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Robust Fault Diagnosis for Systems with Electronic Induced Delays

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  • Université Bordeaux
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Robust Fault Diagnosis for Systems with Electronic Induced Delays

Abstract and Figures

A problem of robust fault diagnosis of digital controlled continuous-time systems with uncertain time-varying input delay is studied in this paper. Two residual-based fault detection and isolation (FDI) schemes are proposed that are robust in terms of time-varying delays induced by the electronic devices and disturbances. The idea of both proposed methods is to transform the uncertainty caused by delays into unknown inputs and decouple them by means of eigenstructure assignment (EA) technique. The first method utilizes a Cayley-Hamilton theorem based transformation and the second relies on a first order Pad'e approximation of the time delay. Finally, the applicability and effectiveness of the proposed methods is illustrated through simulation results from the "high-fidelity" industrial simulator, provided by Thales Alenia Space.
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Robust Fault Diagnosis for Systems with
Electronic Induced Delays
R. Fonod D. Henry E. Bornschlegl ∗∗ C. Charbonnel ∗∗∗
University Bordeaux 1 - IMS-LAPS, Bordeaux, France
{robert.fonod,david.henry}@ims-bordeaux.fr
∗∗ European Space Agency, Noordwijk, The Netherlands
eric.bornschlegl@esa.int
∗∗∗ Thales Alenia Space, Cannes, France
catherine.charbonnel@thalesaleniaspace.com
Abstract:
A problem of robust fault diagnosis of digital controlled continuous-time systems with uncertain
time-varying input delay is studied in this paper. Two residual-based fault detection and
isolation (FDI) schemes are proposed that are robust in terms of time-varying delays induced
by the electronic devices and disturbances. The idea of both proposed methods is to transform
the uncertainty caused by delays into unknown inputs and decouple them by means of
eigenstructure assignment (EA) technique. The first method utilizes a Cayley-Hamilton theorem
based transformation and the second relies on a first-order Pad´e approximation of the time
delay. Finally, the applicability and effectiveness of the proposed methods is illustrated through
simulation results from the ”high-fidelity” industrial simulator, provided by Thales Alenia Space.
Keywords: Fault diagnosis; Uncertainty; Eigenstructure assignment; Electronic induced delay.
ACRONYMS
EA Eigenstructure Assignment
FDI Fault Detection and Isolation
IMU Inertial Measurement Unit
LIDAR LIght Detection And Ranging
MAV Mars Ascent Vehicle
MSR Mars Sample Return
RW Reaction Wheels
STR Star TRacker
SVD Singular Value Decomposition
THR THRuster
UI Unknown Input
1. INTRODUCTION
In recent years, due to the increased complexity, as well
as the need for reliability, safety, and efficient operation
of industrial and aerospace systems, a great deal of at-
tention has been paid to the subject of fault detection
and isolation (FDI) in dynamic systems. A great number
of methods for FDI have been proposed (see Chen and
Patton [1999], Blanke et al. [2006], Ding [2008] and the
references therein). Only a limited results on FDI of time-
delay systems have been developed in recent years.In Yang
and Saif [1998], an unknown input observer is designed for
fault detection of state-delayed systems with known de-
lays. Robust fault detector design problem is investigated
in Karimi et al. [2010] for a class of linear systems with
some nonlinear perturbations and mixed neutral and dis-
crete time-varying delays. Recently, a geometric approach
for FDI of retarded and neutral time-delay systems was
developed in Meskin and Khorasani [2009].
One of the main difficulties in fault detection of systems
subject to uncertain time-varying delay lies in the fact
that influence caused by electronic-induced input delay is
unstructured, therefore robustness to influence caused by
such delays cannot be ensured by applying existing robust
FDI approaches directly. In this paper, by introducing
a Cayley-Hamilton theorem based (see e.g. Wang et al.
[2008]) and Pad´e approximation based transformation,
influence of uncertain time-varying delay is transformed
into unknown inputs, which as shown, greatly facilitates
the above mentioned difficulty.
Two residual generator based FDI schemes are proposed
for a class of linear systems with disturbances. The system
is modelled as a continuous-time one with digital control,
where the control input has a piecewise-continuous delay.
Modelling of continuous-time systems with digital control
and delayed control input was introduced by Mikheev
et al. [1988]. At the end, the disturbance vector and the
unknown input, that models the uncertainty caused by
time-varying delays, are lumped together and decoupled
by means of Eigenstructure Assignment (EA) technique.
