自动控制原理课件 root locus approach to controlsystem design学习资料

Root-locusapproachtocontrol-systemsdesign1Inbuildingacontrolsystem,weknowthatpropermodificationoftheplantdynamicsmaybeasimplewaytomeettheperformancespecifications.This,however,maynotbepossibleinmanypracticalsituationsbecausetheplantmaybefixedandnotmodifiable.Inpractice,theroot-locusplotofasystemmayindicatethatthedesiredperformancecannotbeachievedjustbytheadjustmentofgain(orsomeotheradjustableparameter).Infact,insomecases,thesystemmaynotbestableforallvaluesofgain(orotheradjustableparameter).Thenitisnecessarytoreshapetherootlocitomeettheperformancespecifications.Whyweaddcompensatortoaplant?2IntroductionTheperformanceofafeedbacksystemisofprimaryimportanceOurobjectforsuitablecontrolsystemstableanacceptanceresponsetoinputcommandLesssensitivitytosystemparameterchangesMinimumsteady-stateerrorforinputcommandsReducetheeffectofundesirabledisturbancesThesystemthatprovidesanoptimumperformancewithoutanynecessaryadjustmentsisrareindeed.Compromiseamongthespecificationsisneeded.Itisnotsufficienttoadjustasystemparametertoobtainthedesiredperformance.Thedesignofacontrolsystemisconcernedwitharrangementofthesystemstructureandselectionofsuitablecomponentsandparameters3Acompensatorisanadditionalcomponentorcircuitthatisinsertedintoacontrolsystemtocompensateforadeficientperformance

IntroductioncasacadecompensationFeedbackcompensationOutputcompensationInputcompensation4ApproachestoSystemDesignTheperformanceofacontrolsystem:Thetime-domainperformancemeasures:thepeaktimeTP,maximumovershoot,settlingtimeTsandmaximumallowablesteady-stateerroress

Performancewithtimedomaincanbedefinedintermsofthe

desired

locationofthepoles

oftheclosed-looptransferfunctionT(s)Thelocationofthes-planepolescanbespecified.Thuswecanusethe

rootlocus

methodanddetermineasuitablecompensatornetworktransferfunctionsothatresultantrootlocusyieldsthedesiredclosed-looprootconfiguration52Thefrequencydomainmeasures:thepeakofclosed-loopfrequencyresponseMPω,theresonantfrequencyωr,bandwidthωB,phasemarginandmagnitudemargin.Performancesonfrequencydomainarerelatedwithfrequencyresponse.Thedesignisdevelopedintermsofthefrequencyresponseasportrayedonthepolarplane,theBodediagram,ortheNicholschart.WeusuallyprefertoapproachthefrequencyresponsemethodsbyutilizingtheBodediagram.

