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Preface to the Second Edition

The second edition of Feedback Systems contains a variety of changes that are based on feedback on the first edition, particularly in its use for introductory courses in control. One of the primary comments from users of the text was that the use of control tools for design purposes occurred only after several chapters of analytical tools, leaving the instructor having to try to convince students that the techniques would soon be useful. In our own teaching, we find that we often use design examples in the first few weeks of the class and use this to motivate the various techniques that follow. This approach has been particularly useful in engineering courses, where students are often eager to apply the tools to examples as part of gaining insight into the methods. We also found that universities that have a laboratory component attached to their controls class need to introduce some basic design techniques early, so that students can be implementing control laws in the laboratory in the early weeks of the course.

Changes in the Second Edition

To help emphasize this more design-oriented flow, we have added a new chapter on Feedback Principles that illustrates some simple design principles and tools that can be used to show students what types of problems can be solved using feedback. This new chapter uses simple models, simulations, and elementary analysis techniques, so that it should be accessible to students from a variety of engineering and scientific backgrounds. For courses in which students have already been exposed to the basic ideas of feedback, perhaps in an earlier discipline-specific course, this new chapter can easily be skipped without any loss of continuity.

We have also rearranged some of the material in the final chapters of the book, moving material on fundamental limits from the chapters on frequency domain design and robust performance into a separate chapter on fundamental limits. This new chapter also contains some additional material on techniques for robust pole placement as well as on limits imposed by nonlinearities.

For the electronic versions of the text, we have added a new chapter to the end of the book, focused on control architectures and design. Our intention in this chapter is to provide a systems view that describes how control design is integrated into a larger model-based development framework, motivated in part by our consulting activities with large companies. In this new chapter we also take the opportunity to present some overview material on "bottoms up" and "top down" approaches to control architectures, briefly introducing some of the many additional concepts from the field of control that are in widespread use in applications.

In addition to these relatively large changes, we have made many other smaller changes based on the feedback we have received from early adopters of the text. We have added some material on the Routh--Hurwitz criterion and root locus plots, to at least serve as "hooks" for instructors who wish to cover that material in more detail. We have also made some notational changes throughout, most notably changing the symbols for disturbance and noise signals to and , respectively. The notation in the biological examples has also been updated to match the notation used in the textbook by Del Vecchio and Murray.

Overall, we have tried to maintain the style and organization of the book in a manner that is consistent with our goals for the first edition. In particular, we have targeted the material toward a wide range of audiences rather than any specific discipline. One consequence is that instructors who are teaching department-specific courses may find there are other texts that are better suited to these audiences. Books written over the past few years that are tuned to non-traditional audiences, including Janert computer science), Del Vecchio and Murray (biology), and Bechhoefer (physics). In addition, the textbook Feedback Control for Everyone by Albertos and Mareels provides a readable introduction requiring minimal mathematical background.


Finally, we are indebted to numerous individuals who have taught out of the text and sent us feedback on changes that would better serve their needs. In addition to the many individuals listed in the preface to the first edition, we would like to thank Kalle Åström, Bo Bernhardsson, Karl Berntorp, Constantine Caramanis, Shuo Han, Bjorn Olofsson, Noah Olsman, Richard Pates, Jason Rolfe, Clancy Rowley, and Andre Tits for their feedback, insights, and contributions. Vickie Kearn, our recently-retired editor at Princeton University Press, has continued to serve as an enthusiastic advocate for our efforts and we particularly appreciate her support over the years in our vision for the book and for her advocacy of making the material available for free download.

Karl Johan Åström Richard M. Murray
Lund, Sweden Pasadena, California

Preface to the First Edition

This book provides an introduction to the basic principles and tools for 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 economic systems. To this end, we have chosen to keep the mathematical prerequisites to a minimum while being careful not to sacrifice rigor in the process. Advanced sections, marked by the "dangerous bend" symbol

shown to the right, contain material that is of a more advanced nature and can be skipped on first reading.

This book was originally developed for use in an experimental course at Caltech involving students from a wide variety of disciplines. The course consisted of undergraduates at the junior and senior level in traditional engineering disciplines, as well as first and second year graduate students in engineering and science. This included graduate students in biology, computer science and economics, requiring a broad approach that emphasized basic principles and did not focus on applications in any one given area. A detailed web site has been prepared as a companion to this text:

The web site contains a database of frequently asked questions, supplemental examples and exercises, and lecture materials for a course based on this text. It also contains the MATLAB and other source code for every example in the book, as well as MATLAB libraries to implement the techniques described in the text.