The applicative support of this paper concerns the Mars
Sample Return (MSR) mission. Simulation results from
the MSR ”high-fidelity” industrial simulator demonstrates
the efficiency and capabilities of the proposed schemes.
Notations: Let R,R+and Z+denote the field of real
numbers, the set of non-negative reals and the set of non-
negative integers, respectively. The notation Rm×nis used
for real matrices of dimension m×n. The Euclidean norm
is used for vectors and is written without subscript; e.g.
kxk. By I(or 0) we denote the identity (the null) matrix.
The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)
Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
2. PROBLEM FORMULATION
Consider a continuous-time system given by
˙
x(t) = Ax(t) + Bu(t) + Eff(t) + Ed1d1(t)
y(t) = Cx(t)(1)
where x(t)Rnis the state vector, u(t)Rnuis the
system input vector, y(t)Rnyis the output vector,
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 closed-loop system is controlled by
discrete-time controller, and the sampling time is TR+.
Since there is an electronic-induced delay τkR+,k
Z+, the controller signal uc
kRnu, k Z+generated at
t=tk=kT , k Z+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)
Problem 1. Design a residual generator that is robust
in the presence of uncertain time-varying delay τkand
unknown input d1(t).
In order to solve Problem 1, two approaches are presented
in this paper. The aim of both approaches is to model
the influence of the uncertain time-varying delay as an
unknown input (UI). The first approach uses a Cayley-
Hamilton theorem based transformation and the second
one utilizes a first-order Pad´e approximation. Then, the
unknown inputs are decoupled by means of Eigenstructure
Assignment (EA) technique.
3. TRANSFORMATION INTO POLYTOPIC
UNCERTAINTY
In this section, we assume that the electronic-induced
delay is represented by τk=lT +δk¯τ , kZ+, where
lZ+is a known constant integer, ¯τR+is the upper
bound of τkand δkR+,kZ+is the unknown time
varying part of the delay, bounded by 0 δk< mT with
mZ+being a known integer. In the next, we assume
that m= 1.
Remark 1. The case when the variation part of the delay
is larger than one sampling period, i.e. m > 1, is discussed
in Wang et al. [2008].
Consider the discrete representation of (1) over a sampling
period
xk+1 =¯
Axk+Γδk
0uc
kl+Γδk
1uc
kl1+¯
Effk+¯
Ed1d1
k
yk=¯
Cxk
(3)
where
¯
A=eAT,Γδk
0=
Tδk
Z
0
eAtdtB,¯
Ed1=
T
Z
0
eAtdtEd1
¯
C=C,Γδk
1=
T
Z
Tδk
eAtdtB,¯
Ef=
T
Z
0
eAtdtEf
Let ¯
B=
T
R
0
eAtdtB, then it follows:
Γδk
0+Γδk
1=
T
Z
0
eAtdtB=¯
B(4)
Furthermore, using (3) and (4), and introducing a new
augmented state vector of the form zT
k=xT
k(uc
kl1)T
we 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
time-varying delay part δk. Therefore, the previous system
is an uncertain systems with time-varying uncertainty.
The challenge that remains is to find a transformation of
this uncertainty in order to reformulate (5) into a time-
dependent polytopic uncertainty.
3.1 Expressing uncertainties as polytopes of matrices
In this paper, a Cayley-Hamilton theorem based transfor-
mation is introduced (see Wang et al. [2008]).
Theorem 1. Let Abe a constant matrix with the charac-
teristic polynomial
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), 1 inare 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:
s1(0) = 1
s
1(0) = 0
.
.
.
sn1
1(0) = 0
,
s2(0) = 0
s
2(0) = 1
.
.
.
sn1
2(0) = 0
, ...,
sn(0) = 0
s
n(0) = 0
.
.
.
sn1
n(0) = 1
Proof. The proof can be found in Leonard [1996].
Proposition 1. The Cayley-Hamilton theorem based trans-
formation of Γδk
1can be expressed as the convex matrix
polytopes
Γδk
1=
2n
X
i=1
µk
iUi,
2n
X
i=1
µk
i= 1
µk
i>0,i= 1,...,2n, kZ+
(9)
where Uiand µk
iwill be defined later.
Proof. Using (7), we have
Γδk
1=
T
Z
Tδk
eAtdtB=
n
X
i=1
T
Z
Tδk
si(t)dt
Ai1B
(10)
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Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
Define
smax
i= max
0δkT
T
Z
Tδk
si(t)dt, i = 1,2,...,n
and
smin
i= min
0δkT
T
Z
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 param-
eters satisfying 0 αk
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 0 δkT, that is, they satisfy
the relation
ZT
Tδ1
k
si(t)dt ZT
Tδ2
k
si(t)dt
Liδ1
kδ2
k
for all δ1
kand δ2
kin [0, T ], where |·| denotes the absolute
value and Li, i = 1,2,...,n are the Lipschitz constants.
