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Robust Fault Detection for Systems with Electronic Induced Delays:

Application to the Rendezvous Phase of the MSR Mission

R. Fonod, D. Henry, E. Bornschlegl and C. Charbonnel

Abstract— Two robust fault detection schemes are presented

to detect faults affecting the thrust system of the chaser

spacecraft involved in the rendezvous phase of the Mars Sample

Return (MSR) mission. The idea of both proposed methods

is to transform the unstructured uncertainty caused by the

electronic induced delays into unknown inputs and decouple

them by means of an eigenstructure assignment (EA) technique.

The ﬁrst method utilizes a Cayley-Hamilton theorem based

transformation whereas the second relies on a ﬁrst-order Pad´

e

approximation of the time delay. The performances of the

proposed schemes are compared by a sensitivity/robustness

analysis campaign of 4240 runs within the ”high-ﬁdelity“

industrial simulator provided by Thales Alenia Space.

I. INT RO DU CT ION

The problem of time-delay systems is a research subject

in the Fault Detection and Isolation (FDI) community, see

the surveys [1]–[4] and the references therein. The major

papers in this area deal with networked systems since there

may exist (bounded but unknown) delays in communications.

Only a limited bibliography exist on the research of model-

based fault diagnosis for linear time-delay dynamic systems.

For instance, in [5], an unknown input observer (UIO) is

designed for Fault Detection (FD) of state-delayed systems

with known delays. The well known parity space approach

is extended in [6] for fault detection of retarded time-delay

systems. In [7], [8], the proposed method aims at formulating

the robustness as well as the sensitivity of residual signals

to the unknown inputs as well as to the faults in terms

of L2-gain and, based on it, to formulate the design of

fault detection ﬁlters as an optimization problem. In [9], a

robust fault detection and identiﬁcation approach based on

an adaptive observer is developed for uncertain continuous

linear time-invariant (LTI) systems with multiple discrete

time-delays in both states and outputs. Recently, a geometric

approach for FDI of retarded and neutral time-delay systems

was developed in [10]. Robust fault detector design for a

class of linear systems with some nonlinear perturbations and

mixed neutral and discrete time-varying delays is investigated

in [11] using a descriptor technique, Lyapunov-Krasovskii

functional and a suitable change of variables.

This research work was supported by the European Space Agency (ESA)

and Thales Alenia Space in the frame of the ESA Networking/Partnering

Initiative (NPI) program

R. Fonod and D. Henry are with University of Bordeaux, IMS-lab,

Automatic control group, 351 cours de la lib´eration, 33405 Talence, France.

E-mail: {robert.fonod; david.henry}@ims-bordeaux.fr

E. Bornschlegl is with European Space Agency, ESTEC, Keplerlaan 1,

2200 AG Noordwijk, The Netherlands. E-mail: eric.bornschlegl@esa.int

C. Charbonnel is with Thales Alenia Space, 100 Bd du Midi, 06156

Cannes, France. E-mail: catherine.charbonnel@thalesaleniaspace.com

In this paper, two FD schemes are proposed to cope

with the issue of robust residual generation for a class

of LTI systems with disturbances and input delays. The

system is modelled as a continuous-time one with digital

control and delayed control input. Such modelling approach

was presented in [12]. By introducing a Cayley-Hamilton

theorem based and Pad´e approximation based transforma-

tion, inﬂuence/uncertainty of the time-varying input delay

is transformed into unknown input (UI), which as shown,

greatly facilitates the above mentioned difﬁculty. Finally, the

disturbance and the UI vector are lumped and decoupled by

means of Eigenstructure Assignment (EA) technique.

These two methods have been successfully demonstrated

as applicable for the FD of the Mars Sample Return (MSR)

mission with a ”high-ﬁdelity” industrial simulator, under

realistic conditions, taking into account the effects that the

GNC (Guidance, Navigation, Control) unit has on the FD

performances. The research work draws expertise from ac-

tions undertaken within the European Space Agency, Thales

Alenia Space and the IMS laboratory (Univ. Bordeaux 1),

which develop a collaborative effort to create new robust

on-board FDI technologies that may signiﬁcantly advance

the spacecraft autonomy.

