Feedback Systems:An Introduction for Scientists and EngineersKarl Johan˚Astr¨omDepartment of Automatic ControlLund Institute of TechnologyRichard M. MurrayControl and Dynamical SystemsCalifornia Institute of TechnologyDRAFT v0.915, 26 September 2004c° 2004 Karl Johan˚Astr¨om and Richard MurrayAll rights reserved.This manuscript is for review purposes only and may not be reproduced, in wholeor in part, without written consent from the authors.iiPrefaceThis book provides an introduction to the basic principles and tools fordesign and analysis of feedback systems. It is intended to serve a diverseaudience of scientists and engineers who are interested in understanding andutilizing feedback in physical, biological, information, and economic systems.To this end, we have chosen to keep the mathematical pre-requisites to aminimum while being careful not to sacrifice rigor in the process. Advancedsections, marked by the “dangerous bend” symbol shown to the right indi- Äcate material that is of a more advanced nature and can be skipped on firstreading.This book was originally developed for use in an experimental courseat Caltech involving undergraduates and graduate students from a widevariety of disciplines. The course included undergraduates at the juniorand senior level in traditional engineering disciplines, as well as first andsecond year graduate students in engineering and science. This includedgraduate students in biology, computer science, and economics, requiringa broad approach that emphasized basic principles and did not focus onapplications in a given area.A detailed web site has been prepared as a companion to this text:http://www.cds.caltech.edu/~murray/books/am04The web site contains the MATLAB and other source code for every examplein the book, as well as MATLAB libraries to implement the techniquesdescribed in the text.This book is intended to serve a broad spectrum of audiences and isorganized in a slightly unusual fashion compared to many other books onfeedback in control. In particular, we introduce a number of concepts inthe text which are normally reserved for second year courses on control(and hence often not available to students who are not control majors).This has been done at the expense of certain “traditional” topics, whichwe felt that the astute student could learn on their own (and are ofteniiiivexplored through the exercises). Examples of topics that we have includedare nonlinear behavior, Lyapunov stability, reachability and observability,and fundamental limits of performance and robustness. Topics that wehave de-emphasized include root locus techniques, lead/lag compensation(although this is essentially covered in the Chapter 8, on PID control), anddetailed rules for generating Bode and Nyquist plots by hand.The first half of the book focused almost exclusively on so-called “state-space” control systems. We begin in Chapter 2 with a description of model-ing of physical, biological and information systems using ordinary differentialequations and difference equations. Following this, Chapter 3 looks at thedynamic behavior of models, including definitions of stability and more com-plicated nonlinear behavior. We provide advanced sections in this chapteron Lyapunov stability, because we find that it is useful in a broad array ofapplications (and frequently a topic that is not introduced until much laterin ones studies). Chapter 4 looks at the input/output behavior of dynamicalsystems, focusing primarily on linear systems (for which the concepts arebest defined). Finally, in Chapter 5, we introduce our first feedback systemsby demonstrating how state space control laws can be designed, including ashort description of observers. Chapter 5 introduces the key concepts of con-trollability and observability, which give tremendous insight into the choiceof actuators and sensors, whether for engineered or natural systems.The second half of the book presents tools that are often considered tobe from the field of “classical control”. This includes the transfer function,introduced in Chapter 6, which is a fundamental tool for understandingfeedback systems. Using transfer functions, one can begin to analyze thestability of feedback systems using loop analysis, which allows us to reasonabout the closed loop behavior (stability) of a system from its open loopcharacteristics. This is the subject of Chapter 7, which revolves aroundthe Nyquist stability criterion. In Chapter 8, we again look at the designproblem, focusing on proportional-integral-derivative (PID) controllers. PIDcontrol is by far the most common design technique in control systems and auseful tool for any student. In Chapter 9, we pull together the results fromthe second half of the book to analyze the fundamental tradeoffs betweenrobustness and performance. This is also a key chapter illustrating thepower of the techniques that have been developed. Finally, in Chapter 10 weprovide some insights into the implementation of control systems, primarilyto illustrate how some of the concepts of the course are applied to practicalproblems.The book is designed for use in a 10–12 week course in feedback systemsthat can serve to introduce many of the key concepts that are needed in avvariety of disciplines. For a ten-week course, each chapter can be covered ina week’s time, with some dropping of topics from the final three chapters ofthe book. A more leisurely course, spread out over 12–14 weeks, could spendtwo weeks on modeling (Chapter 2)—particularly for students without muchbackground in ordinary differential equations—and two weeks each on loopanalysis (Chapter 7) and robustness and performance (Chapter 9).In choosing this set of topics and ordering, we necessarily left out sometools which will cause many control systems experts to raise their eyebrows(or choose another textbook). Overall, we believe that the early focus onstate space systems, including the concepts of controllability and observabil-ity, are of such importance to justify trimming other topics to make room forthem. We also included some relatively advanced material on fundamentaltradeoffs and limitations of performance, feeling that these provided suchinsight 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 ex-amples that touch many disciplines, relying on the supplements for morediscipline specific examples and exercises.One
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