CONTROL SYSTEMS TECHNOLOGY Curtis D. Johnson University of Houston Heidar A. Malki University of Houston
Library of Congress Cataloging-in-Publication Data Johnson, Curtis D. Control systems technology / Curtis Johnson & Heidar Malki. p.cm. Includes index. ISBN 0-13-081530-6 1. Process control. I. Malld, Heidar. II. Title. TSI56.8.J6272002 629.8-dc21 20010"2348. Editor in Chief: Stephen Helba Assistant Vice President and Publisher: Charles Stewart Assistant Editor: Delia K. Uherec Production Editor: Tricia L. Rawnsley Design Coordinator: Robin G. Chukes Production Coordination: Carlisle Publishers SeIVices Cover art!photo: Visual Edge Imaging Studios Cover Designer: Linda Fares Production Manager: Matthew Ottenweller This book was set by Carlisle Communications, Ltd. It was printed and bound by R. R. Donnelley & Sons Company. The cover was printed by Phoenix Color Corp. Pearson Education Ltd., London Pearson Education Australia Pty. Limited, Sydney Pearson Education Singapore Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educaci6n de Mexico, S.A de C.V. Pearson Education-Japan, Tokyo Pearson Education Malaysia Pte. Ltd. Pearson Education, Upper Saddle River, New Jersey Copyright 2002 by Pearson Education, Inc., Upper Saddle River, New Jersey 07458. All rights reserved. Printed in the United States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department. 10 9 8 7 6 5 4 3 2 1 ISBN 0-13-081530-6
This text was written to fill a very important educational niche in the broad spectrum of control systems knowledge. That niche lies between the hands-on electromechanical knowledge and skills needed by technicians and the highly abstract and theoretical knowledge required by scholars who research and develop new control strategies. This book focuses on the knowledge required by control systems practitioners to enable them to both understand and evaluate an existing control system and devise and design new control system applications. The text presents classical and digital control systems with an emphasis on careful explanations of the concepts. Many examples illustrate key topics and the operations required to solve problems. The text is an outgrowth of many years of teaching control systems to students in an engineering technology program. It is written for a two-semester course, nominally separated into analog and digital control. The difficulty with this approach is that much of digital control is a spinoff of analog concepts. Therefore, the analog material by itself is more extensive than the digital. In practice, we have found that some of the material on analog control must be delayed to the second course. Although patterned after the course sequence expected for a particular educational program, this text can be adapted to other approaches. For example, Chapter 2 (Measurement) can be omitted by those who prefer to cover sensors and measurement in other courses. Likewise, if Laplace transforms are covered in an independent course, that section in Chapter 3 can be omitted or assigned as review. It would be important to include, however, the last section of Chapter 3, Analog Simulation. The text emphasizes an understanding of control system concepts, but also requires the use of computers to implement practical solutions to problems. There are a number of control and mathematical software packages which are of great value in the study of control systems. Throughout the text, the use of these packages to facilitate solving problems is emphasized, and Mathcad or MATLAB is used to illustrate computer-based mathematical procedures. An attempt has been made to emphasize the use of computers as a tool to implement the mathematical and graphical operations required to solve a problem. A Web page (www.uh.edu/~tech13v/contsystech) will be set up for this text as a means for communication between users and authors, and also for sharing ideas and techniques related to teaching control systems. A solutions manual (ISBN: 0-13-090661-1) is available. It contains examples of physical and simulation experiments that can be conducted to enhance learning. Dr. Malki would like to thank his parents, his wife Layla, and his son Armeen for their support and patience during the long task of writing this book. Dr. Johnson would like to thank his wife Helene and his mother-in-law Lois for their continuing kindness while he undertook this task.
1 INTRODUCTION TO CONTROL SYSTEMS 1 1.1 Purpose 2 1.2 Introduction 2 1.2.1 Control System Strategy 3 1.2.2 Examples of Control Systems 4 1.2.3 Analytical Issues 7 1.3 Analytical Descriptions 7 1.3.1 Block Diagram 8 1.3.2 Transfer Functions 10 1.3.3 Computer Applications Software 13 1.4 Analog and Digital Control 14 1.5 System Design Objectives 17 Summary 19 Problems 20 1.5.1 Dynamic Response 17 1.5.2 Instability 17 2 MEASUREMENT 23 2.1 Purpose 24 2.2 Measurement Principles 24 2.2.1 Sensor 25 2.2.2 Signal Conditioning 31 2.3 Sensors 38 Summary 47 Problems 47 2.3.1 Temperature 38 2.3.2 Displacement 41 2.3.3 Motion 45 vii
