Application of ∞ Fault Detectionand Isolationto a Boeing747-100/200Air craft

AndresMarcos�

SubhabrataGanguli†

GaryBalas‡

University of Minnesota, Minneapolis.

Fault Detectionand Isolation (FDI) methodshave beendevelopedover the past thirty yearsbut only in thelast several yearsthey havebeenapplied to complexrealapplications. This paper presentsa synthesisof

�∞ FDI

filters for the longitudinal motion of the Boeing747-100/200aircraft. The filters are synthesizedto detectandisolatefaults in the elevator actuator and pitch rate sensorwhile attenuating disturbancesand noiseon the faultsignals. They are synthesizedfor open-loopLTI modelsof the aircraft. Closed-loopsimulations with a high-fidelity nonlinear model in the presenceof gust and noiseare performed to validate the disturbance rejectionand robust propertiesof the

�∞ LTI FDI filters.

Intr oductionIn the last thirty yearsthe field of fault detectionand

isolation (FDI) and the associatedfault tolerant control(FTC) have attractedmuch attentionfrom control engi-neers,especiallyin theflight controlcommunity. Methodsto applyFDI andFTC schemesto controlsystemsaboundin theacademicliteratureandsomehave alreadybeenap-plied to realapplications.1–5 Amongthese,observer-basedapproacheshavearisenasoneof themostwidespread,andparticularly � ∞-optimization basedmethodsare increas-ingly of interestdue to the explicit addressof robustnessissues.

The basic ideas behind observer-basedFDI schemesarethe generationof residuals,andthe useof an optimalor adaptive thresholdfunction to differentiatefaults fromdisturbances.5–7 Generally, the residuals, also knownas diagnostic signals, are generatedfrom estimatesofthe system’s measurementsobtainedby a Luenenbergerobserver or a Kalman filter. The thresholdfunction isthenusedto ’detect’the fault by discretisingthe residualsfrom false faults, disturbancesand probablefaults. Forfault isolationthegeneratedresidualhasto includeenoughinformation to differentiate said fault from any other,usually this is accomplishedthroughstructuredresidualsor directionalvectors.Robustnessof theFDI algorithmisdeterminedby its capabilityto de-sensitizethefilters fromdisturbances,errors,andmodeldiscrepancies.7–10

In � ∞-optimization methodsthe filtering problem isformulated so that different performanceindexes are

�Graduate Student, University of Minnesota, Department of

AerospaceEng.andMechanics.E-mail: marcosa@aem.umn.edu†Graduate Student, University of Minnesota, Department of

AerospaceEng.andMechanics.‡Professor, University of Minnesota,Departmentof AerospaceEng.

andMechanics.

either minimized or maximized. The FDI filter hastwomaindesigngoals: to minimize the influenceof non-faultsignals(noise,disturbances,uncertainties,commands)onthe residualsandto maximizeisolationof the faults.5,8,11

These goals are often contradictory since usually atrade-off is required betweenthe residual sensitivenessto faults and its robustnessto non-faults. Thresholdfunctionscan be usedto improve upon the performanceof thefilter sincethey canfurtherde-sensitizetheresidualto non-faults.5 The application of � ∞ techniquestoFDI parallels the emergence of � ∞ theory in controldesign. First, a realizationof its power to solve robustproblemswas appreciated,12 then it was extendedfroma factorization perspective,13 momentum was createdby a numberof researchersthat proposedsolutionsandapplications,7–10,14 andthenmoresophisticatedextensionsof the � ∞ infinity theorywerestartedto be applied. It isonly naturalthento progressinto � ∞ gain scheduledandLinear ParameterVarying (LPV) filters, the latter havebeentreatedrecently.15–17

In this paper, a � ∞ FDI filter at one point of theflight envelopewill be developedbasedon the open-loopaircraftmodel(i.e. independentfrom thecontrollerdesignprocess)with the main designgoal of being capableofdetecting,isolating,andidentifyingimportantfaultsfor thelongitudinalmotion of the Boeing747 aircraft. The FDIschemeis usedto identify faults in the elevator actuatorand the pitch rate sensorin the presenceof uncertaintiessuchasmodelerrors,gustdisturbancesandsensornoise.The approachtaken is an observer-basedfault detectionalgorithmapplying � ∞ filtering techniques.The problemwill beformulatedsuchthat theresidualswill besensitiveto only one of the faults and both as robust as possibleto everything else(model errors,disturbances,noise, �����). In order to analyzethe robustnessand performancecharacteristicsof thedesignin the faceof plantvariations

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the same � ∞ interconnectionand weights will be usedto synthesizeseveral more � ∞ FDI filters throughouttheflight� envelope. Open-loopandclosed-loopanalysiswillbecarriedout for thesetof FDI filters.