II. PROBLE M FOR MUL ATION

Consider a continuous LTI system given by

˙

x(t) = Ax(t) + Bu(t) + Eff(t) + Ed1d1(t)

y(t) = Cx(t)(1)

where x(t)∈Rn,u(t)∈Rnu,y(t)∈Rnyare state, input,

and measurement vectors, respectively. d1(t)∈Rndand

f(t)∈Rnfare the unknown disturbance and the fault vector.

A,B,C,Efand Ed1are known matrices of appropriate

dimensions. The pair (A,C)is assumed to be observable.

Suppose that the system is controlled by a discrete-time

controller with the sampling time Tand that there exists

an upper bounded electronic-induced delay τk∈Rin the

actuation system so that the controller signal uc

k∈Rnu,

generated at t=tk=kT , k = 1,2, ..., arrives at the actuator

at time instant tk+τk. Recalling the fact, every control signal

uc

kis held by a zero-order holder and only valid over the

interval [tk+τk, tk+1 +τk+1), we have

u(t) = uc

k,∀t∈[tk+τk, tk+1 +τk+1)

uc

0,∀t∈[0, τ1)(2)

The problem addressed in this paper can be formulated as:

Problem 1: Design a discrete-time residual generator

r(z) = Hy(z)y(z)+Hu(z)u(z), where Hy/Huare linear

2013 European Control Conference (ECC)

July 17-19, 2013, Zürich, Switzerland.

978-3-952-41734-8/©2013 EUCA 1439

dynamic ﬁlters, so that ris robust against the uncertain time-

varying delay τkand disturbance vector d1.

To solve Problem 1, the inﬂuence of τkis transformed to

unknown input d2and together with d1decoupled by means

of EA technique, Hy/Hubeing observer-based ﬁlters.

III. TRA NS FO RMATI ON IN TO PO LYT OP IC UN CERTAINTY

Assume that τkcan be expressed as: τk=lT +δk≤¯τ,

where lis a known integer, ¯τis the upper bound of τkand

δk∈Ris the unknown varying part of τk, bounded by 0≤

δk< mT , with mbeing a known integer. In the next, assume

that m= 1. (The case when m > 1, is discussed in [13].)

If we assume that d1and fare constant during each time

interval T, that is a reasonable assumption from a practical

point of view, then the discrete representation of (1) is

xk+1 =¯

Axk+Γδk

0uc

k−l+Γδk

1uc

k−l−1+¯

Effk+¯

Ed1d1

k

¯

yk=¯

Cxk

(3)

where

¯

A=eAT,Γδk

0=

T−δk

R

0

eAtdtB,¯

Ed1=

T

R

0

eAtdtEd1

¯

C=C,Γδk

1=

T

R

T−δk

eAtdtB,¯

Ef=

T

R

0

eAtdtEf

Let ¯

B=RT

0eAtdtB, then it follows:

Γδk

0+Γδk

1=ZT

0

eAtdtB=¯

B(4)

Furthermore, using (3) and (4), and introducing a new

augmented state vector zT

k=xT

k(uc

k−l−1)Twe obtain

zk+1 =ˆ

Aτkzk+ˆ

Bτkuc

k−l+ˆ

Effk+ˆ

Ed1d1

k

yk=ˆ

Czk

(5)

where

ˆ

Aδk=¯

AΓδk

1

0 0 ,ˆ

Bδk=¯

B−Γδk

1

I

ˆ

C=¯

C0

0I,ˆ

Ed1=¯

Ed1

0,ˆ

Ef=¯

Ef

0

In this model, Γδk

1is strongly dependent on the uncertain

term δk. Therefore system (5) is an uncertain time-varying

system. The next step is to transform this model to an

uncertain polytopic system. This can be done by means of a

Cayley-Hamilton theorem based transformation.

