Feedback for physicists: A tutorial essay on controlJohn Bechhoefer*Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6,Canada!Published 31 August 2005"Feedback and control theory are important ideas that should form part of the education of a physicistbut rarely do. This tutorial essay aims to give enough of the formal elements of control theory tosatisfy the experimentalist designing or running a typical physics experiment and enough to satisfy thetheorist wishing to understand its broader intellectual context. The level is generally simple, althoughmore advanced methods are also introduced. Several types of applications are discussed, as thepractical uses of feedback extend far beyond the simple regulation problems where it is most oftenemployed. Sketches are then provided of some of the broader implications and applications of controltheory, especially in biology, which are topics of active research.CONTENTSI. Introduction 783II. Brief Review of Dynamical Systems 785III. Feedback: An Elementary Introduction 788A. Basic ideas: Frequency domain 788B. Basic ideas: Feedforward 789C. Basic ideas: Time domain 790D. Two case studies 7921. Michelson interferometer 7922. Operational amplifier 793E. Integral control 794IV. Feedback and Stability 795A. Stability in dynamical systems 795B. Stability ideas in feedback systems 796C. Delays: Their generic origins and effect on stability 797D. Nonminimum-phase systems 798E. MIMO vs SISO systems 799V. I m p l e m e n t a t i o n a n d Some A d v a n c e d To p i c s 8 0 1A. Experimental determination of the transfer function 8011. Measurement of the transfer function 8012. Model building 8023. Model reduction 8024. Revisiting the system 803B. Choosing the controller 8041. PID controllers 8042. Loop shaping 8053. Optimal control 806C. Digital control loops 8091. Case study: Vibration isolation of an atominterferometer 8132. Commercial tools 814D. Measurement noise and the Kalman filter 814E. Robust control 8171. The internal model control parametrization 8172. Quantifying model uncertainty 8183. Robust stability 8194. Robust performance 8205. Robust control methods 821VI. Notes on Nonlinearity 822A. Saturation effects 822B. Chaos: The ally of control? 823VII. Applications to Biological Systems 825A. Physiological example: The pupil light reflex 825B. Fundamental mechanisms 8271. Negative feedback example 8272. Positive feedback example 828C. Network example 829VIII. Other Applications, Other Approaches 830IX. Feedback and Information Theory 831X. Conclusions 832Acknowledgments 833List of Abbreviations 833References 833I. INTRODUCTIONFeedback and its big brother, control theory, are suchimportant concepts that it is odd that they usually findno formal place in the education of physicists. On thepractical side, experimentalists often need to use feed-back. Almost any experiment is subject to the vagariesof environmental perturbations. Usually, one wants tovary a parameter of interest while holding all others con-stant. How to do this properly is the subject of controltheory. More fundamentally, feedback is one of the greatideas developed !mostly" in the last century,1with par-*Electronic address: [email protected] mechanisms regulating liquid level were describedover 2000 years ago, while steam-engine “governors” dateback to the 18th century. #An influential theoretical study ofgovernors was given by Maxwell !1868".$ However, realizationof the broader implications of feedback concepts, as well astheir deeper analysis and widespread application, date to the20th century. Chapter 1 of Franklin et al. !2002" gives a briefhistorical review of the development of control theory.In a more detailed account, Mayr !1970" describes a numberof early feedback devices, from classical examples !Ktesibiosand Hero, both of Alexandria" to a scattering of medievalArab accounts. Curiously, the modern rediscovery of feedbackREVIEWS OF MODERN PHYSICS, VOLUME 77, JULY 20050034-6861/2005/77!3"/783!54"/$50.00 ©2005 The American Physical Society783ticularly deep consequences for biological systems, andall physicists should have some understanding of such abasic concept. Indeed, further progress in areas of cur-rent interest such as systems biology is likely to rely onconcepts from control theory.This article is a tutorial essay on feedback and controltheory. It is a tutorial in that I give enough detail aboutbasic methods to meet most of the needs of experimen-talists wishing to use feedback in their experiments. It isan essay in that, at the same time, I hope to convince thereader that control theory is useful not only for the en-gineering aspects of experiments but also for the con-ceptual development of physics. Indeed, we shall seethat feedback and control theory have recently foundapplications in areas as diverse as chaos and nonlineardynamics, statistical mechanics, optics, quantum com-puting, and biological physics. This essay supplies thebackground in control theory necessary to appreciatemany of these developments.The article is written for physics graduate studentsand interested professional physicists, although most ofit should also be understandable to undergraduate stu-dents. Beyond the overall importance of the topic, thisarticle is motivated by a lack of adequate alternatives.The obvious places to learn about control theory—introductory engineering textbooks !Dutton et al., 1997;Franklin et al., 1998, 2002; Goodwin et al., 2001"—arenot very satisfactory places for a physicist to start. Theyare long—800 pages is typical—with the relevant infor-mation often scattered in different sections. Their ex-amples are understandably geared more to the engineerthan to the physicist. They often cloak concepts familiarto the physicist in unfamiliar language and notation.And they do not make connections to similar conceptsthat physicists will have likely encountered in standardcourses. The main alternative, more mathematical texts#e.g., in order of increasing sophistication, the books byÖzbay !2000", Morris !2001", Doyle et al. !1992", andSontag !1998"$, are terse but assume the reader alreadyhas an intuitive understanding of the subject.At the other end of the intellectual spectrum, the firstreal exposure of many experimentalists to control theorycomes when they are faced with having to use or modifya PID !proportional-integral-derivative" control loopthat regulates some experimental quantity, such as tem-perature or