反馈控制系统的性能

反馈控制系统的性能第1页/共52页DefineandmeasuretheperformanceofcontrolsystemsStabilityTransientresponse：theresponsethatdisappearswithtimeSteady-stateresponse:theresponsethatexistsforalongtimefollowinganinputsignalinitiationDesignspecifications:foraspecifiedinputcommandTimeresponseindicesDesiredsteady-stateaccuracyEffectivecompromise5.1Introduction第2页/共52页StepinputRampinputParabolicinput5.2TestInputSignals第3页/共52页UnitimpulseTheunitimpulsefunctionδ(t)hasthefollowingproperties:

Theimpulseinputisusefulwhenweconsidertheconvolutionintegralfortheoutputy(t)intermsofaninputr(t),whichiswrittenasThegeneralformofstandardtestsignals:5.2TestInputSignals第4页/共52页Iftheinputisaunitimpulsefunction,wehaveTheintegralhasavalueonlyatτ=0;therefore,theimpulseresponseofthesystemG(s).Theimpulseresponsetestsignalcanoftenbeusedforadynamicsystembysubjectingthesystemtoalarge-amplitude,narrowwidthpulseofareaA.ThestandardtestsignalsareofthegeneralformandtheLaplacetransform5.2TestInputSignals第5页/共52页5.2TestInputSignals第6页/共52页Withaunitstepinput5.3PerformanceofaSecond-orderSystem第7页/共52页TheresponseasafunctionofζandtimeisalsoshowninFigure5.5(b)forastepinput.Asζdecreases,theclosed-looprootsapproachtheimaginaryaxis,andtheresponsebecomesincreas-inglyoscillatory.5.3PerformanceofaSecond-orderSystem第8页/共52页WithanimpulsefunctioninputR(s)=15.3PerformanceofaSecond-orderSystem第9页/共52页TheswiftnessoftheresponseismeasuredbytherisetimeTr

andthepeaktimeTp.Forunder-dampedsystemswithanovershoot,the0-100%risetimeTr

isausefulindex.Ifthesystemisover-damped,thenthepeaktimeisnotdefined,andthe10-90%risetimeTr1

isnormallyused.TheactualresponsematchingthestepinputismeasuredbythepercentovershootandsettlingtimeTs.ThepercentovershootisdefinedaswhereMpt

isthepeakvalueofthetimeresponse,andfvisthefinalvalueoftheresponse.Standardperformancemeasures5.3PerformanceofaSecond-orderSystem第10页/共52页Thesettlingtime,Ts,isdefinedasthetimerequiredforthesystemtosettlewithinacertainpercentageδoftheinputamplitude.Forthesecond-ordersystemwithclosed-loopdampingconstantζω„,Tsforwhichtheresponseremainswithin2%ofthefinalvalueisHence,wewilldefinethesettlingtimeasfourtimeconstants(thatis,r=l/ζω„)ofthedominantrootsofthecharacteristicequation.5.3PerformanceofaSecond-orderSystem第11页/共52页Notice:Thetransientresponseofthesystemmaybedescribedintermsoftwofactors:Theswiftnessofresponse,asrepresentedbytherisetimeandthepeaktime.Theclosenessoftheresponsetothedesiredresponse,asrepresentedbytheovershootandsettlingtime.Asnaturewouldhaveit,thesearecontradictoryrequirements,andacompromisemustbeobtained.5.3PerformanceofaSecond-orderSystem第12页/共52页Letdy(t)/dt=0,weobtainωnβt=π.5.3PerformanceofaSecond-orderSystemCalculationofthemeasures第13页/共52页5.3PerformanceofaSecond-orderSystem第14页/共52页Tr1versusξ5.3PerformanceofaSecond-orderSystem第15页/共52页Whenξissetto0.2Whenωn

