如何应用Ziegler-Nichols(Z-N)法整定PID调节器的参数

Page1of4TheDesignofPIDControllersusingZieglerNicholsTuningBrianRCopelandMarch20081.IntroductionPIDcontrollersareprobablythemostcommonlyusedcontrollerstructuresinindustry.Theydo,however,presentsomechallengestocontrolandinstrumentationengineersintheaspectoftuningofthegainsrequiredforstabilityandgoodtransientperformance.ThereareseveralprescriptiverulesusedinPIDtuning.AnexampleisthatproposedbyZieglerandNicholsinthe1940'sanddescribedinSection3ofthisnote.Theserulesarebyandlargebasedoncertainassumedmodels.2.PIDControllerStructureThePIDcontrollerencapsulatesthreeofthemostimportantcontrollerstructuresinasinglepackage.TheparallelformofaPIDcontroller(seeFigure1)hastransferfunction:)11()(sTsTKsKsKKsCdipdip++=++=(1)where:Kp:=ProportionalGainKi:=IntegralGainTi:=ResetTime=Kp/KiKd:=DerivativegainTd:=Ratetimeorderivativetime=Kd/KpKpKdssKi+E(s)U(s)Figure1:ParallelFormofthePIDCompensatorTheproportionalterminthecontrollergenerallyhelpsinestablishingsystemstabilityandimprovingthetransientresponsewhilethederivativetermisoftenusedwhenitisnecessarytoimprovetheclosedloopresponsespeedevenfurther.Conceptuallytheeffectofthederivativetermistofeedinformationontherateofchangeofthemeasuredvariableintothecontrolleraction.Themostimportantterminthecontrolleristheintegratortermthatintroducesapoleats=0intheforwardloopoftheprocess.Thismakesthecompensated1openloopsystem(i.e.originalsystemplusPIDcontroller)atype1systematleast;ourknowledgeofsteadystateerrorstellsusthatsuchsystemsarerequiredforperfectsteadystatesetpointtracking.Thisismoreformallystatedinthefollowingtheorem:Figure2Theorem1FortheunityfeedbacksystemofFigure2,perfectsetpointcontrolcanonlybeachievedforthecontroller,C(s),ifandonlyif1.Theopenloopforwardgainhasasteadystategainofinfinityi.e.1ThePIDcontrollerisoftenconsideredasonememberofafamilyofcompensatorsi.e.,devicesthatcanbeaddedtoanopenloopsystemtochange(compensate)forcharacteristicswhichmaketheachievementofthecontrolobjectivedifficultorimpossible.AsisthecaseforthePIDcontroller,compensatorsareusuallycascadedtotheinputofanexistingplantbeforefeedbackisapplied.Thisintroducesanewsetofpolesand/orzerostothepicture.Controldesignproceedsbytreatingthecascadedpairasthenewsystemtobecontrolled.C(s)G(s)R(s)Y(s)E(s)Notethesedefinitions!Page2of4∞=→)()(lim0ssGsC2.Thesystemisclosedloopstable3.NeitherC(s)norG(s)havezerosattheoriginThefinalconditionofthetheoremeliminatesthepossibilityofthelooptransferfunctioncancellingtheeffectoftheintegratorpole.Bothformsof(1)areusedwiththesecondbeingthemostcommon.Theresetandratetimesareofspecialsignificanceinthisregard:•TheresettimeisthetimetakenfortheintegratortermoutputtoequaltheproportionaltermoutputinresponsetoastepchangeininputappliedtoaPIcontroller.PITermEKpEKpETi•TheratetimeisthetimetakenfortheproportionaltermoutputtoequalthederivativetermoutputinresponsetoarampchangeinputappliedtoaPDcontroller.PDTermKpTdETdKpTdEu(t)=EtInaddition,theProportionalgain,Kp,isoftenexpressedasaproportionalband(%):pKPB1=(2)PBisactuallythefractionalerrorchange,relativetotheerrorrange,requiredtoproducea100%(fullrange)changeintheproportionaltermoutput.Mathematically,wehave:⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎬⎫==←ΔΔ=ΔΔΔΔΔΔ==ΔΔ=PBeoPBeoeeooeoKp11.usuallyislasttermThisx1x//ChangeInputChangeOutputmaxmaxmaxmaxmaxmax(3)InpracticePBisexpressedasapercentagesopKPB100%=(4)ThusaPBof5%⇔Kp=20.Wealsonotethat:1.Theindustryjargonisclearlymorepracticalandusefulinthiscase.Theconceptofgain,usuallyquiteusefulinanalysis,isgenerallyhardertograspthanthedegreeofsignalvariation(absoluteorrelative)requiredtoobtainfulloutputswing.2.Inaddition,mostcontrollersoperateonrelative(percentageorperunit)units.Inthisregard,quantitiesarescaledrelativetotheirmaximumrange.Thismakesiteasytotranslatefromoneunitbasistoanother.Forexample,ifourPVisatemperatureintherange0o-100omappedtoacurrentrangeof4-20mA,thenaPV20%translatestoanabsolutetemperatureof20oandanequivalentcurrentsignalreadingof4+16*.2=7.2mA.NB:Ifanalysisisperformedonanysystemtheunitsusedwilldeterminethegainsinthevarioustransferboxes.3.Popularvariationsof(1)areusedtogeneratethefollowingcontrollers:Page3of4ControllerTypeKpKiKdC(s)P(Proportional)≠0zerozeroKpI(Integral)zero≠0zerosKiPI(ProportionalplusIntegral)≠0≠0zerosKKsKsKsKpipip⎟⎟⎠⎞⎜⎜⎝⎛+=+PD(ProportionalplusDerivative)≠0zero≠0)(dpdpdKKsKKsK+=+PID(Proportional+Integral+Derivative)≠0≠0≠0sKKsKKsKsKsKsKdidpdipd⎟⎟⎠⎞⎜⎜⎝⎛++=++22Table1:CommonPIDControllerVariationsTheactualvariationuseddependsonthespecificationstobemet.Purelyderivativeorintegralplusderivativevariationsarealmostneverused.Inallcasesexceptproportionalcontrol,thePIDcompensatorintroducesonepoleandatleastonezero.3.Ziegler-NicholsTuningIn1942ZieglerandNichols,bothemployeesofTaylorInstruments,describedsimplemathematicalprocedures,thefirstandsecondmethodsrespectively,fortuningPIDcontrollers.Theseproceduresarenowacceptedasstandardincontrolsystemspractice.Bothtechniquesmakeaprioriassumptionsonthesystemmodel,butdonotrequirethatthesemodelsbespecificallyknown.Ziegler-Nicholsformulaeforspecifyingthecontrollersarebasedonplantstepresponses.3.1TheFirstMe