Su Dents Guide 2 Classical Control

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utomatic control is a miracle tech-nology that exploits feedback toimprove the performance of a tremendousrange of technological systems, from thesteam engine to the space station. Watts

governor tamed the steam engine andmade the Industrial Revolution possible.However, feedback is used by everyone:if the shower water is too hot, open up thecold water faucet. Today, feedback con-trol is used in radios (amplifiers), CDplayers (laser t racking), automobiles(cruise control , engines, and suspen-sions), flight control (autopilots and sta-bility augmentation), spacecraft (attitudecontrol and guidance ), machine tools, ro-bots, power plants, materials processing,and many other applications. In manycases automatic control is an enablingtechnology since the system cannot oper-ate without it. Feedback is also used toregulate virtually every system in the hu-man body and is constantly at work in eco-logical systems (but lets not go too fa rafield here). At first sight, control engi-neering looks simple and straightforward.It is not. While automatic control is apo w-erful technology, the subject is amazinglysubtle and intricate in both theory andpractice. This subtlety is mainly becausechanges cannot be effected instantane-ously in a dynamica l system-when theshower is too hot to the touch, it is alreadytoo late to shut down the hot water. Anotherwise correct control decision appliedat the wrong tim e could result in catastro-phe. Accounting for this and numerousother effects is what control engineeringis about.The author is with the Aerospace Engi-neering Department, The University ofMichigan, Ann Aubor, MI 48109,[email protected].

Because automatic control is such anintricate subject it is extremely easy forstudents to miss the forest for the trees.This guide is intended to be of help to bothundergraduate and gradua te students. Un-dergraduates may skim this guide at thebeginning of a first course on control toget arough roadmap of the subject and ter-minology a nd later refer to it periodicallyduring the course. At the end of thecourse, these notes c an be useful for re-viewing the course while studying for thefinal exam. For graduate students em-barking on a course in modern control,these notes can provide a quick review atthe beginning of the course to help placetheir prior knowledge in perspective.

1. Feedback is pervasive. The inter-action of any pair of systems almost al-ways involves feedback. System l reactsto System 2 and vice versa. It is cascade(one-way interaction) that is the excep-tional case. Newtons third law is a state-ment of feedbac k: The force applied to Aby B is counteracted by the force appliedto B by A.When two electrical circuits arewired together, each affects the other, andKirchoffs laws determine what that inter-action is.

2. Block diagrams are not circuitdiagrams.Block diagrams are helpful foranalyzing feedback. H owever, block dia-grams should not be confused with circuitdiagrams. It is useful to b e able to translatecircuit diagrams into block diagrams.This translation requires representing im-pedances an d admittances as systems withinputs and outputs that are voltages andcurrents. Remember that the lines con-necting the boxes are not wires. Analo-gously, it is useful to be able to represent

mechanical systems by block diagramswith forces, positions, velocities, and ac-celerations as inputs and outputs and withthe blocks representing kinematics anddynamics.3. Determine the equilibrium points

and linearize.An equilibrium point is astate in which the system would rem ain ifit were unperturbed by external distur-bances. An equilibrium point can be un-stable (an egg s tanding on i ts end),neutrally stable (a mass connected to aspring), or stable (a book lying on a table).For a system under feedback control, anequilibrium point is called an operatingpoint.

Real systems are nonlinear. How-ever, a linearized model can be used toapproximate a nonlinear system near anequilibrium point of the nonlinear sys-tem by a procedure called linearization.The resulting linear system has an equi-librium point at zero that corresponds tothe equilibrium point of the nonlinearsystem. While linearized models are onlyan approximation to the nonlinear sys-tem, they are convenient to analyze andthey give considerable insight into thebehavior of the nonlinear system near theequilibrium point. For example, if thezero equilibrium point of the linear sys-tem is stable, then the equilibrium of thenonlinear system is locally stable. Un-less stated otherwise, we will consideronly the approximate linearized model ofthe system.

