Content uploaded by Robert Fonod
Author content
All content in this area was uploaded by Robert Fonod on Oct 18, 2017
Content may be subject to copyright.
Position and Attitude Model-Based Thruster Fault
Diagnosis: A Comparison Study
Robert Fonod∗and David Henry†
University of Bordeaux, 33405 Talence, France
Catherine Charbonnel‡
Thales Alenia Space, 06156 Cannes la Bocca, France
Eric Bornschlegl§
European Space Agency, 2200 AG Noordwijk, The Netherlands
This paper deals with performance and reliability evaluation of a fault diagnosis scheme
based on two distinct models to detect and isolate a single thruster fault affecting a chas-
ing spacecraft during rendezvous with a passive target in a circular orbit. The analysis
is conducted in the frame of a terminal rendezvous sequence of the Mars Sample Return
mission. A complete description of a robust residual generation design approach based
on eigenstructure assignment is presented.Unknowntime-varying delays, induced by the
thruster drive electronics and uncertainties on thruster rise times, are considered as un-
known inputs. Particular novelty of the work is a new method for estimating the unknown
input directions used to enhance the robustness properties of the diagnosis scheme. Monte
Carlo results from a high-fidelity industrial simulator and carefully selected performance
and reliability indices allows us to evaluate the effectiveness of both schemes. The ob-
tained results reveal that the proposed fault diagnosis scheme based on a position model
is a justified competitor to the conventionally used attitude model-based scheme.
Nomenclature
A(bold capital letters) = matrices
a= radius of the target’s circular orbit (m)
c(normal lowercase letters) = scalars
C= torque vector (N·m)
F= force vector (N)
f= additive fault vector
I,0= identity and null matrix with appropriate dimensions
In=n×nidentity matrix
Ichs = inertia matrix (kg·m2)
M= thruster configuration matrix
m= mass (kg)
ˆmloss,ˆmleak = thrust loss size and maximum leakage size
n= orbital rate of the target (rad·s−1)
∗Ph.D. Candidate, Automatic Control Group, IMS (Intégration du Matériau au Système) Laboratory - UMR CNRS 5218,
351 Cours de la Libération; robert.fonod@ims-bordeaux.fr (Corresponding Author).
†Professor, Automatic Control Group, IMS (Intégration du Matériau au Système) Laboratory - UMR CNRS 5218, 351 Cours
de la Libération;
‡GNC Senior Research Engineer, Research Department - DRT/SO, 5 allée des Gabians.
§GNC System and Avionics Engineer, ESTEC (European Space Research and Technology Centre), TEC-ECN, Keplerlaan 1,
P.O. Box 299.
1
p, q, r = roll, pitch, and yaw rate (rad·s−1)
ˆ
Qtgt,ˆ
Qchs = quaternion estimates describing the orientation of the target and chaser
with respect to the Mars-centered inertial frame
r,r= vector of residuals and scalar residual
T= sampling interval (s)
Toni= scaled open duration of the ith thruster
~
X,~
Y,~
Z= mutually perpendicular (orthogonal) unit vectors
x(bold lowercase letters) = vectors
ξ, η, ζ = Cartesian component of the relative position (m)
ϕ, θ, ψ = roll, pitch, and yaw angle (rad)
Λ(A)= set of all eigenvalues of a matrix A
τ(t)= time-varying delay (s)
R,Z+= field of real numbers and set of non-negative integers
Rl,Ri,Rb= local (target), Mars-centered inertial, and chaser body-fixed
reference frame
k·k,k·kF= Euclidean norm (vectors) and Frobenius norm (matrices)
Subscripts
a= attitude model
p= position model
f= outlines the faulty case
Superscripts
M= Mars planet
f= outlines the faulty case
I. Introduction
Many space exploration missions require critical autonomous proximity operations. Mission safety is
usually guaranteed through a hierarchical implementation of fault/failure detection, isolation and recovery
(FDIR) approach with several levels of fault containments defined from local component/equipment up to
global system, i.e., through various equipments (sensors, thrusters, reaction wheels etc.) redundancy paths
and ground intervention [1–3]. In the case of the Mars Sample Return (MSR) mission, the hierarchical
implementation of FDIR is concerned at three levels [4, 5]: (i) based exclusively on sensor measurements
in the fault detection and isolation (FDI) function which are mainly signal-based techniques, (ii) relying
on both actuator commands and sensor measurements in the model-based FDI function, and (iii) based on
navigation outputs for the monitoring of the trajectory in the safety monitoring function. Through this
hierarchical strategy, it is assumed that sensor faults are diagnosed and recovered at the first level. This
paper focuses on the second level of this hierarchy. The goal is to develop a robust model-based FDI unit
for faults occurring in the spacecraft involved in the MSR mission.
Common FDIR techniques may be not sufficient in some cases, especially for faulty situations that
cause quickly abnormal dynamics deviation in critical space operations. This is especially the case during
rendezvous and docking/capture proximity operations. Advanced model-based FDI techniques should be
particularly developed to safely accommodate on-board (and on-line) the necessary robustness/stability of
the spacecraft control, the necessary trajectory dynamics and the vehicle nominal operation.
To ensure normal operation, real-time fault diagnosis is essential to provide information for the spacecraft
to accommodate the fault in time. Numerous model-based FDI techniques are reported in the literature and
have potential to be applied [6–9]. The problem of thruster faults is however less studied. Among other
examples, methods based on the so-called unknown input observer (UIO) technique were applied to the Mars
Express mission [11, 12]; an iterative learning observer (ILO) was designed to achieve estimation of time-
varying thruster faults [10]; H∞/H−filters were used to addresses the problem of thruster fault diagnosis in
the Microscope satellite [13] and also the problem of faults affecting the micro-Newton colloidal thrust system
of the LISA Pathfinder experiment [14]. Both H∞/H−filters are based on a residual generator, that is robust
2
against spatial disturbances (e.g., J2disturbances, atmospheric drag and solar radiation), measurement noises
and sensor misalignment phenomena, whilst guaranteeing some fault sensitivity performance.
On the other hand, only limited results on FDI of time-delay systems have been developed in recent
years in the literature. Among the contributions, an UIO was designed for fault detection of state-delayed
systems with known delays [15]. The well known parity space approach was extended for fault detection
of retarded time-delay systems [16]. Two-objective optimization approach was considered for linear time-
invariant (LTI) systems with constant time delays aiming at formulation of the optimization problem as:
enhancing sensitivity of the residual to faults and at the same time suppressing the undesirable effects of
unknown inputs and uncertainties in L2-gain sense [17, 18]. Robust fault diagnosis approach based on an
adaptive observer was developed for uncertain continuous LTI systems with multiple discrete time-delays
in both states and outputs [19]. Recently, a geometric approach for FDI of retarded and neutral time-
delay systems was developed [20]. Problem of robust fault detector design for a class of LTI systems with
some nonlinear perturbations and mixed neutral and discrete time-varying delays was investigated using a
descriptor technique, Lyapunov-Krasovskii functional and a suitable change of variables [21].
Often, uncertain time-varying delays occurs in the input channel of the system. Considering spacecraft’s
thrusters, such delays may be induced by the propulsion drive electronics. One of the main difficulty
in fault diagnosis of systems with uncertain input delays lies in the fact that uncertainty caused by this
delay is unstructured. Therefore, robustness cannot be achieved by applying existing robust unknown input
(disturbance) decoupling approaches directly. There is an important assumption for all such approaches that
the disturbance distribution matrix must be known, but a generalized approach to obtain the matrix is still
lacking [22, 23].
One possible solution to this problem is to transform the unstructured input uncertainty to unknown
input by means of time-dependent Padé approximation [24]. In this paper, we have taken an approach similar
to [25], where by introducing a Cayley-Hamilton theorem-based and h-order Taylor series expansion-based
polytopic transformations, the influence/uncertainty of the time-varying delay on the system is summarized
as an unknown input. By assigning the eigenstructure of the observer, the residuals can be decoupled from
these unknown inputs [22]. The solutions investigated in this paper follow this strategy. Two FDI schemes
are proposed based on two different models of the spacecraft dynamics, i.e., the so-called position model
scheduled by judiciously chosen parameters, which exhibit the translation dynamic of the spacecraft in a
judiciously chosen frame, and the attitude model.
The position model expressed in the local reference frame is well known and mastered for control, but
rarely used for FDI of thrusters. The reason is quite evident, the attitude model seems to be more sensitive to
thruster faults. The innovation that we pursue with this study is concerning the judiciously chosen position
model for FDI and its comparison to the attitude model-based solution.
Simulation results from “high-fidelity” MSR industrial simulator, under realistic conditions and taking
into account the effects that the GNC (Guidance, Navigation, Control) unit has on the FDI performances,
demonstrate the efficiency and capabilities of the proposed methodology. These two FDI schemes have been
successfully demonstrated as applicable for FDI of the chaser spacecraft involved in the MSR mission within
the critical dynamics and operation constraints of the last terminal translation of the rendezvous/capture
phase. The research work presented in this paper draws expertise from actions undertaken within the
European Space Agency (ESA), Thales Alenia Space and the IMS laboratory (University of Bordeaux). The
aim is to demonstrate the benefits of novel robust on-board FDI technologies, that may significantly enhance
the spacecraft autonomy.
The paper is organized as follows: Section II briefly describes the MSR mission and introduces the
modeling of the chaser spacecraft propulsion and control system. Necessities of modeling the spacecraft
dynamics during the rendezvous phase are recalled in Sec. III. Section IV addresses the theoretical basements
and computational procedure of the FDI scheme design. For a detailed assessment of the FDI performances,
results of the Monte Carlo simulation campaign incorporating different fault scenarios are presented in Sec.
