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A Class of Nonlinear Unknown Input Observer for Fault Diagnosis: Application to Fault Tolerant Control of an Autonomous Spacecraft


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In this paper, the problem of Nonlinear Unknown Input Observer (NUIO) based Fault Detection and Isolation (FDI) scheme design for a class of nonlinear Lipschitz systems is studied. The proposed FDI method is applied to detect, isolate and accommodate thruster faults of an autonomous spacecraft involved in the rendezvous phase of the Mars Sample Return (MSR) mission. Considered fault scenarios represent fully closed thruster and thruster efficiency loss. The FDI scheme consists of a bank of NUIOs with adjustable error dynamics, a robust fault detector that is based on judiciously chosen frame and an isolation logic. The bank of observers is in charge of confining the fault to a subset of possible faults and the isolation logic makes the final decision about the faulty thruster index. Finally, a thruster fault is accommodated by re-Allocating the desired forces and torques among the remaining healthy thrusters and closing the associated thruster valve. Monte Carlo results from 'high-fidelity' MSR industrial simulator demonstrate that the proposed fault tolerant strategy is able to accommodate thruster faults that may have effect on the final rendezvous criteria.
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A Class of Nonlinear Unknown Input Observer for
Fault Diagnosis: Application to Fault Tolerant
Control of an Autonomous Spacecraft
Robert Fonod,DavidHenry
, Catherine Charbonneland Eric Bornschlegl
Université de Bordeaux, IMS UMR CNRS 5218, Automatic Control Group, Talence, France
Thales Alenia Space, RT/SO Research Department, Cannes la Bocca, France
European Space Agency, ESTEC, TEC-ECN, Noordwijk, Netherlands
Contact: {robert.fonod, david.henry}
Abstract—In this paper, the problem of Nonlinear Unknown
Input Observer (NUIO) based Fault Detection and Isolation
(FDI) scheme design for a class of nonlinear Lipschitz systems
is studied. The proposed FDI method is applied to detect, isolate
and accommodate thruster faults of an autonomous spacecraft
involved in the rendezvous phase of the Mars Sample Return
(MSR) mission. Considered fault scenarios represent fully closed
thruster and thruster efciency loss. The FDI scheme consists
of a bank of NUIOs with adjustable error dynamics, a robust
fault detector that is based on judiciously chosen frame and an
isolation logic. The bank of observers is in charge of conning
the fault to a subset of possible faults and the isolation logic
makes the nal decision about the faulty thruster index. Finally,
a thruster fault is accommodated by re-allocating the desired
forces and torques among the remaining healthy thrusters and
closing the associated thruster valve. Monte Carlo results from
"high-delity" MSR industrial simulator demonstrate that the
proposed fault tolerant strategy is able to accommodate thruster
faults that may have effect on the nal rendezvous criteria.
In the recent decades, due to the increased complexity, as
well as, the need for reliability, safety, and efcient opera-
tion, a great deal of attention has been paid to the subject
of Fault/Failure Detection Isolation and Recovery (FDIR) in
space systems, see for instance [1], [2]. Literature reports
that conventional FDIR approaches suffer from signicant
shortcomings, like increased mass and system complexity,
often missing on-board isolation of the faults, ground inter-
vention is not always possible due to large communication
delays or visibility issues, and knowledge about the operational
capabilities of the system is not present on-board. Existing
FDIR techniques used in space systems are industrially well
mastered but may be not sufcient in some cases, e.g. when a
dynamic deviation in critical/proximity space operation could
possibly lead to mission loss. This fact motivates the European
Space Agency (ESA) to lead studies for the development of
fully autonomous on-board solutions that shall cope with all
the possible faults, that may occur and endanger mission.
Advanced Fault Detection and Isolation (FDI) approaches
should be specically developed to safely conjugate the nec-
essary robustness/stability of the spacecraft control, trajectory
dynamics and the vehicle nominal performance. In order to
ensure the normal operation, real-time fault detection and
isolation is necessary to provide information for the spacecraft
to accommodate the fault in time. The presented work is
a result of a research collaboration between ESA, Thales
Alenia Space and IMS Laboratory with the aim of promoting
Fault-Tolerant Control (FTC) strategies to advance spacecraft
autonomy. 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. The Guidance, Navigation and Control (GNC) system
may not compensate, e.g. J2 disturbances and/or may lose
attitude and/or position of the sample container (target). The
problem becomes highly critical during the last 20 meters of
the rendezvous phase. During this phase, the chaser spacecraft
must be correctly positioned in the approach corridor in order
to successfully capture the target, as well, as the chaser’s
attitude need to be maintained in the sensors’ eld of view.
Growing interest for potential applications of model-based
FDI algorithms in spacecraft systems is demonstrated by recent
publications, see e.g. [4]–[7]. In terms of FTC techniques, the
interested reader may refer to the excellent bibliographical
review of Zhang and Jiang [8], who explain the existing
approaches on this topic. A special class of observer based FDI
approaches is the so-called Unknown Input Observer (UIO).
Under certain conditions, UIO is able to estimate the state,
when exogenous unknown inputs are present in the system.
This property may be sometimes very useful for FDI scheme
design. Linear UIO algorithms [9], [10] are extended for
various classes of nonlinear systems in [11]–[13]. In [12],
Hoptimization based Nonlinear Unknown Input Observer
(NUIO) design is presented. The observer is called dynamic
UIO which offers an extra degree of design freedom, but
increases the system’s order. In [13], a NUIO is presented for
a class of nonlinear systems. The design procedure is based
on Linear Matrix Inequalities (LMIs).