4. Check stabilityfirst.Once a linear-ized model has been obtained, stabilitymust be checked. Stability can be asked ofany system, whether the system is uncon-trolled (with the contro l turned off) or thesystem is in closed-loop operation (withthe control determined by feedback). If

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the system is represented by a transferfunction, then the roots of the denom ina-tor polynomial (called the poles of thesystem) determine whether the equilib-rium or operating point is stable. Stabil-ity can be tested explicitly by computingthe roots of this polynomial or implicitlyby using the Routh criterion. If the sys-tem is in state spaceform, hen the eigen-values of the dynamics matrix can becomputed as an explicit test of stability.Stability can also be tested by forming thecharacteristic polynomial of the dyn amicsmatrix and applying the Routh criterion.Howe ver, this approach is inconvenient ifthe system is of high order.

5. Stable systems have a frequencyresponse. M u ch o f co n t ro l - s y s t emanalysis involves the frequency re-sponse of linear systems. The meaningof the frequency response can be under-stood by keeping in mind the funda-mental theorem of linear systems:Suppose hat a sinusoidal (or harmonic)input (such as a forcing) with frequencyw is applied to a stable linear system G(s). Then the output of the system ap-proaches a sinusoidal motion whose fre-quency is the same as the frequency ofthe input. This limiting sinusoidal mo-t ion is cal led the harmonic steady-state response. Note, howev er, that theoutput of the system does not have alimit in the usual mathematical senses ince the harmonic s teady-s ta te re -sponse does not approach a constantvalue. The transient behavior of the sys-tem before its response reaches har-monic steady sta te depends on the polesand zeros of the system as well as on theinitial conditions of the internal states.The ratio of the amplitude of the har-monic steady-state response to the am-plitude of the sinusoidal inpu t is equal tothe absolute value or gain of the transferfunct ion eva lua ted a t the inpu t f re -quency w, that is, 1/2G u w) l / 2 , wh i lethe phase shift or phase of the harmonicsteady-state response relat ive to thephase of the input is given by the phaseangle of the transfer function evaluated atthe input frequency, that is, tan-(Im G(iw)/Re G (iw)). Bode plots show thegain and phase of G (jw) for each fre-quency w in a range of frequencies. Al-though the fundamenta l theorem oflinear systems do es not apply to neutrallystable and unstable systems, Bode plotscan be draw n for these systems as well,although the gain may be infinite at cer-

tain frequencies. For convenien ce we re-fer to these plots as the frequency re-sponse even when the system is not stable.6. Remember the loop transfer func-

tion. Breaking the loop of a closed-loopsystem reveals the loop transfer func-tion, which is the product of all of thetransfer functions in the loop, includingthe plant, the sensor, the controller, andthe actuator. Alway s be aware of whetherthe loop transfer function of a closed-loopsystem is stable or not, and be sure to notewhether the closed-loop system involvespositive or negative feedback. The gainand phase of the loop transfer function at agiven frequency are called the loop gainan d loop phase, respectively, and thesequantities are defined whether or not theloop transfer unction is stable. Note thatif the loop transfer function is stable thenit does not necessarily follow that the c or-responding closed-loop system is stable,and that if the loop transfer function is un-stable then it does not n ecessarily followthat the corresponding closed-loop sys-tem is unstable. Stability of a closed-loopsystem can be tested directly by applyingthe Routh criterion to the closed-loop sys-tem or by computing the poles of theclosed-loop transfer function. In the caseof negative feedback, the closed-looptransfer function involves the sensitivityfunction which is one over one plus theloop transfer unction.

While the characteristics of the looptransfer function (such as its frequency re-sponse and its poles and zeros) may bewell understood, the closed-loop systemis relatively complicated. Thus it is muchmore convenient to analyze stability indi-rectly in terms of the loop transfer func-tion. The Nyquist criterion an d rootlocus procedure allow you to do ust that.

While it is intuitively clear from theform of the sensitivity function that thefrequency response of the loop transferfunction of a stable system must never beequal to -1 if the closed-loop system is tobe stable, the Nyquist criterion showsprecisely how the frequency response ofthe loop transfer function must avoid thisvalue. Specifically, the Nyquist criterionsays that the closed-loop system withnegative feedback is stable if and only ifthe polar plot of the loop transfer function,with its argument following a contour thatincludes the imaginary axis, avoids polesof the loop transfer function on the imagi-nary axis, and encloses the right halfplane, encircles the critical point - l + j O as

many times counterclockwise as there areunstable (open right-half plane) poles inthe loop transfer function , whether theseunstable poles arise from the plant or fromthe controller. Note that the Ny quist plotcan only enc ircle the critical point - l + j O ifthe loop gain is greater than unity in som efrequency range. Thus stabilization im-poses a minimal requireme nt on the loopgain in certain frequency ranges. This re-quirement thus imposes a constraint onthe gain of the controller that dependsupon the gain of the plant.

