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Optimal_1

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Optimal_1

    Optimal

ChineseJournalofChemicalEngineering,16(2)235240(2008)

    ;OptimalIterativeLearningControlforBatchProcessesBasedon ;LinearTime.varyingPerturbationModel

    ;xIoNGZhihua(熊智华),zHANGJieandDONGJin(董进)

    ;DepartmentofAutomation,TsinghuaUniversity,Beijing100084,China

    ;ScIio01ofChemicalEngineeringandAdvancedMaterials,UniversityofNewcastle,Ne

    wcastleuponTyne,NE17RU,

    ;SupplyChainManagement&Logistics,IBMChinaResearchLab,Beijing100094,C

    hina

    ;1INTRoDUCTIoN

    ;Batchprocessesaresuitableforthemanufactur. ;ingofhighvalueaddedproducts,suchas,special ;polymers,specialchemicals,andpharmaceuticals1.

    ;Therepetitivenatureofbatchprocessoperationsa1. ;1owsthattheinformationofpreviousbatchrunscan ;beusedtoimprovetheoperationofthenextbatch.Of ;lateiterativelearningcontrol(ILC1hasbeenusedin ;thebatch..to..batchcontrolofbatchprocessestodi.. ;rectlyupdatetheinputtrajectory1,2.Refinementof

    ;controlsignalsbasedonILCcansignificantlyenhance ;theperformanceoftrackingcontrolsystems. ;Bristoweta1.3havepresentedasurveyofthe

    ;majorresultsbasedonlinearmodelsintheILCanalY. ;sisanddesignoverthepasttwodecades.AstheILC ;iswelldevelopedinlinearmodels,mostofthe ;ILC.based.batch.to.batchcontrolschemesarebased ;onsomekindoflinearmodel,forexample,thelinear ;timeinvariantsystem[41.NonlinearmodelbasedILC

    ;schemeshavealsorecentlybeenproposed.Xuand ;Hou5havereviewedtherecentadvancesinILC, ;basedontheLyapunovmethodsofthenonlinearsys. ;tems.OptimalILCisoneoftheimportantmethodsfor ;designinganiterativelearninglaw.inwhichtheILC ;lawisderivedfromaquadraticobjectivefunction3.

    ;Owenseta1.6,71haveproposedanoptimalILC,

    ;basedontheoptimizationprinciple,bycombiningthe ;RiccatifeedbackcontrolandthetypicalILC ;feedforwardcontro1.Gaoeta1.8.9havestudiedthe

    ;optimalILCfurtherforaninjectionmoldprocessand

    ;thenproposedageneraldesignframeworkfortheILC ;ofabatchprocessbasedonatwo.dimensionalf2D1 ;system.Leeeta1.inseveralrelatedarticles2,1014

    ;haveproposedthequadraticcriterion.basedILC ;(O.ILC)approachfortrackingthecontrolfortem. ;peratureofbatchprocesses,basedonalinear ;time.varying(LTV)trackingerrortransitionmode1. ;Leeeta1.l1combinetheadvantagesofILCand

    ;modelpredictivecontrolfMPC)intoasingleframe. ;work.AbatchMPCfBMPC)techniqueanditsexten. ;sionfortrackingcontrolareproposedbyincorporat? ;ingthecapabilityofrea1.timefeedbackcontrolinto ;thequadraticcriterion.basedILCfQ.ILC).Thepro. ;posedapproachisalsoappliedtothetrackingcontrol ;fortemperatureofbatchprocesses.

    ;ILCcanupdatethecontroltrajectoryforthenext ;batchrunbyusingtheinformationfromtheprevious

     ;batchrunssothattheoutputtrajectoryconvergesas

    ;ymptoticallytothedesiredreferencetrajectory. ;Therefore,theconvergenceofaniterativelearning ;lawisanimportantissueinthedesignandapplication ;OfILC.Intheauthors’previousstudies15,161,an