In this paper, we consider a NUIO based FDI scheme
design problem for a class of nonlinear Lipschitz systems.
We extend the results presented in [14] by constraining the
observer error dynamics in a prescribed LMI region. The
observer synthesis is achieved by solving a LMI feasibility
problem together with a pole assignment in LMI regions. Thus,
a bank of NUIOs can be designed such that the error dynamics
of each NUIO is easily tunable. This bank is used to identify a
subset of thrusters that are most likely faulty. Blocked-closed
and loss of efciency thruster faults are considered. Once a
fault is isolated, the remaining N1healthy thrusters are used
to control the spacecraft. This fault accommodation strategy is
achieved by control re-allocation technique. By this, the nom-
inal (in-placed and certied) control laws remain unchanged
which is a prior condition from an industrial perspective.
2014 UKACC International Conference on Control
9th - 11th July 2014, Loughborough, U.K
978-1-4799-5011-9/14/$31.00 ©2014 IEEE
Chilali and Gahinet [15] showed, that a convex set that
represents the desired constraints on the eigenvalues of a real
matrix can be expressed as LMIs. We recall here the denition
of an LMI region and the pole placement LMI constraints.
Denition 1 (LMI region [15]): A subset Dof the com-
plex plane is called an LMI region if there exist two symmetric
matrices α=[αkl ]Rp×pand β=[βkl ]Rp×p, such that
where fD(z)is called the characteristic function.
Theorem 1: Eigenvalues of a real matrix Xlie in D,if
and only if there exists a symmetric positive denite matrix
P>0, such that
αP+β(AP )+βT(AP )T<0(2)
where stands for the Kronecker product of two matrices.
Examples of LMI regions from denition 1 are:
Left-half plane delimited by a vertical line a,a>0
Disk with center at (q, 0) and radius rwith q>0
Conic region with center at the origin and with inner
angle 0<θ<π/2pointing left
fD(z)=sin θ(zz)cosθ(z¯z)
cos θ(z¯z)sinθ(zz)(5)
LMIs can be easily obtained from (3)-(5) using Theorem 1.
A. Problem statement
Let us consider the following nonlinear system given by
x(t)=Ax(t)+Bu(t)+f(x(t)) + Ed(t)(6)
where xRnstands for the state vector, yRnmis the
output, uRnris the input, and dRnqis the unknown
input (disturbance or fault) vector. A,B,C,andEare known
matrices of appropriate dimensions. Without loss of generality,
it is assumed that Eis of full column rank.
The known function f(x)Rncontains the nonlinearities
of the system. Assume that f(x)is globally Lipschitz or at
least locally Lipschitz in a region S, i.e. it satises:
x1,x2Rnglobally Lipschitz
x1,x2∈S locally Lipschitz
where κ>0stands for the Lipschitz constant and ·is the
Euclidian norm. Many nonlinearities satisfy (8) at least locally.
The goal which is pursued is to design a NUIO of the
following structure [13]:
z(t)=Nz(t)+Gu(t)+Ly(t)+Mf (ˆ
x(t)) (9)
where ˆ
xRnis an estimate of x,zRnis an auxiliary
signal and the matrices N,G,L,Mare designed as
N=MA KC,G=MB (11)
L=K(I+CH)MAH (12)
M=I+HC (13)
and Kand Hare designed subsequently.
Dene the state estimation error as
Taking the time derivative of the estimation error yields
e(t)=Ne(t)+(NM+LC MA)x(t)MEd(t)
Denition 2 (Adjustable NUIO): In this paper, an observer
of the form (9)-(10) is referred to as an adjustable NUIO for
the system (6)-(7) if the estimation error tends asymptotically
to zero despite the presence of an unknown input d(t)=0
and if all eigenvalues of the observer dynamics matrix Nlies
in a prescribed region Dof the complex left-half plane.
This denition accommodates the observer given in [13], such
that the state estimation error has an adjustable error dynamics.
The sufcient condition under which the observer given by (9)-
(10) is an adjustable NUIO is given in Theorem 2.
Theorem 2 (Sufcient condition): Let Dbe an LMI region
contained in the complex left-half plane and with (1). If there
exist two matrices Hand Kand a positive denite matrix
P=PT>0such that
HCE =E(15)
NTP+PN +κPMMTP+κI<0(16)
then the adjustable NUIO given by (9)-(10) can make e(t)
tend to zero asymptotically for any e(0) and all eigenvalues
of the observer dynamics matrix Nwill belong to D.
Proof: Using (11)-(13) equalities NM+LCMA =0
and GMB =0are satised, and if His chosen such that
(15) holds, then the condition (15) can be rewritten as
ME =(I+HC)E=0(18)
and therefore the error dynamics is be governed by
x(t)) f(x(t))(19)
Consider a quadratic Lyapunov function V(t)=e(t)TPe(t),
then it follows from (8) and (19) that
Obviously, ˙
V<0holds if (16) is satised, thus lim
t→∞ e(t)=0
for any e(0). Moreover, if (16)-(17) are satised at the same
time, Theorem 1 implies that all eigenvalues of Nare in D.
Remark 1: The LMI (17) itself does not impose stability
of (19), even if all eigenvalues of Nlies in a stable region D.
The necessary condition for HCE =Eto have a solu-
tion is that CE is of full column rank, i.e. rank(CE)=nq,
and the solution is given in a generalized form by
H=U+YV (20)
where Ycan be chosen arbitrarily, Uand Vare given by
and (CE)+denotes the generalized pseudo-inverse of the
matrix CE given by (CE)+=((CE)T(CE))1(CE)T.
