. In the table,

“NV” means negative very big, “NB” means negative big, “NM” means negative

middle, “N” means negative, “NS” means negative small, “ZE” means zero, “PS”

means positive small, “P” means positive, “PM” means positive middle, “PB”

means positive big, “PV” means positive very big. A typical type-1 FLC (TFC)

designed in Huang & Chao (2000) is reconstructed to compare with the proposed

method and the passive suspension system. In the TFC, the inputs of fuzzy con-

troller were the vehicle body displacement and velocity, their crisp membership

functions are chosen as the center of the interval type-2 fuzzy MFs in Fig. 4.6(a).

The TFC output is the actuator force fa and its MFs are chosen as the center of

the interval type-2 fuzzy MFs in Fig. 4.7.

Generally, the scaling factors are chosen based on the range of related param-

eters of passive suspension obtained from numerical simulations. Here, the S1,

S2, S3 are 10, 3, 12 in this section, respectively.

92

Chapter3/Chapter3Figs/EPS/it2mflmsout.eps


Table 4.2: The rules of fuzzy controller

zb zb ∆z fa zb zb ∆z fa zb zb ∆z fa

NB NB N PV P NB - ZE NB Z P PS

NB N N PB P N - NS NB P Z ZE

NB Z N PM P Z - NM NB P P NS

NB P N PS P P - NB NB PB Z NS

NB PB N ZE P PB - NV NB PB P NM

N NB - PV PB NB N PS PB NB Z ZE

N N - PM PB N N ZE PB NB P ZE

N Z - PM PB Z N NS PB N Z NS

N P - PS PB P N NM PB N P NS

N PB - NS PB PB N NB PB Z Z NB

Z NB - PM NB NB Z PB PB Z P NB

Z N - PS NB NB P PM PB P Z NV

Z Z - ZE NB N Z PM PB P P NV

Z P - NS NB N P PM PB PB Z NV

Z PB - NM NB Z Z PS PB PB P NV

93


In this section, firstly, the C class road as the road roughness 2.56 × 10−4

m3/cycle is used to test the non-linear control capability of proposed method.

Secondly, a persistent total 15s simulation with three different road surfaces is

carried out to test the adaptive ability of proposed controller. The B class road

surface with roughness G0 = 6.4 × 10−5m3/cycle is implemented from 0 to 5 s,

the C class road surface is used from 6 to 10 s, and the D class road surface with

roughness G0 = 1.024 × 10−3m3/cycle is utilized from 11 to 15 s. Thirdly, the

±50% changes of nominal mass mb and ±10% changes of nominal spring stiffness

Ks1 are implemented to test the robust performance of proposed method. The

vehicle speed is 20 m/s.

Generally speaking, the ride comfort related to frequency sensitive (vehicle

body acceleration frequency response), and the handling performance related to

the tyre dynamic load. Additionally, from ISO 2361, the human body is very

sensitive to vertical vibration in the frequency range 4-8 Hz. Then the simulation

results are shown by frequency response and RMS values.

The first testing condition:

With the C class road surface, the frequency response of body acceleration

with the proposed IT2 FLC are compared with passive and the TFC shown in

Fig. 4.8. The result has shown that the proposed adaptive IT2 FLC has achieved

a significant decrease of body acceleration in the system low syntonic frequency

8.63Hz.

Fig. 4.9 shows the tyre dynamic loads of the quarter-vehicle suspension system

with passive, proposed method and TFC. With smaller tyre dynamic loads, the

quarter-vehicle suspension by the proposed method can obtain better handling

performance.

The RMS values of vehicle body accelerations and dynamic load are shown in

Table 4.3. It is clear that the proposed active suspension control system achieves

better time response.

The second testing condition:

With a persistent total 15 seconds simulation with three different road sur-

faces, the adaptive ability of proposed controller to variant road conditions is

tested. The frequency responses of body accelerations compared in three parts

94


100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Accele

ration [m

/s2]

passive

AFC

TFC

Figure 4.8: The frequency response of vehicle body acceleration zb

100

101

102

103

0

0.5

1

1.5

2

2.5

Frequency [Hz]

Tyre

dyn

am

ic lo

ad

[K

N]

passive

AFC

TFC

Figure 4.9: The frequency response of tyre dynamic load

Table 4.3: The RMS values comparison in time domain

Control performance passive TFC IT2 FLC

Acceleration[m/s2] 0.4912 0.3202 0.2825

Dynamic Load [KN ] 0.4730 0.3616 0.3298

95

Chapter3/Chapter3Figs/EPS/acce.eps

Chapter3/Chapter3Figs/EPS/tyld.eps


are shown in Fig. 4.10, Fig. 4.11 and Fig. 4.12. From the comparison of fre-

quency response, the proposed IT2 FLC significantly reduced body acceleration

in the low syntonic frequency 8.63Hz, especially on the B class and C class road.

100

101

102

103

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency [Hz]

Bo

dy a

cce

lera

tio

n [

m/s

2]

passive

TFC

AFC

Figure 4.10: The frequency response of vehicle body acceleration on B class road

surface

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

Frequency [Hz]

Bo

dy a

cce

lera

tio

n [

m/s

2]

passive

TFC

AFC

Figure 4.11: The frequency response of vehicle body acceleration on C class road

surface

The RMS values of accelerations and tyre dynamic loads are shown in Table

4.4, Table 4.5 and Table 4.6. The comparisons of RMS values also show the

96

Chapter3/Chapter3Figs/EPS/chvrac05.eps



100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Bo

dy a

cce

lera

tio

n [

m/s

2]

passive

AFC

TFC

Figure 4.12: The frequency response of vehicle body acceleration on D class road

surface

proposed IT2 FLC can hold the best ride comfort and handling performance on

changing road surface than the conventional TFC and the passive suspension.

Table 4.4: The RMS values comparison on B class road surface

RMS values (1-5 s) passive TFC IT2 FLC

Acceleration[m/s2] 0.2309 0.1449 0.1338

Tyre dynamic load[N ] 230.96 159.62 152.61

The third testing condition:

For testing the adaptive and robust properties of the active suspension with

proposed controller, the case for different sprung mass and the spring coefficient

are considered. The sprung mass changes ± 50 percent of nominal value. The

system low syntonic frequency changes to 7.046Hz and 12.205Hz. The frequency

control performances are shown in Fig. 4.13, and the RMS values comparison of

body acceleration and tyre dynamic load are shown in Table 4.7 and Table 4.8.

According to the comparison of body acceleration and tyre dynamic loads, the

97



Table 4.5: The RMS values comparison on C class road surface


Acceleration[m/s2] 0.5882 0.4504 0.3963

Tyre dynamic load[KN ] 0.5611 0.4464 0.4056

Table 4.6: The RMS values comparison on D class road surface


Acceleration[m/s2] 0.6780 0.5267 0.5011

Tyre dynamic load[KN ] 0.6606 0.5382 0.5117

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Bo

dy a

cce

lera

tio

n [

m/s

2]

1

2

passive

TFC

AFC

Figure 4.13: The frequency response of vehicle body acceleration (1: sprung mass

+50%, 2: sprung mass -50%)

98

Chapter3/Chapter3Figs/EPS/chmac.eps


Table 4.7: The RMS comparison of body acceleration in time domain

RMS values[m/s2] passive TFC IT2 FLC

Body mass +50% 0.4654 0.2994 0.2663

Body mass -50% 0.6489 0.5329 0.4870

Ks1 +10% 0.5553 0.4009 0.3871

Ks1 -10% 0.4596 0.3452 0.3427

Table 4.8: The RMS comparison of tyre dynamic loads in time domain

RMS values[KN ] passive TFC IT2 FLC

Body mass +50% 0.5309 0.3854 0.3460

Body mass -50% 0.3209 0.2751 0.2575

Ks1 +10% 0.5553 0.4009 0.3871

Ks1 -10% 0.4596 0.3452 0.3427

99


proposed method achieves better control performance than the passive suspension

and the TFC suspension.

The value of Ks1 changes ± 10 percent of nominal value. The system low

syntonic frequency changes to 9.051Hz and 8.187Hz. Fig. 4.14 and Table 4.7

all show the robust and adaptive ability of the proposed controller is stronger

and also has a better trade-off between riding comfort and handling performance

when the Ks is changed.

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Frequency [Hz]

Body a

ccele

ration [m

/s2]

1

2

passive

TFC

AFC

Figure 4.14: The frequency response of vehicle body acceleration (1: Ks1 +10%,

2: Ks1 -10%)

The simulation results have demonstrated the proposed adaptive IT2 FLC

can well deal with the uncertainty and non-linearity of an quarter-vehicle active

suspension system than a conventional fuzzy controller and a passive suspension

system.

4.4.2 Adaptive IT2 FLC with the PSO method

In this section, considering the purpose of not only riding comfort and handling,

but also energy consumption or reducing the suspension travel, the proposed

adaptive IT2 FLC with PSO optimizer is used to control the non-linear quarter-

vehicle suspension system. By designing the additional optimal performance, the

100

Chapter3/Chapter3Figs/EPS/chkac.eps


proposed method aims to find a new trade-off point which can improve more

suspension performance to satisfy more expectations of customers.

With the same fuzzy inputs, output and fuzzy rules, an optimal performance

for reducing the suspension travel is designed as below.

Γ = minfa∈(y∗,y∗)

[1

m2b

f 2a +

K2s

m2b

(zb − zw)2

](4.32)

The C class road is used to test the control capability of proposed method.