    ;ILCstrategyforthetrackingcontrolofproductquality ;inbatchprocesseswasproposedbasedonanLTVP ;mode1.Inpractice.thePmodelofproductquality ;canbeobtainedbylinearizingthenonlinearmode1. ;withrespecttothenominaltrajectories.Furthermore,to ;addresstheproblemofmode1.plantmismatches.the ;modelpredictionerrorsinthepreviousbatchrunare ;addeddirectlytothemodelpredictionsinthecurrent ;batchrun.Onthebasisofseveralsimulationresults,it ;showsthatthetrackingerrorconvergesnominallyas ;thebatchnumbertendstoinfinite.Inthisarticle,the ;authorspresentarigoroustheoremtoverifythat,when ;thereisnomodelingerror,aperfecttrackingperform

    ;ancecanbeobtained,inthesensethatboththetracking ;errorsandthechangesofthecontrolpolicyconvergeto ;zeroasthebatchindexnumbertendstoinfinite. ;2BATCH.To.BATCHITERATIVELEARN.

    ;INGCoNTRoL

    ;Thebatch?wiseLTVPmodelbasedILCdeveloped

    ;Received20061130.accepted20071028.

    ;SupportedbytheNationalNaturalScienceFoundationofChina(60404012,60674064),U

    KEPSRC(GR/N13319and

;GR

    /R10875):theNationalHighTechnologyResearchandDevelopmentProgramofChina(200

    7AA04Z193),NewStarof

    ;ScienceandTechnologyofBeijingCity(2006A62).andIBMChinaResearchLab2007UR

    Program.

    mail:zhxiong@tsinghua.edu.cn ;T0whomcorrespondenceshouldbeaddressed.E

    ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.2,April2008 ;byXiongandZhang15isreviewedinthissection.

    ;Batchprocesseswherethebatchrunlength(t3isfixed ;andconsistsofNsamplingintervalsareconsidered, ;whereallbatchesrunfromthesameinitialconditions. ;Theproblemofthebatch.to..batchcontrolistoma.. ;nipulatethewholecontrolprofilesothattheproduct ;qualityvariablesfollowthespecificdesiredreference ;trajectories.Itwouldbeconvenienttoconsidera ;batchwisestaticfunctionrelatingthecontrolprofile ;tOtheproductqualityprofileoverthewholebatch ;duration.Itcanbewritteninmatrixformas ;=

    ;F(Yo,)

    ;wherethesubscriptkdenotesthebatchindex, ;:

    ;[(1),(2),…,(?)]T(?n,,z?1),isthe

    ;productqualityvariableandcanbeobtainedonlineor ;offline,Y0istheinitialvalue,=【瓦(1),(2),…,

    ;(N1)I(?R,m:1inthisstudy)isthema

    ;nipulatedvariable.andF(?)representsthenonlinear ;staticfunctions.respectively.Subtractingthetime

    ;varyingnominaltrajectoriesfromtheprocessopera

    ;tiontrajectoriesremovesamajorityoftheprocess ;nonlinearityandallowslinearmodelingmethodsto ;performwellontheresultingperturbationvariables17].

    ;2.1LTVPmodel

    ;Bylinearizingthenonlinearbatchprocessmodel ;describedbyEq.(1)withrespecttocontrolsequence ;aroundthenominaltrajectories,thefollowingcanbe ;obtained

    ;=rs+

    ;(Yo,)

    ;aE,

    ;()+d

    ;whereUisthenominalcontroltrajectory,risthe ;nominalproductqualitytrajectory,and(0)=Y0,

;anddkisasequenceofmodelerrorsbecauseoflin

    ;earization(i.e.,becauseofneglectingthehigheroder ;terms).respectively.ThenthebatchwiseLTVPmodel

    ;canbeobtainedas

    ;=GU+d(3)

    ;whereG:l(v)/~vJ,uuu,

    ;and=rsaredefinedasperturbationvariables ;ofcontrolandproductqualityvariables,andy(0)=0, ;respectively.Gsisbatchwiselineartimevarying,in

    ;thesensethatitvarieswithU,whichusuallyvaries ;frombatchtobatch.Onaccountofthecausality(1.e., ;theproductqualityvariablesattimetareonlyfunc

    ;tionsofallcontrolactionsuptotimef),thestructureof ;Gisrestrictedtothefollowinglower-blocktriangular

    ;form:

    ;G=

    ;g100

    ;g2og21

    ;0

    ;0

    ;gNOgN1’..gNN1

    ;(4)

    ;whereg?R.ThebatchwiseLTVPmodelGscan

    ;U

    ;befoundbylinearizinganonlinearmodelalongthe ;nominaltrajectoriesorthroughdirectidentification ;fromtheprocessoperationaldata.Availablemethods ;foridentifyingGrangefromsimplestaticlinearre

    ;gression,suchas,theleastsquaresanditsvariants ;(e.g.,partialleastsquares,PLS)[15,18],tomore ;elaborateoptimaldynamicestimationmethodssuchas ;theextendedKalmanfiltering(EKF12,12.

