卡尔曼滤波

3.TheKalmanfilterisapowerfulrecursivedatafusiontoolbasedonestimationtheory.ItworksasanestimatorforastatevectorcontainingoneormorerandomvariablesthatareassumedtobeGaussiandistributed.TheunderlyingprincipleofaKalmanfilterisapredictor-correctorstructure.Inthepredictioncycle,thestatevectordynamicsaredescribedbyastatespacemodel.Thetransitionfromonestatetothenextoneatatimeisexpressedbyastatetransitionmatrixor,inthenonlinearcase,asanonlinearfunction.Assignedtothestatevectorisacovariancematrix,whichdescribesthestatisticaldispersionoftherandomstatevectorcomponents.Thecovariancematrixalsorunsthroughthepredictioncycle,wherethequalityofthestatespacemodelisdescribedbyanadditivenoisecomponentQ.AnimportantpreconditionforusingaKalmanfilteristhatatleastoneoftheelementsinthestatevectorisobservableandthatthereisadditionalinformationavailablefortheobservedelement.Thisadditionalinformationcanconsistofe.g.measurementsorknownconstraintsandiscombinedinameasurementvector.Theobservationisusedforthecorrectioncycle(oralso:filterupdate).LeftmultiplicationofthepredictedstatevectorwiththeobservationmatrixCyieldsavectorthatissubtractedfromthemeasurementvectortoobtainaresidual.AfterweightingbytheKalmangain,theresidualisaddedtothepredictedstatevectortoobtainafilterestimate.TheKalmangainincludesthepredictionerrorcovarianceoftheobservedelementsinthestatevectorandthemeasurementnoiseR.FromtheformulaforthecalculationoftheKalmangainitcanbeseenthattheresidualwillbeweightedlessthehigherthemeasurementnoiseis.Inotherwords,iftheaccuracyofthepredictionishigh(lowcovariancevalues)comparedtothemeasurementnoise,theKalmanfilterwill“follow”ratherthestatespacemodelthanthemeasurements.Iftheaccuracyofthepredictionislowbutthequalityofthemeasurementsishigh,theoppositecaseoccurs.AdetailedexplanationoftheKalmanfilteralgorithmanditscompletederivationcanbefoundin[LO90a],[LO90b].a)LinearKalmanFilterIfthestatetransitionandtheobservationarelinear,theKalmanfiltercanbedescribedbyasimplesetofformulas.Predictioncycle:xˆ−=A⋅xˆ+−k+1=A⋅P+⋅AT+Qwithxˆ:n-elementstatevectorA:n×nstatetransitionmatrixP:covarianceassignedtothestatevectorQ:drivingnoiseKalmangainandresidual:(B.1.1)K=P−·CT⋅C⋅P−−1⋅C+Rk+1k+1k+1resk+1=yb(k+1)−C⋅xˆk+1withK:Kalmangainres:residual(B.1.2)y:m-elementmeasurementvectorbUpdatecycle:C:PkTm×nobservationmatrixR:measurementnoise+−k+1k+1k+1k+1+−−k+1k+1k+1k+1(B.1.3)Thesuperscript“–”denotesthepredictedvaluesandthesuperscript“+”indicatesthefilterestimateanditscovariance.b)ExtendedKalmanFilterIfthestatetransitionand/ortheobservationarenotlinear,theycannotbewritteninmatrixform.Inthiscaseitisnecessarytolinearizeitbydifferentiationwithrespecttotheelementsofthestatevector,becausethecovariancecalculationintheKalmanfilteralgorithmusesmatrixcalculus.Forthecalculationofthepredictioncovariance,asquarestatetransitionmatrixisneeded.ThelinearizedobservationmatrixHmustbeconditionedinawaythattheresultoftheterminvertible.H⋅P−⋅HT+R(innovation,seecalculationoftheKalmangain)isThenecessarymatricesaregeneratedbydifferentiationwithrespecttothevariablesinthestatevector.ThefollowingexampleshowsthecalculationofalinearizedstatetransitionmatrixA:xˆ=x1;xˆ−=f(x)f1(x1,x2)+=xk+1xˆkf(x,x)2212xˆ+ddA=df(x)dx=1f1(x1,x2)dx2f1(x1,x2)(B.1.4)dx+dfxxdfxxxˆkdx12(1,2)dx12(1,2)xˆkThecalculationofalinearizedobservationmatrixisdoneanalogously,butthentheobservationvectoryisafunctionofx.Thisisforexamplethecase,whenthestatescannotbeobserveddirectlybutviaintermediatestatesthatdependnonlinearlyonthestatesoftheKalmanfilter.Anexampleforthiscasecanbefoundinchapter2,section2.2.1.4:PointofEllipsoidalIntersection.Theintermediatestatesare:Here,fisanonlinearfunctionofx.y=f(x)(B.1.5)Thelinearizedobservationmatrixiscalculatedbydifferentiatingfo(x)withrespecttox.Here,thetwodimensionalcaseisshown:dfxxdfxxo,1(1,2)o,1(1,2)H=df(x)=dx1dx2(B.1.6)dxo−ddxˆkdx1fo,2(x1,x2)dx1fo,2(x1,x2)xˆkTheresultofthedifferentiationmustbeevaluatedforthepredictedstates.ThecompleteFilteralgorithmis:ˆx−=A⋅ˆx+P−=A⋅P+−AT+qT−T−1Kk+1=Pk+1⋅H·H⋅Pk+1⋅H+R(B.1.7)+−ˆxk+1=xˆk+1+Kk+1⋅yb−yPk+1=Pk+1−Kk+1⋅H⋅Pk+1koReferences[EN02]Ender,J.H.G,Berens,P,Brenner,A.R,Rossing,L,Skupin,andU,Multi-channelSAR/MTIsystemdevelopmentatFGAN:fromAERtoPAMIR,presentedatIEEEInternationalGeoscienceandRemoteSensingSymposium,IGARSS'02,2002.[MA01]MassonnetandD,Capabilitiesandlimitationsoftheinterferometriccartwheel,GeoscienceandRemoteSensing,IEEETransactionson,vol.39,pp.506-520,2001.[HE04]Hein,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-