The parameters of PSO are chosen as: λ = 0.75, c1 = c2 = 2 and ω = 0.9. The

size of particle D is 20.

The frequency response of body acceleration with the proposed IT2 FLC is

compared with the passive and the TFC as shown in Fig. 4.15. The result has

verified that the proposed adaptive IT2 FLC has achieved a significant decrease

of body acceleration in the system low syntonic frequency 8.63Hz.

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Accele

ration (

m/s

2)

passive

TFC

Proposed(PSO)

Figure 4.15: The frequency response of vehicle body acceleration zb

Fig. 4.16 shows the tyre dynamic loads of the quarter-vehicle suspension

system with the passive method, the proposed method and TFC. With smaller

tyre dynamic loads, the quarter-vehicle suspension using the proposed method

can obtain better handling performance.

The RMS values of vehicle body accelerations, dynamic load, suspension travel

and control force are shown in Table 4.9. It is clear that the proposed active sus-

pension control system can not only achieve better riding comfort and handling

101

Chapter3/Chapter3Figs/EPS/qracpso.eps


100

101

102

103

0

0.5

1

1.5

2

2.5

Frequency (Hz)

Tyre

dyn

am

ic lo

ad

(K

N)

passive

TFC

proposed(PSO)

Figure 4.16: The frequency response of tyre dynamic load

performance, but also decrease the suspension travel which will reduce the pos-

sibility of the strut hitting phenomenon. Meanwhile, compared with TFC, the

proposed method need more active control forces.

Table 4.9: The RMS values comparison in time domain

Control performance passive TFC IT2 FLC

Acceleration[m/s2] 0.4912 0.3202 0.2931

Dynamic Load [KN ] 0.4730 0.3616 0.3599

Suspension travel [m] 0.006498 0.004522 0.004035

Control force [KN ] - 0.1653 0.2038

For testing the adaptive and robust properties of the active suspension with

proposed controller, the case for different sprung mass and the spring coefficient

are considered. The sprung mass changes ± 50 percent of its nominal value and

102

Chapter3/Chapter3Figs/EPS/qrtdpso.eps


the spring stiffness changes ±10%. The RMS values of vehicle body acceleration,

dynamic load, suspension travel and control force are shown in Table 4.10-4.13.

Table 4.10: The comparison of RMS values of body acceleration

Condition passive TFC IT2 FLC

Body mass (1+50%)mb 0.4654 0.2994 0.2845

Body mass (1-50%)mb 0.6489 0.5329 0.5067

Spring stiffness (1+10%) Ks1 0.5849 0.3935 0.3901

Spring stiffness (1-10%) Ks1 0.4728 0.3370 0.3228

Table 4.11: The comparison of RMS values of tyre dynamic load


Body mass (1+50%)mb 0.5309 0.3854 0.3650

Body mass (1-50%)mb 0.3209 0.2751 0.2641



It is clear that the proposed method with the PSO optimizer obtained smaller

suspension travel without losing better riding comfort and handling performance

under these different suspension parameters. The force comparison results showed

that the proposed method need higher control forces.

103


Table 4.12: The comparison of RMS values of suspension travel


Body mass (1+50%)mb 0.009252 0.005318 0.005227

Body mass (1-50%)mb 0.004146 0.003558 0.003450



Table 4.13: The comparison of RMS values of control force

Condition TFC IT2 FLC

Body mass (1+50%)mb 152.14 180.29

Body mass (1-50%)mb 174.26 186.16

Spring stiffness (1+10%) Ks1 173.26 211.17

Spring stiffness (1-10%) Ks1 157.14 201.46

104

4.5 Simulations on the Half-vehicle Model


The proposed adaptive IT2 FLC in Section 4.3 is implemented into the non-linear

half-vehicle active suspension system as shown in Chapter 3 whose mathematical

model is given in Section 3.3. Parameters of the model are provided in Table 3.1

and Table 3.2.

For the purposes of minimizing the vehicle body acceleration and reducing

the tyre dynamic loads, the vehicle body displacement zbf and zbr, velocity zbf

and zbr and pitch angle θ are chosen as the input variables. Scaling factors are

used for these five inputs to appropriately map them to the respective universe of

discourses and are respectively called Szbf, Szbr

, Szbf, Szbr

and Sθ. Their MFs are

designed as the IT2 fuzzy membership functions which are shown in Fig. 4.17.

(a) The membership functions of vehicle

body displacements zbf , zbr

(b) The membership functions of vehicle

body velocities zbf ,zbr and the pitch angle

θ

Figure 4.17: The interval type-2 fuzzy membership functions of five inputs

The output is the actuator force Ua = [faf , far]. Considering the stability

of the control system and computational cost, a T-S fuzzy model is used as the

consequent of the fuzzy controller. For example, one of the fuzzy rules can be

presented as:

Ri : If zbf is PB and zbr is PB and zbf is P and zbr is N and θ is N, then Ua

=−K0Xhalf ;

where Xhalf is the measurement matrix and K0 is one of local linear control

gains, which can be solved by using LQG control strategy on piecewise linear

105

Chapter3/Chapter3Figs/EPS/it2mf_half_0.eps

Chapter3/Chapter3Figs/EPS/it2mf_half_1.eps


models (Cao et al., 2007). In this section, with five local operation points, the

non-linear half-vehicle suspension model in Chapter 3 is piecewise linearized to

five local linear models and the local control gains are solved as follows.

K0 =

[−63.854−5966

28.493 −7265.3 5276.2 −55.9585935.3 −84.927 33.582 −34321

1430.2 −103100 295110 −1384.4 −192180196960 −84.892 2315.8 −161790 −2248.2

] (4.33)

K1 =

[−63.854−5966

28.493 −7265.3 5276.2 −55.9585935.3 −84.927 33.582 −30598

1430.2 −89731 281740 −1384.4 −192180193240 −84.892 2315.8 −161790 −2248.2

] (4.34)

K2 =

[−63.854−5966

28.493 −7265.3 5276.2 −55.9585935.3 −84.927 33.582 −26875

1430.2 −76366 268380 −1384.4 −192180189520 −84.892 2315.8 −161790 −2248.2

] (4.35)

K3 =

[−63.854−5966

28.493 −7265.3 5276.2 −55.9585935.3 −84.927 33.582 −23152

1430.2 −63001 255010 −1384.4 −192180185790 −84.892 2315.8 −161790 −2248.2

] (4.36)

K4 =

[−63.854−5966

28.493 −7265.3 5276.2 −55.9585935.3 −84.927 33.582 −19429

1430.2 −49636 241650 −1384.4 −192180182070 −84.892 2315.8 −161790 −2248.2

] (4.37)

With similar consideration of vertical dynamics in the quarter-vehicle sus-

pension system, additional pitch dynamics and coupling information in the half-

vehicle suspension system, a rule base is developed for the half-vehicle suspension

system. There are total 225 fuzzy rules for the half-vehicle suspension control sys-

tem as shown in Appendix B.

The typical type-1 T-S FLC (TFC) designed by Feng (2006) is modified to

compare with the proposed method and the passive suspension system. The

inputs of the TFC were the same as the proposed adaptive IT2 FLC; their crisp

membership functions are chosen as the center of interval type-2 fuzzy MFs in

Fig. 4.17. The TFC T-S consequent is same as the proposed method.

106


The simulations of the proposed adaptive IT2 FLC with the LMS method are

presented in Section 4.5.1. With similar testing conditions, the adaptive IT2 FLC

with PSO method is demonstrated and simulation results are shown in Section

4.5.2.

4.5.1 The adaptive IT2 FLC with the LMS method

For testing the adaptive ability of proposed controller to random road conditions,

a poor road surface (C class in ISO 2361) G0 = 2.56 × 10−4m3/cycle is used.

The ride comfort and handling performance of the half-vehicle suspension system

are tested from frequency and time domain in comparison with the TFC and the

passive suspension system. Furthermore, with the improved suspension model,

the response to different vehicle forward speed is investigated.

First testing condition: constant vehicle speed V = Vcr = 20m/s

The frequency responses of the front and rear body accelerations are shown

in Fig. 4.18 and Fig. 4.19. From the comparison of frequency response, the

proposed IT2 FLC significantly reduced the front and rear body acceleration in

the human sensitive frequency band. The RMS values of front and rear vehicle

body accelerations are compared in Table 4.14. With the TFC and proposed IT2

FLC, the front body acceleration has been reduced by 37.4% and 51.8% compared

with the passive suspension system respectively. By the TFC and proposed IT2

FLC, the rear body acceleration has been reduced by 54.6% and 71.8% over the

passive suspension system .

Fig. 4.20 and Fig. 4.21 showed the frequency responses of the front and

rear tyre dynamic loads. The comparisons clearly verified the proposed method

achieved better handling performance in the frequency domain. From the time

domain, the RMS values of front and rear tyre dynamic loads were compared in

Table 4.14. In terms of the front tyre dynamic loads, the TFC and proposed IT2

FLC showed a 36.2% and 52.3% reduction over the passive suspension system.