A. Expressing the Uncertainty as Polytopes of Matrices

Let us ﬁrst consider the following theorem (see [14]):

Theorem 1: The characteristic polynomial of matrix Ais

p(λ) = det(λI−A) = λn+cn−1λn−1+...+c1λ+c0(6)

then eAtcan be written as

eAt=s1(t)I+s2(t)A+...+sn(t)An−1(7)

where si(t),1≤i≤nare solutions to the nth order

homogenous scalar differential equation

s(n)(t) + cn−1s(n−1)(t) + ...+c1s′(t) + c0s(t) = 0 (8)

satisfying the following initial conditions

s(i−1)

i(0)= 1, s(j)

i(0)= 0 for j6=i−1,0≤j≤n−1

Proof: The proof can be found in [14].

Based on theorem 1, proposition given in [13] is considered:

Proposition 1: The Cayley-Hamilton theorem based

transformation of Γδk

1can be expressed as follows

Γδk

1=

2n

P

i=1

µk

iUi(9)

where Uiare constant matrices and

2n

P

i=1

µk

i= 1, µk

i>0.

Proof: Using (7), we have

Γδk

1=

T

Z

T−δk

eAtdtB=

n

X

i=1

T

Z

T−δk

si(t)dt

Ai−1B

(10)

Deﬁne

smax

i= max

0≤δk≤TZT

T−δk

si(t)dt, i = 1,2,...,n

smin

i= min

0≤δk≤TZT

T−δk

si(t)dt, i = 1,2,...,n

then (10) can be rewritten as

Γδk

1=

n

X

i=1 αk

i,0smin

i+αk

i,1smax

iAi−1B(11)

where αk

i,0and αk

i,1are two time-varying unknown parame-

ters satisfying 0≤αk

i,0≤1,0≤αk

i,1≤1, and αk

i,0+αk

i,1= 1.

It can be veriﬁed that RT

T−δksi(t)dt, i = 1,2,...,n are

Lipschitz-continuous on 0≤δk≤T, i.e. they satisfy

ZT

T−δ1

k

si(t)dt −ZT

T−δ2

k

si(t)dt

≤κiδ1

k−δ2

k

∀δ1

k, δ2

k∈[0, T ], where κi, i = 1,2, ..., n are the Lipschitz

constants. Setting µk

2i−1=αk

i,0/n,µk

2i=αk

i,1/n,U2i−1=

nsmin

iAi−1B,U2i=nsmax

iAi−1B, then from (10) and

(11) proposition 1 yields.

Following proposition 1, it can be seen that the system (5)

can be rewritten to a polytopic uncertain system as follows

zk+1 = ˆ

A0+

2n

X

i=1

µk

iˆ

Ai!zk+

+ ˆ

B0+

2n

X

i=1

µk

iˆ

Bi!uc

k−l+ˆ

Effk+ˆ

Ed1d1

k

yk=ˆ

Czk

(12)

where

ˆ

A0=¯

A0

0 0 ,

ˆ

B0=¯

B

I,

ˆ

Ai=0Ui

0 0

ˆ

Bi=−Ui

0

and the rest of the parameters are the same with those in (5).

1440

B. Expressing the Polytopic Uncertainty as an UI

The uncertain parts of (12), where ˆ

Aiand ˆ

Biare known

constant matrices, µk

iis an unknown time-varying scalar

factor, can be approximated by a disturbance term as in [2]

2n

X

i=1

µk

iˆ

Aizk+

2n

X

i=1

µk

iˆ

Biuc

k−l=ˆ

Ed2d2

k(13)

where

ˆ

Ed2=hˆ

A1,..., ˆ

A2n,ˆ

B1,..., ˆ

B2ni

d2

k=µk

1zk, . . . , µk

2nzk, µk

1uc

k−l, . . . , µk

2nuc

k−lT

Now, the two unknown inputs d1

kand d2

kcan be lumped

together, and deﬁned to be dk. Correspondingly, the UI

distribution matrix ˆ

Ed. That is

dk=(d1

k)T(d2

k)TT,ˆ

Ed=ˆ

Ed1

ˆ

Ed2(14)

Taking the above notation into account, the design model is

expressed in terms of lumped unknown inputs as

zk+1 =ˆ

A0zk+ˆ

B0uc

k−l+ˆ

Effk+ˆ

Eddk

yk=ˆ

Czk

(15)

This model represents the discrete-time model of the orig-

inal system (1), that takes into account both disturbances

d1

kand uncertainties caused by electronic-induced delays

represented as an additional unknown input d2

k.

