control systems technology

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 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 Pearson Pearson Pearson Pearson Pearson Pearson Pearson Pearson Education Ltd., London Education Australia Pty Limited, Sydney Education Singapore Pte Ltd Education North Asia Ltd., Hong Kong Education Canada, Ltd., Toronto Educaci6n de Mexico, S.A de C.V Education-Japan, Tokyo Education Malaysia Pte Ltd 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 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 (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 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 INTRODUCTION TO CONTROL SYSTEMS 1.1 Purpose 1.2 Introduction 1.2.1 1.3 Control System Strategy 1.2.2 Examples of Control Systems 1.2.3 Analytical Issues Analytical Descriptions 1.3.1 Block Diagram 1.3.2 Transfer Functions 1.3.3 Computer Applications Software 10 1.4 Analog and Digital Control 1.5 System Design Objectives Summary 1.5.1 Dynamic Response 1.5.2 Instability 13 14 17 17 17 19 Problems 20 MEASUREMENT 2.1 2.2 2.3 Purpose 23 24 Measurement Principles 2.2.1 Sensor 25 2.2.2 Signal Conditioning Sensors 2.3.1 Temperature Displacement 2.3.3 Motion 45 47 Problems 47 31 38 2.3.2 Summary 24 38 41 vii viii I CONTENTS LAPLACE TRANSFORMS Purpose 3.2 Introduction 3.3 Definition of the Laplace Transform 3.3.1 52 Computer Applications 3.5 Inverse Laplace Transform 3.5.1 Partial-fraction 3.5.2 Convolution 66 75 77 The Operational Amplifier 77 3.6.2 Simulation of Physical Systems 3.6.3 Simulation of Control Systems 87 Problems 89 59 65 Expansion Analog Simulation Summary 54 58 Properties of Laplace Transforms 3.6.1 80 85 CONTROL SYSTEM MODELS 4.1 Purpose 4.2 Transfer Functions 4.3 94 Block Transfer Functions 4.2.2 Transfer Function Properties 4.2.3 Graphing 100 113 Canonical Form 114 4.3.2 Block Diagram Reduction 4.3.3 Multiple Inputs 123 Mason's Gain Formula 4.5 95 106 Block Diagrams 4.4 B 94 4.2.1 4.3.1 52 3.4 3.6 51 3.1 Controller/Compensator 115 126 Transfer Functiom 128 4.5.1 Proportional, Integral, and Derivative Contro~ 129 4.5.2 Lead and Lag Compensation Summary 132 Problems 133 STATIC AND DYNAMIC RESPONSE 5.1 Purpose 5.2 Static Response 140 141 5.2.1 Steady-State Error 141 5.2.2 Disturbance Error 148 131 13! CONTENTS 5.3 Dynamic Response of First- and Second-Order Plants 5.3.1 First-Order Plant 150 5.3.2 Second-Order Plant 153 5.4 Characteristics of Dynamic Response 5.5 Steady-State Error Versus Stability Summary 167 Problems 168 174 Formal Definition of Stability Routh-Hurwitz 6.3.1 165 174 Definitions of Stability 6.2.1 6.3 177 Stability Criterion 184 Problems 185 FREQUENCY RESPONSE ANALYSIS 7.1 Purpose 7.2 Basic Principles 188 7.2.1 Frequency Response Analysis 7.2.2 Exact Analysis 7.2.3 Bode Plot 191 189 190 7.3.1 Manual Construction 7.3.2 Computer Construction 193 194 Bode Plot Applications Summary 187 188 Control System Bode Plots 7.4 178 Special Cases 182 Summary 7.3 150 173 Purpose 6.2 207 210 7.4.1 Gain and Phase Margins 212 7.4.2 Transportation Delay 214 216 Problems 217 ix 158 STABILITY 6.1 I ROOT LOCUS 221 8.1 Purpose 8.2 Introduction to Root Locus 222 8.2.1 Closed-Loop Poles 222 8.2.2 Root Locus Graph 224 222 x I CONTENTS Root Locus Construction 8.3 228 8.3.1 Manual Construction 229 8.3.2 Computer Construction Root Locus Applications 8.4 Summary 240 240 8.4.1 Gain and Phase Margin 240 8.4.2 Transient Response 242 249 Problems 249 STATE-SPACE ANALYSIS 253 9.1 Purpose 9.2 State-Space Definition 9.3 Solving State-Space Equations 9.4 254 255 259 9.3.1 Laplace Transform Solutions 9.3.2 Series Expansion Solution 264 259 9.3.3 Computer Simulation Solution 266 Simulation Diagrams and State-Space Equations 9.5 9.6 Summary 9.4.1 Simulation Diagram Definition 9.4.2 Generalized Rules of Simulation Diagram Construction 270 Transfer Function in State Space 275 Controllability and Observability 277 9.6.1 Controllability 9.6.2 Observability 269 272 277 279 281 Problems 282 10 INTRODUCTION TO DIGITAL CONTROL SYSTEMS 10.1 Purpose 10.2 Definition of a Digital Control System 10.3 286 10.2.1 Digital Control System Hardware 10.2.2 Digital Control System Software 10.2.3 Simulations Using Computers The Difference Equation 291 294 294 10.3.1 Finding the Difference Equation 10.3.2 Solution of the Difference Equation Summary 302 Problems 303 286 287 295 298 285 CONTENTS 11 