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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 T∈R+.

Since there is an electronic-induced delay τk∈R+,∀k∈

Z+, the controller signal uc

k∈Rnu, 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 , ∀k∈Z+, where

l∈Z+is a known constant integer, ¯τ∈R+is the upper

bound of τkand δk∈R+,∀k∈Z+is the unknown time

varying part of the delay, bounded by 0 ≤δk< mT with

m∈Z+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

k−l+Γδk

1uc

k−l−1+¯

Effk+¯

Ed1d1

k

yk=¯

Cxk

(3)

where

¯

A=eAT,Γδk

0=

T−δk

Z

0

eAtdtB,¯

Ed1=

T

Z

0

eAtdtEd1

¯

C=C,Γδk

1=

T

Z

T−δk

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

k−l−1)T

we 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

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(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:

s1(0) = 1

s′

1(0) = 0

.

.

.

sn−1

1(0) = 0

,

s2(0) = 0

s′

2(0) = 1

.

.

.

sn−1

2(0) = 0

, ...,

sn(0) = 0

s′

n(0) = 0

.

.

.

sn−1

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, ∀k∈Z+

(9)

where Uiand µk

iwill be deﬁned later.

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)

The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)

Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012

Deﬁne

smax

i= max

0≤δk≤T

T

Z

T−δk

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

and

smin

i= min

0≤δk≤T

T

Z

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 param-

eters 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, that is, they satisfy

the relation

ZT

T−δ1

k

si(t)dt −ZT

T−δ2

k

si(t)dt

≤Liδ1

k−δ2

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

2i−1=αk

i,0/n,

µk

2i=αk

i,1/n,

U2i−1=nsmin

iAi−1B

U2i=nsmax

iAi−1B(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

k−l+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!Ai−1eAT!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

A−1V(eJiT−eJi(T−δk))V−1B

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

k−l=

=hˆ

A1,..., ˆ

A2n,ˆ

B1,..., ˆ

B2ni

| {z }

ˆ

Ed2

µk

1zk

.

.

.

µk

2nzk

µk

1uc

k−l

.

.

.

µk

2nuc

k−l

| {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

k−l+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(t−τ(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) = e−τ(t)s(18)

The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)

Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012

This transfer function is irrational and it is necessary to

substitute e−τ(t)swith an approximation in form of a ra-

tional transfer function. The most common approximation

is the Pad´e approximation

e−τ(t)s.

=1−k1s+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:

e−τ(t)s.

=1−τ(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 a∈Rand b∈Rbe two real scalars,

where a6= 0 and a+b6= 0, then

(a+b)−1=a−1−a−1b

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) = 0−BC τ0

d

0−Aτ0

d∆τ∗(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) = 0−BC τ0

d

0−Aτ0

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

A0−Lˆ

C)ˆ

zk+ˆ

B0uc

k+Lyk

rk=Q(yk−ˆ

Cˆ

zk)(30)

where rk∈Rnpis the residual vector and ˆ

zkis the

state estimation. The matrix Q∈Rnp×nyis the residual

weighting matrix.

Deﬁning the state estimation error ek=zk−ˆ

zk, the

residual generator is governed by

ek+1 = ( ˆ

A0−Lˆ

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(zI−ˆ

A0+Lˆ

C)−1ˆ

Ef(33)

Grd(z) = H(zI−ˆ

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.

The 10th European Workshop on Advanced Control and Diagnosis (ACD 2012)

Technical University of Denmark, Kgs. Lyngby, Denmark, Nov. 8-9, 2012

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(zI−ˆ

Ac)−1ˆ

Ed=

n

X

i=1

Υi

z−λi

(35)

where Υi=HvilT

iˆ

Ed,viand lT

iare the right and left

eigenvectors of ˆ

Ac=ˆ

A0−Lˆ

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 (n−1) 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(I−ˆ

Cˆ

Ed(ˆ

Cˆ

Ed)+) (38)

where Q1∈Rnp×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 n−npleft 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(λiI−ˆ

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(λiI−ˆ

A)−1}, i.e. a vector wi

exists such that

lT

i=wT

iKi, i = 1,...,np(41)

where

Ki=−ˆ

C(λiI−ˆ

A0)−1, i = 1,...,np(42)

The projection of liin the subspace span{Ki}is denoted

by:

l◦T

i=w◦T

iKi, i = 1,...,np(43)

where

w◦T

i=lT

iKT

i(KiKT

i)−1, i = 1,...,np(44)

If lT

i=l◦T

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 l◦T

i.

The remaining n−npeigenvalues 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=P−1W(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

a⇒n=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−→

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=n2ξ+ 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

(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. M∈R3×8refers 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) = (I8−Ψ(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

Ed∈R9×144 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

l−rj)(uthri

l−uthri), 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

0 200 400 600 800 1000 1200 1400

0

5

10

15

20

25

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

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.

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