Improved Kalman filter with unknown inputs based on data fusion of partial acceleration and displacement measurement文档格式.docx
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Improved Kalman filter with unknown inputs based on data fusion of partial acceleration and displacement measurement文档格式.docx
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//dx.doi.Org/10.12989/sss.2016.17.6.903
ImprovedKalmanfilterwithunknowninputsbasedondata
fusionofpartialaccelerationanddisplacementmeasurements
LijunLiu,JiajiaZhu,YingSuandYingLei*
DepartmentofCivilEngineering,XiamenUniversity,Xiamen361005,China
(ReceivedJanuary5,2016,RevisedApril12,2016,AcceptedApril16,2016)
Abstract.TheclassicalKalmanfilter(KF)providesapracticalandefficientstateestimationapproachforstructuralidentificationandvibrationcontrol.However,theclassicalKFapproachisapplicableonlywhenexternalinputsareassumedknown.Overtheyears,someapproachesbasedonKalmanfilterwithunknowninputs(KF-UI)havebeenpresented.However,theseapproachesbasedsolelyonaccelerationmeasurementsareinherentlyunstablewhichleadspoortrackingandso-calleddriftsintheestimatedunknowninputsandstructuraldisplacementinthepresenceofmeasurementnoises.Eitheron-lineregularizationschemesorpostsignalprocessingisrequiredtotreatthedriftsintheidentificationresults,whichprohibitsthereal-timeidentificationofjointstructuralstateandunknowninputs.Inthispaper,itisaimedtoextendtheclassicalKFapproachtocircumventtheabovelimitationforrealtimejointestimationofstructuralstatesandtheunknowninputs.BasedontheschemeoftheclassicalKF,analyticalrecursivesolutionsofanimprovedKalmanfilterwithunknownexcitations(KF-UI)arederivedandpresented.Moreover,datafusionofpartiallymeasureddisplacementandaccelerationresponsesisusedtopreventinrealtimetheso-calleddriftsintheestimatedstructuralstatevectorandunknownexternalinputs.Theeffectivenessandperformanceoftheproposedapproacharedemonstratedbysomenumericalexamples.
Keywords:
Kalmanfilter;
unknowninputs;
inputestimation;
responseprediction;
datafusion
1. Introduction
Thestateestimationofadynamicsysteminastochasticframeisimportantforstructuralhealthmonitoringandvibrationcontrol(Azametal.2015).Inpracticalcases,itisimpossibletomeasureallstructuralresponses;
henceastateestimationofpartiallyobserveddynamicsystemisessential.Inthisregard,theKalmanfilter(KF),whichwasproposedbyR.E.Kalmanintheearlysixties(Kalman1960),providesaparticularlypracticalandefficientstateestimationalgorithmwithpartialmeasurementsofstructuralresponses.Moreover,KFhastheabilitytoinherentlytaketheuncertaintyinthemodelintoaccount,whichisnotpossibleinthedeterministicapproaches(Naetsetal.2015).However,intheclassicalKFapproach,theexternalinputforcesareassumedeitherknownorbroadband,sothattheycanbemodeledasazeromeanstationarywhiteprocess.Inmanycases,nomeasurementsoftheinputforcesareavailableorthebroadbandassumptionisviolated.
♦Correspondingauthor,Professor,E-mail:
ylei@
Copyright©
2016Techno-Press,Ltd.
http:
//www.techno-press.org/?
journal=sss&
subpage=8 ISSN:
1738-1584(Print),1738-1991(Online)
904 LijunLiu,JiajiaZhu,YingSuandYingLei
Overtheyears,someresearchershaveproposedvariousimprovedKFwithunknowninputstocircumventtheabovelimitationoftheclassicalKFapproach,e.g.,GillijnsandMoor(2007)derivedarecursivefdterwiththestructureoftheKalmanfdterforjointinputandstateidentificationusinglinearminimum-varianceunbiasedestimationforoptimalcontrolapplications;
Panetal.(2010)alsoderivedaKalmanfilterwithunknowninputsapproachbytheweightedleast-squaresestimationmethod.Theleast-squaresestimatorsforstatesandunknowninputsareproveninherentlyoptimalintheminimum-varianceandunbiasedsense.Wuetal.(2009)employedtheKalmanfiltertoestablisharegressionmodelbetweentheresidualinnovationandtheinputexcitation.Basedontheregressionmodel,arecursiveleast-squaresestimatorisproposedtoidentifytheinputexcitationforces.Linetal.(Linetal.2010,Maetal.2003)alsostudiedinputforceestimationoflinearandnonlinearstructuralsystemsbasedontheKalmanfilter(KF)witharecursiveestimator,inwhichtheKFgeneratestheresidualinnovationsequencesandtheestimatorusesaleast-squaresalgorithmtoevaluatethetimehistoriesoftheexcitingforces;
Lourensetal.(2012)developedanaugmentedKalmanfilter(AFK)forforceidentificationinstructuraldynamics,inwhichtheunknownforcesareincludedinthestatevectorandestimatedinconjunctionwiththestates.Dingetal.(2013)presentedadiscreteforceidentificationmethodbasedonaverageaccelerationdiscretealgorithm.Themethodisformulatedinstatespaceandtheexternalexcitationactingonastructureisestimatedwithregularizationmethod;
Liuetal.(2014)transferredtheimplicitNewmark-algorithmtoanexp
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