经典kalman滤波PPT.ppt

1,IntroductiontoKalmanFilters,MichaelWilliams5June2003,2,Overview,TheProblemWhydoweneedKalmanFilters?WhatisaKalmanFilter?ConceptualOverviewTheTheoryofKalmanFilterSimpleExample,3,TheProblem,SystemstatecannotbemeasureddirectlyNeedtoestimate“optimally”frommeasurements,MeasuringDevices,Estimator,MeasurementErrorSources,SystemState(desiredbutnotknown),ExternalControls,ObservedMeasurements,OptimalEstimateofSystemState,SystemErrorSources,System,BlackBox,4,WhatisaKalmanFilter?,RecursivedataprocessingalgorithmGeneratesoptimalestimateofdesiredquantitiesgiventhesetofmeasurementsOptimal?ForlinearsystemandwhiteGaussianerrors,Kalmanfilteris“best”estimatebasedonallpreviousmeasurementsFornon-linearsystemoptimalityisqualifiedRecursive?Doesntneedtostoreallpreviousmeasurementsandreprocessalldataeachtimestep,5,ConceptualOverview,SimpleexampletomotivatetheworkingsoftheKalmanFilterTheoreticalJustificationtocomelaterfornowjustfocusontheconceptImportant:PredictionandCorrection,6,ConceptualOverview,Lostonthe1-dimensionallinePositiony(t)AssumeGaussiandistributedmeasurements,y,7,ConceptualOverview,SextantMeasurementatt1:Mean=z1andVariance=z1Optimalestimateofpositionis:(t1)=z1Varianceoferrorinestimate:2x(t1)=2z1Boatinsamepositionattimet2-Predictedpositionisz1,8,ConceptualOverview,Sowehavetheprediction-(t2)GPSMeasurementatt2:Mean=z2andVariance=z2Needtocorrectthepredictionduetomeasurementtoget(t2)Closertomoretrustedmeasurementlinearinterpolation?,prediction-(t2),measurementz(t2),9,ConceptualOverview,CorrectedmeanisthenewoptimalestimateofpositionNewvarianceissmallerthaneitheroftheprevioustwovariances,measurementz(t2),correctedoptimalestimate(t2),prediction-(t2),10,ConceptualOverview,Lessonssofar:,Makepredictionbasedonpreviousdata-,-,Takemeasurementzk,z,Optimalestimate()=Prediction+(KalmanGain)*(Measurement-Prediction),Varianceofestimate=Varianceofprediction*(1KalmanGain),11,ConceptualOverview,Attimet3,boatmoveswithvelocitydy/dt=uNaveapproach:ShiftprobabilitytotherighttopredictThiswouldworkifweknewthevelocityexactly(perfectmodel),(t2),NavePrediction-(t3),12,ConceptualOverview,BettertoassumeimperfectmodelbyaddingGaussiannoisedy/dt=u+wDistributionforpredictionmovesandspreadsout,(t2),NavePrediction-(t3),Prediction-(t3),13,ConceptualOverview,Nowwetakeameasurementatt3NeedtoonceagaincorrectthepredictionSameasbefore,Prediction-(t3),Measurementz(t3),Correctedoptimalestimate(t3),14,ConceptualOverview,Lessonslearntfromconceptualoverview:Initialconditions(k-1andk-1)Prediction(-k,-k)Useinitialconditionsandmodel(eg.constantvelocity)tomakepredictionMeasurement(zk)TakemeasurementCorrection(k,k)UsemeasurementtocorrectpredictionbyblendingpredictionandresidualalwaysacaseofmergingonlytwoGaussiansOptimalestimatewithsmallervariance,15,TheoreticalBasis,Processtobeestimated:,yk=Ayk-1+Buk+wk-1,zk=Hyk+vk,ProcessNoise(w)withcovarianceQ,MeasurementNoise(v)withcovarianceR,KalmanFilter,Predicted:-kisestimatebasedonmeasurementsatprevioustime-steps,k=-k+K(zk-H-k),Corrected:khasadditionalinformationthemeasurementattimek,K=P-kHT(HP-kHT+R)-1,-k=Ayk-1+Buk,P-k=APk-1AT+Q,Pk=(I-KH)P-k,16,BlendingFactor,Ifwearesureaboutmeasurements:Measurementerrorcovariance(R)decreasestozeroKdecreasesandweightsresidualmoreheavilythanpredictionIfwearesureaboutpredictionPredictionerrorcovarianceP-kdecreasestozeroKincreasesandweightspredictionmoreheavilythanresidual,17,TheoreticalBasis,18,QuickExampleConstantModel,MeasuringDevices,Estimator,MeasurementErrorSources,SystemState,ExternalControls,ObservedMeasurements,OptimalEstimateofSystemState,SystemErrorSources,System,BlackBox,19,QuickExampleConstantModel,Prediction,k=-k+K(zk-H-k),Correction,K=P-k(P-k+R)-1,-k=yk-1,P-k=Pk-1,Pk=(I-K)P-k,20,QuickExampleConstantModel,21,QuickExampleConstantModel,ConvergenceofErrorCovariance-Pk,22,QuickExampleConstantModel,LargervalueofRthemeasurementerrorcovariance(indicatespoorerqualityofmeasurements),Filterslowertobelievemeasurementsslowerconvergence,23,References,Kalman,R.E.1960.“ANewApproachtoLinearFilteringandPredictionProblems”,TransactionoftheASME-JournalofBa