edu Kalman and Extended Kalman Filtering brPage 2br Kalman Filter Introduction Recursive LS RLS was for static data estimate the signal better and better as more and more data comes in eg estimating the mean intensity of an object from a video sequen ID: 29948
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KalmanFilterandExtendedKalmanFilterNamrataVaswani,namrata@iastate.edu KalmanandExtendedKalmanFiltering Thesignalandobservationnoisesareassumeduncorrelated(witheachotherandovertime).Theyarealsouncorrelatedwiththeinitialstatex0 DenoteYi,fy1;y2;:::yigGoal:getthebest(minimummeansquareerror)estimateofxifromYi,fy1;y2;:::yigwhereMeansquareerrorisgivenbyJ(^xi)=E[(xi^xi)2jYi]Minimizeristheconditionalmean^xi=E[xijYi]Note:ThisisalsotheMAPestimate,i.e.^xialsomaximizesp(xijYi)(p(xijYi)isGaussian(willbeshown)andforGaussianpdfs,mean=MAP). KalmanandExtendedKalmanFiltering from(2)).ThusZ1andZ2arejointlyGaussianwithZ1,xijYi1»N(^xiji1;Piji1)(followsfrom(3))Z2jZ1,yijxi;Yi1=yijxi»N(hixi;Ri)(followsfrom(2)) (4)UsingBayesrule,onecancomputetheconditionaldistributionofZ1jZ2=xijYi(whichwillalsobeGaussian).ApplyingtheformulasfromPg155(equationIV.B.49)ofPoor'sbook(AnIntroductiontoSignalDetectionandEstimation),wehave^xi,E[xijYi]=^xiji1+Ki(yiHi^xiji1)Pi,Var(xijYi)=(IKiHi)Piji1;whereKi=Piji1HTi(Ri+HiPiji1HTi)1(5) KalmanandExtendedKalmanFiltering ExampleApplications:KalmanFilterv/sRecursiveLS Kalmanlter:Trackamovingobject(estimateitslocationandvelocityateachtime),assumingthatvelocityatcurrenttimeisvelocityatprevioustimeplusGaussiannoise).Useasequenceoflocationobservationscominginsequentially.RecursiveLS:Keepupdatingestimateoflocationofanobjectthatisstatic.Useasequenceoflocationobservationscominginsequentially.RecursiveLSwithforgettingfactor:objectnotstaticbutdriftsveryveryslowly. KalmanandExtendedKalmanFiltering SummarizingtheExtendedKF Fi=@fi @x(^xi1)^xiji1=i(^xi1)Piji1=FiPi1FTi+iHi=@hi @x(^xiji1)Ki=Piji1HTi(Ri+HiPiji1HTi)1^xi=^xiji1+Ki(zihi(^xiji1))Pi=(IKiHi)Piji1 KalmanandExtendedKalmanFiltering