Remark 2. Note that the Lipschitz constants are not
unique, they can be any finite constants satisfying the
above inequality. Therefore, when smax
iand smin
ican not
be obtained analytically, reliable Lipschitz global opti-
mization algorithms (e.g., Piyavskii’s algorithm), which
can guarantee a global convergence for all Lipschitz-
continuous functions in a closed interval Pint´er [1996],
can be adopted to find smax
iand smin
ino matter si(t),
i= 1,2,...,n are convex or not.
Setting
µk
2i1=αk
i,0/n,
µk
2i=αk
i,1/n,
U2i1=nsmin
iAi1B
U2i=nsmax
iAi1B(12)
From (10) and (11), with the notation (12) Proposition 1
is proved.
Therefore, considering Proposition 1, the system (5) can
be rewritten as a polytopic uncertain system
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
(13)
where
ˆ
A0=¯
A0
0 0 ,
ˆ
B0=¯
B
I,
ˆ
Ai=0Ui
0 0
ˆ
Bi=Ui
0
and the rest parameters are the same with those in (5).
Remark 3. Other transformation can be found in the
literature, for instance, in Hetel et al. [2006] makes use
of a Taylor series expansion of the uncertainty Γδk
1, i.e.:
Γδk
1=
X
i=1
(δk)i
i!Ai1eAT!B
Other methods, as the ones in Olaru and Niculescu [2008],
are based on the Jordan normal form, i.e. A=V JV 1,
when Γδk
1is expressed as:
Γδk
1=
n
X
i=1
A1V(eJiTeJi(Tδk))V1B
Remark 4. From the derivation above, it can be concluded
that the number of vertices of the polytopic representation
is 2n, which is a linear function of the system order.
3.2 Expressing polytopic uncertainty as an unknown input
The time-varying parts of (13), where ˆ
Aiand ˆ
Biare
known constant matrices, µk
iis an unknown scalar time-
varying factor, in this case, can be approximated by the
disturbance term as in Chen and Patton [1999] by:
ˆ
Ed2d2
k=
2n
X
i=1
µk
iˆ
Aizk+
2n
X
i=1
µk
iˆ
Biuc
kl=
=hˆ
A1,..., ˆ
A2n,ˆ
B1,..., ˆ
B2ni
| {z }
ˆ
Ed2
µk
1zk
.
.
.
µk
2nzk
µk
1uc
kl
.
.
.
µk
2nuc
kl
| {z }
d2
k
Now, the two unknown inputs d1
kand d2
kcan be lumped
together, and defined to be dk. That is
dk=(d1
k)T(d2
k)TT(14)
Correspondingly, the UI distribution matrix is
ˆ
Ed=ˆ
Ed1ˆ
Ed2(15)
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
(16)
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.
4. PAD´
E APPROXIMATION
In this section, we assume that the piecewise-constant de-
lay τkis represented by 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)) (17)
where uc(t) = uc
k,t[tk, tk+1) is the control signal.
The transfer function of the time delay is
H(s) = eτ(t)s(18)
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This transfer function is irrational and it is necessary to
substitute eτ(t)swith an approximation in form of a ra-
tional transfer function. The most common approximation
is the Pad´e approximation
eτ(t)s.
=1k1s+k2s2+...±knsn
1 + k1s+k2s2+...+knsn(19)
where nis the order of the approximation and the coeffi-
cients kiare functions of n.
In this paper, a first-order Pad´e approximation of the
time-varying delay τ(t) is used, when k1=τ(t)
2and
ki= 0, i = 2,...,n, that is:
eτ(t)s.
=1τ(t)
2s
1 + τ(t)
2s(20)
If we consider all system inputs, the transfer function (20)
is equivalent with the following state space representation
˙
xd(t) = Ad(t)xd(t) + Bduc(t)
u(t) = Cd(t)xd(t) + Dduc(t)(21)
where xd(t)Rnuis the delayed state and Ad(t) =
2
τ(t)I,Bd=I,Cd(t) = 4
τ(t)I,Dd=Iare matrices with
appropriate dimension.