Th e root locus procedure shows thelocation of the closed-loop poles for eachvalue of a constant feedback gain. Th eclosed-loop poles are located near thepoles of the loop transfer function forsmall values of the feedback gain, and, asthe feedback gain goes to infinity, con-verge to the locations of the loop transferfunction zeros and move toward infinityalong asymptotes. Hence a nonminimumphase system, that is, a system with atleast one zero in the open right half plane,can be destabilized by large constantfeedback gains. Furthermore, it can beseen from the form of the asym ptotes thata system that has relative degree (pole-zero excess) of at least three can also bedestabilized by large constant feedbackgains.

7. After stability, performance iseverything.Once stability is settled, per-formance s everything. In fact, for plantsthat are open-loop stable, Performance isth e only reason for using feedback con-trol. Basic criteria for performance in-clude the ab ility to track or reject signalssuch as steps, ramps, sinusoids, and noise(random signals). Good performancegenerally requires high loop gain whichcorresponds to sm all gain of the sensitiv-ity function and thus tracking error reduc-t i o n o r d i s t u r b a n c e a t t e n u a t i o n .Unfortunately, Bodes integral theoremtells us that for any real control system thesensitivity function cannot have gain lessthan unity at all frequencies.

8. Perfect performance is asymp-totically possible. In the extreme case,asymptotically perfect tracking or distur-bance rejection can be obtained by meansof infinite loop gain and thus zero sensi-tivity at the disturbance frequencies. Thisis the case when an integral controller isused to provide infinite gain at DC (zerofrequency) and thus asymptotically tracka step command or reject a step distur-bance. The same idea can be used to as-ymptotically track or reject a sinusoidal

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signal by using infinite controller gain atthe disturbance frequency (which we areassuming is known). Since the controllerincorporates a model of the command ordisturbance, its operation is a special caseof the internal model principle. Dontforget that closed-loop stability must stillbe checked for this infinite-gain controller.

9. Assure nominal stability. If th esystem is uncertain, as least assure nomi-nal stability. Everything that has been saidso far with regard to stability and perform-ance is based upon the assu mption that thelinearized model of the plant is an accu-rate representation of the system near theequilibrium. While all real systems arenonlinear, it is convenient to view the lin-ear approximation itself as possessingmodeling uncertainty, so that the modelwe have been discussing is merely anominal modelof the plant. All of the sta-bility tests, wh ether direct or indirect, canbe applied to the nominal model to guar-antee nominal stability.

10. What guarantees robust stabil-ity? Robust stability refers to stabilityfo r all linear models in the range of th emodeling uncertainty. Robust stabilitycan be guaranteed in principle by apply-ing any stability test to all possible modelsof the uncertain system. Unfortunately,this is hard work. However, the Nyquistcriterion can be used to determine robuststability with a frequency domain un-certainty model, which measures thelevel of modeling uncertainty at each fre-quency. In g eneral, uncertainty tends to begreater at higher frequencies, and loopgain is generally known better than loopphase.

11.The Nyquist criterion can deter-mine robust stability.The number of en -circlements of the critical point -l+jO bythe Nyquist plot of the loop transfer func -tion is the crucial frequency domain testfor nominal stability. Once the number ofencirclements is correct for nominal sta-bility, the distance from the Nyquist plotof the loop transfer function to the criticalpoint -l+jO determ ines robust stability interms of a frequency domain uncertaintymodel. This distance is the reciprocal ofthe gain of the sensitivity. Thus, smallsensitivity at a given frequency corre-sponds to large distance from the Nyquistplot of the loop transfer function to thecritical point -l+jO and hence robust sta-bility at that frequency. The gain marginmeasures robust stability for frequenciesat which the phase of the loop transfer

