Adaptation in Dynamical Systems

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Notational conventions......Page 17
Part I Introduction and preliminaries......Page 21
1 Introduction......Page 23
1.1.1 Example: quantitative modeling in biophysics and neuroscience......Page 25
1.1.2 Example: adaptive classification in neural networks......Page 27
1.1.3 Preliminary statement of the problem......Page 29
1.2 Regulation problems......Page 30
1.3 Summary......Page 34
2.1 Attracting sets and attractors......Page 35
2.2 Barbalat\'s lemma......Page 40
2.3 Basic notions of stability......Page 43
2.4 The method of Lyapunov functions......Page 48
2.5 Linear skew-symmetric systems with time-varying coefficients......Page 52
3.1 Logical principles of adaptation......Page 64
3.1.1 Searching-based adaptation and extremal systems......Page 65
3.1.2 Principles of adapt ation beyond searching-based optimization strategies......Page 66
3.2 Formal definitions of adaptation and mathematical statements of the problem of adaptation......Page 70
3.2.1 The formulation ofthe problem of adapt ation by R. Bellman and R. Kalaba......Page 71
3.2.2 Adaptivity according to L. Zadeh......Page 72
3.2.3 The problem of adapt ation according to V. A. Yakubovich......Page 73
3.3 Adaptive control for nonlinear deterministic dynamical systems......Page 75
3.3.1 Velocity gradient......Page 76
3.3.2 Adaptive integrator back-stepping......Page 81
3.3.3 Minimax and domination-based algorithms of adaptive regulation......Page 91
3.4 Applicability issues of conventional methods of adaptive control and regulation......Page 94
3.5 Summary......Page 99
Part II Theory......Page 101
4 Input–output analysis of uncertain dynamical systems......Page 103
4.1 Operator description of dynamical systems......Page 104
4.2 Input–output and input–state characterizations of stable systems......Page 109
4.3 Input–output and input–state analysis of uncertain unstable systems......Page 114
4.3.1 Realizability ofinterconnections of systems with locally bounded operators......Page 117
4.3.2 Functional synthesis of adaptive systems: the separation principle......Page 125
4.4 Asymptotic properties of systems withlocally bounded input–output and input–state mappings......Page 129
4.5 Asymptotic properties of a class of unstable systems......Page 132
4.5.1 Small-gain theorems for the analysis of non-uniform convergence......Page 137
4.5.2 Estimates of Milnor attracting sets in the system’s state space......Page 142
4.5.3 Systems with separable dynamics in space-time......Page 145
A4.1 Proof of Theorem 4.1......Page 150
A4.2 Proof of Theorem 4.4......Page 152
A4.3 Proof of Theorem 4.5......Page 154
A4.5 Proof of Theorem 4.7......Page 156
A4.7 Proof of Lemma 4.3......Page 161
A4.8 Proof of Corollary 4.2......Page 162
A4.9 Proof of Corollary 4.3......Page 163
A4.10 Proof of Corollary 4.4......Page 165
A4.11 Proof of Corollary 4.5......Page 168
5 Algorithms of adaptive regulation and adaptation in dynamical systems in the presence of nonlinear parametrization and/or possibly unstable target dynamics......Page 171
5.1 Problems of adaptive control of nonlinear systems in the presence of nonlinear parametrization......Page 172
5.2.1 Virtual adaptation algorithms. Sufficient conditions of realizability......Page 183
5.2.2 Embedding problem......Page 190
5.2.3 Direct adaptive control for systems with lower-triangular structure......Page 194
5.3.1 Systems with parametric uncertainties and nonlinear parametrization......Page 208
5.3.2 Systems with signal uncertainties and linear parametrization......Page 209
5.4 Adaptive control of interconnected dynamical systems......Page 212
5.4.1 Systems with unmodeled dynamics......Page 213
5.4.2 Decentralized adaptive control......Page 215
5.5 Non-dominating adaptive control for dynamical systems with nonlinear parametrization of a general kind......Page 222
5.6.1 Systems with monotone nonlinear parametrization......Page 227
5.6.2 Observer-based state and parameter reconstruction for systems with Lipschitz parametrization......Page 233
A5.1 Proof of Theorem 5.1......Page 238
A5.3 Proof of Theorem 5.2......Page 243
A5.4 Proof of Theorem 5.3......Page 244
A5.5 Proof of Lemma 5.1......Page 245
A5.6 Proof of Theorem 5.4......Page 251
A5.8 Proof of Theorem 5.5......Page 256
A5.10 Proof of Theorem 5.7......Page 263
A5.11 Proof of Theorem 5.8......Page 268
A5.12 Proof of Corollary 5.3......Page 269
A5.13 Proof of Theorem 5.9......Page 271
A5.14 Proof of Theorem 5.10......Page 273
Part III Applications......Page 283
6 Adaptive behavior in recurrent neural networks with fixed weights......Page 285
6.1 Signals to be classified......Page 286
6.2 The class of recurrent neural networks......Page 288
6.3 Assumptions and statement of the problem......Page 289
6.4 The existence result......Page 292
6.5 Summary......Page 312
7 Adaptive template matching in systems for processing of visual information......Page 314
7.1 Preliminaries and problem formulation......Page 320
7.2 A simple adaptive system for invariant template matching......Page 327
7.2.1 Invariant template matching by Milnor attractors......Page 330
7.2.2 Conditions for synchronization of coincidence detectors......Page 342
Extension to frequency-encoding schemes......Page 353
Multiple representations of uncertainties......Page 354
Multiple time scales for different modalities in vision......Page 355
Rotation-invariant matching in images with single objects......Page 356
Rotation-invariant matching in images with multiple objects......Page 360
The effect of differences in time scales......Page 364
7.3.2 Tracking disturbances in scanning microscopes......Page 365
7.4 Summary......Page 368
8 State and parameter estimation of neural oscillators......Page 370
8.1 Observer-based approaches to the problem of stateand parameter estimation......Page 373
8.1.1 Local observability ofneural oscillators......Page 375
8.1.2 Bastin–Gevers canonical form......Page 378
8.1.3 Marino–Tomei canonical form......Page 380
8.2 The feasibility of conventional adaptive-observer canonical forms......Page 382
8.2.1 Parameter-independent time-invariant transformations......Page 384
8.2.2 Parameter-dependent and time-varying transformations......Page 386
Bastin–Gevers adaptive observer......Page 387
The Marino–Tomei observer......Page 390
8.3 Universal adaptive observers for conductance-based models......Page 394
8.3.1 Observer definition and assumptions......Page 396
8.3.2 Asymptotic properties ofthe observer......Page 398
8.4 Examples......Page 399
8.5 Summary......Page 404
Appendix. The Meyer–Kalman–Yakubovich lemma......Page 407
References......Page 415
Index......Page 429