Control Engineering. Editorial Advisory Board Okko Bosgra Delft University The Netherlands. William Powers Ford Motor Company (retired) USA

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2 Control Engineering Series Editor William S. Levine Department of Electrical and Computer Engineering University of Maryland College Park, MD Editorial Advisory Board Okko Bosgra Delft University The Netherlands Graham Goodwin University of Newcastle Australia Petar Kokotovic University of California Santa Barbara Manfred Morari ETH Zurich Switzerland William Powers Ford Motor Company (retired) Mark Spong University of Illinois Urbana-Champaign lori Hashimoto Kyoto University Kyoto Japan

3 Guillermo J. Silva Aniruddha Datta S.R Bhattacharyya PID Controllers for Time-Delay Systems Birkhauser Boston Basel Berlin

4 Guillermo J. Silva Aniruddha Datta IBM Department of Electrical Engineering Burnet Road Texas A&M University Austin, TX College Station, TX S.P. Bhattachaiyya Department of Electrical Engineering Texas A&M University College Station, TX AMS Subject Classifications: 30-02, 37F10, 65-02, 93D99 Library of Congress Cataloging-in-Publication Data Silva, G. J., PID controllers for time-delay systems / GJ. Silva, A. Datta, S.P. Bhattacharyya. p. cm. - (Control engineering) ISBN (alk. paper) 1. PID controllers-design and construction. 2. Time delay systems. I. Datta, Aniruddha, II. Bhattacharyya, S. P (Shankar P), III. Title. IV. Control engineering (Birkhauser) TJ223.P55S '3-dc ISBN Printed on acid-free paper Birkhauser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, Inc., c/o Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY, ), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to property rights. Printed in the United States of America. (SB) SPIN www. birkhauser. com

5 THIS BOOK IS DEDICATED TO My wife Sezi for her loving support and endless patience, and my parents Guillermo and Elvia. G. J. Silva My wife Anindita and my daughters Apama and Anisha. A. Datta The memory of my friend and mentor, the late Yakov Z. Tsypkin, Russian control theorist and academician whose many contributions include the first results, in 194-6, analyzing the stability of time-delay systems. S. P. Bhattacharyya

6 Contents Preface xi 1 Introduction Introduction to Control The Magic of Integral Control PID Controllers Some Current Techniques for PID Controller Design The Ziegler-Nichols Step Response Method The Ziegler-Nichols Frequency Response Method PID Settings using the Internal Model Controller Design Technique Dominant Pole Design: The Cohen-Coon Method New Tuning Approaches Integrator Windup Setpoint Limitation Back-Calculation and Tracking Conditional Integration Contribution of this Book Notes and References 18 2 The Hermite-Biehler Theorem and its Generalization Introduction The Hermite-Biehler Theorem for Hurwitz Polynomials Generalizations of the Hermite-Biehler Theorem 27

7 ii Contents No Imaginary Axis Roots Roots Allowed on the Imaginary Axis Except at the Origin No Restriction on Root Locations Notes and References 37 PI Stabilization of Delay-Free Linear Time-Invariant Systems Introduction A Characterization of All Stabilizing Feedback Gains Computation of All Stabilizing PI Controllers Notes and References 56 PID Stabilization of Delay-Free Linear Time-Invariant Systems Introduction A Characterization of All Stabilizing PID Controllers PID Stabilization of Discrete-Time Plants Notes and References 75 Preliminary Results for Analyzing Systems with Time Delay Introduction Characteristic Equations for Delay Systems Limitations of the Pade Approximation Using a First-Order Pade Approximation Using Higher-Order Pade Approximations The Hermite-Biehler Theorem for Quasi-Polynomials Applications to Control Theory Stability of Time-Delay Systems with a Single Delay Notes and References 106 Stabilization of Time-Delay Systems using a Constant Gain Feedback Controller Introduction First-Order Systems with Time Delay Open-Loop Stable Plant Open-Loop Unstable Plant Second-Order Systems with Time Delay Open-Loop Stable Plant Open-Loop Unstable Plant Notes and References 134 PI Stabilization of First-Order Systems with Time Delay Introduction 135

8 Contents ix 7.2 The PI Stabilization Problem Open-Loop Stable Plant Open-Loop Unstable Plant Notes and References PID Stabilization of First-Order Systems with Time Delay Introduction The PID Stabilization Problem Open-Loop Stable Plant Open-Loop Unstable Plant Notes and References Control System Design Using the PID Controller Introduction Robust Controller Design: Delay-Free Case Robust Stabilization Using a Constant Gain Robust Stabilization Using a PI Controller Robust Stabilization Using a PID Controller Robust Controller Design: Time-Delay Case Robust Stabilization Using a Constant Gain Robust Stabilization Using a PI Controller Robust Stabilization Using a PID Controller Resilient Controller Design Determining fc, T, and L from Experimental Data Algorithm for Computing the Largest Ball Inscribed Inside the PID Stabilizing Region Time Domain Performance Specifications Notes and References Analysis of Some PID Tuning Techniques Introduction The Ziegler-Nichols Step Response Method The CHR Method The Cohen-Coon Method The IMC Design Technique Summary Notes and References PID Stabilization of Arbitrary Linear Time-Invariant Systems with Time Delay Introduction A Study of the Generalized Nyquist Criterion Problem Formulation and Solution Approach Stabilization Using a Constant Gain Controller Stabilization Using a PI Controller 253