    ;2.2Optimaliterativelearningcontrol

    ;Themodelpredictioninthekthbatchrunisob

    ;tainedas

    ;=

    ;GU(5)

    ;whereGistheestimatedLTVPmodelofG.The ;modelpredictionerrorisdefinedas=.In

    ;thisstudy,thepredictionerroroftheperturbation ;modelisassumedtobeboundbyacertainsmallposi

    ;tiveconstantB,’l,thatis,II<.Theprediction

    ;errorboundBisameasuretorepresentthedeviation ;offrom.ThehigherthevalueofB,’lis,the

    ;poorertheidentifiedmodelis.Aftercompletionofthe ;kthbatchrun,theproductqualitiesaremeasuredor ;analyzedofnineandthepredictionerrorcanbecalcu

    ;lated.

    ;Toaddresstheproblemofmode1..plantmis.. ;matchesfortheLTVPmode1.theauthorshaveutilized ;themodelerrorsoftheimmediatepreviousbatchrun, ;tomodifythemodelpredictions.Basedonthepredic

    ;tionerrorsofthekthbatchrun.themodifiedpredic ;tionoftheperturbationmodelinthe(k+1)thbatchrun ;isobtainedas

    ;=+l+

    ;Correspondingly,themodifiedpredictionerrorisdefined ;as+1=+l+l,anditholds+1=+1.

    ;ConsideringthattheobjectiveofILCistotrack ;thedesiredreferencetrajectoriesofproductquality, ;thetrackingerrorsoftheprocess,model,andmodified ;modelpredictionaredefinedrespectivelyas ;e=,=,ek:(7)

    ;where=,andarethespecifiedrefer

    ;encetrajectoriesandareassumedtobesetreasonably ;here.

    ;Fromallthesedefinitions,therelationships ;amongthesethreetrackingerrorsare

    ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.2,April2008 ;ee

    ;ee一一

    ;1

    ;fromwhichthefollowingiterativerelationships ;thebatchindexkcanbeobtained[15:

    ;+=

    ;()(+)

    ;=

    ;G(+,u)

    ;+l=+(+l)(一一1)

    ;=

    ;G(+u)

    ;e+l:e(u+1u)+l

    ;ek+1:+l(e)

    ;=eG(+U)

    ;(8)

    ;along

    ;(10)

;(11)

    ;(12)

    ;Giventheabove..mentionedbatch..wiseerror ;transitionmode1.theobjectiveoftheILCistodesign ;alearningalgorithmtomanipulatethecontrolpolicy ;sothattheproductqualitiesfollowthespecificdesired ;referencetrajectoriesfrombatchtobatch.Bythe

    equivalenceprinciple[101,theauthorscon ;certainty

    ;sidersolvingthefollowingoptimalquadraticobjec

    ;tivefunction,basedonthetrackingerrorofthemodi. ;fiedmodelpredictionuponthecompletionofthekth ;batchrun.toupdatethecontroltrajectoryforthe ;(k+1)thbatchrun

    ;J(U)=+++

    ;=1Q+l+(+uk)R(U+U)

    ;(13)

    ;whereDandgarepositivedefinitivematrices.The ;weightingmatricesQandgaffectthecontributionsof ;thetrackingerrorsandcontrolchangesinEq.f131and ;mustbeselectedcarefully.Arelativelylargeweight,g, ;onthecontrolchange.willleadtomoreconservative ;adjustmentsofcontrolactionsandtoaslowercon. ;vergence.Arelativelysmallgwillleadtolargead. ;justmentsofcontrolactions,butthiscanleadtoun

    ;convergence,asalargechangeincontrolactionsmay ;resultinthelinearizedmodelbeinginvalid.Qandg ;areusuallyselectedasdiagonalmatricesanditisalso ;possibletoweighttrackingerrorsandcontrolactions ;atdifferenttimestagesdifferently.IfQandgarese. ;1ectedasdiagonalmatrices.itcanbeobservedinEq. ;f131matthecontrolperformanceactuallydependson ;therelativevalueoftheweightingmatrices.Forthe ;sakeofsimplicity,andRareselectedinthisstudy ;asQ=I?andg=-I?.

