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 December 6, 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
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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.
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