Stable Adaptive Systems

, by Narendra, Kumpati S.; Annaswamy, Anuradha M.

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ISBN: 9780486442266 | 0486442268
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Copyright: 5/31/2005

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This graduate-level text focuses on the stability of adaptive systems, and offers a thorough understanding of the global stability properties essential to designing adaptive systems. Its self-contained, unified presentation of well-known results establishes the close connections between seemingly independent developments in the field. Prerequisites include a knowledge of linear algebra and differential equations, as well as a familiarity with basic concepts in linear systems theory.

Preface

1 Introduction

(41)

1.1 Introduction

(1)

1.2 Systems Theory

(3)

1.3 Adaptive Systems

(8)

1.3.1 What Is an Adaptive System?,

(3)

1.3.2 Parameter Adaptive and Structurally Adaptive Systems,

(1)

1.4 Direct and Indirect Control

(8)

1.4.1 Parameter Perturbation Method,

(4)

1.4.2 Sensitivity Method,

(3)

1.5 Model Reference Adaptive Systems and Self-Tuning Regulators

(7)

1.5.1 Identification Model,

(1)

1.5.2 Reference Model,

(2)

1.5.3 Reference Model with Inputs,

(1)

1.5.4 Model Reference Adaptive Control,

(3)

1.5.5 Self-Tuning Regulators,

(1)

1.6 Stable Adaptive Systems

(3)

1.6.1 Error Models,

(1)

1.7 Applications

(1)

1.8 Summary

(2)

1.9 Scope of the Book

(1)

Problems

(4)

References

(3)

2 Stability Theory

(57)

2.1 Introduction

(2)

2.2 Notation

(1)

2.3 Linear Systems

(5)

2.3.1 Linear Time-Invariant Systems,

(1)

2.3.2 Almost Time-Invariant Systems,

(3)

2.4 Lyapunov Stability

(13)

2.4.1 Definitions,

(3)

2.4.2 Lyapunov's Direct Method,

(7)

2.4.3 V(x, t) > 0, V(x, t) less than or equal to 0, 56

2.4.4 LTI Systems and Lyapunov Stability,

(1)

2.4.5 Linear Time-Varying Systems,

(2)

2.5 Positive Real Functions

(2)

2.5.1 Positive Real Transfer Matrices,

(1)

2.6 Kalman-Yakubovich Lemma

(2)

2.7 Input-Output Stability

(4)

2.8 Stability of Adaptive Systems

(17)

2.8.1 Two Differential Equations That Arise in Adaptive Systems,

(6)

2.8.2 Growth Rates of Signals in Dynamical Systems,

(9)

2.8.3 Stability in Terms of a Parameter Vector theta,

(2)

2.9 Other Stability Concepts

(6)

2.9.1 Total Stability,

(3)

2.9.2 BIBO Stability,

(2)

2.10 Summary

(1)

Problems

(2)

References

(3)

3 Simple Adaptive Systems

(41)

3.1 Introduction

(1)

3.2 Algebraic Systems

100

(2)

3.2.1 Identification,

100

(2)

3.3 Dynamical Systems

102

(24)

3.3.1 The Identification Problem: Scalar Case,

102

(7)

3.3.2 The Control Problem: Scalar Case (Direct Control),

109

(8)

3.3.3 The Control Problem: Scalar Case (Indirect Control),

117

(4)

3.3.4 Indirect Control: Dynamic Adjustment of Control Parameters,

121

(2)

3.3.5 Examples,

123

(3)

3.4 State Variables Accessible

126

(10)

3.4.1 The Identification Problem: Vector Case,

126

(2)

3.4.2 The Control Problem,

128

(8)

3.5 Summary

136

(1)

Problems

136

(3)

References

139

(1)

4 Adaptive Observers

140

(41)

4.1 Introduction

140

(1)

4.2 The Luenberger Observer

141

(3)

4.3 Adaptive Observers

144

(18)

4.3.1 Statement of the Problem,

144

(1)

4.3.2 Minimal Realization of Adaptive Observers,

145

(6)

4.3.3 Nonminimal Realization of Adaptive Observers,

151

(7)