Thelayoutof thepaperis asfollows. First,a brief theo-reticalbackgroundis given.Then,thenonlinearandlinearaircraft modelsare introduced. The LTI controllersusedfor closed-loopsimulationarepresentedin the followingsection. Next, The FDI filter design,interconnectionandselectionof theadequateweightsis covered. Finally, con-clusionsaredrawn andfutureresearchdirectionsaregiven.

�∞ Robust Fault DiagnosisProblem

Thissectionpresentsabrief introductionto themaindef-initionsandgoalsof � ∞ faultdetectionandisolationfilters.Figure1 illustratesthe � ∞ robust fault diagnosisproblem.P is a nominalLTI plant, andF is the desiredstable � ∞filter. The vectors f d u correspondto the fault, distur-banceandcontrolinputsrespectively. Theestimationerroris givenby e, andis thedifferencebetweenthefaultandtheresidualvector, res. Theoutputfrom theplant is givenbythey vector, andw z denotethefictitious input andoutputof theuncertainmodel,∆.

∆

u

y

e+

−

res

u

d

z wf

P

F

Fig. 1 ∞ filter problem with uncertain model.

Let xp betheplantstatevector, andx f thestatesfor thefilter. Thentheplantstatespacerepresentationis givenby:

xp � Ap xp � Bp u � R1p f � E1p d � Q1 z (1)

y � Cp xp � Dp u � R2p f � E2p d � Q2 z (2)

andthatof thefilter andtheestimationerrorby

x f � A f x f � B1f u � B2f y (3)

res � C f x f � D1f u � D2f y (4)

e � M f I res (5)

whereall thematricesareof appropriateorder. All thesys-temmatricesareknown exceptfor thosepertainingto the

filter (A f B1f B2f C f D1f D2f ). The perturbationout-put, z, in Figure 1, can be consideredas a specialtypeof disturbance.18 The � ∞ robust fault diagnosisproblemis then given by equations1 � 5. Combinethe distur-bance,d, andtheperturbationoutput,z, into a generalizeddisturbancevector, d ��� z d � . Then a modified problemcanbeobtainedwhich is in thestandard� ∞ configurationparadigm,19 Figure2.

y_

d−

u

fe

res

F

P_

Fig. 2 Modified ∞ filter problem.

Theobjectiveof the � ∞ filter synthesisis to find astablefilter that minimizesthe transferfunction from the distur-bancesto theerrors(i.e min ��� TFe d ��� ∞) while maximizingthe faultseffect on the errors(i.e. max ��� TFe f ��� ∞) whered ��� d u � , seeReferences.7,10

Air craft ModelsIn this section the aircraft nonlinear and linear time

invariant(LTI) modelsfor thelongitudinalaxisandthetur-bulencemodelarepresented.A moredetailedpresentationof the nonlinearmodel can be found in references.20,21

The LTI modelsare usedin the designstagewhile thehigh-fidelity nonlinearmodel is usedin the closed-loopsimulations.

Theaircraftmodelusedin this paperis theBoeing747series100/200. The Boeing 747 is an inter-continentalwide-bodytransportwith four fan jet enginesdesignedtooperatefrom internationalairports. Someof its perfor-mancecharacteristicsarea rangeof 6,000nauticalmiles,acruisingspeedgreaterthan965kilometersperhour andadesignceilingof 13,716meters.

The focus of the paperis on the longitudinal motionof the aircraft. A movablehorizontalstabilizerwith fourelevator segments(i.e. two inboardsand two outboards)and the four enginesthrustareusedto control the longi-tudinal axis motion. The horizontalstabilizer, δs, is onlyusedfor trimming purposes.Assumingnormaloperationof the aircraft, the inboardand outboardelevatorsmovetogether, hencefor ourpurposesthemodelwill beassumed

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to have one elevator surface, δe. The aerodynamicdatafor the Boeing 747-100/200aircraft was obtainedfromreferences.� 22,23 This data was simplified through ananalyticalstudyof eachstability derivative andopen-loopsimulationsof the resultinglow-fidelity nonlinearmodel.The detailsof thesestudiesarefound in reference.20 Thenonlinearlow fidelity model is thenusedto obtain lineartime invariant(LTI) models.

