PIDControlTheoryPIDControlThePIDcontrolleristhemostcommonformoffeedback.Itwasanessentialelementofearlygovernorsanditbecamethestandardtoolwhenprocesscontrolemergedinthe1940s.Inprocesscontroltoday,morethan95%ofthecontrolloopsareofPIDtype,mostloopsareactuallyPIcontrol.PIDcontrollersaretodayfoundinallareaswherecontrolisused.Thecontrollerscomeinmanydifferentforms.TheusefulnessofPIDcontrolsliesintheirgeneralapplicabilitytomostcontrolsystems.Inparticular,whenthemathematicalmodeloftheplantisnotknownandthereforeanalyticaldesignmethodscannotbeused,PIDcontrolsprovetobemostuseful.1Introduction1.Proportionalcontroller(PC)2.Integralcontroller(IC)3.Derivativecontroller(DC)Iftheplantissocomplicatedthatitsmathematicalmodelcannotbeeasilyobtained,thenananalyticalorcomputationalapproachtothedesignofaPIDcontrollerisnotpossible.ThenwemustresorttoexperimentalapproachestothetuningofPIDcontrollers.Theserulescan,ofcourse,beappliedtothedesignofsystemswithknownmathematicalmodels2IntegralControlTherewillbenosteadystateerrorwithacontrollerthathasintegralaction.PControlDrawback:processvariableoftendeviatesfromitsreferencevalue3DerivativeControlwhichpredictstheerrorTdtimeunitsahead4Thepastbytheintegraloftheerror(theI-term),thepresent(theP-term)andthefuturebyalinearextrapolationoftheerror(theD-term).Theterm

isalinearpredictionoftheerrorTdtimeunitsinthefuture.Theparametersofthecontrollerarecalled:proportionalgaink,integraltimeTi,andderivativetimeTd.PIDControl5PIDControllerUsingOperationalAmplifiers6EffectsProportionalcontrolActionSimulationofaclosed-loopsystemwithproportionalcontrol.TheprocesstransferfunctionisThecontrollerisgivenbywithTi=infandTd=0.1.Thereisalwaysasteadystateerrorinproportionalcontrol2.

Theerrorwilldecreasewithincreasinggain3.Thetendencytowardsoscillationwillalsoincrease7Effectofproportionalintegralcontrolaction1.ThestrengthofintegralactionincreaseswithdecreasingintegraltimeTi2.Thesteadystateerrordisappearswhenintegralactionisused3.

ThetendencyforoscillationalsoincreaseswithdecreasingTi8EffectofproportionalIntegralDerivativeControl1.

Dampingincreaseswithincreasingderivativetime,butdecreasesagainwhenderivativetime

becomestoolarge.2.Theperiodofoscillationincreaseswhenderivativetimeisincreased.3.

Derivativeactioncanbeinterpretedasprovidingpredictionbylinearextrapolationoverthetime

Td.Usingthisinterpretationitiseasytounderstandthatderivativeactiondoesnothelpifthe

predictiontimeTdistoolarge.DerivativeactionsceasestobeeffectivewhenTdislargerthan

onesixthoftheperiod.9EffectsofCoefficients

BecausemostPIDcontrollersareadjustedon-site,manydifferenttypesoftuningruleshavebeenproposedintheliterature.Usingthesetuningrules,delicateandfinetuningofPIDcontrollerscanbemadeon-site.Also,automatictuningmethodshavebeendevelopedandsomeofthePIDcontrollersmaypossesson-lineautomatictuningcapabilitiesHowever,differentsystemshavedifferentbehavior,differentapplicationshavedifferentrequirements,andrequirementsmayconflictwithoneanotherPIDtuningisadifficultproblem,eventhoughthereareonlythreeparametersandinprincipleissimpletodescribe,becauseitmustsatisfycomplexcriteriawithinthelimitationsofPIDcontrol.Thereareaccordinglyvariousmethodsforlooptuning10Tuning