ApproachestoSystemDesign6Controlsystemdesignbyfrequency-responseapproachTherearebasicallytwoapproachesinthefrequency-domaindesign.OneisthepolarplotapproachandtheotheristheBodediagramapproach.Whenacompensatorisadded,thepolarplotdoesnotretaintheoriginalshape,and,therefore,weneedtodrawanewpolarplot,whichwilltaketimeandisthusinconvenient.Ontheotherhand,aBodediagramofthecompensatorcanbesimplyaddedtotheoriginalBodediagram,andthusplottingthecompleteBodediagramisasimplematter.Also,iftheopen-loopgainisvaried,themagnitudecurveisshiftedupordownwithoutchangingtheslopeofthecurve,andthephasecurveremainsthesame.Fordesignpurposes,therefore,itisbesttoworkwiththeBodediagram.7BasicCharacteristicsofLead,Lag,andLag–LeadCompensationLeadcompensationessentiallyyieldsanappreciableimprovementintransientresponseandasmallchangeinsteady-stateaccuracy.Itmayaccentuatehigh-frequencynoiseeffectsLagcompensation,ontheotherhand,yieldsanappreciableimprovementinsteady-stateaccuracyattheexpenseofincreasingthetransient-responsetime.Lagcompensationwillsuppresstheeffectsofhigh-frequencynoisesignals.Theuseofaleadorlagcompensatorraisestheorderofthesystemby1Lag–leadcompensationcombinesthecharacteristicsofbothleadcompensationandlagcompensation.Theuseofalag–leadcompensatorraisestheorderofthesystemby2.whichmeansthatthesystembecomesmorecomplexanditismoredifficulttocontrolthetransient-responsebehavior8DesignbyRoot-LocusMethodThedesignbytheroot-locusmethodisbasedonreshapingtherootlocusofthesystembyaddingpolesandzerostothesystem’sopen-looptransferfunctionandforcingtherootlocitopassthroughdesiredclosed-looppolesinthesplane.Thecharacteristicoftheroot-locusdesignisitsbeingbasedontheassumptionthattheclosed-loopsystemhasapairofdominantclosed-looppoles.Oncetheeffectsontherootlocusoftheadditionofpolesand/orzerosarefullyunderstood,wecanreadilydeterminethelocationsofthepole(s)andzero(s)ofthecompensatorthatwillreshapetherootlocusasdesired.Therootlociofthesystemarereshapedthroughtheuseofacompensatorsothatapairofdominantclosed-looppolescanbeplacedatthedesiredlocation.9EffectsoftheAdditionofPolesTheadditionofapoletotheopen-looptransferfunctionhastheeffectofpullingtherootlocustotheright,tendingtolowerthesystem’srelativestabilityandtoslowdownthesettlingoftheresponse.(a)Root-locusplotofasingle-polesystem(b)root-locusplotofatwo-polesystem(c)root-locusplotofathree-polesystem10EffectsoftheAdditionofZerosTheadditionofazerototheopen-looptransferfunctionhastheeffectofpullingtherootlocustotheleft,tendingtomakethesystemmorestableandtospeedupthesettlingoftheresponse(a)Root-locusplotofathree-polesystem(b),(c),and(d)root-locusplotsshowingeffectsofadditionofazerotothethree-polesystem11LeadorLagNetworksUsingOperationalAmplifiersOperational-amplifiercircuitThetransferfunctionforthiscircuit1213LeadCompensationTechniquesBasedontheRoot-LocusApproach14156.Onceacompensatorhasbeendesigned,checktoseewhetherallperformancespecificationshavebeenmet.Ifthecompensatedsystemdoesnotmeettheperformancespecifications,thenrepeatthedesignprocedurebyadjustingthecompensatorpoleandzerountilallsuchspecificationsaremet.Ifalargestaticerrorconstantisrequired,cascadealagnetworkoraltertheleadcompensatortoalag–leadcompensator.7.Iftheselecteddominantclosed-looppolesarenotreallydominant,oriftheselecteddominantclosed-looppolesdonotyieldthedesiredresult,itwillbenecessarytomodifythelocationofthepairofsuchselecteddominantclosed-looppoles.16ConsiderthepositioncontrolsystemshowninFigure.Theclosed-looptransferfunctionforthesystemis17(1)(2)18(3)(4)(5)19(6)(7)20(8)(9)212223Unit-stepresponsecurvesofdesignedsystemsandoriginaluncompensatedsystemnum1=[12.28723.876];den1=[15.64616.93323.876];num=[10];den=[1110];t=0:0.05:5;c1=step(num1,den1,t);c=step(num,den,t);plot(t,c1,'-',t,c,'x')grid24Unit-rampresponsecurvesofdesignedsystemsnum1=[12.28723.876];den1=[15.64616.93323.8760];t=0:0.05:5;c1=step(num1,den1,t);plot(t,c1,'-',t,t,'-')grid25Anotherdeignmethodofthephase-lead2627LeadcompensatorusingtherootlocusThecompensatorThegoalofthedesignSettlingtimePercentovershootfora stepinput28Thereforethedampingratioζ>0.32andTs=4.Wecanobtainζ=0.32andωn=3.125.Ifchoosetherootsarehence,ζ=0.45.Designmethod:desiredrootsplacethezeroofthecompensatordirectlybelowthedesiredlocationats=-z=-1calculatethedesiredphaseangledrawnalineatanangleθp

=38°intersectingthedesiredrootlocationandtherealaxis,thepointofintersectionwiththerealaxisiss=-p=-3.6.Therefore,thecompensatoris

29Ifwesetthezeroz=-1,thenpolep=3.6.ThetransferfunctionofthecompensatorThegainK1is8.1whens=-1+j2and|GH(s)Gc(s)|=1.K1=(2.23)^2*3.25/2=8.1Finally,theerrorconstantsofthissystemareevaluated.zerosteady-stateerrorforastepandrampinputsignaltheaccelerationconstantisKa=2.25.Ka=8.1/3.6=2.25Whenwecomparethecompensationnetworkevaluatedbythes-planemethodwiththenetworkobtainedbyusingtheBodediagramapproach,wefindthatthemagnitudesofthepolesandzerosaredifferent.Acomputersimulationofthesystemresultedinanovershootof46%andasettlingtime(towithin2%ofthefinalvalue)of3.8secondsforastepinput.Thesevaluescomparemoderatelywellwiththespecifiedvaluesof35%and4seconds,andtheyjustifytheuseofthedominantrootspecifications.30ElectronicLagCompensatorUsingOperationalAmplifiers31LagCompensationTechniquesBasedontheRoot-LocusApproachCompensationinthiscaseessentiallyconsistsofincreasingtheopen-loopgainwithoutappreciablychangingthetransient-responsecharacteristics.

