Feedback Systems_An Introduction for Scientists and Engineers.pdf

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print.tex, v1.27 2008/01/25 21:53:54 (murray) Feedback Systems An Introduction for Scientists and Engineers Karl Johan ˚Aström Richard M. Murray PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

print.tex, v1.27 2008/01/25 21:53:54 (murray) 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. 2007061033 TJ216.A78 2008 629.8 3–dc22 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

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) Contents Preface Chapter 1. Introduction 1.1 What Is Feedback? 1.2 What Is Control? 1.3 1.4 1.5 1.6 Feedback Examples Feedback Properties Simple Forms of Feedback Further Reading Exercises Chapter 2. System Modeling 2.1 Modeling Concepts 2.2 State Space Models 2.3 Modeling Methodology 2.4 Modeling Examples 2.5 Further Reading Exercises Bicycle Dynamics Chapter 3. Examples 3.1 Cruise Control 3.2 3.3 Operational Ampliﬁer Circuits 3.4 Computing Systems and Networks 3.5 3.6 Drug Administration Population Dynamics 3.7 Exercises Atomic Force Microscopy Chapter 4. Dynamic Behavior Solving Differential Equations 4.1 4.2 Qualitative Analysis 4.3 4.4 4.5 Stability Lyapunov Stability Analysis Parametric and Nonlocal Behavior ix 1 1 3 5 17 23 25 25 27 27 34 44 51 61 61 65 65 69 71 75 81 84 89 91 95 95 98 102 110 120

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) vi 4.6 Further Reading Exercises Chapter 5. Linear Systems 5.1 5.2 5.3 5.4 5.5 Basic Deﬁnitions The Matrix Exponential Input/Output Response Linearization Further Reading Exercises Chapter 6. State Feedback 6.1 Reachability 6.2 6.3 6.4 6.5 Stabilization by State Feedback State Feedback Design Integral Action Further Reading Exercises Chapter 7. Output Feedback State Estimation 7.1 Observability 7.2 7.3 Control Using Estimated State 7.4 7.5 7.6 Kalman Filtering A General Controller Structure Further Reading Exercises Chapter 8. Transfer Functions Frequency Domain Modeling 8.1 8.2 Derivation of the Transfer Function 8.3 8.4 8.5 8.6 Block Diagrams and Transfer Functions The Bode Plot Laplace Transforms Further Reading Exercises Chapter 9. Frequency Domain Analysis The Loop Transfer Function The Nyquist Criterion Stability Margins Bode’s Relations and Minimum Phase Systems 9.1 9.2 9.3 9.4 9.5 Generalized Notions of Gain and Phase 9.6 Further Reading CONTENTS 126 126 131 131 136 145 158 163 164 167 167 175 183 195 197 197 201 201 206 211 215 219 226 226 229 229 231 242 250 259 262 262 267 267 270 278 283 285 290

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) CONTENTS Exercises Chapter 10. PID Control 10.1 Basic Control Functions 10.2 Simple Controllers for Complex Systems 10.3 PID Tuning 10.4 Integrator Windup 10.5 Implementation 10.6 Further Reading Exercises Chapter 11. Frequency Domain Design 11.1 Sensitivity Functions 11.2 Feedforward Design 11.3 Performance Speciﬁcations 11.4 Feedback Design via Loop Shaping 11.5 Fundamental Limitations 11.6 Design Example 11.7 Further Reading Exercises Chapter 12. Robust Performance 12.1 Modeling Uncertainty 12.2 Stability in the Presence of Uncertainty 12.3 Performance in the Presence of Uncertainty 12.4 Robust Pole Placement 12.5 Design for Robust Performance 12.6 Further Reading Exercises Bibliography Index vii 290 293 293 298 302 306 308 312 313 315 315 319 322 326 331 340 343 344 347 347 352 358 361 369 374 374 377 387

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) 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

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) 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

Frontmatter.tex, v1.80 2008/01/30 07:21:04 (murray) PREFACE xi and observability, which give tremendous insight into the choice of actuators and sensors, whether for engineered or natural systems. The second half of the book presents material that is often considered to be from the ﬁeld of “classical control.” This includes the transfer function, introduced in Chapter 8, which is a fundamental tool for understanding feedback systems. Using transfer functions, one can begin to analyze the stability of feedback systems using frequency domain analysis, including the ability to reason about the closed loop behavior of a system from its open loop characteristics. This is the subject of Chapter 9, which revolves around the Nyquist stability criterion. In Chapters 10 and 11, we again look at the design problem, focusing ﬁrst on proportional-integral-derivative (PID) controllers and then on the more general process of loop shaping. PID control is by far the most common design technique in control systems and a useful tool for any student. The chapter on frequency domain design introduces many of the ideas of modern control theory, including the sensitivity function. In Chapter 12, we combine the results from the second half of the book to analyze some of the fundamental trade-offs between robustness and performance. This is also a key chapter illustrating the power of the techniques that have been developed and serving as an introduction for more advanced studies. The book is designed for use in a 10- to 15-week course in feedback systems that provides many of the key concepts needed in a variety of disciplines. For a 10-week course, Chapters 1–2, 4–6 and 8–11 can each be covered in a week’s time, with the omission of some topics from the ﬁnal chapters. A more leisurely course, spread out over 14–15 weeks, could cover the entire book, with 2 weeks on modeling (Chapters 2 and 3)—particularly for students without much background in ordinary differential equations—and 2 weeks on robust performance (Chapter 12). The mathematical prerequisites for the book are modest and in keeping with our goal of providing an introduction that serves a broad audience. We assume familiarity with the basic tools of linear algebra, including matrices, vectors and eigenvalues. These are typically covered in a sophomore-level course on the sub- ject, and the textbooks by Apostol [10], Arnold [13] and Strang [187] can serve as good references. Similarly, we assume basic knowledge of differential equations, including the concepts of homogeneous and particular solutions for linear ordinary differential equations in one variable. Apostol [10] and Boyce and DiPrima [42] cover this material well. Finally, we also make use of complex numbers and func- tions and, in some of the advanced sections, more detailed concepts in complex variables that are typically covered in a junior-level engineering or physics course in mathematical methods. Apostol [9] or Stewart [186] can be used for the basic material, with Ahlfors [6], Marsden and Hoffman [146] or Saff and Snider [172] being good references for the more advanced material. We have chosen not to in- clude appendices summarizing these various topics since there are a number of good books available. One additional choice that we felt was important was the decision not to rely on a knowledge of Laplace transforms in the book. While their use is by far the most common approach to teaching feedback systems in engineering, many stu-

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