Adaptive Systems

1 Adaptive Systems.- 1.1 Introduction.- 1.2 Adaptive systems: examples.- 1.2.1 Adaptive control.- 1.2.2 Adaptive signal processing.- 1.2.3 Adaptive systems versus classical techniques.- 1.3 General structure of adaptive control systems.- 1.3.1 Introduction.- 1.3.2 The general structure.- 1.3.3 The error signal.- 1.3.4 The tuner.- 1.3.5 Certainty equivalence.- 1.3.6 Design and analysis.- 1.4 Illustrating the concepts.- 1.4.1 The MIT rule for adaptive control: feedforward case.- 1.4.2 The MIT rule for adaptive control: feedback problem.- 1.4.3 An adaptive pole placement scheme.- 1.4.4 A universal controller.- 1.4.5 Echo cancelling.- 1.5 Summary of chapter.- 1.6 Notes and references.- 1.7 Exercises.- 2 Systems And Their Representations.- 2.1 Introduction.- 2.2 Notation.- 2.3 The behavior.- 2.4 Latent variables.- 2.5 Equivalent representations.- 2.6 Controllability.- 2.7 Observability.- 2.8 Stability.- 2.9 Elimination of Latent variables.- 2.10 The ring ?[?,??1].- 2.11 An example.- 2.12 A word about the notation.- 2.13 Summary of chapter.- 2.14 Notes and references.- 3 Adaptive systems : principles of identification.- 3.1 Introduction.- 3.2 Object of interest and model class.- 3.2.1 Object of interest.- 3.2.2 Model class.- 3.3 Identification criterion and algorithms.- 3.3.1 Least squares identification.- 3.3.2 Recursive Least Squares (RLS).- 3.3.3 Projection algorithm.- 3.3.3.1 Basic projection algorithm.- 3.3.3.2 Normalized Least Mean Square (NLMS).- 3.3.3.3 Projection with dead zone.- 3.3.3.4 Least Mean Square Algorithm (LMS).- 3.4 Data model assumptions.- 3.4.1 Stable data filter.- 3.4.2 Data in model class.- 3.4.3 Information content of data.- 3.4.4 Data do not fit model class.- 3.5 Analysis of identification algorithms.- 3.5.1 Properties of recursive least squares.- 3.5.1.1 Consistency for RLS.- 3.5.1.2 Consistency with model errors for RLS.- 3.5.2 Properties of the NLMS algorithm.- 3.5.2.1 With NLMS the equation error converges.- 3.5.2.2 Consistency for NLMS.- 3.5.2.3 Consistency with model errors for NLMS.- 3.5.3 Projection algorithm with dead zone.- 3.5.4 Tracking properties.- 3.5.4.1 NLMS algorithm can track.- 3.5.4.2 RLS algorithm cannot track.- 3.5.5 Incorporating prior knowledge in algorithms.- 3.6 Persistency of excitation.- 3.7 Summary of chapter.- 3.8 Notes and references.- 3.9 Exercises.- 4 Adaptive Pole Assignment.- 4.1 Introduction.- 4.2 Preliminaries.- 4.3 The system and its representations.- 4.4 Equilibrium analysis.- 4.4.1 The error model.- 4.4.2 How much can be learned, and how much must be learned?.- 4.5 An algorithm for adaptive pole assignment.- 4.5.1 The adaptive system.- 4.6 Analysis of the algorithm.- 4.6.1 Nonminimal representation.- 4.6.2 Minimal representation.- 4.7 Filtered signals.- 4.7.1 Filter representation of i/o systems.- 4.7.2 Application to adaptive pole assignment.- 4.8 Modification of the projection algorithm.- 4.9 Summary of chapter.- 4.10 Notes and references.- 4.11 Exercises.- 5 Direct Adaptive Model Reference Control.- 5.1 Introduction.- 5.2 Basic problem definition.- 5.3 Model reference control: nonadaptive solution.- 5.4 Error model construction.- 5.5 Equilibrium analysis.- 5.6 Adaptive algorithm.- 5.6.1 Adaptive model reference control algorithm.- 5.7 Analysis of the adaptive system.- 5.7.1 Stability of the adaptive system.- 5.7.2 Parameter convergence?.- 5.8 Adaptive model reference control with disturbance rejection.- 5.8.1 The Internal Model Principle.- 5.8.2 Model reference control with disturbance rejection.- 5.8.3 Adaptive model reference control with known disturbance rejection.