Adaptive three-step Kalman filter for air data sensor fault detection and diagnosis

Delft University of Technology

Adaptive three-step Kalman filter for air data sensor fault detection and diagnosis

Lu, P; van Eykeren, L; van Kampen, EJ; de Visser, CC; Chu, QP

DOI

10.2514/1.G001313

Publication date

2016

Document Version

Accepted author manuscript

Published in

Journal of Guidance, Control, and Dynamics: devoted to the technology of dynamics and control

Citation (APA)

Lu, P., van Eykeren, L., van Kampen, EJ., de Visser, CC., & Chu, QP. (2016). Adaptive three-step Kalman

filter for air data sensor fault detection and diagnosis. Journal of Guidance, Control, and Dynamics: devoted

to the technology of dynamics and control, 39(3), 590-604. https://doi.org/10.2514/1.G001313

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Sensor Fault Dete tion and Diagnosis P. Lu 1 , L. Van Eykeren 2 , E. van Kampen 3 , C.C. de Visser 4 , Q.P. Chu 5 Delft University of Te hnology, P.O. Box 5058, 2600 GB Delft, The Netherlands

AirData Sensor(ADS ) Fault Dete tion and Diagnosis(FDD )is importantfor the safety of air raft. In this paper, rst an extensionof the Robust Three-Step Kalman Filter (RTS-KF) to nonlinear systems is made by proposing a Robust Three-Step Uns ented Kalman Filter (RTS-UKF). The RTS-UKF isfound to be sensitive to the initial onditionerrorwhendealingwithADSfaultestimation. Atheoreti alanalysisof thissensitivityispresentedanda novelAdaptiveThree-StepUns entedKalmanFilter (ATS-UKF ) isproposedwhi hisableto opewith notonlytheestimationoftheADS faultsbutalsothedete tionandisolationoffaults. TheATS-UKF ontainsthreesteps: timeupdate,faultestimationandmeasurementupdate. Thisapproa h anredu ethe sensitivity to the initial ondition error. Finally, the ADS FDD performan e of the ATS-UKF is validated using simulated air raft data. Additionally, its performan e is further validated using real ight test data to demonstrate its performan e under realisti un ertaintiesanddisturban es. Theresultsusingboththesimulateddataand real ight test data demonstrate the satisfa tory FDD performan e of the ATS-UKF and verify that it an be appliedin pra ti e to enhan e the safetyof air raft.

1

Ph.D.Student,ControlandSimulationDivision;P.Lu-1tudelft.nl. 2

Ph.D.Student,ControlandSimulationDivision;L.VanEykerentudelft.nl 3

AssistantProfessor,ControlandSimulationDivision;e.vankampentudelft.nl 4

AssistantProfessor,ControlandSimulationDivision; . .devissertudelft.nl 5

Ax

,

Ay

,

Az

= lineara elerationsalongthebody axis,m

/

s

2

Axm

,

Aym

,

Azm

= measurementsoflineara elerationsalongthebody axis,m

/

s

2

T

= thresholdsfordete tingfaults

V

= trueairspeed,m

/

s

Vm

,

αm

,

βm

= airdatasensormeasurements

f

= outputfaults

ˆ

f

= estimationofoutputfaults

fV

,

fα

,

fβ

= faultsintheairdatasensors

p, q, r = roll,pit handyawratealongthebody axis,rad

/

s

pm

,

qm

,

rm

= measurementsofroll,pit handyawratealongthebody axis,rad

/

s

α

,

β

= angleofatta k,sideslipangle,rad

αvm

,

βvm

= angleofatta k,sideslipanglemeasurementsfromthevane,rad

γ

= innovationofthelter

φ

,

θ

,

ψ

= roll,pit handyawanglesalongthebodyaxis,rad

φm

,

θm

,

ψm

= measurementsofroll,pit handyawanglesalongthebodyaxis,rad

L

,

l

,

m

,

p

= dimensionsofthestate,input,outputandoutputfaults,respe tively

ˆ

x

,

P

= stateestimateanditserror ovarian ematrixofthelter

I. Introdu tion

P

resently, Fault Dete tion and Isolation (FDI ) has an important role in a hieving fault-toleran e ofair raft [1℄. Duringthe pastfew de ades, manyapproa heshavebeenproposed for sensor or a tuator FDI [24℄. In aerospa e engineering, the FDI of sensors and a tuators for xed-wingair raftis widelystudied, as anbefoundin Patton[1℄,Marzat et. al[5℄, and Hajiyev and Caliskan [6℄. Investigationof theFDI for UnmannedAerial Vehi les analsobefound [5, 7℄. Forre entadvan es, thereaderis referredto Goupil [8℄ andZolghadri[9,10℄. TheAir Data Sen-sors (ADS s) measure the dynami pressure, airspeed, angle of atta kand angle of sideslipof the air raft,providingessentialinformationontheair raftstatestothepilot[11℄. TheADS sareusually

resultin faults su h asblo kagefaults [12℄. Thesefaults maynegativelyinuen e theinformation providedtothepilot,whi h anleadto atastrophi a idents. Inthere entpast,therehavebeen ommer ial air raft a idents aused by ADS faults. Due to faults in the ADS s, the ight rew of AustralLineasAeroeas Flight2553 improperlyreferen ed theairspeedindi ator andindu eda stru ture failure by ex eeding safeairspeedlimits[13℄. More re ently, thenal report of the Air Fran eFlight447a identstatedthaterroneousairspeedmeasurementsfromthepitotprobeswere a ontributing fa tor [14℄. Sin e 2003, ommer ial air raft havehad more than 35 re orded in i-dentsofmultipleADSfaults[13℄. Therehavealsobeena identsofmilitaryair raft ausedbyADS faults. The rashof aB-2bomberis dueto alargebias totheADS swhi his ausedbymoisture intheporttransdu erunits[14℄. Thesefa ts indi atestheimportan eofthefaultdete tionofthe ADS s.

The fault dete tion of ADS s has been investigated in a number of studies [12, 15℄. Some resear hers propose to use alternative air data sensing systems su h as a ush air-data sensing system[15,16℄. Nebulaet. alproposeavirtualairdatasystemagainstADSfailures[17,18℄. Looye andJoos[19℄proposetousethedatafromanavigationsystemtodeterminetheairdatainformation. On theother hand, thefaults of the ADS s anbe dete ted. Hou k andAtlas [11℄ are one of the rst to analyze ADS faults. The limitation of their approa h is that independent stati pressure measurementsarenotalwaysavailableinUnmannedAerialVehi le(UAV )appli ations[13℄. Cervia etal. [20℄andEubanketal. [13℄dete tthefaultsusingamultiple-redundan yairdatasystem. The airdatasystemstudiedbyCerviaetal. isbasedonpseudo-quadruplexredundan ywhi hemploys four self-aligningairdata probes. Freemanet al. [12℄investigateanalyti alredundan yinsteadof hardwareredundan yfortheADSfault dete tion. Theyusealongitudinaldynami smodelofthe air raftandtwolinear

H∞

ltersaredesignedtodete tthefaultsandproviderobustnesstomodel errors.

Alternatively,thekinemati model anbeusedtodete tthefaultsintheADSs,therebyredu ing theinuen eofmodelun ertainties ausedbythe al ulationoftheaerodynami for esandmoments [21, 22℄. Van Eykerenand Chu [23℄use an adaptiveExtendedKalman Filter to dete tthe faults

[24℄, aSele tive-ReinitializationMultiple-Model AdaptiveEstimationapproa hisproposedforthe ADSFaultDete tionandDiagnosis (FDD ). Theapproa himprovedtheFDD performan e ofthe Multiple-Model-based approa hes. However, the omputational load of the approa h is intensive whendealingwithsimultaneousfaults.

