Nonlinear observer design via convex optimization

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ProceedingsoftheAmericanControlConferenceAnchorage,AKMay8-10,2002NonlinearObserverDesignviaConvexOptimizationAdamHowellJ.KarlHedrickMechanicalEngineeringDepartmentMechanicalEngineeringDepartmentUniversityofCalifornia,Berkeley94720UniversityofCalifornia,Berkeley94720ahowell@vehicle.me.berkeley.edukhedrick@vehicle.me.berkeley.eduAbstractandnumericallytractablebecausetheycaneasilybeformulatedasaconvexoptimizationproblem[2].Thispaperpresentsanonlinearobserverdesigntech-niquebasedonLyapunov’ssecondmethodwhichpro-DespitethislonghistoryofinterestinthecontrolofvidesanobservergainmatrixthatstabilizestheerrorLur’esystems,verylittleofthisworkhasbeenapplieddynamicsforaclassofnonlinearsystems.Itwillalsototheobserverdesignofsuchsystems.Thispaperwillbeshownhowtheobservergainmatrixcanbeopti-developanonlinearobserverdesigntechniquebasedmallychosen,viaconvexoptimization,withrespecttoonLyapunov’ssecondmethodwhichprovidesanob-threedifferentcosts;specifically,themaximumsingu-servergainmatrixthatstabilizestheerrordynamicslarvalueofthegainmatrix,thedecayrateoftheerrorforaclassofnonlinearsystems.Itwillalsobeshowndynamics,andthe%,normbetweendisturbancesandhowtheobservergainmatrixcanbeoptimallychosen,theestimationerrors.Furthermore,thepaperwilldis-viaconvexoptimization,withrespecttothreedifferentcusshowthesedifferentoptimizationcriteriacanbecosts;specifically,themaximumsingularvalueofthecombinedtoprovideParetooptimalobservergainma-gainmatrix,thedecayrateoftheerrordynamics,andtrices.Simulationresultsforarepresentativeproblemthe%,normbetweendisturbancesandtheestima-willalsobegiven.tionerrors.Furthermore,thepaperwilldiscusshowtheseoptimizationcriteriacmbecombinedtoprovideParetooptimalobservergainmatrices.1IntroductionTheremainderofthepaperisdividedasfollows;Sec-tion2willdiscusstheclassofproblemsthatwillbeThedesignofnonlinearobservershasachievedconsid-considered,whileSection2willpresentthederivationerableattentionrecently,andasaresultawidevarietyofsufficientconditionsfortheexistenceofanobserverofdesigntechniquesfornonlinearobserversexistinthegainmatrixwhichstabilizestheerrordynamics.Next,literature.Manyofthesearebasedonthelineofrea-theseresultswillbeusedtoconstructamethodologysoningpresentedoriginallybyThau[SI.Forexample,foroptimizingthechoiceoftheobservergainmatrixRaghavan161andRajamani[7]providedconstructivewithrespecttothedifferentcostfunctionsinSection4.methodsofdesigningobserversforLipschitznonlin-Simulationresultsforarepresentativeproblem,aswellearsystemsbyiterativelysolvingaseriesofRiccatiascomparisontothedesigntechniquein[7],willbeequations.Similarly,Kokotovic[l]developedadesignpresentedinSection5.Finally,someconcludingre-methodologyforaspecificclassofsector-boundednon-marksandadiscussionofpotentialfutureworkwillbelinearsystemsusingalinearmatrixinequality(LMI)coveredinSection6.conditiontodetermineastabilizingobservergainma-trix.Foradetailedsurveyofthesetechniques,.andmanyothers,theinterestedreaderisreferredto[5].2ProblemFormulationAllofthesemethods,infact,shareacommonstructurefortheerrordynamicsofthenonlinearsystem.TheseFortheremainderofthepaper,thefollowingclassoferrordynamicscanberepresentedasalinearsystemnonlinearsystemswillbeconsidered:withasectorboundednonlinearityinfeedback.Thecontrolofsystemshavingthiscommonstructurehavebeenstudiedquiteextensivelyintheliterature,typi-callyreferredtoastheLur’eproblem[3]orabsolutestabilitytechniquessuchastheCircleorPopovcrite-wheref(Cfx)representsamultivariablenonlineardriftrion[4].Thesetypesofproblemsareboththeoreticallytermandthedimensionofthestatevectorisn.,.The0-7803-7298-0/02/$17.0002002AACC2088 