1

A Geometric Approach to Fault Detection and
Isolation of Two-Dimensional (2D) Systems

arXiv:1507.00943v4 [] 16 Sep 2016

Amir Baniamerian1 , Nader Meskin2 and Khashayar Khorasani1

Abstract
In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time twodimensional (2D) systems that are represented by the Fornasini-Marchesini model II (FMII). This is
accomplished by generalizing the basic invariant subspaces including unobservable, conditioned invariant
and unobservability subspaces of 1D systems to 2D models. These extensions have been achieved and
facilitated by representing a 2D model as an infinite dimensional (Inf-D) system on a Banach vector
space, and by particularly constructing algorithms that compute these subspaces in a finite and known
number of steps. By utilizing the introduced subspaces the FDI problem is formulated and necessary
and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the
FDI problem for 2D systems using both deadbeat and LMI filters are also developed. Moreover, the
capabilities and advantages of our proposed approach are demonstrated by performing an analytical
comparison with the currently available 2D geometric methods in the literature. Finally, numerical
simulations corresponding to an approximation of a hyperbolic partial differential equation (PDE) system
of a heat exchanger, that is mathematically represented as a 2D model, have also been provided.

Index Terms
2D systems, Fornasini-Marchesini model, infinite dimensional systems, fault detection and isolation,
geometric approach, invariant subspaces, LMI-based observer design, deadbeat observer.

*This publication was made possible by NPRP grant No. 4-195-2-065 from the Qatar National Research Fund (a member of
Qatar Foundation). The statements made herein are solely the responsibility of the authors
1

A. Baniamerian and K. Khorasani are with the Department of Electrical and Computer Engineering, Concordia University,

Quebec, Canada am_bani@encs.concordia.ca and kash@ece.concordia.ca
2

N.

Meskin

is

with

the

Department

of

Electrical

Engineering,

Qatar

University,

Doha,

Qatar

nader.meskin@qu.edu.qa

September 20, 2016

DRAFT

2

I. I NTRODUCTION
Over the past few decades, the problem of fault detection and isolation (FDI) of dynamical
systems has increasingly received larger interest and attention from the control community [1]
(c.f. to the references therein). The increasing complexity of human-made machines and devices,
such as gas turbine engines and chemical processes, has necessitated the need for development of
more complex and sophisticated FDI algorithms. Not with standing this requirement, development
of FDI algorithms for systems that are governed by partial differential equations (PDE) has not
been fully addressed and received extensive attention in the literature. One approach to investigate
the FDI problem of PDEs relies on obtaining an approximate model of the system. First, the
PDE system is approximated by a simple model (such as an ordinary differential equation (ODE)
model), and then sufficient conditions for solvability of the FDI problem are derived based on
this approximate model.
It is well-known that parabolic PDE systems can be approximated by ODE representations.
These systems can be approximated through application of finite element methods where sufficient conditions can then be derived by using singular perturbation theory [2]. The FDI problem
of parabolic PDEs has been addressed by using the corresponding approximate models in [3], [4].
On the other hand, by discretizing through spatial coordinates, one can approximate hyperbolic
PDE systems by ODE models. However, unlike parabolic PDE systems, one cannot apply model
decomposition, order reduction and singular perturbation theory to hyperbolic PDE systems
[5]. Moreover, the order of the resulting approximate ODE systems can be dramatically high.
Therefore, researchers have investigated hyperbolic PDE systems by using other formal methods
such as the theory of semigroups [6] and backstepping methods [7].
As shown in [8], a single hyperbolic PDE system can be approximated by using two-dimensional
(2D) systems. In [9], we have shown that this approximation is also applicable to a system of
hyperbolic PDEs. Moreover, as shown in [10], parabolic PDE systems can be approximated by
three-dimensional (3D) Fornasini and Marchesini model II (FMII) representations [11]–[14]. We
have shown in the Appendix that it is straightforward to extend the results of our work to 3D
systems as well (these results are not included here due to space limitations). Consequently, our
proposed methodology in this work can also be applied to parabolic PDE systems. There are
only a few results on FDI of 2D systems in the literature, such as those by using dead-beat

September 20, 2016

DRAFT

3

observers [15] and parity equations [16], [17]. On the other hand, the FDI of hyperbolic PDE
systems (by using even an approximate model) has not yet been adequately addressed in the
literature.
It should be noted that 2D system theory has other applications in the control field. For
example, a class of discrete-time linear repetitive processes can be modeled by 2D systems.
These processes play important roles in tracking control and robotics, where the controlled
system is required to perform a periodic task with high precision (refer to [18] for more details
on repetitive systems). One of the main approaches to control linear repetitive processes is the
iterative learning control (ILC) [18]. Since the ILC problem can be formulated as a control
problem in the 2D system theory [19], 2D systems have been increasingly applied to spatiotemporal and repetitive process control problems in the literature.
The FDI of 1D linear, time-invariant systems has been extensively investigated during the past
few decades [1] (and the references therein). The geometric approach [20]–[22] has provided
a valuable tool for investigating the FDI problem of a large class of dynamical systems such
as linear time-invariant [22], Markovian jump systems [23], time-delay systems [24], linear
impulsive systems [25], and parabolic PDE systems [3]. Moreover, the geometric approach is
also extended to affine nonlinear systems in [26]. Furthermore, hybrid geometric FDI approaches
for linear and nonlinear systems have also been provided in [27] and [28], where a set of residual
generators are equipped with a discrete event-based system fault diagnoser to solve the FDI
problem.
Motivated by the above discussion, in this paper we investigate the FDI problem of 2D systems
and apply the results to a 2D approximate model of a hyperbolic PDE system. As stated earlier, a
hyperbolic PDE system can be approximated by 1D systems, where the order of the approximate
1D system will significantly increase by decreasing the gridding size [5]. We also provide an
example where faults in the 1D approximate model are not isolable, whereas one can detect and
isolate faults by applying the 2D approximate model and by using our proposed methodology.
Recently, the geometric theory of 2D systems has attracted much interest, where basic concepts
such as conditioned invariant and controllable subspaces are studied in detail for the Fornasini
and Marchesini model I (FMI) [29], [30]. The hybrid 2D systems have also been investigated
from the geometric point of view in [31]. The geometric FDI approach for 2D systems, for the
first time, was addressed in [9], where invariant subspaces of the Roesser model are defined and
September 20, 2016

DRAFT

4

the FDI problem is formulated based on these subspaces.
Two-dimensional (2D) systems have been extensively investigated from a system theory point
of view [11]–[14]. Particularly, system theory concepts such as stability [13], [32], controllability
[33], observability [31], and state reconstruction [34] have been investigated in the literature.
However, due to complexity of 2D systems, unlike one-dimensional (1D) systems, there are
various definitions that are introduced for controllability and observability properties. Not surprisingly, the duality between the observability and controllability does not hold in 2D systems.
In this paper, we investigate observability of 2D systems from a new geometric point of view
which has its roots in system theory of infinite dimensional (Inf-D) systems.
Compared to the results reported in [9] and [35], we have specific generalization and novel
contributions in this work. We first investigate the Fornasini and Marchesini model II (FMII) as
an Inf-D system that allows us to deal with Inf-D subspaces (albeit with a finite dimensional (FinD) representation). The invariant subspaces and the corresponding algorithms that are introduced
in [9] for the Roesser model are then generalized to the 2D FMII systems. It is worth noting
that although the introduced subspaces in this work are Inf-D, the corresponding algorithms for
constructing the subspaces converge in a finite and known number of steps.
In addition, in [9] only sufficient conditions for solvability of the FDI problem of the Roesser
model were provided. As shown in [35], the invariance property of an unobservable subspace is
a generic property of 2D systems. We first derive a single necessary and sufficient condition for
detectability and isolability of faults (that is formally introduced in the next section). It is shown
that this condition also necessary for solvability of the FDI. Furthermore, in [9] by utilizing
the existence of an LMI-based observer only sufficient conditions for solvability of the Roesser
model FDI problem were provided. In other words, the procedure to design the observer gains
is not provided in [9], [35]. However, in our paper, we derive both necessary and sufficient
conditions, where sufficient conditions are based on (a) an ordinary, (b) a delayed deadbeat, and
(c) an LMI-based 2D Luenberger filters. Moreover, we develop a procedure to design LMI-based
filter gains.
It must be noted that recently related work has appeared in [36] and [37]. These two papers
investigate the FDI problem of three-dimensional (3D) FMII models. Although, a geometric FDI
methodology is also developed in [36], our work is distinct and unique from [36] in three main
perspectives:
September 20, 2016

DRAFT

5

1) The approach proposed in [36] is based on the results of [21], whereas our approach is
based on the generalization of the results of [21] as reported in [22] (the results in [22]
are more general than [21]) for 2D models (for more information refer to Remark 3).
2) In [36], necessary and sufficient conditions for solvability of the FDI problem were derived
for a subclass of detection filters where it was assumed that the output map of the detection
filters and that of the system are identical. However, in our work here we consider a
general class of detection filters for the residual generation and relax this condition.
3) As shown in Section IV-B, the observability property of the 2D model is a fundamental
requirement and assumption in [36] (although it is stated in [36] that this assumption was
made for simplicity of their presentation). However, our proposed solution does not require
this condition and assumption, and consequently our approach leads to a less restrictive
solution.
Another approach that was developed in the literature [15], [17] has its roots in 2D deadbeat
observers [38]. In [15], the FDI problem is investigated by using polynomial matrices and
unknown input deadbeat observers, where the right zero primeness of the 2D Popov-BelevitchHautus (PBH) matrix (this is reviewed in Subsection II-D) is a necessary condition for solvability
of the FDI problem. In [17], this condition was relaxed and necessary and sufficient conditions
that are based on an extended parity equation approach were obtained.
To provide a fair and comprehensive comparison with the currently available result in [36], in
this work it is shown that on one hand the solvability of the FDI problem by using the method
in [36] is also sufficient to accomplish the FDI task by using our proposed approach. On the
other hand, there are certain 2D systems that are not solvable by the approach in [36], however
our proposed approach can both detect and isolate the faults. Also, by comparing our proposed
results with the algebraic-based methods in [15], [17], where they can solve and derive necessary
and sufficient conditions for solvability of the FDI problem, we will highlight and emphasize
two important considerations as follows:
1) The algebraic methods, in contrast to our geometric approach, need a closed-form and
analytical solution to certain polynomial matrix equations (in two variables). However,
our proposed approach is derived and solved by using commonly available and relatively
straightforward numerical methods.

September 20, 2016

DRAFT

6

2) As shown subsequently in Section IV-B, there are certain examples, where the necessary conditions in [15], [17] are not satisfied. However, our proposed approach can
both detect and isolate these fault scenarios.
To summarize, the main contributions of this paper can be highlighted as follows:
1) By reformulating 2D models as Inf-D systems, the invariance property of the unobservable
subspace is investigated (this is provided in Section III), where an Inf-D unobservable
subspace is also introduced. This result enables one to formally address the solvability of
the FDI problem without restriction on the initial conditions (unlike in [9], [35] where
a restrictive assumption that the unobservable subspace is Fin-D is imposed). Since 2D
systems are Inf-D dynamical system, our proposed Inf-D representation and framework
enables one to address the FDI problem in its most general scenario than that in [9] and
[35].
2) Two important Inf-D invariant subspaces (namely, the conditioned invariant and the unobservability) are introduced for the FMII models. Although, these subspaces are Inf-D, we
provide explicit algorithms that can be invoked to compute these subspaces in a finite and
known number of steps.
3) The FDI problem of 2D systems is formulated in terms of the above introduced invariant
subspaces, and necessary and sufficient conditions for its solvability are derived and
formally analyzed.
4) A novel procedure is developed for designing an observer (also known as a detection filter)
by utilizing the linear matrix inequalities (LMI) technique.
5) Three sets of sufficient conditions for solvability of the FDI problem by utilizing the
ordinary, the delayed deadbeat, as well as our proposed LMI-based observers (detection
filters) are also provided.
6) Analytical comparisons between our proposed approach and the one in [36] are presented.
We show that the sufficient conditions in [36] are also sufficient for solvability of the
FDI problem by using our approach. However, an example is provided that shows our
method can both detect and isolate faults, whereas the approach in [36] cannot be used. In
other words, it is shown that if the method in the above literature can accomplish the FDI
task, our proposed approach can also accomplish this task. However, if our scheme cannot

September 20, 2016

DRAFT

7

achieve the fault detection and isolation goals for a given system, then it is guaranteed
that the schemes in [36] cannot also achieve these goals. Moreover, there are 2D systems
where our approach can achieve the FDI objectives whereas the results in the literature
cannot solve the FDI problem.
7) Our proposed methodology and strategy is applied to an important application area of a
heat exchanger (a hyperbolic PDE system), where it is shown that one can simultaneously
detect and isolate two different faults namely, the leakage and the fouling faults.
The remainder of the paper is organized as follows. The preliminary results including the Inf-D
representation, the FDI problem formulation, the 2D deadbeat observer and the 2D Luenberger
observers (detection filters) are presented in Section II. The unobservable subspaces of the
FMII 2D model are introduced in Section III. The geometric property of these subspaces
and the invariant concept of the FMII model are also presented in Section III. In Section IV,
necessary and sufficient conditions for solvability of the FDI problem are derived and developed.
Analytical comparisons between our proposed approach and the available geometric methods
in the literature, namely [36] and [37] are provided in this section. Furthermore, numerical
comparisons with both geometric and algebraic methods in [15], [17], [36], [37] are presented
in this section. Simulation results for the FDI problem of a heat exchanger that is expressed as
a PDE system are conducted in Section V. Finally, Section VI concludes the paper and provides
suggestions for future work.
Notation: In this work, A , B, ... are used to denote subspaces. For a given vector L, the
subspace span{L} is denoted by L . The inverse image of the subspace

