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Similarity measure application to fault detection of flight system

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Similarity measure application to fault detection of flight systemto,of,fault,Fault

    Similarity measure application to fault

    detection of flight system

    J.Cent.SouthUniv.Techno1.(2009116:07890793

    DoI:10.1007/s11771-009-01312

    Similaritymeasureapplicationtofaultdetectionofflightsystem

    KIMJH,LEESH,WANGHongmei(~洪梅)

    Springer

    (SchoolofMechatronics,ChangwonNationalUniversity,9Sarim

    dong,Changwon,Gyeongnam,641-773,Korea)

    Abstract:Faultdetectiontechniqueisintroducedwithsimilaritymeasure.Thecharacteristicsofconventiona1similaritymeasure

    basedonfuzzynumberarediscussed.Withthehelpofdistancemeasure.similaritymeasureisconstructedexplicitly.Thedesigned

    distance

    basedsimilaritymeasureisapplicabletogeneralfuzzymembershipfunctionsincludingnonconvexfuzzymembership

    function.whereasfuzzynumber-basedsimilaritymeasurehaslimitationtocalculatethesimilarityofgeneralfuzzymembership

    functions.Theapplicabilityoftheproposedsimilaritymeasuretogeneralfuzzymembershipstructuresisprovenbyidentifyingthe

    definition.Todecidefaultdetectionofflightsystem.theexperimentaldata(pitchingmomentcoefficientsandlitcoefficients)are

    transformedintofuzzymembershipfunctions.Distance

    basedsimilaritymeasureisappliedtotheobtainedfuzzymembership

    functions.andsimilaritycomputationandanalysisareobtainedwiththefaultandnormaloperationcoefficients.

    Keywords:similaritymeasure;fuzzynumber,distance;nonconvexmembershipfunction

1Introduction

    Dataclassificationstudyhasbeenappliedin decisionmaking[1-2],faultdetection[3],andpa~em classification4-51.Classificationorclustering

    algorithmwasdevelopedonthebasisofdistancebased

    measuresbetweendifferentdatagroups.Generally,each datawithinthesamegrouphashigherdegreeof similarityascomparedtodataindifferentgroups.Hence, thesimilaritymeasureisappropriateforclassifyingdata withsimilarcharacteristicsorforselectingdifierent typesofdatafromtheentiredata.Unti1nowsimilarity measureismainlybasedonfuzzynumberanddistance measures6l1].Theexplicitsimilaritymeasure

    presentsthesimilarvaluebetweendatagroupsordatato dataset.Generally,thesimilarvaluemeansthedegreeof similarity.Moststudiesemphasizeonthedesignof similaritymeasureonthebasisoffuzzynumberand centerofgravity(COG)[67].withtheconventional

    similaritymeasure.itiseasytocalculatethedegreeof similarity.However,fuzzynumberisresictedto

    triangularandtrapezoidalmembershipfunctionsofa fuzzyset.Thismeansthatthesimilaritymeasurecannot beappliedtogeneralfuzzymembershipfunctions. Whereas.thedistancebasedsimilaritymeasureis

    applicabletogeneralfuzzymembershipfunctions, includingnon.convexfuzzymembershipfunctions. However,thissimilaritymeasurerequiresmoretimeand consideration.Herethesimilaritymeasureisconsidered fromtheviewpointofcertaintyanduncertainty;andthe measuresofcertaintyanduncertaintyarecalculatedby

usingthedistancemeasure.

    Accuratefaultdetectionisessentialinthecaseof aircraftstuckduringnormalaircraftoperation.Theflight controlsystemisgenerallyequippedwithalotOf featuresinordertoincreasethesafetyoftheaircraft. particularlyincaseswheretheaircraflisdamageddueto malfunctioningofthecontrolsurfacesystem.Ifthe extentofthedamageisdeterminedaftertheoccu~ence ofthefailure.thefaulttolerantcontrolsystemiscapable ofovercomingvariousfaultsinrealtime.Therefore,the pilotortheflightcontrolsystemaccomplishesthe missionorreturnstothesafetyregion.BYtheflighttest results.thedecidingcoecientsareobtained.These

    parametersareusedtodecidethefaultornormal operationsbytransformingthesecoefficientsintofuzzy membershipfunction.Theproposeddistance.based similaritymeasureisappliedtocomputingthedegreeof similaritywithrespecttofaultornormaldataset. 2Flightsystemsandflighttest

    Theestimationproceduresareintroducedwiththe helpofwellknownairplanedynamics.Discussiononthe controlsurfacestruckconditionandtheflighttest procedurearealsoincluded.

    2.1Flightsystems

    Considerthewellknownaircraftdynamicsalonga combinationofthelongitudinalandlateraldirections combiningaircraftdynamicsasfollows[12l4]:

    x(t,=ax(t)+Bu(t)

    Foundationitem:ProjectsupposedbytheSecondStageofBrainKoreaandKoreaResearchF

    oundation

Receiveddate:20081125;Accepteddate:20090318

    Correspondingauthor:LEESH,PhD;Tel:+82552133884;E

    mail:leehyuk@changwon.ackr

    790J.Cent.SouthUniv.Techno1.f2009)16:07890793

    y(t)=Cx(t)+Du(t)

    wherethestatevectoris[""q0Pr.

