Robustdissipativefilteringforcontinuous-timepolytopicuncertain
time-delaysystems
2
LVLing-ling1，，DUANGuang-ren1，WUAi-guo1
吕灵灵，段广仁，吴爱国
（1．CenterforControlTheoryandGuidanceTechnology，HarbinInstituteofTechnology，Harbin150001，China，lingling_lv@163．com；
2．InstituteofElectricPower，NorthChinaInstituteofWaterConservancyandHydroelectricPower，Zhengzhou450011，China）
Abstract：Thispaperfocusesontheproblemofdissipativefilteringforlinearcontinuous-timepolytopicuncer-taintime-delaysystems．Toobtainastableandproperlinearfiltersuchthatthefilteringerrorsystemisstrictly
dissipativeforalladmissibleuncertainties，anewdissipativitycriterionwhichrealizesseparationbetweentheLyapunovmatricesandthesystemdynamicmatricesisfirstlyprovidedintermsoflinearmatrixinequalities（LMI）．ThenanLMIsufficientconditionfortheexistenceofarobustfilterisestablishedandadesignproce-dureisproposedforthistypeofsystems．Onenumericalexampledemonstrateslessconservativenessofthepro-posedcriterion，theothernumericalexampleillustratesthevalidityoftheproposedfilterdesign．Keywords：dissipativity；filtering；polytopicuncertainty；LMICLCnumber：TP273Documentcode：AArticleID：1005-9113（2010）04-0511-05
Thenotionofdissipativesystemhasplayedanim-circuit，networkandcontrolthe-portantroleinsystem，
ory．Dissipativityisageneralizationoftheconceptofpassivityinelectricalnetworkandotherdynamicalsys-temswhichdissipateenergyinsomeabstractsense［1－2］．Itcontainssmallgainandpassivityasspe-cialcasesandismainlyusedforstabilityanalysisforlinearornonlinearsystems．Inrecentyears，manypa-pershaveaddressedthesynthesisproblemforL2gain，positiverealorpassivity［3－6］．
Timedelayexistscommonlyindynamicsystemsduetomeasurement，transmissionandtransportlags，compu-tationaldelays，orunmodeledinertiasofsystemcompo-nents，whichhasbeengenerallyregardedasamainsourceofinstabilityandpoorperformance．Therefore，inrecentyearsconsiderableattentionhasbeendevotedtotheanalysisandsynthesisfortime-delaysystems．
Inordertoreducetheconservativenessoftradi-tionalLyapnovfunctionmethods，Feron，Apkarian，andGahinetproposedanewapproach，knownastheparameter-dependentLyapunovmethod，tothestudyof
robuststabilityofsystemswithparametricuncertain-ties．Therehasbeenaconsiderableamountofworksonextensionsandapplicationsofparameter-dependent
Lyapunovmethodtotheanalysisanddesignofrobustfiltersandcontrollers［7－10］．
Astorobustdissipativity，therearesomepapersconsideringnorm-boundeduncertainsystems［11－13］．
Butdissipativityresearchesonpolytopicuncertainde-
layedsystemshavenotbeenseen．Inthispaper，anewcriterionexhibitingakindofseparationbetweenthesystemmatricesandtheLyapunovmatricesisderivedandafilterdesignmethodispresented．Twonumericalexamplesarepresented．Oneshowsthattheproposedperformancecriterionislessconservativethantheonebasedonquadraticframework；theotherillustratesthefeasibilityofthedesignedfilter．
Notation1Throughoutthepresentpaper，thesuperscriptTstandsformatrixtransposition，RndenotesthendimensionalEuclideanspace，Rn×misthesetofalln×mrealmatrices，Iistheidentitymatrix，thenota-T
tionP=P＞0，forP∈Rn×nmeansthatPissymmet-ricandpositivedefinite，andthesymbol"*"denotestheelementdeducedbythesymmetricproperty．1
ProblemFormaulation
Considerthefollowingcontinuous-timedelayedsystem：
x（t）=Ax（t）+Adx（t－d（t））+Bw（t）y（t）=Cx（t）+Cdx（t－d（t））+Dw（t）
，（1）
z（t）=Lx（t）+D1w（t）x（t）=φ（t），t∈［－ρ，0］
n
wherex（t）∈Risthestatevector，w（t）∈Rpisthe
y（t）∈Rmisthemeasurementoutputandnoiseinput，
z（t）∈Rpisthesignaltobeestimated．Forthissys-tem，weassumethatthedelayistime-varyingandsat-（t）≤δ＜1，isfies0≤d（t）＜ρ，0≤dwhereρandδ
arerealscalars，0］istheinitialvalueφ（t）∈L2［－ρ，
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Received2008－04－11．
SponsoredbytheNationalNaturalScienceFoundationofChina（GrantNo．60710002，60974044）andSelf-plannedTaskofStateKeyLaboratoryofRo-boticsandSystem（GrantNo．SKLRS200801A03）．
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