V with respect
A 0
 is denoted by
to the operator A is denoted by A−1 V . The block diagonal matrix 
0 B
diag(A, B). The real, complex, integer and positive integer numbers are denoted by R, C, Z and
N, respectively. N denotes the set N ∪ {0}. In this paper, we deal with infinite dimensional (InfD) subspaces and vectors. An Inf-D vector is designated by the bold letters x, y, · · · . The Inf-D
T
T
T
subspace · · ·⊕V ⊕V ⊕· · · is denoted by ⊕V , where V ⊆ Rn . Let x = (· · · , xT
−1 , x0 , x1 , · · · ) ⊆
P
⊕V and |x|∞ = sup|xi |, where xi ∈ V . The vector space V∞ =
V is defined as {x|x ∈
i∈Z

⊕V and |x|∞ < ∞}. It can be shown that V∞ is a Banach (but not necessarily Hilbert)
space. Let i, j ∈ Z ∪ {−∞, ∞} and j ≥ i. The Inf-D vector xji ∈ ⊕V is expressed as xji =
T
T
∞
[· · · , 0, 0, xT
i , · · · , xj , 0, 0, · · · ] , where x` ∈ V for all i ≤ ` ≤ j, and associated with x−∞ we

September 20, 2016

DRAFT

8

simply use x. The other notations are provided within the text of the paper as appropriate.
II. P RELIMINARY R ESULTS
In this section, we first review 2D systems and their various representational models. Subsequently, a 2D system is expressed as an infinite dimensional (Inf-D) system that allows one
to geometrically analyze the unobservable subspaces (this is to be defined and specified in the
next section). The FDI problem is also formulated in this section. Moreover, we review the
2D Popov-Belevitch-Hautus (PBH) matrix and 2D deadbeat observers in this section. Finally,
an LMI-based approach is introduced to design a 2D Luenberger observer (also known as a
detection filter) for 2D systems.
A. Discrete-Time 2D Systems
2D models can be used for representing a large class of problems such as approximating hyperbolic PDE systems [8], [9], image processing and digital filtering [39]. System theory concepts
such as observability, controllability and feedback stabilization have also been investigated in the
literature for 2D systems [9], [11], [14], [30], [31]. There are various models that are adopted in
the literature for 2D systems including the Rosser model [39], the Fornasini-Marichesini model
I (FMI) and model II (FMII) [11], [14]. The FMI can be formulated as a Roesser model and
the Roesser model is a special case of the FMII model [11]. In this work, we consider and
concentrate on the FMII model, and consequently our results are also derived for this general
class of 2D systems.
Consider the following FMII model [14],
x(i + 1, j + 1) = A1 x(i, j + 1) + A2 x(i + 1, j) + B1 u(i, j + 1) + B2 u(i + 1, j)
+

p
X
k=1

L1k fk (i, j

+ 1) +

p
X

L2k fk (i + 1, j),

(1)

k=1

y(i, j) = Cx(i, j), i, j ∈ Z,
where x ∈ Rn , u ∈ Rm , and y ∈ Rq denote the state, input and output vectors, respectively. The
fault signals and the corresponding fault signatures are designated byfk , L1k and L2k , respectively.
Also, p denotes the number of faults in the system. Since in this work all the introduced invariant
subspaces are based on the operators A1 , A2 and C, we designate the system (1) by the triple
(C,A1 ,A2 ).
September 20, 2016

DRAFT

9

Remark 1. Note that system (1) represents and captures the presence of both actuator and
component faults. To represent sensor faults, one can augment the sensor dynamics and model
the sensor faults as actuator faults in the augmented system (for a complete discussion on this
issue refer to [20] - Chapters 3 and 4). Also, it should be pointed out that the fault signal fk
affects the system through two different fault signatures L1k and L2k . An alternative fault model
could have been expressed according to the following representation,
x(i + 1, j + 1) =A1 x(i, j + 1) + A2 x(i + 1, j) + B1 u(i, j + 1) + B2 u(i + 1, j) +

p
X

Lk gk (i, j),

k=1

y(i, j) =Cx(i, j).

(2)

Model (1) is more general than the one given by equation (2). This is due to the fact that by
denoting fk (i + 1, j) = gk (i, j) for all k = 1, · · · , p, one can represent the model (2) as in the
model (1).
Let us now consider the Roesser model [39] which is expressed as

 

  
p
X
r(i + 1, j)
A11 A12
r(i, j)
B

 =

 +  11  u(i, j) +
Lk fk (i, j),
s(i, j + 1)
A21 A22
s(i, j)
B21
k=1
(3)


r(i, j)
,
y(i, j) =C 
s(i, j)
h
iT
and where rT sT ∈ Rn represents the state, and the variables u, y, fk and Lk are defined
as in equation (1). By defining
 




 
 
r
A11 A12
0
0
B
0
 , A2 = 
 , B1 =  11  , B2 =   ,
x =   , A1 = 
s
A21 A22
B21
0
0
0

(4)

one can formulate the Roesser model (3) as in equation (1). In this paper, we assume that A1
and A2 in model (1) are not necessarily commutative (i.e. A1 A2 6= A2 A1 ), and hence, the results
that are subsequently developed can also be applied to the Roesser model (3). It should be noted
that the commutativity of A1 and A2 is a strong condition that renders the results in [31] (where
it is assumed that A1 and A2 are commutative) not applicable to the system (3).
In this work, we will investigate and develop FDI strategies for the model (1). It is assumed
that A1 and A2 in model (1) are not necessarily commutative (i.e. A1 A2 6= A2 A1 ), and hence,

September 20, 2016

DRAFT

10

the results that are subsequently developed can also be applied to the Roesser model. It should
be emphasized that the commutativity of A1 and A2 is a strong condition that renders the results
in [31] (where A1 and A2 are assumed to commutate) not applicable to Roesser systems.
B. Infinite Dimensional (Inf-D ) Representation
In this subsection, we reformulate the 2D model (1) as an Inf-D system that will be used to
derive the invariance property of unobservable subspaces (for details refer to Section III-A).
Consider the fault free system (1), that is with fk ≡ 0, and with zero input (we are mainly
interested in the unobservable subspaces and do not need to be concerned with the control inputs
in the FDI problem). By considering x(k) = (· · · , x(−1+k, 1)T , x(k, 0)T , x(1+k, −1)T , · · · )T ∈
P n
R , it can be shown that under the above conditions the system (1) can be represented as,
x(k + 1) = Ax(k),

k∈N

(5)

y(k) = Cx(k),
where x(k) ∈ X =

P

Rn , y(k) = (· · · , y(−1 + k, 1)T , y(k, 0)T , y(1 + k, −1)T , · · · )T ∈

P

Rq ,

and A is an Inf-D matrix with A1 and A2 as diagonal and upper diagonal blocks, respectively,
with the remaining elements set to zero, and C = diag(· · · , C, C, · · · ). In other words, we have,




..
.. ..
.
. ···
.
···
· · ·
· · ·






· · · 0 C 0 · · ·
· · · 0 A1 A2 0 · · ·

,C = 
(6)
A=




· · · 0 0 C · · ·
· · · 0 0 A1 A2 · · ·




..
.. ..
.
.
.
···
···
···
···
Note that since we invoke an Inf-D representation to investigate an unobservable subspace,
and where this subspace is defined by only A and C, therefore for sake of presentation simplicity,
an Inf-D system is used that has no fault and zero input.
There are various formulations for the initial conditions of the FMII model (1). These are
based on the separation set that is introduced in [40]. There are two separation sets that are
commonly used in the literature. In the first formulation the initial conditions are denoted by
P n
x(0) = (· · · , x(−1, 1)T , x(0, 0)T , x(1, −1)T , · · · )T ∈
R [14] (this is compatible with the
model (5)). The second formulation is expressed as x(i, 0) = h1 (i) and x(0, j) = h2 (j), where
h1 (i), h2 (j) ∈ Rn and i, j ∈ N [11]. The second formulation is more compatible with applications
(particularly, in case that the system (1) is an approximate model of a PDE system - refer to
September 20, 2016

DRAFT

11

Section V). It will be shown subsequently that since we derive the conditions based on invariant
unobservable subspace (this is formally defined in the next section), our proposed methodology is
applicable to both initial condition formulations. In other words, we use the Inf-D representation
to only show the results and evaluate the developed algorithms. However, to apply our results
there is no need to deal with Inf-D systems and subspaces, and therefore, one can apply our
proposed methods to 2D systems corresponding to both initial condition formulations.
We start with the first formulation of the initial conditions subject to the boundedness assumpP
tion (this is, x(0) ∈ Rn ). However, as shown in Section III-B, our proposed results also hold
for the second initial condition formulation.
As stated in the Notation section, it can be shown that X defined for equation (5) is an Inf-D
Banach space. The system theory corresponding to Inf-D systems is more significantly challenging than Fin-D system theory (1D systems) (refer to [41]). However, as shown subsequently, the
operator A is bounded and consequently, one can readily extend the result of 1D systems to the
system (5) [6], [41]. Let us first define the notion of bounded operators.
Definition 1. [6] Consider the operator A : X1 → X2 , where X1 and X2 are Banach vector
spaces with the norms | · |1 and | · |2 , respectively. The operator A is bounded if there exists a
real number G such that |Ax|2 ≤ G|x|1 for all x ∈ X1 .
Lemma 1. The operator A as defined in the Inf-D system (5) is bounded.
Proof: Let G = 2 max(|A1 |, |A2 |), where |Ai | denotes the norm of Ai and x = (xk )k∈Z ∈ X .
It follows readily that |Ax|∞ = sup|A1 xk + A2 xk+1 | ≤ supG max(|xk |, |xk+1 |) = Gsup|xk |.
k∈Z

k∈Z

k∈Z

Therefore, |Ax|∞ ≤ G|x|∞ . This completes the proof of the lemma.
The above lemma enables one to now formulate the unobservable subspace of the 2D system
(1) in a geometric framework (for details refer to Section III) based on the operator A (and
consequently, in terms of A1 and A2 ).
Remark 2. Although, in [46] all the results such as the controlled invariant subspaces are
presented on Rn , the developed approach in [46] has its roots in the theory of systems over
rings. In this paper, we propose an alternative approach that is based on Inf-D systems that
are defined on a Banach vector space. Similar to [46], our proposed methodology including the
algorithms and the conditions for solvability of the FDI problem can also be addressed in a
September 20, 2016

DRAFT

12

Fin-D scheme. However, as shown in the literature the duality property does not hold for 2D
systems [14], [48]. Therefore, by simply invoking duality the results of this paper cannot be
derived from those in [46].
C. The FDI Problem of 2D FMII Model
In this subsection, we formulate the FDI problem for the 2D system (1). In this paper, without
loss of any generality, it is assumed that the system (1) is subject to two faults, and therefore
we construct two residuals such that each one is sensitive to only one fault and is decoupled
from the other.
More precisely, consider the faulty FMII model (1). The solution to the FDI problem of the
2D FMII system can be stated as that of generating two residuals rk (i, j), k ∈ {1, 2} such that,
∀u, f2 and f1 = 0 then r1 → 0,
and if f1 6= 0 then r1 6= 0,

(7a)

∀u, f1 and f2 = 0 then r2 → 0,
and if f2 6= 0 then r2 6= 0.