Textbook scope

This book is intended to serve a broad spectrum of audiences and 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 on their own (and are often explored through the exercises). Examples of topics that we have included are nonlinear dynamics, Lyapunov stability, reachability and observability, and fundamental limits of performance and robustness. Topics that we have de-emphasized include root locus techniques, lead/lag compensation (although this is essentially covered in Chapters 10 (PID Control) and 11 (Loop Shaping), and detailed rules for generating Bode and Nyquist plots by hand.

Overview of the contents

The first half of the book focuses almost exclusively on so-called "state-space" control systems. We begin in Chapter 2 (System Modeling) with a description of modeling of physical, biological and information systems using ordinary differential equations and difference equations. Chapter 3 (Examples) presents a number of examples in some detail, primarily as a reference for problems that will be used throughout the text. Following this, Chapter 4 (Dynamic Behavior) 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, because we find that it is useful in a broad array of applications (and frequently a topic that is not introduced until much later in one's studies).

The remaining three chapters of the first half of the book focus on linear systems, begining with a description of input/output behavior in Chapter 5 (Linear Systems). In Chapter 6 (State Feedback), we formally introduce feedback systems by demonstrating how state space control laws can be designed. This is followed in Chapter 7 (Output Feedback) by material on output feedback and estimators. Chapters 6 and 7 introduce the key concepts of reachability 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 (Transfer Functions), which is a fundamental tool for understanding feedback systems. Using transfer functions, one can begin to analyze the stability of feedback systems using loop analysis, which allows us to reason about the closed loop behavior (stability) of a system from its open loop characteristics. This is the subject of Chapter 9 (Loop Analysis), which revolves around the Nyquist stability criterion.

In Chapters 10 (PID Control) and 11 (Loop Shaping), we again look at the design problem, focusing first on proportional-integral-derivative (PID) controllers and then on the more general process of loop synthesis. PID control is by far the most common design technique in control systems and a useful tool for any student. The chapter on loop synthesis introduces many of the ideas of modern control theory, including the sensitivity function. In Chapter 12 (Robust Performance), we pull together the results from the second half of the book to analyze the fundamental tradeoffs between robustness and performance. This is also a key chapter illustrating the power of the techniques that have been developed.

Use in courses

The book is designed for use in a 10-15 week course in feedback systems that can serve to introduce many of the key concepts that are needed in a variety of disciplines. For a 10 week course, Chapters 1-6 and 8-11 can each be covered in a week's time, with some dropping of topics from the final chapters. A more leisurely course, spread out over 14-15 weeks, could cover the entire book, with two weeks on modeling (Chapter 2) -- particularly for students without much background in ordinary differential equations -- and two weeks on loop analysis (Chapter 9) or robustness and performance (Chapter 12).

In choosing the set of topics and ordering for the main text, we necessarily left out some tools which will cause many control systems experts to raise their eyebrows (or choose another textbook). Overall, we believe that the early focus on state space systems, including the concepts of reachability and observability, are of such importance to justify trimming other topics to make room for them. We also included some relatively advanced material on fundamental tradeoffs and limitations of performance, feeling that these provided such insight into the principles of feedback that they could not be left for later. Throughout the text, we have attempted to maintain a balanced set of examples that touch many disciplines, relying on the supplements for more discipline specific examples and exercises. Additional notes covering some of the "missing" topics are available on the web.

One additional choice that we felt was very important was the decision not to make use of Laplace transforms in this book. While this is by far the most common approach to teaching feedback systems in engineering, many students in the natural and information sciences may lack the necessary mathematical background. Since Laplace transforms are not required in any essential way, we have only made a few remarks to tie things together for students with that background. Of course, we make tremendous use of transfer functions, which we introduce through the notion of response to exponential inputs, an approach we feel is much more accessible to a broad array of scientists and engineers.


The authors would like to thank the many people who helped during the preparation of this book. The idea for writing this book came in part from a report on future directions in control Template:Bibcite to which Stephen Boyd, Roger Brockett, John Doyle and Gunter Stein were ma jor contributers. Kristi Morgenson and Hideo Mabuchi helped teach early versions of the course at Caltech on which much of the text is based and Steve Waydo served as the head TA for the course taught at Caltech in 2003-04 and provide numerous comments and corrections. Finally, we would like to thank Caltech, Lund University and the University of California at Santa Barbara for providing many resources, stimulating colleagues and students, and a pleasant working environment that greatly aided in the writing of this book.

Karl Johan Åström Richard M. Murray
Lund, Sweden Pasadena, California