observertobeconsideredforthisclassofsystemswillobserver.gainmatrixLwhichquadraticallystabilizeshavethefollowingform,theerrordynamicsshowninEquation(4)if2=AP+B,u+Bff(Cf2)+L(y-C2)(3)3P>Oandr203wheretheobservergainmatrixListobedesignedTP+PA-CTCtoensureconvergenceoftheestimateftothetrue[ABFP+$-Kefstatex.TostudytheconvergenceandperformanceTheresultingobservergainmatrixisL=iP-'CT.ofthisobserver,wewilllookatthedynamicsoftheestimationerrordefinedbye=2-2.Bydifferentiatingtheerror,andsubstitutingthesystemandobserverProof:Toseethisresult,considerthefollowingdynamicsintothedifferentialequation,theresultingquadraticLyapunovcandidatefunctionerrordynamicsare.V(t)=eT(t)Pe(t)e=(A-LC)e+BjQ(Cfe,t)(4)whereP>0andhastheappropriatedimensions.ThewhereQ(Cfe,t):=f(Cfx)-f(CfP)=f(Cfx)-sufficientconditionforstabilityoftheerrordynamicsf(Cj(x-e))andtheexplicittimedependenceofQisthattheLyapunovfunctioncandidatedecreases"%aisduetothestatex.functionoftimealongalltrajectoriesofthestatee,ie.V(t)<0,Vt20.Fortheremainderoftheproof,theFurthermore,thenonlineartermQ(Cfe,t)isassumedexplicitdependenceontimewillbedroppedtosimplifytosatisfyamultivariablesectorcondition.Morespecif-notation.Therefore,usingtheerro?dynamicsderivedically,thenonlinearitymustsatisfytheconstraintinEquation(4),theconditiononV(t)canbewrittenas+(ere,tIT(Q(Cfe,t)-KCfe)L0,(5)VeERnm,andVt20eT((A-LC)TP+P(A-LC))e+#TBfTPe+eTPBj#<0forsomematrixK>0,wherethisnotationsignifieswhereQ=Q(Cfe,t)hasbeenusedforsimplicity.No-thatthematrixKissymmetricpositivedefinite.Al-ticethatthestabilityoftheerrordynamicsisnowrepthoughthisclassofsystemsseemslimited,itisquiteresentedbyaquadraticinequalityineandQ.commoninmechanicalsystems,forexampleinactu-atorsaturationnonlinearities.Inaddition,systemsTherefore,theproblemistofindaP>0andLthatwhichsatisfythemoregeneralmultivariableconstraintresultinV50foralleandqsatisfyingthesectorshownin[4]canalsobeconvertedintothisformusingconstraintshowninEquation(5).Afterperformingaalooptransformation.changeofvariablessuchthatS=LTP,theconditionsonVforstabilityandthesectorconstraintofthenon-Therefore,thegeneralproblemconsideredinthispa-linearitycanberewrittenas:peristochoosetheobservergainmatrgLsuchthattheerrordynamicshaveastableequilibriumattheori-ATP+PA-CTS-STCgininthefaceofallpossibleperturbationscausedbyBfTPthenonlinearityQ(Cfe,t)satisfyingthemultivariablesectorconstraintstatedinEquation(5).3ExistenceofQuadraticallyStabilizingGainMatrixNotethatbothLMI'sholdforalleandQsinceQ(0,t)=0,Vt20bytheassumedsectorconstraintonQ.Asufficientconditionfortheexistenceofagainma-The$-procedurecanthenbeusedtogiveasufficienttrixLwhichsolvesthegivennonlinearobserverdesignconditionforaquadraticconstrainttobesatisfiedgivenproblemcanbeexpressedasaconvexfeasibilityprob-thatsomeotherquadraticconstraintsarealsosatisfied.lemusingLyapunovstabilitytheory.ThesolutiontoTheS-procedurecanbesummarizedasfollows:Giventhisproblemisstatedinthefollowingtheoremandde-matricesPifori=0,...,p,thentheconditionrivedinthesubsequentproof.Theorem1.Fortheclassofsystemsandobserverx~P~x<0,andxTPix50,formsdescribedinEquations(1,2,5,3)thereexistsanVx#0andi=1,...,p2089 