经典kalman滤波PPT.ppt

1 IntroductiontoKalmanFilters MichaelWilliams5June2003 2 Overview TheProblem WhydoweneedKalmanFilters 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 linearsystemoptimalityis qualified Recursive Doesn tneedtostoreallpreviousmeasurementsandreprocessalldataeachtimestep 5 ConceptualOverview SimpleexampletomotivatetheworkingsoftheKalmanFilterTheoreticalJustificationtocomelater fornowjustfocusontheconceptImportant PredictionandCorrection 6 ConceptualOverview Lostonthe1 dimensionallinePosition y t AssumeGaussiandistributedmeasurements y 7 ConceptualOverview SextantMeasurementatt1 Mean z1andVariance z1Optimalestimateofpositionis t1 z1Varianceoferrorinestimate 2x t1 2z1Boatinsamepositionattimet2 Predictedpositionisz1 8 ConceptualOverview Sowehavetheprediction t2 GPSMeasurementatt2 Mean z2andVariance z2Needtocorrectthepredictionduetomeasurementtoget t2 Closertomoretrustedmeasurement linearinterpolation prediction t2 measurementz t2 9 ConceptualOverview CorrectedmeanisthenewoptimalestimateofpositionNewvarianceissmallerthaneitheroftheprevioustwovariances measurementz t2 correctedoptimalestimate t2 prediction t2 10 ConceptualOverview Lessonssofar Makepredictionbasedonpreviousdata Takemeasurement zk z Optimalestimate Prediction KalmanGain Measurement Prediction Varianceofestimate Varianceofprediction 1 KalmanGain 11 ConceptualOverview Attimet3 boatmoveswithvelocitydy dt uNa veapproach ShiftprobabilitytotherighttopredictThiswouldworkifweknewthevelocityexactly perfectmodel t2 Na vePrediction t3 12 ConceptualOverview BettertoassumeimperfectmodelbyaddingGaussiannoisedy dt u wDistributionforpredictionmovesandspreadsout t2 Na vePrediction t3 Prediction t3 13 ConceptualOverview Nowwetakeameasurementatt3NeedtoonceagaincorrectthepredictionSameasbefore Prediction t3 Measurementz t3 Correctedoptimalestimate t3 14 ConceptualOverview Lessonslearntfromconceptualoverview Initialconditions k 1and k 1 Prediction k k Useinitialconditionsandmodel eg constantvelocity tomakepredictionMeasurement zk TakemeasurementCorrection k k Usemeasurementtocorrectpredictionby blending predictionandresidual alwaysacaseofmergingonlytwoGaussiansOptimalestimatewithsmallervariance 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 khasadditionalinformation themeasurementattimek 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 QuickExample ConstantModel MeasuringDevices Estimator MeasurementErrorSources SystemState ExternalControls ObservedMeasurements OptimalEstimateofSystemState SystemErrorSources System BlackBox 19 QuickExample ConstantModel 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 QuickExample ConstantModel 21 QuickExample ConstantModel ConvergenceofErrorCovariance Pk 22 QuickExample ConstantModel LargervalueofR themeasurementerrorcovariance indicatespoorerqualityofmeasurements Filterslowerto believe measurements slowerconvergence 23 References Kalman R E 1960 ANewApproachtoLinearFilteringandPredictionProblems TransactionoftheASME Journa