(7b)

The above residuals are to be constructed by employing fault detection filters. For the 2D
system (1), we consider the following FMII-based fault detection filter,
ω̂(i + 1, j + 1) = F1 ω̂(i, j + 1) + F2 ω̂(i + 1, j) + K1 u(i, j + 1) + K2 u(i + 1, j) + E1 y(i, j + 1) + E2 y(i + 1, j),
rk (i, j) = M ω̂(i, j) − Hy(i, j),

(8)

where ω̂(i, j) denotes the state of the filter and is used to define the residual signal rk (i, j). The
solution to the FDI problem is now reduced to that of selecting the filter gains F1 , F2 , K1 , K2 ,
E1 , E2 , M and H corresponding to the filter (8).
Remark 3. The detection filter (8) can be selected as a full-order (H = I) or as a partialorder (kerH 6= 0) 2D Luenberger observer. As shown subsequently in Section IV, this level of
generality allows one to analytically compare our proposed methods with those results reported
in [36].
Remark 4. In this paper, we investigate the FDI problem by employing two main steps, namely
(i) decoupling the faults, and (ii) designing filter gains for each fault. The first step for decoupling
September 20, 2016

DRAFT

13

f1 addresses the existence of three maps D1 , D2 and H, such that the fault f2 signatures L12 and
L22 are members of the unobservable subspace (defined in the next section) of the system (Hk C,
A1 + D1 C, A2 + D2 C). The same terminology is used to decouple f2 . Moreover, the second step
is mainly concerned with existence of the filter (8) such that stability of the error dynamics is
guaranteed. In this paper, if the first step is solvable for the fault fi we say that fi is detectable
and isolable. Finally, it is stated that there is a solution to the FDI problem if for all the fault
signals fi both steps above are solvable.
D. Deadbeat Observers
In Section IV, necessary and sufficient conditions for solvability of the FDI problem are
derived. We provide sufficient conditions for accomplishing the FDI task by using a delayed
deadbeat detection filter and an ordinary (i.e., without a delay) deadbeat observer (refer to the
subsequent Corollaries 2 and 3). Towards these end, in this section we formally define a (delayed)
deadbeat filter. For a comprehensive discussion on 2D deadbeat observers refer to [17], [38].
Consider the system (1) under the fault free situation. A (delayed) deadbeat observer is
constructed according to,
z(i + 1, j + 1) =F1 z(i, j + 1) + F2 z(i + 1, j) + K1 u(i, j + 1) + K2 u(i + 1, j) + E1 y(i, j + 1) + E2 y(i + 1, j),
x̂(i, j) =L1 z(i, j) + L2 y(i, j),

(9)

where u and y are the input and output signals as defined in the system (1). Note that since
the output is assumed to not be directly affected by the input signal, x̂(i, j) is only a linear
combination of z(i, j) and y(i, j). If there exists a number N0 such that x̂(i, j) = x(i, j) for
all i + j > N0 , the filter (9) is designated as an ordinary (without a delay) deadbeat observer.
On the other hand, if there exist non-negative integers n1 and n2 such that n1 + n2 > 0 and
x̂(i, j) = x(i − n1 , j − n2 ), the observer (9) is designated as a delayed deadbeat observer. The
necessary and sufficient conditions for existence of a (delayed) deadbeat observer are specified
in the following theorem [17], [38].
Theorem 1. Consider the 2D system (1) under a fault free situation and the following 2D
Popov-Belevitch-Hautus (PBH) matrix,


I − z1 A1 − z2 A2

P BH(z1 , z2 ) = 
C
September 20, 2016

(10)

DRAFT

14

where A1 , A2 and C are defined as in equation (1), and z1 , z2 ∈ C. Then,
(i) there exists a delayed deadbeat observer if and only if rank(P BH(z1 , z2 )) = n for all
z1 , z2 ∈ C − {0} (that is, P BH(z1 , z2 ) is right monomic) [17] (Section 3) and
(ii) there exists an ordinary deadbeat observer if and only if rank(P BH(z1 , z2 )) = n for all
z1 , z2 ∈ C (that is, P BH(z1 , z2 ) is right zero prime) [38].
We use the above theorem in Section IV to show the existence of deadbeat filters to derive
sufficient conditions for solvability of the FDI problem (the subsequent Corollaries 2 and 3).
E. LMI-based Observer (Detection Filter) Design
As shown in [38], design of a deadbeat observer (9) requires that one works with polynomial
matrices (this is not always a straightforward process). In this subsection, we address the design
process for the FMII system observer, or the detection filter gains, by using linear matrix
inequalities (LMI). These results will be used to explicitly design a 2D Luenberger detection
filter (that can also be formulated as in equation (8)) subsequently in Section IV for the purpose
of accomplishing the solution to the FDI problem.
In order to show the asymptotic stability of the state estimation error dynamics, one needs to
apply the following stability lemmas.
Lemma 2. [32] The 2D FMII system (1) (under the fault free situation) is asymptotically stable
if there exist two symmetric positive definite matrices R1 , R2 ∈ Rn×n such that,
Ac , AT (R1 + R2 )A − R < 0,

(11)

where A = [A1 A2 ] and R = diag(R1 , R2 ).
Lemma 3. [42] Consider the LMI condition Φ + P T ΛT Q + QT ΛP < 0, where Φ ∈ Rn×n ,
P ∈ Rp×n and Q ∈ Rq×n . There exists a matrix Λ ∈ Rq×p satisfying the previous LMI condition
if and only if WpT ΦWp < 0 and WqT ΦWq < 0, where the columns of Wp and Wq are bases of
the ker P and ker Q, respectively.
Now consider the 2D system (1) under the fault free situation and the corresponding state

September 20, 2016

DRAFT

15

estimation observer as given by,
x̂(i + 1, j + 1) =(A1 + Do1 C)x̂(i, j + 1) + (A2 + Do2 C)x̂(i + 1, j) + B1 u(i, j + 1)
− Do1 y(i, j + 1) − Do2 y(i + 1, j) + B2 u(i + 1, j),
ŷ(i, j) =C x̂(i, j).

(12)

It follows readily that the state estimation error dynamics, as defined by e(i, j) = x(i, j)− x̂(i, j),
is governed by,
e(i + 1, j + 1) =(A1 + Do1 C)e(i, j + 1) + (A2 + Do2 C)e(i + 1, j).

(13)

The following theorem and corollary provide an LMI-based condition for existence of the state
estimation observer gains Do1 and Do2 such that the error dynamics (13) is asymptotically stable.
Theorem 2. Consider the 2D system (1) under the fault free situation. There exist two maps
Do1 , Do2 : Rq → Rn and two symmetric positive definite matrices R1 and R2 such that the
LMI (11) is satisfied for A1 + Do1 C and A2 + Do2 C if and only if R1 and R2 satisfy the LMI
condition WcTd Ac Wcd < 0, where Wcd = diag(Wc , Wc ) and the columns of Wc ∈ Rn×(n−q) are
the basis of ker C.
Proof: Note that without loss of any generality, it is assumed
that C is full row

 rank, and
−1
−(R1 + R2 )
A
. By using
n > q that is equivalent to partial state measurement. Let Φ = 
AT
−R
the Schur complement lemma, we have WcTd Ac Wcd < 0 if and only if,

 
 

−1
−(R1 + R2 )
AWcd
I 0
I 0

=
Φ
 < 0.
(14)
T T
T
T
Wcd A
−Wcd RWcd
0 Wcd
0 Wcd


h
i
0n×2n
 = −R < 0 if and only if R > 0 (or R1 > 0 and
It follows that 02n×n I2n×2n Φ 
I2n×2n
h
i
h
i
R2 > 0). By defining P = 02q×n Cd and Q = In×n 0n×2n , where Cd = diag(C, C),
and using
Lemma
h
i 3 the LMI condition (14) is satisfied if and only if there exits a matrix
Λ = Do1 Do2 ∈ Rn×2q such that,


h
0n×2q
 ΛT In×n
Φ+
CdT

September 20, 2016


i

In×n

0n×2n + 
02n×n


h
 Λ 02q×n

Cd

i


(R1 + R2 )−1
=
GT


G
 < 0,
R

(15)

DRAFT

16

h
i
where G = A1 + Do1 C A2 + Do2 C . Again, by using the Schur complement lemma, we have
GT (R1 + R2 )G − R < 0. This completes the proof of the theorem.
An important corollary to the above theorem and Lemma 2 can be stated as follows.
Corollary 1. Consider the 2D system (1) under the fault free situation and the state estimation
observer (12). If there are two symmetric positive definite matrices R1 and R2 satisfying the LMI
condition WcTd Ac Wcd < 0, then there exists two maps Do1 and Do2 such that the error dynamics
(13) is asymptotically stable.
Proof: Follow directly from Theorem 2 and Lemma 2, and the details are omitted for sake
of brevity.
Remark 5. Note that by solving the LMI condition WcTd Ac Wcd < 0, one can obtain symmetric
positive definite matrices R1 and R2 . Hence, the state estimation observer gains Do1 and Do2
are computed by solving the equation (15) (which is an LMI condition in terms of the gains Do1
and Do2 ). Therefore, Corollary 1 not only provides sufficient conditions for existence of a state
estimation observer, but also provides an approach for computing the observer gains Do1 and
Do2 .
The results of this section will now be used subsequently in Sections III-A and IV to address
the unobservable subspace of the system (1) as well as to provide sufficient conditions for
solvability of the FDI problem, respectively.
III. I NVARIANT S UBSPACES FOR 2D FMII M ODELS
As described earlier, 2D systems can be represented as Inf-D systems (i.e. the initial condition
is a vector of an Inf-D subspace). In this section, we first use the Inf-D representation (5) to
formally define and construct an unobservable subspace. Next, we define a subspace of the
unobservable subspace (this we called as an invariant unobservable subspace) of the 2D system
(1) that can be represented as an infinite sum of the same finite dimensional subspaces. Therefore,
one can compute the invariant unobservable subspace (that is, the Inf-D subspace) in a finite
number of steps. Also, it is shown that the invariant unobservable subspace enjoys an important
geometric property that is crucial for solving the FDI problem.

September 20, 2016

DRAFT

17

A. Unobservable Subspace
As described in the previous section, 2D systems can be represented as Inf-D systems. In
this subsection, the Inf-D representation (5) is utilized to formally define and construct an
unobservable subspace.
The unobservable subspace of the system (5) (and consequently of the system (1)) is defined
as,
Ng =

∞
\

ker CAk ,

(16)

k=0

where A and C are defined as in equation (5). Note that we define the above unobservable
subspace by following along the steps in [43], the results in [41] (Chapter I), and the fact that
the operator A in equation (5) is bounded (refer to Lemma 1).
One of the main difficulties in geometric analysis of Inf-D systems is the convergence of any
developed algorithm that involves computation of certain set of subspaces in a finite number of
steps. For example, consider the unobservable subspace (16). In Fin-D systems, the algorithm for
computing the unobservable subspace converges in a finite number of steps [44]. Moreover, one
is generally interested in investigating the FMII models in a Fin-D representation (1). Motivated
by the above, below two important subspaces of Ng that are denoted by N∞ and Ns,∞ are
introduced. The subspaces N∞ and Ns,∞ can be computed in a finite number of steps and also
allows one to derive necessary and sufficient conditions for solvability of the
 FDI problem.
 u ; i=j=0
0
Consider the initial condition x(0) = (· · · , 0, x0 , 0, · · · ) and u(i, j) =
,
 0 ; otherwise
where u0 ∈ Rm . One can show that the state solution of the model (1) under the fault free
situation is given by [14],
(i,j)

x(i, j) = A(i,j) x0 + AB u0 ,

(17)

(i,j)

where the matrices A(i,j) ’s and AB ’s are defined by the following recursive expressions,
A(i,j) = A1 A(i−1,j) + A2 A(i,j−1) , A(i,j) = 0
(i,j)

AB

if i or j < 0,
(18)

= A(i−1,j) B1 + A(i,j−1) B2 , A(0,0) = I.

Based on the solution that is given by equation (17), and considering that u0 = 0, a finite
observability matrix (given that its null space is a finite dimensional subspace) can be defined
as follows,
h
iT
O = C T , (CA1 )T , (CA2 )T , · · · , (CA(i,j) )T , · · · .
September 20, 2016

(19)
DRAFT

18

Let N = ker O =

T

i,j≥0


ker CA(i,j) . Since dim(N ) ≤ n < ∞, we designate N as the finite

unobservable subspace of the system (1). Also, recall from the 2D Cayley-Hamilton theorem
P
[14] that for all i + j ≥ n, one sets A(i,j) = h+k<n ζh,k A(h,k) , where ζh,k ’s are real numbers.
T
Therefore, for all i + j ≥ n, h+k<n ker CA(h,k) ⊆ ker CA(i,j) , and consequently N can be
computed in a finite number of steps as,
\

N = ker O =


ker CA(i,j) .

(20)

i,j≥0, i+j<n

Now, we consider the following subspace,
N∞ =

X

N

(21)

It follows that if x0 = (· · · , x−1 , x0 , x1 , · · · ) ∈ N∞ , then xi ∈ N for all i ∈ Z, and given the
zero input assumption one gets y(k) = 0 for all k ∈ N (in equation (5)). By considering Ak ,
where A is defined as in equation (5) and k ∈ N, it can be shown that N∞ ⊆ Ng . Also, note
that although N∞ is an Inf-D subspace, it can be computed in a finite number of steps (one
only needs to compute N ). However, as explained in [9], [35] the invariance property (this is
addressed in the next subsection) of N is not lucid. Therefore, in the following a subspace of
N is introduced such that it enjoys this geometric property. To define the subspace Ns one
needs the following notation.
Let us express Aα to denote the sequence of multiplications of A1 and A2 , where α is a
multi-index parameter that specifies the sequence of the multiplication. For example, consider
Aα = A2 A1 A1 A2 A1 , where we have α = (2, 1, 1, 2, 1). The notation ||α|| denotes the number
of all A1 and A2 that are involved in the corresponding multiplication (for the above example,
we have ||α|| = 5). Now, consider the following subspace (for more details on Ns refer to [9]),
\
Ns =
ker CAα
(22)
||α||<n

The following lemma shows that the subspace that is used in [30], [36], [37] as the unobservable (non-observable) subspace is indeed Ns .
Lemma 4. The subspace Ns can be computed in a finite number of steps according to the
following algorithm,
−1
V0 = ker C Vk = A−1
1 Vk−1 ∩ A2 Vk−1 ∩ ker C.

September 20, 2016

(23)
DRAFT

19
−1
−1
Proof: First, note that V1 = A−1
ker C ∩
1 ker C ∩ A2 ker C ∩ ker C and V2 = V1 ∩ (A1 A2 )
T
(A2 A1 )−1 ker C ∩ (A21 )−1 ker C ∩ (A22 )−1 ker C. In other words, Vk = ||α||≤k (Aα )−1 ker C. Note

that for every pair of operators C : Rn → Rq and F : Rn → Rn , one can show that ker CF =
F −1 ker C. Therefore, it follows that Vn = Ns . This completes the proof of the lemma.
P
Now, we set Ns,∞ =
Ns . Note that although dim(Ns,∞ ) = ∞, one can compute it in a
finite number of step (by computing Ns ).
B. A1,2 -Invariant Subspaces
As stated in the Subsection II-B, the 2D system (1) can be represented as an Inf-D system
(5). In order to formulate the corresponding Inf-D invariant subspaces one needs the next two
definitions.
Definition 2. [41] Consider the Inf-D system (5), where the operator A is bounded (according
P
to Lemma 1). The closed subspace V∞ ∈ X = Rn is called A-invariant if AV∞ ⊆ V∞ .
Definition 3. [45] The subspace V ⊂ Rn is said to be an A1,2 -invariant subspace for the 2D
system (1) if A1 V + A2 V ⊆ V , where A1 and A2 are defined as in equation (1).
Note that V is A1,2 -invariant if and only if it is invariant with respect to A1 and A2 (i.e. A1 V ⊆
V and A2 V ⊆ V ). The following theorem provides the connection between the Definitions 2
and 3.
Theorem 3. Consider the 2D system (1) and the Inf-D system (5). Let V∞ =

P

V , where

V ⊆ Rn . The subspace V∞ is A-invariant if and only if V is A1,2 -invariant.
P
k
k
Proof: First, note that every x ∈ V∞ can be expressed as x = ∞
k=−∞ xk , where xk =
h
iT
∈ V∞ and xk ∈ V . Therefore, one only needs to show the result for xkk .
· · · , 0, 0, xT
k , 0, 0 · · ·
(If part): Assume V is A1,2 -invariant. Consider the Inf-D vector xkk . It follows that Axkk =
[· · · , 0, (A2 xk )T , (A1 xk )T , 0, 0 · · · ]T . Since V is A1,2 -invariant, it follows that Axkk ∈ V∞ .
(Only if part): Let AV∞ ⊆ V∞ and x00 ∈ V∞ . Consequently, x0 ∈ V . Since Ax00 = [· · · , 0,
(A2 x0 ))T , (A1 x0 )T , 0, 0, · · · ]T ∈ V∞ , it follows that A1 x0 ∈ V and A2 x0 ∈ V , and consequently
V is A1,2 -invariant. This completes the proof of the theorem.
Consider the subspaces V ⊆ Rn and C ⊆ Rn . If V is the largest A1,2 -invariant subspace
that is contained in C , we denote V =< C |A1,2 >. We have shown in [9] that Ns ⊆ N ,
September 20, 2016

DRAFT

20

and it is the largest A1,2 -invariant subspace that is contained in ker C. Therefore, one can write
Ns =< ker C|A1,2 >. Since Ns is A1,2 -invariant, by Theorem 3, Ns,∞ is A-invariant. Therefore,
if x(0) = (· · · , x−1 , x0 , x1 , · · · ) ∈ Ns,∞ (that is, xi ∈ Ns for all i ∈ Z) and zero input,
x(k) ∈ Ns,∞ for all i ∈ Z and y(k) = 0 for all k ∈ N (in equation (5)). We designate Ns,∞ as
the invariant unobservable subspace.
Remark 6. As stated in Subsection II-B, there are two different types of initial condition
formulations. In this paper, we use the first formulation that is compatible with the Inf-D system
(5). Recall that the second formulation is expressed as x(i, 0) = h1 (i) and x(0, j) = h2 (j),
where i, j ∈ N. Now, let x(i, 0) ∈ Ns and x(0, j) ∈ Ns . The A1,2 -invariance property of Ns
verifies that y(i, j) = 0. In other words, Ns,∞ is also the invariant unobservable subspace
of system (1) with the second formulation of the initial conditions. Therefore, without loss of
any generality, one can apply our proposed approach to both initial condition formulations as
P
provided in Section II-A. Moreover, Ns,∞ is the largest A-invariant in the form Ns,∞ =
Ns
that is contained in ker C, where C is defined in (5).
C. Conditioned Invariant Subspaces
Another important subspace in the geometric FDI toolbox is the conditioned invariant (i.e.,
the (C, A1,2 )-invariant) subspace that is defined next. This definition is an extension of the one
that has appeared and presented in [45] and [46].
Definition 4. The subspace W∞ =

P

W (where W ⊆ Rn ) is said to be the conditioned invariant

subspace for the 2D system (1) if there exist two output injection maps D1 , D2 : Rq → Rn such
that (A1 + D1 C)W ⊆ W and (A2 + D2 C)W ⊆ W . In other words, W is [A + DC]1,2 -invariant
(i.e., invariant with respect to A1 +D1 C and A2 +D2 C). We designate W as the finite conditioned
invariant subspace (since dim(W ) < ∞) of the 2D system (1).
Similar to 1D systems, one can now state the following result.
Lemma 5. The following statements are equivalents.
(i) The subspace W∞ is conditioned invariant.
(ii) A1 (W ∩ ker C) + A2 (W ∩ ker C) ⊆ W .
(iii) A(W∞ ∩ ker C) ⊆ W∞ .
September 20, 2016

DRAFT

21

where W∞ =

P

W.

Proof: (i) ⇔ (ii) and (i) ⇔ (iii): By definition, there exists two maps D1 and D2 such
that W is [A + DC]1,2 -invariant. By utilizing Theorem 3 W∞ is Ad -invariant, where


..
..
.
.
···
· · ·



· · · 0 A1 + D1 C A2 + D2 C
0
· · ·


Ad = 

· · · 0
0
A1 + D1 C A2 + D2 C · · ·


...
...
···
···

(24)

By following along the same lines as in Lemma 1, one can show that Ad is bounded. Consequently, the result of 1D system is also valid for the Inf-D system (5). Hence, we have
A(W∞ ∩ ker C) ⊆ W∞ (that shows (i) ⇔ (iii)). By considering the structure of A and C it
follows that A1 (W ∩ ker C) + A2 (W ∩ ker C) ⊆ W .
(iii) ⇔ (i): Since A is bounded, the domain of A is equal to X =

P

Rn , and therefore, the

result of 1D system is also valid for the Inf-D system (5). Therefore, there exists a bounded
operator D such that W∞ is A + DC-invariant. By considering the structure of W∞ and ker C,
it is easy to show that one solution for D is

...


· · · 0
D=

· · · 0

···

given by
..
. ···
D1 D2
0
···

D1



· · ·

0 · · ·


D2 · · ·

... ...

(25)

Hence, by using Theorem 3, the subspace W is [A + DC]1,2 -invariant and consequently W∞ is
a conditioned invariant subspace. This completes the proof of the lemma.
In the geometric FDI approach, one is interested in conditioned invariant subspaces that are
containing a given subspace [22]. Let us define all the conditioned invariant subspaces containing
P
a subspace L∞ =
L (L ⊆ Rn ) as Q(L ) = {W∞ | ∃ D1 and D2 ; (Ai + Di C)W ⊆
W and W ⊇ L }. It can be shown that for a given subspace L∞ (or L ), the set Q(L ) is closed
under intersection, and hence the set Q(L ) has a minimal member as W∞∗ (L ). The minimal
P
conditioned invariant subspace containing a given subspace L∞ =
L (that is, W∞∗ (L )) is
obtained by invoking the following non-decreasing algorithm that is provided below,
W 0 = L,

(26)


W k = L + A1 (W k−1 ∩ ker C) + A2 (W k−1 ∩ ker C) ,
September 20, 2016

DRAFT

22

and W∞∗ =

P

W ∗ , where W ∗ = W k0 , k0 ≤ n.

Note that the above algorithm converges in a finite number of steps. Also, let W be a finite
conditioned invariant subspace. The set of all maps D = [D1 , D2 ] such that W is [A + DC]1,2 invariant is designated by D(W ).
D. Unobservability Subspace
The unobservability subspace [20], [24] is the cornerstone of geometric FDI approach in 1D
systems. The following definition generalizes and extends this concept to the FMII 2D models.
The Fin-D notion of this definition, for the first time in the literature, was introduced and utilized
in [9] for the FDI problem of 2D Roesser systems.
Definition 5. The subspace S∞ is said to be an unobservability subspace for the 2D system (1)
P
if there are three maps D1 , D2 , H such that S =< ker HC|[A + DC]1,2 > and S∞ =
S. We
designate S as the finite unobservability subspace of the 2D system (1).
Note that S∞ is also conditioned invariant subspace and a generic unobservable subspace
of the system (HC, A1 + D1 C, A2 + D2 C). For accomplishing the gaol of the FDI task, one
first computes an unobservability subspace and then obtains the matrix H [20]. Therefore, it
is necessary to compute the unobservability subspace without having any knowledge of H. By
following along the same lines as those used in [9] (Section IV.D), one can show that the limit
∗
of the following algorithm is the smallest unobservability subspace S ∗ (and consequently S∞
)

that contains a given subspace L , that is,

k−1
k−1
Z 0 = Rn and Z k = W ∗ + A−1
∩ A−1
∩ ker C ,
1 Z
2 Z
∗
where W ∗ is the minimal finite conditioned invariant subspace containing L , S∞
=

(27)
P

S ∗ and

S ∗ = Z n.
∗
Finally, we set S∞
=

P

S ∗ , where S ∗ = Z n . Finally, it should be noted that since W ∗ +

ker C = S ∗ + ker C (this follows by applying Z n+1 = Z n = S ∗ in the algorithm (27)), one
obtains D(W ∗ ) ⊆ D(S ∗ ).
In this section, we first defined the invariance property of the 2D system (1) that are Inf-D
subspace of the Inf-D system (5). Next, an invariance unobservable subspace (that is generically

September 20, 2016

DRAFT

23

equivalent to an unobservable subspace) Ns,∞ =

P

Ns was introduced. Moreover, the con-

ditioned and unobservability subspaces that are crucial in determining the solution to the FDI
problem have been introduced. By utilizing the above results necessary and sufficient conditions
for solvability of the FDI problem are subsequently derived and provided.
IV. N ECESSARY AND S UFFICIENT C ONDITIONS FOR S OLVABILITY OF THE FDI PROBLEM
In this section, we first present sufficient conditions for detectability and isolability of faults.
Next, by employing three different filters (namely, ordinary, delayed deadbeat, and LMI-based
filters) sufficient conditions for solvability of the FDI problem are presented.
Consider the faulty FMII model (1) (i.e., the system is subjected to two faults f1 and f2 ) and
the detection filter (8) designed to detect and isolate the fault f1 . By augmenting the detection
filter dynamics (8) with the faulty 2D model (1), one obtains,
xe (i + 1, j + 1) = Ae1 xe (i, j + 1) + Ae2 xe (i + 1, j) + B1e u(i, j + 1) + B2e u(i + 1, j) + L12,e f2 (i, j + 1)
+ L22,e f2 (i + 1, j) + L11,e f1 (i, j + 1) + L21,e f1 (i + 1, j),

(28)

r1 (i, j) = Ce xe (i, j),

h
iT
h
i
e
n
o
T
T
where xe = x ω
∈ X = R ⊕ R (o refers to the dimension of ω), Ce = H1 C M1
and,


A1

0





A2

0





B1



 , Ae2 = 
 , B1e =   ,
Ae1 = 
E1 C F1
E2 C F2
K1
 
h
iT
B2
B2e =   , Lji,e = (Lji )T 0 , i, j ∈ {1, 2}.
K2

(29)

In this section, by assuming that the unobservable subspace of the above augmented system is
Ae1,2 -invariant, it is shown that the sufficient condition is also necessary for solvability of the FDI
problem. Moreover, an analytical comparison between our proposed approach and the method
developed in [36] is also provided to highlight the strength and capabilities of our proposed
methodology when compared to the literature.
A. Main Results
The following lemma provides an important property for the invariant unobservable subspace
e
Ns,∞
(and Nse ) that is associated with the system (28).