z- TRANSFORM AND THE DIFFERENCE EQUATION 11.1 11.2 Purpose 308 Definition of the z- Transform 11.3 Properties of z- Transforms 11.4 I xi 307 308 316 11.3.1 Linearity 11.3.2 Advance Theorem (Shift Left) 316 316 11.3.3 Delay Theorem (Shift Right) 317 11.3.4 Final Value Theorem 11.3.5 Initial Value Theorem Inverse z- Transform 11.4.1 Partial-Fraction 318 319 321 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 z- Transforms by Software 330 Difference Equation Solution 330 11.4.5 Inverse 11.5 Summary 334 Problems 335 12 DISCRETE CONTROL SYSTEMS 12.1 12.2 12.3 337 Purpose 338 Discrete Transfer Function 338 Open-Loop Transfer Functions 12.4 Closed-Loop Transfer Functions 12.5 Static and Dynamic Response 12.5.1 Static Response 12.5.2 Dynamic Response 347 349 355 355 359 Summary 365 Problems 366 13 STABILITY OF DISCRETE CONTROL SYSTEMS 13.1 Purpose 13.2 Conditions for Stability 13.3 Stability Tests 371 372 372 374 13.3.1 Routh-Hurwitz Test with the Bilinear Transformation 13.3.2 Jury's Stability Test 376 374 xii I CONTENTS 13.4 381 Discrete System Root Locus 13.4.1 Root Locus Construction Rules 381 Summary 387 Problems 388 14 391 DISCRETE STATE SPACE 14.1 14.2 Purpose 392 State-Space Equations in the Discrete Domain 14.2.1 Discrete State Equations 14.2.2 Solution by Recursion 14.3.1 14.4 14.5 396 z- Transforms 398 Discrete State-Space Transfer Function 14.2.3 14.3 392 Solution by Generation of State-Vector Equations Observability and Controllability 400 401 403 406 Discrete Simulation Diagrams Summary 410 Problems 410 APPENDIX A: COMPLEX NUMBERS APPENDIX B: MATRICES GLOSSARY 413 417 423 SOLUTIONS TO SELECTED ODD PROBLEMS INDEX 459 427 392 CONTROL SYSTEMS TECHNOLOGY Acceleration error constant, 146 Accelerometer, 45 ADC,338 Aliasing, 345 Analog control system, 14 Analog: linearization, 35 signal conditioning, 31 simulation, 70 Analog-to-digital conversion (ADC), 32, 338 Analog-to-digital converter (ADC)/sampler,288 Angle of asymptotes, 237 Angle of departure, 237 Auxiliary equation, Routh-Hurwitz, 183 Controller action: derivative, 130 proportional, 129 integral, 129 Controller/compensator, 9, 292 transfer functions, 128 Convolution, 75, 262 discrete, 393 Critically damped response, 156 Crystal puller control, DAC,338 Damping coefficient, 244 Damping ratio, 154,243,360 Dc motor, 98 Delta or impulse function, 55 Derivative theorem, 60 Difference equation, 294, 301, 331, 395 Impulse response, 178, 248, 299, 330 Initial value theorem, 62 Instability, 17, 174,236,246 Integral action, 129 Integral theorem, 61 Interpolation, 35 Inverse Laplace transform, 65 Inverse z-transform, 321 direct method, 329 Bilinear transformation, 374 Block diagram, 8, 113 Difference equation solution, 298 Differential equation, 13,52,77-87,256 Jury's stability test, 376 canonical, 114 reduction, 115 unity feedback, 120 Block diagram reduction, Mason's Rule, 123 Bode plot, 191 applications, 210 computer construction, 207 manual construction, 194, 261, 265 Break-in points, 232 Break-out points, 232, 235 Digital control system, 14 definition, 286 hardware, 287-288 Digital linearization, 35 Digital PI controller, 296 Digital PID controller, 297 Digital signal conditioning, 32 Digital-to-analog converter (DAC)/hold, 289 Discrete simulation diagram, 407 Discrete state equations, 382 Disturbance, 123 error, 148 Dynamic response, 17,28, 140,359 first order system, 28 second order system, 30 Dynamic symbols, 270 Canonical form, 114,222,245,270 Characteristic equation, 103, 115, 175 Closed loop system, Closed loop transfer function, discrete, 349 Compensator (also see controller), lag, 131 lead, 131 lead-lag, 132 Complex numbers, see Appendix A, 413 Controllability, 277, 404 discrete systems, 403 Controlled variable, Engine speed control, Error detection, 291 Error; steady state, 141 Exponential function, 56 Feedback, Final value theorem, 62 First-order systems, discrete, 359 Frequency response analysis, 189 Frequency, undamped natural, 154 Gain margin, 212, 240 Geometric series, 311 Graphing, 106 Homogeneous solution, 260 Laplace transforms, 13, 259 definition, 54 table (functions), 57 theorems, 64 sampled, 339 properties, 59 Linear operation, 10 Linear time invariant systems, 256 Linearity theorem, 59 Linearization, 33 Mapping, 310 z-plane,372 Marginal stability, 110, 177 Mason's gain formula, 126 Mathcad, 14,58,72,74,207,242,298, 330 MATLAB, 