It is clear that there is no systematic way to obtain the
adjustable NUIO parameters directly from Theorem 2. This
motivates us to reformulate (16)-(17) as LMIs.
B. LMI formulation
For sake of simplicity, let Dbe a LMI region dening
a disk with a center (q, 0) and a radius r>0.SinceH
can be computed using (20), the only unknown parameters in
(11)-(13) are Kand Y. The following theorem shows that the
sufcient condition of existence given by Theorem 2 can be
reformulated as LMIs to design the NUIO parameters.
Proposition 1 (LMI based design): Assume that CE is of
full column rank and that the following LMIs
where denotes the symmetric item in a symmetric matrix,
and Q1,Q2and Q3are dened as
Q2=κ[P(I+UC)+ ¯
have feasible solutions for ¯
Kand P=PT>0. Moreover,
if (22) and (23) are fullled simultaneously with the same ¯
Kand P, then the adjustable NUIO given by (9)-(10) can
be designed with Y=P1¯
Kmaking all
eigenvalues of Nlying inside a disk centered at (q, 0) with
radius rand the estimation error e(t)=ˆ
to zero asymptotically for any initial value of e(0).
Proof: It is straightforward to show that (22) is equivalent
to (16) if we let H=U+YV and M=I+HC,and
if we substitute Ngiven by (11) into (16) and use Schur’s
complement and assignment Y=P1¯
Yand K=P1¯
Thus, asymptotical stability yields. The LMI region with
characteristic function (4) gives the following LMI (see [15]):
Using the fact that the eigenvalues of any square matrix X
are equal to the eigenvalues of its transpose XT, then by the
following notation XT=N=A+UCA+YVCAKC,
it follows that (24) implies (23). All eigenvalues of Nwill
therefore lie inside a disk of radius rand center (q, 0).Note
that H=U+YV implies HCE =E, i.e. all conditions
required by Theorem 2 are met and the theorem is proved.
In the following, it is shown how a bank of NUIOs can
be used to isolate actuator faults in the chaser spacecraft of
the MSR mission. The chaser is equipped with a chemical
propulsion system composed of N=12 thrusters. The consid-
ered thruster conguration in this paper is not a baseline MSR
conguration but a special one designed by Thales Alenia
Space to study active FTC strategies. The thrusters are physi-
cally organised in four clusters and are in charge of producing
force FR3and torque TR3vectors expressed in
the chaser body-xed reference frame Fb={Ob;ˆ
Let Sall ={1,2,...N}denote the set of all thruster indices.
Thrusters have xed directions diR3,i∈S
all and each
one is able to produce a maximum thrust of FN= 22N. The
Chemical Propulsion Drive Electronics (CPDE), that drives
the thrusting actuators, is initiating the opening of the thruster
valve for the commanded duration 0ui(t)1,i∈S
The propulsion system is obviously a source of uncertainty
in the system. The irrational transfer H(s)=eτ(t)saims to
model the effect of the unknown time-varying delays τ(t)0
induced by the CPDE and the uncertainties on the thruster
rise times. Let ui(tτ(t)) be the commanded open rate of the
ith thruster delayed by τ(t), then the net forces and torques
generated by thrusters are
F(t)=BFu(tτ(t)),T(t)=BTu(tτ(t)) (25)
where u(t)=[
u1(t)... u
bF1... bF12 ],BT=[
bT1... bT12 ](26)
are the thruster sensitivity (conguration) matrices with
all (27)
where ×denotes the cross product. RMR3is the position
vector of the Center of Mass (CoM), and RiR3,i∈S
are the position vectors of the thrusters, both expressed in the
chaser body-xed frame Fb.
By analysing the conguration matrices BFand BTin
terms of directional properties, the following can be concluded:
thruster indices inside the sets STi,i =1,...,5have similar
torque directions and are dened as
In terms of force directions, the following is revealed
bF1=bF11 ,bF4=bF8,bF3=bF12
bF2=bF10 ,bF5=bF7,bF6=bF9
which means that the thruster pairs given by STi,i =1,...4
produce exactly opposite forces. The last thruster group, i.e.
ST5, has the following orthogonal property
bF3·bF6=0,bF9·bF12 =0 (30)
where ·denotes the dot product.
Assuming no simultaneous faults, the considered thruster
faults can be modeled in a multiplicative way according to:
12(t)) (31)
where 0
all are unknown. ψiis about to
model closed fault types, i.e. ith thruster blocked-closed (ψi=
1) and/or loss of efciency (0
i<1)oftheith thruster.
A. Fault detection based on position model
The proposed fault detector design is based on the relative
position model of the chaser and target expressed in the local
(target) reference frame Fl={Ol;ˆ
zl}. The concerned
reader can found further details on modeling the relative
dynamics of two spacecrafts in the available space literature,
see for instance [16]. A linear 6th order state space model with
state vector x=[xyz ˙x˙y˙z]Tmodeling the chaser relative
motionexpressedinFl, both in fault free (i.e. Ψ=0)and
faulty (i.e. Ψ=0) situations is given by
qc(t))BFuf(tτ(t)) (32)
where the rotation matrix R(ˆ
qc)is calculated from the
quaternion estimates of the chaser ˆ
qcR4and the target
qtR4attitude, and rotates the force vector from Fbinto Fl.
These estimates come from the navigation unit. The output
vector y=[xyz]Tis the relative position in Flmeasured by
a Light Detection and Ranging (LIDAR) device.