V. Finally, some concluding remarks are given in Sec. VI.
II. Background
II.A. Overview of the MSR mission
The MSR mission is one of the most exciting challenges in the international effort on the Solar System
exploration. The mission concepts have been studied for years by NASA [26], French National Space Agency
3
(CNES) [27], and ESA. Its main goal is to collect samples of Martian rocks, soils and atmosphere, and to
return these samples safe and intact back to Earth for analysis.
Figure 1. Illustration of the MSR mission.
This mission consists of two spacecrafts directly injected towards Mars by launchers [28]. The descent
module is released on the Martian atmosphere (Entry phase), lands on the Mars surface and a rover vehicle
is released. Once the rover finished the collecting procedure, the samples are put into the sample container
and loaded on the Mars ascent vehicle (MAV) which is then launched, by means of rockets, into a low
Mars orbit. Meanwhile the second module, the Earth-return vehicle (chaser), is injected towards the Mars
planet to rendezvous with the sample container (target) and bring it back to Earth. The chaser achieves the
sample capture as soon as it is released by the MAV, which performs the last maneuver in order to avoid
any interference with the rendezvous operation. Finally, after successful capture, the sample container is
inserted into an Earth re-entry capsule (ERC) inside the chaser vehicle and the chaser starts its interplanetary
cruise towards the Earth. It should be noted, that this autonomous capture maneuver belongs to the non-
cooperative rendezvous since the target is not equipped with any actuation system. Figure 1 provides an
overview of the mission. The work addressed in this paper concerns the last 20m of the rendezvous phase.
II.B. Chaser Spacecraft Configuration
Figure 2 shows the general setup of the GNC system in the chaser vehicle (for confidential reasons, the
numerical values with regards to the spacecraft geometry and characteristics are omitted).
The selected attitude and orbital control system dedicated to spacecraft mode for the rendezvous phase
corresponds to 6 degree of freedom (DOF) control. The 6 DOF control ensures the application of both
commanded torque and force using thrusters only.
The control system relies on the following sensors:
•2 3-axis inertial measurement unit (IMU) in hot redundancy;
•2 star trackers (STR) in cold redundancy;
•2 short-range rendezvous sensors with a functional hot redundancy, i.e., a light detection and ranging
(LIDAR) sensor and a radio frequency sensor (RFS) as back-up.
The IMU unit is in charge of measuring the angular velocities ω= [p, q, r]T. The STR device gives
the attitude measurement Θ= [ϕ, θ, ψ]T. The LIDAR unit is in charge of the measurement of the relative
4
6 DOF
Control loop
Guidance Propulsion
system
Navigation
(NAV)
Position
dynamics
Attitude
dynamics
LIDAR
RFS
STRs
IMUs
Disturbances Noises
Chaser dynamics Sensors
+
−
Figure 2. General setup of the chaser’s GNC system during the rendezvous.
position ρ= [ξ, η, ζ ]Tbetween the chaser and target. The RFS sensor is used to monitor the chaser trajectory,
and it can trigger a collision avoidance maneuver, if necessary.
The role of the navigation (NAV) unit is to perform an estimate ˆ
Θ,ˆω,ˆρof Θ,ω,ρ, respectively, by
removing the misalignment phenomena, sensor bias and noises on these measurements. The NAV unit also
provides an estimate of the target attitude quaternion ˆ
Qtgt, that will be used later for the design of the FDI
unit. However, we assume that the NAV unit is not perfect and, thus, that there still exists time delays and
noises on ˆ
Θ,ˆω,ˆρ, and ˆ
Qtgt, respectively.
In terms of actuators, the chaser vehicle is equipped with a very precise chemical propulsion system
composed of 16 thrusters (THR) in full redundancy. They are organized in two sets, the nominal set ‘A’ is
used for the nominal vehicle control and the redundant set ‘B’ is reserved for the recovery actions after a
fault has been detected in the active set. The thruster configuration is illustrated in Fig. 3.
Z
X
Y
1A
1B
2A
2B
4A
4B
5A
5B
6A
6B
7A
7B
8A
8B
Figure 3. Thruster configuration of the chaser spacecraft.
II.C. Modeling the Control Loop and the Propulsion System
To carry out its mission and to ensure the mission performance, the chaser is equipped with 6 DOF control
law, see Fig. 4 for an illustration. It consists of two dynamic controllers: Kpos (s)that is in charge of
controlling the position dynamics and Katt(s)that aims to regulate the chaser’s attitude. The controller
outputs, the desired force Fdand the desired torque Cd, are sent to the thruster modulator unit (TMU)
that integrates the small commanded pulses which are below the minimum impulse bit (MIB) and releases
a pulse ( e
Fdor/and e
Cd) when the total reaches a momentum threshold. The TMU is widely used to
compensate the actuator nonlinearities and improve the control accuracy [4, 29]. The purpose of on-board
thruster management function (TMF) is to select specific thrusters and compute their firing times Tonito
realize force e
Fdand torque e
Cdcommand impulses coming from the TMU. The TMF algorithm relies on a
simplified approach with respect to the Simplex and thrusters’ non-linearities (minimum On/Off times) [4].
As mentioned earlier, the propulsion system uses only 8 of the 16 thrusters in the nominal case thus,
in the following, we are focusing on these 8 thrusters. The thrusters have fixed directions and each one
is able to produce a maximum thrust of 22N. The chemical propulsion drive electronics (CPDE), that
5
Thruster
modulator
unit
(TMU)
Thruster
management
function
(TMF)
H(s)I8M
Kpos(s)
Katt(s)
Fd
Cd
e
Fd
e
Cd
Ton1
Ton8
epos
eatt
u1
Propulsion system
u8
CPDE Thrusters
F
C
.
.
..
.
.
6DOF control law
Figure 4. Control architecture of the thruster-based propulsion system of the chaser vehicle.
drives the thrusting actuators, is initiating the opening of the thruster valve for the commanded duration
0≤Toni≤1, i = 1,...,8.Toniare scaled on-times. The scaling is done versus the sampling period Tof the
control unit and is defined according to Toni=¯
Toni/T ,¯
Tonibeing the actual/real firing time.
The propulsion system is a source of uncertainty in the system. The linear parameter-varying transfer
function
H(s) = e−τ(t)s(1)
aims to model the effect of the unknown time-varying delays τ(t)∈Rinduced by the CPDE electronic device
and the uncertainties on the thruster rise times [30]. In the following, it is assumed that each thruster open
duration Toni, i = 1,...,8is delayed with the same delay τ(t). This is a reasonable assumption from the
practical point of view, since for all nominal thrusters the same CPDE device is used to control the opening
of the thruster valve.
Let Tonit−τ(t)be the commanded open duration of the ith thruster delayed by τ(t), then the force
vector Fthr = [Fξ, Fη, Fζ]Tand the torque vector Cthr = [Cϕ, Cθ, Cψ]T, generated by the thrusters, are
given by
Fthr(t)
Cthr(t)!=M T ont−τ(t)(2)
where Ton = [Ton1, . . . , Ton8]Tand M∈R6×8is the thruster configuration matrix. The elements of Mare
the influence coefficients defining how each thruster affects each component of Fthr and Cthr , respectively.
II.D. Fault Considerations
It is obvious, that if a thruster fault occurs, for instance a hardover-type failure (thruster stuck open), it
could lead to a drastic increase of the propellant consumption which is already very constrained by the travel
to Mars. Dramatic consequences can occur, e.g., already in-placed GNC may not compensate such faults,
possibly leading the chaser to lose the attitude and/or the position of the sample container. The problem
becomes highly critical during the last 20 meters of the rendezvous phase when the chaser shall be correctly
positioned in the rendezvous corridor (see Fig. 5 for illustration) in order to successfully capture the sample
container as well as the chaser’s attitude need to be maintained in the rendezvous sensor field of view.
Figure 5. The MSR rendezvous corridor.
6
Such faulty situations obviously cannot be diagnosed by ground support using telemetry information,
due to the potential lack of communication between the chaser and the ground stations or due to signifi-
cant communication delay. This motivates ESA to manage studies for the development of on-board fully
autonomous FDI solutions that shall cope with all the failures which may occur and endanger the mission.
A quick detection and isolation of the fault is the first step towards an efficient recovery action that has no
impact on the mission success.
Through monitoring the sensor outputs, most of the faults in the chaser spacecraft can be detected
at sensor data processing level (health status, operational mode, continuity checks, etc.) and recovery
from these faults is usually quick enough, i.e., switching to a redundant sensor in hot redundancy (IMU
case), or propagation of last valid measurement during the switch-on of the redundant equipment in cold
redundancy (STR case). Most space agencies (NASA, ESA, CNES) have already identified LIDAR as a
preferred candidate instrument for autonomous rendezvous. It seems to be a justified and robust device for
the mission success [31]. Moreover, the problem of fault diagnosis of multiple sensor faults occurring in the
chaser spacecraft has been already addressed in the literature [32]. Thus, the focus of this study concerns
only thruster fault diagnosis.
Obviously, the mission can be endangered if a thruster fault occurs since it may have critical impact on
the GNC system, i.e., it may lead to a degraded GNC performances or even the GNC become unstable.