IV. PAD ´

EAPP ROXIM ATION

This section addresses another method to model the effect

of the time-varying delay τk. Only the necessary develop-

ments from [15] are recalled here. Let us assume that τk

can be represented by a time-varying piecewise continuous

(continuous from the right) delay τ(t) = τk,∀t∈[tk, tk+1).

In this sense, the system input (2) is expressed as

u(t) = uc(t−τ(t)) (16)

where uc(t) = uc

k,∀t∈[tk, tk+1)is the control signal.

A ﬁrst-order Pad´e approximation of the time-varying delay

τ(t)is given by

e−τ(t)s.

=1−τ(t)

2s

1 + τ(t)

2s(17)

An equivalent state space representation of (17) is thus

˙

xd(t) = Ad(t)xd(t) + Bduc(t)

u(t) = Cd(t)xd(t) + Dduc(t)(18)

where xd(t)∈Rnuis the delayed state vector and Ad(t) =

−2

τ(t)I,Bd=I,Cd(t) = 4

τ(t)I,Dd=−I.

The augmented state-space description of the system (1)

and delayed inputs (18) is:

˙

z(t) = ˆ

A(t)z(t) + ˆ

B0uc(t) + ˆ

Eff(t) + ˆ

Ed1d1(t)

y(t) = ˆ

Cz(t)(19)

ˆ

A(t) =A BCd(t)

0Ad(t),ˆ

B0=BDd

Bd,ˆ

C=C0

z(t) =x(t)

xd(t),ˆ

Ef=Ef

0,ˆ

Ed1=Ed1

0

Using (18), the uncertainty is present only in ˆ

A(t).

A. Expressing the Time-Varying Uncertainty as an UI

To proceed, consider that τ(t)can be expressed as

τ(t) = τ0+ ∆τ(t) : |∆τ(t)| ≤ ¯ε(20)

where τ0is the nominal delay, ∆τ(t)is the variation around

τ0, and ¯εis the upper bound.

Proposition 2: The time-varying matrix ˆ

A(t)can be de-

composed in two parts

ˆ

A(t) = ˆ

A0+ ∆ ˆ

A(t)(21)

so that ˆ

A0is a constant matrix, and ∆ˆ

A(t)z(t)will play the

role of an UI. Expression (21) holds with

ˆ

A0=A BCτ0

d

0Aτ0

d,∆ˆ

A(t)= 0−BCτ0

d

0−Aτ0

d∆τ∗(t)(22)

where Aτ0

d=−2

τ0I,Cτ0

d=4

τ0I, and ∆τ∗(t) = ∆τ(t)

τ0+∆τ(t).

Proof: The proof of proposition 2 can be found in [15].

Finally, the time-varying uncertainty is expressed as an

unknown input d2(t), entering (19) through ˆ

Ed2, that is

∆ˆ

A(t)z(t) = ˆ

Ed2d2(t)(23)

ˆ

Ed2=−BCτ0

d

−Aτ0

d,d2(t) = ∆τ∗(t)xd(t)

Now, the two unknown inputs d1(t)and d2(t)and their

distribution matrices ˆ

Ed1and ˆ

Ed2can be lumped together

similarly as in (14). The system (19) can be now written as

˙

z(t) = ˆ

A0z(t) + ˆ

B0uc(t) + ˆ

Eff(t) + ˆ

Edd(t)

y(t) = ˆ

Cz(t)(24)

Note that (24) has the same structure as (15). The only

difference is in the way how the time-varying uncertainty

is handled in terms of unknown inputs.

V. DE SI GN O F THE RO BU ST FAU LT DET EC TIO N SCH EME

A residual generator for discrete-time system (15) is pre-

sented in the next1. The procedure is based on eigenvectors

and eigenvalues assignment. This technique is well tackled

in the FDI community, see [2]–[4] for more details. In order

to avoid duplicating materials, only the main principles are

recalled in the following.