The augmented state-space description of the system (1)
and the delayed inputs (21) is:
˙
z(t) = ˆ
A(t)z(t) + ˆ
Bu(t) + ˆ
Eff(t) + ˆ
Ed1d1(t)
y(t) = ˆ
Cz(t)
(22)
where
ˆ
A(t) = A BC d(t)
0Ad(t),ˆ
B=BDd
Bd,ˆ
C= [ C0]
z(t) = x(t)
xd(t),ˆ
Ef=Ef
0,ˆ
Ed1=Ed1
0
It can be seen, that the uncertainty is present only in ˆ
A(t),
thus the task is to decompose this matrix into the constant
and time-varying part and to model the uncertainty part
as an UI.
4.1 Expressing uncertainty as an unknown input
Problem 2. Decompose the matrix ˆ
A(t) into two parts:
ˆ
A(t) = ˆ
A0+ ∆ ˆ
A(t) (23)
where ˆ
A0is a constant matrix and ˆ
A(t) is the time-
varying part of ˆ
A(t).
Consider, that τ(t) can be expressed as
τ(t) = τ0+ ∆τ(t) : |τ(t)| ≤ ¯ε(24)
where τ0is the nominal delay, ∆τ(t) is the variation
around τ0, and ¯εis the upper bound.
Proposition 2. Let aRand bRbe two real scalars,
where a6= 0 and a+b6= 0, then
(a+b)1=a1a1b
a+b(25)
Proof. Using some basic arithmetic operations, it can be
shown, that (25) holds.
Using Proposition 2, we can write
1
τ(t)=τ0+ ∆τ(t)1=1
τ0
1
τ0
τ(t) (26)
where ∆τ(t) = τ(t)
τ0+∆τ(t).
Problem 2 is solved using (26), that is
ˆ
A0=A BC τ0
d
0Aτ0
d,ˆ
A(t) = 0BC τ0
d
0Aτ0
dτ(t) (27)
where Aτ0
d=2
τ0Iand Cτ0
d=4
τ0I.
Similarly, as in previous section, the time-varying uncer-
tainty is expressed as an unknown input d2(t), which
enters the augmented dynamics (22) through ˆ
Ed2, that
is
ˆ
Ed2d2(t) = ∆ ˆ
A(t)z(t) = 0BC τ0
d
0Aτ0
d
| {z }
ˆ
Ed2
τ(t)z(t)
| {z }
d2(t)
(28)
Now, the two unknown inputs d1(t) and d2(t) can be
lumped together as in (14). Similarly for ˆ
Ed1and ˆ
Ed2
as in (15). Furthermore, the system described by (22) can
be rewritten as
˙
z(t) = ˆ
A0z(t) + ˆ
Buc(t) + ˆ
Eff(t) + ˆ
Edd(t)
y(t) = ˆ
Cz(t)(29)
Note that (29) has the same structure as (16). The only
difference is in the way how the time-varying uncertainty
is handled in terms of UIs.
5. ROBUST RESIDUAL GENERATOR DESIGN BY
USING EIGENSTRUCTURE ASSIGNMENT
In this section we focus on UI decoupling for discrete-
time systems (16), however the same procedure can be
used for decoupling of the UIs in continuous-time systems
(29), with only difference that the observer eigenvalues will
belong to a different set of stable eigenvalues.
In order to solve Problem 1, we define 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)(30)
where rkRnpis the residual vector and ˆ
zkis the
state estimation. The matrix QRnp×nyis the residual
weighting matrix.
Defining the state estimation error ek=zkˆ
zk, the
residual generator is governed by
ek+1 = ( ˆ
A0Lˆ
C)ek+ˆ
Effk+ˆ
Eddk
rk=Hek
(31)
where H=Qˆ
C. The Z-transformed residual response to
faults and UIs is thus
r(z) = Grf (z)f(z) + Grd (z)d(z) (32)
where
Grf (z) = H(zIˆ
A0+Lˆ
C)1ˆ
Ef(33)
Grd(z) = H(zIˆ
A0+Lˆ
C)1ˆ
Ed(34)
Once ˆ
Edis known, the remaining problem is to find the
matrices Land Qto satisfy Grd (z) = 0. The assignment
of the observer’s eigenvectors and eigenvalues is a direct
way to solve this design problem.
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5.1 Unknown input decoupling by assigning left eigenvectors
Lemma 1. The transfer function Grd (z) can be expanded
in terms of the eigenstructure as
Grd(z) = H(zIˆ
Ac)1ˆ
Ed=
n
X
i=1
Υi
zλi
(35)
where Υi=HvilT
iˆ
Ed,viand lT
iare the right and left
eigenvectors of ˆ
Ac=ˆ
A0Lˆ
Cassociated with eigenvalue λi
.