September 20, 2016

DRAFT

24
e
Lemma 6. Consider the 2D system (28) and its invariant unobservable subspace Ns,∞
. Then
e
Q−1 Ns,∞
is an unobservability subspace of the 2D system (1), with Q = diag(· · · , Q, Q, · · · )

and Q representing the embedding operator into X e (i.e., Q : Rn → X e and Qx = [xT , 0]T ).
e
Proof: First, recall that Ns,∞
=

P

Nse . Note that, Q−1 Nse = S = {x| [ x0 ] ∈ Nse }, and

assume that [ x0 ] ∈ Nse . According to the fact that Nse is Ae1,2 -invariant [9], we have Ae1 [ x0 ] ∈ Nse ,
and if x ∈ ker C then A1 x ∈ S . It follows that A1 (S ∩ker C) ⊆ S . By following along the same
P
e
lines as those above one can show that A2 (S ∩ker C) ⊆ S . Therefore, S∞ = Q−1 Ns,∞
=
S
is a conditioned invariant subspace. Moreover, given that Nse is contained in ker Ce , we have
S ⊆ ker H1 C. Therefore, the subspace S is a finite conditioned invariant subspace contained
in ker H1 C. Since Nse is the largest Ae1,2 -invariant subspace in ker Ce , it follows that S is the
largest [A + DC]1,2 invariant subspace in ker H1 C (i.e., S is a finite unobservability subspace
of the 2D system (1)). In other words, S is a finite unobservability subspace of the system (1).
This completes the proof of the lemma.
The following theorem provides a single necessary and sufficient condition for detectability
and isolability of faults (i.e., the existence of a subsystem such that it is decoupled from all
faults but one - refer to Subsection II-C for more details).
Theorem 4. Consider the 2D system (1) that is subject to two faults f1 and f2 . The fault f1
is detectable and isolable (in the sense of Remark 4) if and only if the following condition is
satisfied,
(L11 + L12 ) 6⊆ S1∗

(30)

where S1∗ is the smallest finite uobservability subspace (refer to Definition 5) of the 2D system
(1) containing L21 + L22 (this represents the limit of the algorithm (27), where one sets L =
L21 + L22 in the algorithm (26)).
Proof: (If part): Let H1 , D11 and D21 be the corresponding operators to S1∗ (i.e. S1∗ =<

September 20, 2016

DRAFT

25

ker H1 C|A + DC >). Also, consider the following detection filter, governed according to
ω1 (i + 1, j + 1) =F1 ω1 (i, j + 1) + F2 ω1 (i + 1, j)
+ P1 B1 u(i, j + 1) + P1 B2 (i + 1, j),
r1 (i, j) =M1 ω1 (i + 1, j) − H1 y(i, j),

(31)

where P1 : Rn → Rn / S1∗ is the canonical projection of Rn on Rn /S1∗ , Fk = Akp + Do,k M1 ,
k = 1, 2 where Ap1 P1 = P1 (A1 + D1 C) and Ap2 P1 = P1 (A2 + D2 C), Do,k are filter gains and
M1 is the unique solution to M1 P1 = H1 C. Now, by defining e(i, j) = P1 x(i, j) − ω(i, j), one
obtains
e(i + 1, j + 1) = F1 e(i, j + 1) + F2 e(i + 1, j)
+ P1 f1 (i, j) + P1 f1 (i, j),
r1 (i, j) = M1 e(i, j).

(32)

Note that P1 L12 = P1 L22 = 0 and by the condition (30), at least for k = 1 and/or k = 2, we have
P1 Lk1 6= 0. Therefore, the residual signal r1 is decoupled from f2 and is sensitive to f1 . In other
words, f1 is detectable and isolable.
(Only if part): We show the necessary part by contradiction. Let f1 be detectable and isolable,
and (L11 + L12 ) ⊆ S1∗ . Hence, the residual signal r1 is decoupled from f2 and sensitive to f1 .
However, from Lemma 6 one obtains Q(L11 + L12 ) ⊆ N e . Consequently, the fault f1 is not
detectable by using the residual r1 in (32), which is in contradiction with the solvability of the
FDI problem.
As stated in Remark 4, the FDI problem has two main steps. Theorem 4 provides the necessary
and sufficient condition for the first step (that is, detectability and isolability of f1 ). Therefore,
condition (30) is also necessary for solvability of the FDI problem. For the second step, we need
to determine Do,k , and consequently Fk , k = 1, 2 in the filter (31) such that the residual dynamics
(32) is asymptotically stable. In what follows, we provide the conditions that are based on three
different observers (namely, the ordinary, the delayed deadbeat, and the LMI-based filters). We
first need to present the following theorem. However, one first needs the following notation.
Consider the PBH matrix (10). The matrix N (z1 , z2 ) = [N1 (z1 , z2 ), N2 (z1 , z2 )] is the minimal
left annihilator of the PBH matrix if N (z1 , z2 )P BH(z1 , z2 ) = 0, N (z1 , z2 ) is full row-rank, and
September 20, 2016

DRAFT

26

any other left annihilator Nl (z1 , z2 ) can be expressed as Nl = N3 N , where N3 is a polynomial
matrix of z1 and z2 .
Theorem 5. [17] Consider the 2D system (1). The FDI problem is solvable (by using the
approach that is proposed in [17]) if and only if,


1
2
z1 L + z2 L
) = p , ∀z1 , z2 ∈ C − {0}
rank(N (z1 , z2 )] 
(33)
0q×p
h
i
where p is the number of faults, N (z1 , z2 ) = N1 , N2 denotes the minimal left annihilator of
i
h
the PBH matrix (10), and Li = Lk1 , · · · , Lkp for k = 1, 2.
The following corollary provides the sufficient conditions for solvability of the FDI problem
by utilizing our proposed method as well as a (delayed) deadbeat filter.
Corollary 2. Consider the 2D system (1) that is subject to two faults f1 and f2 . Provided
condition (30) is satisfied and condition of Theorem 5 is satisfied for the quotient subsystem
(H1 C,Ap1 ,Ap2 ), where H1 , Ap1 and Ap2 are defined in (31), then the FDI problem is solvable.
Proof: By invoking Theorem 4, the residual signal r1 in (31) is decoupled from all faults but
f1 . Now, given that the condition (33) is satisfied for the subsystem (H1 C,Ap1 ,Ap2 ), by determining
the observer gains Do,1 and Do,2 , one can design a delayed deadbeat observer such that r1 (i −
n1 , j − n2 ) = f1 (i, j) (refer to [17, Theorem 1]). Hence, the FDI problem is solvable. This
complete the proof of the corollary.
The following corollary provides a sufficient condition for solvability of the FDI problem by
utilizing an ordinary (without delay) deadbeat detection observer.
Corollary 3. Consider the 2D system (1) that is subject to two faults f1 and f2 . Let the condition
(30) be satisfied and the PBH matrix of the quotient subsystem (H1 C,Ap1 ,Ap2 ) be right zero prime,
where H1 , Ap1 and Ap2 are defined in (31). Consequently, the FDI problem is solvable by using
a deadbeat (without a delay) observer.
Proof: By invoking Theorem 4, the residual signal in (32) is only affected by f1 . Therefore,
the detection and isolation of f1 is reduced to that of designing Do,1 and Do,2 gains. Since the
PBH matrix of the subsystem (H1 C,Ap1 ,Ap2 ) is right zero prime, by invoking Theorem 1 one
September 20, 2016

DRAFT

27

can determine the gains such that the error dynamics (32) is asymptotically stable and the FDI
problem is solvable. This complete the proof of the corollary.
As shown in Theorem 4, under certain conditions one can obtain a residual signal that is
decoupled from all faults but one. Therefore, design of a residual generator to detect and isolate
the fault f1 in the 2D system (1) is reduced to that of detecting this fault in the system (31) by
using an observer. The Corollaries 2 and 3 provide sufficient conditions for solvability of the FDI
problem by using delayed and ordinary deadbeat filters, respectively. However, as pointed out
in [38], design of an observer for FMII models is based on polynomial matrices. This method is
unfortunately not always numerically or analytically straightforward to develop and therefore, in
this work we also develop another set of sufficient conditions for solvability of the FDI problem
by using a 2D Luenberger observer.
The next corollary provides sufficient conditions for solvability of the FDI problem by using
a 2D Luenberger observer.
Corollary 4. Consider the 2D model (1), where the condition (30) is satisfied. The FDI problem
is solvable if there exist two symmetric positive definite matrices R1 and R2 such that,


 
p T
R1 0
(A )
 −  1  (R1 + R2 )[Ap1 , Ap2 ])Wm < 0,
WmT (
(Ap2 )T
0 R2

(34)

where Ap1 and Ap2 are defined in equation (31) and the columns of Wm are the basis of ker M .
Proof:
By invoking Theorem 4, the fault f1 is detectable and isolable, and consequently the residual
signal r1 in (31) exists (and is decoupled from all faults but f1 ). By considering the LMI condition
(34) and invoking results from Corollary 1, the error dynamics (32) is asymptotically stable. If
f1 (·, ·) = 0, the residual signal r1 (·, ·) converges to zero as i + j → ∞. Otherwise, the residual
has a value that is different from zero. Therefore, the condition (7a) is satisfied and the FDI
problem is solvable. By following along the same procedure as those above one can also design
another state estimation observer to detect and isolate the fault f2 (·, ·). Therefore, this completes
the proof of the corollary.
Remark 7. It is worth noting that one can directly work with Ng (as defined in (16)) and derive
necessary and sufficient conditions by following along the same steps as those that have been
September 20, 2016

DRAFT

28

proposed in [47]. However, there are two main drawbacks associated with this approach that
are as follows:
1) The invariant subspaces are not necessarily computed in a finite number of steps.
2) By factoring out Ng , the resulting subsystem does not necessarily have a Fin-D representation. For more clarification on 2D realization, refer to the work in [48].
To summarize, Theorem 4 provides a single necessary sufficient condition for detectability
and isolability of the faults (refer to Subsection IV) that is also necessary for solvability of the
FDI problem. Three sets of sufficient conditions for solvability of the FDI problem by utilizing
(i) a (delayed) deadbeat observer, (ii) an ordinary deadbeat observer, and (ii) a 2D Luenberger
observer are derived in Corollaries 2, 3 and 4, respectively.
Table I summarizes the main results that are developed and presented in this subsection.
TABLE I: Pseudo-algorithm to detect and isolate the fault fi in the 2D system (1).
1) Compute the minimal conditioned invariant subspace W ∗ containing all Lj1 + Lj2
P
1
subspaces such that j 6= i (by invoking the algorithm (26), where L =
j6=i (Lj +
Lj2 )).
2) Compute the unobservability subspace Si∗ containing

1
j6=i (Lj

P

+ Lj2 ) (by using the

algorithm (27)).
3) Compute the operators D1 , D2 such that W ∗ is the minimal conditioned invariant
subspace of the 2D system (1).
4) Find the operator H such that ker HC = Si∗ + ker C.
5) If Si∗ ∩ Li1 6= 0 or Si∗ ∩ Li2 6= 0, then the fault fi is detectable and isolable (refer
to Remark 4 and Theorem 4) and
•

If the conditions of the Corollary 2 are satisfied, the FDI problem is
solvable by using a (delayed) deadbeat filter.

•

If the conditions of the Corollary 3 are satisfied, the FDI objective can
be accomplished by using a deadbeat (without a delay) observer.

•

If the conditions of Corollary 4 are satisfied, there exist an LMI-based
observer for detection and isolation of the fault fi .

6) The output norm of any of the above detection filters is the residual that
satisfies the condition (7).

September 20, 2016

DRAFT

29

B. Comparisons with the Other Available Approaches in the Literature
In this subsection, our proposed approach is now compared and evaluated with respect to the
existing geometric methods in the literature [36], [37]. We first show that if the FDI problem is
solvable by using the approach in the above literature, our approach can also detect and isolate
the faults. Furthermore, we provide a numerical example where it is shown that our approach
is capable of detecting and isolating a fault, however, the necessary conditions provided in the
algebraic methods in [15], [17] as well as [36] are not satisfied.
The equivalent 2D version of the necessary condition (as derived in [36], Theorems 2 and 3)
to detect and isolate the fault f1 can be summarized as follows: The fault f1 in the 2D system (1)
is detectable and isolable according to [36] if CW1∗ ∩ CW2∗ = 0, where W1∗ and W2∗ denote the
minimal finite conditioned invariant subspace that contain L11 + L12 and L21 + L22 , respectively
(refer to [36], Theorem 2). It should be pointed out that the observability assumption of (C,
A1 , A2 ) is a fundamental requirement and condition in [36] (although it was stated in [36] that
this assumption was made for simplicity of the presentation). The main reason for the above
limitation lies on and is due to the fact that the proposed approach in [36] is based on the results
of [21]. However, as stated in [21] the observability assumption is quite a crucial and critical
condition. For further illustration and clarification for the above serious concerns consider the
following 2D system,




 
 
0 1
0 0
1
0
 x(i, j + 1) + 
 x(i + 1, j) +   f1 +   f2 ,
x(i + 1, j + 1) = 
0 0
0 1
0
1
h
i
y(i, j) = 0 1 x(i, j).

