Adaptive Control: Stability, Convergence and Robustness

Adaptive Control: Stability, Convergence and Robustness

by Shankar Sastry, Marc Bodson

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Overview

With a focus on linear, continuous time, single-input, single-output systems, this volume surveys the major results and techniques of analysis in the field of adaptive control. The authors offer a clear, conceptual presentation of adaptive methods, enabling a critical evaluation of these techniques and suggesting avenues of further development.
A brief historical overview of adaptive control is followed by a review of mathematical preliminaries and the development of several adaptive identification algorithms. Succeeding chapters examine averaging techniques, the robustness of adaptive schemes, and advanced topics — including the use of prior information and multivariable adaptive control — followed by a concise introduction to the control of a class of nonlinear systems. The treatment is largely self-contained, assuming only some graduate-level background in basic control systems and in linear systems theory.

Product Details

ISBN-13: 9780486482026
Publisher: Dover Publications
Publication date: 09/14/2011
Series: Dover Books on Electrical Engineering Series
Pages: 400
Product dimensions: 6.00(w) x 8.90(h) x 0.90(d)

Table of Contents

Preface xiii

Chapter 0 Introduction 1

0.1 Identification and Adaptive Control 1

0.2 Approaches to Adaptive Control 4

0.2.1 Gain Scheduling 4

0.2.2 Model Reference Adaptive Systems 5

0.2.3 Self Tuning Regulators 8

0.2.4 Stochastic Adaptive Control 9

0.3 A Simple Example 11

Chapter 1 Preliminaries 17

1.1 Notation 17

1.2 Lp Spaces, Norms 18

1.3 Positive Definite Matrices 19

1.4 Stability of Dynamic Systems 20

1.4.1 Differential Equations 20

1.4.2 Stability Definitions 23

1.4.3 Lyapunov Stability Theory 25

1.5 Exponential Stability Theorems 28

1.5.1 Exponential Stability of Nonlinear Systems 28

1.5.2 Exponential Stability of Linear Time-Varying Systems 32

1.5.3 Exponential Stability of Linear Time Invariant Systems 38

1.6 Generalized Harmonic Analysis 39

Chapter 2 Identification 45

2.0 Introduction 45

2.1 Identification Problem 52

2.2 Identifier Structure 53

2.3 Linear Error Equation and Identification Algorithms 57

2.3.1 Gradient Algorithms 58

2.3.2 Least-Squares Algorithms 61

2.4 Properties of the Identification Algorithms-Identifier Stability 63

2.4.1 Gradient Algorithms 63

2.4.2 Least-Squares Algorithms 66

2.4.3 Stability of the Identifier 69

2.5 Persistent Excitation and Exponential Parameter Convergence 71

2.6 Model Reference Identifiers-SPR Error Equation 76

2.6.1 Model Reference Identifiers 76

2.6.2 Strictly Positive Real Error Equation and Identification Algorithms 82

2.6.3 Exponential Convergence of the Gradient Algorithms with SPR Error Equations 85

2.7 Frequency Domain Conditions for Parameter Convergence 90

2.7.1 Parameter Convergence 90

2.7.2 Partial Parameter Convergence 95

2.8 Conclusions 98

Chapter 3 Adaptive Control 99

3.0 Introduction 99

3.1 Model Reference Adaptive Control Problem 103

3.2 Controller Structure 104

3.3 Adaptive Control Schemes 110

3.3.1 Input Error Direct Adaptive Control 111

3.3.2 Output Error Direct Adaptive Control 118

3.3.3 Indirect Adaptive Control 123

3.3.4 Alternate Model Reference Schemes 127

3.3.5 Adaptive Pole Placement Control 129

3.4 The Stability! Problem in Adaptive Control 130