Inthispaper,anewly-developedRobustThree-StepKalmanFilter(RTS-KF)[25℄is ombined withthekinemati modeltoestimatetheADSfaults. First,theRTS-KFisextendedto opewith nonlinearsystemsbyproposing anovelRobustThree-Step Uns entedKalmanFilter (RTS-UKF). The RTS-UKF is able to redu e linearization error. However, it is found that the RTS-UKF is sensitiveto theinitial onditionerrors. Se ond,thesensitivityofthis three-stepKalmanFilter to the initial ondition erroris analyzed theoreti ally. It is provedthat theRTS-UKF doesnot use someofthemeasurementsto updatethestateestimationwhi h ausesthesensitivitytotheinitial onditionerror.

Finally,anovelAdaptiveThree-Step Uns entedKalmanFilter(ATS-UKF) isproposedwhi h does not only estimate the ADS faults, but also dete t and isolate the faults. The ATS-UKF ontainsthree steps: time update,fault estimation andmeasurementupdate. Thefault dete tion is performed before the fault estimation. This approa h also redu es its sensitivity to the initial ondition. Thefaultdete tionisperformedby he kingtheinnovationvarian es. Inthepresen eof faults,theinnovationvarian ein reases. Iftheinnovationvarian eex eedsapre-denedthreshold, thenthefaultalarmistriggered. TheFDDperforman eoftheATS-UKFistestedusingsimulated air raft data with the obje tiveof dete ting, isolating and estimating ADS faults. Two dierent fault s enarios(multiple faults and simultaneous faults) are implemented to test theperforman e andtheresultsdemonstratethesatisfa toryperforman eoftheATS-UKF. Thefaulttypes ontain notonlybiasanddriftfault,but alsoos illatoryfaults.

Furthermore,theFDD performan eoftheATS-UKFisvalidated usingrealighttestdata of aCessnaCitationIIair raft. Thesensormeasurementsfrom therealighttest ontainbiasesand un ertaintiesandaresuitablefortestingtheperforman eoftheATS-UKF. Dierentfaults enarios are generated and the faults are inje ted into the real ight data. The ADS FDD results of the

thesafetyoftheair raft.

The stru tureof the paperis asfollows: InSe tion II,theADS FDD problem is formulated. Thekinemati modelin ludingADSfaultsis introdu ed. Se tionIIIextends theRTS-KFto ope withnonlinearsystemsbyproposingtheRTS-UKF. TheRTS-UKFisappliedtoestimatetheADS faults, whi h turnsouttobesensitivetotheinitial ondition. Thesensitivityproblem isanalyzed theoreti ally and anovel ATS-UKF is proposed to deal with notonly theestimation of the ADS faults,butalsothedete tionandisolationofthefaults. Theperforman eistestedusingasimulated air raftmodel. InSe tion IV,theperforman eoftheATS-UKFisfurthervalidated usingthereal ight data of the Cessna Citation II air raft. The performan e is shown and some remarks are given. Finally,the on lusionsaremadeinSe tionV.

II. AirDataSensor FDDusing thekinemati model

Theobje tiveofthispaperistheFDDoftheair raftADS s. However,model-basedapproa hes are sensitive to model un ertainties. In order to make the proposed approa h more robust, the kinemati model of air raft, whi h does not involve the omputation of aerodynami for es and moments,isusedinsteadoftheaerodynami model.

A. Air raftkinemati modelwithADSfaults

Thekinemati modeloftheair raftin ludingADSfaultsisdes ribedas

˙x(t) = ¯

f (x(t), um(t), t) + G(x(t))w(t)

(1)

y(t) = h(x(t), um(t), t) + v(t) + F (t)f (t) t = ti, i = 1, 2, ...

(2)

where

x ∈ R

L

representsthesystemstates,

um

∈ R

l

themeasuredinput,

y ∈ R

m

themeasurement. The fun tions

f

¯

and

h

are nonlinear fun tions.

G

and

F

are the noise distribution matrix and outputfaultdistributionmatrix. Thefun tion

f ∈ R

p

x = [V α β φ θ ψ]

T

(3)

um

= [Axm

Aym

Azm

pm

qm

rm

]

T

_{= [Ax}

_Ay

_Az

_{p q r ]}

T

_{+ w}

(4)

y = [Vm

αm

βm

φm

θm

ψm]

T

(5)

w = [wx

wy

wz

wp

wq

wr

]

T

(6)

v = [vV

vα

vβ

vφ

vθ

vψ]

T

(7)

f = [fV

fα

fβ]

T

(8)

wheretheinput

um

istheInertialMeasurementUnit(IMU )measurementwhi hmeasuresthelinear a elerations(

Ax

,

Ay

and

Az

)and angular rates(rollrate

p

, pit h rate

q

, and yawrate

r

)of the air raft.

y

is the outputmeasurementwhi h measures the airdata information(true airspeed

V

, angle of atta k

α

, and angle of sideslip

β

) and Euler angles(roll angle

φ

, pit h angle

θ

, andyaw angle

ψ

).

[fV

fα

fβ]

T

are thefaultsof theADS s,i.e.

fV

,

fα

and

fβ

arethefaults in thevelo ity sensor, angle of atta k sensor, and angle of sideslipsensor, respe tively. It isassumed that there are nofaultsin theAttitudeand HeadingReferen eSystemwhi hmeasures theEulerangles and theinuen eof hangingwindsu hasturbulen eislimited. Therefore,theinputnoiseve tor

w(t)

anbeassumedtobea ontinuoustime whitenoisepro esswhiletheoutputnoiseve tor

v(t)

an beassumedto beadis rete timenoisesequen e.

E[w(t)] = 0

E[w(t)w

T

_{(tτ)] = Qδ(t − τ) , Q =}

diag

(σ

2

w

x

, σ

2

w

y

, σ

2

w

z

, σ

2

w

p

, σ

2

w

q

, σ

2

w

r

),

(9)

E[v(t)] = 0

E[v(ti)v

T

(tj)] = Rδ(ti

− tj) , R =

diag

(σ

2

v

V

, σ

2

v

α

, σ

2

v

β

, σ

2

v

φ

, σ

2

v

θ

, σ

2

v

ψ

),

(10)

E[w(t)v

T

_{(ti)] = 0 , t = ti}

_{, i = 1, 2, ...}

(11)

˙

V = (Axm

− wAx

− g sin θ

cos α cos β + (Aym

− wAy

+ g sin φ cos θ) sin β

+ (Azm

− w

Az

+ g cos φ cos θ) sin α cos β

(12)

˙α =

1

V cos β

 − (Axm

− wAx) sin α + (Azm

− wAz) cos α + g cos φ cos θ cos α

+ g sin θ sin α + qm

− wq

− [(pm

− wp) cos α + (rm

− wr) sin α] tan β

(13)

˙

β =

1

V

 − (Axm

− wAx

− g sin θ) cos α sin β + (Aym

− wAy

+ g sin φ cos θ) cos β

− (Azm

− wAz

+ g cos φ cos θ) sin α sin β + (pm

− wp) sin α − (rm

− wr) cos α

(14)

˙

φ = (pm

− wp) + (qm

− wq

) sin φ tan θ + (rm

− wr) cos φ tan θ

(15)

˙θ = (qm

− wq) cos φ − (rm

− wr) sin φ

(16)

˙

ψ = (qm

− wq)

sin φ

_{cos θ}

+ (rm

− wr)

cos φ

_{cos θ}

(17)

and

G(x(t))

isdened as:

G(x(t)) =









































− cos α cos β

− sin α cos β

− sin α cos β

0

0

0

sin α/(V cos β)

0

− cos α/(V cos β) cos α tan β

−1

sin α tan β

cos α sin β/V

− cos β/V

sin α sin β/V

− sin α

0

cos α

0

0

0

−1

− sin φ tan θ − cos φ tan θ

0

0

0

0

− cos φ

sin φ

0

0

0

0

− sin φ/ cos θ − cos φ/ cos θ









































(18)

Therefore,themeasurementmodelin ludingtheADSfaultsis

Vm

= V + fV

+ vV

(19)

αm

= α + fα

+ vα

(20)

βm

= β + fβ

+ vβ

(21)

φm

= φ + vφ

(22)

θm

= θ + vθ

(23)

ψm

= ψ + vψ

(24)

Timeinterval Sensor Faulttype Faultmagnitude Faultunit 10s

< t <

20s

V

bias

2

[m/s℄ 30s

< t <

40s

α

drift

0.01t

[rad/s℄ 50s

< t <

60s

β

os illatory

−2π sin

(

πt)/180

[rad℄

Themeasurementmodel anberewritteninto

y(t) = x(t) + F (t)f (t) + v(t),

t = ti, i = 1, 2, ...

