Table Of ContentAdvances in Industrial Control
Springer-Verlag London Ltd.
Other titles published in this Series:
Control ofM odern Integrated Power Systems
E. Mariani and S.S. Murthy
Advanced Load Dispatch for Power Systems: Principles, Practices and Economies
E. Mariani and S.S. Murthy
Supervision and Control for Industrial Processes
Bjorn Sohlberg
Modelling and Simulation ofH uman Behaviour in System Control
Pietro Carlo Cacciabue
Modelling and Identification in Robotics
Krzysztof Kozlowski
Spacecraft Navigation and Guidance
Maxwell Noton
Robust Estimation and Failure Detection
Rami Mangoubi
Adaptive Internal Model Control
Aniruddha Datta
Price-Based Commitment Decisions in the Electricity Market
Eric Allen and Marija Ilic
Compressor Surge and Rotating Stall: Modeling and Control
Jan Tommy Gravdahl and Olav Egeland
Radiotheraphy Treatment Planning: New System Approaches
Olivier Haas
Feedback Control Theory for Dynamic Traffic Assignment
Pushkin Kachroo and Kaan Ozbay
Control and Instrumentation for Wastewater Treatment Plants
Reza Katebi, Michael A. Johnson and Jacqueline Wilkie
Autotuning ofPID Controllers
Cheng-Ching Yu
Robust Aeroservoelastic Stability Analysis
Rick Lind and Marty Brenner
Performance Assessment of Control Loops: Theory and Applications
Biao Huang and Sirish L. Shah
Data Mining and Knowledge Discovery for Process Monitoring and Control
XueZ. Wang
Advances in PID Control
Tan Kok Kiong, Wang Quing-Guo and Hang Chang Chieh with Tore J. Hagglund
Advanced Control with Recurrent High-order Neural Networks: Theory and
Industrial Applications
George A. Rovithakis and Manolis A. Christodoulou
Aniruddha Datta, Ming-Tzu Ho
and Shankar P. Bhattacharyya
Structure and Synthesis
of PID Controllers
With 105 Figures
, Springer
Aniruddha Datta, PhD
Department ofElectrical Engineering, Texas A&M University, College Station,
TX 77843-3128, USA
Ming-Tzu Ho, PhD
Engineering Science Department, National Cheung Kung University,
1 University Road, Tainan 701, Taiwan, Republic of China
Shankar P. Bhattacharyya, PhD
Department ofElectrical Engineering, Texas A&M University, College Station,
TX 77843-3128, USA
ISBN 978-1-84996-889-8
British Library Cataloguing in Publication Data
Datta, Aniruddha, 1963-
Structure and synthesis of PID controllers. -(Advances in
industrial contral)
l.PID controllers
I.Titie lI.Ho, Ming-Tzu III.Bhattacharyya, S.P.
629.8
ISBN 978-1-84996-889-8 ISBN 978-1-4471-3651-4 (eBook)
DOI 10.1007/978-1-4471-3651-4
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the
publishers, or in the case of reprographie reproduction in accordance with the terms of licences issued by
the Copyright Lieensing Agency. Enquiries concerning reproduction outside those terms should be sent
to the publishers.
© Springer-Verlag London 2000
Originally published by Springer-Verlag London Limited in 2000
Softcover reprint of the hardcover 1s t edition 2000
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a
specific statement, that such names are exempt from the relevant laws and regulations and therefore free
for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the
information contained in this book and cannot accept any legal responsibility or liability for any errors or
omissions that may be made.
