Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches

431 Pages·2006·8.442 MB·English

by  Jay A. Farrell, Marios M. Polycarpou

ADAPTIVE APPROXIMATION
BASED CONTROL
ADAPTIVE APPROXI MATlON
BASED CONTROL
Unifying Neural, Fuzzy and Traditional
Adaptive Approximation Approaches
Jay A. Farrell
University of California Riverside
Marios M. Polycarpou
University of Cyprus and University of Cincinnati
WILEY-
INTERSCI ENCE
A JOHN WILEY 81 SONS, INC., PUBLICATION
Copyright 0 2006 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Farrell, Jay.
Adaptive approximation based control : unifying neural, fuzzy and traditional adaptive
approximation approaches / Jay A. Farrell, Marios M. Polycarpou.
p. cm.
Includes bibliographical references and index.
ISBN-I 3 978-0-471-72788-0 (cloth)
ISBN-I0 0-471-72788-1 (cloth)
1. Adaptive control systems. 2. Feedback control systems. I. Polycarpou, Marios. 11. Title.
TJ217.F37 2006
629.8'3Wc22
2005021 385
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
To our families and friends.
CONTENTS
...
Preface Xlll
1 Introduction 1
1. I Systems and Control Terminology 1
1.2 Nonlinear Systems 3
1.3 Feedback Control Approaches 4
1.3.1 Linear Design 4
1.3.2 Adaptive Linear Design 6
1.3.3 Nonlinear Design 9
1.3.4 Adaptive Approximation Based Design 11
1.3.5 Example Summary 13
1.4 Components of Approximation Based Control 15
1.4.1 Control Architecture 15
1.4.2 Function Approximator 16
1.4.3 Stable Training Algorithm 17
1.5 Discussion and Philosophical Comments 18
1.6 Exercises and Design Problems 19
23
2 Approximation Theory
2.1 Motivating Example 24
2.2 Interpolation 29
vii
Viii CONTENTS
2.3 Function Approximation 30
2.3.1 Offline (Batch) Function Approximation 31
2.3.2 Adaptive Function Approximation 33
2.4 Approximator Properties 39
2.4.1 Parameter (Non)Linearity 39
2.4.2 Classical Approximation Results 43
2.4.3 Network Approximators 46
2.4.4 Nodal Processors 48
2.4.5 Universal Approximator 50
2.4.6 Best Approximator Property 52
2.4.7 Generalization 54
2.4.8 Extent of Influence Function Support 56
2.4.9 Approximator Transparency 65
2.4.10 Haar Conditions 66
2.4.1 1 Multivariable Approximation by Tensor Products 67
2.5 Summary 68
2.6 Exercises and Design Problems 69
3 Approximation Structures 71
3.1 Model Types 72
3.1.1 Physically Based Models 72
3.1.2 Structure (Model) Free Approximation 73
3.1.3 Function Approximation Structures 74
3.2 Polynomials 75
3.2.1 Description 75
3.2.2 Properties 77
3.3 Splines 78
3.3.1 Description 78
3.3.2 Properties 83
3.4 Radial Basis Functions 84
3.4.1 Description 84
3.4.2 Properties 86
3.5 Cerebellar Model Articulation Controller 87
3.5.1 Description 88
3.5.2 Properties 89
3.6 Multilayer Perceptron 93
3.6.1 Description 93
3.6.2 Properties 95
3.7 Fuzzy Approximation 96
3.7.1 Description 96
3.7.2 Takagi-Sugeno Fuzzy Systems 104
3.7.3 Properties 105
CONTENTS ix
3.8 Wavelets 106
3.8.1 Multiresolution Analysis (MU) 108
3.8.2 MR4 Properties 110
3.9 Further Reading 112
3.10 Exercises and Design Problems 112
4 Parameter Estimation Methods 115
4.1 Formulation for Adaptive Approximation 1 I6
4. I.1 Illustrative Example 116
4.1.2 Motivating Simulation Examples 118
4.1.3 Problem Statement 124
4.1.4 Discussion of Issues in Parametric Estimation 125
4.2 Derivation of Parametric Models 127
4.2.1 Problem Formulation for Full-State Measurement 128
4.2.2 Filtering Techniques 129
4.2.3 SPR Filtering 131
4.2.4 Linearly Parameterized Approximators 131
4.2.5 Parametric Models in State Space Form 133
4.2.6 Parametric Models of Discrete-Time Systems 134
4.2.7 Parametric Models of Input-Output Systems 136
4.3 Design of Online Learning Schemes 138
4.3.1 Error Filtering Online Learning (EFOL) Scheme 138
4.3.2 Regressor Filtering Online Learning (RFOL) Scheme 140
4.4 Continuous-Time Parameter Estimation 141
4.4.1 Lyapunov-Based Algorithms 143
4.4.2 Optimization Methods 148
4.4.3 Summary 154
4.5 Online Learning: Analysis 154
4.5.1 Analysis of LIP EFOL Scheme with Lyapunov Synthesis Method 155
4.5.2 Analysis of LIP RFOL Scheme with the Gradient Algorithm 158