(25)

where

F = [I3

03×3]

T

(26)

Theobje tiveoftheADSFDDproblemistodete t,isolateandestimate

f = [fV

fα

fβ]

T

. This paperassumesthat thereare nofaultsin theIMUsensors. Ifthere arefaultsin theIMU sensors, they anbedete tedandestimated byothermethods usinganothersetofkinemati model[26℄.

B. Faults enarios forthe ADSFDD

Inthispaper,twodierentfaults enariosareusedto testtheperforman e oftheapproa hes. The fault s enario for multiple ADS faults is given in Table 1 while that for simultaneous ADS faults isgivenin Table2. Thefaulttype,magnitudeand unitaregiveninthetable. Theunits of thedriftfaultsaregivenbytheunits ofthedriftrates. It an beseenthatthefaulttypesnotonly ontainsbiasfaultsbutalsodriftfaultsandos illatoryfaults.

C. State observability and faultre onstru tibility

Thisse tion he ktheobservabilityofthesystemdes ribedbyEqs.(1)and (2). The observ-abilityanalysisofthesystem anbeperformedby he kingtherankofthefollowingobservability

Timeinterval Sensor Faulttype Faultmagnitude Faultunit 10s

< t <

20s

V

os illatory

2 sin

(

πt)

[m/s℄

α

drift

0.01t

[rad/s℄

β

drift

−0.01t

[rad/s℄ 30s

< t <

40s

V

drift

−0.2t

[m/s

2

℄

α

bias

−2π/180

[rad℄

β

os illatory

−2π sin

(

πt)/180

[rad℄

matrix:

O =

























δxh

δx(L

_f

¯

h)

. . .

δx(L

L−1

_f

¯

h)

























(27)

wheretheLiederivativeisdenedasfollows:

L

_f

¯

h = δxh · ¯

f

. .

. (28)

L

L−1

_f

¯

h = δx(L

L−2

_f

¯

h) · ¯

f

It anbe readily he kedthat

O

is offull rank. Therefore, thesystemstateis observable. In orderto re onstru tthefaults, additional onditionsarerequiredwhi haregivenin (29) .

III. Extensionofthe RobustThree-Step Kalman Filter

Thisse tionextendstheRTS-KFtoestimateoutputfaults. First,inSe tionIIIA,theRTS-KF isextended tononlinearsystemsbyproposingaRTS-UKF. ThisRTS-UKFisapplied totheADS faultestimationproblemandisfoundtobesensitivetotheinitial onditionerrors. Thissensitivity problemisanalyzedtheoreti allyin Se tionIIIB.Then,inSe tionIIIC ,anATS-UKFisproposed whi h andete t,isolateandestimatethefaults. Finally,theATS-UKFisappliedtotheADSFDD problemin Se tionIIID todemonstrateitsFDDperforman e.

TheRTS-KF[25℄ anbeusedforoutputFDD . Considertheair raftkinemati modeldes ribed byEqs.(1) and (2). Forthissystem,sin ethesystemstateisobservable,theexisten e ondition ofaRTS-KFis[25℄:

m ≥ p,

rank

Fk

= p

(29)

Inthisstudy,

m = 6

,

p = 3

andrank

Fk

= 3

. Therefore,aRTS-KF anbedesignedtoestimatethe ADSfaults.

However, the RTS-KFis designed for linear systems while the kinemati model is nonlinear. Therefore, the RTS-KF needs to be extended to ope with nonlinear systems. The Uns ented Kalman Filter (UKF) is a nonlinear lter whi h an a hieve a better level of a ura y than the ExtendedKalmanFilter(EKF )[27,28℄. Thisse tionextends theRTS-KFtononlinearsystemsby proposingaRTS-UKF.

A ordingtothete hniqueinLuetal. [22℄,theRTS-UKF anbederivedasfollows:

Step1 Sigmapoints al ulationandtimeupdate

X

0,k−1

= ˆ

xk−1|k−1

(30a)

Xi,k−1

= ˆ

x

k−1|k−1

− (

q

(L + γ0)P

k−1|k−1)i,

i = 1, 2, ..., L

(30b)

Xi,k−1

= ˆ

x

k−1|k−1

+ (

q

(L + γ0)P

k−1|k−1

)i,

i = L + 1, L + 2, ..., 2L

(30 )

w

₀

(m)

= γ0/(L + γ0)

(31a)

w

(c)

0

= γ0/(L + γ0

) + (1 − α

2

0

+ β0)

(31b)

w

_i

(m)

= w

(c)

_i

= 1/{2(L + γ

0

)},

i = 1, 2, ..., 2L

(31 )

with

Xi,k−1

the sigmapoints ofthe states(dimension

L

)at step

k − 1

.

w

(m)

i

and

w

(c)

i

are

theweightsasso iatedwiththe

i

thpointwithrespe tto

x

ˆ

k−1|k−1

and

P

k−1|k−1

,respe tively.

γ0

= α

2

_{0(L + κ) − L}

isas alingfa tor,

α0

determinesthespreadofthesigmapointsaround

ˆ

x

k−1|k−1

,

κ

isase ondarys alingfa tor,

β0

isusedtoin orporatethepriorknowledgeofthe

meanand ovarian eare omputedasfollows

Xi,k|k−1

= Xi,k−1

+

Z

k

k−1

¯

f (Xi,k−1

, u(t), t)dt

(32)

ˆ

xk|k−1

=

2L

X

i=0

w

(m)

i

X

i,k,k−1

(33)

Pk|k−1

=

2L

X

i=0

w

(c)

_i

[Xi,k|k−1

− ˆxk|k−1

][Xi,k|k−1

− ˆxk|k−1

]

T

_{+ Q}

(34)

X

i,k|k−1

∗

= [X0:2L,k|k−1

X0,k|k−1

− νpQ X0,k|k−1

+ νpQ]i

(35)

Y

∗

i,k|k−1

= h(X

i,k|k−1

∗

)

(36)

ˆ

yk

=

2L

a

X

i=0

w

_i

∗(m)

Y

i,k|k−1

∗

(37)

Pxy,k

=

2L

a

X

i=0

w

_i

∗(c)

[Xi,k|k−1

− ˆxk|k−1

][Yi,k|k−1

− ˆyk]

T

(38)

Pyy,k

=

2L

a

X

i=0

w

_i

∗(c)

[Yi,k|k−1

− ˆyk][Yi,k|k−1

− ˆyk]

T

_{+ R}

(39)

where

L

a

_{= 2L, ν =}

√

_{L + γ0}

_{, w}

∗(m)

i

and

w

∗(c)

i

are al ulatedsimilar to Eq. (31)with the

repla ementof

L

by

L

a

,

Qd

isapproximatedby

G(ˆ

xk|k−1)QG

T

_(ˆ

_{xk|k−1)∆t}

where

∆t = tk

−tk−1

Step2 Estimationofthefaults

γk

= (yk

− ˆyk)

(40)

Nk

= (F

T

k

P

−1

yy,k

Fk

)

−1

_F

T

k

P

−1

yy,k

(41)

ˆ

fk

= Nkγk

(42)

P

_k

f

= (F

T

k

P

−1

yy,k

Fk

)

−1

(43)

where

γk

is theinnovation,

fk

ˆ

is theestimation of

fk

and

P

f

k

isits error ovarian ematrix.