More precisely, we consider the following thruster fault scenarios:
•Case 1: stuck open valve - provides maximum thrust regardless of the demand;
•Case 2: thruster closing itself (blocked-closed) - thruster does not generate any thrust regardless
of the demanded command by the TMF;
•Case 3: bi-propellant leakage - residual propellant leakage of size mleak (t), starting from 0 and
reaching the maximum leakage size ˆmleak >0with a given slope ms>0, i.e., mleak (t) = min{ms(t−
tf),ˆmleak}, where tfdenotes the time of fault occurrence;
•Case 4: thrust loss - loss of efficiency of a particular thruster by a value ˆmloss >0;
Assuming no simultaneous faults, the considered thruster fault scenarios can be modeled in a multiplicative
way according to (index “f00 is used to outline the faulty case)
Ff
thr(t)
Cf
thr(t)!=M(I8−Ψ(t)) Ton (t−τ(t)) (3)
with Ψ(t) = diagψ1(t), . . . , ψ8(t), where 0≤ψi(t)≤1,i= 1, ..., 8are unknown. The health status of the
ith thruster is modeled by ψi(t)as follows
ψi(t) = (0if healthy
1−ϕi(t)/Toni(t)if faulty (4)
where ϕi(t)allows to consider all four fault cases, mentioned earlier, as follows:
ϕi(t) = (max{Toni(t), mleak (t)}if case 1 or 3
(1 −ˆmloss)Toni(t)if case 2 or 4 (5)
where 0< mleak(t)<1is the leakage size and 0<ˆmloss <1is the efficiency loss size. It is obvious
that mleak(t) = 1,∀tmodels a fully open (stuck open valve) and ˆmloss = 1 a blocked-closed thruster fault,
respectively.
III. Modeling the Chaser Dynamics During the Rendezvous Phase
In this section, a linear position model and an attitude model of the chaser spacecraft dynamics are
introduced for FDI purpose. The two proposed models are able to describe the dynamics of the chaser in
both, fault-free and faulty situations. For the sake of brevity, we recall only the necessary developments
about modeling the spacecraft’s dynamics, available in the extensive space literature [33, 34].
7
III.A. Position Model
The translation motion of the chaser is derived from the 2nd Newton law. To proceed, let a,m,Gand mM
denote the radius of the circular orbit of the target, the mass of the chaser during rendezvous, the universal
gravitational constant and the mass of Mars. Then, the orbit of the rendezvous being circular, the velocity
of the target is given by the relation rµ
a(6)
where µ=G · mM. Let Rl={OT;~
Xl,~
Yl,~
Zl}be the local reference frame fixed at the center of target OT,
with its ~
Zlaxis be perpendicular to the ~
Xland ~
Ylaxis and oriented as shown in Fig. 6.
Figure 6. The Mars rendezvous orbit with the associated frames.
The linear velocity of the target is given by the relation a.n, where n= ˙νstands for the uniform angular
speed of the target. This velocity is given in the inertial frame Ri={OM;~
Xi,~
Yi,~
Zi}, which is attached
to the center of Mars OM. From the Kepler’s third law it follows:
a.n =rµ
a⇒n=rµ
a3(7)
During the rendezvous phase, it is assumed that the chaser motion is due to the four following forces, all
given in Rl:
•the Mars attraction force Fa=−mµ
((a+ξ)2+η2+ζ2)3/2(a+ξ)~
Xl+η~
Yl+ζ~
Zl, where ξ, η, ζ denote the
elements of the three dimensional relative position vector ρ= [ξ, η, ζ ]Tof the chaser from the origin
of the target frame OT, expressed in Rl;
•the centripetal force Fe=mn2(a+ξ)~
Xl+n2η~
Yl+ 0~
Zl;
•the Coriolis force Fc=m2n˙η~
Xl−2n˙
ξ~
Yl+ 0~
Zl;
•the non-gravitational (chemical thrust, perturbations) force Fd=Fdξ ~
Xl+Fdη ~
Yl+Fdζ ~
Zl.
Then, from the 2nd Newton law, it follows
¨
ξ=n2(a+ξ)+2n˙η−µ
(a+ξ)2+η2+ζ23/2(a+ξ) + Fdξ
m
¨η=n2η−2n˙
ξ−µ
(a+ξ)2+η2+ζ23/2η+Fdη
m
¨
ζ=−µ
(a+ξ)2+η2+ζ23/2ζ+Fdζ
m
(8)
8
Because the distance between the target and the chaser, during the rendezvous, is much smaller than
the orbit, i.e., kρk a, it is possible to derive the so called Hill-Clohessy-Wiltshire (HCW) equations from
Eq. (8) by means of a first order approximation [35]. Thus, the motion of the chaser can be modeled in the
target (local) frame Rl, in both fault free (i.e., Ψ(t)=0) and faulty (i.e., Ψ(t)6= 0) situations, according to
a linear 6th order state space model with state vector x= [ξ η ζ ˙
ξ˙η˙
ζ]T, i.e., from Eq. (8) it follows
(˙
x(t) = Apx(t) + BpRˆ
Qtgt(t),ˆ
Qchs(t)Ff
thr(t) + Epw w(t)
y(t) = Cpx(t)(9)
where ˆ
Qtgt ∈R4and ˆ
Qchs ∈R4denote the attitude quaternion estimate of the target, and the chaser,
respectively. The attitude estimate ˆ
Qtgt and ˆ
Qchs describe the orientation of the target body frame and the
chaser body frame with respect to Ri, respectively. These estimates are provided by the NAV unit. The
quaternions dependent rotation matrix R(·)performs the projection of the three-dimensional force vector
Ff
thr from the chaser’s frame on to the target frame Rl. The measurement vector y= [ξ η ζ]Tis expressed in
Rland measured by LIDAR. Spatial disturbances (solar radiation pressure, gravity gradient and atmospheric
drag) are denoted by w∈R3.Ap,Bp,Cpand Epw are matrices of appropriate dimension.
Considering Eq. (2) and Eq. (3), a new input vector u∈R3is defined according to
u(t) = Rˆ
Qtgt(t),ˆ
Qchs(t)Fthr (t)(10)
and the fault model is approximated in terms of an additive fault vector f∈R3as follows:
f(t) = −Rˆ
Qtgt(t),ˆ
Qchs(t)MFΨ(t)Ton t−τ(t)(11)
where MF∈R3×8is the upper block of the thruster configuration matrix MT= [MT
FMT
C], related to the
forces F, see Fig. 4 if necessary. This type of approximation is widely used in the literature [6, 36].
Finally, the overall model of the position dynamics that takes into account both, the attitude Qchs, and
the relative position ρ, can be derived from Eq. (9) using Eqs. (10) and (11) as follows
(˙
x(t) = Apx(t) + Bpu(t) + Epf f(t) + Epww(t)
y(t) = Cpx(t)(12)
where Epf =Bp. This model is now suitable for the FDI filter design proposed in the next section.
III.B. Attitude Dynamics
The attitude control system works in a target pointing mode, which means that the chaser keeps one face
of the spacecraft pointed to the target and maintains it during the whole rendezvous phase (see Fig 7 for
illustration).
The equations for the rotational motion of the chaser spacecraft in the body-fixed reference frame Rb=
{OB;~
Xb,~
Yb,~
Zb}(the center of this frame is fixed to the center of mass of the chaser) are derived from the
Euler’s equations of motion [37]
˙ω(t) = I−1
chsCthr (t) + Cdis(t)−ω(t)×Ichsω(t)(13)
where ×denotes the cross product of vectors and Ichs ∈R3×3is the inertia matrix. Cthr and Cdis is the
external torque about the center of mass due to the propulsion and disturbances, respectively. ω= [p, q, r ]T
is the chaser rotational velocity vector, expressed in body frame Rb.
Using the individual rotation matrices from Euler(3,2,1) rotation we can express the relation between
the rotational velocities ωand the rate of the Euler angles Θ= [ϕ, θ, ψ]T. The inverse relationship becomes
[33]
˙
Θ(t) = 1
cos(θ)
cos(θ) sin(ϕ) sin(θ) cos(ϕ) sin(θ)
0 cos(ϕ) cos(θ)−sin(ϕ) cos(θ)
0 sin(ϕ) cos(ϕ)
ω(t)(14)
Finally, noting that the chaser is controlled around the equilibrium point Θ=0and ω=0, one can
derive from Eqs.(13-14) a linear model by means of a first-order approximation of the nonlinear equations
9
Figure 7. Chaser target pointing.
around the equilibrium point. Let the system’s state vector be x= [ϕ, θ, ψ, p, q, r]Tand the measurement
vector y= [ϕ, θ, ψ]T, it boils down to a 6 order linear, pure attitude, model having the same structure as
the position model given in Eq. (12), i.e.,
(˙
x(t) = Aax(t) + Bau(t) + Eaf f(t) + Eaww(t)
y(t) = Cax(t)(15)
where w∈R3denotes the spatial disturbances. The input uand the fault fvector are given by
u(t) = Cthr(t)(16)
f(t) = −MCΨ(t)Tont−τ(t)(17)
with MC∈R3×8being the lower block of the thruster configuration matrix M. The rest of the parameters
Aa,Ba,Eaf and Eaw are the result of the linearization procedure.
IV. Design of the FDI Scheme
This section is dedicated to the FDI task of thruster faults despite the presence of the delay τ(t)in the
actuation system. Two fault diagnosis units based on the eigenstructure assignment (EA) approach of an
observer-based fault detector are designed. The first is based on the position model (12) and the second one
uses the attitude model (15). The isolation strategy is based on the evaluation of a cross-correlation like
criterion between the component of the residuals and the thruster opening times.