function is 1 80 degrees, which quantifiesrobust stability for a pure gain perturba-tion of the loop transfer function, whileth e phase margin quantifies robust sta-bility for frequencies at which the loopgain is unity (these are the crossover fre-quencies),which quantifies robust stabil-ity for a pure phase perturbation of theloop transfer function. The Nyquist crite-rion and the root locus procedure can bothbe used to determine gain and phase m ar-gins. Note that a closed-loop system w ithnonminimum phase loop transfer func-tion has limited gain margin (this wasseen from the root locus procedure), wh ilea closed-loop system with loop gaingreater than unity at some frequency haslimited phase margin (this follows fromthe Nyquist criterion and Lesson 15 be -low). Never forget that the distance fromthe Nyquist plot to the critical point -l+jOis i rrelevant i f the number of encir-clements is w rong, that is, if nom inal sta-b i l i t y d o e s n o t h o l d . T h a t i s w h yindiscriminate use of large control lergains is not a viable control strategy.

12.Always conserve phase. With thecritical point sitting at -l+jO, the Nyquistcriterion shows that any closed-loop sys-tem is never more than 180 degrees frominstability at every crossover frequency.And 180 degrees is not a whole lot, espe-cially at high frequenc ies where the plantis more difficult to accurately model andidentify and thus the loop phase is morelikely to be uncertain. Every degree ofloop phase at crossover frequencies is pre-cious and must be maintained by carefulengineering. This issue especially arisesin digital controlwhere analog-to-digital(sampling) and digital-to-analog (zero-order-hold) devices can cause phase lag.

13. Beware of lightly damped poles.Dont forget that every lightly dampedpair of complex c onjugate poles in theplant (or controller) entails high loopgain near the resonance frequency aswell as a whopping 180 degrees of phaseshift over a small frequency band. Anotch in the control ler can reduce theloop gain at the resonance frequency tohelp avoid an inadvertent chang e in thenumber of Nyquist enc ircleme nts due tomodel ing uncer ta in ty . However , th i stechnique will only work if you actuallyknow the frequency of the resonance.Note that a notch in the co ntroller entailssome perform ance loss in the frequencyband of the notch. An antinotch can beused to increase the loop gain and thus

improve performance in a frequency bandwhere the plant is well known.14. High controller gain has lots of

benefits.If the controller gain is high andthe plant gain is not too small, then theloop gain will be high, which impliessmall sensitivity and thus good tracking orgood disturbance rejection. Also, assum-ing that nominal stability holds, high loopg a i n m ean s l a rg e d i s t an ce f ro m t h eNyquist plot of the loop transfer functionto the critical point -l+jO and thus somemeasure of robust stability. Thus it wouldseem that high gain gives both good per-formance and robust stability. However,there are (at least) three hidden catches tohaving the best of both worlds, namely,rolloff, saturation, and noise.15. Practice safe rolloff.Dont forget(and we cant stress this enough) that highloop gain is useless if the number ofNyquist encirclements is wrong for nomi-

nal stability. Thus, a s already noted, indis-criminate use of high controller gain is notrecommended. If nominal stability doeshold, then there is still the problem of rol-loff, since the gain of a real system, andthus the loop gain, always goes tozero as-ymptotically at high frequency. As th eloop gain drops below unity at the cross-over frequency, gain and phase marginsmust be maintained to provide adequatedistance from the critical point -l+jO inorder to assure robust stability againstmodeling uncertainty. In general, as thefrequency increases you will have to rolloff the loop gain more quickly than theplant uncertainty grows. Also rem emberthat achieving good rolloff isnt as easy asadding poles to the controller to roll offthe loop gain since , as the loop gain rollsoff, the loop phase lags (that is, the loopphase decreases) with 90 degrees of lagincurred at high frequency for each pole.Hence good rolloff requires that the loopgain decrease adequately without accu-mulating excessive loop-phase lag. An-other Bode integral theorem shows thatmost of the loop-phase lag is due to therolloff rate at the crossover frequencywith steeper rolloff causing greater phaselag. Lead-lag compensators are usefulfo r shaping the loop gain and loop phaseto achieve high gain and safe rolloff.16 . Saturation can rob you. As ifthats not all the trouble high loop gain ca nget you into, high loop gain is useless ifthe actuators cannot deliver the specifiedcontrol signal. If the controller asks forfour New tons and the actuator can deliveronly two Newtons, then you have a seri-