Thenonlinearbody-axeslongitudinalmotionof theBoe-ing 747,not includingflexible effects,canbedescribedbythefollowing differentialequations.

α � � Fx � sα � Fz � cα �m �VTAS

� q (6)

q � c7 � My (7)

θ � q (8)

˙VTAS � 1m ��� Fx � cα � Fz � sα � (9)

he � VTAS � cα � sθ VTAS � sα � cθ � VTAS � sinγ (10)

Thebody-axesaerodynamicforcesandmomentsaregivenby

Fx � qS ���CD � cα CL � sα � � ∑i � 1 � 4

Tni mg � sθ (11)

Fz � qS ���CD � sα � CL � cα �� 0 � 0436� ∑i � 1 � 4

T ni � mg � cθ

(12)

My � qSc ��� Cm 1c ��� CD � sα � CL � cα � � xcg � CD � cα CL � sα � � zcg �

� cαVTAS

�Cmα xcg

c �CLα � cα �"! � ∑i � 1 � 4

T ni � zengi

(13)

The LTI modelsareobtainedthroughJacobianlineari-sationsof the nonlinearmodel at different points in theflight envelopeof interest(he # [4000,10000]m, Vtas #[150,250]m/s). TheseLTI modelswill beusedin theFDIfilter designstage. Table 1 shows the trim points in theflight envelopeusedfor designandLTI simulation.

point altitude, m airspeed,m/s Mach number

1 4000 184 0.5672 4000 232 0.713 9250 125 0.714 9250 247 0.815 7000 241 0.77

Table1 Trim points.

The longitudinal axis LTI aircraft model usedfor thefilter design has five states: pitch rate q (rad/s), trueairspeedVtas (m/s),angle-of-attackα (rad),pitch angleθ(rad), andaltitudehe (m). Therearethreecontrol inputs:elevator deflectionδe (rad), stabilizerdeflectionδs (rad)

(always set to the correspondingtrim value) and thrustTn (N).The measurementsavailable are flight path angle(FPA) γ (rad), accelerationV (m/s2 $ g), pitch angle(rad),pitch rate(rad/s),velocity (m/s),andaltitudehe (m).

The deflectionand rate limits for the elevator are 23to 17 deg and % 37 deg/srespectively asspecifiedin refer-ence.23 For the stabilizerthe positionand rate limits are 12 to 3 deg and0 � 5 deg/srespectively.23 Takingtheratelimits into account,theelevatorandstabilizeraremodeledas simple first-order transferfunctions: 37$ � s � 37� and0 � 5$ � s � 0 � 5� respectively. The engine dynamics aremodeledas 0 � 5$ � s � 0 � 5� basedon the enginetransientcharacteristicsprovided in reference.23 The sensorsareconsideredto be ideal,hencetheplant inputsarefed backdirectly to the filter or the controller corruptedonly bynoiseduringsimulation.

The closed-loopsimulationswill be corruptedby noisein the sensorsand by a turbulence model entering thenonlinearaircraft model throughthe stability derivatives.The implementationof a realistic turbulence model iscarried out by using two-dimensional autocovariancefunctions.Thesefunctionsrepresentthestatisticalproper-ties of the atmosphericturbulencefor a two dimensionalstationary, homogeneousand isotropicfield of flow. It isan enhancementover the typical one-dimensionalflow inthat thegustvelocity changesin thehorizontalplanebothalong the Xearth and Yearth axes (the Zearth is considerednegligible dueto the relative sizeof theaircraftalongthisaxis). Basically, turbulenceis simulatedby feedingwhitenoise through stable, minimum-phaseDryden Spectrafilters to thesystem.For the longitudinal(symmetric)andlateral (asymmetric)motionsof an aircraft in turbulencethe model developedat Delft University of Technologywas followed (seereference24 for an excellent report onthe characteristicsof turbulence and its implications toaircraftflight).