    ;FromEq.f121andthroughstraightforwardma ;nipulation,thefollowingILClawisobtained[15

    ;U+1:U+Ke

    ;whereKisthelearningrate

    ;K=IGQG+glGQ(15)l

    J

    ;ItcanbeobservedintheILClawstatedearlier ;thatthechangeofcontroltrajectory(U+lU)is

    ;directlyupdatedbytheactualtrackingerroreofthe ;process,ratherthanthetrackingerrorekofthemodi-

;fledmodelprediction.

    ;3CoNVERGENCEoFTRACKINGERROR

    ;Inthepreviousstudies[15,16],theILClawstated ;earlierhasbeenpresentedandsimulationstudieshave ;alsoshownthatthetrackingerrore}nominallycon

    ;vergesaccordingtothebatchindexk.HoweveLthe ;convergenceoftheILClawstatedherehasnotbeen

     ;analyzedtheoretically.Inthisarticle,arigoroustheo

    ;remhasbeenusedtoprovethatthetrackingerrore}, ;undertheoptimalILClaw,Eq.(14),canconvergeto ;zero.

    ;TheoremLet{e}}bethetrackingerrorse

    ;quencebasedontheLTVPmodelundertheoptimal ;ILClaw,Eq.(14).Ifthereisnomodelingerror(.e., ;G=G)andthekernelofGiszero,then

    ;lime=0(16)

    ;ProofAsU+11stheoptimalcontroltrajectory

    ;oftheoptimizationproblemEq.(13),thefollowing ;inequalityistrue:

    ;J(U)?J(Uk+1)

    ;=

    ;+】】UkUk

    ;:

    ;I1G(u+u)l+Iv:+ukl1

    ;(17)

    ;where+lisanycontroltrajectoryofthe(k+1)th ;batchrnn.Becausethereisnomodelingerror,thatis, ;G=G,itfollowsthat

    ;.0=(e)(1e1)=

    ;Andif:+1ischosentobeUt,thenitholds ;J(U+)?(+:):1(19)

    ;AccordingtoEq.(13),italsofollowsthat ;J(V+)=f++llv+uk?1+(20)

    ;FromEq.(19)andEq.(20),thefollowingcanbeob

    ;tained

    ;?J(V)?

    ;Thisimpliesthatthelimitofimexists.Conse

    ;

    ;‘oo

    ;quently,.1imJ(U)alsoexistsandequalsto

    ;lim},thatis,

    ;l

    ;

    ;im

;..

    ;liell=l

    ;im

    ;..

    ;J(U+)(22)

    ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.2,April2008 ;thenaccordingtoEq.(13),itfollowsthat

    [.,(). ;+厂熙

    ;J(+?)一牌=o(23)o.—o.”

    ;AsRisapositivedefinitivematrix,Eq. ;(23)resultsin

    ;.1im(UU)=0(24)

    ;AccordingtoEq.(11),Eq.(14),andEq.(18)(.e., ;=

    ;0),itwasfoundthat

    ;川卜)(25)

    ;=

    ;,Kd)(vu)

    ;Becausethefollowingequationholds

    =,Qd+RQd ;,

    ;f261

    ;=

    ;[GQd+R]R

    ;Eq.(25)canberewrittenas

    ;(U川一U):RTQd+R]Ke

    ;=

    ;RGQek+l(27)

    ;FromEq.(24)andEq.(27),itfollowsthat ;.

    ;1imRG=0(28)

    ;BecausethekernelofmodelGiszeroandRandQ ;arepositivedefinitematricesintheobjectivefunction,

    ;Eq.(28)means

    ;lime=0(29)

    ;Thiscompletestheproofofthetheorem. ;Theoretically,thistheoremprovesthatwhenthere ;isnomodelingerror,perfecttrackingwillbeobtained, ;thatise0and(U?U)0askoo.