It is well known that, a given left eigenvector lT
iof
ˆ
Acis always orthogonal to the right eigenvectors vj
corresponding to the remaining (n1) eigenvalues λjof
ˆ
Ac, where λi6=λj.
Theorem 2. (Chen and Patton [1999]). If Qˆ
Cˆ
Ed= 0 and
all rows of the matrix H=Qˆ
Care left eigenvectors of ˆ
Ac
corresponding to npeigenvalues of ˆ
Ac, then Grd(z) = 0is
satisfied.
Proof. If the rows of Hare npleft eigenvectors (li, i =
1,...,np) of ˆ
Ac, i.e.
H=l1l2... lnpT(36)
then Hvi=0and Υi= 0 for i=np+ 1,...,n. If further
we have Qˆ
Cˆ
Ed=Hˆ
Ed=0, i.e. lT
iˆ
Ed=0and Υi= 0
for i= 1,2,...,np. Thus Grd(z) = 0.
The first step for the design of an UI decoupled residual
generator (30) is to compute the weighting matrix Qwhich
must satisfy the following necessary condition (Chen and
Patton [1999])
Qˆ
Cˆ
Ed=Hˆ
Ed=0(37)
The necessary and sufficient condition for solution (37) to
exist is rank( ˆ
Cˆ
Ed)< ny. If ˆ
Cˆ
Ed=0, any weighting
matrix can satisfy this necessary condition. A general
solution is
Q=Q1(Iˆ
Cˆ
Ed(ˆ
Cˆ
Ed)+) (38)
where Q1Rnp×nyis an arbitrary matrix and ( ˆ
Cˆ
Ed)+is
the pseudo-inverse of ( ˆ
Cˆ
Ed).
The second step is to determine the eigenstructure of the
observer. The rows of Hmust be the npleft eigenvectors
of ˆ
Ac. The remaining nnpleft eigenvectors can be chosen
without restraint. For the given (stable) eigenvalue spec-
trum Λ( ˆ
Ac) = {λi, i = 1,...,n}, the following relation
holds
lT
i(λiIˆ
A) = lT
iLˆ
C=wT
iˆ
C, i = 1,...,n (39)
where
wT
i=lT
iL(40)
The assignability condition says, that for each λi, the
corresponding left eigenvector lT
ishould lie in the column
subspace spanned by {ˆ
C(λiIˆ
A)1}, i.e. a vector wi
exists such that
lT
i=wT
iKi, i = 1,...,np(41)
where
Ki=ˆ
C(λiIˆ
A0)1, i = 1,...,np(42)
The projection of liin the subspace span{Ki}is denoted
by:
lT
i=wT
iKi, i = 1,...,np(43)
where
wT
i=lT
iKT
i(KiKT
i)1, i = 1,...,np(44)
If lT
i=lT
i,lT
iis in span{Ki}and is assignable. Otherwise,
an approximative procedure must be taken, i.e. to replace
lT
iby it’s projection lT
i.
The remaining nnpeigenvalues and corresponding eigen-
vectors can be chosen freely from the assignable subspace
and assigned using some eigenstructure assignment tech-
nique, e.g. SVD decomposition. Then, the observer gain
matrix Lcan be computed as follows:
L=P1W(45)
where
W=w
1... w
npwnp+1 ... wnT
P=l
1... l
nplnp+1 ... lnT
It is obvious, that the first npeigenvalues corresponding
to the required eigenvectors lT
i, i = 1,...,npmust be real
because all these eigenvectors are real-valued.
Remark 5. The remaining design freedom, after UI de-
coupling has been satisfied, can be used to optimize other
performance indices such as fault sensitivity.
6. APPLICATION TO THE MSR MISSION
The applicative support of this paper concerns the Mars
Sample Return (MSR) mission. It is a future exploration
mission undertaken jointly by the National Aeronautics
and Space Administration (NASA) and the European
Space Agency (ESA). The goal is to take sample of
the surface, the rocks and the atmosphere of Mars and
to return these samples safe and intact back to Earth
for analysis. Five spacecrafts are involved in this space
mission: an Earth/Mars transfer vehicle, an ascent/descent
module, a chaser spacecraft and an Earth re-entry vehicle.
Once the samples are collected, they are loaded on the
Mars Ascent Vehicle (MAV) which is then launched into
the Martian orbit around the planet to rendezvous with
the chaser spacecraft. To a success of the rendezvous
mission, the chaser vehicle uses Inertial Measurement
Units (IMUs), a Star TRacker (STR), a LIght Detection
And Ranging (LIDAR) sensor, a Radio Frequency Sensor
(RFS), a Coarse Sun Sensor (CSS), and a very precise
propulsion system composed of THRusters (THRs) and
reaction wheels (RWs). The rendezvous mission can be in
danger if a fault occurs in these thrusters, since the chaser
may not compensate, for example, J2 disturbances and/or
may lose the attitude and/or the position of the sample
container containing the Mars’s samples.