In Table2, a summaryof theturbulenceparametersval-ues(varianceof x, σ2

x , andturbulencescale,Lg) aregivenfor light, moderateandsevereturbulencefor mediumandhighaltitudeflight (seereferences25,26).

turb ulence altitude σ (m/s) Lg (m)

Light 600 & he & 2800 1.55 Lg ' 5302800 & he & 5100 2 ( 32 ) 0 ( 000274* he idemhe + 5100 0.92 idem

Moderate 600 & he & 3400 3.05 idemhe , 3400 3 ( 84 ) 0 ( 000234* he idem

Severe 600 - he - 1400 3 ( 04 . 0 ( 00244* he idem1400 & he & 5800 6.45 idemhe , 5800 8 ( 40 . 0 ( 000336* he idem

Table2 TurbulenceParametersfor mediumto high altitudes.

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ControllerFor closed-loopsimulationsLTI � ∞ controllersobtained

at the� sametrim pointsasthe � ∞ FDI filters areused(seeTable 1). This sectionoutlinesa brief sketch of the LTIcontrolsynthesis,a morethoroughpresentationis giveninreference.27

In the aforementionedreference,a reconfigurableLPVcontroller for the longitudinal motion of the Boeing747-100/200is presented. The LPV controller has theparticularity of schedulingon true airspeed,altitude andthefault signalproducedby theFDI filter presentedin thispaper. For closed-loopsimulationof the FDI filter, LTIcontrollersareobtainedfrom theLPV controllerat severalpoints in the flight envelope for the non-fault case(i.e.thereis no reconfigurationdueto theoccurrenceof faults).

The controller objectives are to achieve de-coupledtracking of flight-path angle (FPA) and velocity withsettling times of 15 secand 45 secrespectively with theelevator surfacefully functional,andthe rejectionof gustdisturbancesfor theUp-and-Awayflight envelope.

The controller hasfive measurementsavailable: flightpath angle(FPA) γ (rad), accelerationV (m/s2 $ g), pitchangle(rad), pitch rate (rad/s),and velocity (m/s). Thereare two control outputs: elevator deflectionδe (rad), andthrustT n (N). Thestabilizerδs is alwayssetto trim in thenon-fault case.

FDI Filter DesignThis sectionpresentsthe designof the � ∞ FDI filters.

First, theformulationof thefiltering problemis presented.Then,a detaileddescriptionof theweightsusedto achievethe performance and robustness objectives is given.Finally, the open-loop and closed-loopsimulations arepresented.

FDI Filter Formulation

It is well known, see references,7,10,17 that the FDIfiltering problemcanbe posedasa standard� ∞ filteringproblem.Hence,by translatingperformanceobjectivesintofiltering objectivesandrobustnessissuesinto theclassicaldisturbancerejection(whetherthesearenoiseor modelingerrors)it is possibleto usethe available informationandexperiencefrom controldesignin filtering design.

The main philosophy behind the filter design for-mulation in this paper is that of model matching withtracking(detectionfor theFDI filter) requirements.Theseobjectives are in accordancewith standardrequirementsfor FDI filters.Thefilters mustbeableto detectandisolatedifferent failure signals(i.e. elevator actuatorand pitchrate sensor)which is similar to a de-coupledtrackingproblemfrom the control’s point of view. An important

characteristicof a FDI filter is its capabilityto distinguishbetweenthefailuresignaturesandany othersignal(noise,gust, control inputs, ����� ), thus the FDI filters shouldinclude disturbanceattenuationproperties. It is possibleto view the disturbancerejectionasa robustnessproblemin the sensethat the filter is robust to the influenceofnon-fault signals.It canbearguedthatmodelingerrorsfallinto this category, and hencethat the filter is effectivelyrobust. The faultsareassumedto enterthe systemin anadditivemanner:e.g.δe � uactuator � fa.

The filter design is carried out for one point in theflight envelope(specificallyfor point number1, low speedand low altitude, in Table 1). Then, the sameintercon-nectionstructureand weightsare usedto synthesizetheother points in the afore mentionedTable. Open-loopand closed-loopsimulationsare obtainedfor the set ofpointsin orderto analyzetheperformanceandrobustnesscharacteristicsin thefaceof plantvariations.

TheFDI � ∞ filtering interconnectionis shown in Figure3. Thevectory is formedby thefeedbacksignalsfrom theplant (i.e. γ, V , θ, q, Vtas and he); the disturbancevec-tor is given by d andconsistsof noisefor eachfeedbackchannel.Thecontrol inputsaretheelevatordeflectionandthrust,u �/� δe T n � . Two faultsaremodeled,onefor theelevatoractuator, f a andtheotherin thepitch ratesensor,f s, hencef �0� f a f s � ; andequivalentlytheresidualvectoris thengivenby res �0� resact ressen � . Theuncertaintyinputandoutputchannelsarerepresentedrespectively by w andz. Selectionof theweightsis oneof themostcritical steps,throughthemis possibleto shapethe frequency spectrumof thesignalsandthusmakethesystem(in thepresentcasethefilter) behaveasdesired.As seenin theinterconnectionshown in Figure3, therearefour differentsetsof weights:uncertaintyWunc, disturbanceWdist , fault Wf ault and esti-matederrorWerror.