9 X Contents 11.6 Stabilization Using a PID Controller Notes and References Algorithms for Real and Complex PID Stabilization Introduction Algorithm for Linear Time-Invariant Continuous-Time Systems Discrete-Time Systems Algorithm for Continuous-Time First-Order Systems with Time Delay Open-Loop Stable Plant Open-Loop Unstable Plant Algorithms for PID Controller Design Complex PID Stabilization Algorithm Synthesis of Hoc PID Controllers PID Controller Design for Robust Performance PID Controller Design with Guaranteed Gain and Phase Margins Notes and References 295 A Proof of Lemmas 8.3, 8.4, and A.l Preliminary Results 297 A.2 Proof of Lemma A.3 Proof of Lemma A.4 Proof of Lemma B Proof of Lemmas 8.7 and B.l Proof of Lemma B.2 Proof of Lemma C Detailed Analysis of Example References 323 Index 329

10 Preface This monograph presents our recent results on the proportional-integralderivative (PID) controller and its design, analysis, and synthesis. The focus is on linear time-invariant plants that may contain a time delay in the feedback loop. This setting captures many real-world practical and industrial situations. The results given here include and complement those published in Structure and Synthesis of PID Controllers by Datta, Ho, and Bhattacharyya [10]. In [10] we mainly dealt with the delay-free case. The main contribution described here is the efficient computation of the entire set of PID controllers achieving stability and various performance specifications. The performance specifications that can be handled within our machinery are classical ones such as gain and phase margin as well as modern ones such as Hoo norms of closed-loop transfer functions. Finding the entire set is the key enabling step to realistic design with several design criteria. The computation is efficient because it reduces most often to linear programming with a sweeping parameter, which is typically the proportional gain. This is achieved by developing some preliminary results on root counting, which generalize the classical Hermite-Biehler Theorem, and also by exploiting some fundamental results of Pontryagin on quasi-polynomials to extract useful information for controller synthesis. The efficiency is important for developing software design packages, which we are sure will be forthcoming in the near future, as well as the development of further capabilities such as adaptive PID design and online implementation. It is also important for creating a realistic interactive design environment where multiple performance specifications that are appropriately prioritized can be overlaid and intersected to telescope down to a small and satisfactory

11 xii Preface controller set. Within this set further design choices must be made that reflect concerns such as cost, size, packaging, and other intangibles beyond the scope of the theory given here. The PID controller is very important in control engineering apphcations and is widely used in many industries. Thus any improvement in design methodology has the potential to have a significant engineering and economic impact. An excellent account of many practical aspects of PID control is given in PID Controllers: Theory, Design and Tuning by Astrom and Hagglund [2], to which we refer the interested reader; we have chosen to not repeat these considerations here. At the other end of the spectrum there is a vast mathematical literature on the analysis of stability of timedelay systems which we have also not included. We refer the reader to the excellent and comprehensive recent work Stability of Time-Delay Systems by Gu, Kharitonov, and Chen [15] for these results. In other respects our work is self-contained in the sense that we present proofs and justfications of all results and algorithms developed by us. We believe that these results are timely and in phase with the resurgence of interest in the PID controller and the general rekindling of interest in fixed and low-order controller design. As we know there are hardly any results in modern and postmodern control theory in this regard while such controllers are the ones of choice in applications. Classical control theory approaches, on the other hand, generally produce a single controller based on ad hoc loop-shaping techniques and are also inadequate for the kind of computer-aided multiple performance specifications design applications advocated here. Thus we hope that our monograph acts as a catalyst to bridge the theory-practice gap in the control field as well as the classicalmodern gap. The results reported here were derived in the Ph.D. theses of Ming-Tzu Ho, Guillermo Silva, and Hao Xu at Texas A&M University and we thank the Electrical Engineering Department for its logistical support. We also acknowledge the financial support of the National Science Foundation's Engineering Systems Program under the directorship of R. K. Baheti and the support of National Instruments, Austin, Texas. Austin, Texas College Station, Texas College Station, Texas G. J. Silva A. Datta S. P. Bhattacharyya October 2004

12 PID Controllers for Time-Delay Systems