- 5.8.4 Adaptive model reference and disturbance rejection control.- 5.9 Summary of chapter.- 5.10 Notes and references.- 5.11 Exercises.- 6 Universal Controllers.- 6.1 Introduction.- 6.2 Existence of solutions.- 6.3 The first order case.- 6.3.1 Sign b known.- 6.3.2 The Nussbaum controller: sign b unknown.- 6.3.3 The Willems&Byrnes controller: sign b unknown.- 6.4 Higher order systems.- 6.4.1 High gain feedback.- 6.4.2 Willems-Byrnes controller: sign of qn?1 known.- 6.4.3 Willems-Byrnes controller: sign qn?1 unknown.- 6.5 Martensson's algorithm.- 6.5.1 The adaptive control problem.- 6.5.2 The main result.- 6.5.3 Dense curves.- 6.5.4 A dense curve based on an enumeration of ?N.- 6.6 Summary of chapter.- 6.7 Notes and references.- 6.8 Exercises.- 7 The pole/zero cancellation problem.- 7.1 Introduction.- 7.2 The pole/zero cancellation problem in adaptive control.- 7.3 Combining direct and indirect adaptive control.- 7.3.1 The first order case.- 7.3.1.1 Problem statement and reparametrization.- 7.3.1.2 Equilibrium analysis.- 7.3.1.3 Adaptive algorithm.- 7.3.2 The higher order case.- 7.3.2.1 Problem statement and reparametrization.- 7.3.2.2 Equilibrium analysis.- 7.3.2.3 Adaptive algorithm.- 7.4 Adaptive Excitation.- 7.4.1 The first order case.- 7.4.1.1 Problem statement.- 7.4.1.2 Adaptive algorithm.- 7.4.2 The higher order case.- 7.4.2.1 Problem statement.- 7.4.2.2 Adaptive algorithm.- 7.5 A more fundamental viewpoint.- 7.5.1 The connection with tunability.- 7.5.2 Alternative parametrizations.- 7.6 Conclusions.- 7.7 Summary of chapter.- 7.8 Notes and references.- 7.9 Exercises.- 8 Averaging Analysis For Adaptive Systems.- 8.1 Introduction.- 8.2 Averaging.- 8.2.1 An illustration.- 8.2.2 Some notation and preliminaries.- 8.2.3 Finite horizon averaging result.- 8.2.4 Infinite horizon result.- 8.3 Transforming an adaptive system into standard form.- 8.4 Averaging approximation.- 8.5 Application: the MIT rule for adaptive control.- 8.5.1 System description.- 8.5.2 Frozen system for MIT rule.- 8.5.3 Averaging for MIT rule.- 8.5.4 Interpretation of averaged system.- 8.5.4.1 Case I: Reference model equals plant Zm ? Zp.- 8.5.4.2 Case II: Constant reference signal.- 8.5.4.3 Case III: General problem.- 8.5.4.4 How slow is slow adaptation?.- 8.6 Application: echo cancellation in telephony.- 8.6.1 Echo cancellation.- 8.6.2 System description and assumptions.- 8.6.3 Analysis.- 8.6.3.1 The frozen system.- 8.6.3.2 The averaged update equation.- 8.6.3.3 Analysis of the averaged equation.- 8.6.3.4 DEC system behavior.- 8.6.3.5 General observations.- 8.7 Summary of chapter.- 8.8 Notes and references.- 8.9 Exercises.- 9 Dynamics of adaptive systems: A case study.- 9.1 Introduction.- 9.2 The example.- 9.3 Global analysis and bifurcations.- 9.4 Adaptive system behavior: ideal case.- 9.5 Adaptive system behavior: undermodelled case.- 9.5.1 Parameter range.- 9.5.2 Equilibria.- 9.5.3 Beyond period 1 bifurcations.- 9.5.4 Summary d ? 0.- 9.5.5 Flip bifurcation revisited.- 9.6 Discussion.- 9.7 Summary of chapter.- 9.8 Notes and References.- 9.9 Exercises.- Epilogue.- A Background material.- A.1 A contraction result.- A.2 The Comparison Principle.- A.2.1 Bellman-Gronwall Lemma.- A.2.2 Perturbed linear stable systems.- A.3 Miscellaneous stability results.- A.3.1 Stability Definitions.- A.3.2 Some Lyapunov stability results.- A.4 Detectability.- A.5 An inequality for linear systems.- A.6 Finite horizon averaging result.- A.7 Maple code for solving Lyapunov equations.- A.8 Maple code for fixed points and two periodic solutions.