Sensors

res

W_unc

+

W_fault

+

−

W_error e

fs

PlantActuators

fa

f

u

d

Filter

W_dist +

y

u

zw∆

Fig. 3 ∞ Filter Inter connection.

A multiplicative uncertainty setup is used to try toaddressuncertaintyin the augmentedplant (plant, actu-ator and sensors). The weightsare chosenas constantsrepresentinga 50 percentuncertaintyat actuatorinput:Weleunc � Wthrustunc � 0 � 5. A high-passtypeof weightwasalso testedbut it was found that the gain of the weightat high frequencieswas forcing the γ-iteration for the

4 OF 9

� ∞ synthesisto stopat that value, i.e. γ � Wunc � ∞ � , andconsequentlytheFDI filter couldnot beimprovedfurther.

The disturbanceweights are used to attenuatethedisturbanceeffects on the fault estimation. The finalweightswere achieved in a heuristicmannerby startingwith typical shapesof theSensitivity function,S, for flightcontrol systemsand thenanalyzingthe effect on the LTIand nonlinearclosedloop. Sincethe faults are assumedto enter in an additive manner, the q weight is criticalin shapingthe responseof the FDI filter. Its gain andcorner frequencieswere found to be directly connectedto the responseof the sensorresidual. Figure4 show thefrequency responsesof the different disturbanceweights,andthecorrespondingtransferfunctionsaregivenbelow.

Wγdist � 0 � 157� 3

s0 1 01 � 1

s50 � 1

WVdist � 0 � 1s

0 1 01 � 1s

50 � 1(14)

Wθdist � 0 � 157� 3

s0 1 09 � 1

s40 � 1

Wqdist � 0 � 5 s � 1s

100 � 1(15)

WVdist � 0 � 05s

0 1 1 � 1s

50 � 1Whedist � 100

s0 1 01 � 1

s50 � 1

(16)

10−2

10−1

100

101

102

103

104

10−3

10−2

10−1

100

101

102

103

104

105

106

Noise Weights

γVdotθqVtashe

Fig. 4 DisturbanceWeights.

The faults are filtered before entering the estimationerror equation, this is becoming a common fixture in� ∞ FDI techniques,seereferences.7,10,17,28 It is akin tothe well-known model matchingproblemin control.19,29

As such, theseweights are the knobs usedby the filterdesignerto shapethe responseof the residuals. Thechoiceof weights is driven by the needto provide gooddetectioncharacteristicsfor theresidualsanda sufficientlyfast responseto fault signals. They correspondto a risetime of 3 to 6 secondsfor theactuatorandsensorresidualrespectively. The gains are fine tunedafter selectingthenoiseweightsso that therewill be no steadystateerrorin the fault detection. As mentionedbefore the gain

for the sensorresidualis directly tied to the gain of thepitch ratenoiseweight. Onecanachieve fasterresponses(down to milliseconds)at the expenseof the disturbancerejectionpropertiesof the filter and to coupling betweentheresiduals.Hence,thereis adirecttrade-off betweenthefault detectiontime domaincharacteristicsasrisetime andsteadystateerror(filter performance)andfaultdisturbancerejection(filter robustness).The weightsaregiven belowand the correspondingfrequency responsesare shown inFigure5.

Wact f ault � 2s

300 � 1s2 � 1

Wsen f ault � 1 � 25s

300 � 1

s � 1(17)

10−2

10−1

100

101

102

103

104

10−3

10−2

10−1

100

101

Fault and Estimation Error Weights

Wact

fW

senf

Wact

eW

sene

Fig. 5 Filter edFault and Estimation Err or Weights.

