An Equation-Free Approach to Nonlinear Control Coa

arXiv:nlin/0412006v1[nlin.CG]1Dec2004AnEquation-FreeApproachtoNonlinearControl:CoarseFeedbackLinearizationWithPole-PlacementConstantinosI.Siettos1DepartmentofChemicalEngineering,PrincetonUniversity,Princeton,NJ08544Electronicmailaddress:[e-mail]2DepartmentofChemicalEngineering,DepartmentofMathematicsandPrograminComputationalandAppliedMathematics,PrincetonUniversity,Princeton,NJ08544Electronicmailaddress:[e-mail]rtmentofChemicalEngineering,WorcesterPolytechnicInstitute,Worcester,MA01609-2280,Electronicmailaddress:[e-mail]plicationofequation-freecomputationtothecoarse-grainedfeedbacklinearizationproblemofnonlinearsystemsdescribedbymicroscopic/stochasticsimulators.Feedbacklinearizationwithpoleplacementrequiresthesolutionofafunctionalequationinvolvingthemacroscopic(coarse-grained)systemmodel.Intheabsenceofsuchaclosed-formmodel,short,appropriatelyinitializedburstsofmicroscopicsimulationaredesignedandperformed,andtheirresultsusedtoestimateondemandthequantitiesrequiredforthenumericalsolutionofthe(explicitlyunavailable)functionalequation.OurillustrativeexampleisakineticMonteCarlorealizationofasimplifiedheterogeneouscatalyticreactionscheme.1Currently:SchoolofAppliedMathematicsandPhysics,NationalTechnicalUniversityofAthens,ZografouCampus,Athens,GR15780,Greece2Authortowhomcorrespondenceshouldbedirected1IntroductionAfundamentalprerequisiteforthedesignofcontrolsystemsistheavailabilityofreasonablyaccurateclosedformdynamicalmodels.Typically,suchmodelsariseintheformofevo-lutionequations(ordinarydifferential,differentialalgebraic,partialdifferential,possiblyintegrodifferentialequations).Suchequationsaretypicallyderivedfromconservationlaws(e.g.mass,momentumandenergybalances)closedthroughconstitutiveequations(e.g.Newtonianstressesinfluidflow,ormass-actionkineticsexpressionsforchemicalreactions);systemidentificationmayalsoplayaroleinobtainingand/orclosingsuchmacroscopicmodels.Manyreal-worldproblemsofcurrentengineeringinterestarecharacterized-duetotheirstochastic/microscopicnature,andnonlinearcomplexity-bythelackofsuchgoodexplicit,coarse-grainedmacroscopicevolutionequations.Instead,theunderlyingphysicsdescriptionmaybeavailableatamuchfiner,moredetailedlevel:theevolutionrulesmaybegivenintheformofmoleculardynamics,kineticMonteCarlo,Markov-chainorhybridschemes.Whenthisisthecase,conventionalcontinuumalgorithmscannotbeuseddirectlyforsystemslevelanalysisandcontrollerdesign.Bridgingsystematicallytheenormousgapbetweenmicroscopicspaceandtimescalesofacomplexphysical/materialsystemdescrip-tionandthemacroscopiconesatwhichwewanttodesignandcontrolitsbehaviorisagrandchallengeformodelingandcomputation.Overthepastfewyearswehavedemon-stratedthatanequation-freeapproach(basedoncoarsetimesteppers)[Theodoropoulosetal.,2002;Makeevetal.,2002;Kevrekidisetal.,2003;Siettosetal.,2003b;Kevrekidisetal.,2004],canestablishalinkbetweentraditionalcontinuumnumericalanalysisandmicro-scopic/stochasticsimulation.Thisisamathematics-assistedcomputationalmethodology,inspiredfromcontinuumnumericalanalysis,systemidentificationandlargescaleitera-tivelinearalgebra,whichenablesmicroscopic-levelcodestoperformsystem-levelanalysisdirectly,withouttheneedtopassthroughanintermediate,coarse-grained,macroscopic-level,“conventional”descriptionofthesystemdynamics.Thebackboneofthemethodistheon-demandidentificationofthequantitiesrequiredforcontinuumnumerics(coarseresiduals,theactionofcoarseslowJacobians,eigenvalues,Hessians,etc).Theseareob-tainedbyrepeated,appropriatelyinitializedcallstoanexistingfinescaletime-steppingroutine,whichistreatedasaninput-outputblackbox.Thekeyassumptionisthatde-terministic,macroscopic,coarsemodelsexistandclosefortheexpectedbehaviorofafewmacroscopicsystemobservables,yettheyareunavailableinclosedform.Theseobservables(coarse-grainedvariables)aretypicallyafewlowmomentsofmicroscopicallyevolvingdis-tributions(e.g.surfacecoverages,thezerothmomentsofspeciesdistributionsonalattice2modelofasurfacereaction).Thepresentworkaimsatdevelopingasystematicapproachtothefeedbackregulatorsynthesisproblem,whereboththeclosed-loopdynamicslinearizationandpole-placementobjectivesaresimultaneouslyattainedbyusingtheequation-freetimesteppermethodology.Thefeedbacklinearizationandthepole-placementobjectivesfortheunavailablecoarse-graineddynamicsaremetinasingle-step,circumventingthelackofanexplicitdynamicprocessmodel.Theproposedapproachisillustratedthroughtheuseofacoarsetime-stepperbasedonakineticMonteCarlorealizationofasimplifiedsurfacereactionschemeforthedynamicsofNOoxidationbyH2onPtandRhsurfaces.Thepresentpaperisorganizedasfollows:Insection2webrieflydiscussthetraditionalnonlinearcontrolmethodologiesthatrelyonthenotionoffeedbacklinearizationalongwiththeassociatedrestrictionsencounteredattheimplementationstage.Insection3wesuccinctlyreviewarecentlyproposedapproachthatallowstheattainmentofboththefeedbacklinearizationandpoleplacementobjectivesinasinglestep,effectivelyovercomingtherestrictivecondi-tionsassociatedwiththe