Lecture for Kalman filter.pdf

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1、STCLectureSeriesAnIntroductiontotheKalmanFilterGregWelchandGaryBishopUniversityofNorthCarolinaatChapelHillDepartmentofComputerSciencehttp://www.cs.unc.edu/~welch/kalmanLinks.htmlUNCChapelHillComputerScienceSlide1WhereWe’reGoing•Introduction&Intuition•TheDiscreteKalmanFilter•ASimpleExample•Variation

2、softheFilter•RelevantApplications&ReferencesUNCChapelHillComputerScienceSlide2TheKalmanfilter•SeminalpaperbyR.E.Kalman,1960•Setofmathematicalequations•Optimalestimator–minimummeansquareerror•Versatile–Estimationpredictpredictcorrectcorrect–Filtering–Prediction–FusionUNCChapelHillComputerScienceSlid

3、e3WhyaKalmanFilter?•Efficient“least-squares”implementation•Past,presentandfutureestimation•Estimationofmissingstates•Measureofestimationquality(variance)•Robust–forgivinginmanyways–stablegivencommonconditionsUNCChapelHillComputerScienceSlide4SomeIntuitionUNCChapelHillComputerScienceSlide5FirstEstim

4、ateConditionalDensityFunction2z1,sz1xˆ=z11N(z,sz2)1122sˆ1=sz1-202468101214UNCChapelHillComputerScienceSlide6SecondEstimateConditionalDensityFunction2z2,sz22N(z,sz)2xˆ=...?222sˆ2=...?-202468101214UNCChapelHillComputerScienceSlide7CombineEstimates222222xˆ2=[]sz2()sz1+sz2z1+[]sz1()sz1+sz2z2=xˆ1+K2[]z2

5、-xˆ1where222K2=sz()sz+sz112UNCChapelHillComputerScienceSlide8CombineVariances2221s2=()1sz+()1sz12UNCChapelHillComputerScienceSlide9CombinedEstimateDensityConditionalDensityFunctionN(x,ˆsˆ2)xˆ=xˆ222sˆ=s2-202468101214UNCChapelHillComputerScienceSlide10AddDynamicsdx/dt=v+wwherevisthenominalvelocitywis

6、anoiseterm(uncertainty)UNCChapelHillComputerScienceSlide11PropagationofDensityUNCChapelHillComputerScienceSlide12SomeDetailsxxA=+AxxwwzzH=Hxx........UNCChapelHillComputerScienceSlide13DiscreteKalmanFilterMaintainsfirsttwostatisticalmomentsprocessstate(mean)zyxerrorcovarianceUNCChapelHillComputerSci

7、enceSlide14DiscreteKalmanFilterTheIngredients•Adiscreteprocessmodel–changeinstateovertime–lineardifferenceequation•Adiscretemeasurementmodel–relationshipbetweenstateandmeasurement–linearfunction•ModelParame