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Sliding Mode Control:
Theory and Applications
Christopher Edwards
and
Sarah K. Spurgeon
»8 -
Contents
Series Introduction XII
Preface xiv
1 An Introduction to Sliding Mode Control 1
1.1 Introduction 1
1.2 Properties of the Sliding Motion 6
1.3 Different Controller Designs 11
1.4 Pseudo-Sliding with a Smooth Control Action 15
1.5 A State-Space Approach 17
1.6 Notes and References 18
2 Multivariable Systems Theory 19
2.1 Introduction 19
i
2.2 Stability of Dynamical Systems 19
2.2.1 Linear Time Invariant Systems 20
2.2.2 Quadratic Stability 21
2.3 Linear Systems Theory 25
2.3.1 Controllability and Observability 25
2.3.2 Invariant Zeros 27
2.3.3 State Feedback Control 28
2.3.4 Static Output Feedback Control 28
2.3.5 Observer-Based Control 29
2.4 Notes and References 30
Vlll CONTENTS
3 Sliding Mode Control 31
3.1 Introduction 31
3.2 Problem Statement 32
3.3 Existence of Solution and Equivalent Control 33
3.4 Properties of the Sliding Motion 35
3.5 The Reachability Problem 41
3.5.1 The Single-Input Case 41
3.5.2 Single-Input Control Structures 43
3.5.3 An Example: The Normalised Pendulum Revisited 46
3.5.4 The Multivariable Case 47
3.6 The Unit Vector Approach 50
3.6.1 Existence of an Ideal Sliding Mode 52
3.6.2 Description of the Sliding Motion 53
3.6.3 Practical Considerations 54
3.6.4 Example: Control of a DC Motor 55
3.6.5 Concluding Remarks 59
3.7 Continuous Approximations 59
3.8 Summary 63
3.9 Notes and References 63
4 Sliding Mode Design Approaches 65
4.1 Introduction • 65
4.2 A Regular Form Based Approach 65
4.2.1 Robust Eigenstructure Assignment 68
4.2.2 Quadratic Minimisation 72
4.3 A Direct Eigenstructure Assignment Approach 74
4.4 Incorporation of a Tracking Requirement 77
4.4.1 A Model-Reference Approach 78
4.4.2 An Integral Action Approach 82
4.5 Design Study: Pitch-Pointing Flight Controller 85
4.5.1 Model-Reference Design 88
4.5.2 Integral Action Based Design 90
4.6 Summary 91
4.7 Notes and References 92
CONTENTS IX
5 Sliding Mode Controllers Using Output Information 93
5.1 Introduction 93
5.2 Problem Formulation 93
5.3 A Special Case: Square Plants 94
5.4 A General Framework 98
5.4.1 Hyperplane Design 99
5.4.2 Control Law Synthesis 105
5.4.3 Example 1 106
5.5 Dynamic Compensation 108
5.6 Dynamic Compensation (Observer Based) 111
5.6.1 Control Law Construction 113
5.6.2 Design Example 1 116
5.6.3 Design Example 2: Inverted Pendulum 118
5.7 A Model-Reference System Using Only Outputs 121
5.7.1 Aircraft Example 122
5.8 Summary 125
5.9 Notes and References 125
6 Sliding Mode Observers 127
6.1 Introduction 127
6.2 Sliding Mode Observers 127
6.2.1 An Utkin Observer 127
6.2.2 Example 1 129
6.2.3 A Modification to Include a Linear Term 131
6.2.4 A Walcott-Zak Observer 131
6.3 Synthesis of a Discontinuous Observer 133
6.3.1 A Canonical Form for Observer Design 134
6.3.2 Existence Conditions 136
6.4 The Walcott-Zak Observer Revisited 142
6.4.1 Example 2: Pendulum 145
6.4.2 Pendulum Simulation 146
6.5 Sliding Mode Observers for Fault Detection 147
6.5.1 Reconstruction of the Input Fault Signals 148
6.5.2 Detection of Faults at the Output 149
6.5.3 Example: Inverted Pendulum 150
X CONTENTS
6.5.4 Simulations of Different Fault Conditions 151
6.6 Summary 153
6.7 Notes and References 154
7 Observer-Based Output Tracking Controllers 155
7.1 Introduction 155
7.2 System Description and Observer Formulation 155
7.3 An Integral Action Controller 156
7.3.1 Nonlinear Observer Formulation (For Square Plants) 157
7.3.2 State Feedback Integral Action Control Law (Reprise) 159
7.3.3 Closed-Loop Analysis 160
7.3.4 Design and Implementation Issues 166
7.4 Example: A Temperature Control Scheme 170
7.4.1 Observer Design 170
7.4.2 Controller Design 171
7.4.3 Design of the Nonlinear Gain Function 172
7.4.4 Furnace Simulations 173
7.5 A Model-Reference Approach 175
7.5.1 Example: L-1011 Fixed-Wing Aircraft 179
7.6 Summary • 181
7.7 Notes and References 181
8 Automotive Case Studies 183
8.1 Introduction 183
8.2 Automotive Actuator with Stiction 183
8.3 Robust Control of an Automotive Engine 188
8.3.1 Controller Design Issues 190
8.3.2 Engine Controller Design 191
8.3.3 Implementation Results 192
8.4 Summary 197
8.5 Notes and References 197
9 Furnace Control Case Study 199
9.1 Introduction 199
9.2 Observer Design 202
9.3 Controller Design 203
CONTENTS XI
9.4 Implementation Results 204
9.5 Summary 206
9.6 Notes and References 206
Appendices
A Mathematical Preliminaries 207
A.I Mathematical Notation 207
A.2 Linear Algebra 208
A.2.1 Vector Spaces and Linear Maps 208
A.2.2 Properties of Linear Maps (Matrices) 210
A.2.3 Rank and Determinant 211
A.2.4 Eigenvalues, Eigenvectors and Singular Values 213
A.2.5 QR Decomposition 214
A.2.6 Norms, Inner Products and Projections 215
A.2.7 Quadratic Forms 217
1
A.3 Notes and References 218
B Assorted mfiles 219
B.I A Variation on the place Command 219
B.2 Eigenstructure Assignment: The Complex Case 220
224
B.3 World Wide Web Site
225
References
233
Index