Therootlocusintheneighborhoodofthedominantclosed-looppolesshouldnotbechangedappreciably,buttheopen-loopgainshouldbeincreasedasmuchasneededToavoidanappreciablechangeintherootloci,theanglecontributionofthelagnetworkshouldbelimitedtoasmallamount,saylessthan5°Thiscanbeaccomplishedifalagcompensatorisputincascadewiththegivenfeed-forwardtransferfunction.Toassurethis,weplace

thepoleandzeroofthelagnetworkrelativelyclosetogetherandneartheoriginofthesplane.Thentheclosed-looppolesofthecompensatedsystemwillbeshiftedonlyslightlyfromtheiroriginallocations.Hence,thetransient-responsecharacteristicswillbechangedonlyslightly.32Ifgainofthelagcompensatorissetequalto1,thealterationinthetransient-responsecharacteristicswillbeverysmall,despitethefactthattheoverallgainof

theopen-looptransferfunctionisincreasedbyafactorofbeta,wherebeta>1.Ifthepoleandzeroareplacedveryclosetotheorigin,thenthevalueofbetacanbemadelarge.3334Theprocedurefordesigninglagcompensators355.Determinethepoleandzeroofthelagcompensatorthatproducethenecessaryincreaseintheparticularstaticerrorconstantwithoutappreciablyalteringtheoriginalrootloci6.Drawanewroot-locusplotforthecompensatedsystem.Locatethedesireddominantclosed-looppolesontherootlocus7.Adjustgainofthecompensatorfromthemagnitudeconditionsothatthedominantclosed-looppoleslieatthedesiredlocation36abProblem37GoalUncompensatedsystem38(1)(2)39Sincethepoleandzeroofthelagcompensatorareplacedclosetogetherandarelocatedveryneartheorigin,thereeffectontheshapeoftheoriginalrootlocihasbeensmall.However,thevalueofthestaticvelocityerrorconstantofthecompensatedsystemisnearly10timesgreaterthanthatoftheuncompensatedsystems.numc=[10.05];denc=[13.0052.0150.010];num=[1.06];den=[1320];rlocus(numc,denc)holdonrlocus(num,den)40(3)(4)41(5)(6)42(7)Remark43numc=[1.02350.0512];denc=[13.0052.0151.03350.0512];num=[1.06];den=[1321.06];t=0:0.1:40;c1=step(numc,denc,t);c2=step(num,den,t);plot(t,c1,'-',t,c2,'.')grid44numc=[1.02350.0512];denc=[13.0052.0151.03350.05120];num=[1.06];den=[1321.060];t=0:0.1:50;c1=step(numc,denc,t);c2=step(num,den,t);plot(t,c1,'-',t,c2,'.',t,t,'--')grid45Simplifiedstepsfordesignofphase-lagnetworkinthes-plane46Example10.6Designofaphase-lagcompensatorThegoalofdesignare:ζ=0.45andKv=20Theuncompensatedrootlocusisaverticallineats=-1andresultsinarootontheζ=0.45lineats=-1±j2.Measuringthegainatthisroot,wehaveK=5.TheuncompensatedsystemKv=2.5.Set|z/p|=20/2.5=8.Thedifferenceoftheanglesfrompandzatthedesiredrootisapproximately1°;s=-1±j2isstillthelocationofthedominantroots.Theopen-loopTF:47Thesummaryforphase-leadandphase-lagcompensatorBothphase-leadandphase-lagcompensatornetworkcanprovidethebetterspecificationtoasystembyalternativemethodsLeadcompensationbasicallyspeedsuptheresponseandincreasesthestabilityofthesesystemLagcompensationimprovesthesteady-stateaccuracyofthesystem,butreducesthespeedoftheresponseIfimprovementsinbothtransientresponseandsteady-stateresponsearedesired,the