    ;However,forproductqualitytrackingcontrolin ;anonlinearbatchprocess,modelplantmismatchal

    ;waysexistsbetweenanestimatedLTVPmodelandan ;actualprocess.Inpractice.theIVPmodelcanbe ;foundbylinearizinganonlinearmodelalongthe

    ;nominaltrajectoriesorthroughdirectidentification ;fromtheoperationaldataoftheprocess16],inwhich

    ;thelinearizedmodelingerrorcannotbeneglected. ;Therefore,becauseofthemodelingerror,thetracking ;errorwillchangetoe77,askoo,where77isa ;smallpositiveconstant.

    ;Becausethepredictionerroroftheperturbation ;modelexists(.e.,?0)andisbound(i…e

    ;}ek}<Sm),themodifiedpredictionerror+1isalso ;boundby2B,asfollows

    ;l+.l=lek+.ekl<l+Il+lekl<2(30)

    ;SubstitutingtheILCupdatelaw,Eq.(14),toEq.(11), ;itfollowsthat

    ;1?()一反+(31)

    ;=

    ;[,GK]e+1

    ;Becauseoftheboundederror+l’thetrackingerror

    ;ekwillconvergeatasmallvalue,thatis, ;.

    ;1ime:77.

    ?o. ;K

    ;Italsomeansthatthetrackingperformancewillde

    ;pendontheaccuracyoftheIVPmode1.

    ;4SIMULATIoNoNABATCHREACToR

    ;Thisexampleisatypicalnonlinearbatchreactor, ;withtemperatureasthecontrolvariable19].Theop

    ;erationobjectiveofthereactoristomaximizethe ;product(B)afterafixedreactiontime(fF1.Oh).The ;reactionschemeisALB?C.andtheequa

    ;tionsdescribingthebatchprocessare”

    ;d

    ;Q

    ;Xl

    ;=-

    ;k1exp(E./”T~ef)(32)

    ;dt=k1exp(Ell”)--k2exp(/”Tref)X2

    ;(33)

    ;wherex1andx2representthedimensionlessconcen

    ;trationsofAandBrespectively,”=T/.fisthedi—

    ;mensionlesstemperatureofthereactorandTrefisthe ;reference~mPerature.Theparetersaresetat

    ;k1:4.Oxl()J,k2=6.2xl05,El:2.5x10,E,=5.OxlO, ;andTref:348,respectively.Theinitialconditionsare ;x1(O):1andx2(O)=0,andthereactortemperatureis

;constrainedto298K?r?398K.

    ;Inchisstudy.themechanisticmodelstatedearlier, ;Eq.(32)andEq.(33),isassumedtobenotavailable. ;Becausetheobjectiveofthereactoristomaximizethe ;product(),anLTVPmodelisbuilttomodeltherela- ;tionshiPbetweenY=x.and.Thedesiredproduct ;referencetrajectoryhasbeenselectedfromthe

    .TWelvebatchesofprocessoperations, ;literature20

    ;underdifferenttemperatureprofiles.weresimulated ;fromthemechanisticmodelandusedashistorical ;processdatasets.These12temperatureprofileswere ;generatedrandomlybydeviatingfromthenominal ;temperaturetrajectory,U.Subsequently,basedon ;thesehistoricalprocessdatasetsandtheselectedU ;and,theparametersoftheLTVPmodelGwere ;identifiedusingtheleast-squareregressionmethod151.

    ;HerethebatchlengthisdividedintoNequal ;stagesandtwovaluesofNarestudied.N=10and ;N=5.Asmentionedearlier,theweightingmatrices ;QandRintheILClawaresetasQ=,and

    ;R=?,…Thusthecontrolperformanceactually

    ;dependsontherelativevalue2roftheweightingma

    ;tricesQandR.Alarger2rontheinputchangewill ;leadtomoreconservativeadjustmentsandslower ;convergence.Severalvaluesof2rarestudiedand ;comparedintermsofthetrackingperformance,such ;as,rootmeanofsquareerrors(RMSE),andfinally,Q ;andRareselectedasQ=,andR=0.011

    ;ResultsofthetrackingperformanceofILCunder ;twodifferentschemesareshowninFig.1,inwhich ;theRMSEofthetrackingerrorofproductqualitye}’

    ;atdifferenttimestagesareshown.Itcanbeseenfterabouteightbatchruns.AlthoughtI1epa-

    ;rameterstobeestimatedwhenN:10aremorethan ;whenN=5.them?

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