The last setof weightscorrespondto the estimationer-rors. Recall, that the estimationerror, equation5, is thedifferencebetweenthe filtered fault andthe residual.Thede-couplingof theresidualsis achievedby penalizingbotherrors. The estimationerror weightsparallel the perfor-manceweightsin the standardcontrol set-up.The ideaisto minimizetheerrorat low frequenciesandrelaxthecon-straintsat higherfrequencies.Hencelow passweightsareselectedfor the actuatorandsensorerrors. Interestingly,it wasnoticedthat the zerosof the weightsshowed up aspolesof thefilter andhencethey wereselectedto besmallto avoid high frequency poles. The weightsareshown inFigure5.

Wacterror �s

30 � 1s

0 1 3 � 1Wsenerror �

s10 � 1s

0 1 01 � 1(18)

Sensorfailures are intrinsically easier to detect andisolate7 since they enter directly into the filter. Thus itis possibleto get away often with constantgain or lessstringentfrequency weightsenablingthe filter designertofocus in the actuatorfaults and the disturbancerejection.This was especially true in the presentcasesince the

5 OF 9

sensorswere assumedto be ideal which simplified ourtask.Theweightselectionwasdrivenby theactuatorfaultresidual2 andits disturbancecharacteristics.

In selectingthe input channelsto the filter we startedassumingthe samechannelsas for the controller (γ, V

g ,θ, q andVtas) plus the controlleroutputs(δe andTn) inthe assumptionthat in a real implementationit will beeasierandeconomicallymoremeaningfulto useavailablesensors.Even thoughit waspossibleto obtainan accept-able level of performancefor the FDI filters with theseinputs the robustnesscharacteristicswere much poorer.Introducing altitude as an additional input to the filterimprovedits robustness.This is reasonablesincethemoreinformation the filter hasof the plant behavior the betterits performance,especiallysincealtitudeandvelocity aretheprimaryparametersthataffectplantdynamics.

Simulationsand Analysis

In the following simulations we will refer to theopen/closedloop as LTI whenever a LTI plant model isusedandasnonlinearopen/closedloopwhenthenonlinearaircraftmodelis beingused.Also, notethatfor theclosed-loop case,whetherLTI or nonlinear, thecontrollerusedisa LTI � ∞ controllerobtainedat the samedesignpoint asfor the � ∞ FDI filter usedin thatparticularsimulation.

First, the LTI open-loopbehavior of the FDI filters isanalyzed.Figure6 shows the residualresponsesof the 5FDI filters (obtainedat the5 trim pointsgiven in Table1)to a 2 deg stepinput of the elevator at t � 25 secand a300Newtonsstepin the thrustchannelat t � 30 sec.Thefaults consideredin this open-loopsimulation(the samewill beusedin thesequel)arenot realisticin thesensethatis unlikely that a fault in the actuatoror the sensorwillshow this typeof behavior, they areshown in Figure6 bythe solid red line. This convolutedsetof faults inputs isintroducedin orderto differentiatetheeffect of eachfaultin theplantoutputswhile testingthecouplingbetweenthefaults.

The previous LTI open-loopsimulationsareperformedwithout disturbances(neither gust nor noise inputs) ormodelingerrors (they are LTI models). Since there areno modelingerrorsor uncertaintiesthe responsesof the 5filters areindistinguishablefrom oneanother(introducinggust and noise did not changedthe similarity in theresponses).This is expectedsincein the LTI open-loopsimulation all the issuesrelated to changesin the trimpoints and to uncertainty are discardedby means ofthe LTI nature of the simulation. The filters have thedesireddynamicbehavior: rise times of 3 to 6 seconds,de-couplingof the residuals,detectionof the faults, andrejectionof disturbances(in this casethecommands).

Figure 7 shows the plant responses,plant inputs and

0 50 100 150 200 250 300−3

−2

−1

0

1

2

3

δ ele, d

eg

Actuator failure

actuator faultresidual 1

0 50 100 150 200 250 300−1.5

−1

−0.5

0

0.5

1

Time (sec)

q, d

eg/s

sensor failure

sensor faultresidual 2

Fig. 6 Plant inputs - LTI OpenLoop.

