Kalman Filters and Its Realization with MatlabWord下载.docx
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Kalmanfilter;
Matlab;
LTIsystem.
1Kalmanfilters
1.1Introduction
ThecelebratedKalmanfilter,rootedinthestate-spaceformulationoflineardynamicalsystem,providesarecursivesolutiontothelinearoptimalfilteringproblem.Itappliestostationaryaswellasnon-stationaryenvironments.Thesolutionisrecursiveinthateachupdatedestimateofthestateiscomputedfromthepreviousestimateandthenewinputdata,soonlythepreviousestimaterequiresstorage.Inadditiontoeliminatingtheneedforstoringtheentirepastobserveddata,theKalmanfilteriscomputationallymoreefficientthancomputingtheestimatedirectlyfromtheentirepastobserveddataateachstepofthefilteringprocess.
Consideralinear,discrete-timedynamicalsystemdescribedbytheblockdiagramshowninFigure1.1.theconceptofstateisfundamentaltothisdescription.Thestatevectororsimplystate,denotedby,isdefinedastheminimalsetofdatathatissufficienttouniquelydescribetheunforceddynamicalbehaviorofthesystem;
thesubscriptdenotesdiscretetime.Inotherwords,thestateistheleastamountofdataonthepastbehaviorofthesystemthatisneededtopredictitsfuturebehavior.Typically,thestateisunknown.Toestimateit,weuseasetofobserveddata,denotedbythevector.
Inmathematicalterms,theblockdiagramofFigure1.1embodiesthefollowingpairofequations.
Figure1.1Signal-flowgraphrepresentationofalinear,discrete-timedynamicalsystem
1.1.1processequation
whereisthetransitionmatrixtakingthestatefromtimektotimek+1.Theprocessnoiseisassumedtobeadditive,white,andGaussian,withzeromeanandwithcovariancematrixdefinedby
wherethesuperscriptTdenotesmatrixtransposition.ThedimensionofthestatespaceisdenotedbyM.
1.1.2measurementequation
whereistheobservableattimekandisthemeasurementmatrix.Themeasurementnoiseisassumedtobeadditive,white,andGaussian,withzeromeanandwithcovariancematrixdefinedby
Moreover,themeasurementnoiseisuncorrelatedwiththeprocessnoise.thedimensionofthemeasurementspaceisdenotedbyN.
TheKalmanfilteringproblem,namely,theproblemofjointlysolvingtheprocessandmeasurementequationsfortheunknownstateinanoptimummannermaynowbeformallystatedasfollows:
Usetheentireobserveddata,consistingofthevectors,,…,tofindforeachk1theminimummean-squareerrorestimateofthestate.
Theproblemiscalledfilteringifi=k,predictionifik,andsmoothingif1ik.
1.2OptimumEstimates
BeforeproceedingtoderivetheKalmanfilter,wefinditusefultoreviewsomeconceptsbasictooptimumestimation.Tosimplifymatters,thisreviewispresentedinthecontextofscalarrandomvariables;
generalizationofthetheorytovectorrandomvariablesisastraightforwardmatter.Supposewegiventheobservable
whereisanunknownsignalandisanadditivenoisecomponent.Letdenotetheaposterioriestimateofthesignal,giventheobservations,,….Ingeneral,theestimateisdifferentfromtheunknownsignal.Toderivethisestimateinanoptimummanner,weneedacostfunctionforincorrectestimates.Thecostfunctionshouldsatisfytworequirements:
●Thecostfunctionisnonnegative.
●Thecostfunctionisanon-decreasingfunctionoftheestimationerrordefinedby
Thesetworequirementsaresatisfiedbythemean-squareerrordefinedby
WhereEistheexpectationoperator.Thedependenceofthecostfunctionontimekemphasizesthenon-stationarynatureoftherecursiveestimationprocess.
Toderiveanoptimalvaluefortheestimate,wemayinvoketwotheoremstakenfromstochasticprocesstheory[1,4]:
Theorem1.1ConditionalmeanestimatorIfthestochasticprocessesandarejointlyGaussian,thentheoptimumestimatethatminimizesthemean-squareerroristheconditionalmeanestimator:
Theorem1.2PrincipleoforthogonalityLetthestochasticprocessesandbeofzeromeans;
thatis
Then:
(i)ThestochasticprocessandarejointlyGaussian;
or
(ii)Iftheoptimalestimateisrestrictedtobealinearfunctionoftheobservablesandcostfunctionisthemean-squareerror.
(iii)Thentheoptimumestimate,giventheobservables….,istheorthogonalprojectionofonthespacespannedbytheseobservables.
Withthesetwotheoremsathand,thederivationoftheKalmanfilterfollows.
1.3KalmanFilter
Supposethatameasurementona