María M. Seron Julio H. Braslavsky Graham C. Goodwin Fundamental Limitations in Filtering and Control With 114 Figures This book was originally Published by Springer-Verlag London Limited in 1997. The present PDF file fixes typos found until February 2, 2004. Springer Copyright Notice María M. Seron, PhD Julio H. Braslavsky, PhD Graham C. Goodwin, Professor School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, New South Wales 2308, Australia Series Editors B.W. Dickinson • A. Fettweis • J.L. Massey • J.W. Modestino E.D. Sontag • M. Thoma ISBN 3-540-76126-8 Springer-Verlag Berlin Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Apart from any fair dealing for the purposes of research or private study, or criti- cism 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 reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquires concerning reproduction outside those terms should be sent to the publishers. c  Springer-Verlag London Limited 1997 Printed in Great Britain 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 Printed and bound at the Athenæum Press Ltd, Gateshead 69/3830-543210 Printed on acid-free paper Preface This book deals with the issue of fundamental limitations in filtering and control system design. This issue lies at the very heart of feedback theory since it reveals what is achievable, and conversely what is not achievable, in feedback systems. The subject has a rich history beginning with the seminal work of Bode during the 1940’s and as subsequently published in his well-known book Feedback Amplifier Design (Van Nostrand, 1945). An interesting fact is that, although Bode’s book is now fifty years old, it is still extensively quoted. This is supported by a science citation count which remains comparable with the best contemporary texts on control theory. Interpretations of Bode’s results in the context of control system design were provided by Horowitz in the 1960’s. For example, it has been shown that, for single-input single-output stable open-loop systems having rel- ative degree greater than one, the integral of the logarithmic sensitivity with respect to frequency is zero. This result implies, among other things, that a reduction in sensitivity in one frequency band is necessarily accom- panied by an increase of sensitivity in other frequency bands. Although the original results were restricted to open-loop stable systems, they have been subsequently extended to open-loop unstable systems and systems having nonminimum phase zeros. The original motivation for the study of fundamental limitations in feedback was control system design. However, it has been recently real- ized that similar constraints hold for many related problems including filtering and fault detection. To give the flavor of the filtering results, con- sider the frequently quoted problem of an inverted pendulum. It is well vi Preface known that this system is completely observable from measurements of the carriage position. What is less well known is that it is fundamentally difficult to estimate the pendulum angle from measurements of the car- riage position due to the location of open-loop nonminimum phase zeros and unstable poles. Minimum sensitivity peaks of 40 dB are readily pre- dictable using Poisson integral type formulae without needing to carry out a specific design. This clearly suggests that a change in the instrumenta- tion is called for, i.e., one should measure the angle directly. We see, in this ex- ample, that the fundamental limitations point directly to the inescapable nature of the difficulty and thereby eliminate the possibility of expend- ing effort on various filter design strategies that we know, ab initio, are doomed to failure. Recent developments in the field of fundamental design limitations in- clude extensions to multivariable linear systems, sampled-data systems, and nonlinear systems. At this point in time, a considerable body of knowledge has been assem- bled on the topic of fundamental design limitations in feedback systems. It is thus timely to summarize the key developments in a modern and comprehensive text. This has been our principal objective in writing this book. We aim to cover all necessary background and to give new succinct treatments of Bode’s original work together with all contemporary results. The book is organized in four parts. The first part is introductory and it contains a chapter where we cover the significance and history of design limitations, and motivate future chapters by analyzing design limitations arising in the time domain. The second part of the book is devoted to design limitations in feed- back control systems and is divided in five chapters. In Chapter 2, we summarize the key concepts from the theory of control systems that will be needed in the sequel. Chapter 3 examines fundamental design limita- tions in linear single-input single-output control, while Chapter 4 presents results on multi-input multi-output control. Chapters 5 and 6 develop cor- responding results for periodic and sampled-data systems respectively. Part III deals with design limitations in linear filtering problems. After setting up some notation and definitions in Chapter 7, Chapter 8 covers the single-input single-output filtering case, while Chapter 9 studies the multivariable case. Chapters 10 and 11 develop the extensions to the re- lated problems of prediction and fixed-lag smoothing. Finally, Part IV presents three chapters with very recent results on sen- sitivity limitations for nonlinear filtering and control systems. Chapter 12 introduces notation and some preliminary results, Chapter 13 covers feed- back control systems, and Chapter 14 the filtering case. In addition, we provide an