卡尔曼滤波美国北卡大学

IntroductiontoKalmanFilterGregWelch&GaryBishopIntroductiontoKalmanFilterGregWelch&GaryBishopASimpleExampleforSingleSensorDataProcessing—BatchProcessingModel12,,,nXXXarenmeasurementsforavariableSupposetheprecisionofeachmeasurementisthesame,whyisbetterthan?iXninXY22ˆnXXXYnn21ˆnYˆASimpleExampleforSingleSensorDataProcessing—RecursiveProcessingModel11ˆXY1ststep:2121221ˆ212ˆXYXXYnnnnXnYnnnXXXY1ˆ1ˆ1212ndstep:3rdstep:nthstep:32321331ˆ323ˆXYXXXYASimpleExampleforSingleSensorDataProcessing—RecursiveProcessingModel11ˆXY121212ˆ21ˆ2ˆYXYXXY2323213ˆ31ˆ3ˆYXYXXXY1121ˆ1ˆˆnnnnnYXnYnXXXY1ststep:2ndstep:3rdstep:nthstep:ASimpleExampleforSingleSensorDataProcessing—RecursiveProcessingModel221ˆ2212122221ˆ21ˆ1ˆK2222232331ˆ32ˆ1ˆK2212121ˆ1ˆ1ˆnnnKnnnn1ststep:2ndstep:3rdstep:nthstep:WhyKalmanFilter?Whatiftheprecisionforeachmeasurementisdifferent?Solution:KalmanFilter!!!AnIntuitiveExampleforKalmanFilterConsideraquantity(x),forexamplealength,thatismeasuredtwicewiththesameorwithdifferentmeasurementequipment,forexampleamechanicalrulerandalasersystem.Thetwomeasurementsarenotedand;theyarecharacterizedbyGaussianprobabilitydistributionswithmeansandandstandarddeviationsand:12221-exp21iiiiixxxp1x2x1x2xAnIntuitiveExampleforKalmanFilterThetwomeasurementsarecombinedtogiveanestimateofthelength:21)1(ˆxwwxxBecauseandhaveGaussiandistributions,hasalsoaGaussiandistributionwithstandarddeviationgivenby(andareindependent)1x2xxˆˆ2222122)1(ˆww1x2xAnIntuitiveExampleforKalmanFilterHence:222122optw22221211222122ˆxxx222122212ˆAnIntuitiveExampleforKalmanFilterToestimateisequaltominimizethesumofthedistancestoand,weightedbytherespectivestandarddeviations:1x2xxˆ222211xminargˆxxxxxAnIntuitiveExampleforKalmanFilterSupposethetwomeasurementsbecomeavailablesequentially.Attimestep1measurementbecomesavailable.Sincethisistheonlyinformation,thestateestimateanditsvarianceare:11ˆxx11ˆAnIntuitiveExampleforKalmanFilterThen,attimestep2,measurementbecomesavailable,andtheestimateis:2x2122212122ˆˆˆ1ˆ1222212112ˆˆˆˆˆxxxxAnIntuitiveExampleforKalmanFilterAssuming:222121ˆˆK2122ˆ1ˆKWehave:K:updategain;12ˆxx:innovation1212ˆˆˆxxKxxAnIntuitiveExampleforKalmanFilterPeterSMaybeck—Stochasticmodels,estimation,andcontrolStochasticProcessModelforKalmanFilterEdwardV.StansfieldStochasticProcessModelforKalmanFilterEdwardV.StansfieldNotationsforKalmanFilterEdwardV.StansfieldKalmanFilterEquationsEdwardV.StansfieldLinearUnbiasedEstimatorEdwardV.StansfieldMinimumVarianceEstimatorEdwardV.StansfieldMinimumPredictionErrorCovarianceEdwardV.StansfieldMinimumEstimationErrorCovarianceEdwardV.StansfieldKalmanFilterProcessModelGregWelch&GaryBishopKalmanFilterEstimatorModelEdwardV.StansfieldKalmanFilterProcessModelEdwardV.StansfieldWhatisKalmanFilter?AKalmanfilterisalinear,modelbased,stochastic,recursive,weighted,leastsquaresestimator.Estimator:theKFestimatesthestateofasystem,orpartofit,basedontheknowledge(measurement)ofthesysteminputsandoutputs.Modelbased&linear:TheKFisbasedonasystemmodelconsistingofastateequationandanoutput(measurement)equation,whicharealllinear.Leastsquares:TheKFthenprovidestheestimatethattriestominimizetheinconsistencieswithallpiecesofinformationintheleastsquaressense.Inthisrespect,theKFisanoptimalestimator.WhatisKalmanFilter?AKalmanfilterisalinear,modelbased,stochastic,recursive,weighted,leastsquaresestimator.Weighted:Whenminimizingthesumoftheirleastsquares,theinconsistencieswiththedifferentpiecesofinformationareweightedwithameasureofthecertaintyoftheinformation.Uncertaininformationisgivenlowweight,whereashighlycertaininformationisgivenaveryhighweight.Recursive:Whenallinformationisavailableatonce,itcanbeprocessedinbatch.If,however,theinformationbecomesavailableincrementally,asisthecaseforanon-lineestimator,arecursiveformulationoftheestimationprocessisnecessary.TheKFdoesnothingmorethanthat.WhatisKalmanFilter?AKalmanfilterisalinear,modelbased,stochastic,recursive,weighted,leastsquaresestimator.Stochastic:Theconfidenceaboutpiecesofinformationisexpressedintermsofprobabilitydistributions.TheKFworkswithGaussiandistributionsforbothmeasurementsandstateestimates.AKalmanfilterisastochastic,recursiveestimator,whichestimatesthestateofasystembasedontheknowledgeofthesysteminput,themeasurementofthesystemoutput,andamodeloftherelationbetweeninputandoutput.AnExample:EstimatingaRandomConstantAttempttoestimateascalarrandomconstant,avoltageforexample.Assumethatthemeasurementsarecorruptedbya0.1voltRMSwhitemeasurementnoise(e.g.ouranalogtodigitalconverterisnotveryaccurate).kxkwxx1kxkvxzStateEquationObservationEquation0,1fφ1HAnExample:EstimatingaRandomConstant10PGregWelch&GaryBishop0QAnExample:EstimatingaRandomConstant10PGregWelch&GaryBishop0QAnExample:EstimatingaRandomConstantGregWelch&GaryBishopAnExample:EstimatingaRandomConstantGregWelch&GaryBishopExampleforMulti-sensorDataFusion