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Optimal_2

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Optimal_2

    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

    ;sidersolvingthefollowingoptimalqua