Consider the following residual generator based on full-

order observer

ˆ

zk+1 = ( ˆ

A0−Lˆ

C)ˆ

zk+ˆ

B0uc

k+Lyk

rk=Q(yk−ˆ

Cˆ

zk)(25)

where rk∈Rnp,ˆ

zk∈Rn+nuis the residual and the state

estimation vector, respectively. Qis the residual weighting

1The same procedure can be applied for the continuous system (24), but

the observer eigenvalues will belong to a different set of stable eigenvalues.

1441

matrix of appropriate dimension. The Z-transformed residual

response to faults and unknown inputs is

r(z) = Grf (z)f(z) + Grd (z)d(z)(26)

where Grf (z)and Grd (z)denote the transfers between f(z)

and r(z), and d(z)and r(z), respectively.

Once ˆ

Edis known, the remaining task is to ﬁnd matrices

Land Qto satisfy Grd (z) = 0. The assignment of the

observer (25) eigenvectors and eigenvalues is a direct way

to solve this design problem.

Note that, because the EA technique does not consider

a sensitivity constraint in the design procedure, the fault

sensitivity performance of the proposed FD scheme can only

be veriﬁed a posteriori. Specially the subspace of considered

faults should not intersect the subspace of decoupled distur-

bances, i.e. Im(ˆ

Ef)6⊂ Im(ˆ

Ed), see [2].

For the decision making algorithm, the idea is to test the

variance σ2of the jth residual signal rj

kaccording to the χ2

statistic given by Lapin [4]:

SN

k:= (N−1)s2

σ2

0

, s2=

k

P

i=k−N

(rj

i−¯rj

k)2

N−1(27)

where ¯rj

kis the mean value of rj

i, i =k−N, ... , k and σ2

0

is the variance of rjin fault free situation. The statistic (27)

has the standard χ2sampling distribution with the degree

of freedom equal to N−1. Thus, for a given α > 0(the

signiﬁcance level), the threshold is determined by (using the

standard χ2distribution table)

Jth =χ2

α, prob{χ2> χ2

α}=α(28)

The decision rule is thus

SN

k=≤Jth, H0(σ2≤σ2

0)is accepted

> Jth, H1(σ2> σ2

0)is accepted (29)

For the given constant α > 0, the change in variance can be

detected with a false alarm rate smaller than α.

VI. AP PLI CATION TO T HE MSR M ISS IO N

The MSR mission is a future exploration mission under-

taken jointly by NASA and ESA. The goal is to return

samples from Mars atmosphere and ground to Earth for

analysis. It is obvious that the rendezvous phase of this

mission can be in danger if a fault occurs in the chaser’s

systems since the GNC system may not compensate, for

example, J2disturbances and/or may lose the attitude and/or

the position of the sample container. This problem becomes

specially critical during the last 20 meters of the rendezvous

phase, since the chaser must be correctly positioned in the

rendezvous corridor in order to successfully capture the

sample container. This motivates ESA to manage studies

for the development of on-board FDI solutions. The robust

FD schemes presented in the previous sections are potential

candidates since one of the critical system in the chaser’s

actuation system is the thruster modulator unit (TMU), an

electronic device with an unknown (but bounded) time-

varying delay that manage the actuators (thrusters) used for

−→

Xi

−→

Yi

−→

Xl

−→

Yl

−→

Zi=−→

Zl

θ

ξ

η

the rendezvous

orbit

target

chaser

a

Fig. 1. The Mars rendezvous orbit and the associated frames

the control of both the position and the attitude of the chaser

spacecraft during the rendezvous phase.

