503503CHAPTER13CHAPTER13CONTROLSYSTEMCONTROLSYSTEM PERFORMANCEMODIFICATIONPERFORMANCEMODIFICATIONSuhadaJayasuriyaSuhadaJayasuriya DepartmentofMechanicalEngineeringDepartmentofMechanicalEngineering TexasAhencetheGMisalways.Forhigherordersystems,itispossible tohavemorethanonecrossingoftheunitamplitudecircleandmorethanonecrossingof the180line.InsuchsituationstheGMandPMaresomewhatmisleading.Furthermore, non-minimum-phasesystemsexhibitstabilitycriteriathatareoppositetothosepreviouslydefined.2GainandPhaseMargin505505Figure2Figure2Phasemargin.2.3Gain-PhasePlots2.3Gain-PhasePlotsThegraphicalrepresentationofthefrequencyresponseofthesystemG(s)usingeitherG(j)G(s)ReG(j)jImG(j)sjorj()G(j)G(j)ewhere()G(j)/isknownasthepolarplot.Thecoordinatesofthepolarplotaretherealandimaginaryparts ofG(j),asshowninFig.3.Example1Example1ObtainthepolarplotofthetransferfunctionKG(s)s(s1)ThefrequencyresponseisgivenbyKG(j)j(j1)ThenthemagnitudeandthephasecanbewrittenasKG(j)221and506506ControlSystemPerformanceModificationFigure3Figure3Polarrepresentation.1G(j)tan2IfG(j)andarecomputedfordifferentfrequencies,anaccurateplotcanbeob-G(j)/ tained.Aquickideacan,however,begainedbysimplydoingalimitinganalysisat0, ,andthecornerfrequency1/. WenotethatG(j)G(j)for0 /2 G(j)0G(j)for/K31G(j)G(j)for/4 2 ThepolarplotisshowninFig.4. Again-phaseplotiswherethefrequencyresponseinformationisgivenwithrespectto aCartesianframewithverticalaxisforgainandhorizontalaxisforphase.2.4PolarPlotasaDesignToolintheFrequencyDomain2.4PolarPlotasaDesignToolintheFrequencyDomainAsadesigntoolitsbestuseisindeterminingrelativestabilitywithrespecttoGMandPM. Iftheuncertaintyinthetransferfunctioncanbecharacterizedbyboundsonthegain-phase plot,thenitallowsonetodeterminewhattypeofcompensationneedstobeprovidedfor thesystemtoperforminthepresenceofsuchuncertainties.Asanexampleconsiderthe closed-loopsystemshowninFig.5. Supposethegain-phaseplotisknowntolieintheshadedregionintheG(j)plane, asshowninFig.6. Sincetheshadedregionincludesthe(1,0)point,andthetruegain-phaseplotforthe plantcanlieanywhereintheshadedregion,thesystemcanpotentiallybeunstable.If2GainandPhaseMargin507507Figure4Figure4PolarplotforG(s)K/s(s1).Figure5Figure5Closed-loopsystem.stabilizationinthepresenceofuncertaintyistheprimarydesignissue,thenwewouldrequire aGc(s)sothatitwouldreshapethehigh-frequencypartofthepolarplotwithareduced bandofuncertainty.Thereductioninthebandofuncertaintyisarequiredfeatureofany soundfeedbacksystemdesign.Qualitativelyonewouldexpecttoreshapethepolarplotto somethingthatlookslikewhatisshowninFig.6b.Moreover,knowingtheimportantfre- quencyrangeswillallowonetobemoreconcernedwithrelevantportionsofthegain-phase plotforreshaping.Tofurtherillustratethebasicphilosophyofatypicaldesigninthefre- quencydomain,considertheplanttransferfunctionKG(s)ps(1s)(l0.0125s)inthefeedbackconfigurationshowninFig.7. Itisrequiredthatwhenarampinputisappliedtotheclosed-loopsystem,thesteady- stateerrorofthesystemdoesnotexceed1%oftheamplitudeoftheinputramp.Usingsteady-stateerrorcomputationswefindthattheminimumKshouldbesuchthat11Steady-stateerrorelim0.01sssG(s)Ks0pthatis,K100.ItcanbeeasilyverifiedthatwithGc(s)1thesystemisunstableforK81,implying thatacontrollerGc(s)mustbedesignedtosatisfythesteady-stateperformanceandrelative stabilityrequirements.Puttingitanotherway,thecontrollermustbeabletokeepthezero-508508ControlSystemPerformanceModificationFigure6Figure6Polarplotsforanuncertainsystem.Figure7Figure7Closed-loopsystemwithG(s)K/s(s1) (0.0125s1).frequencygainofsGpGc(s)effectivelyat100whilemaintainingaprescribeddegreeofrel- ativestability.Theprincipleofthedesigninthefrequencydomainisbestillustratedbythe polarplotofGp(s)showninFig.8.Inpractice,theBodediagramispreferredfordesign purposesbecauseitissimplertoconstruct.Thepolarplotisusedmainlyforanalysisand addedinsight. AsshowninFig.8,whenK100,thepolarplotofGp(s)enclosesthe(1,0)point, andtheclosed-loopsystemisunstable.LetusassumethatwewishtorealizethatPM 30.ThismeansthatthepolarplotmustpassthroughpointA(withmagnitude1andphase 150).IfKistheonlyadjustableparametertoachievethisPM,thedesiredvalueK3.4, asshowninFig.8.But,Kcannotbesetto3.4sincetheramperrorconstantwouldonlybe3.4s1,andthesteady-stateerrorrequirementwillnotbesatisfied. Sincethesteady-stateperformanceofthesystemisgovernedbythecharacteristicsof thetransferfunctionatlowfrequency,andthedampingorthetransientbehaviorofthe systemisgovernedbytherelativelyhigh-frequencycharacteristics,asFig.8shows,tosi- multaneouslysatisfythetransientandthesteady-staterequirements,thefrequencylocusof Gp(s)hastobereshapedsothatthehigh-frequencyportionoftheplotfollowstheK3.4 trajectoryandthelow-frequencyportionfollowstheK100trajectory.Thesignificanceof thisreshapingofthefrequencylocusisthatthecompensatedlocusshowninFig.8willbe coincidentwiththehigh-frequencyportionyieldingPM30,whilethezero-frequency gainismaintainedat100tosatisfythesteadystatereq