Feedback Systems: An Introduction for Scientists and Engineers

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Feedback Systems

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Feedback Systems
An Introduction for Scientists and Engineers
Karl Johan ˚
Aström
Richard M. Murray
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD

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Copyright © 2008 by Princeton University Press
Published by Princeton University Press
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Åström, Karl J. (Karl Johan), 1934-
Feedback systems : an introduction for scientists and engineers / Karl Johan
Åström and Richard M. Murray
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-691-13576-2 (alk. paper)
ISBN-10: 0-691-13576-2 (alk. paper)
1. Feedback control systems. I. Murray, Richard M., 1963-. II. Title.
TJ216.A78 2008
629.8 3–dc22
2007061033
British Library Cataloging-in-Publication Data is available
This book has been composed in LATEX
The publisher would like to acknowledge the authors of this volume for providing
the camera-ready copy from which this book was printed.
Printed on acid-free paper. ∞
press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1

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Contents
Preface
ix
Chapter 1. Introduction
1
1.1
What Is Feedback?
1
1.2
What Is Control?
3
1.3
Feedback Examples
5
1.4
Feedback Properties
17
1.5
Simple Forms of Feedback
23
1.6
Further Reading
25
Exercises
25
Chapter 2. System Modeling
27
2.1
Modeling Concepts
27
2.2
State Space Models
34
2.3
Modeling Methodology
44
2.4
Modeling Examples
51
2.5
Further Reading
61
Exercises
61
Chapter 3. Examples
65
3.1
Cruise Control
65
3.2
Bicycle Dynamics
69
3.3
Operational Amplifier Circuits
71
3.4
Computing Systems and Networks
75
3.5
Atomic Force Microscopy
81
3.6
Drug Administration
84
3.7
Population Dynamics
89
Exercises
91
Chapter 4. Dynamic Behavior
95
4.1
Solving Differential Equations
95
4.2
Qualitative Analysis
98
4.3
Stability
102
4.4
Lyapunov Stability Analysis
110
4.5
Parametric and Nonlocal Behavior
120

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vi
CONTENTS
4.6
Further Reading
126
Exercises
126
Chapter 5. Linear Systems
131
5.1
Basic Definitions
131
5.2
The Matrix Exponential
136
5.3
Input/Output Response
145
5.4
Linearization
158
5.5
Further Reading
163
Exercises
164
Chapter 6. State Feedback
167
6.1
Reachability
167
6.2
Stabilization by State Feedback
175
6.3
State Feedback Design
183
6.4
Integral Action
195
6.5
Further Reading
197
Exercises
197
Chapter 7. Output Feedback
201
7.1
Observability
201
7.2
State Estimation
206
7.3
Control Using Estimated State
211
7.4
Kalman Filtering
215
7.5
A General Controller Structure
219
7.6
Further Reading
226
Exercises
226
Chapter 8. Transfer Functions
229
8.1
Frequency Domain Modeling
229
8.2
Derivation of the Transfer Function
231
8.3
Block Diagrams and Transfer Functions
242
8.4
The Bode Plot
250
8.5
Laplace Transforms
259
8.6
Further Reading
262
Exercises
262
Chapter 9. Frequency Domain Analysis
267
9.1
The Loop Transfer Function
267
9.2
The Nyquist Criterion
270
9.3
Stability Margins
278
9.4
Bode’s Relations and Minimum Phase Systems
283
9.5
Generalized Notions of Gain and Phase
285
9.6
Further Reading
290

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CONTENTS
vii
Exercises
290
Chapter 10. PID Control
293
10.1 Basic Control Functions
293
10.2 Simple Controllers for Complex Systems
298
10.3 PID Tuning
302
10.4 Integrator Windup
306
10.5 Implementation
308
10.6 Further Reading
312
Exercises
313
Chapter 11. Frequency Domain Design
315
11.1 Sensitivity Functions
315
11.2 Feedforward Design
319
11.3 Performance Specifications
322
11.4 Feedback Design via Loop Shaping
326
11.5 Fundamental Limitations
331
11.6 Design Example
340
11.7 Further Reading
343
Exercises
344
Chapter 12. Robust Performance
347
12.1 Modeling Uncertainty
347
12.2 Stability in the Presence of Uncertainty
352
12.3 Performance in the Presence of Uncertainty
358
12.4 Robust Pole Placement
361
12.5 Design for Robust Performance
369
12.6 Further Reading
374
Exercises
374
Bibliography
377
Index
387

