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基于Delta算子的时滞系统故障检测.docx

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基于Delta算子的时滞系统故障检测 Introduction Deltaoperatorisamathematicaltoolthathelpsindiscretizingcontinuous-timesystems.Itallowsustomodeltime-delaysystemsinthediscretedomainandiswidelyusedinvariousfieldslikecontroltheory,signalprocessing,andcommunicationengineering.However,thepresenceoftimedelaysinthesystemcanleadtoinstability,reducedperformance,andevensystemfailure.Therefore,accurateandtimelydetectionoffaultsintime-delaysystemsiscrucialtoensuretheirproperfunctioning.Inthispaper,wediscussthefaultdetectionoftime-delaysystemsbasedontheDeltaoperator. WhatisaTimeDelaySystem? Beforeunderstandingthefaultdetectionapproach,itisessentialtoknowwhatatimedelaysystemis.Atimedelaysystemisasystemwheretheoutputataparticulartimedependsnotonlyonthecurrentinputbutalsoontheinputatsomeearliertime.Inotherwords,thepresentoutputisdelayedcomparedtotheinput.Thisdelaycanoccurduetoseveralreasons,likesignaltransmission,networkdelays,computationdelays,andsoon. DeltaOperator Thedeltaoperatorisdefinedastheforwarddifferenceoperator.ItisrepresentedbythesymbolΔ.Mathematically,itisdefinedas: Δf(k)=f(k+1)-f(k) Wecanalsodefinethebackwarddifferenceoperatorbyusingthesymbolδ.Itisrepresentedas: δf(k)=f(k)-f(k-1) Thedeltaoperatorcanbeusedtoapproximatethederivativeofafunction.Forexample,wecanapproximatethefirstderivativeofafunctionf(t)as: df(t)/dt≈Δf(k)/T whereTisthesamplingtime. DeltaOperatorBasedFaultDetection Todetectfaultsintime-delaysystems,wecanusetheDeltaoperatortomodelthesysteminthediscretedomain.Thebasicideaistocomparetheexpectedoutputofthesystemwiththeactualoutputtodetectanychanges.Wecanusevariousmethodslikemodel-based,data-driven,orhybridapproachestodetectfaultsbasedontheDeltaoperator.Let'sdiscussthesemethodsindetail. Model-BasedApproach Inthemodel-basedapproach,weuseananalyticalmodelofthesystemtodetectfaults.Wecanmodelthetime-delaysystembasedontheDeltaoperatorbyusingthefollowingequation: Y(k)=G(q)U(k-d) whereY(k)istheoutput,U(k-d)istheinputdelayedbydsamplingperiods,qistheshiftoperator(q^-1U(k)=U(k-1)),andG(