(35)

We have W1∗ = L1 = ker C, W ∗ = L2 , and consequently CW1∗ ∩ CW2∗ = 0. Therefore,
the sufficient condition for solvability of the FDI problem under the zero initial condition in
[36] (Theorem 2) is satisfied. However, it is easy to verify that L1 ⊆ N , and consequently
f1 is not even detectable (in other words, f1 has no effect on the output signal). Our proposed
methodology does not suffer from the above limitation and restriction.
We are now in a position to state the following theorem.
Theorem 6. Consider the 2D system (1) and assume that the FDI problem is solvable by using
the approach that is proposed in [36]. Then the approach that we have proposed in this work
can also detect and isolate the faults in the system (1).
September 20, 2016

DRAFT

30

Proof: According to Theorem 3 in [36], CW1∗ ∩ CW2∗ = 0, and W1∗ and W2∗ are internally/externally stabilizable. Therefore, there exist two maps D1 and D2 such that W1∗ and W2∗ are
both [A+DC]1,2 -invariant, and the pair (A1 +D1 C, A2 +D2 C) is stable (that is, the corresponding
2D system is asymptotically stable). Since CW1∗ ∩ CW2∗ = 0, there exists a map H such that
H1 CW1∗ = CW1∗ and H1 CW2∗ = 0. It follows that Nsh ∩ W1∗ = 0, where Nsh is the invariant
unobservable subspace of (H1 C, A1 + D1 C, A2 + D2 C). Note that Nsh is an unobservability
subspace of (C, A1 , A2 ) containing L21 + L22 , and since S1∗ is the smallest unobservability
subspace containing L21 + L22 , it follows that S1∗ ∩ (L11 + L12 ) = 0. Also, since (A1 + D1 C,
A2 + D2 C) is stable, it can be shown that (Ap1 , Ap2 ) is also stable, where P [A1 + D1 C] = Api P
and P is the canonical projection of S1∗ . Therefore, one can construct an observer for the residual
system (31) to detect and isolate the fault f1 (by choosing Do1 = Do2 = 0). By following along
the same procedure, one can detect and isolate the fault f2 . This completes the proof of the
theorem.
Remark 8. Theorem 6 shows that our proposed approach can detect and isolate faults that
are detectable and isolable by using the geometric method in [36]. However, as shown below
an example is provided where this approach fails whereas our proposed approach can still
detect and isolate the faults. Also, we show that in this example the methods in [15], [17]
are also not applicable.
1) Illustrative Example (Limitations of the Methods in [15], [17] and [36]):
Consider the 2D system (1) that is subjected to two faults f1 and f2 . As stated earlier, in this
work our main concern is on actuator faults and it is assumed that the output signals are not
affected by faults (refer to Remark 1).
two faults f1 and f2 where,

0
 , L21 = L22 = 0, B1 = B2 = 0
I


1 0 0 0

L11 = [0, 0, 0, 1]T , L12 = [0, 0, −1, 1]T , C = 
0 0 0 1

Consider the 2D system (1) that is subjected to



0 0
02×2
0 0.5 0.5I 
A1 = 
, A2 = 0.5 
0 02×2
I

September 20, 2016

(36)

DRAFT

31

The FDI problem is solvable by using the approach in [15] if and only if,


1
1
I − z1 A1 − z2 A2 z1 [L1 , L2 ]
) = n , ∀z1 , z2 ∈ C
rank(
C
0

(37)

The full rankness of the PBH matrix (10) is the necessary condition for the above property [15]
(page 231-item (i)). However, for the 2D system (36), one gets rank(P BH(1, 0)) = 3 < 4.
Therefore, the FDI problem is not solvable by using the method in [15].
To check the conditions of Theorem 5, let us first obtain N (z1 , z2 ). Consider [a, b, c, d, e, f ]
as a row of N (z1 , z2 ). Since N (z1 , z2 )P BH(z1 , z2 ) = 0, it follows that,


 a − 0.5z2 c + e = 0
 0.5z1 b + (1 − 0.5z2 )d + f = 0
,
.


− 0.5z1 a + (1 − 0.5z2 )c = 0
(1 − 0.5z1 )b − 0.5z2 d = 0

(38)

From the first set of equations, one choice for a, c and e is given by a = 2 − z2 , c = z1
and e = 0.5z2 z1 + z2 − 2, respectively. Also, based on the second
set one can write
b = z2 ,


a 0 c 0 e 0
. Therefore,
d = 2−z1 and f = 2−z2 −z1 . It can be shown that N (z1 , z2 ) = 
0 b 0 d 0 f






1
1
1
1
z [L , L ]
0
z1
z1 [L1 , L2 ]
 = z1 
. It is easy to check that the rank(N (z1 , 2)  1 1 2 ) <
N (z1 , z2 ) 
0
0
2 − z1 2 − z1
2 for all z2 ∈ C and z1 = 2. Hence, the necessary condition that is proposed in [17] (as given
by equation (33)) is not satisfied.
Furthermore, the necessary condition to detect and isolate the fault f1 by using the approach in
[36] is CW1∗ ∩ CW2∗ = 0. Since L11 , L21 6∈ ker C, by invoking the algorithm (26), one obtains
W1∗ = L1 and W2∗ = L2 . It follows that, CW1∗ ∩ CW2∗ = span{[0, 1]T } . Therefore, the
necessary condition in [36] is also not satisfied. In other words, the fault f1 cannot be detected
and isolated by using the detection filter (8), if one restricts the filter to the case with M = C
(or H = I), according to the required results in [36].
Finally, it is now shown and demonstrated that one can detect and isolate both faults f1 and
f2 by using our proposed methodology. Towards this end, by invoking the algorithm (27), one
1
can write S1∗ = L21 (that is, the finite unobservability
subspace
containing

T
 L2 ) that satisfies
T
0
0
0 0
0 0 0
0
 and D2 = 
 ,
the condition (30). By considering D1 = 
0.5 −0.5 0 0
0 0 0.5 −0.5
∗
∗
W2 is [A + DC]1,2 -invariant. Also, since ker C + S1 = ker H1 C and M P2 = HC, one gets

H1 = [1, 0] and M1 = [1, 0, 0]. Hence, the residual (32) that is only affected by the fault f1
September 20, 2016

DRAFT

32



0

√

0


is given by Ap1 = 0 0.5
0

Ap2 )

0

2
2

0

0

0

0



 p

, A2 =  0

0

0


p
 and B1 = B2 = 0. Since the pair (A1 ,





√

0

2
2

√

2
2

0.5

is stable, by considering Do1 = Do2 = 0, the detection filter for the fault f1 (as given by

equation (8)) is now obtained according to,
ω(i + 1, j + 1) = Ap1 ω(i, j + 1) + Ap2 ω(i + 1, j),

(39)

r1 (i, j) = M1 ω(i, j) − H1 y(i, j).
By following along the same procedure, one can also design a detection filter to detect and
isolate the fault f2 . Therefore, our proposed approach can accomplish the FDI objectives while
the approaches that are proposed in [15], [17] and [36] cannot achieve this goal.
Remark 9. All the conditions for solvability of the FDI problem in the literature (for both 1D
and 2D systems) and also our proposed conditions are generic, although this fact is not explicitly
mentioned. For clarification, consider the faulty model (1), where x ∈ R2 , A1 = A2 = 0.4 ∗ I,
B11 = L11 = [1, 1]T , B22 = L22 = [0, 1] and C = [1, −1]T . Let the initial condition x(0) = 0
and f1 (i, j) = 1 for all i + j ≥ 0. It follows that y ≡ 0, and consequently f1 is not detectable.
However, below we show that sufficient conditions in the literature [15], [17], [36] as well as
our proposed conditions are still all satisfied.
1) (Conditions
in [17] and [15]).

 It follows that N (z1 , z2 )P HB(z1 , z2 ) = 0, where N (z1 , z2 ) =
−1 0 0.4(z1 + z2 ) − 1

, and consequently the condition in Theorem 5 is also satisfied.
0 1 0.4(z1 + z2 ) − 1
2) (Conditions in [36]). By following the algorithm (26) we obtain W1∗ = L1 and W2∗ = L2 .
It follows that CW1∗ ∩ CW2∗ = 0, and consequently the condition in [36] is also satisfied.
3) (Our proposed Conditions) By following the algorithm (27) we obtain S1∗ = L1 and
S2∗ = L2 . It follows that S1∗ ∩ L2 = 0, and consequently the condition (30) is also
satisfied.
To summarize, in this section we have developed and presented a solution to the FDI problem
of 2D systems by invoking an Inf-D framework for the first time in the literature and by utilizing
invariant subspaces and derived necessary and sufficient conditions for solvability of the problem.
It was shown that if the sufficient conditions for solvability of the FDI problem that are provided
in [36], [37] are satisfied, then our proposed approach can also detect and isolate the faults.
September 20, 2016

DRAFT

33

However, as shown above there are certain systems that the methods in [15], [17], [36] are not
applicable and capable of detecting and isolating faults, whereas our proposed approach can
solve this problem successfully.
V. S IMULATION R ESULTS
In this section, we apply our proposed FDI methodology to a two-line parallel heat exchanger
system [8], [49]. Specifically, we verify the necessary and sufficient conditions that are derived
in the previous section, and also design a set of filters to detect and isolate faults under both
full and partial state measurement scenarios (this is to be realized by an appropriate selection
of the output matrix C).
The heat exchanger is usually subject to two different types of faults, namely the fouling
and the leakage [49]. The mathematical model of a typical heat exchanger is governed by the
following hyperbolic PDEs,
∂Tf
∂Tf
= −αf
− β(Tf − Tg ) − f2 (z, t),
∂t
∂z
(40)
∂Tg
∂Tg
= −αg
− β(Tg − Tf ) + f1 (z, t) + f2 (z, t),
∂t
∂z
where Tf and Tg denote the temperature of the cold (fuel) and the hot (exhaust gas) sections,
respectively. The coefficients αf and αg are proportionally dependent on the speed of the fluid
and the gas, respectively, and the coefficient β is related to the heat transfer coefficient of the
wall [49]. Moreover, f1 and f2 denote the leakage and the fouling effects, respectively. Finally,
it is assumed that the boundary conditions (the inlet temperature) Tf (t, 0) and Tg (t, 0) are given
and only the outer section (i.e. Tg ) is subject to the leakage.
A. The Approximation of Hyperbolic PDE Systems by 2D Models
Let us first illustrate and demonstrate how one can approximate a general hyperbolic PDE
system by using 2D models and representations. Consider the following hyperbolic PDE system
p
X
∂ x̃
∂ x̃
= Ã1
+ Ã2 x̃ + B̃u +
L̃k f˜k ,
∂t
∂z
k=1

(41)

where z denotes the spatial coordinate, x̃(z, t) ∈ Rn , u(z, t) ∈ Rq and f˜k (z, t) ∈ R denote the
state, input and fault signals, respectively. Also, the operators Ã1 , Ã2 , B̃ and L̃k ’s are real matrices

September 20, 2016

DRAFT

34

with appropriate dimensions. Note that every hyperbolic PDE system with constant coefficients
can be represented in the form (41) with a diagonalizable Ã1 [50] (Chapter 1, Detention 1.1.1).
By applying the finite difference method to the system (41), one obtains
x̃(i∆z, (j + 1)∆t) − x̃(i∆z, j∆t)
x̃(i∆z, j∆t) − x̃((i − 1)∆z, j∆t)
=Ã1
+ Ã2 x̃(i∆z, j∆t)
∆t
∆z
p
X
+ B̃ ũ(i∆z, j∆t) +
L̃k f˜k (i∆z, j∆t).

(42)

k=1



x̃((i − 1)∆z, j∆t)
, we can now write
By setting x(i, j) = 
x̃(i∆z, j∆t)
x(i + 1,j + 1) = A1 x(i, j + 1) + A2 x(i + 1, j) + B1 u(i + 1, j) +

p
X

Lk fk (i + 1, j),

(43)

k=1



0 I
 and
where u(i + 1, j) = ũ(i, j) for all i and j, A1 = 
0 0
 

 

0
0
0
0
 , B1 =   , Lk =   , f (i, j) = f˜(i + 1, j).
A2 = 
∆t
∆t
L̃k
− ∆z
B̃
Ã1 (I + ∆z
Ã1 + ∆tÃ2 )
Therefore, the PDE system (41) can be approximated by the FMII 2D model (43).
Remark 10. As shown in [10], the parabolic PDE systems can be approximated by FMII 3D
models. As shown in Appendix A, our proposed FDI methodology for 2D systems can be extended
to 3D systems, therefore the extension of the results of our proposed FDI methodology can also
be applied to parabolic PDE systems.
For the purpose of conducting simulations, the parameters in equation (40) are taken as αf =
αg = β = 1. Also, by considering ∆z = ∆t = 0.1 one can discretize the system (40) as,

02×2
x(i + 1, j + 1) = 
02×2


0
 x(i, j + 1) +  2×2
02×2
I2×2
I2×2



02×2


 x(i + 1, j) + L11 f1 (i + 1, j)

−0.1 0.1
0.1 −0.1

+ B21 u(i + 1, j) + L12 f2 (i + 1, j),

(44)

y(i, j) =Cx(i, j),

where L11 = [0, 0, 0, 1]T , L12 = [0, 0, −1, 1]T and x(i, j) = [Tg ((i−1)∆z, j∆t), Tf ((i−1)∆z, j∆t),
T
Tg (i∆z,
 j∆t), Tf(i∆z, j∆t)] . Finally, by following along the same steps as in [9], one obtains

B2 = 

02×2

1.1 −0.1
−0.1 1.1

September 20, 2016

, u(0, j) = [Tf (0, ∆t), Tg (0, ∆t)]T and u(i, j) = u(0, j) for all i.