A. Modeling the Chaser’s Dynamics During the Rendezvous

From [16], [17] we only consider the modeling of the

relative position of two spacecrafts on a circular orbit around

the planet. The motion of the chaser is derived from the 2nd

Newton law. To proceed, let a,m,Gand mMdenote the

orbit of the target, the mass of the chaser, the gravitational

constant and the mass of the planet Mars. Then, the orbit of

the rendezvous being circular, the velocity of any object (e.g.

the chaser and the target) is given by the relation pµ

a, where

µ=G.mM. Let Rl: (Otgt ,−→

Xl,−→

Yl,−→

Zl)be the frame attached

to the target and oriented as shown in Fig. 1. Because the

linear velocity of the target is given by the relation a˙

θin the

inertial frame Ri: (OM,−→

Xi,−→

Yi,−→

Yi)(see Fig. 1), it follows:

a. ˙

θ=rµ

a⇒n=rµ

a3(30)

The chaser motion is due to the following four forces:

•the Mars attraction force −→

Fagiven in Rlby:

−→

Fa=−mµ

((a+ξ)2+η2+ζ2)3/2(a+ξ)−→

Xl+η−→

Yl+ζ−→

Zl;

•the centripetal force −→

Fe=mn2ξ−→

Xl+n2η−→

Yl+ 0−→

Zl;

•the Coriolis force −→

Fcin Rlis given by:

−→

Fc=m2n˙η−→

Xl−2n˙

ξ−→

Yl+ 0−→

Zl;

•and the forces due to the thrusters:

−−→

Fthr =Fξ

−→

Xl+Fη

−→

Yl+Fζ

−→

Zl.

Then, from the 2nd Newton law, it follows

¨

ξ=n2(a+ξ) + 2n˙η−µ

((a+ξ)2

+η2

+ζ2)3/2(a+ξ) + Fξ

m

¨η=n2η−2n˙

ξ−µ

((a+ξ)2

+η2

+ζ2)3/2η+Fη

m

¨

ζ=−µ

((a+ξ)2

+η2

+ζ2)3/2ζ+Fζ

m

(31)

where ξ, η, ζ denote the three dimensional position of the

chaser (assumed to be a punctual mass) in Rl.

Because the distance between the target and the chaser

is smaller than the orbit a, it is possible to derive the

1442

so called Hill-Clohessy-Wiltshire equations from (31) by

means of a ﬁrst order approximation. This boils down to a

linear six order state space model with input vector u(t) =

(FξFηFζ)T, output vector y(t) = (ξ η ζ )Tand state vector

x(t) = (ξ η ζ ˙

ξ˙η˙

ζ)T, i.e. from (31) it follows

˙

x(t)= Ax(t)+ BR(ˆ

Qtgt(t),ˆ

Qchs(t))M uthr (t)+Eww(t)

y(t)= C x(t)+ v(t)

where ˆ

Qtgt(t)∈R4and ˆ

Qchs(t)∈R4denote the attitude’s

quaternion of the target and the chaser, respectively. These

quaternions are estimates from the navigation module (NAV).

M∈R3×8refers to the thruster conﬁguration matrix,

uthr(t)∈R8are the thruster inputs. The relative position

y(t)∈R3is measured by means of a LIDAR unit that is

corrupted by a measurement noise v(t)∈R3,w(t)∈R3

refers to orbital disturbances entering the system through

Ewmatrix. The quaternion dependent rotation matrix R(·)

performs the projection of the three-dimensional thrust force

vector from the chaser’s frame on to the target frame Rl.

The considered thrusters faults can be modeled in a

multiplicative form according to

uthr

f(t) = (I8−Ψ(t))uthr(t)(32)

where Ψ(t) = diag{ψi(t)}: 0 ≤ψi(t)≤1, i = 1,...,8

models thruster faults, e.g. a locked-in-placed fault can be

modeled by Ψi(t) = 1 −c

uthri(t)where cdenotes a constant

value (the particular values c={1,0}allows to consider

open/closed faults) whereas a ﬁx value of Ψi(t)models a

loss of efﬁciency of the ith thruster. Ψ(t) = 0 ∀tmeans that

no fault occurs in the thrusters.

Taking into account some unknown but bounded delays

induced by the electronic devices, and uncertainties on the

thruster rise times due to the TMU that is modeled here as

an unknown time-varying delay τ(t) = τ0+ ∆τ(t)with a

(constant) nominal delay τ0and upper bounded variation part

|∆τ(t)| ≤ ¯ε, the overall model of the chaser dynamics that

takes into account both the attitude Qchs(t)and the relative

position (ξ η ζ)of the chaser and target can be written as

˙

x(t) = Ax(t) + Bu(t−τ(t)) + Eff(t) + Eww(t)

y(t) = Cx(t) + v(t)

(33)

by considering R(ˆ

Qtgt(t),ˆ

Qchs(t))M uthr (t)as the delayed

input vector u(t−τ(t)) and approximating the fault model

−R(ˆ

Qtgt(t),ˆ

Qchs(t))MΨ(t)uthr (t)in terms of additive

faults f(t)∈R3acting on the state via a constant distri-

bution matrix Ef(then Ef=B).