IV.A. Robust Residual Generator Design
Consider the following general description of the continuous LTI system
(˙x(t) = Ax(t) + Bu(t) + Eff(t)
y(t) = Cx(t)(18)
where x∈Rnx,u∈Rnu,y∈Rny, and f∈Rnfis state, input, measurement, and unknown fault
vector, respectively. The quadruplet {A,B,C,Ef}represents the state-space matrices either of the position
Eq. (12), or the attitude Eq. (15) model. It is assumed that all considered faults fare detectable (see [38] for
more details on fault delectability) and that the pair (A,C)is observable. The spatial disturbances ware
omitted from Eq. (18), because they have the same directional properties as those of faults, i.e., Ef=Ew,
and thus, exact spatial disturbance decoupling cannot be achieved.
10
The TMF generates the thrusters’ opening times Ton equidistantly with a fixed sampling interval T > 0.
The control signal Ton(k), generated at time t=k T, k ∈Z+, arrives at the actuator at time instant
t=kT +τ(k). Assuming that the time-varying delay τ(k)is unknown but upper bounded, i.e., τ(k)≤
¯τ , ∀k∈Z+, then the system’s input, affected by delays, is given by
u(t) =
uc(k−1), t ∈kT, k T +τ(k)
uc(k), t ∈kT +τ(k),(k+ 1)T(19)
where ucdepends on the selected model according to
uc(k) =
Rˆ
Qtgt(k),ˆ
Qchs(k)MFTon (k),if position model is used
MCTon(k),if attitude model is used (20)
Our objective is to design discrete-time residual generator r(z) = Hy(z)y(z) + Hu(z)u(z), so that ris
robust against the uncertain delay τ(k),Hy&Hubeing observer-based filter transfer functions. To achieve
this goal, the influence of the uncertainty τ(k)is first transformed to unknown input acting on the system
model and then decoupled by means of eigenstructure assignment technique.
To proceed, assume τ(k)can be expressed as: τ(k) = lT +δ(k)≤¯τ, where lis a known integer, and
δ(k)∈Ris the unknown varying part of τ(k), bounded by 0≤δ(k)< mT , with mbeing a known integer.
Following, assume m= 1, which means that the variation part of the delay is less than one sampling interval.
The case when m > 1is discussed in [39].
If we assume that fis constant during each sampling interval T, what is a reasonable assumption from
a practical point of view (the counterpart of this assumption is that the sampling interval Tis chosen
adequately so that Eq. (21) holds), then the discrete representation of Eqs. (18) and (19) is
(x(k+ 1) = ¯
Ax(k) + Γ0(δ(k))uc(k−l) + Γ1(δ(k))uc(k−l−1) + ¯
Eff(k)
y(k) = ¯
Cx(k)(21)
where
¯
A=eAT,Γ0(δ(k)) =
T−δ(k)
R0
eAtdtB,¯
Ef=
T
R0
eAtdtEf
¯
C=C,Γ1(δ(k)) =
T
R
T−δ(k)
eAtdtB
It’s obvious that the following holds
¯
B=Γ0(δ(k)) + Γ1(δ(k)) = ZT
0
eAtdtB(22)
Using (21) and (22), and introducing a new augmented state vector zT(k) = hxT(k)uT
c(k−l−1)i,
we obtain
z(k+ 1) = ˆ
A0+ˆ
A(δ(k))z(k) + ˆ
B0+ˆ
B(δ(k))uc(k−l) + ˆ
Eff(k)
y(k) = ˆ
Cz(k)
(23)
where
ˆ
A0="¯
A0
0 0 #,
ˆ
B0="¯
B
I#,
ˆ
A(δ(k)) = "0 Γ1(δ(k))
0 0 #,
ˆ
B(δ(k)) = "−Γ1(δ(k))
0#,
ˆ
C=h¯
C0i
ˆ
Ef="¯
Ef
0#
The system given in Eq. (23) is a time-varying system with unstructured uncertainty, where Γ1(δ(k)) is
strongly dependent on the uncertain term δ(k). The remaining task is to transform this model to an uncertain
polytopic system for which structured properties can be extracted in terms of unknown inputs. The polytopic
system is then rewritten as a LTI system subject to an unknown input with a suitable distribution matrix.
Two polytopic transformations are used for this purpose. The first uses a Cayley-Hamilton theorem based
transformation [39] and is introduced in Proposition 1.
11
Proposition 1.The Cayley-Hamilton theorem based transformation of Γ1(δ(k)) can be expressed as the
convex matrix polytope
Γa
1(δ(k)) =
2nx
P
i=1
µa
i(k)Ua
i(24)
where µa
i(k)>0, i = 1, ..., 2nx,∀k∈Z+are uncertain scale factors satisfying
2nx
P
i=1
µa
i(k)=1,∀k∈Z+, and
Ua
i, i = 1,...,2nxare known constant matrices given in Appendix A.
The second transformation is based on the h-order Taylor series expansion [40] and is given in Proposi-
tion 2.
Proposition 2.The h-order Taylor series approximation of Γ1(δ(k)) can be expressed as the convex matrix
polytope
Γb
1(δ(k)) =
h+1
P
i=1
µb
i(k)Ub
i(25)
where µb
i(k)>0, i = 1, ..., h + 1,∀k∈Z+are uncertain scale factors satisfying
h+1
P
i=1
µb
i(k) = 1,∀k∈Z+, and
Ub
i, i = 1, . . . , h + 1 are known constant matrices given in Appendix B.
Taking into account the structure of the uncertain matrices ˆ
A(δ)and ˆ
B(δ)in Eq. (23) and the two
transformations of Γ1(δ)introduced in Eq. (24) and Eq. (25), the influence of the uncertain scalar factors
µa
iand µb
ion the state xcan be approximated in terms of unknown inputs as
2nx
X
i=1
µa
i(k)Ua
iuc(k−l−1) −uc(k−l)=Ea
dda(k)(26)
h+1
X
i=1
µb
i(k)Ub
iuc(k−l−1) −uc(k−l)=Eb
ddb(k)(27)
where
da(k) = µa
1(k)uT
c(k−l−1) −uT
c(k−l), ..., µa
2nx(k)uT
c(k−l−1) −uT
c(k−l)T
db(k) = µb
1(k)uT
c(k−l−1) −uT
c(k−l), ..., µb
h+1(k)uT
c(k−l−1) −uT
c(k−l)T
Ea
d=Ua
1, ..., Ua
2nx,Eb
d=hUb
1, ..., Ub
h+1i
Now, the mean value of the two unknown inputs, daand db, is considered. The augmented distribution
matrix ˆ
Edand the augmented unknown input dtake the following forms
ˆ
Ed="¯
Ed
0#="Ea
dEb
d
0#,d(k) = 1
2"da(k)
db(k)#(28)
The elements (columns) ˆ
Eddefine the directions how each component of daffects the augmented state. This
kind of approach gains advantage of combining two techniques to model the effect of the complex uncertainty
δon the state.
Finally, the augmented model with lumped unknown inputs can be expressed as
(z(k+ 1) = ˆ
A0zk+ˆ
B0uc(k−l) + ˆ
Eff(k) + ˆ
Edd(k)
y(k) = ˆ
Cz(k)(29)
This model is a quasi-equivalent representation of the augmented system given in Eq. (23), in other words,
using two polytopic transformations, the influence of the uncertainty Γ1(δ)on the augmented state zis
approximated in terms of unknown input d.
By closer examining the structure of ˆ
A0,ˆ
B0,ˆ
Ef,ˆ
Edin Eq. (29), one can see that only the upper state
of z, i.e., the system state x, is influenced by fand dand, that there is no coupling between the lower and
12
upper state. This allows us to consider only the upper state xin Eq. (29) for residual generator design. The
following observer-based residual generator with matrices ¯
A,¯
B,¯
C,¯
Ef, and ¯
Edis thus considered:
(ˆx(k+ 1) = ( ¯
A−L¯
C)ˆx(k) + ¯
Buc(k−l) + Ly(k)
r(k) = Qy(k)−¯
Cˆx(k)(30)
where r∈Rnr,ˆx∈Rnxis the residual, and the state estimation vector, respectively. The matrix Q∈Rnr×ny
is the residual weighting matrix.
The Z-transformed residual response to faults and unknown inputs is
r(z) = Grf (z)f(z) + Grd (z)d(z)(31)
where Grf (z)and Grd (z)denote the transfers between f(z)and r(z), and d(z)and r(z), respectively.
Once ¯
Edis known, the remaining problem is to find matrices Land Qso that (¯
A−L¯
C)is stable, and
Grd(z) = 0holds. The assignment of the observer eigenvectors and eigenvalues is a direct way to solve this
design problem. Note that, because this technique does not consider a sensitivity constraint in the design
procedure, therefore the fault sensitivity performance of the proposed FDI scheme can only be verified a
posteriori. Especially, the subspace of considered faults should not intersect the subspace of decoupled
disturbances [41], i.e., Im(¯
Ef)6⊂ Im(¯
Ed).
The following lemmas and theorems give the solution to the design problem.
Lemma 1.The transfer function Grd(z)can be expanded in terms of the eigenstructure as
Grd(z) = H(zI−¯
Ac)−1¯
Ed=
nx
X
i=1
HvilT
i¯
Ed
z−λi
(32)
where H=Q¯
C,viand lT
iare the right and left eigenvectors of ¯
Ac=¯
A−L¯
Cassociated with eigenvalue
λi.
Lemma 2.A given left eigenvector lT
iof ¯
Acis always orthogonal to the right eigenvectors vjcorresponding
to the remaining (nx−1) eigenvalues λjof ¯
Ac, where λi6=λj.
Theorem 1 (Unknown input decoupling using left EA).If the necessary condition
Q¯
C¯
Ed=H¯
Ed=0(33)
holds and all rows of the matrix Hare left eigenvectors of ¯
Accorresponding to nreigenvalues of ¯
Ac, then
Grd(z) = 0is satisfied. The proof of this theorem can be found in [42].