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ou s problem. You can think of the Nyquistplot as being squashed dow n due to thissaturation effect. Margins will be re-duced, and, even worse, the Nyquist en-circlemen t count can ch ange and theclosed-loop sy stem will be unstable. It isextremely importan t to remem ber that theinability of the actuators to deliver thespecified control signal is not just the faultof the controller gain being too high, butrather is also due to both the size of theplant gain and the amplitude of the distur-bance signal. If the disturbance signal haslarge amplitude, then the actuator maysaturate and you will have no choice but toreduce the gain of the controller and thussacrifice some of the performance youwant (and which was the reason for usingfeedback control in the fi rs t p lace!).Therefore, saturation can rob you of bothstability and performance. Although wediscussed saturation in terms of linear sta-bility analysis, saturation is actually a spe-cial kind of nonlinearity.

17. High gain amplifies noise. If yourhigh gain controller survives the rolloffdragon and the saturation beast, it may yetsuccumb to the noise monster. While anintegrator (l/s) tends to smooth and at-tenuate noise, a differentiator (s) tendsto amplify noise. Every pole in a transferfunction is like an integrator, whileever y zer o is likea differentiator. As theplant gain rolls off, you may wish to in-clude zeros in your controller in order toincrease the loop gain for better perform-ance while adding phase lead to the looptransfer function in order to increase yourphase margin for robust stability. Zeroswill do both of these things quite nicelyfor you. Howe ver, the resulting high con-troller gain will now amplify noise in themeasurements, and this amplificationmay outw eigh the performan ce and stabil-ity benefits of the high loop gain andloop-phase lead. This amplification ofnoise was to be expected since the zeros,after all, act like differentiators.

18. Time delays can be deadly. Atime delay can be deadly-think of theTrojan horse or the A IDS virus. A time de-lay in the feedback loop corresponds to atransfer function that has unit magnitudeat all frequencies as well as phase lag thatincreases linearly with frequency. Thisphase lag w ill warp the Nyquist plot, espe-cially at high frequencies. Thus, if aclosed-loop system has an unmodeledtime delay in the loop, then the number ofNyquist encirclements by the loop trans-

fer function of the actual system may bedifferent from what you expect, and theclosed-loop system may be unstable. Ifyou know about the delay in advance, thenyou may b e able to counteract the effect ofthe delay through careful loop-shapingdesign. However, although you can (atleast in theory) design stabilizing control-lers in the presence of even large time de-lays, the closed-loop performance willmost likely be poor. Imagine trying tomake decisions using old information.

19. Respect the unstable. As Gun-ter Stein discussed in his classic 1989Bode lecture (watch the video!), control-ling unstable systems can be a dangerousundertaking. Real unstable plants (exceptfor rigid body motion) are alw ays nonlin-ear, and the ability to stabilize them re-quires a minimal amount of actuatorbandwidth and stroke. A disturbance mayperturb the state of a nonlinear system out-side of its domain of attraction so thatthe actuators are unable to move the stateback to the equilibrium and thus recoverfrom the disturbance. In addition, actuatorfailure can lead to disaster and thus cannotbe tolerated.

20. Multi-loop control is nontrivial.Multi-loop control is much more chal-lenging than single-loop control. Every-thing that has been said so far applies tosingle-loop control . Mult iple controlloops are needed whenever a plant hasmultiple sensors or ac tuators. In this casethe interaction of every feedback loopwith every other feedback loop must beaccounted for. While many single-loopconcepts hold in principle in the multi-loop case, the technicalities are muchmore involved. The performance benefitsof multi-loop control, however, are oftenfar more than one would expect from acollection of single-loop controllers.

21. Nonlinearities are always pres-ent. Almo st everything that has been saidso far applies to linearized plant models.Real system s, howev er, have all kinds ofnonlinearities: deadband, backlash, Cou-lomb friction, hysteresis, quantization,saturation, kinematic nonlinearities, andmany others. Thus a controller designedfor a linear plant model to satisfy per-formance specificat ions may performpoorly when applied to the actual plant!The tradeoff here is between mathemati-

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cal tractability of the linearized m odel andgreater validity of a nonlinear m odel.22. Peoples lives may be at stake.Control-system engineers must accountfor all of these issues in designing andanalyzing control systems that work.They must also specify, design, analyze,build, program, test, and maintain the

electromechanical hardware, processors,and software needed to implement controlsystems. Real control systems must be ex-tremely reliable, especially if peopleslives depend on them. These are challeng-ing and rewarding engineering tasks thatwill keep lots of control-system engineersbusy for a long time to come.