Meanwhile, with the TFC and proposed IT2 FLC, the rear tyre dynamic loads

have been reduced by 46.1% and 57.1% over the passive suspension system re-

spectively. The frequency responses of pitch angle acceleration are compared in

Fig. 4.22. From the comparison result, the proposed control method reduced the

107


100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Acce

lera

tio

n (

m/s

2)

Passive

TFC

Proposed

Figure 4.18: The frequency response of vehicle front body acceleration

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Frequency (Hz)

Acce

lera

tio

n (

m/s

2)

Passive

TFC

Proposed

Figure 4.19: The frequency response of vehicle rear body acceleration

108

Chapter3/Chapter3Figs/EPS/fr-ac-lms.eps

Chapter3/Chapter3Figs/EPS/re-ac-lms.eps


100

101

102

103

0

0.5

1

1.5

2

2.5

3

Frequency (Hz)

Tyre

dyn

am

ic lo

ad

(K

N)

Passive

TFC

Proposed

Figure 4.20: The frequency response of front tyre dynamic load

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Tyre

dyn

am

ic lo

ad

(K

N)

Passive

TFC

Proposed

Figure 4.21: The frequency response of rear tyre dynamic load

109

Chapter3/Chapter3Figs/EPS/fr-td-lms.eps

Chapter3/Chapter3Figs/EPS/re-td-lms.eps


pitch vibration and achieved better attitude performance. From the comparison

of RMS value in Table 4.14, with the TFC and proposed IT2 FLC, it has been

reduced by 61.8% and 72.2% over the passive suspension system respectively.

100

101

102

103

0

1

2

3

4

5

6x 10

−3

Frequency (Hz)

Pitch

an

gle

(ra

d)

Passive

TFC

Proposed

Figure 4.22: The frequency response of pitch angle acceleration

Second testing condition: different vehicle forward speeds

By using different vehicle rolling speeds (i.e., Vcr=16m/s, 20m/s and 24m/s),

the pitch angle dynamics were tested by the improved non-linear half-vehicle

model with the proposed IT2 FLC method. The simulation time was 4 seconds.

The road input was same as the first testing condition. The testing aim was to

verify the effect of the vehicle acceleration or deceleration on vehicle body pitch

dynamics. Fig. 4.23 showed the comparison of pitch angle accelerations with

the proposed IT2 FLC. It was clear that different vehicle rolling speed brought

different vehicle accelerations and different pitch dynamics. These comparisons

showed the improved model presented more accurate suspension dynamics and it

will be helpful to support the application of proposed model and control methods.

The RMS values of pitch angle acceleration were compared in Table 4.15.

When the vehicle accelerated (i.e., Vcr=16m/s), the pitch angle acceleration in-

creased 38.0% over the constant vehicle speed(i.e., V =20m/s). When the vehicle

decelerated (i.e.,Vcr=24m/s), there was 18.7% reduction over the constant vehicle

speed in the pitch angle acceleration. Because of the non-linear relation between

110

Chapter3/Chapter3Figs/EPS/angle-lms.eps


Table 4.14: The RMS values comparison with constant vehicle speed

RMS values passive TFC IT2 FLC

Front body acceleration[m/s2] 0.6739 0.4217 0.3248

Rear body acceleration[m/s2] 0.5271 0.2395 0.1486

Front tyre dynamic load[KN ] 0.6812 0.4346 0.3248

Rear tyre dynamic load[KN ] 0.4044 0.2179 0.1733

Pitch angle acceleration [rad/s2] 0.2744 0.1046 0.0764

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

time(s)

pitch a

ngle

accele

ration(r

ad/s

2)

Vcr

=16m/s

Vcr

=24m/s

Vcr

=20m/s

Figure 4.23: The different pitch dynamics with different vehicle speed

111

Chapter3/Chapter3Figs/EPS/chang-angle-lms.eps


vehicle speed and pitch angle, the rate of increase was not same as the rate of

decrease.

Table 4.15: The RMS values comparison with changing vehicle speed

Vcr pitch angle acceleration (rad/s2)

16 m/s 0.06966

24 m/s 0.04101

20 m/s 0.05047

4.5.2 The IT2 FLC with the PSO method

Under similar testing conditions as in Section 4.5.1, the proposed IT2FLC with

the PSO method was implemented on the non-linear half-vehicle suspension sys-

tem and its control performance was investigated. An optimal performance for

constraining the pitch vibration was designed for the PSO optimizer in equation

4.38.

Γhalf = minfaf∈(y∗

f,y∗

f),far∈(y∗

r,y∗

r)

[(1

mb+

d2f

Ib

)f 2

af+

df−dr

Ib(zbf − zbr)

2 +(

1mb

+ d2r

Ib

)f 2

ar

] (4.38)

The parameters for the PSO method were chosen as: λf = 0.75, λr = 0.75,

c1f = c2f = 2 ,c1r = c2r = 2, ωf = 0.9 and ωr = 0.9. The size of particles Df and

Dr was 20. They are all chosen by initial tests.

First testing condition: constant vehicle speed

The frequency responses of front and rear body accelerations are shown in

Fig. 4.24 and Fig. 4.25. From the comparison of frequency response, the pro-

posed IT2 FLC significantly reduced the front and rear body acceleration in the

human sensitive frequency band. The RMS values of front and rear vehicle body

accelerations are compared in Table 4.16. With the TFC and proposed IT2 FLC,

112


the front body acceleration has been reduced by 37.4% and 51.4% over the pas-

sive suspension system respectively. With the TFC and proposed IT2 FLC, the

rear body acceleration has been reduced by 54.6% and 71.3% over the passive

suspension system respectively.

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Acce

lera

tio

n (

m/s

2)

Passive

TFC

Proposed

Figure 4.24: The frequency response of vehicle front body acceleration

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Frequency (Hz)

Acce

lera

tio

n (

m/s

2)

Passive

TFC

Proposed

Figure 4.25: The frequency response of vehicle rear body acceleration

Fig. 4.26 and Fig. 4.27 showed the frequency responses of front and rear tyre

dynamic loads. The comparisons clearly verified the proposed method achieved

better handling performance in the frequency domain. From the time domain,

the RMS values of front and rear tyre dynamic loads were compared in Table

113

Chapter3/Chapter3Figs/EPS/fr-ac-pso.eps

Chapter3/Chapter3Figs/EPS/re-ac-pso.eps


4.16. By the controller of TFC and proposed IT2 FLC, the front tyre dynamic

loads have been reduced by 36.2% and 49.9% over the passive suspension system

respectively. Meanwhile, the rear tyre dynamic loads have been reduced by 46.1%

and 57.6% over the passive suspension system respectively.

100

101

102

103

0

0.5

1

1.5

2

2.5

3

Frequency (Hz)

Tyre

dyn

am

ic lo

ad

(K

N)

Passive

TFC

Proposed

Figure 4.26: The frequency response of front tyre dynamic load

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (Hz)

Tyre

dyn

am

ic lo

ad

(K

N)

Passive

TFC

Proposed

Figure 4.27: The frequency response of rear tyre dynamic load

The frequency responses of the pitch angle acceleration are compared in Fig.

4.28. From the comparison results, the proposed control method reduced the pitch

vibration and achieved better attitude performance. From the comparison of

114

Chapter3/Chapter3Figs/EPS/fr-td-pso.eps

Chapter3/Chapter3Figs/EPS/re-td-pso.eps


RMS values in Table 4.16, by the TFC and proposed IT2 FLC, it has been reduced

by 61.8% and 76.2% over the passive suspension system respectively. Compared

100

101

102

103

0

1

2

3

4

5

6x 10

−3

Frequency (Hz)

Pitch

an

gle

(ra

d)

Passive

TFC

Proposed

Figure 4.28: The frequency response of pitch angle acceleration

with the results in Section 4.5.1, the ride comfort and handling performance of the

suspension system were similar. But the pitch angle acceleration by the proposed

IT2 FLC with the PSO was reduced by 14.5% over the proposed IT2 FLC with

LMS because of the further optimization.

Second testing condition: different vehicle forward speed

By using different vehicle rolling speeds (i.e., Vcr=16m/s, 20m/s and 24m/s),

the pitch angle dynamics were tested by the improved non-linear half-vehicle

model with the proposed IT2 FLC method with PSO. The simulation time was

4 seconds. The road input was same as the first testing condition. The testing

aim was to verify the effect of vehicle moving acceleration or deceleration on

vehicle body pitch dynamics. Fig. 4.29 showed the comparison of pitch angle

accelerations with the proposed IT2 FLC with PSO. Similar conclusions in Section

4.5.1 were verified.

The RMS values of pitch angle acceleration were compared in Table 4.15.

When the vehicle accelerated (i.e., Vcr=16m/s), the pitch angle acceleration in-

creased 42.75% over the constant vehicle speed(i.e., V = Vcr=20m/s). When

the vehicle decelerated (i.e.,Vcr=24m/s), there was a 21.06% reduction over the

constant vehicle speed in the pitch angle acceleration. Because of the non-linear

115

Chapter3/Chapter3Figs/EPS/angle-pso.eps


Table 4.16: The RMS values comparison with constant vehicle speed

RMS values passive TFC IT2 FLC

Front body acceleration[m/s2] 0.6739 0.4217 0.3275

Rear body acceleration[m/s2] 0.5271 0.2395 0.1511

Front tyre dynamic load[KN ] 0.6812 0.4346 0.3411

Rear tyre dynamic load[KN ] 0.4044 0.2179 0.1716

Pitch angle acceleration [rad/s2] 0.2744 0.1046 0.0653

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−4

time(s)

pitch a

ngle

accele

ration(r

ad/s

2)

Vcr

=16m/s

Vcr

=24m/s

Vcr

=20m/s

Figure 4.29: The different pitch dynamics with changing vehicle speed

116

Chapter3/Chapter3Figs/EPS/chang-angle-pso.eps

4.6 Summary

relation between vehicle speed and pitch angle, the rate of increase was not same

as the rate of decrease.

