自动控制理论 I_ch3

1,Chapter3.FeedbackControlSystemCharacteristic,3.1Open-andClose-loopControlSystems3.2TheTransientResponseofControlSystemwithStepInput3.3PerformanceofaSecond-orderSystem3.4Thesteadystateerroroffeedbackcontrolsystem3.5Examples,2,3.1Open-andClose-loopControlSystems,Open-loopsystemsAnopen-loop(direct)systemoperateswithoutfeedbackanddirectlygeneratestheoutputinresponsetoaninputsignal.,TheopenloopsystemishighlysensitivetodisturbancesandtochangesofparametersinG(s).,3,Close-loopsystems,Decreasedsensitivitytovariationsintheparameters.Improvedrejectionofthedisturbance.Improvedmeasurementnoiseattenuation.Improvedreductionofthesteady-stateerror.Easycontrolofthetransientresponseofthesystem.,4,ErrorSignalAnalysis,Consider,MakingGc(s)tohavealargemagnitudeatlowfrequencies(generallyassociatedwiththefrequencyofdisturbances),canrejectdisturbances.MakingGc(s)tohaveasmallmagnitudeathighfrequencies(generallyassociatedwithmeasurementnoise),canattenuatemeasurementnoise.,5,SensitivityofControlSystemstoParametervariations,Theclosed-looptransferfunction:,Systemsensitivityistheradioofthechangeinthesystemtransferfunctiontothechangeofprocesstransferfunction(orparameter)forasmallincrementalchange:,6,Anopen-loopsystemssensitivityis1.Aclosed-loopsystemssensitivityisreducedbytheloopgainGc(s)G(s),andallowsG(s)tobelessaccurate.,7,TheCostofFeedback,Increasethenumberofcomponentsandcomplexityinthesystem.Lossofgain.Possibilityofinstability.,Theadvantagesfaroutweighthedisadvatages.,Causesmorechallengesfordesigners.,8,3.2TheTransientResponseofControlSystemwithStepInput,Thetransientresponseofcontrolsystemwillbeaffectedbyfollowingfactors:(1)Transferfunction(2)InputsignalIftwosystemshavedifferentstructure,asameinputsignalaregiveninthetwosystemsrespectively,theoutputresponsewillbedifferent.Iftwosystemshavesamestructure,butdifferentinputsignalisgiveninthesystemsrespectively,theoutputresponsewillbedifferenttoo.,9,Therearetwopartsofsolutioninthelineardifferentialequation:GeneralSolutionandSpecialSolution.TheGeneralsolutionwillbedeterminedbytherootsofcharacteristicequation(eigenvalue),itstandfortheresponsetransientmotionofsystemoutput.TheSpecialsolutionstandforthesteadyinput.,10,Thetransientresponseistheresponsethatdisappearswithtime.GeneralSolutionThesteady-stateresponseistheresponsethatexitsforalongtimefollowinganinputsignalinitiation.SpecialSolution,11,表3-1、不同极点与运动模态,极点运动模态单实数极点sm重实数极点s一对复数极点s+jwm重复数极点s+jw,Themodeofmotion,12,TestInputSignal,13,t,r(t),0,1(t),(a)UnitStep,r(t),0,t,(b)UnitImpulse,r(t),t,0,t,（c）UnitRamp,r(t),0,t,（d）UnitParabolic,(e)SineFunction,14,TheResponseofFirst-orderSystemwithStepInput,15,TheResponseofSecond-orderSystemwithStepInput,isdampingration,isnaturalfrequency,16,Whenthe,thenistworealroots.,Iftheinputsignalisstep,sotheoutputequationisasfollowing,17,Accordingtopartialfraction,thereare,18,When,therootsofCharacteristicEquationarecomplexandconjugative,theresponseofthesystemisasfollowing:,19,When,thetwoeigenvaluelocatedinimaginaryaxis,20,Transientresponseofasecond-ordersystem,21,22,3.3PerformanceofaSecond-orderSystem,Percentovershoot(P.O.%)andPeaktimeTp,Let,thepeaktimeTpwillbegottenas,23,24,SettlingtimeTs,Becausetheresponsecurveisenvelopedbyapairofenvelopecurves.Whenthesystemgettinginthesteadystate,theoutputwillbeapproachingthefinalvaluey().Assumingthesteadystateerrorhave5%or2%toleranceband(d)intheenvelopecurvesrespectively,theoutputofsecondordersystemwillgetthesteadystatevaluebetweentheenvelopecurves.,25,26,RiseTimeTr,27,总结：1、Tp只与极点虚部有关，极点虚部绝对值越大，Tp就越小，振荡越激烈；2、Ts取决于极点实部，实部绝对值越大，过渡时间越小，稳定得越快；3、超调量（PercentovershootP.O.