issetto5Foragivenξ,theresponseisfasterforlargerωn.Theovershootisindependentofωn.Foragivenωn,theresponseisfasterforlowerξ.Theswiftnessoftheresponse,however,willbelimitedbytheovershootthatcanbeaccepted.5.3PerformanceofaSecond-orderSystem第16页/共52页5.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponseWhen|1/γ|≥10|ζωn|,theperformanceindicescanberepresentedbytheonesofthesecondordersystem.Inthecase,thepolesofthesecondordersystemarecalleddominantpolesofthesystem.第17页/共52页Notice:theaboveresultsisonlyforatransferfunctionwithoutfinitezeros.5.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponseSimulationresultsforζ=0.45第18页/共52页5.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponseWhenthetransferfunctionhasazero,第19页/共52页5.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponse第20页/共52页Example5.1ParameterselectionForthegivensystem,selectthegainKandtheparameterpsothatthetime-domainspecificationswillbesatisfied.P.O.≤5%Ts≤4sζ=0.707,P.O.=4.3%Ts=4/ζωn≤4,ζωn≥1Choser12=-1±j,thenP.O.=4.3%Ts=4sζ=0.707,ωn=1/ζ=1.4145.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponse第21页/共52页Example5.2DominantpolesofT(s)Ifa>>ζωnandτ<<1/ζωn,thepoleandzerowillhavelittleeffectonthestepresponse.P.O.=55%accordingtoFig5.13(a)Ts=4/3=1.33s5.4EffectsofaThirdPoleandaZeroontheSecond-orderSystemResponseUsingacomputersimulationfortheactualthird-ordersystem,wefindthatthepercentovershootisequalto38%andthesettlingtimeis1.6seconds.Thus,theeffectofthethirdpoleofT(s)istodampentheovershootandincreasethesettlingtime(hencetherealpolecannotbeneglected).第22页/共52页AccordingtothepercentovershootP.O.AccordingtothenumberofcyclesofthedampedsinusoidduringTs5.5EstimationoftheDampingRatioThefrequencyofthedampedsinusoidaltermforζ<1isThenumberofcyclesin1secondisω/2π.Thetimeconstantfortheexponentialdecayisτ=l/ζωn

inseconds.Thenumberofcyclesofthedampedsinusoidduringonetimeconstantis

Assumingthattheresponsedecaysinnvisibletimeconstants.

Forthesecond-ordersystem,theresponseremainswithin2%ofthesteady-statevalueafterfourtimeconstants(4τ,i.e.,n=4)第23页/共52页Example:examinetheresponseshowninFigureforζ

=EstimationoftheDampingRatioUsey(t)=0asthefirstminimumpointandcount1.4cyclesvisible(untiltheresponsesettleswith2%ofthefinalvalue).Thenweestimate第24页/共52页5.6TheS-planeRootLocationandtheTransientResponse第25页/共52页Ea(s),actuatingsignal,whichisameasureofthesystemerror.E(s)=R(s)-Y(s),theactualsystemerror.WhenH(s)=1,5.7TheSteady-stateErrorofFeedbackControlSystems第26页/共52页Niscallthetypeofsystems.N=0,type-0system;N=1,type-1system;N=2,type-2system;Foratype-0systemDefineaspositionerrorconstant.ForN≥15.7TheSteady-stateErrorofFeedbackControlSystemsStepinput第27页/共52页ForthesystemwithN=0,thesteady-stateerrorisinfinite.Forthetype-1system,isdefinedasvelocityerrorconstant.Thusthesteady-stateerrorexists.Forthetype-2system,thesteady-stateerroriszero.Rampinput5.7TheSteady-stateErrorofFeedbackControlSystems第28页/共52页ForN=0and1,ess=∞;ForN=2Accelerationerrorconstant5.7TheSteady-stateErrorofFeedbackControlSystemsAccelerationinput第29页/共52页ForastepinputWhenK2=0WhenK2>0Forarampinput5.7TheSteady-stateErrorofFeedbackControlSystemsExample5.3Mobilerobotsteeringcontrol