Such faulty situations cannot obviously be diagnosed by
ground operators using telemetry information due to the
potential lack of communication between the chaser and
the ground stations or due to significant delays. This
has motivated the development of onboard fault diagnosis
solutions. As a result, two robust fault detection schemes
presented in previous sections are now considered for the
detection and isolation of faults occurring in the chaser’s
thrusters unit.
The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)
Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
target
a
MARS
rendezvous
orbit of the
Xi
Yi
Xl
Yl
Zi=
Zl
ν
ξ
η
chaser
Fig. 1. The rendezvous orbit and associated frames
6.1 Modeling the chaser dynamics during the rendezvous
From the exhaustive space literature we only consider the
necessary backgrounds on modeling the relative position
of two spacecrafts on a circular orbit around a planet. The
interested reader can refer to e.g. Wie [1998], Wertz and
Larson [1999], Wisniewski [1999] for more details.
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 Figure 1. Because the
linear velocity of the target is given by the relation a˙
θ
in the inertial frame Ri: (OM,
Xi,
Yi,
Yi) (those that is
attached to the center of Mars), it follows:
a. ˙
θ=rµ
an=rµ
a3(46)
During the rendezvous phase, it is assumed that the chaser
motion is due to the four following 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=n2ξ+ 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
(47)
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
so called Hill-Clohessy-Wiltshire equations from equations
(47) by means of a first order approximation. This boils
down to a linear six order state space model whose input
vector is u(t) = (FξFηFζ)Tand state vector x(t) =
(ξ η ζ ˙
ξ˙η˙
ζ)T. Now, projecting the thrust forces due to
the eight thrusters that equip the chaser into the frame
Rl, it follows from (47):
˙
x(t) = Ax(t) + BR(ˆ
Qtgt(t),ˆ
Qchs(t))M uthr (t) + Eww(t)
y(t) = Cx(t) + v(t)
where ˆ
Qtgt(t)R4and ˆ
Qchs(t)R4denote the atti-
tude’s quaternion of the target and the chaser, respec-
tively. These quaternions are estimates from the naviga-
tion module. MR3×8refers to the (static) allocation
module, uthr (t)R8refers to the thrusters input and
R(ˆ
Qtgt(t),ˆ
Qchs(t)) is the quaternion dependent rotation
matrix. x(t)R6is the state vector defined previously,
y(t)R3refers to the three-dimensional (relative) po-
sition measured by a LIDAR unit and w(t)R3refers
to the spatial disturbances. The considered disturbances
in this study are solar radiations, gravity gradient and
atmospheric drag. v(t)R3denotes the measurement
noise assumed to be a white noise with very small variance
due to the technology used for the design of the LIDAR.
A,B,Cand Eware matrices of adequate dimension.
The considered thrusters faults can be modeled in a
multiplicative form according to
uthr
f(t) = (I8Ψ(t))uthr(t) (48)
where
Ψ(t) = diag{ψi(t)}: 0 ψi(t)1, i = 1,...,8
models the 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={0,1}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 t
means that no fault occurs in the thrusters.
Taking into account some unknown but bounded delays
induced by the electronic devices and the uncertainties on
the thruster rise times due to the thruster modulator unit
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.
Furthermore, considering R(ˆ
Qtgt(t),ˆ
Qchs(t))M uthr (t) as
the input vector u(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 distribu-
tion matrix Ef(then Ef=B), it follows that 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 can be written in the form (1), i.e.:
˙
x(t) = Ax(t) + Bu(t) + Eff(t) + Eww(t)
y(t) = Cx(t) + v(t)(49)
The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)
Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
6.2 Design of the FDI schemes
The both FDI schemes use the above derived model (49)
to construct the residual generator of the form (30).
The uncertainty caused by the unknown time-varying
delay τkis handled as an unknown input entering the
augmented system’s dynamics, (16) resp. (29), through
the distribution matrix ˆ
Ed2. The sampling period of the
navigation module is T= 0.1sand the numerical values
of the nominal time delays have been determined to be
one sampling period, i.e. 0.1sfor the input vector u.
Since the spatial disturbances whave the same directional
properties as the faults, i.e. Ew=B, the residual signal
rkcannot be decoupled from w, thus the disturbance
decoupling is not considered here, i.e. ˆ
Ed=ˆ
Ed2.