Nk

isthegainmatrixwhi h an a hieveanunbiasedestimationof

fk

.

Step3 Measurementupdate

Kk

= Pxy,k

P

_yy,k

−1

(44)

ˆ

xk|k

= ˆ

xk|k−1

+ Kk(yk

− ˆyk

− Fk

fk)

ˆ

(45)

for omparisonandqui kreferen e:

Kk

= Pxy,kP

_yy,k

−1

(47)

ˆ

xk|k

= ˆ

xk|k−1

+ Kk

(yk

− ˆyk)

(48)

P

k|k

= P

k|k−1

− Kk

Pyy,kK

k

T

(49)

It anbe seenthat the measurement update of thenormal UKF, asgiven by Eqs. (47)-(49), does not takethe fault estimation and error ovarian e into a ount. Alsonote that the normal UKFdoesnotestimatethefaults,whi hmeansthat itdoesnot ontainEqs.(40)-(43).

TheADSfaultestimationusingtheRTS-UKFisshowninthefollowing.

ADSfaultestimationusingtheRTS-UKF

Theperforman eoftheRTS-UKFwillbedemonstratedunderdierentinitial onditions. The simulation data is taken from the simulationmodel of aCessna Citation II air raft. During10 s

< t <

17s. there isa3-2-1-1 ommandontheaileron. Thefaults enarioisgiveninTable1. The

trueinitialstate

x0

isasfollows:

x0

= [90, 0.056, 0, 0, 0.0037, 0]

T

(50)

First,thetrueinitial ondition(50)isusedastheinitialguess

x0

ˆ

inthelter.

P0

= 10

−3

_{· I6}

. The standarddeviationsofthemeasurementnoisesare:

σw

x

= σw

y

= σw

z

= 0.001

m/s

2

σw

p

= σw

q

= σw

r

= 0.000018

rad/s

σv

V

= 0.1

m/s

, σv

α

= σv

β

= 0.0018

rad

σv

φ

= σv

θ

= σv

ψ

= 0.0018

rad

Therefore,

Q

and

R

anbeinferredfrom Eqs.(9)and(10) . TheresultsareshowninFig.1. The estimation errors of

V

,

α

and

β

, as shown in Fig. 1(a), are lose to zero-mean. The estimationerrorsof

φ

,

θ

and

ψ

using theRTS-UKFaregivenin Fig.1(b). It anbeseenthatthe estimationerrorsarezero-meanex eptduring theperiodwhenthereisamaneuver(10s

< t <

17

0

10

20

30

40

50

−0.1

0

0.1

∆

V (m/s)

10

20

30

40

50

−0.01

0

0.01

∆α

(rad)

10

20

30

40

50

−0.01

0

0.01

∆β

(rad)

time (s)

(a)Errorofestimationof

V

,

α

and

β

10

20

30

40

50

−2

0

2

x 10

−3

∆φ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆θ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆ψ

(rad)

time (s)

(b)Errorofestimationof

φ

,

θ

and

ψ

0

10

20

30

40

50

60

−5

0

5

f

V

(m/s)

0

10

20

30

40

50

60

−0.1

0

0.1

0.2

f

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

f

β

(rad)

estimation

true

time (s)

( )Estimationof

f

V

,

f

α

and

f

β

0

10

20

30

40

50

−0.5

0

0.5

∆

f

V

(m/s)

0

10

20

30

40

50

−0.01

0

0.01

∆

f

α

(rad)

0

10

20

30

40

50

−0.01

0

0.01

∆

f

β

(rad)

time (s)

(d)Estimationerrorof

f

V

,

f

α

and

f

β

Fig.1: ResultofstateandADSfaultestimationusingtheRTS-UKFapproa handinitial ondition(50)inthepresen eofmultiplefaults

s). However,duringthisperiodtheestimationerrorsaresmall,e.g.,themaximumestimationerror of

φ

islessthan2

×10

−3

rad.

Theestimationof

fV

,

fα

and

fβ

isgiveninFig.1( ). As anbeseen,allthefaultsareestimated in anunbiasedsense. Theestimationerrors anbefoundinFig.1(d).

Next, the performan e with two dierent initial onditions for

x0

ˆ

is tested. The two initial onditionsareasfollows:

ˆ

x0

= [90, 0, 0, 0, 0, 0]

T

,

(51)

ˆ

x0

= [1, 0, 0, 0, 0, 0]

T

_.

(52)

ondition (51) slightlydeviates from ondition(50).

P0

isthe samewith the previoussimulation andis

10

−3

_{· I}

6

.

Thestateestimation errorsoftheRTS-UKFusing theinitial onditionEq.(51)are shownin Fig.2(a)and 2(b) . As anbeseenfrom Fig.2(a), theestimation errorsof

V

,

α

and

β

arelarger than those shown in Fig. 1(a). The estimation errors of

φ

,

θ

and

ψ

, shown in Fig.2(b) , are the sameasthoseshownin Fig.1(b).

Thestateestimation errorsoftheRTS-UKFusing theinitial onditionEq.(52)are shownin Fig.2( )and2(d). Theestimationerrorsof

V

,

α

and

β

,shownin Fig.2( ),aresigni antlyworse thanthose shown inFig.1(a)and Fig.2(a). However,theestimation errorsof

φ

,

θ

and

ψ

,shown in Fig.2(d),arestillzero-mean.

Theestimatesof

fV

,

fα

and

fβ

usingtheinitial onditionEqs.(51)and (52)aredemonstrated inFig.2(e)and 2(f)respe tively. As anbeseenfromFig.2(e),whentheinitial

x0

deviatesfrom thetruestate,theestimatesofthefaultsalsodeviatefromtheirtruemagnitudesespe iallythatof

fα

. Whentheinitial onditiondeviatessigni antlyfromthetrueinitial ondition,theperforman e

be omessigni antlyworse,as anbeseeninFig.2(f) .

Based onthe abovesimulationresults, it is seenthat theRTS-UKFis sensitiveto the initial onditionerrors. Thissensitivityproblem willbeanalyzedtheoreti allyinthefollowingse tion.

B. Problemanalysis oftherobust three-step lter

Inthepreviousse tions,itwasshownthattheperforman eoftheRTS-UKFisinuen edbythe giveninitial ondition. Thisse tionanalyzestheproblem ofthesensitivitytotheinitial ondition.

RewriteEq.(45)into

ˆ

x

k|k

= ˆ

x

k|k−1

+ Lkγk

(53)

where

Lk

isdenedas

Lk

:= Kk(I − Fk

Nk)

(54)

0

10

20

30

40

50

60

−0.5

0

0.5

∆

V (m/s)

0

10

20

30

40

50

60

0.04

0.05

0.06

∆α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

∆β

(rad)

time (s)

(a)Errorofestimationof

V

,

α

and

β

using ondition(51 )

10

20

30

40

50

−2

0

2

x 10

−3

∆φ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆θ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆ψ

(rad)

time (s)

(b)Errorofestimationof

φ

,

θ

and

ψ

using ondition(51 )

0

10

20

30

40

50

60

−100

0

100

∆

V (m/s)

0

10

20

30

40

50

60

−1

0

1

∆α

(rad)

0

10

20

30

40

50

60

−2

0

2

∆β

(rad)

time (s)

( )Errorofestimationof

V

,

α

and

β

using ondition(52 )

10

20

30

40

50

−2

0

2

x 10

−3

∆φ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆θ

(rad)

10

20

30

40

50

−1

0

1

x 10

−3

∆ψ

(rad)

time (s)

(d)Errorofestimationof

φ

,

θ

and

ψ

using ondition(52 )

0

10

20

30

40

50

60

−5

0

5

f

V

(m/s)

