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503CHAPTER13CONTROLSYSTEMPERFORMANCEMODIFICATIONSuhadaJayasuriyaDepartmentofMechanicalEngineeringTexasA&MUniversityCollegeStation,Texas1INTRODUCTION5032GAINANDPHASEMARGIN5042.1GainMargin5042.2PhaseMargin5042.3Gain-PhasePlots5052.4PolarPlotasaDesignToolintheFrequencyDomain5063HALLCHART5093.1Constant-MagnitudeCircles5103.2Constant-PhaseCircles5103.3Closed-LoopFrequencyResponseforNonunityFeedbackSystems5113.4Closed-LoopAmplitudeRatio5124NICHOLSCHART5134.1Closed-LoopFrequencyResponsefromThatofOpenLoop5144.2SensitivityAnalysisUsingtheNicholsChart5165ROOTLOCUS5175.1AngleandMagnitudeConditions5185.2Time-DomainDesignUsingtheRootLocus5235.3Time-DomainResponseversuss-DomainPoleLocations5316POLELOCATIONSINTHEz-DOMAIN5326.1StabilityAnalysisofClosed-LoopSystemsinthez-Domain5356.2PerformanceRelatedtoProximityofClosed-LoopPolestotheUnitCircle5366.3RootLocusinthez-Domain5377CONTROLLERDESIGN538REFERENCES540BIBLIOGRAPHY5401INTRODUCTIONChapter12presentsawidevarietyoftoolsforanalyzingclosed-loopcontrolsystems.Thischapterfocusesonthedevelopmentofadditionalanalyticaltoolsforclosed-loopcontrolsystems,toolsaimedatperformancemodicationandimprovement.Eachsuccessivetoolpresentedinthischapterisusefulinitsabilitytopredictsystemperformanceandtopinpointtheappropriatemodicationssothattheclosed-loopsystemwillmeetitsrequiredperform-anceobjectives.Chapter14alsoaddressestheproblemofsystemmodications,buttheretheemphasisistoworkwiththeadjustableparametersofadevicecalledtheservocontrollertoachievethenecessaryresult.ReprintedfromInstrumentationandControl,Wiley,NewYork,1990,bypermissionofthepublisher.Mechanical Engineers Handbook: Instrumentation, Systems, Controls, and MEMS, Volume 2, Third Edition.Edited by Myer KutzCopyright 2006 by John Wiley & Sons, Inc.504ControlSystemPerformanceModicationFigure1Gainmargin.2GAINANDPHASEMARGINTheNyquiststabilitycriterionmaybeconvenientlyusedtodenecertainmeasuresofrel-ativestabilityorrobustness.Wenotethatthe(1,0)pointintheGHplaneplaysacrucialroleindeterminingtheclosed-loopstabilityofasystem.Ifasystemsstabilitystatusisknown,onemightbeinterestedinknowinghowstablethesystemisduetochangesinparameters.Forexample,ifthesystemremainsstabledespitelargechangesinparameters,thenitissaidtopossessahighdegreeofrelativestabilityorrobustness.Gainmarginandphasemarginaretwomeasurestypicallyemployedtocharacterizethisrobustness.TheycharacterizehowclosetheNyquistplotistoencirclingthe(1,0)pointintheGHplane.2.1GainMarginThisisameasureofhowmuchtheloopgaincanberaisedbeforeclosed-loopinstabilityresults.Thebasicdenitionofgainmargin(GM)isapparentfromFig.1.2.2PhaseMarginThephasemargin(PM)isthedifferencebetweenthephaseofGH(j)and180whentheGH(j)crossesthecirclewithunitmagnitude.ApositivePMcorrespondstoacasewheretheNyquistlocusdoesnotencirclethe(1,0)point.ThisisshowninFig.2.Astablesystemcorrespondstogainandphasemarginsthatarepositive.Insomecases,however,thePMandGMnotionsbreakdown.Forrst-andsecond-ordersystems,thephasenevercrossesthe180line;hencetheGMisalways.Forhigherordersystems,itispossibletohavemorethanonecrossingoftheunitamplitudecircleandmorethanonecrossingofthe180line.InsuchsituationstheGMandPMaresomewhatmisleading.Furthermore,non-minimum-phasesystemsexhibitstabilitycriteriathatareoppositetothosepreviouslydened.2GainandPhaseMargin505Figure2Phasemargin.2.3Gain-PhasePlotsThegraphicalrepresentationofthefrequencyresponseofthesystemG(s)usingeitherG(j)G(s)ReG(j)jImG(j)sjorj()G(j)G(j)ewhere()G(j)/isknownasthepolarplot.ThecoordinatesofthepolarplotaretherealandimaginarypartsofG(j),asshowninFig.3.Example1ObtainthepolarplotofthetransferfunctionKG(s)s(s1)ThefrequencyresponseisgivenbyKG(j)j(j1)ThenthemagnitudeandthephasecanbewrittenasKG(j)221and506ControlSystemPerformanceModicationFigure3Polarrepresentation.1G(j)tan2IfG(j)andarecomputedfordifferentfrequencies,anaccurateplotcanbeob-G(j)/tained.Aquickideacan,however,begainedbysimplydoingalimitinganalysisat0,andthecornerfrequency1/.WenotethatG(j)G(j)for0/2G(j)0G(j)for/K31G(j)G(j)for/42ThepolarplotisshowninFig.4.Again-phaseplotiswherethefrequencyresponseinformationisgivenwithrespecttoaCartesianframewithverticalaxisforgainandhorizontalaxisforphase.2.4PolarPlotasaDesignToolintheFrequencyDomainAsadesigntoolitsbestuseisindeterminingrelativestabilitywithrespecttoGMandPM.Iftheuncertaintyinthetransferfunctioncanbecharacterizedbyboundsonthegain-phaseplot,thenitallowsonetodeterminewhattypeofcompensationneedstobeprovidedforthesystemtoperforminthepresenceofsuchuncertainties.Asanexampleconsidertheclosed-loopsystemshowninFig.5.Supposethegain-phaseplotisknowntolieintheshadedregionintheG(j)plane,asshowninFig.6.Sincetheshadedregionincludesthe(1,0)point,andthetruegain-phaseplotfortheplantcanlieanywhereintheshadedregion,thesystemcanpotentiallybeunstable.If2GainandPhaseMargin507Figure4PolarplotforG(s)K/s(s1).Figure5Closed-loopsystem.stabilizationinthepresenceofuncertaintyistheprimarydesignissue,thenwewouldrequireaGc(s)sothatitwouldreshapethehigh-frequencypartofthepolarplotwithareducedbandofuncertainty.Thereductioninthebandofuncertaintyisarequiredfeatureofanysoundfeedbacksystemdesign.QualitativelyonewouldexpecttoreshapethepolarplottosomethingthatlookslikewhatisshowninFig.6b.Moreover,knowingtheimportantfre-quencyrangeswillallowonetobemoreconcernedwithrelevantportionsofthegain-phaseplotforreshaping.Tofurtherillustratethebasicphilosophyofatypicaldesigninthefre-quencydomain,considertheplanttransferfunctionKG(s)ps(1s)(l0.0125s)inthefeedbackcongurationshowninFig.7.Itisrequiredthatwhenarampinputisappliedtotheclosed-loopsystem,thesteady-stateerrorofthesystemdoesnotexceed1%oftheamplitudeoftheinputramp.Usingsteady-stateerrorcomputationswendthattheminimumKshouldbesuchthat11Steady-stateerrorelim0.01sssG(s)Ks0pthatis,K100.ItcanbeeasilyveriedthatwithGc(s)1thesystemisunstableforK81,implyingthatacontrollerGc(s)mustbedesignedtosatisfythesteady-stateperformanceandrelativestabilityrequirements.Puttingitanotherway,thecontrollermustbeabletokeepthezero-508ControlSystemPerformanceModicationFigure6Polarplotsforanuncertainsystem.Figure7Closed-loopsystemwithG(s)K/s(s1)(0.0125s1).frequencygainofsGpGc(s)effectivelyat100whilemaintainingaprescribeddegreeofrel-ativestability.TheprincipleofthedesigninthefrequencydomainisbestillustratedbythepolarplotofGp(s)showninFig.8.Inpractice,theBodediagramispreferredfordesignpurposesbecauseitissimplertoconstruct.Thepolarplotisusedmainlyforanalysisandaddedinsight.AsshowninFig.8,whenK100,thepolarplotofGp(s)enclosesthe(1,0)point,andtheclosed-loopsystemisunstable.LetusassumethatwewishtorealizethatPM30.ThismeansthatthepolarplotmustpassthroughpointA(withmagnitude1andphase150).IfKistheonlyadjustableparametertoachievethisPM,thedesiredvalueK3.4,asshowninFig.8.But,Kcannotbesetto3.4sincetheramperrorconstantwouldonlybe3.4s1,andthesteady-stateerrorrequirementwillnotbesatised.Sincethesteady-stateperformanceofthesystemisgovernedbythecharacteristicsofthetransferfunctionatlowfrequency,andthedampingorthetransientbehaviorofthesystemisgovernedbytherelativelyhigh-frequencycharacteristics,asFig.8shows,tosi-multaneouslysatisfythetransientandthesteady-staterequirements,thefrequencylocusofGp(s)hastobereshapedsothatthehigh-frequencyportionoftheplotfollowstheK3.4trajectoryandthelow-frequencyportionfollowstheK100trajectory.ThesignicanceofthisreshapingofthefrequencylocusisthatthecompensatedlocusshowninFig.8willbecoincidentwiththehigh-frequencyportionyieldingPM30,whilethezero-frequencygainismaintainedat100tosatisfythesteadystaterequirement.WhenweinspectthelociofFig.8,weseethatthereareatleasttwoalternativesinarrivingatthecompensatedlocus:1.StartingfromtheK100locusandreshapingthelocusintheregionnearthegaincrossoverfrequencygwhilekeepingthelow-frequencyregionofGp(s)relativelyunaltered3HallChart509Figure8Polarplotforopen-loopsystemtransferfunctionofFig.7.2.StartingfromtheK耠3.4locusandreshapingthelow-frequencyportionofGp(s)toobtainanerrorconstantof100whilekeepingthelocusnear耠耠耠grelativelyunchangedIntherstapproach,thehigh-frequencyportionofGp(s)ispushedinthecounterclockwise(CCW)direction,whichmeansthatmorephaseisaddedtothesysteminthepositivedi-rectionintheproperfrequencyrange.Thisschemeisbasicallyphase-leadcompensationandcontrollersusedforthispurposeareoftenofthehigh-passltertype.Thesecondapproachapparentlyinvolvestheshiftingofthelow-frequencypartoftheK耠3.4trajectoryintheclockwise(CW)direction,oralternativelyreducingthemagnitudeofGp(s)withK耠100atthehigh-frequencyrange.Thisschemeisoftenreferredtoasphase-lagcompensationsincemorephaselagisintroducedtothesysteminthelow-frequencyrange.Thecontrollersusedforthispurposeareoftenreferredtoaslow-passlters.3HALLCHARTIntypicalfrequencyresponsedesignonlytheopen-looptransferfunctionisplotted.There-foreitisusefultoknowhowtheclosed-loopperformanceisrelatedtotheopenloop.Hallchartsprovideaconvenientwayofcarryingoutafrequencyresponsedesignwithclosed-loopperformancespecications.Oneimportantconsiderationisthemaximumclosed-loopgain.Anotheristheclosed-loopphase.AHallchartprimarilyconsistsofconstantclosed-loopgainlociandconstantclosed-loopphaseloci.Adesignwouldthenproceedbydrawingtheopen-looppolarplotontheHallchart.Foraunitynegative-feedbacksystemasshowninFig.9theclosed-looptransferfunc-tionisC(s)G(s)耠(1)R(s)1耠G(s)InthefollowingdiscussionweassumethatthepolarplotofG(j耠)isknown.510ControlSystemPerformanceModicationFigure9Unitynegative-feedbacksystem.3.1Constant-MagnitudeCirclesThelocionwhichtheclosed-loopmagnitudeC(s)G(s)颠颠M颠constR(s)1颠G(s)arereferredtoasconstant-magnitudeloci.InfacttheselociarecirclesintheG(j颠)-plane.ThiscanbeestablishedbynotingatypicalpointontheG(j颠)plotasX颠jY.Then颠X颠jY颠M颠颠1颠X颠jY颠and22X颠Y2M颠22(1颠X)颠YHence222MM22X颠X颠颠Y颠022M颠1M颠1whichcanbewrittenas222MM2X颠颠Y颠(2)颠颠222M颠1(M颠1)Equation(2)istheequationofacirclewithcenteratX颠颠M2/(M2颠1),Y颠0andwithradius颠M/(M2颠1)颠.Afamilyofconstant-McirclesisshowninFig.10.GivenapointP颠(X1,Y1)onanopen-looppolarplotG(j颠),thecorrespondingclosed-loopmagnitudecanbedeterminedbylocatingtheMcirclepassingthroughthatpoint.GraphicallytheintersectionoftheG(j颠)plotandtheconstant-MlocusgivesthevalueofMatthefrequencydenotedontheG(j颠)curve.Ifitisdesiredtokeepthevalueofthemaximumclosed-loopgainMrlessthanacertainvalue,theG(j颠)curvemustnotintersectthecorrespondingMcircleatanypointandatthesametimemustnotenclosethe(颠1,j0)point.Theconstant-McirclewiththesmallestradiusthatistangenttotheG(j颠)curvegivesthevalueofMr,andtheresonantfrequency颠risreadoffatthetangentpointontheG(j颠)curve.3.2Constant-PhaseCirclesThelociofconstantphaseoftheclosed-loopsystemcanalsobedeterminedintheG(j颠)-planebyamethodsimilartothatusedforconstant-Mloci.WithreferencetoEq.(1)thephaseoftheclosed-loopsystemcorrespondingtothepointP颠X颠jYiswrittenas3HallChart511Figure10Familyofconstant-Mcircles.YY11筠筠tan筠tan(3)X1筠XTakingthetangentonbothsidesofEq.(3)andrearrangingyields2221111X筠筠Y(4)22N42NwhereN筠tan筠.Equation(4)representsafamilyofcircleswithcenterat(筠1/2,1/2N)andwithradiusTheconstant-phaselociareshowninFig.11.2筠1/4筠1/(2N).Theuseofconstant-magnitudeandconstant-phasecirclesenablesonetondtheentireclosed-loopfrequencyresponsefromtheopen-loopfrequencyresponseG(j筠)withoutcal-culatingthemagnitudeandphaseoftheclosed-looptransferfunctionateachfrequency.TheintersectionsoftheG(j筠)locusandtheMcirclesandNcirclesgivethevaluesofMandNatfrequencypointsontheG(j筠)locus.3.3Closed-LoopFrequencyResponseforNonunityFeedbackSystemsTheconstant-Mandconstant-Ncirclesarelimitedtoclosed-loopsystemswithunitynegativefeedback,whosetransferfunctionisgivenbyEq.(1).Whenasystemhasnonunityfeedback,theclosed-looptransferfunctionisC(s)G(s)筠(5)R(s)1筠G(s)H(s)512ControlSystemPerformanceModicationFigure11Familyofconstant-Ncircles.andconstant-Mlociderivedearliercannotbedirectlyapplied.However,withaslightmod-icationconstant-Mandconstant-Nlocicanstillbeappliedtosystemswithnonunityfeed-back.WemodifyEq.(5)asC(s)1G(s)H(s)艠R(s)H(s)1艠G(s)H(s)ThemagnitudeandphaseangleofG1(s)/1艠G1(s),whereG1(s)艠G(s)H(s),maybeobtainedeasilybyplottingtheG1(j艠)locusandreadingthevaluesofMandNatvariousfrequencypoints.Theclosed-loopfrequencyresponseC(j艠)/R(j艠)maythenbeobtainedbymultiplyingG1(j艠)/1艠G1(j艠)by1/H(j艠).3.4Closed-LoopAmplitudeRatioInobtainingsuitableperformance,theadjustmentofgainisusuallytherstconsideration.Theadjustmentofgainisusuallybasedonthemaximumclosed-loopgainortheresonantpeak.ThatisthegainKwhichmustbechosensothatovertheentirefrequencyrangetheclosed-loopamplituderatioMrisnotexceeded.ConsiderrstisolatingthecirclecorrespondingtoMrasshowninFig.12.ThenatangentlinetotheMrcircleisdrawnfromtheorigin,whichmakesanangle艠withtherealline.4NicholsChart513Figure12Mcircle.IfMr1,then2M/(M1)1rrsin22M/(M1)MrrrItcanbeshownthatthelinedrawnfromPperpendiculartothenegativerealaxisintersectsthisaxisatthe(1,0)point.Thesetwofacts,namelysin1/MrandthatthenormalfromPpassesthrough(1,0),canbeusedtodeterminetheappropriategainK.Example2ConsiderthesystemshowninFig.13a:DetermineKsothatMr1.4.FirstsketchthepolarplotofG(j)1Kj(1j)asshowninFig.13b.ThevalueofcorrespondingtoMr1.4isobtainedfrom1111sinsin45.6M1.4rThenextstepistodrawalineOPthatmakesanangle45.6withthenegativerealaxis.ThendrawthecirclethatistangenttoboththeG(j)/KlocusandthelineOP.TheperpendicularlinedrawnfromthepointPintersectsthenegativerealaxisat(0.63,0).ThenthegainKofthesystemisdeterminedasfollows:1K1.580.634NICHOLSCHARTBoththegainandphaseplotsaregenerallyrequiredtoanalyzetheperformanceofaclosed-loopsystem.Amajordisadvantageinworkingwithpolarplotsisthatthecurvenolonger514ControlSystemPerformanceModicationFigure13(a)Closed-loopsystem;(b)determinationofthegainKusinganMcircle.retainsitsoriginalshapewhenasimplemodicationsuchasthechangeoftheloopgainismadetothesystem.Indesign,however,notonlytheloopgainmustbealteredbutoftenseriesorfeedbackcontrollersaretobeaddedtotheoriginalsystemthatrequirethecompletereconstructionoftheresultingopen-looptransferfunction.FordesignpurposesitismoreconvenienttoworkwithBodediagramsorgain-versus-phaseplots.ThelatterrepresentationwithcorrespondingMandNcirclessuperimposedonitisreferredtoastheNicholschart.Inagain-versus-phaseplottheentireG(j)isshiftedupordownverticallywhenthegainisaltered.ANicholschartisshowninFig.14.Thischartissymmetricaboutthe180axis.TheMandNlocirepeatforevery360,andthereissymmetryatevery180interval.TheMlociarecenteredaboutthecriticalpoint(0dB,l80).4.1Closed-LoopFrequencyResponsefromThatofOpenLoopItisquiteeasytodeterminetheclosed-loopfrequencyresponsefromthatoftheopenloopbyusingtheNicholschart.Iftheopen-loopfrequencyresponsecurveissuperimposedontheNicholschart,theintersectionsoftheopen-loopfrequencyresponsecurveG(j)andtheMandNlocigivethemagnitudeMandphaseangleoftheclosed-loopfrequencyresponseateachfrequencypoint.IftheG(j)locusdoesnotintersecttheMMrlocusbutistangenttoit,thentheresonantpeakvalueoftheclosed-loopfrequencyresponseisgivenbyMr.Theresonantfrequencyisgivenbythefrequencyatthepointoftangency.Asanexampleconsidertheunitynegative-feedbacksystemwiththefollowingopen-looptransferfunction:4NicholsChart515Figure14Nicholschart.KG(s)K1s(s1)(0.5s1)Tondtheclosed-loopfrequencyresponsebyuseoftheNicholschart,theG(j)locusisrstconstructed.(ItiseasytorstconstructtheBodediagramandthentransfervaluestotheNicholschart.)Theclosed-loopfrequencyresponsecurves(gainandphase)maybeconstructedbyreadingthemagnitudeandphaseanglesatvariousfrequencypointsontheG(j)locusfromtheMandNlociasshowninFig.15.SincetheG(j)locusistangenttotheM5-dBlocus,thepeakvalueoftheclosed-loopfrequencyresponseisMr5dB,andtheresonantfrequencyis0.8rad/s.Thebandwidthoftheclosed-loopsystemcaneasilybefoundfromtheG(j)locusintheNicholschart.Thefrequencyattheintersectionof