The first FDI scheme is based on a polytopic transfor-
mation of the uncertainty caused by the influence of
the unknown time-varying delay τk. First, the model
(49) is transformed into the discrete representation
(16), with l= 1 and m= 1. It practically means, that
the unknown delay τkis assumed to be in the closed
interval [T , 2T). The obtained distribution matrix
ˆ
EdR9×144 has rank( ˆ
Ed) = 6 and a large number
of columns. Thus, a full column rank factorization
is performed using SVD decomposition. Finally, the
obtained distribution matrix is used in the residual
generator design using left eigenvector assignment
(see section 5.1).
The second FDI technique is formulated using a
first order Pad´e approximation of the input time
delay. The necessary theoretical developments were
presented in section 4. The distribution matrix ˆ
Ed
is computed as in (28), with τ0= 0.1s. That ba-
sically means, that after UI decoupling is achieved,
the resulted residual generator (30), using this tech-
nique, is robust against the time variations ∆τ(t)
(uncertainty) around the nominal delay τ0. Finally,
the residual generator (30) is converted to discrete-
time using a Tustin approximation and implemented
within the nonlinear simulator of the MSR mission.
Remark 6. In order to compare the proposed approaches,
for the second design method (Pad´e), the assigned eigen-
values were chosen to be close to ≈ −0.5. Then, after
the discretization of the continuous residual generator, the
obtained closed-loop eigenvalues (discrete) were used for
the eigenvalues assignment of the first (polytopic) method.
6.3 The fault isolation method
The proposed isolation strategy is based on the follow-
ing normalized cross-correlation criterion between the jth
residual signal rj
kand the associated controlled thrusters
open rate uthri
k
σj
k= arg
i
min1
N
k+N
X
l=k
(rj
lrj)(uthri
luthri), i = 1...8 (50)
where rjand uthri, i = 1 ...8, j ∈ {1,2,3}denote the
mean values of the rjand uthri. For real-time reason, this
criterion is computed on a N-length sliding-window. The
resulting index σj
krefers to the identified faulty thruster
using the jth residual signal.
6.4 Simulation results
The two FDI schemes are then implemented within the
MSR ”high-fidelity” industrial simulator, provided by
Thales Alenia Space. The simulations are carried out all
during the last 20m of the rendezvous phase. The naviga-
tion unit is not considered to deliver “perfect” measure-
ments. We also assume time delays induced by the thruster
modulator unit and spatial disturbances (i.e. gravity gra-
dient, atmospheric drag, and solar radiation pressure).
The simulated fault scenarios correspond to a single
thruster opening at 100%, thruster closing itself (locked-
closed) and monopropellant leakage. To make a final deci-
sion about the fault, a simple threshold based decision test
is applied to ||r(k)|| and implemented within the simulator.
The isolation strategy is computed according to (50) using
j= 1. The strategy works as follows: as soon as the fault is
declared, the cross-correlation criterion (50) is computed.
Figure 2 and 3 illustrate the behaviour of the residual
norm krkkand the isolation criteria σk, for some faulty
situations. For each simulation, the fault occurs at t=
1100sand lasts 50s. As it can be seen from the figures,
after a small transient behaviour, all considered faults are
successfully detected and (quite well) isolated by both
FDI units. To compare the performance of the proposed
FDI schemes ”isolation time” (time from fault occurrence
to fault isolation) was considered (see Table 1). The
results in the table are almost identical, but the Pad´e has
an improved isolation performance towards locked-closed
fault situation. Note, that the occurrence of incipient
or small size thruster faults (e.g. small monopropellant
leakage) may be covered by control actions, and the early
detection/isolation of them is clearly more difficult.
