Algebraic differential equations and rational cont

ALGEBRAICDIFFERENTIALEQUATIONSANDRATIONALCONTROLSYSTEMS1YuanWangMathematicsDepartment,FloridaAtlanticUniversity,BocaRaton,Fl33431(407)367-3317,E-mail:[e-mail]atics,RutgersUniversity,NewBrunswick,NJ08903(908)932-3072,E-mail:[e-mail]isshownbetweenrealizabilityofinput/outputoperatorsbyrationalcontrolsystemsandhighorderalgebraicdierentialequationsforinput/outputpairs.Thisgeneralizes,tononlinearsys-tems,theequivalencebetweenautoregressiverepresentationsandnitedimensionallinearrealizability.1.Introduction.Inthispaperweproveanequivalencebetweenrealizabilityofinput/outputoperatorsbyrationalcontrolsystemsandtheexistenceofhighorderalgebraicdierentialequationsrelatingderivativesofinputsandoutputs.Inmanyexperimentalsituationsinvolvingsystems,itisoftenthecasethatonecanmodelsystembehaviorthroughdierentialequations,whicharereferredtoasinput/output(\i/o)equationsinthiswork,ofthetypeEu(t);u0(t);u00(t);:::;u(r)(t);y(t);y0(t);y00(t);:::;y(r)(t)=0(1)whereu(
)andy(
)aretheinputandoutputsignalsrespectively,andEisapolynomial.Ani/ooperatorF:u(
)7!y(
)issaidtosatisfytheequation(1)iftheequationholdsforeachsucientlydierentiableinputuandthecorrespondingoutputy=F[u]ofF.(Precisedenitionswillbegivenlater.)ThefunctionalrelationEisusuallyestimated,forinstancethroughleastsquarestechniques,ifaparametricgeneralform(e.g.polynomialsofxeddegree)ischosen.Forexample,inlinearsystemstheoryoneoftendealswithdegree-onepolynomialsE:y(k)(t)=a1y(t)+:::+aky(k−1)(t)+b1u(t)+:::+bku(k−1)(t)(2)(ortheirfrequency-domainequivalent,transferfunctions;thedierenceequationanalogueissometimescalledan\autoregressivemovingaveragerepresentation).Inthelinearcase,suchrepresentationsformthebasisofmuchofmodernsystemsanalysisandidenticationtheory.State-spaceformalismsaremorepopularthani/oequationsinnonlinearcontrol,however.There,oneassumesthatinputsandoutputsarerelatedbyasystemofrstorderdierentialequationsx0(t)=f(x(t))+G(x(t))u(t);y(t)=h(x(t))(3)wherethestatex(t)isnowavector,andnoderivativesofcontrolsareallowed.Thesedescriptionsarecentraltothemodernnonlinearcontroltheory,astheypermittheapplicationoftechniquesfromdierentialequations,dynamicalsystems,andoptimizationtheory.Thusabasicquestionisthatofdecidingwhenagiveni/ooperatoradmitsarepresentationofthisform.Thisistheareaofrealizationtheory,whichiscloselyrelated,especiallywhenstochasticeectsareincluded,tosystemsidentication.Roughlyspeaking,ifsuchastatespacedescriptiondoesexistforagiveni/ooperator,thenwesaythatthei/ooperatorisrealizable.Moreprecisely,weshallbeinterestedinrealizationsinwhichtheentriesoffandG,aswellasthefunctionh,canbeexpressedintermsofrationalfunctionsofthestate,butduetothetechnicalproblemsthatariseinthedenitionbecauseofpossiblepolesoftheserationalfunctions,wewillgivetheprecisedenitionintermsof\singularpolynomialsystemsandwewillalsostudyrealizabilityby(nonsingular)polynomialsystems.Oneknowsthatanequationsuchas(2)canbereduced,byaddingstatevariablesforenoughderivativesoftheoutputy,toasystem(3)ofrstorderequations,withf(x)linearandG(x)constant,1ThisresearchwassupportedinpartbyUSAirForceGrantAFOSR-88-0235.Keywords:Rationalsystems,input/outputequations,identication.AMS(MOS)subjectclassication:Primary:93B15,Secondary:93A25,93B25,93B27,93B29Runninghead:Algebraicequationsandrationalsystems1i.e.,alinearnite-dimensionalsystem.Infrequency-domainterms,rationalityofthetransferfunctionisequivalenttorealizability.(Forreferencesonthelineartheory,seeeg[14],[23]and[32].)Oneofthemethodsforobtainingalinearrealizationfromagivenlineari/oequationreliesonLordKelvin'sprincipleforsolvingdierentialequationsbymeansofmechanicalanalogcomputers(cf.[14]).Theprinciple,whichwassuggestedahundredyearsago,providedawayforsimulatingasystemwithoutusingdierentiators.Fornonlinearsystemsthisreductionpresentsafarharderproblem,onethatistoagreatextentunsolved.Theproblemisbasicallythatofinsomesensereplacinganontrivialequation(1)byasystemofrst-orderequations(3)whichdoesnotinvolvederivativesoftheinputs.Anumberofresultswerealreadyavailableabouttherelationbetween(1)and(3);seeforinstance[4],[12],or[26].Itiseasytoshow,byelementaryargumentsinvolvingnitetranscendencedegree,thatanyi/ooperatorrealizablebyarationalstatespacesystemsatisessomei/oequationoftype(1),withEapolynomial.In[6]itwasremarked|asaconsequenceoftheoremsfromdierentialalgebra,|thatinordertocharacterizethei/obehaviorofastatespacesystemuniquely,oneneedstoaddinequalityconstraintsto(1).In[18]and[27]itwasshownthat,undersomeconstantrankconditions,theoutputsofanobservablesmoothstatespacesystemcanbedescribedbyanequationoftype(1)forwhichEisasmoothfunction,andlocali/oequationswereshowntoexist,forgenericinitialstatesof(3),in[3].1.1.OurApproach.Thediscrete-timeworkreportedin[20]and[21]providedoneapproachtorelatingthesetwotypesofrepresentations|withdierenceequationsappearinginstead,|basedontheideaofdealingwithexistenceofrealizationsseparatelyfromthequestionof\well-posednessoftheequation(inthesensetobedescribe