    Inthestatevector,P,qandraretheangularvelocities;d istheangleofattack;denotesthesideslipangle;"= istheelevatordeflection;and0representrolland pitchangles,respectively.

    ThesystemmatrixAandinputmatrixBthat containthemodelparametersarefoundinRefs.12141.

    InEq.(2),itisgenerallyconsideredthatC=IandD=0. Theestimatedparametersareassumedtobeconstant duringflighttestmaneuvering15].

    ByapplyingFouriertransformtoEqs.(1)and(2), wehave[12-14]

    joY(o))=A~-(co)B~(co)

    y(co)=c()+D()

    wherej?denotesdifferentialoperator

    Theleastsquarescostfunctionis

    =

    (1,~xo).(1,xo)

    =

    [Re(]Re(X?,)

    (3)

    f41

    (5)

    (61

    whereYisthevectortobemeasured;Xisamatrixof

    independentvariablethatdenotesthegivenflightpoint, Y=XO+,andOisaparametervector:siSa

    complexerrorvector;andthediscreteFouriertransfoFITI fortheithsamplingtimeisobtainedasX)=

    l"U)+Xie-j(oi,hereeisconstantforagiven

    frequencyandconstantsamplinginterval,andAtdenotes samplingtime.Furthermore,theestimatedparameter covariancematrixis

    coy(O)=(p)()]=Re(]

    wheretheequationerrorvariancecanbeestimated fromtheresidua1:[(~-xo).(Y-XO),pis

    ,Hp

    thenumberofparameterstobeestimated.and,isthe

    numberoffrequencypointsofinterest.

    Frequencyspacingof0.02Hzisfoundtobe

    adequate,whichleadsto50frequenciesbeingevenly distributedintheinterva1of0.02-1.00Hzforeach transformedtimedomainsigna1.

    2.2Controlsurfacestuckconditionandtestprocedure Controlsurfacefailureisrelatedtothefailuresof elevator,rudder,andaileron.Tofixtheelevator,the onepieceelevatorissplitintotwo.Tofixtherudder.one morerudderisadded14].Intentionalcontrolsurface

    failuresareconsidered,suchastherightelevatorfailure, the1eftrudderfailure.theleftaileronfailureanda combinationofthesefailures.Theflighttestis performedbyexcludingtheuncontrOl1abilitvand untrimmabilityoftheaircraftunderpostfailure

    conditionsandtheflighttestisscheduled.Table1shows theflighttestconditionsforcontrolsurfacestuck.

TablelControlsurfacestuckconditions

    ControlsurfacestuckStuckangle

    Rightelevator

    Rightelevatorand

    leftrudder

    10.,5.,0.,5.,10.

    5.,0.,5.and10.,5.,0.,5.,10.

    Inthefirstflighttest.thecontrolinputforthe real?timeparameterestimationisappliedbyusingthe knobandswitchingmethod14].Theflightdataare

    acquiredforexcitingtheaircraftdynamicsbyusingthe abovementionedmethod14-l5].However,itisfound

    thattheappliedtimeintervalisslightlyinaccurate.In ordertorealizeaconstantcontrolinputandtimeinterva1. inthesecondstageoftheflighttest.thecontrolinput deviceismodifiedtouseaRFmodemandanR/C

    transmitter.Dependingontheflightconditionofthe controlsurfacestuck,theflighttestresultshelpengineers toadapttothecontro1surfacestuck.Afterthegroundtest. thepilotmaintainsthetrimmedlevelflightunderthe postfailurecondition,andtheengineerappliesthe controlinputbyusinganewdevice.

    Consideringtheprimarycontrolofsurfacestuck andcombinationstuck,theaircraftmustshow controllabilityandtrimmabilityunderpostfailure

    conditions.Fig.1showsthephotoofaircraftDURUMI

    IIintake.offfortheflighttest.Theflighttestprocedure involvestherecoveryoftheaircraftfromthefaultstate tothenorma1state151.Table2showsthattheaircraft

    DURUMIIIintakeoffisSUCCESSfullytestedunder faultconditions.Furthermore.itshowsthetrimvalueof theavailableprimarycontro1surfaceunderpost.failure conditions.Trimvalueissatisfiedby5.7524whenitis inanorma1mode.