Table 4.17: The RMS values comparison with changing Vcr

Vcr pitch angle acceleration (rad/s2)

16 m/s 0.07593

20 m/s 0.05319

24 m/s 0.04199

4.6 Summary

This chapter has presented the adaptive IT2 FLC with two optimization methods

(i.e., LMS and PSO), which is not only aimed at improving the ride comfort and

handling performance of a vehicle suspension system, but also aimed at reducing

suspension travel and longitudinal vibrations. In comparison with the passive

suspension system and the typical fuzzy control suspension system, the results

showed that the proposed methods are better at constraining vertical vibrations,

pitch angle changes and suspension travels. The simulation results on non-linear

quarter-vehicle and half-vehicle suspension models can be summarized as follows, The simulation results by the adaptive IT2 FLC with LMS method have

shown that the proposed method can significantly reduce the vehicle body

accelerations and tyre dynamic loads on quarter-vehicle suspension model.

Under a C class road surface, the vehicle body acceleration was reduced by

42.5% over passive suspension system and by 11.7% over the TFC on the

quarter-vehicle suspension model. The tyre dynamic load was reduced by

42.5% over the passive suspension system and by 8.8% over the TFC. Under

117

4.6 Summary

different class random road surfaces, using nominal parameters of suspen-

sion components and significant variations from their nominal parameters,

the proposed method has been found to offer a better solution to handle

the trade-off between ride comfort and handling performance. Also its po-

tential adaptive ability in dealing with linguistic uncertainty and non-linear

dynamics was verified to be superior to the typical fuzzy control method. On the half-vehicle suspension model, the simulation results by the adaptive

IT2 FLC with the LMS method have shown the proposed method achieved

better ride comfort and handling performance with random road surfaces.

Furthermore, with the additional optimization part, the longitudinal vibra-

tion was constrained and the pitch angle acceleration was reduced by 61.8%

over passive suspension system and by 26.9% over the TFC method. With non-linear optimal performance functions, the adaptive IT2 FLC with

the PSO method obtained additional expected performance without losing

the ride comfort and handling performance. On the quarter-vehicle model,

the suspension travel was reduced to avoid the strut hitting phenomenon.

On the half-vehicle model, the pitch angle acceleration was reduced to im-

prove the longitudinal stability. Based on the improved half-vehicle suspension model, the simulation re-

sults have presented the effect of vehicle moving acceleration. With the

proposed controller, when the vehicle accelerated, the pitch angle accelera-

tion increased, and vice versa.

With the simulation results on the quarter-vehicle and the half-vehicle sus-

pension model, the proposed method obtained better control performance than

the passive suspension and the typical fuzzy control method. In the next chapter,

the stability of the closed-loop fuzzy control system with the proposed method is

analysed and the stability conditions for the proposed method are given.

118

Chapter 5

Stability Analysis of Closed-loop

Systems

5.1 Introduction

Stability is one of the most important issues in analysis and design of control

systems. Stability analysis of fuzzy control system has been more difficult because

the system is essentially non-linear. Reviewing the existing stability analysis

results of typical fuzzy control systems, T-S fuzzy-model-based control systems

provided great development of systematic approaches to stability analysis and

controller design of fuzzy control systems in view of powerful conventional control

theory and techniques (Feng, 2006). The most acceptable method is to design a

feedback controller for each local model and to construct a global controller from

the local controllers in such a way that global stability with/without various

performance indexes of the closed-loop fuzzy control system is guaranteed (Akar

& Ozguner, 2000; Assawinchaichote et al., 2004; Cao & Frank, 2001; Cao & Lin,

2003; Chen et al., 2007; Feng et al., 1997a; Guerra & Vermeiren, 2004; Kiendl &

Ruger, 1995; Kiriakidis et al., 1998; Kosko, 1998; Lam et al., 2001; Ting, 2006).

The major techniques that have been used include quadratic stabilization, linear

matrix inequalities (LMI), Lyapunov stability theory, bilinear matrix inequalities.

Inspired from the above stability analysis approaches, in this chapter, for

the proposed framework of adaptive IT2 FLC for the vehicle active suspension

system, its closed-loop stability is analysed. Firstly, the typical T-S fuzzy control

119

5.2 Typical T-S Fuzzy Control Systems and the Stability Conditions

system and its stability analysis results are reviewed in Section 5.2. Secondly,

the general form of interval type-2 fuzzy system is built in Section 5.3. With the

proposed framework of control system, the IT2 FLC system is demonstrated. In

Section 5.4, the stability of proposed fuzzy control system is analysed and the

sufficient stability conditions are obtained. Concluding remarks, perspectives and

challenges are discussed in Section 5.5.

5.2 Typical T-S Fuzzy Control Systems and the

Stability Conditions

T-S fuzzy model was proposed by Takagi & Sugeno (1985). This model is based

on using a set of fuzzy rules to describe a global non-linear system in terms of a set

of local linear models which are smoothly connected by fuzzy membership func-

tions. A lot of theoretical results on function approximation, stability analysis,

and controller synthesis have been developed for T-S fuzzy model during recent

decades (Fang et al., 2006; Feng, 2006; Guan & Chen, 2004; Tanaka & Wang,

2001; Taniguchi et al., 2000; Tseng et al., 2001). The research results have shown

that T-S fuzzy model is able to approximate any smooth non-linear functions to

any degree of accuracy in any convex compact region (Ke et al., 2000). In this

section, the T-S fuzzy model is represented and the T-S model-based fuzzy con-

troller by parallel distributed compensation (PDC) method is rebuilt in Section

5.2.1. The main approaches of stability analysis and their stability conditions are

reviewed in Section 5.2.2.

5.2.1 T-S Fuzzy Model and Control System

The T-S fuzzy dynamic model is described by fuzzy IF-THEN rules, which rep-

resent local linear input-output relations of non-linear systems. The ith rule of

T-S fuzzy model is shown as follows.

R(i): IF z1 is M i1 and z2 is M i

2, . . ., and zg is M ig THEN x(t + 1) = Aix(t) +

Biu(t). (i = 1, 2, . . . , m).

where M is the typical fuzzy set, x(t)=[x1(t), x2(t), . . . , xn(t)]T denotes the

state vector , u(t) denotes the input vector, m denotes the number of fuzzy rules,

120


and z1, z2, . . . , zg denote measurable variables.

x(t) ∈ ℜn, u(t) ∈ ℜp, Ai ∈ ℜn×n, Bi ∈ ℜn×p (5.1)

Assume that, g=n, z1=x1(t), . . ., zn=xn(t). For a pair of inputs (x(t), u(t)), the

output of T-S fuzzy system is presented as follows:

x(t + 1) =

m∑i=1

ωi[Aix(t)+Biu(t)]

m∑i=1

ωi

=m∑

i=1

hi [Aix(t) + Biu(t)]

(5.2)

where

ωi =n∏

j=1

Mij(xj(t))

hi = ωim∑

i=1

ωi

(5.3)

here, Mij(xj(t)) is the membership grade of xj(t) in the fuzzy set Mij. For general,

the normalized form of ωi and hi are defined as:

ωi ≥ 0, i = 1, 2, · · · , mm∑

i=1

ωi ≥ 0

hi ≥ 0, i = 1, 2, · · · , mm∑

i=1

hi = 1

(5.4)

The T-S fuzzy controller can be designed by the method PDC. The ith control

rule is described as:

R(i): IF z1 is M i1 and z2 is M i

2, . . ., and zn is M in THEN u(t) = Kix(t).

(i = 1, 2, . . . , m).

The control rules have linear state feedback control laws in its consequent

parts. The final output of this fuzzy controller is:

u(t) =

m∑

i=1

hiKix(t), i = 1, 2, · · · , m (5.5)

Substitute equation 5.5 into equation 5.2, the closed-loop T-S fuzzy control system

is presented as follows.

x(t + 1) =

m∑

i=1

m∑

j=1

hihj (Ai − BiKj)x(t) (5.6)

121


5.2.2 The Stability Conditions with Lyapunov Stability

Theory

As mentioned by Feng (2006), based on the above T-S fuzzy control system in

equation 5.6, the existing main methods for stability analysis include quadratic

stabilization, linear matrix inequalities and bilinear matrix inequalities, and so

on. Most of these methods will require a Lyapunov function V (x) = xT Px (e.g.,

common quadratic Lyapunov function, piecewise quadratic Lyapunov function

and fuzzy Lyapunov function). The basic stability condition for the open-loop

T-S fuzzy system can be presented by the following Theorem 5.1.

Theorem 5.1: The equilibrium of system in 5.2 (with u=0) is asymptotically

stable in the large if there exists a common positive definite matrix P such that

ATi P + PAi < 0 (5.7)

for all subsystems, that is, i=1, 2, . . . , m.

A common Lyapunov function in equation 5.7 can be solved numerically by

convex programming algorithms (e.g., LMI method). More details can be found

in the book by Tanaka & Wang (2001). For the further stability analysis of T-S

fuzzy control system, several stability conditions were summarized by Wang et al.

(1996). These stability conditions are rearranged by using the general T-S fuzzy

control system in the equation 5.6.