B. Design of the Fault Detection Schemes

Both schemes use the above derived model (33) to con-

struct the residual generator (25). The uncertainty τ(t)is

handled as an UI entering the augmented system’s dynamics,

(15) resp. (24), through the distribution matrix ˆ

Ed2. The

difference between the two proposed methods rest in the

different way of treating the uncertainty. The sampling period

of the NAV is T= 0.1sand a reasonable value of τ0was

determined to be exactly one sampling period for the input

vector u, i.e. τ0= 0.1. Since the orbital disturbances whave

the same directional properties as the faults (Ew=Ef), the

residual rkcannot be decoupled from w, thus the disturbance

decoupling is not considered here, i.e. ˆ

Ed=ˆ

Ed2.

•Method 1: ﬁrst, the model (33) is transformed into the

discrete form (15), with l= 1 and m= 1. It practically

means that the unknown delay τ(t)is assumed to be

in the closed interval [T , 2T). The obtained distribution

matrix ˆ

Edhas rank(ˆ

Ed) = 6 and a large number of

columns. Thus, a full column rank factorization is per-

formed using SVD decomposition. Finally, the obtained

distribution matrix is used in the residual generator (25)

design using the left EA technique.

•Method 2: is formulated using a ﬁrst order Pad´e ap-

proximation of the input delay. The distribution matrix

ˆ

Edis computed as in (23), with τ0= 0.1s. That

basically means, that after UI decoupling is achieved,

the resulted residual generator (25), using this method,

is robust against the time variations ∆τ(t)(uncertainty)

around the nominal delay τ0. Finally, the residual gen-

erator (25) is converted to discrete-time (t=k T )using

a Tustin approximation

For both methods, the weighting matrix was determined to

be Q=I3, thus np= 3 and r= (r1, r2, r3)T. The decision

rule was computed according to (29) with N= 10,j= 3 for

the ﬁrst, and j= 1 for the second method, respectively. The

probability of a false alarm has been ﬁxed at 1% (α= 0.01).

Remark 1: In order to compare the proposed approaches,

the assigned eigenvalues (dynamics) for Method 2 were

chosen to be close to ≈ −0.5, and after the discretization of

the continuous residual generator, the obtained closed-loop

eigenvalues were used for Method 1.

C. Simulation Results

The aforementioned FD schemes were implemented

within the MSR ”high-ﬁdelity” industrial simulator provided

by Thales Alenia Space. All simulations are carried out under

realistic conditions, i.e. the NAV is considered to deliver

“non-perfect” measurements. We assume delays induced by

the TMU, orbital disturbances (i.e. solar radiation pressure,

gravity gradient, atmospheric drag) and uncertainties.

To evaluate the performances in terms of detection time

delays (td), non-detection (ˆ

Pnd)and false alarm (ˆ

Pf)rates,

a sensitivity/robustness analysis campaign of 4240 runs has

been performed considering all the aforementioned sources

of disturbances, noises, delays and uncertainties whose con-

sidered variations range are 10% of the chaser inertia and 5%

of the chaser mass. The min. detectable leakage size ( ˆmleak)

and min. detectable thrust loss size ( ˆmloss)are also consid-

ered as performance criteria. In this study, following fault

scenarios were generated on the eight (Thr.No.) thrusters:

•Case 1: single thruster opening at 100%

•Case 2: thruster closing itself (locked-closed)