In [7], a sensitivity/robustness analysis was performed
showing high reliability and efciency (in terms of detection
times) of a fault detector based on a position model in Fl. Here,
an observer-based fault detector is designed that has enhanced
robustness to above mentioned time delay τ(t). This observer
uses the model given in (32) and (33) to generate the state
estimate ˆ
xused to produce the residual r=[r1,r
where Qis a weighting matrix. The design of (34) is based on
theoretical developments given in [3], using the Padé method.
The proposed decision making rule is based on the scalar
valued Generalized Likelihood Ratio (GLR) test given in [17].
The decision test (t)is then dened by:
th fault declared
0S(k)Jth fault not present (35)
with S(k)=3
i=1 wiSi(k),wherewi0,i=1,2,3are the
normalized weight factors used to prioritize certain elements
(axis) of the residual, Si(k)is the estimated likelihood of the
GLR algorithm applied to the ith residual ri(k)evaluated at
time instant t=kTs,k Z+where Tsis the navigation
sampling time, and Jth is a xed threshold.
B. Thruster group isolation using a bank of NUIOs
Recalling the thruster conguration properties given by
(28)-(30), we assume that it is easier to get explicit information
from the angular velocity ωR3measurement than from the
linear position/velocity. Therefore, the model of the attitude
dynamics of a rigid-body spacecraft , i.e.
is used for the design of a bank of NUIOs. In (36), JR3×3
stands for the inertia of the chaser in Fb. The adjustable NUIO,
introduced in section III, has been selected because of its de-
coupling properties, ability to take into account nonlinearities
of the attitude dynamics (36) and adjustable error dynamics.
The attitude model (36) can be represented in the form
of (6) and (7) with the following assignment: x=ω,
One may argue that f(ω)is not globally Lipschitz, because
the Jacobian f/∂ωis not uniformly bounded over R3.
However, f(ω)is continuously differentiable on R3. Thus, it is
locally Lipschitz. This means that the angular velocity shall be
bounded in magnitude which is a reasonable assumption from
a practical point of view. Using a constrained optimization
algorithm, one can nd a Lipschitz constant κover the set
S={ωR3:|ωi|≤¯ω, i =1,2,3},where¯ωis the upper
bound of the angular velocity for each axis.
For each thruster group STi, a dedicated NUIO is thus
designed. Each NUIO is such that it can fully estimate the
angular velocity with all the inputs except those belonging to
all\STi. As a result, the NUIO dedicated to
the group STi will not be affected by faults occurring in the
thrusters belonging to STi, while all the other NUIOs will be.
The proposed method is summarised by Algorithm 1.
Algorithm 1 Design of a Bank of Adjustable NUIOs
Find a Lipschitz constant κsatisfying (8)
for k=1to 5do
Construct B
kwhose columns are bTi,i∈S
Set E=bTi for any arbitrary i∈S
Tk and B=B
Prescribe the desired dynamics using (q, 0) and r
Compute Uand Vaccording to (21)
Solve LMIs dened by (22)-(23) for ¯
Kand P>0
Let Y=P1¯
Yand K=P1¯
K, then the observer
parameters for the kth NUIO are determined by (11)-(20)
end for
This suggests the following isolation procedure: dening
the angular velocity estimation error of the ith observer as
ei(t)= ˆ
ωi(t)ω(t), then the faulty thruster group STi is
identied based on the following rule
where tdis the fault detection time, i.e. the time when the
fault is declared by (t),GT={1,2, ...5}denote the set of all
indices linked with the thruster groups ST1, ..., ST5,andthe
function σg(t):R+→G
Trepresents the identied thruster
group index that is most likely affected by a fault.
C. Thruster isolation logic
Once a thruster group STi is identied by σg(t), the faulty
thruster can be easily isolated by examining the angle of the
vector rgiven by (34) along the thruster directions di,i∈S
If the ith thruster is faulty, then, the vectors rand dishould
be collinear (owing the fault model (31)). Using the directional
cosine approach, the following isolation logic reveals
σ(t) = arg max
which results in the thruster index matching the faulty thruster.
Only thrusters belonging to the (already) identied group
STi are tested in (38). The thruster directions within the
groups STi,i ∈G
Tare either exactly opposite, see (29),
or are orthogonal, see (30), what makes the isolation logic
all very reliable. The proposed FDI
strategy is summarised by algorithm 2.
To avoid initial transition phenomena and to ensure robust-
ness, two conrmation windows are introduced in Algorithm 2,
i.e. δg>0for σg(t)and δ>0for σ(t).
Algorithm 2 Thruster Fault Detection and Isolation
if (t)=1then
Declare the fault presence
if σg(t)=σg(ν),ν(tδg,t]then
Declare the STj group to be faulty, where j=σg(t)
if σ(t)=σ(ν),ν(tδ, t]then
Declare the ith thruster be faulty, where i=σ(t)
end all if
The investigated thruster conguration disposes of an ad-
ditional freedom to achieve fault tolerance, i.e. it is possible
to achieve admissible GNC performance even if only N1
(healthy) thrusters are used to control the spacecraft. The
nominal controller is designed based on certain predetermined
performance criteria. Since it is desirable to keep the nominal
controller in the loop, the proposed solution consists in per-
forming the fault accommodation using control re-allocation.