Firstly, the T-S fuzzy control system 5.6 can be represented in a general form

as follows.

x(t + 1) = G0x(t) +m∑

i=1

m∑j=1

hihj∆Gijx(t)

= G0x(t) +m∑

i=1

hihi∆Giix(t)

+m∑

i<j

hihj∆Fijx(t)

= G0 + W∆(t)Zx(t)

(5.8)

here,

G0 =1

m

m∑

i=1

(Ai + BiKi) (5.9)

∆Gij = Ai + BiKj − G0

∆Gii = QiiΦiiSTii

∆Fij = ∆Gij + ∆Gji = QijΦijSTij , i < j

(5.10)

122


Q and S are unitary matrix, and W ∈ ℜn×γ, ∆(t) ∈ ℜγ×γ , Z ∈ ℜγ×n, γ =

[n × m × (m + 1)]/2, the matrices W ,Z are as follows:

W =[

Q1 Q2 · · · Qm

]

Z =[

S1 S2 · · · Sm

]T

∆(t) = block − diag[

Φe1 Φe

2 · · · Φem

] (5.11)

whereQi =

[Qii Qii+1 · · · Qim

]

Si =[

Sii Sii+1 · · · Sim

]

Φei = block − diag

[eiiΦii eii+1Φii+1 · · · eimΦim

]

eii = hihj

(5.12)

Based on the equation 5.11, the matrices M and N are defined as follows:

M = N = block − diag[

Φd1 Φd

2 · · · Φdm

](5.13)

whereΦd

i = block − diag[

dii

2Φii

dii+1

2Φii+1 · · · dim

2Φim

]

dij = maxhihj

(5.14)

Theorem 5.2: The equilibrium of a general T-S fuzzy control system as given in

equation 5.6 is quadratically stable in the large if and only if one of the following

conditions is satisfied.

C1) There exists a positive definite matrix P such that

P (G0 + WMZ) + (G0 + WMZ)TP+PWNNTWTP + ZTZ < 0

C2) G0 + WMZ is a stable matrix and

∥∥Z(sI − G0 − WMZ)−1WN∥∥∞

< 1

C3) If defined

H =

[G0 + WMZ −WNNTWT

ZTZ −(G0 + WMZ)

]

the condition is

Reλi(H) 6= 0, i = 1, 2, · · · , 2 × n.


[P (G0 + WMZ) + (G0 + WMZ)TP + ZTZ PWN

NTWTP −I

]< 0

123

5.3 The General Interval Type-2 T-S Fuzzy System


P (G0 + WMZ) + (G0 + WMZ)TP PWN ZT

NTWTP −I 0Z 0 −I

< 0

Remark 5.1: The condition C1 is obtained by using the common quadratic

method which finds the positive definite solution P from Riccati equation. The

condition C2 connects the global stability with the H∞ control performance. By

the research of Khargonekar et al. (1990), the conditions C3 - C5 are equivalent

to C1 and C2, however they are described by LMI methods. There have been

some efficient algorithms to check the global stability by LMI methods (Li et al.,

2000; Tanaka & Wang, 2001).

Remark 5.2: All these conditions can be fitted for the deterministic T-S fuzzy

systems, but not for the stochastic T-S fuzzy systems (Feng, 2006). That is means,

if the membership grades are not crisp values ( e.g., uncertain or interval vari-

ables), these above conditions can not directly work. Several authors have made

attempt to address these issues (Chen et al., 2003; Liu & Li, 2005). However, the

existing results are not enough for stability analysis of the proposed adaptive IT2

FLC in this thesis.

5.3 The General Interval Type-2 T-S Fuzzy Sys-

tem

A brief introduction on general interval type-2 T-S fuzzy system is firstly pre-

sented in this section, then the type-2 reasoning methods and proposed adaptive

optimization structure are demonstrated, finally the section is concluded with the

general formulation of proposed IT2 T-S fuzzy control system.

5.3.1 The Interval Type-2 T-S Fuzzy System

Considering an interval type-2 T-S fuzzy model with m rules represented as the

general form:

R(l): IF z1 is F l1 and z2 is F l

2, . . ., and zν is F lν THEN x(t + 1) is gl(X, U).

(l ∈ L := 1, 2, . . . , m).

124


where R(l) denotes the lth fuzzy inference rule, m denotes the number of

fuzzy rules, F lj(j = 1, 2, . . . , ν) denote the interval type-2 fuzzy sets, z(t) :=

[z1, z2, . . . , zν ] denote measurable variables, x(t) ∈ ℜn denotes the state vector,

u(t) ∈ ℜp denotes the input vector, and the T-S consequent terms gl is defined

in equation 5.15.

gl(X, U ; θl) = Alx(t) + Blu(t) + al

l ∈ L := 1, 2, . . . , m(5.15)

where Al, Bl and al are the parameter matrices of the lth local model.

Its firing strength of the lth rule belongs to the following interval set:

ωl(x) ∈ [ωl(x), ωl(x)]; l = 1, 2, · · · , m (5.16)

where

ωl(x) = µF l

1

(x) ∗ µF l

2

(x) · · · ∗ µF l

m

(x) (5.17)

ωl(x) = µF l1(x) ∗ µF l

2(x) · · · ∗ µF l

m(x) (5.18)

in which, µF l

i

(x) and µF li(x) denote the lower and upper membership grades,

respectively. Then the inferred IT2 T-S fuzzy model is defined as

x(t + 1) =m∑

l=1

(α · ωl(x) + β · ωl(x))(Alx + Blu)

=m∑

l=1

ωl(x)(Alx + Blu)(5.19)

whereωl(x) = α · ωl(x) + β · ωl(x) ∈ [0, 1]m∑

l=1

ωl(x) = 1(5.20)

Herein, the values of α and β are both depend on the uncertainty which poten-

tially existed in parameters and fuzzy rules.

As mentioned before, the non-linear suspension model includes parameter and

linguistic uncertainties, then these uncertainties will lead to the uncertainty of

ωl(x) by the two variables α and β. Furthermore, the interval type-2 T-S fuzzy

model is just used as a tool to aid the controller design and it is not necessary

implemented.

125


5.3.2 The Interval Type-2 T-S Fuzzy Control System

In order to control the non-linear system based on the IT2 T-S fuzzy model

described by equation 5.19, an adaptive IT2 T-S fuzzy controller is designed and

its fuzzy rules are given as below,

R(r): IF z1 is F r1 and z2 is F r

2 , . . ., and zν is F rν , THEN u(t) is Krx(t).

(r ∈ L := 1, 2, . . . , m).

where Kr stands for the rth local linear control gain. The output of this

controller is defined as

u(t) =m∑

r=1

f(ωLr (x), ωU

r (x))Kr · x (5.21)

here,

ωLr (x) =

ωr(x)m∑

r=1

(ωr(x) + ωr(x))(5.22)

ωUr (x) =

ωr(x)m∑

r=1

(ωr(x) + ωr(x))(5.23)

ωLr and ωU

r are satisfied with

m∑

r=1

(ωLr (x) + ωU

r (x)) = 1 (5.24)

and the value of f(ωLr (x), ωU

r (x)) depends on the type reduction methods and

belongs to an interval.

In the recent research of Lam & Seneviratne (2008), with the normalized cen-

tral method(i.e., f(ωLr (x), ωU

r (x)) = (ωLr (x) + ωU

r (x))/2), the stability of interval

type-2 T-S fuzzy control system was studied and the stability condition was con-

ducted. However, it can not work on the proposed methods in this thesis because

the type reduction method is different. That is, based on the proposed adaptive

IT2 FLC system, the interval fuzzy outputs were optimised by LMS and PSO

methods with the equation 4.22 and equation 4.28. Then related with the equa-

tion 5.21, a general formula for the control output of proposed method is written

126

5.4 Stability Analysis of the IT2 T-S Fuzzy Control System

as:

u(t) =

m∑

r=1

(αωLr (x) + βωU

r (x))Kr · x (5.25)

For general, the coefficients α and β should be satisfied with the condition: α+β =

1. Since the stability analysis methods for typical fuzzy systems will require the

crisp or precise value of ωl(x), these approaches cannot be directly used to analyse

the stability of IT2 fuzzy control systems. So in this chapter, the stability analysis

approach will be restructured by integrating the lower and upper membership

grades.

5.4 Stability Analysis of the IT2 T-S Fuzzy Con-

trol System

In this section, the stability of the proposed closed-loop IT2 T-S fuzzy control

system is analysed. With the IT2 T-S fuzzy model in equation 5.19 and the

proposed controller in equation 5.25, the closed-loop IT2 T-S fuzzy control system

can be described as follows,

x(t + 1) =m∑

i=1

m∑

j=1

Gij(Ai + BiKj)x(t) (5.26)

where, Gij denotes the fixed membership grade from the IT2 antecedents and

T-S consequent, it is described as,

Gij =[αωL

r (x) + βωUr (x)

]ωj = ωiωj (5.27)

where ωLr (x), ωU

r (x) and ωj are defined in equations 5.22,5.23 and 5.20.

With the closed-loop IT2 T-S fuzzy control system in equation 5.26, a Lya-

punov function candidate is defined as:

V (t) = x(t)TPx(t) (5.28)

here, the matrix P is positive definite, and this function satisfies the following

properties:V (0) = 0, V (x(t)) > 0 for x(t) 6= 0 and V (x(t)) approaches infinity as

‖x(t)‖ → ∞.

