第29卷　第4期2003年7月自　动　化　学　报

ACTAAUTOMATICASINICAVol129,No14July,2003　

MultipleModelPredictiveControlforMIMOSystems1)

LINing　LIShao2Yuan1　XIYu2Geng(InstituteofAutomation,ShanghaiJiaotongUniversity,Shanghai　200030)1(E2mail:syli@sjtu.edu.cn)

Abstract　Amulti2model2basedpredictivecontrol(MMPC)strategydealingwithnonlinearmodel2basedpredictivecontrol(NMPC)forMIMOsystemsisdevelopedinthispaper.Firstlyamulti2modeli2dentificationmethodisgiven.Usingfuzzysatisfactoryclusteringalgorithmpresentedinthispaper,thecomplexnonlinearsystemcanbequicklydividedintomultiplefuzzyparts.Aglobalmodelcanbeob2tainedbysometransformationoftheobtainedmultiplelinearmodels.AnMMPCalgorithmisthereforedesignedfortheglobalMIMOsystemswithsystemperformanceanalysis.TakingapHneutralizationcontrolsystemassimulationexample,thesimulationresultsverifytheeffectivenessofMMPConcom2plexnonlinearsystems.

Keywords　MIMOsystems,multi2model,model2basedpredictivecontrol(MPC),fuzzysatisfactoryclustering,pHneutralizationprocess

1)SupportedbyNationalNaturalScienceFoundationofP.R.China(69934020and60074004)ReceivedMay8,2002;inrevisedformAugust28,2002收稿日期　2002205208;收修改稿日期　20022082281　Introduction

RecentlyModelPredictiveControl(MPC)hasbecomeanattractiveresearchfieldinauto2

maticcontrolforitsadvantagesoverconventionaltechniquesandsuccessfulapplicationsinin2

dustry.MPCalgorithmswereoriginallydevelopedforlinearprocesses,butthebasicideacan

betransferredtononlinearsystems[1,2].Unfortunately,twomajorissueslimititspossibleapplica2

tiontononlinearsystems.Thefirstistheirassumptionofamodelthathastobequiteaccurate;

however,themodelingofindustrialsystemsoftenpresentsproblemsofnonlinearity,strong

coupling,uncertainty,andevenwideoperatingrange,asatisfiedmodelisalwaysdifficultto

obtain.Thesecondisthatanonlinearnon2convexoptimizationproblemmustbesolvedforeach

samplingperiodwithalgorithmswhichareusuallytooslowforreal2timecontrolduetoalarge

amountofcomputation.Thefactshaveforcedthecontrolcommunitytostudysimplificationsof

thisgeneralapproachinordertoremovethesedrawbacks.Usually,thenonlinearmodelislin2

earizediterativelyineachcontrolintervaltosolvetheaboveproblems.Thispaperwillpresenta

newsolutionbasedonmulti2modelapproach.

Multi2modelapproachesareverypropertocontrolindustrialprocesses,especiallychemical

processesfortheirinherentlynonlinearityandlargesetpointchangesorloaddisturbances.

Basedondivide2and2conquerstrategy,multi2modelapproachesdeveloplocallinearmodelsor

controllerscorrespondingtotypicaloperatingregimes,thenfittheglobalsystemthroughcer2

tainintegrationoflocalmodelsorcontrollers.Actually,applyingmulti2modelcontroltononlin2

earortime2varyingsystemshasalonghistory.However,multi2modelapproachforMIMOsys2

temsseldomappearsinliteratures.

Inthispaper,aMulti2ModelPredictiveControl(MMPC)ispresentedtodealwithNMPC

problemofMIMOsystems.Firstly,amulti2modelmodelingmethodusingT2Sstructuremodel

isintroduced.Usingfuzzysatisfactoryclusteringalgorithmgiveninthispaper,acomplexnon2

linearsystemcanbequicklydividedintolocalsystems,andtheglobalsystemcanbedescribed

byintegrationofthelocallinearmodels.Secondly,mergingtheobtainedmultiplelinearmodels

withMIMOGeneralizedPredictiveControl(GPC),anovelMMPCalgorithmisdesignedfor

theglobalsystem.Asamajorbenefitofthemulti2modelstrategy,linearpredictivecontrollerscanbeused.Then,thepapertriestouseMMPCtoregulateatypicalcomplexnonlinearpro2

cess:anMIMOpHneutralizationsystem.

