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Robust Fault Detection for Systems with Electronic Induced Delays: Application to the Rendezvous Phase of the MSR Mission

Authors:
  • Université Bordeaux

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 Padé 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.
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 τkRin the
actuation system so that the controller signal uc
kRnu,
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
δkRis 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
kl+Γδk
1uc
kl1+¯
Effk+¯
Ed1d1
k
¯
yk=¯
Cxk
(3)
where
¯
A=eAT,Γδk
0=
Tδk
R
0
eAtdtB,¯
Ed1=
T
R
0
eAtdtEd1
¯
C=C,Γδk
1=
T
R
Tδk
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
kl1)Twe obtain
zk+1 =ˆ
Aτkzk+ˆ
Bτkuc
kl+ˆ
Effk+ˆ
Ed1d1
k
yk=ˆ
Czk
(5)
where
ˆ
Aδk=¯
AΓδk
1
0 0 ,ˆ
Bδk=¯
BΓδk
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(λIA) = λn+cn1λn1+...+c1λ+c0(6)
then eAtcan be written as
eAt=s1(t)I+s2(t)A+...+sn(t)An1(7)
where si(t),1inare solutions to the nth order
homogenous scalar differential equation
s(n)(t) + cn1s(n1)(t) + ...+c1s(t) + c0s(t) = 0 (8)
satisfying the following initial conditions
s(i1)
i(0)= 1, s(j)
i(0)= 0 for j6=i1,0jn1
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
Tδk
eAtdtB=
n
X
i=1
T
Z
Tδk
si(t)dt
Ai1B
(10)
Define
smax
i= max
0δkTZT
Tδk
si(t)dt, i = 1,2,...,n
smin
i= min
0δkTZT
Tδk
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
iAi1B(11)
where αk
i,0and αk
i,1are two time-varying unknown parame-
ters satisfying 0αk
i,01,0αk
i,11, and αk
i,0+αk
i,1= 1.
It can be verified that RT
Tδksi(t)dt, i = 1,2,...,n are
Lipschitz-continuous on 0δkT, i.e. they satisfy
ZT
Tδ1
k
si(t)dt ZT
Tδ2
k
si(t)dt
κiδ1
kδ2
k
δ1
k, δ2
k[0, T ], where κi, i = 1,2, ..., n are the Lipschitz
constants. Setting µk
2i1=αk
i,0/n,µk
2i=αk
i,1/n,U2i1=
nsmin
iAi1B,U2i=nsmax
iAi1B, 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
kl+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).
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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
kl=ˆ
Ed2d2
k(13)
where
ˆ
Ed2=hˆ
A1,..., ˆ
A2n,ˆ
B1,..., ˆ
B2ni
d2
k=µk
1zk, . . . , µk
2nzk, µk
1uc
kl, . . . , µk
2nuc
klT
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
kl+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(tτ(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
eτ(t)s.
=1τ(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)= 0BCτ0
d
0Aτ0
dτ(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
A0Lˆ
C)ˆ
zk+ˆ
B0uc
k+Lyk
rk=Q(ykˆ
Cˆ
zk)(25)
where rkRnp,ˆ
zkRn+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:= (N1)s2
σ2
0
, s2=
k
P
i=kN
(rj
i¯rj
k)2
N1(27)
where ¯rj
kis the mean value of rj
i, i =kN, ... , 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 N1. 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(σ2σ2
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
an=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
Xl2n˙
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˙ηµ
((a+ξ)2
+η2
+ζ2)3/2(a+ξ) + Fξ
m
¨η=n2η2n˙
ξµ
((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).
MR3×8refers 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) = (I8Ψ(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(tτ(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(tτ(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
1443
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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1444
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