0

10

20

30

40

50

60

−0.1

0

0.1

0.2

f

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

f

β

(rad)

estimation

true

time (s)

(e)Estimationof

f

V

,

f

α

and

f

β

using ondition(51 )

0

10

20

30

40

50

60

−100

0

100

f

V

(m/s)

0

10

20

30

40

50

60

−1

0

1

f

α

(rad)

0

10

20

30

40

50

60

−2

0

2

f

β

(rad)

estimation

true

time (s)

(f)Estimationof

f

V

,

f

α

and

f

β

using ondition(52 )

Fig.2: Resultofstateandfault estimationusingtheRTS-UKFapproa handtwodierentinitial onditionsin thepresen eofmultiple faults

Pxy,k

=









P11

P12

P21

P22









, Fk

=









Ip

0









,

γk

=









γp

γm−p









, x =









xp

xL−p









(55)

Sin e

Pyy,k

isinvertible,itsinverse anbepartitionedasfollows:

P

_yy,k

−1

=









˜

R11

R12

˜

˜

R21

R22

˜









(56) where

R11

˜

∈ R

p×p

,

R12

˜

∈ R

p×(m−p)

,

R21

˜

∈ R

(m−p)×p

and

R22

˜

∈ R

(m−p)×(m−p)

. Therefore,Eq.(41) anbe omputedby

Nk

= [ ˜

R

11

−1

0]









˜

R11

R12

˜

0

0









=



Ip

R

˜

₁₁

−1

R12

˜



(57)

Substituting Eq.(57)intoEq.(54) ,itfollows

Lk

= Pxy,kP

_yy,k

−1









0 − ˜

R

−1

11

R12

˜

0

Im−p









(58)

=









0 L12

0 L22









(59)

where

L12

and

L22

aredened as

L12

:= P12( ˜

R22

− ˜

R21

R

˜

−1

11

R12)

˜

L22

:= P22( ˜

R22

− ˜

R21

R

˜

−1

₁₁

R12)

˜

Therefore,themeasurementupdateoftherobustthree-steplter,denotedin Eq.(53), anbe furtherwritten asfollows:

ˆ

x

k|k

= ˆ

x

k|k−1

+









L12γm−p

L22γm−p









(60)

It anbeseenthat

γp

isnotusedinthemeasurementupdate. Sin e

γp

isnotused,theestimationof

xp

isnotupdatedbymeasurementsof

xp

. Therefore,theestimationof

xp

(

V

,

α

and

β

),issensitive

to theinitial ondition. If theinitial

x0

signi antly deviates from thetrue value, it will notbe orre ted to the true value. However, the estimation of

φ

,

θ

and

ψ

is not inuen ed sin e they areupdatedbythemeasurement. Thisis onsistentwiththeresultshowninFigs.2(b)and 2(d), wheretheestimationof

φ

,

θ

and

ψ

isstillgoodevenwhenthatof

V

,

α

and

β

isnot.

In asethat

p = m

andrank

Fk

= m

,it anbefoundthat

Nk

= F

_k

−1

(61)

Lk

= 0

(62)

Consequently,themeasurementupdate ofthethree-stepKalmanlteris

ˆ

xk|k

= ˆ

xk|k−1

(63)

This means that allthe states are notupdated by their measurements. In this situation, all the stateestimationwillbesensitiveto theinitial ondition.

Throughthe analysis in this se tion and the performan e demonstration of the RTS-UKFin Se tion IIIA, theneed for a modi ation of the RTS-UKFis emphasized. In real life, the exa t initial onditionisdi ulttoobtainduetoun ertaintiesinthesystem(whi h analsobefoundin Se tionV).TheRTS-UKFwillinterprettheinitializationerrorasafault,whi hresultsintowrong faultestimation. Therefore,theRTS-UKF annotbeappliedtotheFDD oftheADS s.

C. NovelAdaptiveThree-Step Uns ented KalmanFilter forADS FDD

Havingfound the auseforperforman edegradationoftheRTS-UKF,this se tionproposesa novelATS-UKF to solvetheADS FDD . The sensitivity to theinitial onditionof theRTS-UKF anbesolvedbyperformingthemeasurementupdateofnormalUKF.

It shouldbenotedthat theRTS-UKFonly onsiderstheestimation ofthefaults. Itdoesnot dete t and isolate the faults. The proposed ATS-UKFdeals with not only the estimation of the faults, butalsothedete tionandisolation.

ompleteFDD systemisintrodu ed.

1. Initialmeasurement update

The solutionto redu e thesensitivityof theRTS-UKF to theinitial onditionis proposed in thissubse tion,whi histousethemeasurementupdate ofnormalUKF(Eqs.(47) -(49))whenthe stateestimationisinuen edbytheinitializationerror. However,whenthe orre tionissu ient, i.e., whenthe measurement update of theUKF is su ient, needsto bedetermined. This paper proposesa riteriawhi h andeterminewhetherthemeasurementupdateoftheUKFissu ient. Thedetailsaregivenasfollows:

Let

Cii,k

,

i = 1, 2, 3

denote the

i

th diagonalelementsof theinnovation ovarian ematrix

Ck

asso iatedwith themeasurementswhi h arenotused in theupdate oftheRTS-UKFat timestep

k

. (i.e.,themeasurementof

V

,

α

and

β

respe tivelyinthispaper).

Denethe hange oftheinnovationvarian e

∆Cii,k

as

∆Cii,k

:= Cii,k

− Cii,k−1

,

i = 1, 2, 3.

(64)

When the following inequality holds, the measurement update anbe regarded as su ient. Theinequalityis

∆Cii,k

< ηi, i = 1, 2, 3.

(65)

where

ηi, i = 1, 2, 3

arepre-dened onstantswhi h anbetunedtostopthemeasurementupdate. Theprin ipleisthatifthereareinitializationerrors,

Cii,k

isnot onstant. Whentheltera hieves steady-state,

Cii,k

isapproximately onstant. Therefore,

∆Cii,k

shouldbesmall. If

∆Cii,k

issmaller than

ηi

, then it indi ates that thelter hasrea hedsteady-state and themeasurementupdate of theUKF issu ient. If

ηi

is hosento be small, thenthe numberofinitial measurement update will be bigger while the inuen e of the initial onditionerror will be less.

Ck

anbe estimated usingthefollowing[29,30℄:

ˆ

Ck

=

1

N

k

X

j=k−N +1

γjγ

j

T

(66)

The fault dete tion is performed by monitoring the innovation varian e of the lter. In the presen eof

i

thfault,

Cii,k

in reases. Thefaultdete tionandisolationlogi attimestep

k

is:

if

Cii,k

> Ti

,

FA

i

= 1

. otherwise

FA

i

= 0

,i=1,2,3. where

FA

= [FA

V

FA

α

FA

β

]

T

arethealarmindi ators.

Ti

arethethresholdswhi haredesigned todete tthefaultsinthe

V

,

α

and

β

sensorsrespe tively. Thesethresholdsaredesignedbasedon thefault-free ase. It anbeseenthatthefaultdete tionandisolationaresimultaneouslyrealized.

Theweightedfaultestimation an be al ulatedasfollows:

¯

fi,k

= FA

i

fi,k,

ˆ

i = 1, 2, 3.

(67)

3. Adaptive Three-StepUns entedKalmanFilter

Whentheinitialmeasurementupdateissu ient,therearetwooptionstoa hieveFDDwhi h areasfollows:

1. Aftertheinitialmeasurementupdate, theFDIs hemeisusedtodete tandisolatethefaults. TheRTS-UKFisusedtoestimatethefaults.

2. Aftertheinitialmeasurementupdate, theFDIs hemeisusedtodete tandisolatethefaults. Ifthereare nofaults dete ted,theUKFisused andthe faultestimation is onsidered tobe zero. If there are faults dete ted, then the RTS-UKF is used for the fault estimation and measurementupdate.

