Generalized-predictive-control—Part-I.-The-basic-a

Automatica,VoL23,No.2,pp.137148,198700051098/87$3.00+0.0~PrintedinGreatBritain.PergamonJournalsLtd.~1987InternationalFederationofAutomaticControlGeneralizedPredictiveControlAlgorithm*PartI.TheBasicD.W.CLARKEt,C.MOHTADItandP.S.TUFFS~:Anewmemberofthefamilyoflong-rangepredictivecontrollersisshowntobesuitablefortheadaptivecontrolofprocesseswithvaryingparameters,dead-timeandmodel-order.KeyWords--Adaptivecontrol;predictivecontrol;LQcontrol;self-tuningcontrol;nonminimum-phaseplant.Akstraet--Currentself-tuningalgorithmslackrobustnesstopriorchoicesofeitherdead-timeormodelorder.Anovelmethod--generalizedpredictivecontrolorGPC--isdevelopedwhichisshownbysimulationstudiestobesuperiortoacceptedtechniquessuchasgeneralizedminimum-varianceandpole-placement.Thisreceding-horizonmethoddependsonpredictingtheplant'soutputoverseveralstepsbasedonassumptionsaboutfuturecontrolactions.Oneassumption--thatthereisacontrolhorizonbeyondwhichallcontrolincrementsbecomezero--isshowntobebeneficialbothintermsofrobustnessandforprovidingsimplifiedcalculations.ChoosingparticularvaluesoftheoutputandcontrolhorizonsproducesassubsetsofthemethodvarioususefulalgorithmssuchasGMV,EPSAC,Peterka'spredictivecontroller(1984,Automatica,20,39-50)andYdstie'sextended-horizondesign(1984,IFAC9thWorldCongress,Budapest,Hungary).HenceGPCcanbeusedeithertocontrolasimpleplant(e.g.open-loopstable)withlittlepriorknowledgeoramorecomplexplantsuchasnonminimum-phase,open-loopunstableandhavingvariabledead-time.InparticularGPCseemstobeunaffected(unlikepole-placementstrategies)iftheplantmodelisoverparameterized.Furthermore,asoffsetsareeliminatedbytheconsequenceofassumingaCARIMAplantmodel,GPCisacontenderforgeneralself-tuningapplications.Thisisverifiedbyacomparativesimulationstudy.1.INTRODUCTIONALTHOUGHSELF-TUNINGandadaptivecontrolhasmademuchprogressoverthepreviousdecade,bothintermsoftheoreticalunderstandingandpracticalapplications,noonemethodproposedsofarissuitableasageneralpurposealgorithmforthestablecontrolofthemajorityofrealprocesses.Tobeconsideredforthisroleamethodmustbeapplicableto:*Received2March1985;revised7July1985;revised3March1986;revised22September1986.TheoriginalversionofthispaperwasnotpresentedatanyIFACmeeting.ThispaperwasrecommendedforpublicationinrevisedformbyAssociateEditorM.GeversunderthedirectionofEditorP.C.Parks.tDepartmentofEngineeringScience,ParksRoad,OxfordOXI3PJ,U.K.AluminumCompanyofAmerica,AlcoaTechnicalCenter,Pittsburgh,PA15069,U.S.A.(1)anonminimum-phaseplant:mostcontinuous-timetransferfunctionstendtoexhibitdiscrete-timezerosoutsidetheunitcirclewhensampledatafastenoughrate(seeClarke,1984);(2)anopen-loopunstableplantorplantwithbadly-dampedpolessuchasaflexiblespacecraftorrobots;(3)aplantwithvariableorunknowndead-time:somemethods(e.g.minimum-varianceself-tuners,/~str6mandWittenmark,1973)arehighlysensitivetotheassumptionsmadeaboutthedead-timeandapproaches(e.g.KurzandGoedecke,1981)whichattempttoestimatethedead-timeusingoperatingdatatendtobecomplexandlackrobustness;(4)aplantwithunknownorder:pole-placementandLQGself-tunersperformbadlyiftheorderoftheplantisoverestimatedbecauseofpole/zerocancellationsintheidentifiedmodel,unlessspecialprecautionsaretaken.Themethoddescribedinthispaper--General-izedPredictiveControlorGPC--appearstoover-cometheseproblemsinonealgorithm.Itiscapableofstablecontrolofprocesseswithvariableparame-ters,withvariabledead-time,andwithamodelorderwhichchangesinstantaneouslyprovidedthattheinput/outputdataaresufficientlyrichtoallowreasonableplantidentification.Itiseffectivewithaplantwhichissimultaneouslynonminimum-phaseandopen-loopunstableandwhosemodelisoverparameterizedbytheestimationschemewithoutspecialprecautionsbeingtaken.Henceitissuitedtohigh-performanceapplicationssuchasthecontrolofflexiblesystems.Hithertotheprincipalappliedself-tuningmethodshavebeenbasedontheGeneralizedMinimum-Varianceapproach(ClarkeandGawthrop,1975,1979)andthepole-placementalgorithm(Wellsteadetal.,1979;/~str6mand137AUT23/2-A1381).W.CLARKketal.Wittenmark,1980).TheimplicitGMVself-tuner(so-calledbecausethecontrollerparametersaredirectlyestimatedwithoutanintermediatecalcul-ation)isrobustagainstmodelorderassumptionsbutcanperformbadlyiftheplantdead-timevaries.Theexplicitpole-placementmethod,inwhichaDiophantineequationisnumericallysolvedasabridgingstepbetweenplantidentificationandthecontrolcalculation,cancopewithvariabledead-timebutnotwithamodelwithoverspecifiedorder.Itsgoodbehaviourwithvariabledead-timeisduetooverparameterizationofthenumeratordynamicsB(q-1);thismeansthattheorderofthedenominatordynamicshastobechosenwithgreatcaretoavoidsingularitiesintheresolutionoftheDiophantineidentity.TheGPCapproachbeingbasedonanexplicitplantformulationcandealwithvariabledead-time,butasitisapredictivemethoditcanalsocopewithoverparameterization.Thetypeofplantthataself-tunerisexpectedtocontrolvarieswidely.Ontheonehand,manyindustrialprocesseshavesimplemodels:loworderwithrealpolesandprobablywithdead-time.Forsomecriticalloops,however,themodelmightbemorecomplex,suchasopen-loopunstable,underdampedpoles,m