4.3.4 Modified Adaptive Observer,

158

(4)

4.4 Design of Adaptive Observers

162

(15)

4.4.1 Representation 1,

163

(2)

4.4.2 Representation 2,

165

(1)

4.4.3 Hybrid Algorithms,

166

(3)

4.4.4 Time-Varying Adaptive Gains,

169

(1)

4.4.5 Integral Algorithm,

170

(1)

4.4.6 Multiple Models,

171

(2)

4.4.7 Adaptive Algorithms: An Overview,

173

(4)

4.5 Summary

177

(1)

Problems

177

(2)

References

179

(2)

5 The Control Problem

181

(57)

5.1 Introduction

181

(1)

5.2 Statement of the Problem

182

(2)

5.3 Adaptive Control of Plants with Relative Degree Unity

184

(15)

5.3.1 Case (i) kp, Unknown,

186

(1)

5.3.2 Case (ii) Unknown Zeros,

187

(3)

5.3.3 Case (iii) Unknown Poles,

190

(2)

5.3.4 General Case (n* = 1),

192

(7)

5.4 Adaptive Control of Plants with Relative Degree greater than 2

199

(13)

5.4.1 Bezout Identity,

199

(3)

5.4.2 n* > or equal to 2: Algebraic Part,

202

(1)

5.4.3 n* = 2: Analytic Part,

203

(1)

5.4.4 n* > or equal to 2: Analytic Part,

204

(2)

5.4.5 kp Known,

206

(3)

5.4.6 kp Unknown,

209

(3)

5.5 Proof of Global Stability of the Adaptive System

212

(10)

5.5.1 kp Known,

213

(5)

5.5.2 kp Unknown,

218

(4)

5.6 Comments on the Control Problem

222

(2)

5.7 Some Remarks on the Controller Structure

224

(2)

5.8 Stable Adaptive Control Using the Indirect Approach

226

(5)

5.8.1 n* = 1,

227

(3)

5.8.2 n* > or equal to 2,

230

(1)

5.9 Combined Direct and Indirect Approach

231

(2)

5.10 Summary

233

(1)

Problems

234

(2)

References

236

(2)

6 Persistent Excitation

238

(35)

6.1 Introduction

238

(1)

6.2 Persistent Excitation in Adaptive Systems

239

(7)

6.3 Definitions

246

(3)

6.3.1 Examples,

248

(1)

6.4 Properties of Persistently Exciting Functions

249

(9)

6.4.1 Algebraic Transformations,

249

(2)

6.4.2 Dynamic Transformations,

251

(5)

6.4.3 Almost Time-Invariant Systems,

256

(1)

6.4.4 Persistent Excitation of Unbounded Signals,

257

(1)

6.5 Applications to Adaptive Systems

258

(8)

6.5.1 Parameter Convergence in Adaptive Observers,

258

(2)

6.5.2 Parameter Convergence in Adaptive Control,

260

(3)

6.5.3 Design Considerations,

263

(3)

6.6 Persistent Excitation and Uniform Asymptotic Stability

266

(2)

6.7 Summary

268

(1)

Problems

269

(2)

References

271

(2)

7 Error Models

273

(18)

7.1 Introduction

273

(1)

7.2 Error Model 1

274

(4)

7.3 Error Model 2

278

(1)

7.4 Error Model 3

279

(3)

7.5 Error Model 4

282

(4)

7.6 Further Comments on Error Models 1-4

286

(1)

7.7 Use of Error Models in Adaptive Systems

287

(1)

Problems

288

(2)

References

290

(1)

8 Robust Adaptive Control

291

(68)

8.1 Introduction

291

(2)

8.2 Adaptive Observers in the Presence of Disturbances

293

(4)

8.2.1 Identifier,

293

(1)

8.2.2 Adaptive Observers,

294

(1)

8.2.3 Error Model,

295

(2)

8.3 Adaptive Control of a First-Order Plant in the Presence of Bounded Disturbances

297

(21)

8.3.1 Exact Cancellation of Disturbance,

297

(4)

8.3.2 Bounded Disturbances: Statement of the Problem,

301

(5)