TheATS-UKFproposedinthispaper,isbasedonthelatteronesin eit anredu ethe ompu-tationalload. Themeasurementupdate oftheATS-UKFswit hes adaptivelybetweenthat ofthe normal UKFand that of the RTS-UKF throughthe FDI s heme. Thespe i three stepsof the ATS-UKFaregivenasfollows:

1. Timeupdate

Thisis thesameas in theUKF, whi h also in ludes thesigmapoint al ulation. Thesteps aredes ribedbyEqs.(30)-(39).

Before estimating the faults, the FDI , whi h has been introdu ed above, is performed. If

FA

= 0

, then

fk

ˆ

= 0

. If

FA

6= 0

,thenthefaults areestimatedusing theRTS-UKF,whi h is

des ribedbyEqs. 40-(43). Thisstepis theFDD .

3. Measurementupdate

Asmentioned,duringtheinitialmeasurementupdate,themeasurementupdateofthenormal UKFisapplied. Whentheinitial measurement updateis done, themeasurementupdate of theATS-UKFisasfollows:

(a) If

FA

= 0

,usethethemeasurementupdateofthenormalUKF.

Inthissituation,therearenofaultsdete tedinthesystem. Toredu ethe omputational load, the measurement update of the UKF is used. This means that the faults are onsidered to be zero, so that the faults estimation and measurement update of the RTS-UKFarenotneeded. Thestepsaredes ribedbyEqs.(47)-(49) .

(b) If

FA

6= 0

,usethemeasurementupdateoftheRTS-UKF.

In this situation, faults are dete ted. Therefore, the measurement update of the RTS-UKFisneededto obtainanunbiasedstateestimation andfaultestimation, whi h annotbea hievedusingthenormalUKF. Thestepsaredes ribedbyEqs.(44)-(46) .

D. ADSFDDusingthe ATS-UKF

Inthisse tion,theFDDaswellasthestateestimationperforman eoftheproposedATS-UKF isdemonstratedusingtwodierentfaults enarios. Theinitial onditionisthesameasinEq.(52). Thethresholdto stoptheinitial measurementupdate is

η = [5 × 10

−3

_{, 2 × 10}

−5

_{, 2 × 10}

−5

_]

T

and thethresholdtodete tthefault is

T = [0.2, 1 × 10

−4

_{, 5 × 10}

−5

_]

T

.

1. MultipleFDD

Inthiss enario, onse utiveADSfaultsaregenerated,whi hareshowninTable1. Theresults usingtheATS-UKFareshowninFig.3.

10

20

30

40

50

−0.05

0

0.05

∆

V (m/s)

10

20

30

40

50

−1

0

1

x 10

−3

∆α

(rad)

0

10

20

30

40

50

60

−1

0

1

x 10

−3

∆β

(rad)

time (s)

(a)Errorofestimationof

V

,

α

and

β

0

10

20

30

40

50

60

−2

0

2

x 10

−3

∆φ

(rad)

0

10

20

30

40

50

60

−2

0

2

x 10

−3

∆θ

(rad)

0

10

20

30

40

50

60

−5

0

5

x 10

−3

∆ψ

(rad)

time (s)

(b)Errorofestimationof

φ

,

θ

and

ψ

0

10

20

30

40

50

60

0

0.5

1

FA

V

0

10

20

30

40

50

60

0

0.5

1

FA

α

0

10

20

30

40

50

60

0

0.5

1

FA

β

time (s)

( )Faultdete tionandisolation

0

10

20

30

40

50

−2

0

2

f

V

(m/s)

0

10

20

30

40

50

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

−0.05

0

0.05

f

β

(rad)

estimation

true

time (s)

(d)Estimationof

f

V

,

f

α

and

f

β

0

10

20

30

40

50

−2

0

2

f

V

(m/s)

0

10

20

30

40

50

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

−0.05

0

0.05

f

β

(rad)

estimation

true

time (s)

(e) Weightedestimationof

f

V

,

f

α

and

f

β

0

10

20

30

40

50

−0.5

0

0.5

∆

f

V

(m/s)

0

10

20

30

40

50

−0.1

0

0.1

∆

f

α

(rad)

0

10

20

30

40

50

−0.02

0

0.02

∆

f

β

(rad)

time (s)

(f)Errorofestimationof

f

V

,

f

α

and

f

β

Fig.3: Resultofstateestimation andADSFDD usingtheproposed ATS-UKFapproa hinthe presen eofmultiple faults

twotimesteps. Theestimationof

V

,

α

and

β

isshowninFig.3(a). Despitethefa tthattheinitial

x0

signi antly deviatesfrom thetrue state, the estimation is still satisfa tory. This meansthat

thesensitivitytotheinitial onditionoftheRTS-UKFista kledbytheATS-UKF. Theestimation of

φ

,

θ

and

ψ

, as shown in Fig.3(b), is also satisfa tory. This demonstrates thestate estimation performan eoftheATS-UKF.

The fault dete tionand isolationis a hieved by he king

FA

V

,

FA

α

and

FA

β

, whi h isshown

in Fig.3( ). Fromthegure,it anbeseenthat

fV

is dete tedinstantaneously. Thedete tionof

fα

takeslonger than that of

fV

. This is be ause

fα

is adrift fault whi h is a slow time-varying

fault.

fβ

isalsodete tedinstantaneously. However,

FA

β

swit hesfrom1to0ninetimes. Thisisas

expe tedsin etheos illatoryfault rosseszeroninetimes. Whenthemagnitudeofthefaultiszero, it anberegardedas no fault. Fromthe gure,it is obviousthat fault isolation is alsoa hieved. Forinstan e,when

FA

V

= 1

,both

FA

α

and

FA

β

areequaltozero,whi hmeansonly

fV

o urs.

Theestimationof

fV

,

fα

and

fβ

isshowninFig.3(d). As anbeseen,allthefaultsareestimated inanunbiasedsense. Theweightedfaultestimation, al ulatedusingEq.(67),isshowninFig.3(e). Theerroroftheestimationof

fV

,

fα

and

fβ

isshowninFig.3(f) . Itisseenthatalltheestimation errorsare zero-mean. Thisdemonstratesthe faultestimation performan eof theATS-UKF. It is alsonoti ed thatwhenthere arenofaults orthefaultsarenotdete ted,theestimatesofthefault are zero and soare the estimation errors. This is due to the fa t that when there are nofaults dete ted,themeasurementupdateoftheUKFisused andthefaults are onsidered tobezero.

The fault dete tion of theos illatory faults, shown in Fig. 3( ), showsa hattering behavior. Todete tthepresen eof os illatoryfailures,thedete tionlogi ofos illatoryfaults inGoupil[31℄ isused. Thebasi ideaisto ountthe rossingsofthefaultestimate(shownin Fig.3(e))through apositiveandnegativethresholdwithinasliding timewindow. Inthispaper,theos illatoryfaults aredete tedifonefullos illationisdete ted. Theresultofdete tingtheos illatoryfaultisshownin Fig.4(a). Inthegure,

OF C

denotesos illatoryfailure ase(OFC).As anbeseen,anos illatory faultis onlydete tedin the

β

sensor. Ifwetakethebiggervalueof

FAi

and

OF Ci

(

i

isasso iated with thesensor of

V

,

α

and

β

),the faultdete tion in ludingthedete tionof theos illatoryfault anbe obtained,whi h is demonstrated in Fig. 4(b). Inthe gure, the red dashedline indi ates

0

10

20

30

40

50

60

0

0.5

1

OFC

V

0

10

20

30

40

50

60

0

0.5

1

OFC

α

0

10

20

30

40

50

60

0

0.5

1

OFC

β

time (s)

(a)Dete tionofos illatoryfaults,

OF C

indi atesthe dete tionofos illatoryfaults

0

10

20

30

40

50

0

0.5

1

FA

V

0

10

20

30

40

50

0

0.5

1

FA

α

0

10

20

30

40

50

0

0.5

1

FA

β

OFC

time (s)

(b)Faultdete tionandisolationin ludingos illatory faultindi ation

Fig.4: Resultof faultdete tionand isolationusingtheproposedATS-UKFapproa hinthe presen eofmultiple faults

that thedete tedfaultisanos illatoryfault.