•Case 3: propellant leakage between 15% and 35%

•Case 4: loss of efﬁciency ranging from 40% to 90%

For each simulation, the fault occurs at t= 1100sand

is maintained. The results shown in Tab. I illustrate the

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TABLE I

PER FO RM AN CE S OF TH E FAU LT DET EC TI ON SC HE ME S BAS E D ON 4 ,240 RUNS

Method 1 Method 2

Thr.No. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Case 1

min(td) [s]1 1.1 1 1.1 1.1 0.9 1.1 0.9 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

max(td) [s]1 1.1 1 1.1 1.1 1 1.1 1 0.7 0.7 0.6 0.6 0.7 0.7 0.7 0.7

med(td) [s]1 1.1 1 1.1 1.1 1 1.1 1 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6

Case 2

min(td) [s]8.7 13.5 9.7 5 25.1 17.6 12.2 7.7 10.3 10.4 8.7 7.7 17.4 13.5 7 8.9

max(td) [s]12.5 18.2 11.3 16.4 33.9 25.3 14.1 14.8 12.1 12.2 10.2 10 23.8 21.8 10.5 13.3

med(td) [s]10.8 14.8 10.4 15.1 28.8 22.7 13 12.4 11.1 11.3 9.3 9.1 20.7 17.7 8.8 11.1

Case 3

min(td) [s]1.4 1.5 1.4 1.6 1.6 1.4 1.6 1.3 1 1 0.9 0.9 0.9 0.9 0.9 0.9

max(td) [s]2.7 2.6 2.8 2.6 2.8 2.6 2.8 2.5 1.8 1.8 2.0 1.9 2.3 2.3 2.1 1.9

med(td) [s]1.7 1.8 1.7 1.8 1.9 1.6 1.9 1.6 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1

ˆmleak [%] 15 15 15 15 15 15 15 15 20 20 15 15 15 20 20 20

ˆ

Pnd 0 0 0 0 0 0 0 0 0.2 0.2 0 0 0 0.01 0.17 0.19

Case 4

min(td) [s]9.4 13.5 9.7 5 25.1 17.6 12.2 7.9 10.3 10.4 8.6 7.7 17.4 13.5 7 9.6

max(td) [s]53.9 100 85.1 43.2 97.7 98.9 77.6 77.7 49.4 35.9 36.2 36 31.2 79.3 51.6 54.6

med(td) [s]11.4 15.6 11 15.35 29.4 24.6 13.3 12.9 11.7 12.1 10 9.7 21.4 18.25 9.7 11.9

ˆmloss [%] 50 60 60 60 70 60 60 60 50 40 40 40 50 50 40 40

ˆ

Pnd 0.15 0.31 0.18 0.29 0.37 0.22 0.31 0.16 0 0 0 0 0.17 0.14 0 0

effectiveness and good reliability characteristics of the pro-

posed methods since no false alarms have been revealed, i.e.

ˆ

Pf= 0 for all fault cases, thrusters and for both methods.

Further, ˆ

Pnd = 0 was observed for case 1 and case 2, again

for all thrusters and both methods (these results are omitted

from Tab. I due to space limitations). It can be seen that

the second method presents a greater sensitivity level to the

leakage type faults (see Fig. 2 for illustration) as well as

handles the other fault types better.

0 200 400 600 800 1000 1200 1400

−10

−5

0

5

magnitude

0 200 400 600 800 1000 1200 1400

−1

0

1

2

magnitude

Method 1¡r3(t)¢

Method 2¡r1(t)¢

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

0

20

40

60

time inseconds

frequency

Method 1

Method 2

Fig. 2. Residual signals and the corresponding histogram of the detection

time tdfor the leakage type of fault affecting the 3rd thruster

VII. C ONCLU SI ONS

In this paper, two residual-based FD schemes were pro-

posed that are robust against unknown time-varying delays

induced by electronic devices. The idea is to transform the

unstructured uncertainty to unknown input and decouple it

by means of an EA technique. The ﬁrst method utilizes a

Cayley-Hamilton theorem based transformation whereas the

second method relies on a ﬁrst-order Pad´e approximation.

Simulation results show that all considered fault scenarios are

covered with the suggested FD schemes. Some performance

indicators allows to demonstrate (in a statistical point of

view) the reliability (no false alarm) and the efﬁciency

(reasonable detection time) of the proposed schemes.

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