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Zdzislaw Bubnicki Modern Control Theory

Zdzislaw Bubnicki Modern Control Theory With 104 figures

Professor Zdzislaw Bubnicki, PhD Wroclaw University of Technology Institute of Information Science and Engineering Wyb. Wyspianskiego 27 50-370 Wroclaw Poland zdzislaw.bubnicki@pwr.wroc.pl Originally published in Polish by Polish Scientiﬁ c Publishers PWN, 2002 Library of Congress Control Number: 2005925392 ISBN 10 3-540-23951-0 Springer Berlin Heidelberg New York ISBN 13 978-3-540-23951-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by the author. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: Medionet AG, Berlin Printed on acid-free paper 89/3141/Yu – 5 4 3 2 1 0

Preface The main aim of this book is to present a unified, systematic description of basic and advanced problems, methods and algorithms of the modern con- trol theory considered as a foundation for the design of computer control and management systems. The scope of the book differs considerably from the topics of classical traditional control theory mainly oriented to the needs of automatic control of technical devices and technological proc- esses. Taking into account a variety of new applications, the book presents a compact and uniform description containing traditional analysis and op- timization problems for control systems as well as control problems with non-probabilistic models of uncertainty, problems of learning, intelligent, knowledge-based and operation systems – important for applications in the control of manufacturing processes, in the project management and in the control of computer systems. Into the uniform framework of the book, original ideas and results based on the author’s works concerning uncertain and intelligent knowledge-based control systems, applications of uncertain variables and the control of complexes of operations have been included. The material presented in the book is self-contained. Using the text does not require any earlier knowledge on the control science. The presentation requires only a basic knowledge of linear algebra, differential equations and probability theory. I hope that the book can be useful for students, re- searches and all readers working in the field of control and information science and engineering. I wish to express my gratitude to Dr. D. Orski and Dr. L. Siwek, my co- workers at the Institute of Information Science and Engineering of Wro- claw University of Technology, who assisted in the preparation of the manuscript. Z. Bubnicki

Contents 1 General Characteristic of Control Systems...............................................1 1.1 Subject and Scope of Control Theory................................................1 1.2 Basic Terms .......................................................................................2 1.2.1 Control Plant...............................................................................4 1.2.2 Controller....................................................................................6 1.3 Classification of Control Systems......................................................7 1.3.1 Classification with Respect to Connection Between Plant and Controller.............................................................................................7 1.3.2 Classification with Respect to Control Goal...............................9 1.3.3 Other Cases...............................................................................11 1.4 Stages of Control System Design ....................................................13 1.5 Relations Between Control Science and Related Areas in Science and Technology .....................................................................................14 1.6 Character, Scope and Composition of the Book..............................15 2 Formal Models of Control Systems ........................................................17 2.1 Description of a Signal ....................................................................17 2.2 Static Plant .......................................................................................18 2.3 Continuous Dynamical Plant ...........................................................19 2.3.1 State Vector Description...........................................................20 2.3.2 “Input-output” Description by Means of Differential Equation24 2.3.3 Operational Form of “Input-output” Description......................25 2.4 Discrete Dynamical Plant ................................................................29 2.5 Control Algorithm ...........................................................................31 2.6 Introduction to Control System Analysis.........................................33 2.6.1 Continuous System ...................................................................35 2.6.2 Discrete System ........................................................................37 3 Control for the Given State (the Given Output)......................................41 3.1 Control of a Static Plant...................................................................41 3.2 Control of a Dynamical Plant. Controllability.................................44 3.3 Control of a Measurable Plant in the Closed-loop System .............47 3.4 Observability....................................................................................50

VIII Contents 3.5 Control with an Observer in the Closed-loop System .....................55 3.6 Structural Approach.........................................................................59 3.7 Additional Remarks .........................................................................62 4 Optimal Control with Complete Information on the Plant .....................65 4.1 Control of a Static Plant...................................................................65 4.2 Problems of Optimal Control for Dynamical Plants........................69 4.2.1 Discrete Plant............................................................................69 4.2.2 Continuous Plant.......................................................................72 4.3 Principle of Optimality and Dynamic Programming .......................74 4.4 Bellman Equation ............................................................................79 4.5 Maximum Principle .........................................................................85 4.6 Linear-quadratic Problem ................................................................93 5 Parametric Optimization .........................................................................97 5.1 General Idea of Parametric Optimization ........................................97 5.2 Continuous Linear Control System..................................................99 5.3 Discrete Linear Control System.....................................................105 5.4 System with the Measurement of Disturbances.............................107 5.5 Typical Forms of Control Algorithms in Closed-loop Systems ....110 5.5.1 Linear Controller.....................................................................111 5.5.2 Two-position Controller .........................................................112 5.5.3 Neuron-like Controller............................................................112 5.5.4 Fuzzy Controller .....................................................................113 6 Application of Relational Description of Uncertainty .........................117 6.1 Uncertainty and Relational Knowledge Representation ................117 6.2 Analysis Problem...........................................................................122 6.3 Decision Making Problem .............................................................127 6.4 Dynamical Relational Plant ...........................................................130 6.5 Determinization .............................................................................136 7 Application of Probabilistic Descriptions of Uncertainty.....................143 7.1 Basic Problems for Static Plant and Parametric Uncertainty........143 7.2 Basic Problems for Static Plant and Non-parametric Uncertainty152 7.3 Control of Static Plant Using Results of Observations..................157 7.3.1 Indirect Approach ...................................................................158 7.3.2 Direct Approach......................................................................164 7.4 Application of Games Theory........................................................165 7.5 Basic Problem for Dynamical Plant...............................................170 7.6 Stationary Stochastic Process ........................................................174

IX 7.7 Analysis and Parametric Optimization of Linear Closed-loop Control System with Stationary Stochastic Disturbances....................178 7.8 Non-parametric Optimization of Linear Closed-loop Control System with Stationary Stochastic Disturbances..............................................183 7.9 Relational Plant with Random Parameter......................................188 8 Uncertain Variables and Their Applications.........................................193 8.1 Uncertain Variables .......................................................................193 8.2 Application of Uncertain Variables to Analysis and Decision Making (Control) for Static Plant ........................................................201 8.2.1 Parametric Uncertainty ...........................................................201 8.2.2 Non-parametric Uncertainty ...................................................205 8.3 Relational Plant with Uncertain Parameter....................................211 8.4 Control for Dynamical Plants. Uncertain Controller .....................216 9 Fuzzy Variables, Analogies and Soft Variables ...................................221 9.1 Fuzzy Sets and Fuzzy Numbers.....................................................221 9.2 Application of Fuzzy Description to Decision Making (Control) for Static Plant ...........................................................................................228 9.2.1 Plant without Disturbances .....................................................228 9.2.2 Plant with External Disturbances............................................233 9.3 Comparison of Uncertain Variables with Random and Fuzzy Variables ..............................................................................................238 9.4 Comparisons and Analogies for Non-parametric Problems ..........242 9.5 Introduction to Soft Variables........................................................246 9.6 Descriptive and Prescriptive Approaches. Quality of Decisions ...249 9.7 Control for Dynamical Plants. Fuzzy Controller ...........................255 10 Control in Closed-loop System. Stability ...........................................259 10.1 General Problem Description.......................................................259 10.2 Stability Conditions for Linear Stationary System ......................264 10.2.1 Continuous System ...............................................................264 10.2.2 Discrete System ....................................................................266 10.3 Stability of Non-linear and Non-stationary Discrete Systems .....270 10.4 Stability of Non-linear and Non-stationary Continuous Systems 277 10.5 Special Case. Describing Function Method.................................278 10.6 Stability of Uncertain Systems. Robustness ................................282 10.7 An Approach Based on Random and Uncertain Variables..........291 10.8 Convergence of Static Optimization Process...............................295 11 Adaptive and Learning Control Systems ............................................299 11.1 General Concepts of Adaptation..................................................299

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