Adaptive Systems

Loosely speaking, adaptive systems are designed to deal with, to adapt to, chang­ ing environmental conditions whilst maintaining performance objectives. Over the years, the theory of adaptive systems evolved from relatively simple and intuitive concepts to a complex multifaceted theory dealing with stochastic, nonlinear and infinite dimensional systems. This book provides a first introduction to the theory of adaptive systems. The book grew out of a graduate course that the authors taught several times in Australia, Belgium, and The Netherlands for students with an engineering and/or mathemat­ ics background. When we taught the course for the first time, we felt that there was a need for a textbook that would introduce the reader to the main aspects of adaptation with emphasis on clarity of presentation and precision rather than on comprehensiveness. The present book tries to serve this need. We expect that the reader will have taken a basic course in linear algebra and mul­ tivariable calculus. Apart from the basic concepts borrowed from these areas of mathematics, the book is intended to be self contained.

Contenu

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 Mårtensson'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 Miscel…