Following the above lemmas and theorem the first step for the design of the unknown input decoupled
residual generator (30) is to compute the weighting matrix Qsuch that it satisfies Eq. (33). The maximum
row rank of Qis ny−rank(¯
C¯
Ed), thus the residual signal dimension should be chosen according to
nr≤ny−rank(¯
C¯
Ed)(34)
The second step is to determine the eigenstructure of the observer. All rows of Hmust be the nrleft
eigenvectors of ¯
Ac. The remaining n−nrleft eigenvectors can be chosen without restraint. For the given
(stable) eigenvalue spectrum Λ( ¯
Ac) = {λi, i = 1, . . . , nx}, the following relation holds
lT
i(λiI−¯
A) = −lT
iL¯
C=−mT
i¯
C, i = 1, . . . , nx(35)
where mT
i=lT
iL. The assignability condition says that for each λi, the corresponding left eigenvector lT
i
should lie in the column subspace spanned by {¯
C(λiI−¯
A)−1}, i.e., a vector mT
iexists such that
lT
i=mT
iKi, i = 1, . . . , nr(36)
where Ki=−¯
C(λiI−¯
A)−1, i = 1, . . . , nr. The projection of liin the subspace span{Ki}is denoted by
l◦T
i=m◦T
iKi, i = 1, . . . , nr(37)
where m◦T
i=lT
iKT
i(KiKT
i)−1, i = 1, . . . , nr.
13
If lT
i=l◦T
i,lT
iis in span{Ki}, the required observer eigenstructure is assignable and perfect decoupling
can be achieved. Otherwise, the eigenvectors must be chosen to be close, e.g., in a least-square sense
klT
i−l◦T
ik, to the desired eigenvectors, i.e., an approximative procedure must be considered in order to
replace lT
iby its projection l◦T
i. In this situation, the residuals have low sensitivity to unknown inputs due
to approximate decoupling [22].
The remaining nx−nreigenvalues and corresponding eigenvectors can be chosen freely from the assignable
subspace, e.g., using singular value decomposition (SVD). Then, the observer matrix Lcan be computed as
follows
L=P−1S(38)
where
S=hm◦
1. . . m◦
nrmnr+1 . . . mnxiT
P=hl◦
1. . . l◦
nrlnr+1 . . . lnxiT
It is obvious, that the first nreigenvalues corresponding to the required eigenvectors lT
i, i = 1, . . . , nrmust
be real because all these eigenvectors are real-valued.
Remark 1.Without loss of generality, it can be assumed that ¯
Edhas a full column rank. When this is not
the case, the following decomposition can be applied: ¯
Edd(k) = ¯
Ed1¯
Ed2d(k), where ¯
Ed1is a full column
rank matrix and ¯
Ed2d(k)can now be considered as a new unknown input.
IV.A.1. Computational Results
First, the position model (Eq. (12)) and the attitude model (Eq. (15)) are transformed into the discrete form
(21) with l= 0 and m= 1. It practically means, that the unknown time-varying delay τ(k)is assumed to be
in the closed interval [0, T ). Using the Cayley-Hamilton theorem given in proposition 1 and the 2nd order
(h= 2) Taylor series expansion given in proposition 2, the uncertain time-varying delay τ(k)is modeled as
an unknown input as in Eq. (28).
The resulting ¯
Edmatrix has a large number of columns and the rank condition given by Eq. (34) cannot
be explicitly satisfied. Choosing the desired residual dimension be equal to one, i.e., nr= 1, the following
low rank factorization is performed for both models:
¯
E∗
d= arg min k¯
Ed−¯
E∗
dk2
F,s.t. nr=ny−rank(¯
C¯
E∗
d)(39)
By this factorization, the most significant directions are kept. Finally, a full column rank decomposition is
performed on ¯
E∗
dusing SVD decomposition.
The obtained distribution matrix is used for the residual generator design given by Eq. (30). The desired
left eigenvectors of the observer are the rows of the matrix H=Q¯
Cwhere the weighting matrix Qis
determined such that Eq. (33) is satisfied. In order to compare the FDI performances of both (position and
attitude) models, the assigned eigenvalues (dynamics of the observer) were selected to be exactly the same
for both models, i.e., Λ( ¯
Ac) = {0.85,0.87,0.89,0.91,0.93,0.95}.
IV.A.2. Comments on Implementation Issues
In order to avoid using an optimization procedure to determine smax
iand smin
iin Eq. (49), the solutions
si(t), i = 1, . . . , nxof the differential equation (48) were found numerically, and therefore, smax
iand smin
ican
be found using a simple iterative method. It is worth noting that other exact unknown input (disturbance)
decoupling methods exist [42]. In our particular case only the left EA technique appeared to be a viable
candidate. Other methods, such as UIO or right EA technique, violated some necessary conditions of the
solution existence.
IV.B. Residual Evaluation and Fault Isolation
Once the residual generation problem is solved, the problem is to make a decision about the fault presence.
The generalized likelihood ratio (GLR) test is used here [43]. The decision is made based on two hypotheses:
H0, the null hypothesis means no fault is present, while H1, the alternative hypothesis, indicates some
14
anomaly in the system considered to be due to the thruster fault. In this case, the decision test %Jth is
defined by
%Jth =(SNd(k)≤Jth, H0is accepted
SNd(k)> Jth, H1is accepted (40)
where Jth is a fixed threshold selected by the designer and SNd(k)is given by
SNd(k) = Ndln(σ0)−Nd
21 + ln(ˆσ2
1(k)) −ˆσ2
1(k)
σ2
0,ˆσ2
1(k) = 1
Nd
k
X
j=k−Nd+1
r2(j)(41)
where ris the residual signal, σ0is the standard deviation of rin fault free situation, and Nd>1represents
the detection sliding window due to on-line realization aspects. The interested reader can refer to the
monograph of Basseville and Nikiforov [44] for details on the threshold determination.
In this paper, a cross-correlation test between the residual rand the associated thruster open duration
Toniis considered for fault isolation purposes. In the thruster configuration of Fig. 3, each thruster has its
partner which provides same torque but force in exactly opposite direction. Therefore, a full coverage of
the isolation problem cannot be solved based on the so-called “sub-space isolation approach” because this
test cannot distinguish between faults in either thruster, only in the thruster pair [38]. Structured observer
schemes could be possible candidates for thruster fault isolation [45]. They are based on making each residual
signal sensitive to a subset of faults while being insensitive to another subset. This, however, requires a bank
of observers to be designed and run in parallel. Due to the on-board computational limitations, the following
considers an alternative solution that requires only one observer.
The proposed isolation strategy is based on minimum σNor maximum ¯σNcross-correlation criterion
between the residual signal rand Toni, i.e.,
σN(k) = (σN(k)if fault case 1 or 3
¯σN(k)if fault case 2 or 4 (42)
where
¯σN(k) = arg
i
max
1
N+ 1
k
X
j=k−N+1
r(j)Toni(j), i = 1...8,∀k∈Z+(43)
σN(k) = arg
i
min
1
N+ ΩN
i(k)
k
X
j=k−N+1
r(j)Toni(j), i = 1...8,∀k∈Z+(44)
ΩN
i(k)=1−
k
X
j=k−N+1
ϕi(j), i = 1,...,8, ϕi(j) = (0if Toni(j)6= 0
1if Toni(j)=0
These cross-correlation functions are statistical quantities that try to find the associated thruster index
that has the smallest/greatest impact on the resulting residual signal. For real-time reason, these criteria
are computed on a N-length sliding-window. An increase in the value of Nresults in elongated isolation
time delay. Hence again, an optimal value of Nhas to be selected. The resulting index σN(k)∈ {1,2,...,8}
refers to the identified faulty thruster at time instance k.
Finally, the resulting thruster index is confirmed at time instant k, if the following holds:
σN(k) = σN(k−1) = . . . =σN(k−Nc+ 1) (45)
where a confirmation window of length Nc>1is introduced in order to avoid initial transition phenomena
and to ensure robustness.
A key feature of this isolation strategy is that it is static, and thus, it has a low computational burden.
Note that if the ith thruster is not used by the TMF, i.e., Toni= 0, the minimum cross-correlation function
will possibly result in σ(k) = i. This fact is taken into account by introducing a penalty function ΩN
i(k)in
(44).
Remark 2.It should be noted that an event resolution algorithm must be implemented for the decision given
by Eq. (42), since it is required to distinguish the fault cases 1-3 from the case 2-4. Fortunately, because
cases 1-3 imply a propellant overconsumption and since the chaser is equipped with a dedicated sensor that
monitors the overall propellant consumption, this problem can be easily solved.
15
−5
0
5
10
15
20
25
magnitude
P ositi on model
0 200 400 600 800 1000 1200 1400
−5
0
5
10
15
20
25
time in seconds
magnitude
Attitude model
Jth
S10(k)
Jth
S10(k)
Figure 8. A set of 200 Monte Carlo simulations for fault-free case.
IV.C. Computational Procedure
In order to ensure robustness, whilst being sensitive to faults, the threshold Jth has to be selected carefully.
A higher value of Jth will increase the non-detection rate while a lower threshold will increase the false alarm
rate. The optimal value of Jth can be selected through Monte Carlo (MC) simulation. This approach is
widely used in the FDI community to analyze the efficiency and performance of the designed algorithm [11].