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Preface
This book provides an introduction to the basic principles and tools for the design
and analysis of feedback systems. It is intended to serve a diverse audience of
scientists and engineers who are interested in understanding and utilizing feedback
in physical, biological, information and social systems. We have attempted to keep
the mathematical prerequisites to a minimum while being careful not to sacrifice
rigor in the process. We have also attempted to make use of examples from a variety
of disciplines, illustrating the generality of many of the tools while at the same time
showing how they can be applied in specific application domains.
A major goal of this book is to present a concise and insightful view of the
current knowledge in feedback and control systems. The field of control started
by teaching everything that was known at the time and, as new knowledge was
acquired, additional courses were developed to cover new techniques. A conse-
quence of this evolution is that introductory courses have remained the same for
many years, and it is often necessary to take many individual courses in order to
obtain a good perspective on the field. In developing this book, we have attempted
to condense the current knowledge by emphasizing fundamental concepts. We be-
lieve that it is important to understand why feedback is useful, to know the language
and basic mathematics of control and to grasp the key paradigms that have been
developed over the past half century. It is also important to be able to solve simple
feedback problems using back-of-the-envelope techniques, to recognize fundamen-
tal limitations and difficult control problems and to have a feel for available design
methods.
This book was originally developed for use in an experimental course at Caltech
involving students from a wide set of backgrounds. The course was offered to
undergraduates at the junior and senior levels in traditional engineering disciplines,
as well as first- and second-year graduate students in engineering and science. This
latter group included graduate students in biology, computer science and physics.
Over the course of several years, the text has been classroom tested at Caltech and
at Lund University, and the feedback from many students and colleagues has been
incorporated to help improve the readability and accessibility of the material.
Because of its intended audience, this book is organized in a slightly unusual
fashion compared to many other books on feedback and control. In particular, we
introduce a number of concepts in the text that are normally reserved for second-
year courses on control and hence often not available to students who are not
control systems majors. This has been done at the expense of certain traditional
topics, which we felt that the astute student could learn independently and are often

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x
PREFACE
explored through the exercises. Examples of topics that we have included are non-
linear dynamics, Lyapunov stability analysis, the matrix exponential, reachability
and observability, and fundamental limits of performance and robustness. Topics
that we have deemphasized include root locus techniques, lead/lag compensation
and detailed rules for generating Bode and Nyquist plots by hand.
Several features of the book are designed to facilitate its dual function as a basic
engineering text and as an introduction for researchers in natural, information and
social sciences. The bulk of the material is intended to be used regardless of the
audience and covers the core principles and tools in the analysis and design of
feedback systems. Advanced sections, marked by the “dangerous bend” symbol
shown here, contain material that requires a slightly more technical background,
of the sort that would be expected of senior undergraduates in engineering. A few
sections are marked by two dangerous bend symbols and are intended for readers
with more specialized backgrounds, identified at the beginning of the section. To
limit the length of the text, several standard results and extensions are given in the
exercises, with appropriate hints toward their solutions.
To further augment the printed material contained here, a companion web site
has been developed and is available from the publisher’s web page:
http://press.princeton.edu/titles/8701.html
The web site contains a database of frequently asked questions, supplemental exam-
ples and exercises, and lecture material for courses based on this text. The material is
organized by chapter and includes a summary of the major points in the text as well
as links to external resources. The web site also contains the source code for many
examples in the book, as well as utilities to implement the techniques described in
the text. Most of the code was originally written using MATLAB M-files but was
also tested with LabView MathScript to ensure compatibility with both packages.
Many files can also be run using other scripting languages such as Octave, SciLab,
SysQuake and Xmath.
The first half of the book focuses almost exclusively on state space control
systems. We begin in Chapter 2 with a description of modeling of physical, biolog-
ical and information systems using ordinary differential equations and difference
equations. Chapter 3 presents a number of examples in some detail, primarily as a
reference for problems that will be used throughout the text. Following this, Chap-
ter 4 looks at the dynamic behavior of models, including definitions of stability
and more complicated nonlinear behavior. We provide advanced sections in this
chapter on Lyapunov stability analysis because we find that it is useful in a broad
array of applications and is frequently a topic that is not introduced until later in
one’s studies.
The remaining three chapters of the first half of the book focus on linear systems,
beginning with a description of input/output behavior in Chapter 5. In Chapter 6,
we formally introduce feedback systems by demonstrating how state space control
laws can be designed. This is followed in Chapter 7 by material on output feed-
back and estimators. Chapters 6 and 7 introduce the key concepts of reachability

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