the nonlinear state space model

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1、Chapter7ThenonlinearstatespacemodelInapplyingtheresultsandconceptsofPartIinthedomainsoftimesseriesorsystemstheory,wehavesofaranalyzedonlylinearmodelsinanydetail,albeitrathergeneralandmultidimensionalones.Thischapterisintendedasarelativelycompletedescriptionofthewayinwhichnonlinearmodelsmaybeana

2、lyzedwithintheMarkoviancontextdevelopedthusfar.Wewillconsiderboththegeneralnonlinearstatespacemodel,andsomespecificapplicationswhichtakeonthisparticularform.Thepatternofthisanalysisistoconsiderfirstsomeparticularstructuralorsta-bilityaspectoftheassociateddeterministiccontrol,orCM(F),modelandthenu

3、nderappropriatechoiceofconditionsonthedisturbanceornoiseprocess(typicallyaden-sityconditionasinthelinearmodelsofSection6.3.2)toverifyarelatedstructuralorstabilityaspectofthestochasticnonlinearstatespaceNSS(F)model.Highlightsofthisdualityare(i)iftheassociatedCM(F)modelisforwardaccessible(aformof

4、controllability),andthenoisehasanappropriatedensity,thentheNSS(F)modelisaT-chain(Section7.1);(ii)aformofirreducibility(theexistenceofagloballyattractingstatefortheCM(F)model)isthenequivalenttotheassociatedNSS(F)modelbeingaψ-irreducibleT-chain(Section7.2);(iii)theexistenceofperiodicclassesforthe

5、forwardaccessibleCM(F)modelisfur-therequivalenttotheassociatedNSS(F)modelbeingaperiodicMarkovchain,withtheperiodicclassescoincidingforthedeterministicandthestochasticmodel(Section7.3).ThuswecanreinterpretsomeoftheconceptswhichwehaveintroducedforMarkovchainsinthisdeterministicsetting;andconverse

6、ly,bystudyingthedeterministicmodelweobtaincriteriaforourbasicassumptionstobevalidinthestochasticcase.InSection7.4.3theadaptivecontrolmodelisconsideredtoillustratehowtheseresultsmaybeappliedinspecificapplications:forthismodelweexploitthefactthat1467.1.Forwardaccessibilityandcontinuouscomponents14

7、7ΦisgeneratedbyaNSS(F)modeltogiveasimpleproofthatΦisaψ-irreducibleandaperiodicT-chain.Wewillendthechapterbyconsideringthenonlinearstatespacemodelwithoutforwardaccessibility,andshowinghowe-chainpropertiesmaythenbeestablishedinlieuo