Digital Control Engineering Analysis and Design Second Edition M. Sami Fadali Antonio Visioli AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK Copyright r 2013 Elsevier Inc. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data Fadali, M. Sami. Digital control engineering : analysis and design / M. Sami Fadali, Antonio Visioli. À Second edition. pages cm Includes bibliographical references and index. ISBN 978-0-12-394391-0 (hardback) 1. Digital control systems. I. Visioli, Antonio. II. Title. TJ223.M53F33 2013 629.809Àdc23 2012021488 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our website at http://store.elsevier.com Printed in the United States of America 121314 987654321 Contents Preface ......................................................................................................................xi CHAPTER 1 Introduction to Digital Control.............................................1 1.1 Why digital control?......................................................................2 1.2 The structure of a digital control system......................................2 1.3 Examples of digital control systems.............................................3 1.3.1 Closed-loop drug delivery system .......................................3 1.3.2 Computer control of an aircraft turbojet engine .................4 1.3.3 Control of a robotic manipulator.........................................4 Resources...............................................................................................6 Problems ................................................................................................7 CHAPTER 2 Discrete-Time Systems ..........................................................9 2.1 Analog systems with piecewise constant inputs...........................9 2.2 Difference equations ...................................................................11 2.3 The z-transform ...........................................................................12 2.3.1 z-Transforms of standard discrete-time signals.................13 2.3.2 Properties of the z-transform .............................................15 2.3.3 Inversion of the z-transform...............................................19 2.3.4 The final value theorem.....................................................28 2.4 Computer-aided design ...............................................................29 2.5 z-Transform solution of difference equations.............................31 2.6 The time response of a discrete-time system .............................32 2.6.1 Convolution summation.....................................................32 2.6.2 The convolution theorem ...................................................34 2.7 The modified z-transform............................................................37 2.8 Frequency response of discrete-time systems.............................39 2.8.1 Properties of the frequency response of discrete-time systems .........................................................42 2.8.2 MATLAB commands for the discrete-time frequency response.............................................................44 2.9 The sampling theorem.................................................................45 2.9.1 Selection of the sampling frequency .................................46 Resources.............................................................................................49 Problems ..............................................................................................49 Computer exercises .............................................................................52 iii iv Contents CHAPTER 3 Modeling of Digital Control Systems..............................55 3.1 ADC model .................................................................................55 3.2 DAC model .................................................................................56 3.3 The transfer function of the ZOH ..............................................57 3.4 Effect of the sampler on the transfer function of a cascade......58 3.5 DAC, analog subsystem, and ADC combination transfer function..........................................................................61 3.6 Systems with transport lag..........................................................69 3.7 The closed-loop transfer function...............................................71 3.8 Analog disturbances in a digital system.....................................74 3.9 Steady-state error and error constants ........................................75 3.9.1 Sampled step input ............................................................77 3.9.2 Sampled ramp input ..........................................................77 3.10 MATLAB commands .................................................................79 3.10.1 MATLAB ........................................................................79 3.10.2 Simulink ..........................................................................80 Resources.............................................................................................85 Problems..............................................................................................85 Computer exercises .............................................................................89 CHAPTER 4 Stability of Digital Control Systems ................................91 4.1 Definitions of stability ................................................................91 4.2 Stable z-domain pole locations...................................................93 4.3 Stability conditions .....................................................................94 4.3.1 Asymptotic stability ..........................................................94 4.3.2 BIBO stability ...................................................................95 4.3.3 Internal stability.................................................................98 4.4 Stability determination..............................................................101 4.4.1 MATLAB ........................................................................101 4.4.2 Routh-Hurwitz criterion ..................................................102 4.5 Jury test .....................................................................................104 4.6 Nyquist criterion .......................................................................109 4.6.1 Phase margin and gain margin........................................114 Resources...........................................................................................123 Problems............................................................................................123 Computer exercises ...........................................................................125 CHAPTER 5 Analog Control System Design........................................127 5.1 Root locus .................................................................................127 5.2 Root locus using MATLAB .....................................................132 5.3 Design specifications and the effect of gain variation.............132 Contents v 5.4 Root locus design ......................................................................135 5.4.1 Proportional control .........................................................137 5.4.2 PD control ........................................................................138 5.4.3 PI control..........................................................................147 5.4.4 PID control.......................................................................153