Here a set of 200 fault free MC simulations were performed using Nd= 10 in Eq. 41, i.e., the notation S10
is used. It can be clearly seen from Fig. 8, that for both models signal S10(k)does not exceed 20 in all the
cases. Therefore, for the simulations in the next section, threshold of Jth = 20 is chosen to ensure (ideally)
a zero false alarm rate. For the isolation function σN(k), a sliding window of N= 10 and confirmation
window of Nc= 15 samples was considered.
V. Simulation Results
The two FDI schemes described in the previous section are next implemented within the MSR “high-
fidelity” industrial simulator, provided by Thales Alenia Space. All simulations are carried out under realistic
conditions, i.e., the NAV unit is not considered to deliver “perfect” state estimates. We also assume constant
time delay between the NAV and the control block, time-varying delays induced by the electronic devices,
spatial disturbances (e.g., solar radiation pressure, gravity gradient, atmospheric drag) and parameter un-
certainties. The simulations are all carried out during the last 20m of the rendezvous phase.
A set of nmc = 1600 Monte Carlo simulations, for each faulty case (4×nmc), have been run in order
to assess the performances of the two proposed FDI schemes. For each run, model parameters, e.g., mass,
center of mass (CoM), etc., were altered within a specific limit, see Table 1 for details. The mass, CoM and
inertia were scattered according to the normal distribution and truncated to the corresponding 3σvalue.
Uncertainties of thruster rise times and thruster misalignment phenomena are modeled by 1% uncertainty
on thruster forces.
Table 1. Parameter uncertainties of the chaser spacecraft (3 sigma)
Parameter Variation range Unit
Mass ±10% [kg]
Inertia ±20% [kg.m2]
Thrusters forces ±1% [N]
CoM ±3 [cm]
16
Thruster faults were uniformly distributed among all of the 8 thrusters. Correspondingly, the leakage and
the thrust loss size were drawn from the uniform distribution with the following range: ˆmleak ∈<10%,30% >
and ˆmloss ∈<40%,90% >. The leakage is implemented as a dynamic lower saturation to the commanded
thruster open rate, where this saturation starts at value 0 and ends at ˆmleak with a slope of ms= 0.1. In
all cases, fault occurs at time 1000sand is maintained.
800 900 1000 1100 1200 1300
0
5
x 104
magnitude
Position model
̺20(k)
S10(k)
800 900 1000 1100 1200 1300
0
2
x 107Attitude model
magnitude
̺20(k)
S10(k)
1001 1001.5 1002 1002.5 1003 1003.5
0
5
T hr.No.
σ10(k)
Nc= 15
1001 1001.5 1002 1002.5 1003 1003.5
0
5
T hr.No.
σ10(k)
Nc= 15
800 900 1000 1100 1200 1300
0
5x 104
magnitude
̺20(k)
S10(k)
800 900 1000 1100 1200 1300
0
2
x 106
magnitude
̺20(k)
S10(k)
1013 1013.5 1014 1014.5 1015 1015.5
0
2
4
time in seconds
T hr.No.
σ10(k)
Nc= 15
1007.5 1008.5 1009.5 1010.5 1011.5 1012.5
0
5
time in seconds
T hr.No.
σ10(k)
Nc= 15
Thr.N.8 100% open Thr.N.8 100% open
Thr.N.3 blocked−closed
Thr.N.3 blocked−closed
Figure 9. Behaviour of the internal signals of the position (on the left) and attitude (on the right) model-based FDI
scheme, respectively.
Figures 9 and 10 illustrate the behaviour of the most important characteristics of the FDI units and their
internal signals. Both the position model-based (left figures on Fig. 9 and 10) and attitude model-based
(right figures on Fig. 9 and 10) FDI units are considered. These characteristics are:
i)the GLR signal S10(k)represented at each sample kand for a detection sliding window of length
Nd= 10 samples, see Eq. (41);
ii)the decision (alarm) signal %20(k)with the defined threshold Jth = 20, see Eq. (40);
iii)the thruster declared to be faulty by an isolation unit which is represented by the signal σ10(k)for a
computation sliding window of length N= 10 samples, see Eqs. (42)-(44).
The confirmation time window of length Nc= 15 samples is also considered. Figures 9 and 10 also illustrate
from top to bottom, the above listed characteristics for the following set of four arbitrary chosen faulty
situations selected from the scattered parameter space listed in Table 1:
•A fault that corresponds to a stuck open valve, occurs in the thruster No.8., i.e., the thruster No. 8 is
fully opened so that it provides a maximum thrust;
•A fully blocked-closed fault occurs in thruster No.3. In this case, the thruster does not generate any
thrust regardless of the command by the TMF;
17
•The third faulty situation corresponds to thruster No.2 suffering from a leakage of size 19.2%;
•The fourth faulty situation corresponds to the case when the thruster No.7 loses its thrust level by a
value of 54.7%.
Figure 9 is concerned with the two first situations whereas Fig. 10 considers the two last cases.
800 900 1000 1100 1200 1300
0
200
400
magnitude
Position model
̺20(k)
S10(k)
800 900 1000 1100 1200 1300
0
1000
2000
3000
Attitude model
magnitude
̺20(k)
S10(k)
1003 1003.5 1004 1004.5 1005 1005.5
0
1
2
3
T hr.No.
σ10(k)
Nc= 15
1002 1003 1004 1005
0
1
2
3
T hr.No.
σ10(k)
Nc= 15
800 900 1000 1100 1200 1300
0
200
400
magnitude
̺20(k)
S10(k)
800 900 1000 1100 1200 1300
0
1000
2000
3000
magnitude
̺20(k)
S10(k)
1019 1020 1021 1022
0
2
4
6
8
time in seconds
T hr.No.
σ10(k)
Nc= 15
1014 1014.5 1015 1015.5 1016 1016.5
0
2
4
6
8
time in seconds
T hr.No.
σ10(k)
Nc= 15
Thr.N.2 Leakage Thr.N.2 Leakage
Thr.N.7 Loss of efficiencyThr.N.7 Loss of efficiency
Figure 10. Behaviour of the internal signals of the position (on the left) and attitude (on the right) model-based FDI
scheme, respectively.
Clearly, the nonlinear simulations show that faults are detected and isolated by the proposed FDI units
within a reasonable time. Moreover, Fig. 10 shows the ability to detect and isolate small (thrust loss) and
incipient (residual leakage) thruster faults.
To evaluate the performance and reliability of the two FDI schemes, some statistical indices have been
used, i.e., the detection time td(time from fault occurrence to fault detection) and isolation time ti(time
from fault occurrence to fault isolation) are computed. The considered indices are listed below.
•mean(td)/mean(ti)- mean detection/isolation time in seconds;
•std(td)/std(ti)- standard deviation of the detection/isolation time in seconds;
•nondetection rate Pnd - number of non-detections divided by nmc;
•false alarm rate Pf- number of wrongly detected faults divided by nmc; and
•true isolation rate Pi- number of correctly isolated thrusters divided by nmc.
These performance indices are calculated for each fault scenario and model separately.
Table 2 presents complete results obtained from the Monte Carlo simulation campaign. As it can be seen
from this table, the two proposed FDI schemes present good reliability characteristics since no false alarms
Pf= 0 and non-detections Pnd = 0 have been revealed. Furthermore, all thruster faults were correctly
18
Table 2. FDI performances based on 4x1600 Monte Carlo runs
FDI based on position model FDI based on attitude model
Criterion CASE 1 CASE 2 CASE 3 CASE 4 CASE 1 CASE 2 CASE 3 CASE 4
mean(td)1.6 16.3 3.6 18.8 1.4 11.7 2.9 12.2
std(td)0.017 3.803 0.167 6.594 0.048 2.895 0.076 3.064
mean(ti)3.4 17.9 5.5 20.4 3.6 15.0 5.5 15.6
std(ti)0.207 3.799 0.399 6.592 0.237 3.231 0.459 3.456
Pnd/Pf/Pi0/0/1 0/0/1 0/0/1 0/0/1 0/0/1 0/0/1 0/0/1 0/0/1
isolated at the end of the campaign, i.e., Pi= 1. These achieved results also demonstrate that each FDI
scheme is able to successfully detect and isolate the chaser thruster faults.
In order to better appreciate the results presented in Table 2, a histogram plot is performed to graph-
ically represent the distribution of the detection tdand isolation titimes, respectively. Figure 11 shows
a comparison of the obtained results in terms of detection times and Fig. 12 in terms of isolation times.
This visual representation allows to evaluate the FDI performances in terms of minimum and maximum
detection/isolation times, as well as to observe the median values.
1.2 1.4 1.6 1.8
0
500
1000
1500
2000 Thruster fully open
frequency
Position model
Attitude model
4 6 8 10 12 14 16 18 20 22 24 26
0
100
200
300
400
500
600 Blocked thruster
frequency
Position model
Attitude model
3 3.5 4 4.5
0
200
400
600
800
1000
1200
1400 Propellant leakage
detection time in seconds
frequency
Position model
Attitude model
10 20 30 40 50
0
200
400
600
800 Loss of efficiency
detection time in seconds
frequency
Position model
Attitude model
Figure 11. Histograms of the detection times for the considered fault cases.
It can be seen from Fig. 11, that (as expected) the FDI unit based on the attitude model presents a
greater sensitivity towards all faulty situations. This can be easily explained by the fact that attitude is
more sensitive to small thruster faults.
Note that the occurrence of incipient or small size thruster faults (e.g., small propellant leakage or thrust
loss) may be covered by (robust) control actions, and the early detection of them is clearly more difficult.
Another problem can arise when a blocked-closed thruster is not commanded and thus a fault detection is
almost impossible. Such behaviour was not observed, since the TMF respects the thruster non-linearities
(minimum On/Off times) of each thruster.
