Global edition d igital Control System analysis and design FoUrth edition Charles L. Phillips • H. Troy Nagle • Aranya Chakrabortty Digital Control System Analysis & Design F E ourth dition Global Edition Charles L. Phillips Auburn University H. Troy Nagle North Carolina State University Aranya Chakrabortty North Carolina State University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director, Engineering and Computer Science: Marcia J. 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Troy Nagle, and Aranya Chakrabortty to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition entitled Digital Control System Analysis and Design, 4th edition, ISBN 978-0-13-293831-0, by Charles L. Phillips, H. Troy Nagle, and Aranya Chakrabortty, published by Pearson Education © 2015. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 10: 1-292-06122-7 ISBN 13: 978-1-292-06122-1 (Print) ISBN 13: 978-1-292-06188-7 (PDF) British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 Typeset in 9/10 Times LT Std by Jouve India Printed and bound by Courier Westford in The United States of America Dedication To Laverne, Susie, Chuck, and Carole Susan, Julia, and Amy My parents ContEntS Preface 9 Chapter 1 IntroduCtIon 11 Overview 11 Digital Control System 12 The Control Problem 15 Satellite Model 16 Servomotor System Model 18 Antenna Pointing System 20 Robotic Control System 21 Temperature Control System 22 Single-Machine Infinite Bus Power System 24 Summary 27 References 27  •  Problems 27 Chapter 2 dIsCrete-tIme systems and the z-transform 35 Introduction 35 Discrete-Time Systems 35 Transform Methods 37 Properties of the z-Transform 40 Addition and Subtraction 40 Multiplication by a Constant 40 Real Translation 41 Complex Translation 43 Initial Value 44 Final Value 44 Finding z-Transforms 45 Solution of Difference Equations 48 The Inverse z-Transform 51 Power Series Method 51 Partial-Fraction Expansion Method 52 Inversion-Formula Method 56 Discrete Convolution 57 Simulation Diagrams and Flow Graphs 59 State Variables 63 Other State-Variable Formulations 71 Transfer Functions 80 4 Contents 5 Solutions of the State Equations 84 Recursive Solution 84 z-Transform Method 86 Numerical Method via Digital Computer 87 Properties of the State Transition Matrix 88 Linear Time-Varying Systems 89 Summary 90 References and Further Readings 90  •  Problems 90 Chapter 3 samplIng and reConstruCtIon 100 Introduction 100 Sampled-Data Control Systems 100 The Ideal Sampler 103 Evaluation of E*(S) 105 Results from the Fourier Transform 108 Properties of E*(S) 110 Data Reconstruction 113 Zero-Order Hold 114 First-Order Hold 118 Fractional-Order Holds 119 Summary 121 References and Further Readings 121  •  Problems 122 Chapter 4 open-loop dIsCrete-tIme systems 126 Introduction 126 The Relationship Between E(Z) and E*(S) 126 The Pulse Transfer Function 127 Open-Loop Systems Containing Digital Filters 133 The Modified z-Transform 136 Systems with Time Delays 139 Nonsynchronous Sampling 142 State-Variable Models 145 Review of Continuous-Time State Variables 146 Discrete-Time State Equations 150 Practical Calculations 154 Summary 156 References and Further Readings 156  •  Problems 156 Chapter 5 Closed-loop systems 167 Introduction 167 Preliminary Concepts 167 6 Contents Derivation Procedure 171 State-Variable Models 178 Summary 187 References and Further Readings 187  •  Problems 188 Chapter 6 system tIme-response CharaCterIstICs 198 Introduction 198 System Time Response 198 System Characteristic Equation 207 Mapping the s-Plane into the z-Plane 208 Steady-State Accuracy 215 Simulation 218 Control Software 223 Summary 223 References and Further Readings 224  •  Problems 224 Chapter 7 stabIlIty analysIs teChnIques 230 Introduction 230 Stability 230 Bilinear Transformation 234 The Routh-Hurwitz Criterion 236 Jury’s Stability Test 239 Root Locus 244 The Nyquist Criterion 248 The Bode Diagram 257 Interpretation of the Frequency Response 259 Closed-Loop Frequency Response 261 Summary 270 References and Further Readings 270  •  Problems 270 Chapter 8 dIgItal Controller desIgn 279 Introduction 279 Control System Specifications 279 Steady-State Accuracy 280 Transient Response 280 Relative Stability 282 Sensitivity 283 Disturbance Rejection 284 Control Effort 285 Compensation 285 Phase-Lag Compensation 287 Contents 7 Phase-Lead Compensation 294 Phase-Lead Design Procedure 295 Lag-Lead Compensation 303 Integration and Differentiation Filters 307 PID Controllers 309 PID Controller Design 313 Design by Root Locus 321 Summary 334 References and Further Readings 334  •  Problems 335 Chapter 9 pole-assIgnment desIgn and state estImatIon 343 Introduction 343 Pole Assignment 343 State Estimation 352 Observer Model 352 Errors in Estimation 354 Error Dynamics 354 Controller Transfer Function 359 Closed-Loop Characteristic Equation 362 Closed-Loop State Equations 363 Reduced-Order