ORBIT- Operating-Regime-Based Modeling and Identif

ORBIT-Operating-Regime-BasedModelingandIdenticationToolkitTorA.JohansenaBjarneA.FossaaDepartmentofEngineeringCybernetics,TheNorwegianUniversityofScienceandTechnology,N-7034Trondheim,Norway.ORBITisaMATLAB-basedtoolkitforblack-boxandgrey-boxmod-elingofnon-lineardynamicsystems.Themodelrepresentationisbasedonmultiplelocalmodelsvalidindierentoperatingregimes,thataresmoothlyblendedintoaglobalnon-linearmodel.ORBITisacomputer-aidedmodelingenvironmentthatsupportstheinteractivedevelopmentofregime-basedmodelsonthebasisofamixtureofempiricaldataandpriorknowledge(similartoTakagi-Sugeno-typefuzzymodels).ORBITcontainsfunctionsforrobustparameteridentication,includingconstrainediden-tication,regularizationandBayesianidentication.Therearefunctionsforstructureidentication,includingselectionofthelocalmodelordersaswellasdecompositionsintooperatingregimesthatgiveagoodttothedata.Modelvalidation,simulation,visualizationandautomaticcodegenerationarealsosupported.Keywords:Softwaretools,systemidentication,nonlinearsystems,fuzzysystems.1IntroductionThereisagrowinginterestinmodelingandidenticationmethodsbasedontheexplicitdecompositionoftheoperatingrangeofdynamicsystemsintooperatingregimesandtheuseofsimplelocalmodelswithineachoperatingregime,seee.g.(JohansenandFoss1997,Murray-SmithandJohansen1997,TakagiandSugeno1985,Jacobsetal.1991).Suchdynamicmodelshavefoundwideapplicabilityinmodelpredictivecontrol(Fossetal.1995,Chowetal.1995),gainschedulingcontrol(HuntandJohansen1997,Banerjeeetal.1995)andfuzzycontrol(TakagiandSugeno1985,Zhaoetal.1995).SINTEFandtheNorwegianUniversityofScienceandTechnologyhavedevelopedaresearchtooltofacilitatefurtherresearchanddevelopmentofthistechnology,aswellasinternalandexternaluseineducationandindustrialresearchprojects.TheMATLAB-basedsoftwaretooliscalledORBIT(Operating-Regime-BasedmodelingandIdenticationToolkit).PreprintsubmittedtoElsevierPreprint18July1998Mathematicalmodelingisthescienceoftransformingtheavailableempiricalandmechanisticdataandknowledgeintoasetofequationsthatareusefulfortheintendedapplication.Theblendingofempiricismwithanunderstandingofthemechanismsofthesystemisoffundamentalimportance.ORBITisaninteractiveenvironmentforcomputer-aidedmodelingandsystemidentica-tion,typicallyleadingtomathematicalmodelsthatcanbeviewedasgrey-boxmodels.Dependingontheapplicationandtheavailabledataandknowledge,ORBITcanbeappliedasanythingbetweenapurelydata-drivenautomaticmodelingtoolandagraphicaluserinterface(GUI)formanualspecicationofthemodel.Advancedmethodsfornon-linearsystemidenticationformthecoreofORBIT.Theregime-basedmodelsdevelopedinORBITarecloselyrelatedtoTakagi-Sugenotypefuzzymodels(TakagiandSugeno1985).AdistinguishedfeatureofORBIT,comparedforexampletotheMATLABFuzzyLogicToolbox,isthestrongfocusontheapplicationratherthanonthefuzzysystemstechnology.InORBITanattempthasbeenmadetohideasmanyaspossibleofthemoretechnicaldetailsoffuzzysystems.Forexample,thereisnoexibilityinORBITwithrespecttotheselectionofthemembershipfunctionparameterizationorthefuzzyoperators.Ontheotherhand,theuserhasbeengivenextensiveexibilityontheapplicationlevel.Forexample,itispossibletoconstrainthemodelparametersindividuallyduringidentication,andtoanalyzeandtunetheindividuallocalmodels.TheauthorsbelievethatthismakesORBITausefultoolfortheapplication-orientedresearcher.Thispaperisstructuredasfollows:InSection2thefundamentalsofoperating-regime-basedmodelingarereviewed.AnoverviewoftheORBITtoolisgiveninSection3.MoredetailsonthealgorithmsandfunctionalityinORBITfollowinSection4.ThepaperendswithsomeconcludingremarksinSection5.Alaboratory-scaleidenticationexampleisusedthroughoutthepapertoillustratetheideas.2Operating-Regime-BasedModelingThepurposeofthissectionistoprovideabriefoutlineofthebasicsofoperating-regime-basedmodeling.Moredetailscanbefoundin,e.g.(Murray-SmithandJohansen1997).22.1LocalmodelsandoperatingregimesAnymodelwillhavealimitedrangeofvalidity.Thisrangemayberestrictedbythemodelingassumptionsforamechanisticmodel,orbytheexperimentalconditionsunderwhichthedatawasloggedforanempiricalmodel.Toem-phasizethisaspect,amodelthathasarangeofvaliditythatissmallerthanthedesiredrangeofvaliditywillbecalledalocalmodel,asopposedtoaglobalmodelthatisvalidinthefullrangeofoperation.Theregioninwhichalocalmodelisvalid,iscalledanoperatingregime.OperatingRegime2OperatingRegime3OperatingRegime4OperatingRegime1z1z2Fig.1.DecompositionofoperatingrangeintoregimesTheoperating-regime-basedmodelingframeworkisconceptuallyillustratedinFig.1.Thesystem’sfullrangeofoperation(characterizedbysomevectorz=(z1;:::;zd))iscoveredbyanumberofpossiblyoverlappingoperatingregimes.Withineachoperatingregimethesystemismodeledbyalocalmodel,andthelocalmodelsareblendedintoaglobalmodelusingweightingfunctions.Noticethattheoperatingregimesarenotreallyregionswithhardboundariesasillustratedinthegure,butratherregionswithsoftboundaries.Thismeansthattherewillbeagradualtransitionbetweenlocalmodelswhenmovingbetweenoperatingregimes.Thisisimplementedasaninterpolationtechnique,whichcanalsobeviewedasafuzzy