By closer examining Fig. 12, it can be further observed that the final performances (isolation times)
of both FDI schemes are only slightly different from each other. Note that the position model-based FDI
unit succeeds thanks to the judiciously chosen linear model, i.e., a model which takes into account both
the rotational and translational motions of the chaser. In other words, the dynamics of the attitude of the
chaser is not modeled, but the chaser’s quaternion is introduced in the residual computation. This allows
19
3 3.5 4 4.5 5
0
100
200
300
400
500
600
700 Thruster fully open
frequency
Position model
Attitude model
7 9 11 13 15 17 19 21 23 25 27
0
100
200
300
400
500 Blocked thruster
frequency
Position model
Attitude model
5 6 7 8
0
100
200
300
400
500
600 Propellant leakage
isolation time in seconds
freq uency
Position model
Attitude model
10 20 30 40 50
0
100
200
300
400
500
600 Loss of efficiency
isolation time in seconds
freq uency
Position model
Attitude model
Figure 12. Histograms of the isolation times for the considered fault cases.
to propose a fault diagnosis solution with a very similar performances to those based on the attitude model.
Moreover, the position model is naturally robust against the model uncertainties, such as center of mass
and inertia whilst the attitude model not. The linearity of the attitude model during the fault presence is
questionable.
VI. Conclusion
Design and implementation of two distinct model-based FDI strategies for thruster fault diagnosis of an
autonomous spacecraft involved in the rendezvous phase of the MSR mission has been presented. Effects
of unknown time-varying delays induced by the propulsion drive electronics and uncertainties on thruster
rise times has been transformed into unknown input vector using Cayley-Hamilton theorem and h-order
Taylor expansion. The estimation of the complex unknown input distribution matrix can be considered as
a contribution to the theory. The robust (in the sense of unknown input decoupling) residual generation for
FDI has been achieved using observer eigenstructure assignment technique, i.e., some left eigenvectors of the
observer have been assigned to be orthogonal to the unknown input directions (columns of the distribution
matrix). Thruster fault isolation has been achieved by cross-correlation test between the residual signal and
the commanded thruster open durations. The core element of this study is the judiciously chosen position
model. This model has been used to design the first FDI scheme and was compared (in terms of well es-
tablished FDI performance indices) to the second, attitude model-based, scheme. A Monte Carlo simulation
campaign, using a “high-fidelity” industrial simulator, has been performed under realistic conditions, con-
sidering imperfect navigation, delays, spatial disturbances and parameter uncertainties. Four different fault
scenarios were injected throughout simulations. Obtained results indicate, that the proposed FDI strategies
are effective and applicable for on-board implementation. Moreover, the selected performance indices reveal,
that the position model-based scheme tends to achieve very similar FDI performances as the scheme based
on the pure attitude model.
Appendix A: Uncertainty Transformation Using Cayley-Hamilton Theorem
Let us first consider the following theorem [46]:
20
Theorem 2.The characteristic polynomial of matrix Ais
p(λ) = det(λI−A) = λnx+cn−1λnx−1+. . . +c1λ+c0(46)
then eAtcan be written as
eAt=s1(t)I+s2(t)A+. . . +snx(t)Anx−1(47)
where si(t),1≤i≤nxare solutions to the nth order homogenous scalar differential equation
s(nx)(t) + cnx−1s(nx−1)(t) + . . . +c1s0(t) + c0s(t)=0 (48)
satisfying the following initial conditions
s(i−1)
i(0) = 1, s(j)
i(0) = 0 for j6=i−1,0≤j≤nx−1
The proof of this theorem can be found in [46].
Using (47), we have
Γa
1(δ(k)) =
T
Z
T−δ(k)
eAtdtB=
nx
X
i=1
T
Z
T−δ(k)
si(t)dt
Ai−1B
(49)
Define
smax
i= max
0≤δ(k)≤TZT
T−δ(k)
si(t)dt, i = 1,2, . . . , nx
smin
i= min
0≤δ(k)≤TZT
T−δ(k)
si(t)dt, i = 1,2, . . . , nx
then (49) can be rewritten as
Γa
1(δ(k)) =
nx
X
i=1 αi,0(k)smin
i+αi,1(k)smax
iAi−1B(50)
where αi,0(k)and αi,1(k)are two time-varying unknown parameters satisfying 0≤αi,0(k)≤1,0≤αi,1(k)≤
1, and αi,0(k) + αi,1(k)=1for ∀k∈Z+. It can be verified that RT
T−δ(k)si(t)dt, i = 1,2, . . . , nxare Lipschitz-
continuous on 0≤δ(k)≤T, i.e., they satisfy
ZT
T−δ1(k)
si(t)dt −ZT
T−δ2(k)
si(t)dt≤κi|δ1(k)−δ2(k)|
∀δ1(k), δ2(k)∈[0, T ], where κi, i = 1,2, ..., nxare the Lipschitz constants.
Setting
µa
2i−1(k) = αi,0(k)/nx,
µa
2i(k) = αi,1(k)/nx,
Ua
2i−1=nxsmin
iAi−1B
Ua
2i=nxsmax
iAi−1B(51)
then Eq. (24) in proposition 1 yields Eq. (50).
Appendix B: Uncertainty Transformation Using Taylor Series Expansion
Theorem 3.Taylor series expansion of the uncertainty Γ1(δ(k)) is given by
Γb
1(δ(k)) = −
∞
X
i=1
(−δ(k))iAi−1
i!eATB
21
The h-order approximation of Taylor expansion for the uncertainty Γ1(δ(k)) can be expressed as a finite
sum of the first helements
Γb
1(δ(k)) = −
h
X
i=1
(−δ(k))iAi−1
i!eATB
Consider ε=δmin,ε=δmax and ε∈[ε, ε]. Now setting
µ1(k)=1−ε(k)−ε
ε−ε,
µi(k) = εi−1(k)−εi−1
εi−1−εi−1−εi(k)−εi
εi−εi, i = 2, . . . , h
Ub
i=hGh, . . . , G1iΦi, i = 1, . . . , h + 1
(52)
where
Gi= (−1)i+1 Ai−1
i!eATB, i = 1, . . . , h
Φ1=hεhIεh−1I, ..., ε2IεIiT
Φ2=hεhIεh−1I, ..., ε2IεIiT
.
.
.
Φh+1 =hεhIεh−1I, ..., ε2IεIiT
then using h-order Taylor expansion, Eq. (52) yields Eq. (25) given in proposition 2. This proposition is
proved in [40].
Acknowledgments
This research work was supported by the European Space Agency (ESA) and Thales Alenia Space France
in the frame of the ESA Networking/Partnering Initiative (NPI) program.
References
[1] Osder, S.,“Practical View of Redundancy Management Application and Theory,” Journal of Guidance,
Control, and Dynamics, Vol. 22, No. 1, 1999, pp. 12–21. doi:10.2514/2.4363
[2] Olive, X., “FDI (R) for satellites: How to deal with high availability and robustness in the space domain?,”
International Journal of Applied Mathematics and Computer Science, Vol. 22, No. 1, 2012, pp. 99–107.
doi:10.2478/v10006-012-0007-8
[3] Zolghadri, A., “Advanced Model-Based FDIR Techniques for Aerospace Systems: Today Chal-
lenges and Opportunities,” Progress in Aerospace Sciences, Vol. 53, No. 3, 2012, pp. 18–29.
doi:10.1016/j.paerosci.2012.02.004
[4] Peuvédic, C. L., Colmenarejo, P., and Guiotto, A., “Autonomous Rendezvous Control System: a High
Fidelity Functional Engineering Simulator Developed for GNC/AMM/FDIR Validation,” Proceeding of
the 7th International ESA Conference on Guidance, Navigation and Control Systems, Tralee, County
Kerry, Ireland, Jun. 2008.
[5] Strippoli, L. et. al., “Integrated vision-based GNC for Autonomous Rendezvous and Capture around
Mars,” 9th International ESA Conference on Guidance, Navigation & Control Systems, Porto, Portugal,
Jun. 2014.
[6] Isermann, R., “Model-Based Fault Detection and Diagnosis - Status and Applications,” Annual Reviews
in Control, Vol. 29, No. 1, 2005, pp. 71–85. doi:10.1016/j.arcontrol.2004.12.002
[7] Venkatasubramanian, V., Rengaswamy, R., Yin, K., and Kavuri, S. N., “A Review of Process Fault De-
tection and Diagnosis: Part I: Quantitative Model-Based Methods,” Computers & Chemical Engineering,
Vol. 27, No. 3, 2003, pp. 293–311. doi:10.1016/S0098-1354(02)00160-6
22
[8] Henry, D., Simani, S., and Patton, R., “Fault Detection and Diagnosis for Aeronautic and Aerospace
Missions,” Fault Tolerant Flight Control: A Benchmark Challenge, Springer Verlag, Berlin/Heidelberg,
2010, pp 91–128.
[9] Zolghadri, A., Henry, D., Cieslak, J., Efimov, D., and Goupil P., Fault Diagnosis and Fault-Tolerant
Control and Guidance for Aerospace Vehicles, Springer Verlag, London, 2014, Chap. 7.