Theprocessofselectingthecontrollerparameterstomeetgivenperformancespecifications1.Manualtuningmethod2.Ziegler–Nicholstuningmethod3.PIDtuningsoftwaremethods111.ManualTuningMethod

Parametersareadjustedbywatchingsystemresponses.

arechangeduntildesiredorrequiredsystemresponseisobtained.KiandKdaresettozero.Kpisincreaseduntiltheoutputofthelooposcillates,afterobtainingoptimumKpvalue,itshouldbesettoapproximatelyhalfofthatvalue.Kiisincreaseduntilanyoffsetiscorrectedinsufficienttimefortheprocess.Kdisincreased,untiltheloopisacceptablyquicktoreachitsreferenceafteraloaddisturbance.OneManualTuningMethodExample

122.Ziegler–Nicholstuningmethod(introducedbyJohnG.ZieglerandNathanielB.Nicholsinthe1940s)

Basedonexperimentalstepresponsesorbasedonthevalueof

thatresultsinmarginalstabilitywhenonlyproportional

controlactionisusedSuggestasetofvaluesofthatwillgiveastableoperationofthesystem.However,theresultingsystemmayexhibitalargemaximumovershootinthestepresponse,whichisunacceptable.Insuchacaseweneedseriesoffinetuningsuntilanacceptableresultisobtained.Weneedseriesoffinetuningsuntilanacceptableresultisobtained.Infact,theZiegler–Nicholstuningrulesgiveaneducatedguessfortheparametervaluesandprovideastartingpointforfinetuning,ratherthangivingthefinalsettingsforandinasingleshot.13Ziegler–NicholsRulesforTuningPIDControllersFirstMethodUnit-stepresponseofaplantThetransferfunctionDelaytimeTimeconstantThismethodappliesiftheresponsetoastep

inputexhibitsanS-shapedcurve.Suchstep-responsecurvesmaybegeneratedexperimentallyorfromadynamicsimulationoftheplant14Ziegler–NicholsTuningRuleBasedonStepResponseofPlant(FirstMethod)15SecondMethodClosed-loopsystemwithaproportionalcontrollerSet

UsingtheproportionalcontrolactiononlyincreaseKpfrom0toacriticalvalueKcratwhichtheoutputfirstexhibitssustainedoscillations.Thus,thecriticalgainKcrandthecorrespondingperiodPcrareexperimentallydetermined.SustainedoscillationwithperiodPcr.16Ziegler–NicholsTuningRuleBasedonCriticalGainKcrandCriticalPeriodPcr(SecondMethod)17CommentsZiegler–Nicholstuningrules(andothertuningrulespresentedintheliterature)havebeenwidelyusedtotunePIDcontrollersinprocesscontrolsystemswheretheplantdynamicsarenotpreciselyknown.Overmanyyears,suchtuningrulesprovedtobeveryuseful.Ziegler–Nicholstuningrulescan,ofcourse,beappliedtoplants

whosedynamicsareknown.(Iftheplantdynamicsareknown,manyanalyticalandgraphicalapproachestothedesignofPIDcontrollersareavailable,inadditiontoZiegler–Nicholstuningrules.)18ConsiderthecontrolsystemshowninFigureinwhichaPIDcontrollerisusedtocontrolthesystem.ThePIDcontrollerhasthetransferfunctionasfollows,applyaZiegler–NicholstuningruleforthedeterminationofthevaluesofparametersPID-controlledsystemThenobtainaunit-stepresponsecurveandchecktoseeifthedesignedsystemexhibitsapproximately25%maximumovershoot.Ifthemaximumovershootisexcessive(40%ormore),makeafinetuningandreducetheamountofthemaximumovershoottoapproximately25%orless.19BlockdiagramofthesystemwithPIDcontrollerdesignedbyuseoftheZiegler–Nicholstuningrule(secondmethod).Theunit-stepresponseofthesystemnum=[6.32231812.811];den=[1611.32231812.811];step(num,den)gridtitle('Unit-StepResponse')20Wefindthatbykeepingkp=18andbymovingthedoublezeroofthePIDcontrollertos=-0.65thatis,usingthePIDcontrollerUnit-stepresponseofthesystemshowninFigurewithPIDcontrollerhavingparameterskp=18,Ti=3.077andTd=0.769221IftheproportionalgainKpisincreasedto39.42,withoutchangingthelocationofthedoublezero(s=–0.65),Unit-stepresponseofthesystemshowninFigurewithPIDcontrollerhavingParametersandKp=39.42,Ti=3.077andTd=0.769222SecondmethodoftheZiegler–NicholstuningruleFinetuningparametersTheimportantthingtonotehereisthattheZiegler–Nicholstuningrulehasprovidedastartingpointforfinetuning.23Forthecasewherethedoublezeroislocatedats=-1.4235,increasingthevalueofKpincreasesthespeedofresponse,butasfarasthepercentagemaximumovershootisconcerned,varyinggainKphasverylittleeffect.Root-locusdiagram