The proposed fault accommodation strategy works as fol-
lows: as soon as the faulty thruster index is clearly isolated by
Algorithm 2, the faulty thruster is turned off using the dedi-
cated thruster latch valve and the desired forces and torques
are re-allocated among the N1healthy thrusters. Here,
the quadratic programming approach is used. This problem
is posed as the following Sequential Least-Squares (SLS)
u=arg min
u∈M Wu(uud)(39)
M=arg min
where BT
T]is the overall conguration matrix,
vdis the vector of the desired forces and torques, and
u=[¯u1, ..., ¯u12]Tare the upper limits dened as: ¯uj=
all\σ(t)and ¯ui=0,i =σ(t). This optimization
problem should be interpreted as follows: given M,thesetof
feasible control inputs that minimize Bauvd(weighted by
Wv), pick the control input that minimizes uud(weighted
by Wu). Here, udis the desired control input and Wu
and Wvare nonsingular weighting matrices. Wuaffects
the control distribution among the thrusters and Wvaffects
the prioritization among the virtual control components when
Bauvdcannot be attained due to, e.g. thruster constraints.
A faster algorithm can be obtained by approximating the SLS
formulation as a Weighted Least-Squares (WLS) problem:
min Wu(uud)2+γWv(Bauvd)2 0u¯
As γ→∞, the two formulations have the same optimal solu-
tion u. An iterative Fixed-Point (FXP) algorithm can be used to
solve the WLS formulation (41), see [18] for implementation
details. This algorithm asymptotically converges to the optimal
solution and the maximum number of iteration Nca can be
considered to reect the maximum computation time available.
The thruster Fault Detection, Isolation and Accommodation
(FDI-A) strategy described in the previous sections is imple-
mented within the MSR “high-delity” industrial simulator
provided by Thales Alenia Space. Following the design steps
given in Algorithm 1, a bank of 5 adjustable NUIOs is designed
1 2 3 4 5 6 7 8 9 1011 12
Thruster indices distribution
020 40 60 80 100
Thrust loss size
0.9 0.95 11.05 1.1
Mass (10%)
0.02 00.02
0.02 00.02
0.02 00.02
0.8 1 1.2
Inertia (Ixx)
0.8 1 1.2
Inertia (Ixy)
0.8 1 1.2
Inertia (Ixz)
0.8 1 1.2
Inertia (Iyx)
0.8 1 1.2
Inertia (Iyy)
0.8 1 1.2
Inertia (Iyz)
0.8 1 1.2
Inertia (Izx)
0.8 1 1.2
Inertia (Izy)
0.8 1 1.2
Inertia (Izz)
Fig. 1. Inertia (top left), mass (middle left), CoM (bottom left), thruster
indices (top right), and thrust loss size (bottom right) distribution, respectively
with κ=0.2,q=0.5and r=0.1. The FXP algorithm is
used for control re-allocation with Wv=I,Wu=I,ud=0,
Nca = 200,andγ=10
6. The remaining design parameters
are chosen as follows: Q=I,Jth = 300,Ts=0.1s,
1000 Monte Carlo (MC) simulations is performed in order to
assess the performance and robustness of the proposed FDI-A
scheme. The navigation is considered to deliver “non-perfect”
estimates. We also assume delays induced by the CPDE unit,
1% uncertainty on the thruster rise times, ±3cm uncertainty
on CoM (thus uncertain BT), 10% uncertainty on mass, 20%
uncertainty on inertia, and spatial disturbances (i.e. gravity
gradient, atmospheric drag, and solar radiation pressure). See
Fig. 1 for an illustration of considered uncertainties. All sim-
ulations are carried out during the last 20m of the rendezvous
phase (capture phase) and are associated with a fault scenario
when the ith thruster lose its effectiveness (thrust loss) of a
size ψi[0%,100%]. The other thrusters are fault-free. Faults
starts at tf= 1000sand are maintained. The two marginal
cases, i.e. ψi=0%and ψi= 100%, represent a nominal
operational and fully closed thruster, respectively.
The effect of small thrust losses (ψi15%) is relatively
small on the system dynamics and shall be compensated by a
robust control law. On the other hand, these faults are very
hard or even impossible to detect and isolate. The aim of
this MC simulation campaign is to show that if the FDI
unit fails to detect or isolate the faulty thruster, the effect
that this fault has on the GNC system and/or on the nal
MSR capture performance requirements is negligible. As seen
in Fig. 2, despite the fact that in some cases the FDI unit
failed, the nal capture requirements in terms of position and
velocities are fully met. The nal attitude and angular rate
error requirements (see Fig. 3) are met in 97.9% and 96.2%
simulation cases, respectively. These results may be further
improved by ne-tuning the FDI scheme (e.g. by adjusting
the NUIOs dynamics). Note that, in some cases, the angular
rate error requirement is not met even if the FDI succeed.
This can be the case when it took too long for the FDI
unit to detect and/or isolate the faulty thruster, thus the fault
accommodation unit has not enough time to fully recover the
faulty system. Figure 4 illustrates that the chaser maintains the
nominal trajectory, i.e. stays inside the rendezvous corridor,
and that the chaser keeps its attitude pointing towards the
target, thus the target remains visible from the sensors.