127


Then

V (t) = x(t)TPx(t) + x(t)TP x(t) (5.29)

Substituting the equation 5.26 into the equation 5.29, we have

V (t) =

(m∑

i=1

m∑j=1

[αωL

j (x) + βωUj (x)

]ωi(Ai + BiKj)x(t)

)T

Px(t)

+x(t)TP

(m∑

i=1

m∑j=1

[αωL

j (x) + βωUj (x)

]ωi(Ai + BiKj)x(t)

)

= xT(t)

(m∑

i=1

m∑j=1

ωi(αωLj + βωU

j )((Ai + BiKj)TP + P (Ai + BiKj)

)x(t)

(5.30)

For using the general formulations by Lam & Seneviratne (2008); Sala & Arino

(2007), the equation 5.30 can be rewritten as

V (t) =m∑

i=1

m∑j=1

[αωL

j (x) + βωUj (x)

]ωi(Ai + BiKj)Z(t)TQijZ(t)

= Z(t)TΨZ(t)(5.31)

where Z(t) = Px(t), Qij = AiP−1 + P−1AT

i + BiKjP−1 + (KjP

−1)TBTi , Ψ =

m∑i=1

m∑j=1

[αωL

j (x) + βωUj (x)

]ωiQij .

Based on the Lyapunov stability theory, if the condition V (t) ≤ 0 is satisfied,

the related IT2 fuzzy control system is asymptotic stable. From equation 5.31,

Ψ ≤ 0 should be satisfied. Considering the well-known expression of stability

analysis (Arino & Sala, 2008; Sala & Arino, 2007; Sala & AriNo, 2008; Takagi &

Sugeno, 1985), let Ξ = −Ψ and Qij = −Qij . If the following condition is proved,

− ˙V (t) = −Z(t)TΨZ(t) = Z(t)TΞZ(t) ≥ 0 (5.32)

V (t) ≤ 0 can be obtained.

Most of existing stability analysis results for typical T-S fuzzy control system

considered the Qij in the case of same membership grades of fuzzy controller and

fuzzy model(i.e., µi = ηi, here, µi is one of membership grade of fuzzy model

and ηi is one of membership grade of fuzzy controller). However, in the proposed

IT2 fuzzy control system, the membership grades of fuzzy model ωi are not same

128


as the membership grades of fuzzy controller αωLj (x) + βωU

j (x). So some new

conditions must be reconsidered.

Since the interval type-2 fuzzy membership grades belong to an interval, there

will be some constraints between these membership grades of fuzzy model and

fuzzy controller.

ηi −ωU

i

ωi

µi = ηi + aiµi ≤ 0 (5.33)

−ηi +ωL

i

ωi

µi = −ηi + biµi ≤ 0 (5.34)

here, ηi = αωLj (x) + βωU

j (x) and µi = ωi.

The conditions in equation 5.33 and 5.34 can be rewritten as,

m∑i=1

(ηi + aiµi) ≤ 0

m∑i=1

(−ηi + biµi) ≤ 0(5.35)

The positive definite matrices Γk(k = 1, 2) are defined as:

Γ1 =m∑

j=1

(µjRj1 + ηjRj1) ≥ 0

Γ2 =m∑

j=1

(µjTj2 + ηjTj2) ≥ 0(5.36)

Multiplying the first condition in equation 5.35 by Γ1, we get a negative-

semidefinite matrix:

H1 =

m∑

i=1

m∑

j=1

(ηiµjRj1 + ηiηjRj1 + aiµiµjRj1 + aiµiηjRj1) ≤ 0 (5.37)

Multiplying the second condition in equation 5.35 by Γ2, we get another

negative-semidefinite matrix:

H2 =m∑

i=1

m∑

j=1

(−ηiµjTj2 − ηiηjTj2 + biµiµjTj2 + biµiηjTj2) ≤ 0 (5.38)

129


Subsequently, it is evident that if Ξ + H1 ≥ 0 and Ξ + H2 ≥ 0 can be proved,

then Ξ ≥ 0 is satisfied. That is,

Ξ + H1 =m∑

i=1

m∑j=1

(µiηjQij + ηiµjRj1 + ηiηjRj1 + aiµiµjRj1 + aiµiηjRj1)

=m∑

i=1

(aiµ2i Ri1 + η2

i Ri1) +∑

i<j≤m

(µiµj (aiRj1 + ajRi1) +

(ηiηj

(Ri1 + Rj1

)))

+m∑

i=1

m∑j=1

µiηj

(Qij + Ri1 + aiRj1

)

(5.39)

Ξ + H2 =m∑

i=1

m∑j=1

(µiηjQij − ηiµjTj2 − ηiηjTj2 + biµiµjTj2 + biµiηjTj2)

=m∑

i=1

(biµ2i Ti2 − η2

i Ti2) +∑

i<j≤m

(µiµj (biTj2 + bjTi2) −

(ηiηj

(Ti2 + Tj2

)))

+m∑

i=1

m∑j=1

µiηj

(Qij − Ti2 + biTj2

)

(5.40)

Defining the matrices Xij=XTji, i, j = 1, . . . , m, it is satisfied with the condi-

tions as follows.

aiRj1 + ajRi1 ≥ Xij + Xji (5.41)

Ri1 + Rj1 ≥ X(i+m)(j+m) + X(j+m)(i+m) (5.42)

Qij + Ri1 + aiRj1 ≥ Xi(j+m) + X(j+m)i (5.43)

biTj2 + bjTi2 ≥ Xij + Xji (5.44)

−Ti2 − Tj2 ≥ X(i+m)(j+m) + X(j+m)(i+m) (5.45)

Qij − Ti2 + biTj2 ≥ Xi(j+m) + X(j+m)i (5.46)

130


X11 · · · X1(2r)...

. . ....

X(2r)1 · · · X(2r)(2r)

> 0 (5.47)

With the conditions in equations 5.41 - 5.43, the equation 5.39 can be extended

as:

Ξ + H1 ≥m∑

i=1

(µ2i Xii + η2

i Xii)

+m∑

i=1

∑i<j≤m

(µiµj (Xij + Xji) + ηiηj

(X(i+m)(j+m) + X(j+m)(i+m)

))

+m∑

i=1

m∑j=1

(µiηj

(Xi(j+m) + X(j+m)i

))(5.48)

With the different conditions in equations 5.44 - 5.46, the equation 5.40 can

be extended as the same formula as equation 5.48.

Hence, with the equation 5.48, Ξ + H ≥ 0. Then Ξ ≥ 0. And based on the

condition 5.32, ˙V (t) ≤ 0. The proposed interval type-2 fuzzy control system is

asymptotic stable.

Furthermore, the conditions of equations 5.41 - 5.46 can be rewritten as:

aiRj1 + ajRi1 + biTj2 + bjTi2 ≥ Xij + Xji (5.49)

Ri1 + Rj1 − Ti2 − Tj2 ≥ X(i+m)(j+m)X(j+m)(i+m) (5.50)

Qij + Ri1 + aiRj1 − Ti2 + biTj2 ≥ Xi(j+m) + X(j+m)i (5.51)

The stability analysis result is summarized in the following theorem.

Theorem 5.3: With the known conditions of equations 5.33 and 5.34, if there

exist matrices Xij=XTji, i, j = 1, . . . , 2m and symmetric definite positive matrices

Rj1, Rj1, Tj2 and Tj2 for all j = 1, . . . , r, which satisfy the conditions in equations

5.49- 5.51 and equation 5.47, the proposed interval type-2 fuzzy control system

in equation 5.26 is asymptotic stable.

Remark 5.3: When all the membership functions in fuzzy control system are

typical membership functions, the closed-loop system is reduced to a typical T-S

fuzzy control system. The constraints of membership grades in equations 5.33 and

131

5.5 Summary

5.34 will be changed to µi = ηj. The process of stability analysis in this section

will be shrunk to the general formulation for typical T-S fuzzy system. So this

stability analysis method is an extension of existing typical stability results to

consider the interval membership grades in type-2 fuzzy system. Furthermore,

the stability analysis can not only be utilized for proposed adaptive IT2 fuzzy

control system, but also for the general interval type-2 fuzzy systems.

Remark 5.4: Since the stability result in this section is deducted from Lya-

punov stability theory, a common quadratic Lyapunov function need to be found

for all the local subsystems in T-S fuzzy model. And this stability conditions are

sufficient conditions. Less conservative methods about interval type-2 T-S fuzzy

control system will be studied in future.

5.5 Summary

With the proposed framework of adaptive IT2 FLC in Chapter 4, a closed-loop

model of interval type-2 fuzzy control system was presented. Its closed-loop sta-

bility has been investigated by quadratic Lyapunov stability theory. Under the

constraints of interval membership grades in the T-S fuzzy model, the proposed

control system has been proved to be asymptotic stable. And the stability con-

ditions have been carried out to guarantee its stability.

According to the vehicle non-linear suspension system, since the proposed

control method has been guaranteed to be stable with required conditions, it

provides a theoretical foundation for further experimental study and industrial

application development.

132

Chapter 6

Conclusions

6.1 Overview

This thesis started by considering three key aims as the main objectives of this

research project. The first aim was to study the non-linear performance of a

vehicle active suspension system and build the proper model to represent the

effect of changing vehicle speed and random road vibrations. This task involves

the modelling of tyres and numerical simulations on an improved half-vehicle

suspension model. The second aim was to design intelligent control strategies

to minimize the vertical and longitudinal vibrations of vehicle body to improve

the ride comfort and handling performance. The uncertainties from road inputs

and suspension parameters were considered. The third aim was to analyse the

closed-loop system stability of the proposed intelligent controller on the vehicle

suspension system.