14,58,72,74,207,233, 239,246,248,261,263, 277,278,279,330 Matrices, dimension, 417 invese,420 review, 417 transpose, 419 459 460 I INDEX Measurement, 3, 9, 24 Multiple-input -multiple-output (MIMO), 254, 258, 276 Multiple inputs, 123 Numerical integration, 266 Observability, 279, 404 discrete systems, 403 Open loop transfer function, discrete, 348 Operational amplifier, 77 differential, 78 integrator, 78-79 inverting, 77 summing, 78-79 Overdamped, 17, 154 Partial fraction expansion, 66, 321 PD controller, 130 Peak time, 160 Percent overshoot, 160, 362 Phase margin, 212, 240 Phase shift, 189 PI control, 130,232,237 PID controller, 130 Plant: first order, 150 second order, 153 Poles, 67 complex, 68 multiple or repeated, 72 real, 68 Position control, 244 Position error, 357 Positional error constant, 144 Positive feedback, liS Process control, 2, Proportional action, 129 Proportional control system, 10, 257 Proportional-derivative action, 130 Proportional-integral action, 130 Rate action, 130 Reaction vessel control, 258 Recursion, 396 Reference (or set-point), parabolic, 146 quadratic, 146 ramp, 145 Refining branches, 233 Regulation, 2, Reset action, 129 Residue theorem, 189 Response: critical, 156 dynamic, ISO first order, 151 overdamped, 154 step, 151 underdamped, 157 dynamic, 140 static, 141 transient, 140 Rise time, 159 Root locus, 222, 381 application, 240 computer control, 240 graph, 224 interpretation, 227 manual construction, 229 rules, 229 stability, 226 discrete systems, 381 Routh-Hurwitz test, 178, 236, 246 discrete, 374 auxiliary equation, 183 Sampler, 338 Sampling, practical, 345 Shannon, 345 Sampling frequency, 343 Sampling theorem, 345 Sampling time, 343 Second-order systems, discrete, 359 Self-regulation, Sensor,S, 7, 24, 38 dynamic response, 141 accelerational and rotation, 45 capacitive, 44 displacement, 41-42 LVDT, 44 motion, 45 potentiometer, 44 resolver, 44 RTD, 38-40 solid-state, 40 spring-mass, 46 strain gauge, 43 tachometers, 46 temperature, 28 thermistor, 40 thermocouple, 41 Series expansion, 264 Servomechanism, 3, Settling time, 161, 362 Shannon's sampling theorem, 345 Shifting theorem, 62 Signal conditioning, 31 Simulation, 80 analog, 81 control systems, 85 Simulation diagram, 406 Sine and cosine function, 56-58 Single-input-single-output system (SISO), 254 s-plane plots, 109 Stability, 174,350, 372 definition, 177 discrete criteria, 374 discrete systems, 350 discrete, 372 frequency domain, 199 Jury Test, 376 sampled systems, 372 time domain, 178 Starred transform, 339 State-space: definition, 255 equation, 255-262 equations construction, 272 equations simulation diagrams, 269-275 equations solutions, 267-268 equations, 392 equations, z-transforms, 398 State-space, difference equations, 395 State-space, discrete, 382 State variable, 269 State- vector equations, 40 I, see also state-space equations Static response, 25 Static response, discrete, 355 Steady-state error, 17 Steady-state error table, 147 Step function, 55 Step response, 244 Superposition, 123 System type, 144,357 System variables, 269 Table, Routh-Hurwitz, 179 Time response, 28 Transfer function, 10, 25, 94 discrete, 338, 400 chemical system, 99 closed loop, liS electrical system, 97 INDEX mechanical system, 98 open loop, 115 properties, 1()() controller/compensator, 128 Transform, starred, 339 Transient, 123 Transient response, 17, 140,242 Transition matrix, 260 Transport delay, 214 Transpose, 419 Type, system, 144 Undamped natural frequency, 360 Underdamped response, 17, 157 Unity feedback, 120 Velocity error constant, 145 Water heater control, Zero-order-hold, 339 Zeros, 61, 67, 80 ZOH, see zero-order-hold z-transform, 308,311, 398 cosine function, 319 exponential function, 314 sine function, 319 properties (theorems), 316 I 461 ... INDEX 459 427 392 CONTROL SYSTEMS TECHNOLOGY INTRODUCTION TO CONTROL SYSTEMS CHAPTER OUTLINE 1.1 Purpose 1.2 Introduction 1.2.1 Control System Strategy 1.2.2 Examples of Control Systems 1.2.3 Analytical... simple view, any control system that does not use a computer is referred to as an analog control system; those that use computers for control are called digital control systems Analog Control System... of analog control systems The knowledge gained in this study is easily transferred to a study of digital control systems Digital Control System When a computer is used as part of a control system,