filter residualsfor the LTI closed loop at the 5 designpoints. The commandsto the controller are a -2 deg γsquareinput from t � 15 to t � 60 sec,anda squareVtasof 15 m/s startingat t � 16 and endingat t � 130 sec.The residualsresponsesare identical to thosefor the LTIopen-loop(Figure 6) due to the non-uncertaintyin thesimulations(seereference30 for a theoreticalexplanation).The sensorfault has a negligible coupling effect on theactuatorresidual,but it hasa muchgreatereffect on someof the plant outputs(especiallyin the flight path angle).This is natural since sensorfaults affects primarily thecontrollerbehavior which usesthepitch rateasoneof thefeedbackchannels(observe how the elevator output fromthe controller shows substantialpeaksat the times thesensorfault is introduced). From Figure 7 this feedbackrelation is easily observed, the momentthe sensorfaultentersin the systemthe controller reactsin an impulsivemanner( i.e. it reactsby commandinga high demandofthe elevator in a very short time) which in turn produceshigh peaksin the θ, q and γ outputs. Sincethe θ and qchannelsareintrinsically relatedthismeansthatthechoiceof thecorrespondingdisturbanceweights,Wθdist andWqdist ,andtheactuatorestimatederrorweight,Wacterror , aregoingto affect eachother. Theactuatorfault hasa similar effect,although less dramatic, in the plant outputsand also asmallnoticeableeffecton thesensorresidual.

The next step is to simulatethe � ∞ FDI filters in thenonlinearclosedloop. As mentionedbefore,in this casethe plant is the nonlinear high-fidelity model (this willenable to test for model discrepancieswith respecttothe linear plantsusedin the designprocedure).Figure8shows theresidualsof thenonlinearclose-loopsimulationfor all the trim points. The manoeuvreperformedis asteadilyacceleratedclimb performedby commandinga 3deg squareflight pathanglefrom t � 15 to t � 95 andavelocity stepcommandof 18 m/sstartingat t � 20 sec. Itis observed that as expectedthe sensorresidualis muchmoreefficient androbust with respectto plant variations.

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0 100 200 300−4

−2

0

2γ,

deg

responsecommand

0 100 200 300−10

0

10

20

Vta

s, m

/s

responsecommand

0 100 200 300−4

−2

0

2

q, d

eg/s

ec

0 100 200 300

−0.1

0

0.1

Vdo

t/g, −

−

0 100 200 300−4

−2

0

2

θ, d

eg

time, sec0 100 200 300

−400

−200

0

200he

, m

time, sec

a) Plant responses.

0 50 100 150 200 250 300−6

−4

−2

0

2

4

δ ele (

deg)

0 50 100 150 200 250 300−1

−0.5

0

0.5

1x 10

5

Thr

ust (

N)

Time (sec)

b) Plant inputs.

0 50 100 150 200 250 300−3

−2

−1

0

1

2

3Actuator failure

δ ele, d

eg

time, s

actuator faultresidual 1

0 50 100 150 200 250 300−1.5

−1

−0.5

0

0.5

1Sensor failure

q, d

eg/s

time, s

sensor faultresidual 2

c) Filter residuals.

Fig. 7 Plant responses,inputs and filter residuals- LTI closedloop all points.

Theactuatorresidualshowsgooddetectioncapabilitiesbutis clearlymoreaffectedby plantvariations,speciallyin thehigh speedandhigh altituderegion. This is alsoexpectedsincethe � ∞ weightswereobtainedfor a low speed,lowaltitude condition (specifically point 1) and then finallyfinetunedto obtainacceptablebehavior at theotherpoints.This is an inherent limitation of LTI techniques,but itcouldbeimprovedby usingmoresophisticatedtechniquesas gain schedulingor LPV synthesis; it could also beimprovedby usingadaptivethresholdsto removetheeffectproducedby thecommands(basicallythechangein trim).Also, the effect of the 2nd squaresensorfault (at t � 250andt � 280) is morepronouncedin the actuatorresidual.This is due to the magnitudeof the sensorfault, a 1.5degreefault in the pitch ratesensoris quite strongalmosta failure. As mentionedbefore,this sensorfault producesa high demandof controlauthorityin theelevatorchannelthat getstransmitteddown to the plant outputsandto thefilter where the residualof the elevator shows relativelybig transientsat thosetimes.

0 50 100 150 200 250 300−3

−2

−1

0

1

2

3

actu

ator

res

idua

l, de

g

residualfault

0 50 100 150 200 250 300

−1.5

−1

−0.5

0

0.5

1

q se

nsor

res

idua

l, de

g/s

Time, sec

residualfault

Fig. 8 Residuals- Nonlinear ClosedLoop for all points.

