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 Scientific Publishers PWN, 2002 Library of Congress Control Number: 2005925392 ISBN 10 ISBN 13 3-540-23951-0 Springer Berlin Heidelberg New York 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 – 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 control 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 processes Taking into account a variety of new applications, the book presents a compact and uniform description containing traditional analysis and optimization 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, researches 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 coworkers at the Institute of Information Science and Engineering of Wroclaw University of Technology, who assisted in the preparation of the manuscript Z Bubnicki Contents General Characteristic of Control Systems .1 1.1 Subject and Scope of Control Theory 1.2 Basic Terms .2 1.2.1 Control Plant .4 1.2.2 Controller 1.3 Classification of Control Systems 1.3.1 Classification with Respect to Connection Between Plant and Controller 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 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 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 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 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 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 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 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 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 X Contents 11.2 Adaptation via Identification for Static Plant 303 11.3 Adaptation via Identification for Dynamical Plant 309 11.4 Adaptation via Adjustment of Controller Parameters 311 11.5 Learning Control System Based on Knowledge of the Plant .313 11.5.1 Knowledge Validation and Updating 314 11.5.2 Learning Algorithm for Decision Making in Closed-loop System 317 11.6 Learning Control System Based on Knowledge of Decisions 319 11.6.1 Knowledge Validation and Updating 319 11.6.2 Learning Algorithm for Control in Closed-loop System 321 12 Intelligent and Complex Control Systems 327 12.1 Introduction to Artificial Intelligence 327 12.2 Logical Knowledge Representation 328 12.3 Analysis and Decision Making Problems 332 12.4 Logic-algebraic Method .334 12.5 Neural Networks 341 12.6 Applications of Neural Networks in Control Systems .346 12.6.1 Neural Network as a Controller 346 12.6.2 Neural Network in Adaptive System 348 12.7 Decomposition and Two-level Control 349 12.8 Control of Complex Plant with Cascade Structure 355 12.9 Control of Plant with Two-level Knowledge Representation 358 13 Control of Operation Systems 363 13.1 General Characteristic 363 13.2 Control of Task Distribution 365 13.3 Control of Resource Distribution .371 13.4 Control of Assignment and Scheduling .375 13.5 Control of Allocation in Systems with Transport 382 13.6 Control of an Assembly Process 386 13.7 Application of Relational Description and Uncertain Variables 391 13.8 Application of Neural Network 398 Conclusions 401 Appendix 405 References 411 Index 419 General Characteristic of Control Systems 1.1 Subject and Scope of Control Theory The modern control theory is a discipline dealing with formal foundations of the analysis and design of computer control and management systems Its basic scope contains problems and methods of control algorithms design, where the control algorithms are understood as formal prescriptions (formulas, procedures, programs) for the determination of control decisions, which may be executed by technical devices able to the information processing and decision making The problems and methods of the control theory are common for different executors of the control algorithms Nowadays, they are most often computer devices and systems The computer control and management systems or wider − decision support systems belong now to the most important, numerous and intensively developing computer information systems The control theory deals with the foundations, methods and decision making algorithms needed for developing computer programs in such systems The problems and methods of the control theory are common not only for different executors of the control algorithms but also − which is perhaps more important – for various applications In the first period, the control theory has been developing mainly for the automatic control of technical processes and devices This area of applications is of course still important and developing, and the development of the information technology has created new possibilities and – on the other hand – new problems The full automatization of the control contains also the automatization of manipulation operations, the control of executing mechanisms, intelligent tools and robots which may be objects of the external control and should contain inner controlling devices and systems Taking into account the needs connected with the control of various technical processes, with the management of projects and complex plants as well as with the control and management of computer systems has led to forming foundations of modern control science dealing in a uniform and General Characteristic of Control Systems systematic way with problems concerning the different applications mentioned here The scope of this area significantly exceeds the framework of so called traditional (or classical) control theory The needs and applications mentioned above determine also new directions and perspectives of the future development of the modern control theory Summarizing the above remarks one can say that the control theory (or wider − control science) is a basic discipline for the automatic control and robotics and one of basic disciplines for the information technology and management It provides the methods necessary to a rational design and effective use of computer tools in the decision support systems and in particular, in the largest class of such systems, namely in control and management systems Additional remarks concerning the subject and the scope of the control theory will be presented in Sect 1.2 after the description of basic terms, and in Sect 1.5 characterizing interconnections between the control theory and other related areas 1.2 Basic Terms To characterize more precisely the term control let us consider the following examples: Control (steering) of a vehicle movement so as to keep a required trajectory and velocity of the motion Control of an electrical furnace (the temperature control), consisting in changing the voltage put at the heater so as to stabilize the temperature at the required level in spite of the external temperature variations Stabilization of the temperature in a human body as a result of the action of inner steering organs Control of the medicine dosage in a given therapy in order to reach and keep required biomedical indexes Control of a production process (e.g a process of material processing in a chemical reactor), consisting in proper changes of a raw material parameters with the purpose of achieving required product parameters Control of a complex manufacturing process (e.g an assembly process) in such a way that the suitable operations are executed in a proper time Control (steering, management) of a complex production plant or an enterprise, consisting in making and executing proper decisions concerning the production size, sales, resource distributions, investments etc., with 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Characteristic of Control Systems 1.1 Subject and Scope of Control Theory The modern control theory is a discipline dealing with formal foundations of the analysis and design of computer control and... Characteristic of Control Systems .1 1.1 Subject and Scope of Control Theory 1.2 Basic Terms .2 1.2.1 Control Plant .4 1.2.2 Controller 1.3 Classification of Control. .. simplest structure of the control system in which the controller C controls the plant CP control C CP Fig 1.1 Basic scheme of control system Remark 1.1 Regardless different names (control, steering,