Loosely speaking, adaptive systems are designed to deal with, to adapt, changing 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 mathematics 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 multivariable calculus. Apart from the basic concepts borrowed from these areas of mathematics, the book is intended to be self contained. The main body is formed by the treatment of three major streams in the theory of deterministic adaptive control systems: adaptive pole placement (Chapter 4), model reference adaptive control (Chapter 5), and the theory of universal controllers (Chapter 6). Clearly, this does not cover the whole area of adaptive control; most notably optimal control and stochastic systems are missing. However, we wanted to present an introductory text, suitable to be covered in a one trimester course at the graduate level. Alternatively part of the book is well suited for a more modest yet advanced undergraduate level course. Apart from Chapter 6, dealing with universal controllers, the text treats the subject in a discrete time setting. Even in the approach of the three major subjects, we have not pursued generality. We strongly feel that the intended audience is better served with a fairly complete and precise treatment of more or less simple algorithms rather than being subjected to an as-general-as-possible treatment. Even if we had tried, we would inevitably have produced a less transparent exposition. Anyway, encyclopedic treatments of adaptive control are already available in the literature. As a general point of view we have chosen to consider situations that are as simple as possible, yet sufficient to illustrate the main issues of adaptation. Upon completing the book, it is our hope that the reader will find it relatively easy to access the more specialized literature. Although the selection of material reflects tradition and the historical development of adaptive systems theory, our presentation of the ideas does not. The different adaptive algorithms are approached from a unified perspective. The no-adaptation or equilibrium analysis plays a key role in gaining an understanding of adaptive algorithms. In essence the same technique, equilibrium analysis followed by transient analysis, is used for all algorithms presented in the text. The equilibrium or no-adaptation analysis is truly a leitmotif throughout the text. Whereas the main part of the book is concerned with the three subjects that we just mentioned, we have included several chapters on both prerequisites and extensions. The prerequisites consist of a chapter on representations of linear deterministic systems (Chapter 2) and a chapter on identification (Chapter 3). Representations play an important role in systems theory in general and in adaptive control in particular. The classical transfer function approach is in our view not quite the appropriate tool since transfer functions cannot take into account autonomous parts, or, almost equivalently, the initial conditions of a system. Therefore we have chosen to use elements of the behavioral approach. In particular we benefit from the theory of equivalent representations and elimination of auxiliary variables within this framework, both of which are of great importance in the exposition of adaptive system theory. We have hesitated as to whether this material should be the contents of a chapter or should be part of the appendix. Since we make essential use of the behavioral approach at several points, and since it is still not quite standard theory, we finally felt that we should provide it right after the introductory first chapter. The chapter on identification introduces the reader to the basic gradient or steepest-descent-like algorithms that minimize some prediction error criteria, such as recursive least squares, normalized least mean square, and projection. The treatment is kept to a minimum, sufficient to introduce the adaptive algorithms. The properties of the identification algorithms are studied with as few restrictions imposed on the input signals as feasible, or, better stated, we derive as many properties as possible without assuming anything about the nature of the inputs at all. The rationale is, of course, that in adaptive control systems, the input is generated in a highly complicated, practically unpredictable way so that it would be difficult to guarantee particular conditions on the input signal. Chapter 7 revisits the basic algorithms and addresses the validity of one of the underlying assumptions in the adaptive system analysis, namely that the control problem is well posed during adaptation. It is shown how the well-posedness can be ensured at the expense of more complicated adaptive rules. Chapter 8 presents averaging techniques for the analysis of adaptive systems. This way we aim to address concerns about another important simplifying assumption in the analysis of adaptive systems, that the plant belongs to the model class. Averaging techniques are a powerful tool in the analysis of the performance of adaptive systems. The basic concepts introduced should enable the reader to gain some insight into the complexity of adaptive systems and get some intuition about the behavior of adaptive systems in general. No prior knowledge of averaging techniques is required. The minimal tools are developed and immediately illustrated in the context of some particular examples of adaptive systems. It transpires from the examples how the averaging analysis actually leads to design guidelines for adaptive systems. Chapter 9 introduces the idea of a global dynamical analysis of adaptive systems operating in a plant- model mismatch situation. This topic is by necessity treated rather superficially. Finally, we have referred some of the technical and preliminary material to the appendix. Every chapter, except Chapter 2 and the appendix, ends with a list of exercises. Some of them are fairly straightforward; others require more work and serve to extend the material covered. We have also included some simulation exercises to illustrate the theory and to allow one to venture beyond the theory. The book is clearly intended as a text book for teaching. It can be used at different levels. Suggestions of sensible selections, for a short course, are: Chapters 1, 2, 3, 4, 5, and 6 for a fairly complete coverage of the basic theory. Chapters 1, 2, 3, 4, 5, 6, and 7 for a slightly extended course. Adding Chapters 8 and 9 would probably not be feasible for a one trimester course. However, skipping one or two of the Chapters 4, 5, 6, and 7 would solve that problem, although skipping both 4 and 5 is strongly dissuaded. We have covered most of the book, say more than eighty percent, in a sixteen hour lecture course. The book can be covered in its entirety in about twenty class room hours. In our course we actively involve the students by requiring them to work out substantial exercises in take home exams. Most of these exams have found their way into the end-of-chapter exercises. As a virtual appendix to the book we provide a homepage, containing additional exercises, simulation problems with Matlab and Mathematica or Maple codes and a regularly updated list of errata. Also, readers are invited to leave comments, remarks and the like. Solutions to all exercises will be made available through a protected part of the homepage; access will be granted on request to teachers only. Finally, it is our sincere wish that the reader, both the practitioner and the theoretician, will find the book useful as an introduction to the intriguing field of adaptive systems. When the audience that we have in mind, graduate students as well as undergraduates, enjoy reading the book, we have achieved our goal. We hope that the learning experience will be a pleasant one. It would be a delight to know that we enticed a few to venture beyond the known.

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