6.2 Contributions

In this thesis, an improved non-linear half-vehicle suspension model has been built

to more accurately represent suspension dynamics. With the proposed model,

this thesis has presented a novel framework to adaptively control the vehicle

active suspension system by using interval type-2 fuzzy logic. The bounded in-

terval fuzzy reasoning results have been obtained to cover more general non-linear

133

6.2 Contributions

and uncertain information in the suspension control system. Two optimization

methods have been integrated into the interval type-2 fuzzy control structure to

self-tune the control forces. Simulations with different road conditions and vary-

ing suspension coefficients have been done to verify the efficiency of the proposed

control methods. The closed-loop stability of the proposed system has been anal-

ysed and the stability conditions have been derived. These are addressed in turn

in the following subsections.

6.2.1 The Improved Models

In Chapter 3, two improved half-vehicle active suspension models which inte-

grated tyre longitudinal dynamics, linear and non-linear spring and damper dy-

namics were proposed. By introducing a rigid tyre dynamic model, the tyre

longitudinal vibration and its effect on pitch motion in the suspension were rep-

resented in the improved system model. The improved model has been evaluated

using the open-loop system (not considering the active force) and the closed-

loop system (with linear controller). The simulation results have shown that the

improved models represent more accurate pitch motion dynamics in comparison

with the existing models. Also the non-linear model represents more accurate

vertical vibrations than the linear models.

6.2.2 The Interval Type-2 Fuzzy Controller with LMS

In Chapter 4, Section 4.3.2, an adaptive IT2 FLC with LMS was proposed to con-

trol the vehicle non-linear suspension system. Compared with existed methods,

the proposed method can not only deal with the non-linear dynamics and uncer-

tain information from road vibration and linguistic expert knowledge, but also

make use of the interval reasoning results to do further optimization to reduce

the system error. In Chapter 4, Section 4.4.1 and Section 4.5.1, the simulation

results have shown that the proposed method can significantly reduce the vehicle

body accelerations and tyre dynamic loads on the quarter-vehicle and half-vehicle

suspension models. Furthermore, on the half-vehicle suspension model, with the

additional optimization structure, the longitudinal vibration was constrained and

134

6.2 Contributions

the pitch angle acceleration was reduced by 61.8% over the passive suspension

system and by 26.9% over the TFC method. Also its potential adaptive ability

of dealing with linguistic uncertainty and non-linear dynamics was verified to be

superior than the TFC method. The results have also clearly showed the rela-

tionship between the pitch motion dynamics and different vehicle forward speeds.

6.2.3 The Interval Type-2 Fuzzy Controller with PSO

In Chapter 4, Section 4.3.3, an adaptive IT2 FLC with PSO was proposed to con-

trol the vehicle non-linear suspension system. Different from the LMS method,

the PSO method can solve non-linear optimisation problem. With the same

framework as the adaptive IT2 FLC, this method has dealt with the required

aim of constraining the suspension travel and reducing the pitch vibration. In

Chapter 4, Section 4.4.2 and Section 4.5.2, the simulation results have shown

that the proposed method can not only improve the riding comfort and handling

performance, but also limit the suspension travel to avoid strut hitting on the

quarter-vehicle suspension model. It achieved the aim of reducing the pitch an-

gle accelerations on the half-vehicle suspension model. Also its adaptive ability

of dealing with linguistic uncertainty and non-linear dynamics was verified to be

superior than the typical fuzzy control method. The effect of different vehicle for-

ward speed on suspension pitch motion dynamics has been verified by simulation

results.

6.2.4 Closed-loop Stability Analysis

In Chapter 5, based on existing stability analysis results on the T-S fuzzy con-

trol system and by extending the constraints of membership grades to interval

domains, the quadratic Lyapunov stability theory has been restructured to anal-

yse the closed-loop stability of the proposed interval type-2 fuzzy control system.

Sufficient conditions have been derived by a theorem in Chapter 5, Section 5.3,

to guarantee that the proposed fuzzy control system was asymptotic stable.

135

6.3 Future Work

6.3 Future Work

6.3.1 Expansion of Type-2 Fuzzy Inference Engine

This thesis is the first to design an interval type-2 fuzzy controller for a vehicle

active suspension system. Since the type-2 fuzzy logic system is still an emerging

area of vigorous research activities, its outstanding feature on handling uncer-

tainty need to be verified from theory and applications. Based on the further

optimization structure on the interval type-2 fuzzy reasoning results, the pro-

posed method indirectly optimise the interval fuzzy membership functions and

inference engine performance. Furthermore, this kind of optimal action can be

transferred to the inference engine and directly optimise the fuzzy membership

functions, fuzzy rules and fuzzy reasoning process. With the proposed frame-

work, the probabilistic and fuzzy qualitative reasoning methods can be merged

to expand the properties of type-2 fuzzy system (Liu, 2008b; Liu et al., 2007,

2008a).

6.3.2 Relaxation on Stability Conditions

Using the well developed quadratic Lyapunov stability theory on T-S fuzzy sys-

tems, sufficient conditions were deduced to guarantee the global stability of pro-

posed control methods. As is usual for this approach, a quadratic Lyapunov

function for all local subsystems in a T-S fuzzy model need to be found and this

is conservative in some cases. Recent work has been done to relax the restrictive

conditions by using piecewise quadratic Lyapunov functions or fuzzy Lyapunov

functions (Chen et al., 2005; Feng, 2004). However, ways of extending these

results to type-2 fuzzy systems have not been well studied.

6.3.3 Generalization

To analyse the benefit of the proposed control framework, it should be generalised

to a number of different situations. In this thesis, the generalisation for this

framework was only tested upon the quarter-vehicle and half-vehicle suspension

136

6.4 Summary

system. For closely studying the coupled states of four quarter-vehicle models,

the whole-vehicle model need to be considered. Furthermore, by integrating other

control subsystems in the vehicle(e.g., anti-brake control, steering control or anti-

roll control), a hybrid model will be studied under the proposed control framework

in the future.

6.3.4 Application

The numerical simulations with the proposed control method on the improved

suspension model have strongly supported its practical applications. However,

before the proposed method is used for a real vehicle active suspension system,

some extended research needs to be done to include actuators and physical limits

into the model. Additionally, a solid model of a vehicle active suspension system

will be designed to verify the numerical simulation results.

6.4 Summary

In this dissertation, an improved suspension model was proposed to integrate

the tyre longitudinal dynamics into suspension motion. An overall adaptive IT2

fuzzy control framework has been built to improve the ride comfort and handling

performance by efficiently overtaking non-linear dynamics and uncertainty from

random road surfaces and expert linguistic knowledge. The simulations have

demonstrated that proposed method outperform the passive and typical fuzzy

control methods. Using quadratic Lyapunov stability theory, the global asymp-

totic stability of the proposed method has been proved and the sufficient stability

conditions have been derived.

137

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Appendix A

Publications

1. Cao, J., Liu, H. and Li, P. An Interval Fuzzy Controller for Vehicle Ac-

tive Suspension Systems (under review), IEEE Transactions on Intelligent

Transportation Systems.

2. Cao, J., Li, P., and Liu, H. An Extended Fuzzy Controller for Vehicle Active

Suspension System (under review), Vehicle System Dynamics.

3. Cao, J., Li, P., Liu, H. An Extended Fuzzy Logic System for Uncertainty

Modelling (accepted), IEEE International Conference on Fuzzy Systems,

Jeju Island, Korea, 2009.

4. Cao, J., Ju, Z., Ji, X. and Liu, H. A Switching Fuzzy Control Method

for the Magnetic Active Suspension System (accepted), IEEE International

Conference on Fuzzy Systems, Jeju Island, Korea, 2009.

5. Cao, J., Liu, H., Li, P. and Brown, D.J. State of the Art in Vehicle Active

Suspension Adaptive Control Systems Based on Intelligent Methodologies,

IEEE Transactions on Intelligent Transportation Systems, 2008, 9(3), 392-

405.

6. Cao, J., Liu, H., Li, P. and Brown, D.J. An Interval Type-2 Fuzzy Logic

Controller for Quarter Vehicle Active Suspensions, Proceedings of the Insti-

154

tution of Mechanical Engineers, Part D, Journal of Automobile Engineer-

ing, 2008, 222(8), 1361-1374.

7. Cao, J., Li, P., Liu, H. and Brown, D.J. Adaptive Fuzzy Logic Controller for

Vehicle Active Suspensions with Interval Type-2 Fuzzy Membership Func-

tions, Proc. 2008 IEEE World Congress on Computational Intelligence,

Hong Kong, China, 2008.

8. Cao, J., Li, P., Liu, H. and Brown, D.J. Adaptive Fuzzy Controller for

Vehicle Active Suspensions with Particle Swarm Optimization, Proc. 7th

International Symposium on Instrumentation and Control Technology, Bei-

jing, China, 2008.

9. Cao, J., Liu, H., Li, P., Brown, D.J. and Dimirovski, G.M. Study on Ac-

tive Suspension Control System Based on an Improved Half-Vehicle Model,

International Journal of Automation and Computing, 2007, 4(3), 236-242.

10. Cao, J., Liu, H., Li, P., Brown, D.J. and Dimirovski, G.M. An Improved

Active Suspension Model for Attitude Control of Electric Vehicles, IEEE

International Conference on Mechatronics and Automation, Harbin, China,

2007.

