SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A Volume 4
Series Editors: Ardeshir Guran & Daniel J Inman
Mathematical Problems of Control Theory
An Introduction
Gennady A. Leonov
Department of Mathematics and Mechanics
St. Petersburg State University
World Scientific
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MATHEMATICAL PROBLEMS OF CONTROL THEORY: An Introduction
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.
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Preface
It is human to feel the attraction of concreteness. The necessity of con-creteness is most pronounced in the mathematical activity. The famous mathematician of the 19th century Karl Weierstrass has pointed to the fact that progress in science is impossible without studying concrete problems. The abandonment of concrete problems in favor of more and more abstract research led to a crisis, which is often discussed now by the mathematicians. The mathematical control theory is no exception. The Wiener idea on a partitioning of a control system into the parts, consisting of sensors, actuators, and control algorithms being elaborated first by mathematicians and being realized then, using the comprehensive facilities of electronics, by engineers, was of crucial importance in the making of cybernetics. However at the same time it has been considered as a contributary factor for arising a tendency for an abandonment of concreteness.
The notorious and very nontrivial thesis of LA. Vyshnegradsky: "there is no governor without friction", and the conclusion of V, Volterra that an average number of preys in the ecological predator-pray system turns out to be constant with respect to the initial data are a result of the solutions of concrete problems only.
The study of concrete control systems allows us to evaluate both the remarkable simplicity of the construction of a damper winding for a synchronous machine, which acts as a stabilizing feedback, and the idea of television broadcasting of two-dimensional pictures via one-dimensional information channels by constructing a system of generators synchronization.
In this book we try to show how the study of concrete control systems has become a motivation for development of the mathematical tools needed
for solving such problems. In many cases by this apparatus far-reaching generalizations were made and its further development exerts a material effect on many fields of mathematics.
A plan to write such a book has arisen from author's perusal of many remarkable books and papers of the general control theory and of special control systems in various domains of technology: energetics, shipbuilding, communications, aerotechnics, and computer technology. A preparation of the courses of lectures: "The control theory. Analysis" and "Introduction to the applied theory of dynamical systems" for students of the Faculty of Mathematics and Mechanics, making a speciality of "Applied mathematics" was a further stimulus.
An additional motivation has also become the following remark of K.J. Astrom [5]: "The introductory courses in control are often very similar to courses given twenty or thirty years ago even if the field itself has developed substantially. The only
difference may be a sprinkling of Matlab exercises. We need to take a careful look at our knowledge base and explore how it can be weeded and streamlined. We should probably also pay more attention to the academic positioning of our field".
The readers of this book are assumed to be familiar with algebra, calculus, and differential equations according with a program of two first years of mathematical departments. We have tried to make the presentation as near to being elementary as possible.
The author hopes that the book will be useful for specialists in the control theory, differential equations, dynamical systems, theoretical and applied mechanics. But first of all it is intended for the students and postgraduates, who begin to specialize in the above-mentioned scientific fields.
The author thanks N.V.Kuznetsov, S.N.Pakshin, I.I.Ryzhakova, M.M.Shumafov, and Yu.K.Zotov for the patient cooperation during the final stages of preparation of this monograph.
The author also thanks prof. A.L. Fradkov and A.S. Matveev who read the manuscript in detail and made many useful remarks.
Finally, the author wishes to acknowledge his great indebtedness to Elmira A. Gurmuzova for her translating the manuscript from Russian into English.
St. Petersburg, April 2001.
Contents
Preface v
Chapter 1 The Watt governor and the mathematical theory
of stability of motion 1
1.1 The Watt fly ball governor and its modifications 1
1.2 The Hermite—Mikhailov criterion 7
1.3 Theorem on stability by the linear approximation 13
1.4 The Watt governor transient processes 25
Chapter 2 Linear electric circuits. Transfer functions and
frequency responses of linear blocks 33
2.1 Description of linear blocks 33
2.2 Transfer functions and frequency responses of linear blocks . . 40
Chapter 3 Controllability, observability, stabilization 53
3.1 Controllability 53
3.2 Observability 62
3.3 A special form of the systems with controllable
pair (A, b) 66
3.4 Stabilization. The Nyquist criterion 67
3.5 The time-varying stabilization. The Brockett problem 72
Chapter 4 Two-dimensional control systems. Phase portraits 93
4.1 An autopilot and spacecraft orientation system 93
4.2 A synchronous electric machine control and phase locked loops 106
4.3 The mathematical theory of populations 126
Chapter 5 Discrete systems 133
5.1 Motivation 133
5.2 Linear discrete systems 140
5.3 The discrete phase locked loops for array processors 148
Chapter 6 The Aizerman conjecture. The Popov method 155
Bibliography 167
Index 171