The robustnessof the filters to external disturbancesis testedby using the turbulencemodelgiven beforeandwhite noiseinputsaddedto the plant outputs(the sensorsareconsideredto beideal).A moderatelevel of turbulenceis selected(Lg � 530m andσ � 3 � 84 he 0 � 000234m/s).The noise is given by a uniformly distributed randomnumbergenerator(a standardSIMULINK c

3block) with

initial seedof 0 and a minimum/maximuminterval of %0.1 deg for the γ and θ channels,% 0.1 deg/secfor thepitch rate, % 0.1/g for the acceleration,% 0.2 m/s for thetrueairspeedand % 10m for thealtitude.

In Figures9 � 11 the nonlinearclosed-loopresponseswith gustandnoisefor the � ∞ FDI filter designat pointnumber5 of the flight envelopeare given. The simula-tion is affectedby moderateturbulenceandnoiseasdefinedabove. Therefore,thedisturbancerejectionpropertiesandthemodelingrobustnesscanbeaddressedthroughthesim-

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ulation analysis. The manoeuvreperformedis the sameas before,a steadilyacceleratedclimb that will take theaircraft4 in this instanceoneflight level up (from 7000mup to 8000m) andaccelerateit from 240 m/s to 258 m/s(from Mach=0.77to Mach=0.84). It canbe seenthat thedisturbancerejectionpropertiesareacceptablealthoughtheactuatorresidualhasdegradedcapabilitieswith respecttothesensorresidual.

0 100 200 300−2

0

2

α, d

eg

0 100 200 300−2

0

2

q, d

eg/s

ec

0 100 200 300240

250

260

270

VT

AS

, m/s

responsecommand

0 100 200 300−2

0

2

4

θ, d

eg

0 100 200 300

7000

7500

8000

Time, sec

he, m

0 100 200 300−2

0

2

4

Time, sec

γ, d

eg

responsecommand

Fig. 9 Plant responses- Nonlinear ClosedLoop with Moder-ategustand noise(point 5).

0 50 100 150 200 250 300−2

0

2

4

6

δ elev

ator

, deg

Time, sec

0 50 100 150 200 250 3000

0.5

1

1.5

2x 10

5

δ Thr

ust, N

ewto

ns

Time, sec

Fig. 10 Plant inputs - Nonlinear ClosedLoop with Moderategustand noise(point 5).

The � ∞ γ - iterationlevel obtainedfor theLTI FDI filterat that designpoint is 1.414 and the filter has17 states.The filter residualsareableto detectand isolatethe faultsignatureswith a high degreeof accuracy distinguishingat the sametime from control inputs and disturbanceseffects. The jittery behavior of the residuals(mainlydue to the noiseand gust effects) is relatively small anddoesnot prevent the filter from detectingthe faults. Anadaptive threshold5–7 could be added to increase thelikelihoodof identifying soft faultsandmissedfaults.The

0 50 100 150 200 250 300

−3

−2

−1

0

1

2

3

actu

ator

res

idua

l, de

g

residualfault

0 50 100 150 200 250 300

−1.5

−1

−0.5

0

0.5

1

q se

nsor

res

idua

l, de

g/s

Time, sec

residualfault

Fig. 11 Residuals- Nonlinear ClosedLoop with Moderategustand noise(point 5).

controller succeedsin providing smoothtracking of theVtas commandas expected,while the flight path angletrackingis muchmoresensitive to theeffectsof faultsbutyet robustenoughto handlenon-criticalfaults.

ConclusionsIn this paper, it hasbeenshown an applicationof � ∞

Fault Detectionand Isolation filter techniquesto a high-fidelity modelof theBoeing747-100/200aircraft. Ad-hocanalysesof the robustnessproperties and disturbancerejectioncapabilitiesof the LTI filters have beenobtainedthroughopen-loopandclosed-loopsimulations. The � ∞FDI filters thus obtainedare capableof detecting andisolating faults in the elevator actuatorand the pitch ratesensor.

The next natural step is to use available LPV controlsynthesistechniquesto synthesizean LPV FDI filter.Parallelismdrawn throughoutthe paperwith standard� ∞controltechniquesandthe � ∞ filtering designenableustoview thisasapromisingpathwhichwehopeto incorporatein a futurepaper.

AcknowledgmentsThis researchwassupportedunderNASA Langley Co-

operative AgreementNo. NCC-1-337,technicalcontractmonitorDr. ChristineBelcastro.Theauthorswish to thankthefacultyof theAerospacedepartmentatDelft Universityin TheNetherlandsfor theavailability of the lecturenotesonturbulenceandtheoriginalsoftwarefor theBoeing747-100/200aircraft.

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