8.3.3 Category 1: Modifications of the Adaptive Law,

306

(6)

8.3.4 Category 2: Persistent Excitation of the Reference Input,

312

(4)

8.3.5 Performance of the Adaptive System,

316

(2)

8.4 Adaptive Control of an nth Order Plant

318

(13)

8.4.1 Robustness without Persistent Excitation;

322

(5)

8.4.2 Robustness with Persistent Excitation,

327

(3)

8.4.3 Comments on Adaptive Algorithms for Bounded Disturbances,

330

(1)

8.5 Hybrid Adaptive Control

331

(6)

8.5.1 Hybrid Error Models,

331

(3)

8.5.2 Hybrid Adaptive Control,

334

(2)

8.5.3 Hybrid Adaptive Control in the Presence of Bounded Disturbances,

336

(1)

8.6 Adaptive Control of Time-Varying Plants

337

(8)

8.6.1 Adaptive Control of a First-Order Plant,

338

(4)

8.6.2 Time-Varying Plants of Higher Order,

342

(3)

8.7 Reduced-Order Controllers

345

(8)

8.7.1 Instability with Reduced Order Controllers: An Example,

346

(1)

8.7.2 Statement of the Problem,

347

(1)

8.7.3 Small Parameter μ in P,

348

(2)

8.7.4 Transfer Function Wm(s),

350

(3)

8.8 Summary

353

(1)

Problems

354

(2)

References

356

(3)

9 The Control Problem: Relaxation of Assumptions

359

(33)

9.1 Introduction

359

(1)

9.2 Assumption (i): Sign of kp

360

(6)

9.3 Assumption (iii): Upper Bound on the Order of the Plant

366

(8)

9.3.1 Regulation of Plant with Relative Degree n* = 1,

366

(2)

9.3.2 n* = 2,

368

(1)

9.3.3 Arbitrary n*,

369

(4)

9.3.4 Universal Controller,

373

(1)

9.4 Assumption (iv): Zeros of Wp(s) in C-

374

(9)

9.4.1 Direct Control,

375

(3)

9.4.2 Indirect Control,

378

(5)

9.5 Command Generator Tracking Method

383

(4)

9.5.1 Statement of the Problem,

383

(4)

9.6 Summary

387

(1)

Problems

388

(1)

References

389

(3)

10 Multivariable Adaptive Systems

392

(28)

10.1 Introduction

392

(1)

10.2 Mathematical Preliminaries

393

(6)

10.3 Multivariable Adaptive Identification

399

(3)

10.3.1 Statement of the Problem,

399

(1)

10.3.2 The Adaptive Observer: Minimal Realization,

400

(1)

10.3.3 The Adaptive Observer: Nonminimal Realization,

401

(1)

10.4 Adaptive Control Problem

402

(4)

10.5 Adaptive Control of Two Input-Two Output Plants

406

(2)

10.6 Dynamic Precompensator for a General Multivariable Plant

408

(9)

10.6.1 Design of the Adaptive Controller,

410

(7)

10.7 Summary

417

(1)

Problems

417

(2)

References

419

(1)

11 Applications of Adaptive Control

420

(46)

11.1 Introduction

420

(1)

11.2 Ship Dynamics

421

(8)

11.3 Process Control

429

(11)

11.3.1 Distillation Columns,

430

(10)

11.4 Power Systems

440

(3)

11.5 Robot Manipulators

443

(11)

11.6 Adaptive Control in Bioengineering

454

(10)

11.6.1 Blood Pressure Control,

455

(1)

11.6.2 Adaptive Control for Ventricular Assist Devices,

456

(4)

11.6.3 Muscular Control,

460

(4)

11.7 Comments and Conclusions

464

(2)

References

466

(7)

A Kalman-Yakubovich Lemma

473

(3)

B Growth Rates of Unbounded Signals

476

(4)

C Properties of the Adaptive System with Bounded Parameters

480

(4)

D Proof of Boundedness in the Presence of Bounded Disturbances

484

(5)

E Some Facts About Transfer Matrices

489

(2)

Index

491

(4)

Errata List

495

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