Chaser spacecraft Y axis
Chaser spacecraft Z axis
Basket aperture
Misalignment requirement
Target center (FDI success)
Target center (FDI failed)
Lateral Y velocity
Lateral Z velocity
Velocity requirement
Target lateral velocity (FDI success)
Target lateral velocity (FDI failed)
Longitudinal X velocity (cm/s)
Nominal velocity
Out of requirement (3 sigma)
Target velocity (FDI success)
Target velocity (FDI failed)
Fig. 2. MSR capture performance: position misalignment on +X face (top
left), lateral velocity (top right) and longitudinal velocity (bottom) errors
Fig. 3. Final attitude misalignments (left) and nal angular rate errors (right)
A method to detect, isolate, and accommodate thruster
faults of an autonomous spacecraft has been studied in this
paper. The FDI unit consists of a bank of nonlinear unknown
input observers with adjustable dynamics, a fault detector that
is based on judiciously chosen position model and an isolation
logic that uses the directional cosine approach. Once a fault
is isolated, a control re-allocation technique redistributes the
desired force and torque vectors among the remaining N1
healthy actuators. This makes the FDI-A without any change
in the nominal controller, without any redundant thruster set
or without any additional valve position sensor. Results from
the MC simulation campaign show that the proposed FDI-A
scheme is able to accommodate thruster faults that may have
effect on the GNC performance and on the rendezvous criteria.
This research work was supported by ESA and Thales Ale-
nia Space in frame of ESA’s Networking/Partnering Initiative.
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... Algorithms to estimate the inputs from the outputs of systems described by ordinary differential equations (ODEs) are an ongoing research topic; see, e.g., Refs. [9][10][11][12][13][14][15]. However, no such algorithm can succeed if the output does not provide sufficient information about the input. ...
... There are several algorithms for estimating the unknown input w from measurement data, ranging from feedback controllers via modifications of the nonlinear Kalman filter to moving horizon estimation [9][10][11][12][13][14][15]. We cannot discuss all these approaches here, but it is instructive to briefly discuss a simple version of the optimization-based approach, where an error functional J½w is minimized with respect to wðtÞ. ...
... Typically, we know which states could, in principle, be measured, and we can define a maximum set Z 0 of potential sensor nodes. If the resulting system with the maximum output set Z 0 is invertible, one can start the acquisition of time series data and feed them into one of the algorithms [9][10][11][12][13][14][15] to infer the input. This approach, though straightforward, would potentially be wasteful or even impractical. ...
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Despite recent progress in our understanding of complex dynamic networks, it remains challenging todevise sufficiently accurate models to observe, control, or predict the state of real systems in biology,economics, or other fields. A largely overlooked fact is that these systems are typically open and receiveunknown inputs from their environment. A further fundamental obstacle is structural model errors causedby insufficient or inaccurate knowledge about the quantitative interactions in the real system. Here, weshow that unknown inputs to open systems and model errors can be treated under the common frameworkof invertibility, which is a requirement for reconstructing these disturbances from output measurements.By exploiting the fact that invertibility can be decided from the influence graph of the system, we analyzethe relationship between structural network properties and invertibility under different realistic scenarios.We show that sparsely connected scale-free networks are the most difficult to invert. We introduce a newsensor node placement algorithm to select a minimum set of measurement positions in the network requiredfor invertibility. This algorithm facilitates optimal experimental design for the reconstruction of inputs ormodel errors from output measurements. Our results have both fundamental and practical implications fornonlinear systems analysis, modeling, and design.
... In the continuous-time versions the framework of UIO is based on the monotone system theory, applying similarity transformations [3]. To solve UIO design task for continuoustime linear parameter varying (LPV) systems, a vector of scheduling parameters has to be exploited [11], [10]. ...
... where A • is defined in (10). Since now N = DP ⊖1 , the final relation leads in the affirmative case to the solution ...
... Closely related to the problem of structural model errors is the theory of fault detection, an important topic in the engineering literature, see for instance Isermann (2011) and Blanke et al. (2016) for textbooks on fault detection, Fonod et al. (2014) and Chakrabarty et al. (2017) for works on unknown input observers. Geometrical and algebraic (Sain and Massey, 1969;Hirschorn, 1979;Fliess, 1988) treatments of the theory behind unknown input observers and fault detection of linear and nonlinear systems have found renewed interest (Martinelli, 2019;Villaverde et al., 2019). ...
... The practically most common approach to fault detection is to utilize unknown input observers. But those make strong assumptions about the system and especially about the ability to precisely understand the interactions and to collect data, see for instance Fonod et al. (2014) and Chakrabarty et al. (2017). These assumptions may be justified for systems which went through a design process, but they become questionable as soon as we work with biological systems like a cell or even an organ, which permanently interact with their exterior, whereas we do not even oversee the vast number of internal processes. ...
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Mathematical modeling is seen as a key step to understand, predict, and control the temporal dynamics of interacting systems in such diverse areas like physics, biology, medicine, and economics. However, for large and complex systems we usually have only partial knowledge about the network, the coupling functions, and the interactions with the environment governing the dynamic behavior. This incomplete knowledge induces structural model errors which can in turn be the cause of erroneous model predictions or misguided interpretations. Uncovering the location of such structural model errors in large networks can be a daunting task for a modeler. Here, we present a data driven method to search for structural model errors and to confine their position in large and complex dynamic networks. We introduce a coherence measure for pairs of network nodes, which indicates, how difficult it is to distinguish these nodes as sources of an error. By clustering network nodes into coherence groups and inferring the cluster inputs we can decide, which cluster is affected by an error. We demonstrate the utility of our method for the C. elegans neural network, for a signal transduction model for UV-B light induced morphogenesis and for synthetic examples.
... Among these FE approaches, the UIO-based FE technique is proven to be an effective strategy to eliminate the disturbances via decoupling [29]. In addition, it has been widely implemented, such as in autonomous spacecraft [30], wind turbines [31], and anaerobic bioreactors [32]. Motivated by the above discussion, a robust grid voltage sensor FTC approach for the rectifier with application in railway traction systems is developed in this work. ...