155

Appendix B

Fuzzy Rules Table

Table B.1: The fuzzy rules for half-vehicle active suspension (part 1)

zbf ˙zbf zbr zbr θ Ua zbf ˙zbf zbr zbr θ Ua

NB P PB P Z -K0Xhalf NB P PM N Z -K1Xhalf

NB Z PB P P -K0Xhalf NB Z PM N Z -K1Xhalf

NB N PB P P -K0Xhalf NB N PM N Z -K0Xhalf

NB P PB Z Z -K1Xhalf NB P Z P Z -K1Xhalf

NB Z PB Z P -K0Xhalf NB Z Z P Z -K1Xhalf

NB N PB Z P -K0Xhalf NB N Z P P -K0Xhalf

NB P PB N Z -K1Xhalf NB P Z Z Z -K1Xhalf

NB Z PB N P -K1Xhalf NB Z Z Z Z -K1Xhalf

NB N PB N P -K0Xhalf NB N Z Z P -K0Xhalf

NB P PM P Z -K1Xhalf NB P Z N Z -K0Xhalf

NB Z PM P Z -K1Xhalf NB Z Z N Z -K1Xhalf

NB N PM P P -K0Xhalf NB N Z N Z -K0Xhalf

156



NB P PM Z Z -K1Xhalf NB P NM P Z -K1Xhalf

NB Z PM Z Z -K1Xhalf NB Z NM P Z -K1Xhalf

NB N PM Z P -K0Xhalf NB N NM P P -K0Xhalf

NB P NM Z Z -K2Xhalf NM P PB Z Z -K1Xhalf

NB Z NM Z Z -K2Xhalf NM Z PB Z Z -K1Xhalf

NB N NM Z Z -K0Xhalf NM N PB Z P -K0Xhalf

NB P NM N Z -K2Xhalf NM P PB N Z -K2Xhalf

NB Z NM N Z -K0Xhalf NM Z PB N P -K1Xhalf

NB N NM N Z -K0Xhalf NM N PB N P -K0Xhalf

NB P NB P Z -K1Xhalf NM P PM P Z -K2Xhalf

NB Z NB P Z -K1Xhalf NM Z PM P Z -K2Xhalf

NB N NB P P -K0Xhalf NM N PM P P -K3Xhalf

NB P NB Z Z -K2Xhalf NM P PM Z Z -K2Xhalf

NB Z NB Z Z -K2Xhalf NM Z PM Z Z -K4Xhalf

NB N NB Z Z -K0Xhalf NM N PM Z P -K2Xhalf

NB P NB N Z -K1Xhalf NM P PM N Z -K2Xhalf

NB Z NB N Z -K1Xhalf NM Z PM N Z -K2Xhalf

NB N NB N Z -K0Xhalf NM N PM N P -K3Xhalf

NM P PB P Z -K0Xhalf Z P PM Z Z -K4Xhalf

NM Z PB P P -K0Xhalf Z Z PM Z Z -K4Xhalf

NM N PB P P -K0Xhalf Z N PM Z Z -K3Xhalf

157



NM P Z P Z -K2Xhalf NM P NB Z Z -K2Xhalf

NM Z Z P Z -K4Xhalf NM Z NB Z Z -K2Xhalf

NM N Z P P -K3Xhalf NM N NB Z Z -K0Xhalf

NM P Z Z Z -K2Xhalf NM P NB N Z -K2Xhalf

NM Z Z Z Z -K4Xhalf NM Z NB N Z -K0Xhalf

NM N Z Z P -K3Xhalf NM N NB N P -K0Xhalf

NM P Z N Z -K2Xhalf Z P PB P Z -K0Xhalf

NM Z Z N Z -K4Xhalf Z Z PB P Z -K1Xhalf

NM N Z N P -K3Xhalf Z N PB P P -K0Xhalf

NM P NM P Z -K2Xhalf Z P PB Z Z -K1Xhalf

NM Z NM P Z -K2Xhalf Z Z PB Z Z -K1Xhalf

NM N NM P P -K3Xhalf Z N PB Z Z -K0Xhalf

NM P NM Z Z -K2Xhalf Z P PB N Z -K1Xhalf

NM Z NM Z Z -K4Xhalf Z Z PB N Z -K1Xhalf

NM N NM Z Z -K3Xhalf Z N PB N Z -K0Xhalf

NM P NM N Z -K2Xhalf Z P PM P Z -K3Xhalf

NM Z NM N Z -K2Xhalf Z Z PM P Z -K3Xhalf

NM N NM N Z -K3Xhalf Z N PM P P -K2Xhalf

NM P NB P Z -K1Xhalf PM P PM P Z -K3Xhalf

NM Z NB P Z -K1Xhalf PM Z PM P Z -K3Xhalf

NM N NB P P -K0Xhalf PM N PM P N -K2Xhalf

158



Z P PM N Z -K4Xhalf Z P NB P Z -K1Xhalf

Z Z PM N Z -K4Xhalf Z Z NB P Z -K1Xhalf

Z N PM N Z -K3Xhalf Z N NB P N -K0Xhalf

Z P Z P Z -K4Xhalf Z P NB Z Z -K1Xhalf

Z Z Z P Z -K4Xhalf Z Z NB Z Z -K1Xhalf

Z N Z P P -K4Xhalf Z N NB Z Z -K1Xhalf

Z P Z Z Z -K4Xhalf Z P NB N N -K0Xhalf

Z Z Z Z Z -K4Xhalf Z Z NB N Z -K0Xhalf

Z N Z Z Z -K4Xhalf Z N NB N Z -K0Xhalf

Z P Z N Z -K4Xhalf PM P PB P Z -K0Xhalf

Z Z Z N Z -K4Xhalf PM Z PB P Z -K0Xhalf

Z N Z N Z -K4Xhalf PM N PB P Z -K1Xhalf

Z P NM P Z -K2Xhalf PM P PB Z Z -K0Xhalf

Z Z NM P Z -K2Xhalf PM Z PB Z Z -K0Xhalf

Z N NM P P -K2Xhalf PM N PB Z Z -K1Xhalf

Z P NM Z Z -K4Xhalf PM P PB N Z -K0Xhalf

Z Z NM Z Z -K4Xhalf PM Z PB N Z -K1Xhalf

Z N NM Z Z -K4Xhalf PM N PB N Z -K1Xhalf

Z P NM N N -K3Xhalf PB P PB N Z -K0Xhalf

Z Z NM N Z -K3Xhalf PB Z PB N Z -K0Xhalf

Z N NM N Z -K2Xhalf PB N PB N Z -K0Xhalf

159



PM P PM Z Z -K2Xhalf PM P NM N N -K3Xhalf

PM Z PM Z Z -K4Xhalf PM Z NM N N -K2Xhalf

PM N PM Z Z -K3Xhalf PM N NM N Z -K3Xhalf

PM P PM N Z -K2Xhalf PM P NB P N -K1Xhalf

PM Z PM N Z -K3Xhalf PM Z NB P N -K1Xhalf

PM N PM N Z -K2Xhalf PM N NB P Z -K0Xhalf

PM P Z P Z -K3Xhalf PM P NB Z N -K1Xhalf

PM Z Z P Z -K4Xhalf PM Z NB Z Z -K1Xhalf

PM N Z P Z -K4Xhalf PM N NB Z Z -K1Xhalf

PM P Z Z Z -K3Xhalf PM P NB N N -K0Xhalf

PM Z Z Z Z -K4Xhalf PM Z NB N N -K0Xhalf

PM N Z Z Z -K4Xhalf PM N NB N N -K0Xhalf

PM P Z N N -K2Xhalf PB P PB P Z -K0Xhalf

PM Z Z N N -K3Xhalf PB Z PB P Z -K0Xhalf

PM N Z N Z -K3Xhalf PB N PB P Z -K0Xhalf

PM P NM P Z -K2Xhalf PB P PB Z Z -K0Xhalf

PM Z NM P Z -K2Xhalf PB Z PB Z Z -K0Xhalf

PM N NM P Z -K2Xhalf PB N PB Z Z -K0Xhalf

PM P NM Z N -K2Xhalf PB P NB N N -K0Xhalf

PM Z NM Z N -K4Xhalf PB Z NB N N -K0Xhalf

PM N NM Z Z -K2Xhalf PB N NB N Z -K0Xhalf

160



PB P PM P Z -K0Xhalf PB P NM Z N -K0Xhalf

PB Z PM P Z -K1Xhalf PB Z NM Z N -K1Xhalf

PB N PM P Z -K2Xhalf PB N NM Z Z -K1Xhalf

PB P PM Z Z -K0Xhalf PB P NM N N -K0Xhalf

PB Z PM Z Z -K1Xhalf PB Z NM N N -K0Xhalf

PB N PM Z Z -K1Xhalf PB N NM N N -K0Xhalf

PB P PM N N -K0Xhalf PB P NB P Z -K0Xhalf

PB Z PM N N -K0Xhalf PB Z NB P N -K1Xhalf

PB N PM N Z -K0Xhalf PB N NB P N -K1Xhalf

PB P Z P Z -K0Xhalf PB P NB Z N -K0Xhalf

PB Z Z P Z -K1Xhalf PB Z NB Z N -K0Xhalf

PB N Z P Z -K1Xhalf PB N NB Z Z -K1Xhalf

PB P Z Z Z -K1Xhalf - - - - - -

PB Z Z Z Z -K1Xhalf - - - - - -

PB N Z Z Z -K1Xhalf - - - - - -

PB P Z N N -K0Xhalf - - - - - -

PB Z Z N N -K0Xhalf - - - - - -

PB N Z N Z -K0Xhalf - - - - - -

PB P NM P N -K0Xhalf - - - - - -

PB Z NM P Z -K1Xhalf - - - - - -

PB N NM P Z -K2Xhalf - - - - - -

161

Modelling and Control of Vehicle Suspension Control Systems

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