## Robust Adaptive Control - Personal World Wide Web Pages

May 2, 2010 - on adaptive systems at the senior undergraduate, or first and second gradu- ... in undergraduate courses on linear systems, differential equations, and auto- ...... y, uf = 1 s2 + 2s + 1 u (1.2.26) are signals that can be generated by filtering. If we now replace Î¸â. 1 and Î¸â. 2 with their estimates Î¸1 and Î¸2 in ...
Contents Preface

xiii

List of Acronyms

xvii

1 Introduction 1.1 Control System Design Steps . . . . . . . . . . . 1.2 Adaptive Control . . . . . . . . . . . . . . . . . . 1.2.1 Robust Control . . . . . . . . . . . . . . . 1.2.2 Gain Scheduling . . . . . . . . . . . . . . 1.2.3 Direct and Indirect Adaptive Control . . 1.2.4 Model Reference Adaptive Control . . . . 1.2.5 Adaptive Pole Placement Control . . . . . 1.2.6 Design of On-Line Parameter Estimators 1.3 A Brief History . . . . . . . . . . . . . . . . . . . 2 Models for Dynamic Systems 2.1 Introduction . . . . . . . . . . . . . . 2.2 State-Space Models . . . . . . . . . 2.2.1 General Description . . . . . 2.2.2 Canonical State-Space Forms 2.3 Input/Output Models . . . . . . . . 2.3.1 Transfer Functions . . . . . . 2.3.2 Coprime Polynomials . . . . 2.4 Plant Parametric Models . . . . . . 2.4.1 Linear Parametric Models . . 2.4.2 Bilinear Parametric Models . v

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26 26 27 27 29 34 34 39 47 49 58

vi

CONTENTS 2.5

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3 Stability 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Norms and Lp Spaces . . . . . . . . . . . . . . . . . . 3.2.2 Properties of Functions . . . . . . . . . . . . . . . . . 3.2.3 Positive Definite Matrices . . . . . . . . . . . . . . . . 3.3 Input/Output Stability . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lp Stability . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The L2δ Norm and I/O Stability . . . . . . . . . . . . 3.3.3 Small Gain Theorem . . . . . . . . . . . . . . . . . . . 3.3.4 Bellman-Gronwall Lemma . . . . . . . . . . . . . . . . 3.4 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Definition of Stability . . . . . . . . . . . . . . . . . . 3.4.2 Lyapunov’s Direct Method . . . . . . . . . . . . . . . 3.4.3 Lyapunov-Like Functions . . . . . . . . . . . . . . . . 3.4.4 Lyapunov’s Indirect Method . . . . . . . . . . . . . . . 3.4.5 Stability of Linear Systems . . . . . . . . . . . . . . . 3.5 Positive Real Functions and Stability . . . . . . . . . . . . . . 3.5.1 Positive Real and Strictly Positive Real Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 PR and SPR Transfer Function Matrices . . . . . . . 3.6 Stability of LTI Feedback Systems . . . . . . . . . . . . . . . 3.6.1 A General LTI Feedback System . . . . . . . . . . . . 3.6.2 Internal Stability . . . . . . . . . . . . . . . . . . . . . 3.6.3 Sensitivity and Complementary Sensitivity Functions . 3.6.4 Internal Model Principle . . . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 67 67 72 78 79 79 85 96 101 105 105 108 117 119 120 126 126 132 134 134 135 136 137 139

4 On-Line Parameter Estimation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 Simple Examples . . . . . . . . . . . . . . . . . . . 4.2.1 Scalar Example: One Unknown Parameter 4.2.2 First-Order Example: Two Unknowns . . . 4.2.3 Vector Case . . . . . . . . . . . . . . . . . .

144 144 146 146 151 156

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