CLOSED-LOOP PERFORMANCE AND BIFURCATION ANALYSIS O

CLOSED-LOOPPERFORMANCEANDBIFURCATIONANALYSISOFACASCADECONTROLLEDEXOTHERMICCSTRLouisP.RussoandB.WayneBequetteHowardP.IsermannDepartmentofChemicalEngineeringRensselaerPolytechnicInstituteTroy,[e-mail]@rpi.eduFax:[phone]Tel:[phone]Presentedatthe1995AnnualAIChEmeetinginMiami,FL.Session68:ChemicalReactorStabilityandDynamics-68g.Itisincreasinglyunderstoodthatlimitationstocontrolsystemperformanceareduetononlineareectsintheopen-loopsystem.Aplethoraofnonlinearbehaviorssuchaslargeamplitudeunforcedoscillations,ignition/extinctionbehavior,andchaoticoscillationsarefoundinchemicalprocesses,particularlychemicalreactors.Duetotheincreasinglyintegratednatureofunitoperations,itisimportanttodesignrobustcontrolstrategiestominimizethesetypesofbehavior.Inthispaperweshowthatthetraditionalmethodsofanalyzingcascadecontrol(basedonseriesorparallelinput/outputrelationships)shouldnotbeusedwhentheprimaryandsecondaryprocessesarecoupled,andtheprocessisopen-loopunstable.Inaddition,wedevelopacascadecontrolsystemmethodologywhichincorporatesaspeciedinner-loopcontrollerintheouter-loopcontrollerdesign.Thiscascadecontroldesignprocedureiscomparedtoaconventionalcascadecontroldesignprocedureandtoanoncascadecontrolstrategy.Thebifurcationbehaviorofthecascadecontrolstructureisexamined,andpreviousresultsareextendedtothecaseofmultipleoutputsbeingfed-back.Anequationrelatingthefeedbackgainsattheonsetofalimitpointinstabilityisderived.Keywords:Cascadecontrol;chemicalreactors;unstableprocesses;bifurcation.1INTRODUCTIONInput/output(transferfunction)analysisisoftenusedtodesignfeedbackcontrolsystemsforchemicalprocesses.Cascadecontrolsystemsareusuallydesignedbyrsttuningaslavecontrolloopbasedonthetransferfunctionmodelofthesecondaryprocess.Iftheslaveloopcanbetuned\fastenough,thenthemasterloopcanbetunedbasedontheprimaryprocesstransferfunction,independentoftheslaveloop.Ifthebandwidthoftheslaveloopisclosetothatoftheprimaryloop,thentheloopsmustbeiterativelytuned,untilacceptableperformanceoftheentireclosed-loopprocessisobtained.Implicitinthetypicalcontrollerdesignprocessistheassumptionthatthesecondaryandprimaryprocessesaredecoupled.Thisisillustratedbythetypicalseriescascadecontrolblockdiagram,showninFigure1.Theoutputofthesecondaryprocessistheinputtotheprimaryprocessinaseriescascadecontrolblockdiagram.Similarly,aparallelcascadecontrolstructure(inwhichthemanipulatedinputhasadirecteectonbothoftheoutputs)isshowninFigure2.Themotivatingproblemiscascadecontrolofanexothermiccontinuousstirredtankreactor(CSTR)operatedatanopen-loopunstableoperatingpoint.Althoughmanynewnonlinearcontroltechniqueshavebeendeveloped(Bequette,1991;McLellanetal.,1990;KravarisandKantor,1990),weuselinearcontrolfortworeasons:Linearfeedbackandcascadecontrollersarestillthedominantstrategiesusedinindustry.Thelineartechniquesarewell-knownandeasytouseforcomparisonpurposes.2CSTRMODELINGEQUATIONSThestandardtwo-stateCSTRmodel(Uppaletal.,1974)describinganexothermicdiabaticirreversiblerst-orderreaction(A!B)isasetoftwononlinearordinarydierentialequationsobtainedfromdynamicmaterialandenergybalances(withtheassumptionsofconstantvolume,perfectmixing,negligiblecoolingjacketdynamics,andconstantphysicalparameters).dCadt=QV(CafCa)k0expEaRTCa(1)dTdt=QV(TfT)UAVCp(TTc)+HCpk0expEaRTCa(2)whereCaandTaretheconcentrationofcomponentAandthetemperatureinthereactor,respectively.AnadditionalenergybalancearoundthecoolingjacketassumingperfectmixingyieldsdTcdt=QcVc(TcfTc)+UAVccCpc(TTc)(3)whereTcisthecoolingjackettemperature.Equations1-3canbewrittenindimensionlessformdx1d=q(x1fx1)x1(x2)(4)dx2d=q(x2fx2)(x2x3)+x1(x2)(5)2dx3d=1[qc(x3fx3)+2(x2x3)](6)wherex1,x2,x3,andqcarethedimensionlessconcentration,reactortemperature,coolingjackettemperature,andcoolingjacketowrate,respectively.Thesedimensionlessvariablesandparameters(forexample:,q,,etc.)aredenedinTable1.RepresentativevaluesoftheparameterscanbefoundinRussoandBequette(1995a,1995b),withCase1andCase2conditionslistedinTable2.Case1isopen-loopstableovertheentireregionofoperationforthetwo-stateCSTRmodel(cf.Figure3)butopen-loopunstableinacertainoperatingregionforthethree-stateCSTRmodel(cf.Figure4).Case2exhibitsignition/extinctionbehaviorforthetwo(cf.Figure5)andthree-statemodels(cf.Figure6).Itisimportanttoconsiderthecoolingjacketinthecontrolsystemdesign,especiallyduetosafetyconsider-ations.TheoutputmultiplicitiesshowninFigure4resultinignition/extinctionbehavior.IftheCSTRisbeingoperatedatanopen-loopunstableoperationpoint(sayx2=2:0)andthecoolingjacketowratehitsaconstraint(thuseectively\openingupthecontrolloop)thenthetwo-statemodelincorrectlypredictsstablebehaviorwhilethethree-statemodelwillcorrectlypredictinstability.Itiswell-knownthattheexponentialrelationshipofreactionratewithrespecttoreactortemperatureisoneofthemajornonlinearitiesoftheCSTR.RussoandBequette(1995a,1995b)studiedboththesteady-statenonlinearities(outputandinputmultiplicities,infeasibleoperationregions)andthedynamicnonlinearities(Hopfbifurcationbehavior)o