Typesetting: Camera ready by authors
69/3830-543210 Printed on acid-free paper SPIN 10712015
Advances in Industrial Control
Series Editors
Professor Michael J. Grimble, Professor ofIndustrial Systems and Director
Professor Michael A. Johnson, Professor of Control Systems and Deputy Director
Industrial Control Centre
Department of Electronic and Electrical Engineering
University of Strathclyde
Graham Hills Building
50 George Street
GlasgowG11QE
United Kingdom
Series Advisory Board
Professor Dr-Ing J. Ackermann
DLR Institut ffir Robotik und Systemdynamik
Postfach 1116
D82230 WeBling
Germany
Professor I.D. Landau
Laboratoire d'Automatique de Grenoble
ENSIEG, BP 46
38402 Saint Martin d'Heres
France
Dr D.C. McFarlane
Department of Engineering
University of Cambridge
Cambridge CB21QJ
United Kingdom
Professor B. Wittenmark
Department of Automatic Control
Lund Institute of Technology
POBox 118
S-221 00 Lund
Sweden
Professor D.W. Clarke
Department of Engineering Science
University of Oxford
Parks Road
Oxford OX1 3PJ
United Kingdom
Professor Dr -Ing M. Thoma
Institut ffir Regelungstechnik
Universitiit Hannover
Appelstr. 11
30167 Hannover
Germany
Professor H. Kimura
Department of Mathematical Engineering and Information Physics
Faculty of Engineering
The University of Tokyo
7-3-1 Hongo
BunkyoKu
Tokyo 113
Japan
Professor A.J. Laub
College of Engineering -Dean's Office
University of California
One Shields Avenue
Davis
California 95616-5294
United States of America
Professor J.B. Moore
Department of Systems Engineering
The Australian National University
Research School of Physical Sciences
GPO Box 4
Canberra
ACT 2601
Australia
Dr M.K. Masten
Texas Instruments
2309 Northcrest
Plano
TX75075
United States of America
Professor Ton Backx
AspenTech Europe B.V.
De Waal 32
NL-5684 PH Best
The Netherlands
THIS BOOK IS DEDICATED TO
My Parents Akhil Chandra Datta and Sheela Datta
A. Datta
My Parents Mei-Ni Wang and Chen-Cheng Ho
M. T.Ho
J. Boyd Pearson
Friend, advisor, mentor and
Outstanding control theorist of the 20th century.
S. P. Bhattacharyya
Series Editors' Foreword
The series Advances in Industrial Control aims to report and encourage
technology transfer in control engineering. The rapid development of control
technology has an impact on all areas of the control discipline. New theory, new
controllers, actuators, sensors, new industrial processes, computer methods, new
applications, new philosophies ... , new challenges. Much of this development
work resides in industrial reports, feasibility study papers and the reports of
advanced collaborative projects. The series offers an opportunity for researchers
to present an extended exposition of such new work in all aspects of industrial
control for wider and rapid dissemination.
It has taken a surprisingly long time for control theorists to turn their
attention to the paradoxical success of three-term control in industrial
applications but some small groups of researchers in Singapore, Sweden, Taiwan
and elsewhere have continued to investigate fundamental issues concerned with
PID and the autotune culture. In this new text, A. Datta, M.T. Ho and S.P.
Bhattacharyya report on their fundamental investigations into three-term
control. They concentrate on determining the set of all stabilising PID controllers
for a given single-input, single-output plant and provide efficient computational
routes for depicting this set. The examples displayed are instructive and
insightful, showing that the topology of the stabilising set can be exceedingly
complicated.
The importance of this PID work is that it creates a framework for
determining the limits of performance of PID control to satisfy other design
specifications. In one sense this research activity in PID control is only just
beginning and there are many unanswered questions remaining; however, this
c.-c.
volume complements that by Yu on Autotuning of PID Controllers: Relay
Feedback Approach and the new volume by K. K. Tan, Q.-G. Wang, C.-C. Hang
and T.J. Hagglund on Advances in PID Control which are both published in the
Advances in Industrial Control series. Industrial control engineers will be
pleased to see the spotlight turned on to the ubiquitous PID controller and will
welcome the insight that these investigations provide. In the academic
community these volumes will be read to see the hidden theoretical depth and
subtly that exists with this commonest form of controller.
M.J. Grimble and M.A. Johnson
Industrial Control Centre
Glasgow, Scotland, UK
Preface
In this monograph we describe some new results on the structure and design
of Proportional-Integral-Derivative (PID) controllers. We believe that these
results could form the beginnings of an attempt to close the significant gap
that has arisen between control theory and control engineering practice since
the late 1950's. This gap may be described as follows.
