卡尔曼滤波

3.TheKalmanfilterisapowerfulrecursivedatafusiontoolbasedonestimationtheory.Itworksas an estimator forastatevector containingoneor morerandomvariablesthat are assumedtobeGaussiandistributed. Theunderlyingprinciple ofaKalman filterisapredictor-correctorstructure.Inthepredictioncycle,thestatevectordynamicsaredescribedb yastatespacemodel. Thetransition from onestatetothenextoneatatimeisexpressedbyastatetransitionmatrixor,inthenonlinearcase,asanonli nearfunction.Assignedtothestatevectorisacovariancematrix,whichdescribesthestatisticaldispersi onoftherandomstatevectorcomponents.Thecovariancematrixalsorunsthroughthepredictioncycle ,wherethequalityofthestatespacemodelisdescribedbyanadditivenoisecomponent Q .AnimportantpreconditionforusingaKalmanfilteristhatatleastoneoftheelementsinthestatevectoris observableandthatthereisadditionalinformationavailablefortheobservedelement.Thisadditionali nformationcanconsistofe.g.measurementsorknownconstraintsandis combinedinameasurement vector. Theobservation is used forthecorrectioncycle(oralso:filterupdate).Leftmultiplicationofthepredictedstatevectorwiththeo bservationmatrix C yieldsavectorthatissubtractedfromthemeasurementvectortoobtainaresidual.A fterweightingbytheKalmangain,theresidualisaddedtothepredictedstatevectortoobtainafilteresti mate.TheKalmangainincludesthepredictionerrorcovarianceoftheobservedelementsinthestatevector andthemeasurement noise R .From the formulaforthecalculationoftheKalmangainitcan beseen thattheresidualwillbeweightedlessthehigherthemeasurementnoiseis.Inotherwords,iftheaccuracyofthepredictionishig h(lowcovariancevalues)comparedtothemeasurementnoise,theKalmanfilterwill “f ollow ”ratherth estatespacemodelthanthemeasurements.Iftheaccuracyofthepredictionislowbutthequality ofthemeasurementsishigh,theoppositecaseoccurs.AdetailedexplanationoftheKalmanfilteralgorithmanditscompletederivationcanbefoundin[LO90a],[LO90b].a) LinearKalmanFilterIfthestatetransitionandtheobservationarelinear,theKalmanfiltercanbedescribedbyasimplesetoff ormulas.Predictioncycle:x ˆ− =A ⋅xˆ+ − k +1 =A ⋅P +⋅A T +Q with x ˆ: n-elementstatevector A : n×nstatetransitionmatrix P : covarianceassignedtothestatevector Q : "drivingnoise"Kalmangainandresidual:(B.1.1)K=P − · C T ⋅⎡C ⋅P − −1⋅C +R ⎤ k +1 k +1 ⎣k +1 ⎦ res k +1 =y b (k +1)−C ⋅x ˆk +1 withK : Kalman gain res : residual(B.1.2)y :m-elementmeasurementvectorbUpdatecycle:C :k +1 k P k T −m ×nobservati onmatrix R : measu rementnoise+ − k +1 k +1 k +1 k +1 + − −k +1k +1k +1k +1(B.1.3)Thesuperscript “–”d enotesthepredictedvaluesandthesuperscript “+”indi catesthefilterestimateand itscovariance.b) ExtendedKalmanFilterIfthestatetransitionand/ortheobservationarenotlinear,theycannotbewritteninmatrixform.Inthisca seitisnecessarytolinearizeitbydifferentiationwithrespecttotheelementsofthestatevector,becauset hecovariancecalculationintheKalmanfilteralgorithmusesxˆ =xˆ +K⋅res P =P −K⋅C ⋅Pmatrixcalculus.Forthecalculationofthepredictioncovariance,asquarestatetransitionmatrixisneed ed.Thelinearizedobservationmatrix H mustbeconditionedinawaythattheresult ofthe terminvertible.H ⋅P − ⋅H T +R (innovation, seecalculation of the Kalmangain) is Thenecessarymatricesaregeneratedbydifferentiationwithrespecttothevariablesinthestatevector.Thefollowingexampleshowsthecalculationofalinearizedstatetransitionmatrix A :x ˆ=⎡ x 1⎤ ; x ˆ−=f (x ) ⎡ f 1(x 1,x 2)⎤ + =⎢ ⎥ xk +1 x ˆk f (x ,x ) ⎣ 2⎦⎣ 2 1 2 ⎦ x ˆ+⎡ d d ⎤ ⇒ A = d ⎡ f (x )⎤ ⎢ dx =⎢ 1 f 1(x 1,x 2) dx 2 f 1(x 1,x 2)⎥ ⎥ (B.1.4) ⎦ ⎣ ⎦ ⎢ ⎥dx + d f x x d f x x x ˆk⎣ dx 12(1, 2)dx 12(1,2)⎥⎦ x ˆkThecalculationofalinearizedobservationmatrixisdoneanalogously,butthentheobservation vector y isafunction of x . Thisisforexamplethe case, when thestatescannotbeobserveddirectlybutviaintermediatestatesthatdependnonlinearlyonthestatesof theKalmanfilter.Anexampleforthiscasecanbefoundinchapter2,section2.2.1.4:PointofEllipsoidal Intersection.Theintermediatestatesare:Here,f isanonlinearfunctionof x .y =f(x )(B.1.5)Thelinearizedobservationmatrixiscalculatedbydifferentiating f o (x )withrespectto x .Here,thetwodi mensionalcaseisshown:⎡ d f x x d f x x ⎤⎢ o ,1(1, 2) o ,1(1, 2)⎥ H = d⎡ f (x )⎤ =⎢ dx 1 dx 2 ⎥ (B.1.6) dx ⎣ o ⎦ − ⎢ d d ⎥x ˆk⎣ dx 1f o ,2(x 1,x 2)dx 1f o ,2(x 1,x 2)⎥⎦ x ˆkTheresultofthedifferentiationmustbeevaluatedforthepredictedstates.ThecompleteFilteralgorithmis:ˆx − =A ⋅ˆx + P − = A ⋅P +−A T +q T − T−1K k +1 =P k +1⋅H · ⎡H ⋅P k +1⋅H +R ⎤(B.1.7)⎣⎦+−ˆx k +1 =xˆk +1+K k +1⋅⎡ y b −y ⎤ P k +1 =P k +1−K k +1⋅H⋅P k +1k +1⎢ k ⎢ + o⎢ − k +1kk +1 k ⎣ +− −References[EN02]Ender,J.H.G,Berens,P,Brenner,A.R,Rossing,L,Skupin,andU,"Multi-channelSAR/MTIs ystemdevelopmentatFGAN:fromAERtoPAMIR,"presentedatIEEEInternationalGeoscienceandRemoteSensingSymposium,IGARSS'02,2002.[MA01]MassonnetandD,"Capabilitiesandlimitationsoftheinterferometriccartwheel,"GeoscienceandRemoteSensing,IEEETransactionson,vol.39,pp.506-520,2001.[HE04]H ein,Achim,ProcessingofSAR-Data-Fundamentals,SignalProcessing,Interferometry,vol.XV:Springer-Verlag,2004.[LO04a]Loffeld,O,Nies,H,Peters,V,Knedlik,andS,"ModelsandusefulrelationsforbistaticSARprocessing,"IEEETransactionsonGeoscienceandRemoteSensing,vol.42,pp.2031-2038,2004.[WG08]Wang,R,Loffeld,O,Qurat,U,etal.,"AnalysisandExtensionofLoffeld’sBistaticFormulainSpaceborne/AirborneConfiguration,"presentedatEusar2008,Friedrichshafen,Germany,2008[LO04b]Loffeld,O,Nies,H,Gebhardt,U,Peters,V,Knedlik,S,Wiechert,andW,"BistaticSAR-SomeReflectionsonRocca'sSmile,"presentedatEusar2004,Ulm,Germany,2004.[WA05a]Walterscheid,I,Brenner,A,Ender,J.H.G,Loffeld,andO,"AbistaticairborneSARexperimentandprocessingresults,"presentedatIGARSSInternationalGeoscienceandRemoteSensingSymposium,Seoul,Korea,2005.[WA05b]Walterscheid,I,Ender,J.H.G,Brenner,A,Loffeld,andO,"BistaticSARProcessingusinganomega-ktypealgorithm,"presentedatIGARSSInternationalGeoscienceandRemoteSensingSymposium,Seoul,Korea,2005.[EN06] Ender,J.H.G,Klare,J,Walterscheid,I,Brenner,A.R,Weiß,M,Kirchner,C,Wilden,H,Loffeld,O,Kolb,A,Wiechert,W,Kalkuhl,M,Knedlik,S,Gebhardt,U,Nies,H,Natroshvili,K,Ige,S,MedranoOrtiz,A,Amankwah,andA,"BistaticExporationUsingSpaceborneandAirborneSARSensors:ACloseCollaborationBetweenFGAN,ZESSandFOMAAS,"presentedatIeeeIgarss,Denver,Colorado,2006.[MI03]M ittermayer,J,Lord,R,Borner,andE,SlidingspotlightSARprocessingforTerraSAR-Xusinganewformulationoftheextendedchirpscalingalgorithm,vol.3.Toulouse,France,2003.[GE05]G ebhardt,U,Loffeld,O,Nies,H,Knedlik,S,Ender,andH.J.G,"Bistaticairborne/spacebornehybridexperiment:BasicConsiderations,"presentedatSPIEInternationalSymposiumonRemoteSensing,Brügge,Belgium,2005.