Now that you have completed the ba-sic course in classical control. you areready to enter the wonderland of moderncontrol. The linear control yo u havel ea rn ed , wh i ch h a s b een l i m i t ed t osingle-input, single-output systems, cannow be expanded to mul t ip le - inpu t ,multiple-output plants. Multivariablecontrol is often studied with state space

(differential equation) m odels and trans-fer function matrices. State space modelsprovide the means for designing controlsystems that are optimal with respect tocertain design criteria. Optimal controlencompasses LQG and H. control theorywhere explicit formulas are used to syn-thesize multivariable feedback control-lers. Robust control seeks controllersthat provide robust stability and perform-ance for uncertain plants. The computerimplementation of these controllers is thesubject of digital control.While the jumpfrom single-variable control to optimalmultivariable control is a major and im-portant step, nonlinear control takesnonlinearities into account and showshow to design nonlinear controllers to ob-tain improvements over linear control.The next, and most exciting, leap is intothe subject of adaptive control, wherecontrollers learn and adapt in response tochanges in the plant and disturbances.With amazing advances in sensing, ac-tuation, and processing as well as a betterunderstanding of learning and adaptation(for example, using neural computers).

automatic control will become the technol-ogy of the next millennium. You are defi-nitely in the right place at the right time!

tory of automatic control in:0.Mayr, The Origins of Feedback Control, MITPress, Cambridge, 1970.S. Bennett, A History of Control Engineering1800-1930,Peter Peregrinus Ltd.,London, 1979.S. Bennett, A History of Control Engineering1930-1955,Peter Peregrinus Ltd., London, 1993.To get a glimpse of the future and what hath Wattwrought, see the exciting book:K. Kelly, O u t of Control,Addison-Wesley, Read-ing, 1994.

I would like to thank an anonymous re-viewer and Steve Yurkovich for numer-ous helpful suggest ions. I especial lythank all of my students, graduate and un-dergraduate, who surely taught me farmore than I taught them. This guide istheir legacy.

1997 IEEE Fellows(continued r o m page 95)curity assessment of power systems, power system operation and control, and application of robustcontrol techniques to power systems. He is the author and co-author of several papers in his field. Hehas also co-authored a textbook entitled Power System Tranisent Stability Assessment Using the Tran-sient Energy Function Method. During 1993-1994 he was the Program Director of the Power System sProgram at the U.S. National Science F oundation. He is a recipient of the 1985 Presidential Young In-vestigator Aw ard. In 1988, he received the NCR Faculty Award of Excellence. He also received the1989 Iowa State University C ollege of Engineering Young Engineering Faculty Research A ward.

Yuan F. ZhengT he Ohio State UniversityFo r contributions to the development of mechanisms fo r coordination o multiple robots and manipu-lams.Yuan F. Zheng received M.S. and Ph.D. degrees in electrical engineering in 19 80 and 1984, respec-tively, from The Oh io State University in Columbus, OH . His B.S. degree was received in 1970 atTsinghua University in Beijing, China. From 1984 to 1989, he w as with the Department of Electricaland Computer Engineering at Clemson U niversity in Clem son, SC. He has been w ith the Ohio StateUniversity since 1989, where is currently a professor and is chairman of the Department of ElectricalEngineering. His research interests include coordination of multiple robots f or both ma nipulators andwalking machines, coordination of human arm and robot, and wavelet transform for image com pres-sion and surface digitizing. He received the Presidential Young Investigator Aw ard in 1987 for his re-search contributions. He is Vice-president for Tec hnical Affairs of he IEEE Robotics and Automa tionSociety, an associate editor of the IEEE Transactions on Robotics and Automation, on the EditorialBoard of Autonomous Robots, an associate editor of the International Journal of Intelligent Autorna-tion and Soft Computing, and on the editorial board of the International Journal o Intelligent Controland Systems. He has served on the program com mittees of many international conferences and will bethe Program Com mittee Chairman of the 1999 International Conference on Robotics and Automation.

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Su Dents Guide 2 Classical Control