obviouslyholdsif3r,20fori=1,...,psuchthateigenvalueofPshouldbemaximizedtoreducethemaximumsingularvalueoftheobservergainmatrixelements.Theorem1canbeeasilymodifiedtoincludeXTPoX-~X'PaZ<0,vx#0themaximizationofXmin(P),andisstatedherewith-i=loutproof.Byapplyingtheseresultstotheinequalitieslistedabove,thefollowingconditionisobtained,Theorem2.FortheclassofsystemsandobserverformsdescribedinEquations(1,%,3,5)thereexistsanATP+PA-CTS-STCPBf+$CFKTtI,720,andG(P)+U(P)S+STU(P)T<0eATP+PA-CTCPBf+$rCfTKT<3~ER3G(P)-DU(P)U(P)~BfTP+$rKCf-7-1130->03G(P)-DU(P)U(P)~whereGandUarematricesdependentonthevariableTheresultangobservergainmatrixisL=$P-'CTP,andcisanewscalarvariable.ByapplyingthisproceduretothepreviousLMI,thefollowingconstraint4.2MaximizationofDecayRateisobtained,AnotherdesirablepropertyforanobserveristhattheATP+PA-aCTCPBf+$CFKTtI,T20,andformsdescribedinEquations(3,5,6,7)thereexistsanobservergainmatrixLwhichquadraticallystabilizesTP+PA-CTC+2aPPBf+iftI,aER,r20,andAnotherdesirablepropertyofanobserveristhatthestateestimatesshouldbeinsensitivetodisturbancesMl1M12M&-TI0inboththeplantandthesensormeasurements.ToMS0-y2I-aD;Ddconsiderthisproperty,thesystemdynamicspresentedinSection2mustbechangedtoincludedisturbanceswherethematricesM11,M12,andM13aredefinedasinboththestatedynamicsandoutputs,ie.5=AX+BUu+Bff(Cj~)+Bdd(6)y=Cx+Ddd(7)wheredEEtndisanunknownexogenousinput.TheerrordynamicsofthesystemandobserverthenbecomeTheresultingobservergainmatrixisL=$P-'CT.e=(A-LC)e+Bf$(Cfe,t)+(Bd-LDd)d(8)NoticethatthistheoremisslightlydifferentthanThe-Thesecondmetricusedtodescribetheperformanceorem1becausethehomogeneityoftheresultingLMIoftheobserveristheinducedL2gainbetweentheislostwhenincludingtheintegralquadraticconstraintdisturbancesdtotheestimationerrore,signifiedasonIIHd-+ellm-IIHd+ellm.Thisgaincanbeexpressedasanintegralquadraticconstraintondande,since4.4CombinedPerformanceusingParetoOpti-mizationAlthoughthepreviousthreesectionspresenteddiffer-entmeasuresofperformancefortheestimator,amoreTherefore,bythedefinitionofthesupremumandtherealisticproblemistochoosetheobservergainmatrixinducedL2norm,byperformingatrade-offbetweenallthreeobjectives.ATTInfact,forfixedvaluesofthedecayratea,andtheeTedt5y21dTddt,VT20IIHd-rellmgain,theperformancemetricscanbecom-binedintoasingleSDPwhichachievesafixeddecayToincludethisconstraintinamannersimilartotherateandIIHd+ellmgainwhilemaximizingXmin(P).previoussections,supposethatThedevelopmentofthiscombinedoptimizationprob-V+eTe-y2dTd50,(9)lemfollowsreadilyfromthepreviousthreesectionsandissummarizedinthefollowingtheorem,Veanddsatisfyingequation(8)Theorem5.FortheclassofsystemsandobserverthentheIIHd-,eIlm5y.Toshowthis,simplyintegrateformsdescribedinEquations(3,5,6,7),akeddecayequation(9)from0toTwiththeassumptionthatratea,ajixedIIHd+ellmgain7,andafixedconstante(0)=0.Then/3E[011,thereexistsanobservergainmatria:LwhichquadraticallystabilizestheerrordynamicsshowninV(T)+lT(eTe-y2dTd)dt50,Equation(4)andhaselementswiththesmallestmag-nitudeiftheremktsasolutiontothefollowingsemi-QT20,Qeanddsatisfyingequation(8)definiteprogram(SDP):andsinceV(T)20thisimpliesthatllHd-,ellm5y[3].SobyminimizingII&+,Ila,theworstcaseeffectoftheman/3u-(1-/3)tdisturbancesupontheestimationerrorwillbemini-subjecttoP>tI,aE8,T20,andmized.Asintheprevioussection,Theorem1canbeeasilymodifiedtoconsiderthisnewoptimizationcrite-TP+PA-aCTC+2aPPBj+rion,andisstatedherewithoutproof.BfTP+iTKCf-TI2091 Ml1MI2MS-rI0]