Table 1. Isolation time in seconds
Fault type Location Pad´e appr. Polytopic
Opening at 100% 8 1.1s 1.1s
Opening at 100% 3 1.2s 1.2s
Locked-closed 6 2.0s 2.7s
Leakage (20%) 4 1.9s 1.9s
7. CONCLUSIONS
In this paper, the problem of fault diagnosis of a linear
continuous-time systems with subject to time-varying in-
put delays is investigated. Two residual-based schemes
were proposed that are robust against the presence of
unknown time-varying delays induced by the electronic
devices, which has not been addressed before to the best
of our knowledge. The idea of both proposed methods
is to transform the uncertainty caused by delays into
unknown inputs and decouple them by means of EA tech-
nique. The first method utilizes a Cayley-Hamilton theo-
rem based transformation when the influence of uncertain
time-varying delay is transformed into polytopic uncer-
tainty, which as shown later, greatly facilitates further
manipulation. The second approach relies on a first-order
Pad´e approximation around the nominal delay, where the
variation part is expressed as an unknown input. Simula-
tion results from the ”high-fidelity” industrial simulator
are presented in order to show the efficiency and capa-
bilities of the proposed methods. Despite the presence
The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)
Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
0 200 400 600 800 1000 1200 1400
0
20
40
60
80
100
Pade
Polytopic
1100 1100.5 1101 1101.5 1102 1102.5 1103
0
2
4
6
8
Pade
Polytopic
Thruster No. 8 opening at 100%
time in second
σk
krkk
time in second
σkkrkk
Thruster No. 6 locked-closed
Fig. 2. Behaviour of the residual norm krkkand the
isolation criteria σkfor some faulty situations
of measurement noises, delays in the thruster modulator
unit and spacial disturbances, the faults are successfully
detected and isolated in a reasonable time.
ACKNOWLEDGEMENT
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.
REFERENCES
M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki.
Diagnosis and Fault-Tolerant Control. Springer, 2006.
J. Chen and R. Patton. Robust model-based fault diagnosis
for dynamic systems. Kluwer Academic Pub, 1999.
S.X. Ding. Model-based fault diagnosis techniques: design
schemes, algorithms, and tools. Springer Verlag, 2008.
L. Hetel, J. Daafouz, and C. Iung. Stabilization of ar-
bitrary switched linear systems with unknown time-
varying delays. IEEE Transactions on Automatic Con-
trol, 51(10):1668–1674, 2006.
H.R. Karimi, M. Zapateiro, and N. Luo. A linear matrix
inequality approach to robust fault detection filter de-
sign of linear systems with mixed time-varying delays
and nonlinear perturbations. Journal of the Franklin
Institute, 347(6):957–973, 2010.
I.E. Leonard. The matrix exponential. SIAM review, 38
(3):507–512, 1996.
0 200 400 600 800 1000 1200 1400
0
20
40
60
80
100
Pade
Polytopic
1100 1100.5 1101 1101.5 1102 1102.5 1103
0
2
4
6
8
Pade
Polytopic
Thruster No. 3 openin g at 100%
time in second
σk
krkk
0 200 400 600 800 1000 1200 1400
0
2
4
6
8
Pade
Polytopic
1100 1100.5 1101 1101.5 1102 1102.5 1103 1103.5 1104
0
2
4
6
8
Pade
Polytopic
time in second
σkkrkk
Monopropellant leakage (20%) of the thruster No. 4
Fig. 3. Behaviour of the residual norm krkkand the
isolation criteria σkfor some faulty situations
N. Meskin and K. Khorasani. Robust fault detection
and isolation of time-delay systems using a geometric
approach. Automatica, 45(6):1567 – 1573, 2009.
Y.V. Mikheev, VA Sobolev, and E. Fridman. Asymptotic
analysis of digital control systems. Automation and
Remote Control, 49(9):1175–1180, 1988.
S. Olaru and S.I. Niculescu. Predictive control for linear
systems with delayed input subject to constraints. In
IFAC World Congress, Seoul, Korea, 2008.
J.D. Pint´er. Global optimization in action: continu-
ous and Lipschitz optimization–algorithms, implemen-
tations, and applications, volume 6. Springer, 1996.
Y. Wang, S.X. Ding, H. Ye, and G. Wang. A new fault
detection scheme for networked control systems subject
to uncertain time-varying delay. IEEE Transactions on
Signal Processing, 56(10), 2008.
J.R. Wertz and W.J. Larson. Space Mission Analysis and
Design. Space Technology Library. Springer, 1999.
B. Wie. Space vehicle dynamics and control(book). Re-
ston, VA: American Institute of Aeronautics and Astro-
nautics, Inc, 1998., 1998.
R. Wisniewski. Lecture Notes on Modeling of a Spacecraft.
Dep. of Control Engineering, Aalborg University, 1999.
H. Yang and M. Saif. Observer design and fault diagnosis
for state-retarded dynamical systems. Automatica, 34
(2):217 – 227, 1998.
The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)
Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012
... The application concerns the rendezvous phase of the Mars Sample Return (MSR) mission. The goal of the mission is to return samples from Mars to the Earth for analysis (see [3] for more details about this mission). It is obvious, that the rendezvous phase can be in danger, if thruster fault occurs. ...
... where Q is a weighting matrix. The design of (34) is based on theoretical developments given in [3], using the Padé method. ...
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