[10] Chen, W., and Saif, M., “Observer-Based Fault Diagnosis of Satellite Systems Subject to Time-Varying
Thruster Faults,” Journal of Dynamic Systems, Measurement and Control, Vol. 129, No. 3, 2007, pp. 352–
356. doi:10.1115/1.2719773
[11] Patton, R., Uppal, F., Simani, S., and Polle, B., “Reliable Fault Diagnosis Scheme for a Space-
craft Attitude Control System” Journal of Risk and Reliability, Vol. 222, No. 2, 2008, pp. 139–152.
doi:10.1243/1748006XJRR98
[12] Patton, R., Uppal, F., Simani, S., and Polle, B., “Robust FDI Applied to Thruster Faults
of a Satellite System,” Control Engineering Practice, Vol. 18, No. 9, 2010, pp. 1093–1109.
doi:10.1016/j.conengprac.2009.04.011
[13] Henry, D., “Fault Diagnosis of Microscope Satellite Thrusters Using H∞/H−Filters,” AIAA Journal of
Guidance, Control, and Dynamics, Vol. 31, No. 3, 2008, pp. 699–711. doi:10.1109/TSMCA.2010.2063022
[14] Falcoz, A., Boquet, F., and Flandin, G., “Robust H∞/H−Thruster Failure Detection and Isolation with
Application to the LISA Pathfinder Spacecraft,” AIAA Guidance, Navigation, and Control Conference,
AIAA Paper 2010–7906, Toronto, Ontario, Aug. 2010.
[15] Yang H., and Saif, M., “Observer Design and Fault Diagnosis for State-Retarded Dynamical Systems,”
Automatica, Vol. 34, No. 2, 1998, pp. 217–227. doi:10.1016/S0005-1098(97)00175-1
[16] Kratz, F., Nuninger, W., and Ploix, S., “Fault Detection for Time-Delay Systems: A Parity Space
Approach,” Proceedings of the American Control Conference, Vol. 4, Philadelphia, PA, Jun 1998, pp. 2009–
2011.
[17] Ding, S. X., Maiying, Z., Bingyong, T., and Zhang, P., “An LMI Approach to the Design of Fault
Detection Filter for Time-Delay LTI Systems with Unknown Inputs,” Proceedings of the American Control
Conference, Vol. 3, Arlington, VA, Jun 2001, pp. 2137–2142. doi:10.1109/ACC.2001.946063
[18] Jiang, B., Staroswiecki, M., and Cocquempot, V., “H∞Fault Detection Filter Design for Linear Discrete-
Time Systems with Multiple Time Delays,” International Journal of System Science, Vol. 34, No. 5, 2003,
pp. 365–373. doi:10.1080/00207720310001601037
[19] Jiang, C., and Zhou, D. H., “Fault Detection and Identification for Uncertain Linear Time-
Delay Systems,” Computers and Chemical Engineering, Vol. 30, No. 2, 2005, pp. 228–242.
doi:10.1016/j.compchemeng.2005.08.012
[20] Meskin, N., and Khorasani, K. “Robust Fault Detection and Isolation of Time-Delay Sys-
tems Using a Geometric Approach,” Automatica, Vol. 45, No. 6, 2009, pp. 1567–1573.
doi:10.1016/j.automatica.2009.02.019
[21] Karimi, H. R., Zapateiro, M., and Luo, N., “A Linear Matrix Inequality Approach to Robust Fault
Detection Filter Design of Linear Systems with Mixed Time-Varying Delays and Nonlinear Perturbations,”
Journal of the Franklin Institute, Vol. 347, No. 6, 2010, pp. 957–973. doi:10.1016/j.jfranklin.2010.03.004
[22] Patton, R. J., and Chen, J., “Robust Fault Detection of Jet Engine Sensor Systems Using Eigenstruc-
ture Assignmen,” Journal of Guidance, Control, and Dynamics, Vol. 15, No. 6, 1992, pp. 1491–1497.
doi:10.2514/3.11413
[23] Patton, R. J., Willcox, S. W., and Winter, J. S., “Parameter-Insensitive Technique for Aircraft Sen-
sor Fault Analysis,” Journal of Guidance, Control, and Dynamics, Vol. 10, No. 4, 1987, pp. 359–367.
doi:10.2514/3.20226
23
[24] Fonod, R., Henry, D., Charbonnel, C., and Bornschlegl, E., “Robust Thruster Fault Diagnosis: Appli-
cation to the Rendezvous Phase of the Mars Sample Return Mission,” 2nd CEAS Specialist Conference
on Guidance, Navigation and Control, Delft, Netherlands, Apr. 2013.
[25] Fonod, R., Henry, D., Bornschlegl, E., and Charbonnel, C., “Robust Fault Detection for Systems with
Electronic Induced Delays: Application to the Rendezvous Phase of the MSR Mission,” European Control
Conference, Zurich, Switzerland, Jul. 2013.
[26] D’Amario, L. A., et al. “Mars Orbit Rendezvous Strategy for the Mars 2003/2005 Sample Return
Mission,” AAS/AIAA, Astrodynamics Specialist Conference, AIAA Paper 99–306, Girdwood, AK, Aug.
1999.
[27] Cazaux, C., et al. “The NASA/CNES Mars Sample Return - A Status Report,” Acta Astronautica,
Vol. 54, No. 8, 2004, pp. 601–617. doi:10.1016/j.actaastro.2003.07.001
[28] Beaty, D., Grady, M., May, L., and Gardini, B., “Preliminary planning for an international Mars Sample
Return mission,” Report of the International Mars Architecture for the Return of Samples (iMARS) Work-
ing Group, Jun. 2008. Available from: <http://mepag.jpl.nasa.gov/reports/iMARS_FinalReport.pdf>
[29] Peuvédic, C. L., Maini, M., Lafontaine, J. D., Kron, A., and Ortega, G., “Key Control Techniques
for GNC Design of Martian Vehicles,” 6th International ESA Conference on Guidance, Navigation and
Control Systems, Loutraki, Greece, Oct. 2005.
[30] Pettazzi, L., Lanzon, A., Theil, S., and Finzi, A. E., “Design of Robust Drag-Free Controllers with
Given Structure,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 5, 2009, pp. 1609–1621.
doi:10.2514/1.40279
[31] Pelletier, F., Golla, D., and Allen, A., “Lidar-Based Rendezvous Navigation for MSR,” AIAA Guidance,
Navigation, and Control Conference, AIAA Paper 2004–4987, Providence, Rhode Island, Aug. 2004.
[32] Issury, I., Henry, D., Charbonnel, C., Bornschlegl, E., and Olive, X., “A Boolean Algebraic-Based Solu-
tion for Multiple Fault Diagnosis: Application to a Spatial Mission,” Aerospace Science and Technology,
Vol. 28, No. 1, 2013, pp. 214–226. doi:10.1016/j.ast.2012.11.002
[33] Wie, B., Space vehicle dynamics and control, AIAA Education Series, Reston, VA, 1998, Chap. 5.
[34] Schaub, H., and Junkins, J. L., Analytical Mechanics of Space Systems, AIAA Education Series, New
York, NY, 2003, Chap. 13.
[35] Clohessy, W. H., and Wiltshire, R. S., “Terminal Guidance System for Satellite Rendezvous,” Journal
of the Aerospace Sciences, Vol. 27, No. 9, 1960, pp. 653–658. doi:10.2514/8.8704
[36] Frank, P. M., Alcorta-Garcia, E., and Köppen-Seliger, B., “Modelling for Fault Detection and Isolation
versus Modelling for Control,” Mathematical and Computer Modelling of Dynamical Systems, Vol. 7, No .1,
2001, pp. 1–46. doi:10.1076/mcmd.7.1.1.3633
[37] Wertz, J. R., Spacecraft Attitude Determination and Control, D. Reidel Publishing Company, Boston,
AM, Vol. 73, 1978, Chap. 16.
[38] Massoumnia, M. A., “A Geometric Approach to the Synthesis of Failure Detection Filters,” IEEE
Transactions on Automatic Control, Vol. 31, No. 9, 1986, pp. 839–846. doi:10.1109/TAC.1986.1104419
[39] Wang, Y., Ding, S. X., Ye, H., and Wang, G., “A New Fault Detection Scheme for Networked Control
Systems Subject to Uncertain Time-Varying Delay,” IEEE Transactions on Signal Processing, Vol. 56,
No. 10, 2008, pp. 5258–5268. doi:10.1109/TSP.2008.928703
[40] Hetel, L., Daafouz, J., and Iung, C., “Stabilization of Arbitrary Switched Linear Systems with Unknown
Time-Varying Delays,” IEEE Transactions on Automatic Control, Vol. 51, No. 10, 2006, pp. 1668–1674.
doi:10.1109/TAC.2006.883030
24
[41] Ding, S. X., Jeinsch, T., Frank, P. M., and Ding, E. L., “A Unified Approach to the Optimization
of Fault Detection Systems,” International Journal of Adaptive Control and Signal Processing, Vol. 14,
No. 7, 2000, pp. 725–745.
[42] Chen J., and Patton, R., Robust model-based fault diagnosis for dynamic systems, Kluwer Academic
Publishers, Dordrecht, 1999, Chap. 4.
[43] Ding, S. X., Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools, Springer
Verlag, New York, 2008, Chap. 10.
[44] Basseville, M., and Nikiforov, I. V., Detection of Abrupt Changes: Theory and Application, Prentice
Hall, Englewood Cliffs, NJ, 1993, Chap. 10.
[45] Frank, P. M., “Fault Diagnosis in Dynamic Systems Using Analytical and Knowledge-Based Redun-
dancy: A Survey and Some New Results,” Automatica, Vol. 26, No. 3, 1990, pp. 459–474. doi:10.1016/0005-
1098(90)90018-D
[46] Leonard, I. E., “The Matrix Exponential,” SIAM review, Vol. 38, No. 3, 1996, pp. 507–512.
doi:10.1137/S0036144595286488
25