    Fig.1PhotoofDURUMII1intakeoff

    3Similarmeasureandapplicationtofault

    detection

    Forcalculatingsimilarityofdata,similaritymeasure isintroducedbytwodifferentapproaches,i.e.fuzzy J.Cent.SouthUniv.Techno1.(2009116:07890793791

    numberanddistancemeasure

    3.1Similarit,rmeasurewithfuzzynumberand distancemeasure

    Thetrapezoidalmembershipfunction?jis

    denotedbyfuzzynumberA=(a,b,c,?)[7].Anew

    methodwaspresentedtocalculatetheCOGpointofa generalizedfuzzynumber,andanewCOGcalculation methodwasderivedbasedontheconceptofthemedium curve[12].TheseCOGpointsplayanimportantrolein thecalculationofthefuzzynumberbasedsimilarity measure.InRefs.7-8],thedegreesofsimilaritieswere derivedbyusingthemembershipfunctionsofthefuzzy numberandCOGandasimilaritymeasuretoovercome thedrawbacksoftheexistingsimilaritymeasureswas proposedasfollows:

    S(3,)=[1~?I

    min(y~,)

    ;)max(yA,

V4](1一圳,

    (7)

    where(,)and~(,denotetheCOGpoints

    offuzzynumbersAandB,respectively;jand areexpressedby04aland=b4~bl,

    respectively.Bs(,sa)issatisfiedbylif+Sg >0,and0ifSj+=0.However,similaritymeasure (Eq.(7))canbeusedonlyfortriangularortrapezoidal fuzzymembershipfunctions.

    Todesignsimilaritymeasureforgeneralfuzzy membershipfunctionincludingnon.convexfuzzy membershipfunctiondata,otherpointofviewis required.Here,withthehelpofaxiomaticdefinitionof similaritymeasureproposedbyLIU[10],thesimilarity betweenthefuzzysetsisconsideredfortwo Gaussian.typemembershipfunctions.

    Theareaoftheshadedregioncanbeconsideredas acomponentofthesimilaritymeasure.Thetwoitems 1(AUB),[1]and1(AuB),[Oh)will

    hereaflerbedenotedasCandD.Inthesetwoitems. [1]xand[0Ixaredenotedby1and0forthewhole universeofdiscourse,respectively.Cdenotesthe commonareaofamembershipfunction;whereasD representsthesetinformationbetweenthetwo membershipfunctions.Hence.apropersimilarity measureisobtainedbycombiningthetwovalues.In Theorem1,thedistancemeasurebetweenfuzzy membershipfunctionsAandBisbasedonthewell knownHammingdistancemeasure.

Theorem1ForanysetsA,B?月,

    (A,B)2(AnB),[1](AUB),[0(8)

    isthesimilaritymeasure.wheredsatisfiestheHamming distance.

    Proofscanbefollowedeasily.Fromthedefinition ofLIU[1O],(S1)referstothecommutativityofsetsA andB,whichisclearfrom(Eq.(8))itselfiFor(S2), (DnD.),[1and(DUD),[0areequalto1

    Hence,(D,D)=0iseasilyobtained.

    ForarbitrarydifferentsetsAandB,theinequality inrS31isprovedasfcIllows:

    (A,B)?2(CnC),[1】一(CUC),[0]=

    s(C,C)

    Finally,VA,B,C?F(),ifAcBc2C

    (A,B)=2A,[1]B,[0]?

    2,[1]C,[0]=s(A,C)

    Similarly,s(B,C)?(A,C)isalsosatisfied

    3.2Similaritymeasureforfaultdetection ThescatterdiagramofcoefficientsCm,Cm

    andC,isshowninFig.2.Inthelongitudinalmode.in ordertoobtainthefailurestatusoftheelevator,the followingdataarerequired:thepitchingmoment coefficientduetothechangesoftheelevatordeflection (),thepitchingmomentcoefficientduetothe changeoftheattackangle(C),andtheliftcoefficient duetothechangeoftheattackangle().For

    constructingthefuzzymembershipfunction,valuesof Cm,CmandCLaredividedintoeightgroups,and thenumberofdatasetsisnormalized.InFig.2.itcanbe observedthatthenormalandfaultvaluesofdonot

    interchange,whereasthoseofCandCLinter- change.Bymeansofscatterdiagrams.normalandfault fuzzymembershipfunctionsarejllustratedinFig.3. FromFig.2.thecontro1surfacestuckisdeterminedby monitoringthevalueofCwhichfacilitatesthe discriminationbetweennormalandfaultconditions.Data 792J.Cent.SouthUniv.Techno1.(2009,16:07890793

    Trueairspeed/0n?S)

    Tx'ueairspeed/(m?S,

    Trueairspeed/(m?S1

    Fig?2ScaUerdiagramsofcoefficientsC

    m&(a),C(b)and

    (c)

    pointsaredescribedforelevatorstruckcase,forexample, epl0meanselevator10.fixed.Forelevatorandruder left(right)stuckcase,e0rl10denoteselevator0.and rudder1eft10.fixedCaSeSf151.Thesimilaritymeasure usingCiSasfollows:

    s(,)2(n),[1lx)a(0%u),[Olx)(9)

    whereFNandFdenotethenormalandfaultfuzzy membershipfunctionsfromFig.3.Furthermore,other similaritymeasuresusingCm

    &

    andCLaredefinedas

    0

    ()

    O

    O

    C"

    %

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