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The main purpose of this work is to address the problem of robust grid voltage sensor fault‐tolerant control for a single‐phase two‐level rectifier. First, considering a rectifier with disturbances and grid voltage sensor fault, the standard switched system model is constructed. Based on identical transformation, the original system is converted into an augmented system. In the augmented system, the information of the grid voltage sensor fault is included in the state vector. Second, an unknown input observer is designed for the augmented system for sensor fault estimation. The observer error is totally robust against independent deterministic disturbances. In addition, the effect of the disturbance caused by unknown disturbances or a modelling error is minimised with respect to a prescribed H∞ performance index. Finally, the sensor fault‐tolerant control is realised by fault compensation technique, which protects the rectifier from serious performance degradation in the presence of the grid voltage sensor fault. The proposed fault‐tolerant control strategy requires no supplementary hardware and all the inputs involved in the algorithm are directly available. Simulation and experimental results are exhibited to illustrate the capability of the proposed method to tolerate the grid voltage sensor fault.
Le travail présenté dans ce mémoire concerne l'élaboration d'approches de commande tolérante aux défauts (FTC) ainsi que l'estimation simultanée des états du système et de défauts capteurs et actionneurs pour différentes classes de systèmes. Les conditions de synthèse décrites dans ce manuscrit améliorent et permettent de réduire le conservatisme des résultats existants dans la littérature. Dans une première étape, nous nous sommes intéressés à la synthèse de la commande FTC à base d'observateur pour les systèmes T-S descripteurs à retard en présence de défauts actionneurs. Les conditions de synthèse dépendantes de la taille du retard, à la fois de l'observateur et du contrôleur sont résolues en une seule étape. Dans une deuxième étape, nous avons proposé une stratégie d'estimation des défauts et de la commande FTC pour les systèmes descripteurs flous de type T-S en présence simultanée de défauts actionneurs et capteurs ainsi que des perturbations extérieures. La troisième et dernière partie de ce travail concerne la synthèse d'une loi de commande robuste FTC basée sur un observateur adaptatif descripteur pour la dynamique du véhicule. La méthode proposée offre des conditions de LMIs moins restrictives que celles établies dans la littérature. L'approche proposée est validée en simulation sur le logiciel Carsim comme étant une application de la dynamique dérive-lacet-roulis du véhicule en présence de défauts capteurs, actionneurs et de perturbations extérieures
Aerospace is considered as one of the most critical areas of application requiring a certain level of precision and security. In fact, in the attitude control system of a satellite, reaction wheels are one of the most commonly used actuators presenting the highest percentage of failures that can appear in a spacecraft. Thus, the main challenge of the present research is to develop a fault diagnosis module for uncertain nonlinear process. An optimized interval fault detection and isolation method based on a Midpoint-Radii Kernel Principal Component Analysis (MR - KPCA) is then designed. Actually, the Kernel Principal Component Analysis is adopted to properly estimate the nonlinear process of the reaction wheels. A generalized squared prediction error denoted SPEMR to interval data is adopted in the detection phase. To improve the performance of the diagnosis module, a merge between the SPE index and the exponentially weighted moving average EWMA filter is proposed. Based on data provided by a ”high fidelity” industrial simulator developed by Thales Alenia Space, the obtained results proved the effectiveness of the proposed interval fault diagnosis method on detecting and isolating reaction wheels’ faults.
The problem of fault estimation for nonlinear systems with Lipschitz nonlinearities is addressed in this work for the estimation of both the system fault and states. In the proposed approach disturbance is regarded to be a function which is nonlinear and coupled with states of the system, and fault to be a function which is additive. In order to diagnose the fault and reduce the disturbances effects by dissipativity theory, Luenberger and two unknown input observers (UIOs) are designed separately. If the system satisfies the matching condition, the first UIO can accurately estimate faults by decoupling the effects of state-coupled disturbances. Otherwise, the second UIO estimates faults by decoupling partial disturbances, and attenuating the disturbances which cannot be decoupled. The essential conditions for all designed observers to exist are stated. Finally, the suggested method is applied to a robot by simulation to analyze its performance.
In this section, the problem of FD is investigated for continuous-time switched systems. A UIO is designed as an FD filter to generate the residual signal. To make the residual signal sensitive to the fault and robust to the unknown disturbances, we proposed designed method by employing the switched Lyapunov function and the ADT techniques. Furthermore, sufficient conditions for the existence of such an FD filter are exploited to guarantee that the error systems under ADT condition are asymptotically stable with an / performance index. Finally, a practical example is provided and simulation results are conducted to demonstrate the effectiveness of the proposed approach.
In this section, the problem of robust FE observer for discrete-time switched systems with unknown input is investigated. Firstly, let the state vector and fault vector of the initial system into the state vector of the augmented system, which can obtain the augmented system. Next, based on the P-radius technique, an interval observer is designed to estimate the state and the fault of the augmented system which can ensure the error system is stable. Then, some assumptions are given in order to simplify the calculation, and the LMI technique and Schur Complement Lemma are used to solve the observer. Finally, an interval hull computation approximation algorithm of FE is obtained for discrete-time switched systems. Compared with the region-based method, this method has more precise boundary and higher efficiency.