2. SimultaneousFDD

Inthiss enario,simultaneousfaultsaregeneratedwhi hareshowninTable2. TheATS-UKF isusedto dete t,isolateandestimatethesefaults. TheresultsaregiveninFig.5.

The estimation of

V

,

α

,

β

and

φ

,

θ

and

ψ

using the ATS-UKF is shown in Fig. 5(a) and Fig. 5(b), respe tively. As an be seen, even in the presen e of simultaneous faults, the state estimationperforman eoftheATS-UKFisstillsatisfa tory.

Thefault dete tionand isolationperforman eisshownin Fig.5( ). As anbeseen,there are nofalse alarms,whi h demonstratesitsgoodperforman e. Forthedete tionof

fα

and

fβ

during 10s

< t <

20s,and

fV

during30s

< t <

40s,therearedete tiondelayssin etherearedriftfaults. Thereddashedlinesinthegureindi ate thatthedete tedfaultsareos illatoryfaults.

Fig.5(d)and 5(e)showtheestimationandweightedestimationof

fV

,

fα

and

fβ

,respe tively. It anbeseenthat thefaultestimation performan e issatisfa tory. All faults areestimatedin an unbiased sensein luding theos illatoryfaults. Theestimationerrorof

fV

,

fα

and

fβ

is shownin Fig.5(f) . It anbeseenthattheerroriszero-mean,whi h onrmsthegoodestimationperforman e

10

20

30

40

50

−0.05

0

0.05

∆

V (m/s)

10

20

30

40

50

−1

0

1

x 10

−3

∆α

(rad)

0

10

20

30

40

50

−1

0

1

x 10

−3

∆β

(rad)

time (s)

(a)Errorofestimationof

V

,

α

and

β

0

10

20

30

40

50

−2

0

2

x 10

−3

∆φ

(rad)

0

10

20

30

40

50

−2

0

2

x 10

−3

∆θ

(rad)

0

10

20

30

40

50

−5

0

5

x 10

−3

∆ψ

(rad)

time (s)

(b)Errorofestimationof

φ

,

θ

and

ψ

0

10

20

30

40

50

0

0.5

1

FA

V

0

10

20

30

40

50

0

0.5

1

FA

α

0

10

20

30

40

50

0

0.5

1

FA

β

OFC

time (s)

( )Faultdete tionandisolation

0

10

20

30

40

50

−5

0

5

f

V

(m/s)

0

10

20

30

40

50

−0.2

0

0.2

f

α

(rad)

0

10

20

30

40

50

−0.2

0

0.2

f

β

(rad)

estimation

true

time (s)

(d)Estimationof

f

V

,

f

α

and

f

β

0

10

20

30

40

50

−2

0

2

f

V

(m/s)

0

10

20

30

40

50

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

−0.1

0

0.1

f

β

(rad)

estimation

true

time (s)

(e) Weightedestimationof

f

V

,

f

α

and

f

β

0

10

20

30

40

50

−1

0

1

∆

f

V

(m/s)

0

10

20

30

40

50

−0.1

0

0.1

∆

f

α

(rad)

0

10

20

30

40

50

−0.01

0

0.01

∆

f

β

(rad)

time (s)

(f)Errorofestimationof

f

V

,

f

α

and

f

β

Fig.5: ResultofstateandADSFDDusingtheproposed ATS-UKFapproa hin thepresen eof simultaneous faults

Inthepreviousse tion,theFDDperforman eoftheATS-UKFistestedusingsimulatedair raft data. Inthisse tion,theFDDperforman eoftheATS-UKFisvalidatedusingrealighttestdata oftheCessnaCitationIIair raft. Airdatainformationsu has

α

and

β

aremeasuredforpostight analysis. Therealightdata ontainsun ertainties su h asbiasesandspikes. Additionally,in real ight,externaldisturban es,su has hangingwind, analsoinuen etheairdatameasurements. Therefore, thereal ightdata poses hallenges to theADS FDD problem andprovidesarealisti validation oftheperforman eofFDDapproa hessu hastheATS-UKF.

Theprimaryobje tiveoftheighttestisaerodynami modelidenti ationwhereanumberof maneuverswereperformedbytheair raftinordertoobtainsu ientex itation. Sin etherewere no faultsduring the ight, sensor faultsare inje ted into thereal ight datato validate theFDD performan eoftheATS-UKF. Besidesthefaults enariospresentedin Tables1and 2,afault-free aseisalsostudied.

The real ight data used in this paper is the same as that in Lu et. al [26℄. In Lu et. al, theestimated windturns outto betimevarying. This antesttheADSFDD performan eofthe ATS-UKFunderthe onditionof winds.

Theupdateratesoftheon-boardsensorsaregiveninTable3.

Table3: Updatefrequen iesofdierentmeasurements

Measurements Unit Updatefrequen y

V

[m/s℄ 100Hz

u

n

,

v

n

,

w

n

[m/s℄ 1Hz

α

,

β

[rad℄ 100Hz

φ

,

θ

[rad℄ 100Hz

ψ

[rad℄ 10Hz

A. Real-life measurementmodel

For simulated air raft data, the measurement model is given in Eqs. (19) - (24). If

f = 0

, themeasurementsareonly orruptedbywhiteGaussiannoises,as anbeseenfromtheequations.

However,this is neverthe asein real life. Inthis ight test, theairdata information, su h as

α

and

β

, ismeasuredby multiple vanes onaboom(shown in Fig.6) whi h ismountedonthenose of the air raft. The angle of atta k and angle of sideslip measured by the vanes are denoted by

αvm

and

βvm

,respe tively. Themeasurements

αvm

and

βvm

isdierentfrom Eqs.(20) and(21),

respe tively. Themeasurementmodelforthereal-lifemeasurementsisgivenasfollows[3234℄:

Vm

= V + vV

(68)

αm

= Cα0

+ (1 + Cup)α +

xαq

V

+ vα

(69)

βm

= Cβ0

+ (1 + Csi)β −

xβr

_V

+

zβp

V

+ vβ

(70)

φm

= φ + vφ

(71)

θm

= θ + vθ

(72)

ψm

= ψ + vψ

(73)

where

xα

,

xβ

and

zβ

arethepositionofthevanesinthebodyframe,

Cα0

,

Cβ0

,

Cup

and

Csi

arethe boom orre tionparameters. In this paper,

zβ

isassumed to be zero. Theparameter estimation anbefoundin Luet al. [34℄. Forthe ADSFDD using realightdata, this measurementmodel is used. However, it should be noted that in this real-life measurement model, boom bending is onsideredto benegligibleforthemaneuversown.

Fig.6: Thevanes ontheboomformeasuringtheangleofatta kand angleofsideslip. Photo reditsbyDaanPool.

Usingthereal-lifemeasurements,un ertaintiesanddisturban essu hasvaryingwinds analso have anegativeinuen e on the FDD performan e, whi h anresult in falsealarms. Under this ondition,theFDDapproa hesshouldnotgivefalsealarms. Inthisse tion,theATS-UKFistested in afault-free asetoverifywhether itgivesfalsealarms.

