Table of Contents

Table of Contents

1 Adaptive Systems

1

    1.1 Introduction

1

    1.2 Adaptive systems: examples

1

      1.2.1 Adaptive control

1

      1.2.2 Adaptive signal processing

3

      1.2.3 Adaptive systems versus classical techniques

3

    1.3 General structure of adaptive control systems

4

      1.3.1 Introduction

4

      1.3.2 The general structure

4

      1.3.3 The error signal

8

      1.3.4 The tuner

9

      1.3.5 Certainty equivalence

11

      1.3.6 Design and analysis

12

    1.4 Illustrating the concepts

13

      1.4.1 The MIT rule for adaptive control: feedforward case

13

      1.4.2 The MIT rule for adaptive control: feedback problem

16

      1.4.3 An adaptive pole placement scheme

19

      1.4.4 A universal controller

21

      1.4.5 Echo cancelling

22

    1.5 Summary of chapter

25

    1.6 Notes and references

25

    1.7 Exercises

25

2 Systems And Their Representations

27

    2.1 Introduction

27

    2.2 Notation

28

    2.3 The behavior

28

    2.4 Latent variables

31

    2.5 Equivalent representations

32

    2.6 Controllability

33

    2.7 Observability

35

    2.8 Stability

37

    2.9 Elimination of Latent variables

39

    2.10 The ring $\@mathbb R [\xi ,\xi ^-1 ]$

43

    2.11 An example

47

    2.12 A word about the notation

48

    2.13 Summary of chapter

49

    2.14 Notes and references

49

3 Adaptive systems : principles of identification

50

    3.1 Introduction

50

    3.2 Object of interest and model class

51

      3.2.1 Object of interest

51

      3.2.2 Model class

53

    3.3 Identification criterion and algorithms

58

      3.3.1 Least squares identification

58

      3.3.2 Recursive Least Squares (RLS)

59

      3.3.3 Projection algorithm

63

        3.3.3.1 Basic projection algorithm

63

        3.3.3.2 Normalized Least Mean Square (NLMS)

64

        3.3.3.3 Projection with dead zone

65

        3.3.3.4 Least Mean Square Algorithm (LMS)

66

    3.4 Data model assumptions

67

      3.4.1 Stable data filter

67

      3.4.2 Data in model class

67

      3.4.3 Information content of data

69

      3.4.4 Data do not fit model class

70

    3.5 Analysis of identification algorithms

71

      3.5.1 Properties of recursive least squares

71

        3.5.1.1 Consistency for RLS

74

        3.5.1.2 Consistency with model errors for RLS

76

      3.5.2 Properties of the NLMS algorithm

78

        3.5.2.1 With NLMS the equation error converges

78

        3.5.2.2 Consistency for NLMS

80

        3.5.2.3 Consistency with model errors for NLMS

82

      3.5.3 Projection algorithm with dead zone

84

      3.5.4 Tracking properties

86

        3.5.4.1 NLMS algorithm can track

87

        3.5.4.2 RLS algorithm cannot track

88

      3.5.5 Incorporating prior knowledge in algorithms

91

    3.6 Persistency of excitation

91

    3.7 Summary of chapter

96

    3.8 Notes and references

96

    3.9 Exercises

97

4 Adaptive Pole Assignment

103

    4.1 Introduction

103

    4.2 Preliminaries

105

    4.3 The system and its representations

107

    4.4 Equilibrium analysis

109

      4.4.1 The error model

110

      4.4.2 How much can be learned,\\ and how much must be learned?

110

    4.5 An algorithm for adaptive pole assignment

114

      4.5.1 The adaptive system

114

    4.6 Analysis of the algorithm

117

      4.6.1 Nonminimal representation

118

      4.6.2 Minimal representation

120

    4.7 Filtered signals

124

      4.7.1 Filter representation of i/o systems

124

      4.7.2 Application to adaptive pole assignment

128

    4.8 Modification of the projection algorithm

133

    4.9 Summary of chapter

135

    4.10 Notes and references

135

    4.11 Exercises

136

5 Direct Adaptive Model Reference Control

139

    5.1 Introduction

139

    5.2 Basic problem definition

140

    5.3 Model reference control: nonadaptive solution

142

    5.4 Error model construction

147

    5.5 Equilibrium analysis

152

    5.6 Adaptive algorithm

155

      5.6.1 Adaptive model reference control algorithm

155

    5.7 Analysis of the adaptive system

156

      5.7.1 Stability of the adaptive system

157

      5.7.2 Parameter convergence?

162

    5.8 Adaptive model reference control with\\disturbance rejection

164

      5.8.1 The Internal Model Principle

164

      5.8.2 Model reference control with disturbance rejection

167

      5.8.3 Adaptive model reference control with\\known disturbance rejection

168

      5.8.4 Adaptive model reference and disturbance rejection control

169