3.5 Analysis of the Model Reference Adaptive Control System 132

3.6 Useful Lemmas 138

3.7 Stability Proofs 142

3.7.1 Stability-Input Error Direct Adaptive Control 142

3.7.2 Stability-Output Error Direct Adaptive Control 149

3.7.3 Stability-Indirect Adaptive Control 151

3.8 Exponential Parameter Convergence 154

3.9 Conclusions 156

Chapter 4 Parameter Convergence Using Averaging Techniques 158

4.0 Introduction 158

4.1 Examples of Averaging Analysis 159

4.2 Averaging Theory-One-Time Scale 166

4.3 Application to Identification 175

4.4 Averaging Theory-Two-Time Scales 179

4.4.1 Separated Time Scales 183

4.4.2 Mixed Time Scales 186

4.5 Applications to Adaptive Control 187

4.5.1 Output Error Scheme-Linearized Equations 188

4.5.2 Output Error Scheme-Nonlinear Equations 192

4.5.3 Input Error Scheme 202

4.6 Conclusions 207

Chapter 5 Robustness 209

5.1 Structured and Unstructured Uncertainty 209

5.2 The Rohrs Examples 215

5.3 Robustness of Adaptive Algorithms with Persistency of Excitation 219

5.3.1 Exponential Convergence and Robustness 221

5.3.2 Robustness of an Adaptive Control Scheme 225

5.4 Heuristic Analysis of the Rohrs Examples 231

5.5 Averaging Analysis of Slow Drift Instability 236

5.5.1 Instability Theorems Using Averaging 236

5.5.2 Application to the Output Error Scheme 241

5.6 Methods for Improving Robustness-Qualitative Discussion 248

5.6.1 Robust Identification Schemes 248

5.6.2 Specification of the Closed Loop Control Objective-Choice of Control Model and of Reference Input 250

5.6.3 The Usage of Prior Information 250

5.6.4 Time Variation of Parameters 251

5.7 Robustness via Update Law Modifications 251

5.7.1 Deadzone and Relative Deadzone 251

5.7.2 Leakage Term (σ-Modification) 253

5.7.3 Regressor Vector Filtering 253

5.7.4 Slow Adaptation, Averaging and Hybrid Update Laws 254

5.8 Conclusions 254

Chapter 6 Advanced Topics in Identification and Adaptive Control 257

6.1 Use of Prior Information 257

6.1.1 Identification of Partially Known Systems 257

6.1.2 Effect of Unmodeled Dynamics 263

6.2 Global Stability of Indirect Adaptive Control Schemes 266

6.2.1 Indirect Adaptive Control Scheme 267

6.2.2 Indirect Adaptive Pole Placement 272

6.2.3 Indirect Adaptive Stabilization-The Factorization Approach 274

6.3 Multivariate Adaptive Control 277

6.3.1 Introduction 277

6.3.2 Preliminaries 278

6.3.2.1 Factorization of Transfer Function Matrices 278

6.3.2.2 Interactor Matrix and Hermite Form 282

6.3.3 Model Reference Adaptive Control-Controller Structure 286

6.3.4 Model Reference Adaptive Control-Input Error Scheme 290

6.3.5 Alternate Schemes 292

6.4 Conclusions 293

Chapter 7 Adaptive Control of a Class of Nonlinear Systems 294

7.1 Introduction 294

7.2 Linearizing Control for a Class of Nonlinear Systems-A Review 295

7.2.1 Basic Theory 295

7.2.2 Minimum Phase Nonlinear Systems 299

7.2.2.1 The Single-Input Single-Output Case 300

7.2.2.2 The Multi-Input Multi-Output Case 305

7.2.3 Model Reference Control for Nonlinear Systems 307

7.3 Adaptive Control of Linearizable Minimum Phase Systems 309

7.3.1 Single-Input Single-Output, Relative Degree One Case 309

7.3.2 Extensions to Higher Relative Degree SISO Systems 312

7.3.3 Adaptive Control of MIMO Systems Decouplable by Static State Feedback 320

7.4 Conclusions 322

Chapter 8 Conclusions 324

8.1 General Conclusions 325

8.2 Future Research 325

Appendix 331

References 359

Index 373

Errata 379

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