ofsystemwhenPID

controllerhasdouble

zeroats=–1.4235ThedominantbranchesofrootlociarealongthelinesforaconsiderablerangeofK,varyingthevalueofK(from6to30)willnotchangethedampingratioofthedominantclosed-looppolesverymuch.24Varyingthelocationofthedoublezerohasasignificanteffectonthemaximumovershoot,becausethedampingratioofthedominantclosed-looppolescanbechangedsignificantlyThischangeintheconfigurationmakesitpossibletochangethedampingratioofthedominantclosed-looppoles25DesignofPIDcontrollerswithfrequency-responseapproachUsingafrequency-responseapproach,designaPIDcontrollersuchthatthestaticvelocityerrorconstantis4sec^-1,phasemarginis50°ormore,andgainmarginis10dBormore.Obtaintheunit-stepandunit-rampresponsecurvesofthePIDcontrolledsystemwithMATLAB.LetuschoosethePIDcontrollertobe26ThusNext,weplotaBodediagramofWeneedthephasemarginofatleast50°andgainmarginof10dBormore.FromtheBodediagramwenoticethatthewcfrequencyisapproximatelyw=1.8rad/sec.27Letusassumethewcfrequencyofthecompensatedsystemtobesomewherebetweenw=1andw=10rad/secProducestheBodediagramofnum=[204];den=[10.0000000000110];w=logspace(-2,1,101);bode(num,den,w)28BysimpleMATLABtrials,wefindb=0.25togivethephasemarginofatleast50°andgainmarginof±∞dBOpen-looptransferfunctionFromitweseethatthestaticvelocityerrorconstantis4sec^1,thephasemarginis55°,andthegainmarginis±∞dB29Theclosed-looptransferfunctionis%*****Unit-stepresponse*****num=[5214];den=[15224];t=0:0.01:14;c=step(num,den,t);plot(t,c)gridtitle('Unit-StepResponseofCompensatedSystem')xlabel('t(sec)')ylabel('Outputc(t)')Program8-530%*****Unit-rampresponse*****num1=[5214];den1=[152240];t=0:0.02:20;c=step(num1,den1,t);plot(t,c,'-',t,t,'--')title('Unit-RampResponseofCompensatedSystem')xlabel('t(sec)')ylabel('Unit-RampInputandOutputc(t)')text(10.8,8,'CompensatedSystem')31ModificationsofPIDcontrolschemesThePIcontrollerThePIcontrollerisalagcompensator.Itpossessesazeroats=-1/Tiandapoleats=0Thus,thecharacteristicofthePIcontrollerisinfinitegainatzerofrequency.Thisimprovesthesteady-statecharacteristicsHowever,inclusionofthePIcontrolactioninthesystemincreasesthetypenumberofthecompensatedsystemby1,andthiscausesthecompensatedsystemtobelessstableorevenmakesthesystemunstableThevaluesofmustbechosencarefullytoensureapropertransientresponse.Byproperlydesigni