Inorder to show theee tiveness of theATS-UKF, theRTS-UKFis alsoapplied to estimate theADSfaults. Theinitial ondition

x0

givenfortheRTS-UKFistherstmeasurementwhi his:

[104.8733, 0.0796, 0.0073, −0.0019, 0.0733, 4.6692]

T

(74)

Theinitial ondition

x0

givenfortheATS-UKFis:

[1, 0, 0, 0, 0, 0]

T

(75)

In this manner, the initial ondition given for the RTS-UKF is lose to the true state whereas that of theATS-UKF signi antlydeviates from thetrue state. The thresholdto stopthe initial measurementupdate is

η = [5 × 10

−3

_{, 2 × 10}

−5

_{, 2 × 10}

−5

_]

T

and thethresholdtodete t thefault is

T = [0.2, 1 × 10

−4

_{, 5 × 10}

−5

_]

T

, whi harethesameasthoseusedinthepreviousse tion. Theestimation of

V

,

α

and

β

using theRTS-UKFis givenin Fig.7(a). Thefault estimation using theRTS-UKFis givenin Fig.7(b). As anbeseen,the estimatedfaults deviatefrom their truemagnitudes. ThisresultshowsthattheRTS-UKFisnotabletobeappliedforrealappli ations unlessmodi ationsaremade.

TheresultsoftheATS-UKFareshowninFigs.7( )-7(f) . Theestimationof

V

,

α

and

β

,and

φ

,

θ

and

ψ

areshowninFig.7( )and7(d)respe tively. Theestimatesof

α

and

β

usingtheATS-UKF

aredierentfromthoseusingtheRTS-UKF. Thefaultdete tionresult,showninFig.7(e),indi ates that there areno faults. This demonstratesthat the ATS-UKFdoesnotgivefalsealarms in the presen e of no faults even when the real ight data is used. The weighted estimates of

fV

,

fα

and

fβ

, shownin Fig.7(f) , are zero-mean. This onrmsthefault estimation performan eof the ATS-UKF.

0

10

20

30

40

50

60

104

106

108

V (m/s)

0

10

20

30

40

50

60

0.06

0.08

0.1

α

(rad)

0

10

20

30

40

50

60

0

0.02

0.04

β

(rad)

time (s)

(a)Estimationof

V

,

α

and

β

usingtheRTS-UKF

0

10

20

30

40

50

60

−0.5

0

0.5

f

V

(m/s)

0

10

20

30

40

50

60

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

60

−0.03

−0.02

−0.01

0

0.01

f

β

(rad)

true

estimation

time (s)

(b)Estimationof

f

V

,

f

α

and

f

β

usingtheRTS-UKF

0

10

20

30

40

50

60

104

106

108

V (m/s)

0

10

20

30

40

50

60

0

0.05

0.1

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

β

(rad)

time (s)

( ) Estimationof

V

,

α

and

β

usingtheATS-UKF

0

10

20

30

40

50

60

−0.1

0

0.1

φ

(rad)

0

10

20

30

40

50

60

0

0.05

0.1

θ

(rad)

0

10

20

30

40

50

60

4.65

4.7

ψ

(rad)

time (s)

(d)Estimationof

φ

,

θ

and

ψ

usingtheATS-UKF

0

10

20

30

40

50

60

0

0.5

1

FA

V

0

10

20

30

40

50

60

0

0.5

1

FA

α

0

10

20

30

40

50

60

0

0.5

1

FA

β

time (s)

(e)Faultdete tionandisolationusingtheATS-UKF

0

10

20

30

40

50

60

−2

0

2

f

V

(m/s)

0

10

20

30

40

50

60

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

f

β

(rad)

true

estimation

time (s)

(f)Weightedestimationof

f

V

,

f

α

and

f

β

usingthe ATS-UKF

Fig.7: Stateestimation andADSFDD ofthereal-lifemeasurementmodeloftheair raftusing theRTS-UKFandtheATS-UKFapproa hin theabsen eoffaults

Inthissubse tion,theADSFDD performan eof theATS-UKFwillbeveriedusingthe real-life measurementmodel inthe presen eofmultiple faults(givenin Table1). Theinitial ondition forRTS-UKFisthesameasin(74)andthatoftheATS-UKFisthesameasin (75).

Theresultsusingthese twoapproa hesareshownin Fig.8.

Theestimationof

V

,

α

and

β

usingtheRTS-UKFisshowninFig.8(a). Thefaultestimationis showninFig.8(b) . Althoughtheinitial onditionoftheRTS-UKFis hosentobethemeasurements, theestimation of thefaultsare stillbiased. Thisshowsthedrawba kof theRTS-UKFwhenit is usedin pra ti ebe ausetheinitial onditionerrorwillbeestimatedasabiasfault.

Theestimation of

V

,

α

,

β

and

φ

,

θ

,

ψ

using theATS-UKFispresentedin Fig.8( )and 8(d) respe tively. It anbeseenthattheestimatesof

α

,

β

areagaindierentfromthoseoftheRTS-UKF showninFig.8(a).

Thefaultdete tionandisolationusingtheATS-UKFisgivenin Fig.8(e). Nofalsealarmsare generatedandtheisolationisalso orre t. It anbeseenthattheperforman eisasgoodasthatin Fig.4(b) wherethe simulationdatais used. Theos illatoryfault isalsodete ted, whi h isshown bythereddashedline.

Theweightedestimation of

fV

,

fα

and

fβ

usingtheATS-UKFispresentedinFig.8(f) . Even thoughthe initial onditionofthe ATS-UKF deviatesfrom thetruestate signi antly,its perfor-man eisnotsensitivetotheinitial ondition. Sin ethefaultsareestimated inanunbiasedsense, theestimatesof

α

,

β

usingtheATS-UKFaremorereliablethanthoseusingtheRTS-UKF.

D. ADSFDDusingreal ight data inthe presen eofsimultaneous faults

Inthis subse tion,simultaneous faults (givenin Table2) areinje ted into thereal ightdata tovalidatetheperforman eoftheATS-UKF. TheresultoftheRTS-UKFisalsopresented,whi h isgiveninFig.9. Fromthis gure,itisseenthefaultestimation oftheRTS-UKFisagainbiased although the initial

x0

is hosen to be the measurements. This highlights the limitation of the RTS-UKFwhenusedinreality.

0

10

20

30

40

50

60

104

106

108

V (m/s)

0

10

20

30

40

50

60

0.06

0.08

0.1

α

(rad)

0

10

20

30

40

50

60

0

0.02

0.04

β

(rad)

time (s)

(a)Estimationof

V

,

α

and

β

usingtheRTS-UKF

0

10

20

30

40

50

60

−5

0

5

f

V

(m/s)

0

10

20

30

40

50

60

−0.2

0

0.2

f

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

f

β

(rad)

true

estimation

time (s)

(b)Estimationof

f

V

,

f

α

and

f

β

usingtheRTS-UKF

0

10

20

30

40

50

60

104

106

108

V (m/s)

0

10

20

30

40

50

60

0

0.05

0.1

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

β

(rad)

time (s)

( ) Estimationof

V

,

α

and

β

usingtheATS-UKF

0

10

20

30

40

50

60

−0.1

0

0.1

φ

(rad)

0

10

20

30

40

50

60

0

0.05

0.1

θ

(rad)

0

10

20

30

40

50

60

4.65

4.7

ψ

(rad)

time (s)

(d)Estimationof

φ

,

θ

and

ψ

usingtheATS-UKF

0

10

20

30

40

50

60

0

0.5

1

FA

V

0

10

20

30

40

50

60

0

0.5

1

FA

α

0

10

20

30

40

50

60

0

0.5

1

FA

β

OFC

time (s)

(e)Faultdete tionandisolationusingtheATS-UKF

0

10

20

30

40

50

60

−2

0

2

f

V

(m/s)

0

10

20

30

40

50

60

−0.1

0

0.1

f

α

(rad)

0

10

20

30

40

50

60

−0.05

0

0.05

f

β

(rad)

true

estimation

time (s)

(f)Weightedestimationof

f

V

,

f

α

and

f

β

usingthe ATS-UKF

Fig.8: Stateestimation andADSFDD ofthereal-lifemeasurementmodeloftheair raftusing theRTS-UKFandtheATS-UKFapproa hin thepresen eofmultiplefaults