4.5.3 Analysis of LIP RFOL Scheme with RLS Algorithm 160
4.5.4 Persistency of Excitation and Parameter Convergence 161
4.6 Robust Learning Algorithms 163
4.6.1 Projection Modification 165
4.6.2 a-Modification 168
4.6.3 c-Modification 169
4.6.4 Dead-Zone Modification 170
4.6.5 Discussion and Comparison 172
4.7 Concluding Summary 173
4.8 Exercises and Design Problems 173
X CONTENTS
5 Nonlinear Control Architectures 179
5.1 Small-Signal Linearization 180
5.1.1 Linearizing Around an Equilibrium Point 181
5.1.2 Linearizing Around a Trajectory 183
5.I .3 Gain Scheduling 186
5.2 Feedback Linearization 188
5.2.1 Scalar Input-State Linearization 188
5.2.2 Higher-Order Input-State Linearization 190
5.2.3 Coordinate Transformations and Diffeomorphisms 193
5.2.4 Input-Output Feedback Linearization 196
5.3 Backstepping 203
5.3.1 Second Order System 203
5.3.2 Higher Order Systems 205
5.3.3 Command Filtering Formulation 207
5.4 Robust Nonlinear Control Design Methods 211
5.4.1 Bounding Control 21 1
5.4.2 Sliding Mode Control 212
5.4.3 Lyapunov Redesign Method 215
5.4.4 Nonlinear Damping 219
5.4.5 Adaptive Bounding Control 220
5.5 Adaptive Nonlinear Control 222
5.6 Concluding Summary 225
5.7 Exercises and Design Problems 226
6 Adaptive Approximation: Motivation and Issues 231
6.1 Perspective for Adaptive Approximation Based Control 232
6.2 Stabilization of a Scalar System 236
6.2.1 Feedback Linearization 231
6.2.2 Small-Signal Linearization 238
6.2.3 Unknown Nonlinearity with Known Bounds 239
6.2.4 Adaptive Bounding Methods 24 1
6.2.5 Approximating the Unknown Nonlinearity 243
6.2.6 Combining Approximation with Bounding Methods 250
6.2.7 Combining Approximation with Adaptive Bounding Methods 252
6.2.8 Summary 252
6.3 Adaptive Approximation Based Tracking 253
6.3.1 Feedback Linearization 253
6.3.2 Tracking via Small-Signal Linearization 253
6.3.3 Unknown Nonlinearities with Known Bounds 256
6.3.4 Adaptive Bounding Design 258
6.3.5 Adaptive Approximation of the Unknown Nonlinearities 262
CONTENTS xi
6.3.6 Robust Adaptive Approximation 264
6.3.7 Combining Adaptive Approximation with Adaptive Bounding 266
6.3.8 Advanced Adaptive Approximation Issues 27 1
6.4 Nonlinear Parameterized Adaptive Approximation 278
6.5 Concluding Summary 280
6.6 Exercises and Design Problems 28 1
7 Adaptive Approximation Based Control: General Theory 285
7.1 Problem Formulation 286
7.1.1 Trajectory Tracking 286
7.1.2 System 286
7.1 .3 Approximator 287
7.1.4 Control Design 288
7.2 Approximation Based Feedback Linearization 288
7.2.1 Scalar System 289
7.2.2 Input-State 294
7.2.3 Input-Output 306
7.2.4 Control Design Outside the Approximation Region 23 308
7.3 Approximation Based Backstepping 309
7.3.1 Second Order Systems 309
7.3.2 Higher Order Systems 316
7.3.3 Command Filtering Approach 323
7.3.4 Robustness Considerations 328
7.4 Concluding Summary 330
7.5 Exercises and Design Problems 33 1
8 Adaptive Approximation Based Control for Fixed-Wing Aircraft 333
8.1 Aircraft Model Introduction 334
8.1.1 Aircraft Dynamics 334
8.1.2 Nondimensional Coefficients 335
8.2 Angular Rate Control for Piloted Vehicles 336
8.2.1 Model Representation 337
8.2.2 Baseline Controller 337
8.2.3 Approximation Based Controller 338
8.2.4 Simulation Results 345
8.3 Full Control for Autonomous Aircraft 349
8.3.1 Airspeed and Flight Path Angle Control 350
8.3.2 Wind-Axes Angle Control 355
8.3.3 Body Axis Angular Rate Control 359
8.3.4 Control Law and Stability Properties 362
8.3.5 Approximator Definition 365
xii CONTENTS
8.3.6 Simulation Analysis 367
8.3.7 Conclusions 371
8.4 Aircraft Notation 371
Appendix A: Systems and Stability Concepts 377
A. 1 Systems Concepts 377
A.2 Stability Concepts 379
A.2.1 Stability Definitions 3 79
A.2.2 Stability Analysis Tools 381
A.2.3 Strictly Positive Real Transfer Functions 391
A.3 General Results 392
A.4 Trajectory Generation Filters 394
AS A Useful Inequality 391
A.6 Exercises and Design Problems 398
Appendix B: Recommended Implementation and Debugging Approach 399
References 40 1
Index 417