Kalman's pathbreaking work on the linear quadratic regulator problem
in 1960, ushered in the era of state space based optimal control as the design
method of choice for linear multivariable feedback control systems. Almost
overnight, research on classical control design methods, which were largely
of an adhoc nature, were abandoned in favor of the new analytically based
optimal designs. This trend has persisted until now with the added element
that the Hoo and II optimal control problems assess performance based on
worst case inputs.
Over the last 40 years, many feedback design problems have all been ele
gantly formulated and solved within the broad framework of optimal control
theory. In each case the resulting feedback controller is of high dynamic or
der, typically comparable to that of the plant. From the early days these high
order controllers have been justified by control theorists on the grounds that
they could be easily implemented using digital computers, since the cost of
memory was decreasing, and the speed of computing increasing, at exponen
tial rates.
In the control industry, however, the situation is quite different. The ma
jority of control systems in the world continue to be operated by PID con
trollers, just as they were in the pre 1960 period. The net effect of computer
technology has been that today's PID's are efficiently implemented digitally
AID I
by microprocessors, and D A converters, and special purpose signal
processing chips. There is a great deal of current research and an extensive
literature on the design of PID controllers. However, modern optimal control
methods are conspicuously absent from this literature. This is the situation
prevalent in most industries including process control, motion control and
aerospace industries.
This severe gap is not healthy for theory or for practice and needs to be
bridged if control theory is to remain a useful or even relevant engineering
discipline. Although it is generally argued that the gap is due to practice
XII
lagging theory, our opinion is exactly the opposite. As it turns out despite
extensive progress, control theory has been unable to solve the robust or
optimal PID control problem in any degree of generality. This means that
there are no known procedures for designing PID controllers using the most
successful modern theories, namely H2, Hoo, l1 or J-l optimal control.
In this environment it is indeed unrealistic to expect an industry to over
haul its existing PID hardware in favor of a complex and high order controller.
Even if optimality under a worst case scenario is accepted as a worthwhile de
sign goal, unless it is shown that the best performance attainable by the PID
controller is indeed inadequate, the argument in favor of a high order optimal
controller is questionable. This is the reason that the aforementioned "gap"
persists till today, several decades after the initial introduction of optimal
control theory.
At this point it is important to clarify, that although modern controls
researchers have, by and large, ignored the PID controller, an important
exception is the work of a small group of researchers prominent among which
is Astrom and his associates. This group and others have developed extensive
results and expertise on the theory, autotuning, adaptation and antiwindup
techniques for the design of PID controllers, based mostly on experimental
methods.
A major obstacle to designing the "best" PID controller, in any sense
whatsoever, has been the difficulty in characterizing the entire set of stabiliz
ing PID controllers. Indeed this difficulty holds for every fixed and low order
controller. The solution of this problem is a necessary first step to optimal,
or at least, rational design based on achievable performance with fixed or
der and structure, and thus the first step in closing the gap. The absence of
results of this type is a significant bottleneck in applying control theory to
practical problems.
The main contribution of the present monograph is an effective solution
of the above problem, in the case of PID control. Specifically, we give a
constructive and computationally efficient characterization of the set of all
stabilizing PID controllers for a given but arbitrary, single input single output
linear time invariant (LTI) plant. This is accomplished by first generalizing
a classical stability result developed in the last century, the Hermite-Biehler
Theorem. This generalization is used to reduce the PID stabilization problem
to a one parameter family of linear programming problems. Thus a definite
yes or no answer to the question of existence of stabilizing PID controllers is
given, and in case stabilization is possible, the entire set of stabilizing PID
controllers is found. This result is applicable to all LTI plants regardless of
order, and regardless of whether it is open loop stable or not.
In general we see, through the various examples presented in the book,
that the set of PID controllers stabilizing a given system has a very compli
cated topology. For example the set is not convex nor is it even connected.
This reinforces our belief that optimum design with prescribed structure and