[GE06]G ebhardt,U,Loffeld,O,Nies,H,Natroshvili,K,Ender,andH.J.G,"BistaticAirborneSpaceHybridExperiment:SimulationandAnalysis,"presentedat6thEuropeanConferenceonSyntheticApertureRadarEUSAR,Dresden,Germany,2006.114 References [WA06]Walterscheid,I,Klare,J,Brenner,A,Ender,H.H.G,Loffeld,andO,"Challengesofabistaticspaceborne/airborneSARexperiment,"presentedat6htEuropeanConferenceonSyntheticApertureRadar,Dresden,Germany,2006. [EN07]Ender,J,"TheDoubleSlidingSpotlightModeforBistaticSAR,"presentedatIRS(InternationalRadarSymposium)2007,Cologne,Germany,2007[NI06]H.Nies,ZentraleunddezentraleEstimationsverfahreninMulti-SensorsystemenundderenAnwendungamBeispielhochgenauerPositionsbestimmung,vol.23.Aachen,Germany:ShakerVerlag,2006.[NA06]Natroshvili,K,Loffeld,O,Nies,H,Ender,andJoachimH.G,"2Dinversescalingbistaticprocessingandthefocusedimagequalitymeasurements,"presentedatInternationalGeoscienceandRemoteSensingSymposium-IGARSS2006,Colorado,2006.[MG00]Montenbruck,O.,Gill,E.,“SatelliteOrbits”,Springer,Corrected3rd Printing2005,ISBN3-540-67280-x[LA95]LaudetP.;DeleuzeM.;GuitardA.;PiuzziA.;ValorgeC.;PreciseOrbitDeterminationwithDORIS;CarrouJ.P.(ed.),SpaceflightDynamics;Cépaduès-Éditions(1995))[KN92]R.Knüfermann,"SimulationerdnaherSatellitenbahnenunterEinbeziehungrealistischerLuftwiderstands-undGravitationsmodellesowiestochastischerStöreinflüsse,"in FachbereichElektrotechnik.Siegen:UniversitätSiegen,1992.[KL83]H.-H.Klinkrad,"AnalytischeBerechnungerdnaherSatellitenbahnenunterVerwendungeinesrealistischenLuftwiderstandsmodells,"in FakultätfütMaschinenbauundElektrotechnik.Braunschweig:TechnischeUniversitätCarolo-WilhelminazuBraunschweig,1983.[KA03]M.Kalkuhl,"ErdnaheOrbitsimulationeinesinterferometrischenCart-Wheels,"inInstitutfürSystemtschnik.Siegen:UniversitätSiegen,2003.[RE]T.Reubelt,"AnalysederSatellitenbewegungsgleichungmitHilfekartesischerBahnkoordinatenamBeispieldesCHAMP-Satelliten,"in GeodätischesInstitut.Stuttgart:UniversitätStuttgart.[AU00]G.R.Austen,Tilo,"RäumlicheSchwerefeldanalyseaussemi-kontinuierlichenEphemeridenniedrigfliegenderGPS-vermessenerSatellitenvomTypCHAMP,GRACEundGOCE,"in GeodätischesInstitut.Stuttgart:UniversitätStuttgart,2000. [Sch96]M.Scheinert(1996)ZurBahndynamikniedrigfliegenderSatelliten UniversitätStuttgart[REI]R eigber,Ch.,Balmino,G.,Schwintzer,P.,Biancale,R.etal.,"AhighqualityglobalgravityfieldmodelfromCHAMPGPStrackingdataandAccelerometry(EIGEN-1S),"GeophysicalResearchLetters,29(14),10.1029/2002GL015064,2002[TA]T apley,B.D.,S.Bettadpur,M.Watkins,andC.Reigber,"Thegravityrecoveryandclimateexperiment:Missionoverviewandearlyresults,"GeophysicalResearchLetters,31,L09607,doi:10.1029/2004GL019920,May2004.References 115 [ALB]A.Albertella,F.Migliaccio,F.Sansò,"GOCE–TheEarthFieldbySpaceGradiometry,"CelestialMechanicsandDynamicalAstronomy83,pp.1–15.[GE04]G ebhardt,U,Loffeld,O,Kalkuhl,M,Nies,H,Knedlik,andS,"OrbitTrackingandInterpolationUsingaRealisticGravitationModel,"presentedatIGARSSInternationalGeoscienceandRemoteSensingSymposium,Anchorage,2004.[LO90a]Loffeld,O,EstimationstheorieI-GrundlagenundstochastischeKonzepte:Oldenbourg-Verlag,1990.[LO90b]Loffeld,O,EstimationstheorieII-Anwendungen-Kalman-Filter:Oldenbourg-Verlag,1990.[KAL60]Kalman,R.E.,“ANewApproachtoLinearFilteringandPredictionProblems”,ResearchInstituteforAdvancedStudy,Baltimore,Md.,1960Author’sremark:ThispapercanbefoundontheWorldWideWeb:/~welch/kalman/media/pdf/Kalman1960.pdf。