DRAFT

35

We first assume that both temperatures Tf and Tg are available for measurement along the
spatial coordinates at discrete points (i.e., Tf (i∆z, j∆t) and Tg (i∆z, j∆t) are available from
sensors). Next, we consider the case where only the outer temperature (that is, Tg ) is available
for measurement. As we shall show subsequently, in the latter case by using the 1D approximate
ODE model (for example, as in [51]), the faults f1 and f2 are not isolable, whereas by using
our proposed 2D FDI methodology one can detect and isolate both faults.
B. FDI of a Heat Exchanger by Using Full State Measurements
Let us assume that both temperatures (namely, T
f and Tg ) are
 available for measurement.
1 0 0 0
.
Therefore, one can select the output matrix as C = 
0 0 0 1
1) Detectability and Isolability Conditions:
By applying the algorithm (26), the minimal finite conditioned invariant subspace containing
L = span{L12 } is obtained as W2∗ = L21 , and by applying the algorithm (27), one obtains
S ∗ = W2∗ = L21 . It follows that L11 ∩ S ∗ = 0. Therefore, the sufficient condition (as given by
Theorem 4) is satisfied. By following along the same lines the necessary and sufficient condition
for detectability and isolability of the fault f2 can also be shown to be satisfied.
2) FDI 2D Luenberger Filter Design:
As stated earlier, we are interested in designing 2D Luenberger detection filters by using the
LMI condition that is proposed in the Subsection II-E. In this part of the paper, we design a filter
for detecting and isolating the fault f1 (without loss of any generality, by following along the
same lines as conducted below one can also design a filter to detect and isolate the fault f2 ). The
2D detection filter must be decoupled from the fault f2 (refer to the conditions in equation (7)).
As stated above, the finite unobservability subspaces containing the subspace L21 is obtained by
using the algorithm (27) and is given by S ∗ = W2∗ = L21 .
The output injection matrices D1 and D2 for W2∗ are to be derived such that W2∗ is [A +
DC]1,2 -invariant. Therefore, one can write (A1 + D1 C)W2c = 0 and (A2 + D2 C)W2c = 0,
where the columns of W2c are the basis of W2∗ ∩ (W2∗ ∩ ker C)⊥ . One solution
to D1 and

 D2 is
T

T

1 0 0 0
0 0 0 0
0 0
0
0

 . Also, let P = 
 and D2 = 
D1 = 
0 1 0 0 (which


1 −1 0 0
0 0 −0.2 0.2
0 0 1 1
∗
is the canonical projector of the subspace S1 ), where P is used in equation (31). By using
September 20, 2016

DRAFT

36

ker H1 C = S1∗ + ker C and M1 P1 = H1 C, we have H = [1, 0, 0] and the output matrix M
becomes M = [1, 0, 0]. Hence, the factored out 2D system is now expressed as,
ω(i + 1, j + 1) =

h 0 0 −1.42 i
00
00

0
0

ω(i, j + 1) +

h

0
0 0
0
0 0
−0.7 −0.7 0

i

ω(i + 1, j) +

h

i

0 0
0 0
0.7 0.7

u(i + 1, j) +

h i
0
1
0

f1 (i + 1, j),

yp (i, j) = [1, 0, 0]ω(i, j),

where ω(i, j) = P1 x(i, j). It is straightforward to show that the positive definite matrices R1 =
diag(0.4, 1, 2.133) and R2 = diag(0.4, 2.15, 0.86) satisfy the inequality WcTd AC Wcd < 0. By
h
iT
using Remark 5, one can obtain Do1 = 0 and Do2 = 0 0 −0.7 . Therefore, the filter to
detect and isolate the fault f1 is given by,
h 0 0 −1.42 i
h 0 0 0i
h
0
0 0 ω(i + 1, j) +
ω(i + 1, j + 1) = 0 0 0
ω(i, j + 1) + −0.7
−0.7 0
0 0

0

0 0
0 0
0.7 0.7

i

u(i + 1, j)

+ Do2 P2 y(i, j + 1),
r1 (i, j) = [1, 0, 0]ω(i, j) − Hy(i, j).
Designing a filter to detect and isolate the fault f2 follows along the same lines as those given
above for the fault f1 . These details are not included for brevity.
3) Threshold Computation:
Due to presence of input and output noise, disturbances, and uncertainties in the model, the
value of the residual rk (i, j) is not exactly equal to zero under the fault free situation. Therefore,
to reduce the number of false alarm flags one needs to apply threshold bands to the residual
signals. In this subsection, we present an approach for determining the thresholds that are needed
for achieving the FDI task. For 1D systems, there are a number of approaches for computing
a threshold, e.g. based on the maximum or the root mean square (RMS) of the residual signal
[52]. In this work, we use the maximum residual norm. However, one can also apply the RMS
approach to 2D systems.
Consider the FMII 2D model (1) subject to the fault free situation. The threshold thk is then
determined from,
thk = Max |rk` (i, j)| , for i, j ≤ N1
`<N0

(45)

where | · | denotes a norm function (in this work we use the norm-2), N0 is the number of the
Monte Carlo simulations (refer to [53]) that are used to determine the threshold, rk` (·, ·) denotes
the signal of k th residual in the `th length and N1 is a sufficiently large number.

September 20, 2016

DRAFT

37

(a) The residual signal r1 for detecting and

(b) The residual signal r2 for detecting and

isolating the fault f1 .

isolating the fault f2 .

Fig. 1: The residual signals for detecting and isolating the fault f1 for the first scenario (a single
fault case) using full state measurement.

By utilizing predefined thresholds that are denoted by thk , k = 1, 2, the FDI logic can be
summarized as follows,
if r1 > th1 ⇒ the fault f1 has occurred.
(46)
if r2 > th2 ⇒ the fault f2 has occurred.
Let us now consider two scenarios. In the first scenario, a single fault f1 with the severity of 1
occurs at (i = 5 and j ≥ 60). In the second scenario, multiple faults f1 and f2 with the severity
of 1 occur at (i = 5 and j ≥ 50) and (i = 5 and j ≥ 70), respectively. The residuals r1 and r2 for
the first scenario are shown in Figure 1. For the second scenario, the results are shown in Figure
2. The thresholds are determined by conducting Monte Carlo simulations [53] corresponding to
the healthy 2D system. As shown in Figures 1 and 2, our proposed methodology can detect and
isolate the faults in both single- and multiple-fault scenarios (according to the FDI logic that is
given by equation (46)).
C. FDI of a Heat Exchanger Using Partial State Measurements
In this section, we assume that only the outer temperature (Tg ) is available for measurement.
This corresponds to a more practical and physically feasible scenario in various applications
(sensing the inner temperature requires a more sophisticated hardware). In this case, we set

September 20, 2016

DRAFT

38

(a) The residual signal r1 for detecting and isolating (b) The residual signal r2 for detecting and isolating
the fault f1 .

the fault f2 .

Fig. 2: The residual signals for detecting and isolating the faults f1 and f2 for the second scenario
(a multiple fault case) using full state measurement.


0 1 0
C =
0 0 0
we demonstrate


0
 (in other words, we measure Tg (i − 1, j) and Tg (i, j)). In this subsection,
1
and illustrate the capabilities of our proposed FDI approach based on the 2D

system modeling, whereas it is shown that by using a 1D approximation of the PDE system (40)
the FDI problem cannot be solved.
1) 1D Approximation of the Heat Exchanger System:
The hyperbolic PDE system (40) can be approximated by applying discretization through the z
coordinates as follows. Let ` denote the length of the heat exchanger that is discretized into N
equal intervals (i.e., ∆z =

`
).
N

By using the approximation

∂Tg (k∆z,t)
∂z

=

Tg (k∆z,t)−Tg ((k−1)∆z,t)
,
∆z

one can represent the PDE system (40) by the following 1D approximate model,
ẋ = Ax + Bu +

N
X

(Lk1 f1k + Lk2 f2k ),

k=1

(47)

y = Cx,

T
iT
1, 0, 0, 0, · · · , 0
 ,
where x(t) = Tg (∆z, t), Tf (∆z, t), · · · , Tg (N ∆z, t), Tf (N ∆z, t) ∈ R2N , B = 
0, 1, 0, 0, · · · , 0
and fik (t) = f1 (k∆z, t) (i = 1, 2). Also, Lk1 = [0, · · · , 0, −1, 1, 0, · · · ]T , and the fault signatures
| {z }
h

(k−1)

Lk2 are 2N -dimensional vectors such that only the k th element is 1 and the rest are zeros.

September 20, 2016

DRAFT

39

Moreover,

A1


A2
A=

0

0

"
in which A1 =

0
A1

0

A2

A1
..
.

0

1

1

− 1+∆z
∆z



1




 , C = 0




0
..
.

···
···
..
.

#

− 1+∆z
∆z



···

0

"
and A2 =

1
− ∆z

0

0

1
− ∆z



0

0

0

···

0

1

0

0

0

0



· · ·

..
.

(48)

#
. Now, consider the faults f1k and f2k .

It can be shown that W2∗ = span{Lk1 , Lk2 } , where W2∗ is the minimal conditioned invariant
∗
∗
subspace containing L2k (from the 1D system perspective). Consequently, S1D
∩ L1k 6= 0 (S1D

denotes the unobservability subspace containing L2k in the 1D system sense). Consequently, the
faults f1k and f2k are not isolable. However, we show below that the faults can be detected and
isolated if one approximates the system (40) by using the 2D model representation.
2) 2D Representation of the Heat Exchanger:
Let us set x1 (i, j) = Tg (i, j) +

∆t
f (i, j)
∆z 1

and x2 (i, j) = Tf (i, j), so that the system (40)

is approximated by the system
" (44) where
# all the operators are defined as before except for
h
i
0 1 0 0
L11 = 0 1 0 0 and C =
. Note that since only the state Tg is assumed to be
0

0

0

1

available for measurement, we sense Tg (i − 1, j) and Tg (i, j). By applying the algorithm (27),
h
iT h
iT
where L = L21 , one gets S ∗ = span{L12 , 1 0 0 0 , 0 0 1 1 } . Since, L11 ∩S ∗ = 0,
the fault f1 is both detectable and isolable (according to Theorem 4). It should be noted that
by applying the approaches in [15], [17], [36], one can also detect and isolate the fault f1 . By
following along the same lines as the ones given earlier one can show that the fault f2 is also
detectable and isolable, where one can design the required detection filters. Figures 3 and 4 depict
the simulation results for the scenarios that presented above. As can be observed from Figures
2 and 4 (particularly, Figures 2b and 4b), it follows that fewer available information (recall that
both Tf and Tg are measurable in Figure 2, whereas in Figure 4, only Tg is measurable) results
in a situation where the spatial coordinate of a fault cannot be estimated accurately.
VI. C ONCLUSIONS
The fault detection and isolation (FDI) problem for discrete-time two-dimensional (2D) systems represented by the Fornasini-Marchesini model II (where other 2D models are subclasses
of this representation) is investigated in this work. In order to derive the necessary and sufficient
September 20, 2016

DRAFT

40

(a) The residual signal r1 for detecting and

(b) The residual signal r2 for detecting and

isolating the fault f1 .

isolating the fault f2 .

Fig. 3: The residual signals for detecting and isolating the fault f1 for the first scenario (a single
fault case) using partial state measurement.

(a) The residual signal r1 for detecting and isolating (b) The residual signal r2 for detecting and isolating
the fault f1 .

the fault f2 .

Fig. 4: The residual signals for detecting and isolating the faults f1 and f2 for the second scenario
(a multiple faults) using partial state measurement.

conditions for solvability of the FDI problem, the notion of the conditioned invariant and
unobservability subspace of 1D systems was generalized to 2D systems by using an infinite
dimensional (Inf-D) framework and representation. Moreover, algorithms for computing and
constructing these subspaces are introduced and provided that converge in a finite and known
number of steps. By applying the linear matrix inequality (LMI) approach, sufficient conditions
for existence of an asymptotically convergent 2D state estimation observer is derived. Necessary

September 20, 2016

DRAFT

41

and Sufficient conditions for solvability of the FDI problem are also provided. Furthermore,
another set of sufficient conditions (that are based on deadbeat observers) are also provided.
It was shown that although the sufficient conditions for applicability of the currently available
geometric results in the literature are also sufficient for our proposed approach to accomplish
the FDI goal, however, there are 2D systems where the geometric approaches in the literature
are not applicable to detect and isolate the faults, whereas our approach can still achieve the FDI
objective and task. Finally, simulation results are provided for the application of our proposed
FDI methodology to an heat exchanger system to demonstrate and illustrate the capabilities and
advantages of our proposed solution as compared to the alternative 1D representation and 1D
FDI approaches.
A PPENDIX
The results that are derived in this paper are dependent on invariant concepts that are provided
in previous sections. Since these invariant subspaces are independent of the order of A1 and
A2, there is a natural extension that is possible to FMII 3D models. In this appendix, we briefly
show how our previous results can be extended to 3D systems. Furthermore, it should be pointed
out that one can also extend these results to nD systems
Consider the following 3D FMII model
x(i + 1, j + 1, k + 1) = A1 x(i, j + 1, k + 1) + A2 x(i + 1, j, k + 1)
+ A3 x(i + 1, j + 1, k) + B1 u(i, j + 1, k + 1)
(49)
+ B2 u(i + 1, j, k + 1) + B3 u(i + 1, j + 1, k),
y(i, j, k) = Cx(i, j, k).