viii I CONTENTS 3 LAPLACE TRANSFORMS 51 3.1 Purpose 52 3.2 Introduction 52 3.3 Definition of the Laplace Transform 54 3.3.1 Computer Applications 58 3.4 Properties of Laplace Transforms 59 3.5 Inverse Laplace Transform 65 3.5.1 Partial-fraction Expansion 66 3.5.2 Convolution 75 3.6 Analog Simulation 77 Summary 87 Problems 89 3.6.1 The Operational Amplifier 77 3.6.2 Simulation of Physical Systems 80 3.6.3 Simulation of Control Systems 85 4 CONTROL SYSTEM MODELS B 4.1 Purpose 94 4.2 Transfer Functions 94 4.2.1 Block Transfer Functions 95 4.2.2 Transfer Function Properties 100 4.2.3 Graphing 106 4.3 Block Diagrams 113 4.3.1 Canonical Form 114 4.3.2 Block Diagram Reduction 115 4.3.3 Multiple Inputs 123 4.4 Mason's Gain Formula 126 4.5 Controller/Compensator Transfer Functiom 128 Summary 132 Problems 133 4.5.1 Proportional, Integral, and Derivative Contro~ 129 4.5.2 Lead and Lag Compensation 131 5 STATIC AND DYNAMIC RESPONSE 13! 5.1 Purpose 140 5.2 Static Response 141 5.2.1 Steady-State Error 141 5.2.2 Disturbance Error 148
CONTENTS I ix 5.3 Dynamic Response of First- and Second-Order Plants 150 5.3.1 First-Order Plant 150 5.3.2 Second-Order Plant 153 5.4 Characteristics of Dynamic Response 158 5.5 Steady-State Error Versus Stability 165 Summary 167 Problems 168 6 STABILITY 173 6.1 Purpose 174 6.2 Definitions of Stability 174 6.2.1 Formal Definition of Stability 177 6.3 Routh-Hurwitz Stability Criterion 178 Summary 184 Problems 185 6.3.1 Special Cases 182 7 FREQUENCY RESPONSE ANALYSIS 187 7.1 Purpose 188 7.2 Basic Principles 188 7.2.1 Frequency Response Analysis 189 7.2.2 Exact Analysis 190 7.2.3 Bode Plot 191 7.3 Control System Bode Plots 193 7.3.1 Manual Construction 194 7.3.2 Computer Construction 207 7.4 Bode Plot Applications 210 Summary 216 Problems 217 7.4.1 Gain and Phase Margins 212 7.4.2 Transportation Delay 214 8 ROOT LOCUS 221 8.1 Purpose 222 8.2 Introduction to Root Locus 222 8.2.1 Closed-Loop Poles 222 8.2.2 Root Locus Graph 224
x I CONTENTS 8.3 Root Locus Construction 228 8.3.1 Manual Construction 229 8.3.2 Computer Construction 240 8.4 Root Locus Applications 240 8.4.1 Gain and Phase Margin 240 8.4.2 Transient Response 242 Summary 249 Problems 249 9 STATE-SPACE ANALYSIS 253 9.1 Purpose 254 9.2 State-Space Definition 255 9.3 Solving State-Space Equations 259 9.3.1 Laplace Transform Solutions 259 9.3.2 Series Expansion Solution 264 9.3.3 Computer Simulation Solution 266 9.4 Simulation Diagrams and State-Space Equations 269 9.4.1 Simulation Diagram Definition 270 9.4.2 Generalized Rules of Simulation Diagram Construction 272 9.5 Transfer Function in State Space 275 9.6 Controllability and Observability 277 Summary 281 Problems 282 9.6.1 Controllability 277 9.6.2 Observability 279 10 INTRODUCTION TO DIGITAL CONTROL SYSTEMS 285 10.1 Purpose 286 10.2 Definition of a Digital Control System 286 10.2.1 Digital Control System Hardware 287 10.2.2 Digital Control System Software 291 10.2.3 Simulations Using Computers 294 10.3 The Difference Equation 294 Summary 302 Problems 303 10.3.1 Finding the Difference Equation 295 10.3.2 Solution of the Difference Equation 298
CONTENTS I xi 11 z-transform AND THE DIFFERENCE EQUATION 307 11.1 Purpose 308 11.2 Definition of the z-transform 308 11.3 Properties of z-transforms 316 11.3.1 Linearity 316 11.3.2 Advance Theorem (Shift Left) 316 11.3.3 Delay Theorem (Shift Right) 317 11.3.4 Final Value Theorem 318 11.3.5 Initial Value Theorem 319 11.4 Inverse z-transform 321 11.4.1 Partial-Fraction Expansion for Real Poles 321 11.4.2 Partial-Fraction Expansion for Complex Poles 325 11.4.3 Partial-Fraction Expansion for Repeated Poles 327 11.4.4 Direct Method: Long Division 329 11.4.5 Inverse z- Transforms by Software 330 11.5 Difference Equation Solution 330 Summary 334 Problems 335 12 DISCRETE CONTROL SYSTEMS 337 12.1 Purpose 338 12.2 Discrete Transfer Function 338 12.3 Open-Loop Transfer Functions 347 12.4 Closed-Loop Transfer Functions 349 12.5 Static and Dynamic Response 355 Summary 365 Problems 366 12.5.1 Static Response 355 12.5.2 Dynamic Response 359 13 STABILITY OF DISCRETE CONTROL SYSTEMS 371 13.1 Purpose 372 13.2 Conditions for Stability 372 13.3 Stability Tests 374 13.3.1 Routh-Hurwitz Test with the Bilinear Transformation 374 13.3.2 Jury's Stability Test 376
xii I CONTENTS 13.4 Discrete System Root Locus 381 13.4.1 Root Locus Construction Rules 381 Summary 387 Problems 388 14 DISCRETE STATE SPACE 391 14.1 Purpose 392 14.2 State-Space Equations in the Discrete Domain 392 14.2.1 Discrete State Equations 392 14.2.2 Solution by Recursion 396 14.2.3 Solution by z- Transforms 398 14.3 Discrete State-Space Transfer Function 400 14.3.1 Generation of State-Vector Equations 401 14.4 Observability and Controllability 403 14.5 Discrete Simulation Diagrams 406 Summary 410 Problems 410 APPENDIX A: COMPLEX NUMBERS 413 APPENDIX B: MATRICES 417 GLOSSARY 423 SOLUTIONS TO SELECTED ODD PROBLEMS 427 INDEX 459
CONTROL SYSTEMS TECHNOLOGY