Inthecaseofthesteeringcontrolsystem,wewanttoincreasethegainfactorKK2inordertoincreaseKvandreducethesteady-stateerror.However,anincreaseinKK2resultsinanattendantdecreaseinthesystem'sdampingratioζandthereforeamoreoscillatoryresponsetoastepinput.Thus,wewantacompromisethatprovidesthelargestKvbasedonthesmallestζallowable.ess=0第30页/共52页5.8TheSteady-stateErrorofofNonunityFeedbackSystemsAnonunityfeedbacksystemAspeedcontrolsystem:K1andK2accountfortheconversionofonesetofunitstoanothersetofunits.TheequivalentblockdiagramwithK1=K2.Aunityfeedbacksystem.第31页/共52页IfK1=K2,thesystemistransformedtothatofFig5.23(forthedcgainorsteady-statecalculation).5.8TheSteady-stateErrorofofNonunityFeedbackSystems第32页/共52页DetermineK1andcalculatethesteady-stateerrorforaunitstepinput.Solution:SelectK1=K2=25.8TheSteady-stateErrorofofNonunityFeedbackSystemsExample5.4Steady-stateerroror5.9%ofthemagnitudeofthestepinput.第33页/共52页AssumewecannotinsertagainK1followingR(s).TheactualerrorisE(s)=[1(s)-T(s)]R(s).TrytodetermineanappropriategainKsothatthesteady-stateerrortoastepinputinminimized.Toachievezerosteady-stateerror,werequirethatThusK=4willyieldazerosteady-stateerror.5.8TheSteady-stateErrorofofNonunityFeedbackSystemsExample5.5FeedbackSystemSolution:第34页/共52页Aperformanceindexisaquantitativemeasureoftheperformanceofasystemandischosensothatemphasisisgiventotheimportantsystemspecifications.ISE:IAE:ITAE:ITSE:

5.9PerformanceIndicesThecalculationoftheintegralsquarederror(ISE).第35页/共52页Forastepinput5.9PerformanceIndicesExample5.6PerformancecriteriaTheperformanceindexITAEprovidesthebestselectivityoftheperformanceindices.ThevalueofthedampingratioζselectedonthebasisofITAEis0.7.Forasecond-ordersystem,thisresultsinaswiftresponsetoastepwitha4.6%overshoot.第36页/共52页SelectK3tominimizetheeffectofthedisturbanceD(s).5.9PerformanceIndicesExample5.7Spacetelescopecontrolsystem第37页/共52页WithK1=0.5,K1K2Kp=2.5andaunitstepdisturbance5.9PerformanceIndicesThenthenaturalfrequencyofthevehicleisK3=3.2andζ=0.5.(ISE)K3=4.2andζ=0.665.(IAE)第38页/共52页Complexsystemswithhigh-ordertransferfunctionslower-orderapproximatemodelMethod1:deleteacertaininsignificantpole,inthemeanwhileretainthesteady-stateresponse.Example:5.10TheSimplificationofLinearSystems第39页/共52页-Method2:frequencyresponsemethodsCriteria:SelectcianddiinsuchawaythatL(s)hasafrequencyresponseveryclosetothatofH(s)q=1,2,……

5.10TheSimplificationofLinearSystemsinwhichthepolesareintheleft-hands-planeandm<n.wherep≤g<n,Kwithoutchange.第40页/共52页Example5.9Asimplifiedmodel5.10TheSimplificationofLinearSystems第41页/共52页Poles:S=-1,-2,-3→→→-1.029,-1.5555.10TheSimplificationofLinearSystems第42页/共52页ChooseKandK1sothat:(1)Thepercentovershootoftheoutputtoastepcommandr(t)≤10%;(2)Thesteady-stateerrortoarampcommandisminimized;(3)Theeffectofastepdisturbanceisreduced.5.11DesignExample:HubbleTelescopePointingControl第43页/共52页(1)SelectKandK1tomeetP.O.≤10%forR(s)=A/s.SetD(s)=0.Whenζ=0.6,P.O.=9.5%.5.11DesignExample:HubbleTelescopePointingControl第44页/共52页(2)Examinethesteady-stateerrorforarampinput.(3)Reducetheeffectofastepdisturbance.Thesteady-stateerrorduetoaunitstepdisturbanceisequalto-1/K.ThetransientresponseoftheerrorduetothestepdisturbanceinputcanbereducedbyincreasingK.(4)Insummary,weneedlargeK,largeK/K1andζ=0.6.

SelectK=25,K1=6,K/K1=4.17;SelectK=100,K1=12,K/K1=8.33.Realistically,wemustlimitKsothatthesystem’soperationremainslinear.5.11DesignExample:HubbleTelescopePointingControl第45页/共52页5.11