2　Multi2modelidentificationbasedfuzzysatisfactoryclustering

Itiswellknownthatclusteringalgorithmsaimtodivideadatasetintoseveralsub2sets.

Therefore,theycanbenaturallyusedforsystemdivisioninmulti2modelapproach.Itis

thoughtthatclusternumbercintheclusteringalgorithmcorrespondstothenumberoflocal

modelsinthemulti2modelapproach.Therefore,todivideaglobalsysteminasatisfactoryway

equalstolookforaproperclusternumber.However,formanykindsofclusteringmethods,in2

cludingGKalgorithm[3],clusteringnumbercisalwaysneededinadvance,whichhampers

clusteringalgorithmstobeused.Thispaperaimstosolvecomplexsystemcontrolproblem.It

isknownthat,forcontrolproblems,themodelingprecisionisthecontraryofmodelnumbers.

Togiveattentiontothem,herewepresentasatisfactoryclusteringalgorithmbasedonGK.

Simplyspeaking,lettheclusteringmethodstartwithc=2(c∈[2,c3]),wherec3isthe

satisfactoryclusternumber.Thendeterminewhetheranewclustercentershouldbeincreased

ornot.Iftheclusteringresultisnotsatisfiedyet,fromthegivendataset,findoutasample

mostdifferentfromtheexistingclustercentersv1～vcasnewcentervc+1.Startwithv1～

vc+1asinitialclustercenters,andcomputethenewNOT2randompartitionmatrixU.Then

repeatGKalgorithmtodividethesetintoc+1parts.Dotheabovestepsagainuntiltheresultis

satisfactory.

ConsideraMISOsystem,whosedatasetZiscomposedofsysteminput2outputdata.De2

fineadatapairaszj=[φj,yj]T∈Rd+1,j=1,…,N,whereφjiscalledasregressionvector

orgeneralizedinputvector,yjissystemoutput.SupposeZisdividedintocclusters.Thatis,

thesystemcanbecomposedofclocalmodels.Theglobalmodelisconstructedbyfuzzyinter2

polationoftheselocalmodels.Multi2modelidentificationbasedonsatisfactoryclustering(Algo2

rithmⅠ)canbedescribedasfollows:

Step1.Setinitialclusternumberc=2.

Step2.UsingGKalgorithm,byinitialpartitionmatrixU0,divideZintocparts{Z1,

Z2,…,Zc}andobtainfuzzypartitionmatrixU=[μi,j]c×N.

Step3.Foreachsubset,identifyitsconsequentparametersusingstable2stateKalmanfil2

termethod[4].Thelocalmodelisthendescribedas

Riifφj,yj∈Zi　then　yi=pi0+pi1φj1+…+pidφjd　i=1,…,c(1)

Step4.Computethesystemoutputy^correspondingtoinputzj

y^=∑c

i=1μijyi/∑c

i=1μij(2)

Topredicttheoutputy～ofanewinputφ～,returnGKalgorithmandusethefollowingequation

tocalculateμ～icorrespondingtoithrule[5],

μ～i(φ～)=1∑c

j=1DAxi(φ～,vxi)/DAxi(φ～,vxj)2/(m-1)(3)wherevxidenotestheprojectionoftheithclustercenterviontothegeneralizedinputspace;

DAxiφ～,vximeasuresthedistanceofthenewinputvectorfromtheprojectionofthecluster

centervxi;m>1isaparameterthatcontrolsfuzzinessofclusters.Thenthepredictedoutputy～

canbecalculatedby(2).

Step5.UseS=RMSEtoevaluatemodelingresults.IfS≤STHissatisfied,whereSTH

isgiventhreshold,modelingisover.Otherwise,gotoStep6.715No.4LINingetal.:MultipleModelPredictiveControlforMIMOSystems