appendix with an almost self-contained re- view of complex variable theory, which furnishes the necessary mathe- matical background required in the book. Preface vii Because of the pivotal role played by design limitations in the study of feedback systems, we believe that this book should be of interest to re- search and practitioners from a variety of fields including Control, Com- munications, Signal Processing, and Fault Detection. The book is self- contained and includes all necessary background and mathematical pre- liminaries. It would therefore also be suitable for junior graduate students in Control, Filtering, Signal Processing or Applied Mathematics. The authors wish to deeply thank several people who, directly or in- directly, assisted in the preparation of the text. Our appreciation goes to Greta Davies for facilitating the authors the opportunity to complete this project in Australia. In the technical ground, input and insight were obtained from Gjerrit Meinsma, Guillermo Gómez, Rick Middleton and Thomas Brinsmead. The influence of Jim Freudenberg in this work is im- mense. [...]... by using the fact that q > p Case (ii) can be shown similarly by combining (1.9) and (1.11) and using the fact that q < p In the following subsection, we illustrate the previous results by analyzing time domain limitations arising in the control of an inverted pendulum This example will be revisited in Chapter 3, where we study frequency domain limitations in the context of feedback control, and in Chapter...xiii Contents IV Limitations in Nonlinear Control and Filtering 243 12 Nonlinear Operators 12.1 Nonlinear Operators 12.1.1 Nonlinear Operators on a Linear Space 12.1.2 Nonlinear Operators on a Banach Space 12.1.3 Nonlinear Operators on a Hilbert Space 12.2 Nonlinear Cancelations 12.2.1 Nonlinear Operators on Extended Banach... interpolation constraints) The above seemingly innocuous constraints actually have profound implications on the achievable performance as we will see below is a characteristic velocity, a characteristic length, the uid density, and Ê Ă Â 4 Here viscosity the 9 1.3 Time Domain Constraints 1.3 Time Domain Constraints In the main body of the book we will carry out an in- depth treatment of constraints for interconnected... loop is stable, linear, and time-invariant (actually, in the text we will relax these latter restrictions and also consider nonlinear and time-varying loops) In the linear scalar case, equations (1.5) and (1.4) encapsulate the key relationships that lead to the constraints The central observation is that we require the loop to be stable and hence we require that, whatever value for the controller transfer... magnitude and phase response in the frequency domain since each was determined uniquely by the other Horowitz (1963) applied Bodes theorems to the feedback control problem, and also obtained some preliminary results for open-loop unstable systems These latter extensions turned out to be in error due to a missing term, but the principle is sound Francis and Zames (1984) studied the feedback constraints imposed... reasonable since the only elements that are being used are the complementarity, interpolation and analyticity constraints introduced in Đ1.2 Recent extensions of the results include multivariable systems, ltering problems, periodic systems, sampled-data systems and, very recently, nonlinear systems We will cover all of these results in the remainder of the book 1.6 Summary This chapter has introduced... and controller can be described in transfer function form by G(s) and K(s), where G(s) = NG (s) , DG (s) and K(s) = NK (s) DK (s) (1.3) 2 It is sufcient here to consider an input-output operator as a mapping between input and output signals 7 1.2 Performance Limitations in Dynamical Systems Disturbance c E i Ă E i E Controller E Plant Ơ Ă Ă Â T Output E Ê Error Ô Reference FIGURE 1.1 Feedback control. .. frequency domain limitations from a ltering point of view 1.3.3 Example: Inverted Pendulum Consider the inverted pendulum shown in Figure 1.4 The linearized model for this system about the origin (i.e., = = y = y = 0) has the following transfer function from force, u, to carriage position, y Y(s) (s q)(s + q) = , U(s) M s2 (s p)(s + p) (1.20) 17 1.3 Time Domain Constraints Ê ÔÂ Ă Ê Ơ FIGURE 1.4 Inverted... used to derived this bound), and (ii) the bound correctly predicts an increase of the undershoot as the difference p q decreases 1.4 Frequency Domain Constraints The results presented in Đ1.3 were expressed in the time domain using Laplace transforms However, one might expect that corresponding results hold in the frequency domain This will be a major theme in the remainder of the book To give the... the origin, unstable poles, and nonminimum phase zeros We will then see that the results below quantify limits in performance as constraints on transient properties of the system such as rise time, settling time, overshoot and undershoot Throughout this subsection, we refer to Figure 1.1, where the plant and controller are as in (1.3), and where e and y are the time responses to a unit step input (i.e., . Braslavsky Graham C. Goodwin Fundamental Limitations in Filtering and Control With 114 Figures This book was originally Published by Springer-Verlag London Limited in 1997. The present PDF file. concepts from the theory of control systems that will be needed in the sequel. Chapter 3 examines fundamental design limita- tions in linear single-input single-output control, while Chapter 4 presents results. variety of fields including Control, Com- munications, Signal Processing, and Fault Detection. The book is self- contained and includes all necessary background and mathematical pre- liminaries. It