TestInputSignalsPerformanceofSecond-OrderSystemsThes-PlaneRootLocationandtheTransientResponseTheSteady-StateErrorofFeedbackControlSystemsSystemResponsePerformanceIndicesDesignExamples-Specifythemeasuresofperformance1DefineandmeasuretheperformanceofcontrolsystemsStabilityTransientresponse：theresponsethatdisappearswithtimeSteady-stateresponse:theresponsethatexistsforalongtimefollowinganinputsignalinitiationDesignspecifications:foraspecifiedinputcommandTimeresponseindicesDesiredsteady-stateaccuracyEffectivecompromise2StepinputRampinputParabolicinput3UnitimpulseTheunitimpulsefunctionδ(t)hasthefollowingproperties:

Theimpulseinputisusefulwhenweconsidertheconvolutionintegralfortheoutputy(t)intermsofaninputr(t),whichiswrittenasThegeneralformofstandardtestsignals:45Withaunitstepinput6TheresponseasafunctionofζandtimeisalsoshowninFigure5.5(b)forastepinput.Asζdecreases,theclosed-looprootsapproachtheimaginaryaxis,andtheresponsebecomesincreas-inglyoscillatory.7WithanimpulsefunctioninputR(s)=18TheswiftnessoftheresponseismeasuredbytherisetimeTr

andthepeaktimeTp.Forunder-dampedsystemswithanovershoot,the0-100%risetimeTr

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

isnormallyused.TheactualresponsematchingthestepinputismeasuredbythepercentovershootandsettlingtimeTs.ThepercentovershootisdefinedaswhereMpt

isthepeakvalueofthetimeresponse,andfvisthefinalvalueoftheresponse.Standardperformancemeasures9Thesettlingtime,Ts,isdefinedasthetimerequiredforthesystemtosettlewithinacertainpercentageδoftheinputamplitude.Forthesecond-ordersystemwithclosed-loopdampingconstantζω?,Tsforwhichtheresponseremainswithin2%ofthefinalvalueisHence,wewilldefinethesettlingtimeasfourtimeconstants(thatis,r=l/ζω?)ofthedominantrootsofthecharacteristicequation.10Notice:Thetransientresponseofthesystemmaybedescribedintermsoftwofactors:Theswiftnessofresponse,asrepresentedbytherisetimeandthepeaktime.Theclosenessoftheresponsetothedesiredresponse,asrepresentedbytheovershootandsettlingtime.Asnaturewouldhaveit,thesearecontradictoryrequirements,andacompromisemustbeobtained.11Letdy(t)/dt=0,weobtainωnβt=π.Calculationofthemeasures1213Tr1versusξ14Whenξissetto0.2Whenωn

issetto5Foragivenξ,theresponseisfasterforlargerωn.Theovershootisindependentofωn.Foragivenωn,theresponseisfasterforlowerξ.Theswiftnessoftheresponse,however,willbelimitedbytheovershootthatcanbeaccepted.15Forthegivensystem,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=1.41416Ifa>>ζωnandτ<<1/ζωn,thepoleandzerowillhavelittleeffectonthestepresponse.Usingacomputersimulationfortheactualthird-ordersystem,wefindthatthepercentovershootisequalto38%andthesettlingtimeis1.6seconds.Thus,theeffectofthethirdpoleofT(s)istodampentheovershootandincreasethesettlingtime(hencetherealpolecannotbeneglected).17Ea(s),actuatingsignal,whichisameasureofthesystemerror.E(s)=R(s)-Y(s),theactualsystemerror.WhenH(s)=1,18Niscallthetypeofsystems.N=0,type-0system;N=1,type-1system;N=2,type-2system;Foratype-0systemDefineaspositionerrorconstant.ForN≥1Stepinput19ForthesystemwithN=0,thesteady-stateerrorisinfinite.Forthetype-1system,isdefinedasvelocityerrorconstant.Thusthesteady-stateerrorexists.Forthetype-2system,thesteady-stateerroriszero.Rampinput20ForN=0and1,ess=∞;ForN=2AccelerationerrorconstantAccelerationinput21ForastepinputWhenK2=0WhenK2>0ForarampinputExample5.3Mobilerobotsteeringcontrol

ess=022Complexsystemswithhigh-ordertransferfunctionslower-orderapproximatemodelMethod1:deleteacertaininsignificantpole,inthemeanwhileretainthesteady-stateresponse.Example:23Criteria:SelectcianddiinsuchawaythatL(s)hasafrequencyresponseveryclosetothatofH(s)inwhichthepolesareintheleft-hands-planeandm<n.wherep≤g<n,Kwithoutchange.2425Poles:S=-1,-2,-3→→→-1.029,-1.55526ChooseKandK1sothat:(1)Thepercentovershootoftheoutputtoastepcommandr(t)≤10%;(2)Thesteady-stateerrortoarampcommandisminimized;(3)Theeffectofastepdisturbanceisreduced.27(1)SelectKandK1tomeetP.O.≤10%forR(s)=A/s.SetD(s)=0.Whenζ=0.6,P.O.=9.5%.28(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.29K=100,ess=B/8