The corresponding Inf-D system is given by
x(k + 1) = Ax(k),

k∈N

(50)

y(k) = Cx(k),
where x(k) ∈ X =

P

Rn , y(k) = (· · · , y(−1 + k, 1)T , y(k, 0)T , y(1 + k, −1)T , · · · )T ∈

P

Rq ,

and A is an Inf-D matrix with A1 , A2 and A3 as diagonal and upper diagonal blocks, respectively,

September 20, 2016

DRAFT

42

with the remaining elements

..
.


· · · 0
A=

· · · 0

···

set to zero, and C = diag(· · · , C, C, · · · ). In other words, we have,



..
..
. ···
.
· · ·
···
· · ·




· · · 0 C 0 · · ·
A1 A2 A3 0 · · ·
,C = 

(51)



· · · 0 0 C · · ·
0 A1 A2 A3 · · ·



... ...
...
···
···
···

The generalization of the Definition 3 can be stated as follows.
Definition 6. The subspace V is called A1,2,3 -invariant if Ai V ⊆ V for all i = 1, 2, 3.
Also, it can be shown that the finite unobservable subspace of the 3D system (49) is given
T
by N = ni+j+k=0 ker CA(i,j,k) ,
where
A(i,j,k) = A1 A(i−1,j,k) + A2 A(i,j−1,k) + A3 A(i,j,k−1) ,
A(0,0,0) = I A(i,j,k) = 0 if i or j or k < 0.
In view of the above, one can now extend the invariant unobservable subspace Ns to Ns =
T
α
||α||>0 ker CA , where, α is a multi-index parameter, and |α| and ||α|| are defined according
to the same manner that were defined for 2D systems (refer to the notation in the Introduction
section). For instance, if α = (3, 1, 3, 2, 1, 1, 1, 3, 3, 3), then |α| = (4, 1, 5) and ||α|| = 10. Also,
P
we have Ns,∞ = Ns .
By following along the same lines as in Theorem 3 and Lemma 5, we have the following
corollaries.
Corollary 5. Consider the 3D system (49) and the Inf-D system (50). Let V∞ =

P

V , where

V ⊆ Rn . The subspace V∞ is A-invariant if and only if V is A1,2,3 -invariant.
Corollary 6. Consider the 3D system (49) and the Inf-D system (50). The following statements
are equivalents.
(i) The subspace W∞ is conditioned invariant.
(ii) A1 (W ∩ ker C) + A2 (W ∩ ker C) + A3 (W ∩ ker C) ⊆ W .
(iii) A(W∞ ∩ ker C) ⊆ W∞ .
P
where W∞ =
W.

September 20, 2016

DRAFT

43

By following along the same steps as above, the invariant subspace and the results provided
in Sections III and IV can be extended to nD systems. Therefore, one can generalize the results
of this paper to nD systems without major barriers and challenges.
R EFERENCES
[1] R. Isermann, Fault-diagnosis systems: an introduction from fault detection to fault tolerance. Springer, 2006.
[2] P. D. Christofides, Nonlinear and robust control of PDE systems: Methods and applications to transport-reaction processes.
Springer, 2001.
[3] A. Baniamerian and K. Khorasani, “Fault detection and isolation of dissipative parabolic PDEs: Finite-dimensional
geometric approach,” in 2012 American Control Conference, pp. 5894–5899, 2012.
[4] N. H. El-Farra and S. Ghantasala, “Actuator fault isolation and reconfiguration in transport-reaction processes,” AIChE
Journal, vol. 53, no. 6, pp. 1518–1537, 2007.
[5] P. Christofides and P. Daoutidis, “Robust control of hyperbolic PDE systems,” Chemical engineering science, vol. 53,
no. 1, pp. 85–105, 1998.
[6] R. F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, vol. 21. Texts in Applied
Mathematics, Springer-Verlag, Springer, New York, 1995.
[7] M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems
with actuator and sensor delays,” Systems & Control Letters, vol. 57, no. 9, pp. 750–758, 2008.
[8] W. Marszalek, “Two-dimensional state-space discrete models for hyperbolic partial differential equations,” Applied
Mathematical Modeling, vol. 8, pp. 11–14, 1984.
[9] A. Baniamerian, N. Meskin, and K. Khorasani, “Geometric fault detection and isolation of two-dimenional systems,” in
2013 American Control Conference, pp. 3547–3554, 2013.
[10] P. Stavroulakis and S. Tzafestas, “State reconstruction in low-sensitivity design of 3-dimensional systems,” IEE Proceedings
of Control Theory and Applications, Part D, vol. 130, no. 6, pp. 333–340, 1983.
[11] T. Kaczorek, Two-dimensional linear systems. Springer-Verlag, 1985.
[12] G. W. Pulford, “The two-dimensional power spectral density: A connection between 2-D rational functions and linear
systems,” IEEE Transactions on Automatic Control, vol. 56, no. 7, pp. 1729–1734, 2011.
[13] S. Knorn and R. H. Middleton, “Stability of two-dimensional linear systems with singularities on the stability boundary
using lmis,” IEEE Transaction on Automatic Control, vol. 58, no. 10, pp. 2579–2590, 2013.
[14] E. Fornasini and G. Marchesini, “Properties of pairs of matrices and state-models for two-dimensional systems. part1:
State dynamics and geometry of the pairs,” Multivariate analysis: Future directions, vol. 5, pp. 131–180, 1993.
[15] M. Bisiacco and M. E. Valcher, “Observer-based fault detection and isolation for 2D state-space models,” Multidimensional
Systems and Signal Processing, vol. 17, pp. 219–242, 2006.
[16] E. Fornasini and G. Marchesini, “Residual generators for detecting failure in 2D systems,” in Integrating Research, Industry
and Education in Energy and Communication Engineering, MELECON’89, pp. 69–72, 1989.
[17] M. Bisiacco and M. E. Valcher, “The general fault detection and isolation problem for 2D state-space models,” Systems
& control letters, vol. 55, no. 11, pp. 894–899, 2006.
[18] E. Rogers, K. Galkowski, and D. H. Owens, Control systems theory and applications for linear repetitive processes,
vol. 349. Lecture Notes in Control and Information Sciences, Springer, 2007.

September 20, 2016

DRAFT

44

[19] D. Meng, Y. Jia, J. Du, and S. Yuan, “Robust discrete-time iterative learning control for nonlinear systems with varying
initial state shifts,” IEEE Transactions on Automatic Control, vol. 54, no. 11, pp. 2626–2631, 2009.
[20] M. A. Massoumnia, A geometric approach to failure detection and identification in linear systems. PhD thesis, MIT, Dep.
Aero. & Astro., 1986.
[21] M. A. Massoumnia, “A geometric approach to the synthesis of failure detection filters,” IEEE Transactions on Automatic
Control, vol. 31, no. 9, pp. 839–846, 1986.
[22] M. A. Massoumnia, G. C. Verghese, and A. Willsky, “Failure detection and identification,” IEEE Transactions on Automatic
Control, vol. 34, pp. 316–321, March 1989.
[23] N. Meskin and K. Khorasani, “A geometric approach to fault detection and isolation of continuous-time Markovian jump
linear systems,” IEEE Transactions on Automatic Control, vol. 55, no. 6, pp. 1343–1357, 2010.
[24] N. Meskin and K. Khorasani, “Fault detection and isolation of distributed time-delay systems,” IEEE Trans. Automatic
Control, vol. 54, pp. 2680–2685, 2009.
[25] N. Meskin and K. Khorasani, “Fault detection and isolation of linear impulsive systems,” IEEE Transactions on Automatic
Control, vol. 56, no. 8, pp. 1905–1910, 2011.
[26] C. De Persis and A. Isidori, “A geometric approach to nonlinear fault detection and isolation,” Transactions on Automatic
Control, vol. 46, no. 6, pp. 853–865, 2001.
[27] N. Meskin, K. Khorasani, and C. A. Rabbath, “A hybrid fault detection and isolation strategy for a network of unmanned
vehicles in presence of large environmental disturbances,” IEEE Transactions on Control Systems Technology, vol. 18,
no. 6, pp. 1422–1429, 2010.
[28] N. Meskin, K. Khorasani, and C. A. Rabbath, “Hybrid fault detection and isolation strategy for non-linear systems in the
presence of large environmental disturbances,” IET control theory & applications, vol. 4, no. 12, pp. 2879–2895, 2010.
[29] L. Ntogramatzidis and M. Cantoni, “Detectability subspaces and observer synthesis for two-dimensional systems,”
Multidimensional Systems and Signal Processing, pp. 1–18, 2012.
[30] L. Ntogramatzidis, “Structural invariants of two-dimensional systems,” SIAM Journal on Control and Optimization, vol. 50,
no. 1, pp. 334–356, 2012.
[31] T. Vasilache and V. Prepelita, “Observability and geometric approach of 2D hybrid systems,” in The International
Conference on Systems, Control, Signal Processing and Informatics, Informatics, (Greece), pp. 207–214, July, 2013.
[32] T. Ooba, “On stability analysis of 2-D systems based on 2-D lyapunov matrix inequalities,” IEEE Transactions on Circuits
and Systems I: Fundamental Theory and Applications, vol. 47, no. 8, pp. 1263–1265, 2000.
[33] E. Rogers, K. Galkowski, A. Gramacki, J. Gramacki, and D. Owens, “Stability and controllability of a class of 2-D linear
systems with dynamic boundary conditions,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and
Applications, vol. 49, no. 2, pp. 181–195, 2002.
[34] Y. Zou, M. Sheng, N. Zhong, and S. Xu, “A generalized Kalman filter for 2D discrete systems,” Circuits, Systems and
Signal Processing, vol. 23, no. 5, pp. 351–364, 2004.
[35] A. Baniamerian, N. Meskin, and K. Khorasani, “Fault detection and isolation of fornasini-marchesini 2D systems: A
geometric approach,” in 2014 American Control Conference, pp. 5527–5533, 2014.
[36] S. Maleki, P. Rapisarda, L. Ntogramatzidis, and E. Rogers, “Failure identification for 3D linear systems,” Multidimensional
Systems and Signal Processing, vol. 26, no. 2, pp. 481–502, 2015.
[37] S. Maleki, P. Rapisarda, L. Ntogramatzidis, and E. Rogers, “A geometric approach to 3D fault identification,” in Proceedings
of the 8th International Workshop on Multidimensional Systems, pp. 1–6, 2013.

September 20, 2016

DRAFT

45

[38] M. Bisiacco, “On the state reconstruction of 2D systems,” Syst. Contr. Lett, vol. 5, pp. 347–353, 1985.
[39] R. P. Roesser, “A discrete state-space model for linear image processing,” IEEE Transaction on Automatic Control, vol. AC20, pp. 1–10, 1975.
[40] E. Fornasini and G. Marchesini, “Stability analysis of 2-D systems,” IEEE Transactions on Circuits and Systems, vol. 27,
no. 12, pp. 1210–1217, 1980.
[41] Geometric Theory for Infinite Dimensional Systems, vol. 115. Springer, 1989.
[42] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H∞ control,” International Journal of Robust and
Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994.
[43] R. F. Curtain, “Invariance concepts in infinite dimensions,” SIAM Journal on Control and Optimization, vol. 24, no. 5,
pp. 1009–1030, 1986.
[44] W. M. Wonham, Linear multivariable control: a geometric approach. Springer-Verlag, second ed., 1985.
[45] G. Conte and A. Perdon, “On the geometry of 2D systems,” in IEEE International Symposium on Circuits and Systems,
pp. 97–100, 1988.
[46] G. Conte and A. Perdon, “A geometric approach to the theory of 2-D systems,” IEEE Transactions on Automatic Control,
vol. 33, pp. 946–950, 1988.
[47] A. Baniamerian, N. Meskin, and K. Khorasani, “Fault detection and isolation of riesz spectral systems: A geometric
approach,” in 2014 European Control Conference, pp. 2145–2152, June 2014.
[48] E. Fornasini and G. Marchesini, “On the problems of constructing minimal realizations for two-dimensional filters,” IEEE
Transactions on Pattern Analysis and Machine Intelligence, no. 2, pp. 172–176, 1980.
[49] G. Szederkenyi, “Simultaneous fault detection of heat exhangers,” Master’s thesis, University of Veszprem, 1998.
[50] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Second Edition. Society for Industrial and
Applied Mathematics, 2th ed., 2004.
[51] A. Maidi, M. Diaf, and J.-P. Corriou, “Boundary geometric control of a counter-current heat exchanger,” Journal of Process
Control, vol. 19, no. 2, pp. 297–313, 2009.
[52] S. Ding, P. Zhang, P. Frank, and E. L. Ding, “Threshold calculation using LMI-technique and its integration in the design
of fault detection systems,” in Proceedings of the IEEE Decision and Control Conference, pp. 469–474, 2003.
[53] R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo method, vol. 707. John Wiley & Sons, 2011.

September 20, 2016

DRAFT