Observers 364 Current Observers 369 Controllability and Observability 374 Systems with Inputs 378 Summary 383 References and Further Readings 384  •  Problems 384 Chapter 10 system IdentIfICatIon of dIsCrete-tIme systems 390 Introduction 390 Identification of Static Systems 391 Identification of Dynamic Systems 394 Black-Box Identification 394 Least-Squares System Identification 401 Estimating Transfer Functions with Partly Known Poles and Zeros 407 Recursive Least-Squares System Identification 409 Practical Factors for Identification 412 Choice of Input 412 Choice of Sampling Frequency 413 Choice of Signal Scaling 413 Summary 414 References and Further Readings 414  •  Problems 414 8 Contents Chapter 11 lInear quadratIC optImal Control 418 Introduction 418 The Quadratic Cost Function 419 The Principle of Optimality 421 Linear Quadratic Optimal Control 424 The Minimum Principle 433 Steady-State Optimal Control 434 Optimal State Estimation—Kalman Filters 440 Least-Squares Minimization 446 Summary 446 References and Further Readings 447  •  Problems 448 Chapter 12 Case studIes 454 Introduction 454 Servomotor System 455 System Model 456 Design 459 Environmental Chamber Control System 461 Temperature Control System 463 Aircraft Landing System 467 Plant Model 468 Design 468 Neonatal Fractional Inspired Oxygen 474 Plant Transfer Function 474 Taube’s PID Controller 476 MATLAB pidtool PIDF Controllers 477 Topology Identification in Electric Power System Models 484 References 488 Appendix I Design Equations 490 Appendix II Mason’s Gain Formula 491 Appendix III Evaluation of E*(s) 496 Appendix IV Review of Matrices 501 Appendix V The Laplace Transform 508 Appendix VI z-Transform Tables 522 Index 525 PrEfACE This book is intended to be used primarily as a text for a first course in discrete-time control systems at either the senior undergraduate or first-year graduate level. Furthermore, the text is suitable for self-study by the practicing control engineer. This book is based on material taught at both Auburn University and North Carolina State University, and in intensive short courses taught in both the United States and Europe. The practicing engineers who attended these short courses have influenced both the content and the direction of this book, resulting in emphasis placed on the practical aspects of designing and implementing digital control systems. Chapter 1 presents a brief introduction and an outline of the text. Chapters 2–11 cover the analysis and design of discrete-time linear control systems. Some previous knowledge of continuous-time control systems is helpful in understanding this material. The mathemat- ics involved in the analysis and design of discrete-time control systems is the z-transform and vector-matrix difference equations; these topics are presented in Chapter 2. Chapter 3 is devoted to the very important topic of sampling signals and the mathematical model of the sampler and data hold. This model is basic to the remainder of the text. The implications and the limitations of this model are stressed. The next four chapters, 4–7, are devoted to the application of the mathematics of Chapter 2 to the analysis of discrete-time systems, emphasis on digital control systems. Classical design techniques are covered in Chapter 8, with the frequency-response Bode technique emphasized. Modern design techniques are presented in Chapters 9–11. Chapter 12 summarizes some case studies in discrete-time control system design. Throughout these chapters, practical computer- aided analysis and design using MATLAB are stressed. In this fourth edition, several changes have been made. We • Added additional MATLAB examples throughout the chapters. • Added a new chapter on system identification (Chapter 11). • Added new problems in many of the chapters. • Renumbered the end-of-chapter problems to reflect their corresponding textbook sections. • Added the MATLAB pidtool design technique in Chapter 8. • Added two new case studies in Chapter 12. • Removed four chapters (formerly Chapters 11–14) and two appendices (formerly Appendices V and VI) on digital filter implementation to reduce the overall page count, thus placing more emphasis on control design. Each end-of-chapter problem has been written to illustrate basic material in the chapter. Generally, short MATLAB programs are given with many of the textbook examples to illustrate the computer calculations of the results of the example. These programs are easily modified for the homework problems. To further assist instructors using this book, a set of PowerPoint slides and a manual con- taining problem solutions has been developed. The authors feel that the problems at the end of the chapters are an indispensable part of the text, and should be fully utilized by all who study this book. Requests for both the problem solutions and PowerPoint slides can be sent directly to the publisher. 9