%）只与阻尼比z有关；4、设计二阶系统时，可先根据对超调量（P.O.%）的要求求出z，再根据Ts等指标确定wn。,28,Example3.1:z=0.6,wn=5rad/s,r(t)=1(t).Calculatetr、tp、ts、P.O.%.,29,Example3.2Afeedbacksystemisshowninthefigure.IfP.O.%=20andTp=1siswanted,determinethevalueofKandKA,andcalculateTrandTs.,30,Example3.2Amechanicalsystemisshowninthefigure.IfF(t)=3N,andP.O.=0.095,Tp=2s,x()=0.01marewanted,determinethevalueofM,K,f.Differentialequationis:TheLaplacetransformyields:,31,32,DCmotor,33,DCmotor,34,3.4Thesteadystateerroroffeedbackcontrolsystem,Steady-stateerrorisdefinedasthedifferencebetweentheinputandoutputofasysteminthelimitastimegoestoinfinity(i.e.whentheresponsehasreachedthesteadystate).Thesteady-stateerrorwilldependonthetypeofinput(step,ramp,etc)aswellasthesystemtype(0,I,orII).,35,Howtocalculatingsteady-stateerrors,36,Thesteady-stateerrorfornon-unityfeedbacksystems,37,TheerrorconstantsofKp,Kv,Ka,TheconstantG(0)isdenotedbyKp,whichiscalledpositionerrorconstant,andisgivenby:,TheconstantsG(0)isdenotedbyKv,whichiscalledvelocityerrorconstant,andisgivenby:,Theconstants2G(0)isdenotedbyKa,whichiscalledaccelerationerrorconstant,andgivenby:,G(s):open-looptransferfunction,38,Thesteadystateerrorwithdifferenttestsignals,UnitStepInput(R(s)=1/s):,UnitRampInput(R(s)=1/s2):,UnitParabolicInput(R(s)=1/s3):,39,Systemtypeandsteady-stateerror,Forn=0,itwillbecalledtype0system.,40,Forn=1,itwillbecalledtype1system.,41,Forn=2,itwillbecalledtype2system.,42,Steadystateerrorwithdifferentinputfordifferentsystemtype,43,Example3.4AunityfeedbacksystemhasopenlooptransferfunctionCalculateasthereferenceinputr(t)=1(t),10t,or3t2.,ItistypeIsystem,soasr(t)=1(t),Asr(t)=3t2,Asr(t)=10t,Note:canalsobecalculatedbyFinalValueTheorem.,44,Thesteady-stateerrorfornon-unityfeedbacksystems,可见型别对于稳态误差的规律同样适用。注：一般H(s)中不会含积分项。,45,Steadystateerrorinthesystemwithdisturbance,46,Itwillbefoundthesteady-stateerrorforastepdisturbanceinputwiththefollowingequation:,WhereR(s)=1/s;Td(s)=1/s.,注：对于干扰Td(s)引起的稳态误差，当积分项位于G(s)中时，并不起减少稳态误差的作用。因此，如果Td(s)为阶跃信号时，要消除余差，必须在Gc(s)中含积分项。,47,Example3.5r(t)=t，f(t)=-1(t),48,积分环节的增加使系统动态特性变差，因此，不能为消除稳态误差而随意增加系统型别。,49,3.5PerformanceIndices,Asystemisconsideredanoptimumcontrolsystemwhenthesystemparametersareadjustedsothattheindexreachesanextremum,commonlyaminimumvalue.,50,Table3.2TheOptimumCoefficientsofT(s)BasedontheITAECriterionforStepInput,Page308,Table5.6,Figure5.30,51,3.6Higher-ordersystemmotion,Whenthecontrolsystemhavemorethan2polesinclosedloop,itwillbetoodifficulttoanalysisistransientresponse.WehavetobymeansoftheDominantpolestoreplaceoriginalcomplexsystem,andthenanalysisitsperformanceindicators.Dominantpolesofsystem:Theclosed-looppolesclosetotheimaginaryaxisdominatethosefartheraway,andinmanycasestheresponsemaybeapproximatedbytheresponseoftheclosed-looppolesclosesttotheimaginaryaxis.Polesandzerosareeitherreal,orformcomplexconjugativepairs.Approximateahigherordersystemwithafirstorsecondorderone.,52,Ifallclosed-looppoleslieinthelefthalfs-plane,therelativemagnitudesoftheresiduesdeterminetherelativeimportanceofthecomponentsintheexpandedformofy(t).,53,Analysis：(1).Ifthereisaclosed-loopzeroclosetoapole,thentheresidueatthispoleissmallandthecoefficientofthetransientresponsetermcorrespondingtothispolebecomessmall.Soapairofcloselylocatedpoleandzerowilleffectivelycanceleachother.,54,(2)、Ifapoleislocatedveryfarfromtheorigin,Thetransientscorrespondingtosucharemotepolearesmallandlast