    5.9 Summary of chapter

169

    5.10 Notes and references

170

    5.11 Exercises

171

6 Universal Controllers

172

    6.1 Introduction

172

    6.2 Existence of solutions

174

    6.3 The first order case

174

      6.3.1 Sign $b$ known

175

      6.3.2 The Nussbaum controller: sign $b$ unknown

178

      6.3.3 The Willems \& Byrnes controller: sign $b$ unknown

183

    6.4 Higher order systems

185

      6.4.1 High gain feedback

186

      6.4.2 Willems-Byrnes controller: sign of $q_n-1 $ known

189

      6.4.3 Willems-Byrnes controller: sign $q_n-1 $ unknown

190

    6.5 M\aa rtensson's algorithm

193

      6.5.1 The adaptive control problem

194

      6.5.2 The main result

195

      6.5.3 Dense curves

197

      6.5.4 A dense curve based on an enumeration of $\@mathbb Q ^N$

198

    6.6 Summary of chapter

198

    6.7 Notes and references

199

    6.8 Exercises

199

7 The pole/zero cancellation problem

204

    7.1 Introduction

204

    7.2 The pole/zero cancellation problem in adaptive control

205

    7.3 Combining direct and indirect adaptive control

207

      7.3.1 The first order case

207

        7.3.1.1 Problem statement and reparametrization

207

        7.3.1.2 Equilibrium analysis

208

        7.3.1.3 Adaptive algorithm

209

      7.3.2 The higher order case

210

        7.3.2.1 Problem statement and reparametrization

210

        7.3.2.2 Equilibrium analysis

212

        7.3.2.3 Adaptive algorithm

218

    7.4 Adaptive Excitation

219

      7.4.1 The first order case

220

        7.4.1.1 Problem statement

220

        7.4.1.2 Adaptive algorithm

220

      7.4.2 The higher order case

223

        7.4.2.1 Problem statement

223

        7.4.2.2 Adaptive algorithm

223

    7.5 A more fundamental viewpoint

225

      7.5.1 The connection with tunability

226

      7.5.2 Alternative parametrizations

227

    7.6 Conclusions

228

    7.7 Summary of chapter

228

    7.8 Notes and references

228

    7.9 Exercises

230

8 Averaging Analysis For Adaptive Systems

232

    8.1 Introduction

232

    8.2 Averaging

233

      8.2.1 An illustration

235

      8.2.2 Some notation and preliminaries

239

      8.2.3 Finite horizon averaging result

241

      8.2.4 Infinite horizon result

244

    8.3 Transforming an adaptive system into standard form

251

    8.4 Averaging approximation

258

    8.5 Application: the MIT rule for adaptive control

260

      8.5.1 System description

260

      8.5.2 Frozen system for MIT rule

261

      8.5.3 Averaging for MIT rule

261

      8.5.4 Interpretation of averaged system

263

        8.5.4.1 Case I: Reference model equals plant $Z_m \equiv Z_p$

263

        8.5.4.2 Case II: Constant reference signal

263

        8.5.4.3 Case III: General problem

264

        8.5.4.4 How slow is slow adaptation?

266

    8.6 Application: echo cancellation in telephony

267

      8.6.1 Echo cancellation

267

      8.6.2 System description and assumptions

268

      8.6.3 Analysis

271

        8.6.3.1 The frozen system

271

        8.6.3.2 The averaged update equation

272

        8.6.3.3 Analysis of the averaged equation

274

        8.6.3.4 DEC system behavior

277

        8.6.3.5 General observations

279

    8.7 Summary of chapter

280

    8.8 Notes and references

281

    8.9 Exercises

282

9 Dynamics of adaptive systems: A case study

286

    9.1 Introduction

286

    9.2 The example

287

    9.3 Global analysis and bifurcations

289

    9.4 Adaptive system behavior: ideal case

291

    9.5 Adaptive system behavior: undermodelled case

294

      9.5.1 Parameter range

296

      9.5.2 Equilibria

296

      9.5.3 Beyond period 1 bifurcations

298

      9.5.4 Summary $d \not =0$

299

      9.5.5 Flip bifurcation revisited

300

    9.6 Discussion

301

    9.7 Summary of chapter

302

    9.8 Notes and References

303

    9.9 Exercises

303

Epilogue

304

A Background material

305

    A.1 A contraction result

305

    A.2 The Comparison Principle

306

      A.2.1 Bellman-Gronwall Lemma

307

      A.2.2 Perturbed linear stable systems

308

    A.3 Miscellaneous stability results

311

      A.3.1 Stability Definitions

311

      A.3.2 Some Lyapunov stability results

312

    A.4 Detectability

313

    A.5 An inequality for linear systems

317

    A.6 Finite horizon averaging result

319

    A.7 Maple code for solving Lyapunov equations

324

    A.8 Maple code for fixed points and two periodic solutions

325

Bibliography

327

Index

336

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