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 |