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The presented work is a result of a research collaboration between European Space Agency, Thales Alenia Space and IMS Laboratory with the aim of promoting fault-tolerant control strategies to advance spacecraft autonomy. A multiple observer based scheme is proposed jointly with an online constrained allocation algorithm to detect, isolate and accommodate a single thruster fault affecting the propulsion system of an autonomous spacecraft. Robust residual generator with enhanced robustness to time delays induced by the propulsion drive electronics and uncertainties on thruster rise times is used for fault detection purposes. A decision test on the residual of the fault detector triggers a bank of nonlinear unknown input observers which is in charge of confining the fault to a subset of possible faults. The faulty thruster isolation is achieved by matching the residual and the thruster force directions using the direction cosine approach. Finally, the fault is accommodated by redistributing the desired forces and torques among the remaining (healthy) thrusters and closing the isolated thruster. Simulation results from the "high-fidelity" industrial simulator, provided by Thales Alenia Space, demonstrate the fault-tolerance capabilities of the proposed scheme.
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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.
Conference Paper
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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.
In this paper, a bibliographical review on reconfigurable fault-tolerant control systems (FTCS) is presented. The existing approaches for fault detection and diagnosis (FDD) and reconfigurable control are considered with emphasis on the reconfigurable/restructurable controller design techniques. Several open problems and current research topics are addressed. 250 references in the open literature are listed to provide an outline of the historical and recent development in the field. The review reported in this paper is in no way to be complete, we apologize in advance if any of the existing works were left out. We encourage readers to communicate with us for any additional information.
This paper presents research activities conjointly led by EADS Astrium Satellites and the European Space Agency on innovative and robust health monitoring system for the next generation of spacecraft. Two robust FDI schemes are presented to detect and isolate faults affecting the micro-Newton colloidal thrust system of the LISA Pathfinder spacecraft. The first FDI strategy is based on a bank of eight H∞/H- residual generators designed according to the Generalized Observer Strategy whereas the second strategy consists of Kalman-based projected observers. The efficiency of the proposed FDI techniques is assessed through non linear simulations performed under realistic conditions (physical parameter uncertainties, disturbances, measurement noises, measurement delays, thruster jet misalignment, ⋯). The results are quite encouraging, illustrate the effectiveness of the proposed techniques and suggest that the solutions could be practical viable candidates.
A most critical and important issue surrounding the design of automatic control systems with the successively increasing complexity is guaranteeing a high system performance over a wide operating range and meeting the requirements on system reliability and dependability. As one of the key technologies for the problem solutions, advanced fault detection and identification (FDI) technology is receiving considerable attention. The objective of this book is to introduce basic model-based FDI schemes, advanced analysis and design algorithms and the needed mathematical and control theory tools at a level for graduate students and researchers as well as for engineers. © 2008 Springer-Verlag Berlin Heidelberg. All rights are reserved.
FDI(R) for satellites: How to deal with high availability and robustness in the space domain? The European leader for satellite systems and at the forefront of orbital infrastructures, Thales Alenia Space, is a joint venture between Thales (67%) and Finmeccanica (33%) and forms with Telespazio a Space Alliance. Thales Alenia Space is a worldwide reference in telecoms, radar and optical Earth observation, defence and security, navigation and science. It has 11 industrial sites in 4 European countries (France, Italy, Spain and Belgium) with over 7200 employees worldwide. Satellite evolution and the wish to design more autonomous missions imply the enhancement of the satellite architecture and special attention paid to fault management (i.e., Fault Detection, Isolation and Recovery, or FDIR, in space). Nevertheless, the constraints on FDIR techniques and strategies remain the same as for standard missions: robustness, reactive detection, quick isolation/identification and validation. This paper gives an introduction to Fault Tolerance (FT) in the space domain and some principles for the coming FT architectures. The current context of FDIR is presented by describing the approach implemented on telecommunication satellites and, more precisely, on one of the most FDIR sensible subsystems: the AOCS (Attitude and Orbit Control System). Following the current state of FDIR in the space domain, some perspectives are given such as a centralized distributed FDIR strategy for the next generation of autonomous satellites as well as some research tracks and hybrid diagnosis.
Description This book provides a comprehensive treatment of dynamics of space systems starting with the basic fundamentals. This single source contains topics ranging from basic kinematics and dynamics to more advanced celestial mechanics; yet all material is presented in a consistent manner. The reader is guided through the various derivations and proofs in a tutorial way. The use of "cookbook" formulas is avoided. Instead, the reader is led to understand the underlying principle of the involved equations and shown how to apply them to various dynamical systems.The book is divided into two parts. Part I covers analytical treatment of topics such as basic dynamic principles up to advanced energy concept. Special attention is paid to the use of rotating reference frames that often occur in aerospace systems. Part II covers basic celestial mechanics treating the two-body problem, restricted three-body problem, gravity field modeling, perturbation methods, spacecraft formation flying, and orbit transfers.A Matlab® kinematics toolbox provides routines which are developed in the rigid body kinematics chapter. A solutions manual is also available for professors. Matlab® is a registered trademark of The MathWorks, Inc.
Conference Paper
The paper presents conditions suitable in design of two types of state observers for a class of continuous-time systems, represented by the Takagi-Sugeno fuzzy model with bilinear rule consequence and the set of measurable premise variables. A Luenberger type observer structure, as well as an unknown input based bilinear observer are explored and subsidiary methods are used for their stability analyze. Giving the notion of linear state estimation error dynamics, and exploiting Lyapunov stability theory, the sufficient conditions are outlined in the terms of linear matrix inequalities, to possess asymptotic stable state estimation, irrespective of the input variables. The method generates an observer for each local bilinear model, and compiles the sub-models by inference through the membership functions. Simulation results illustrate the design procedures and demonstrate the specific performances of the proposed methods.