Sensor fault detection and isolation problems subject to
H
_/
H
_{∞}
, performance are concerned for linear timeinvariant systems with time delay in a state and parametric uncertainties. To that end, a modelbased observer bank approach is pursued. The design conditions for both continuous and discretetime cases are formulated in terms of matrix inequalities, which are then converted to the problems solvable via an algorithm involving convex optimization.
1. Introduction
Fault detection and isolation (FDI) for dynamic systems holds a major field in the modern control theory academia, due to the ever increasing demand from industry for higher safety and reliability standards
[1

3]
. A great deal of research works on FDI has been released from different points of view, among which a successful one is a modelbased observer approach using effective design tools that have been accumulated in the field over decades. See
[2]
and good references therein.
When dealing with singleoutput systems, isolating a fault on a sensor is identical to detecting one. On the other hand, it is more complicated in multioutput systems, because constructing a noninteractive map (diagonal transfer matrix for the linear timeinvariant (LTI) case) from faults to residuals is particularly challenging, even if the system is LTI
[4]
. A simple yet effective alternative could be a scheme via an observer bank, where as many observers as sensors are involved
[5]
. The residual generated by each observer in the bank should be as sensitive to the faults on all sensors except each one and as robust against disturbance, as possible, in a reasonable (for example, an
H
_ and an
H
_{∞}
[6

10]
) sense. Then the fault can be isolated based on a suitable voting scheme
[5
,
11]
. However, time delay and uncertainty which are widely believed to exist in physical model configurations
[12]
could be as the major obstacles against satisfactory FDI.
Paper
[5]
introduces an application of the observer bankbased FDI method to loadfrequency control problem without deliberating time delay, uncertainty, nor any performance index. In
[11]
, a robust FDI for a robot manipulator is developed, where the residual sensitivity to the fault is considered in the
H
_{∞}
sense. A solution to FDI is presented in the frame of
H
_{∞}
modelmatching in
[13]
. However, these referred works do not consider the time delay. Besides, although a fault detector, rather than an isolator, is designed with delays and/or uncertainties in
[1
,
14

16]
, available literatures are relatively few for the FDI of LTI systems with the time delay and the uncertainty in the
H
_/
H
_{∞}
criterion. This motivates our present study.
In this paper, we concern design conditions of the
H
_/
H
_{∞}
sensor FDI observer banks for continuous and discretetime LTI systems subject to the time delay in the state and the parametric uncertainty. The bank is composed of the sensors’ number of observers. Both the observer gain and the residual gain (that is not necessarily symmetric or triangular due to additional manipulation such as the matrix square root or a Cholesky factorization) are taken into account as the design variables. Sufficient conditions to find the gains are developed in terms of matrix inequality. An algorithm involving a convex optimization is presented based on the cone complementary linearization technique
[12]
.
We follow standard notations:
A
=
A^{T}
≺ 0 is a negative definite matrix. ‖
x
‖ stands for a Euclidean norm while ‖
x
‖
_{ℒ2}
means the
ℒ
_{2}
norm. Symbol * denotes a transposed element in a symmetric position. Ellipsis He{
S
} :=
S
+
S^{T}
is used for simplicity. For any vector
y
∈ ℝ
^{m}
and matrix C ∈ ℝ
^{m×n}
, we define as follows:
2. Preliminaries
Consider the following uncertain timedelay LTI system:
where
x
∈ ℝ
^{n}
is the state;
y
∈ ℝ
^{m}
is the sensor output;
d
∈ [0,
d_{u }
],
d_{u}
∈ ℝ
_{>0}
, is the known timevarying delay; and
w
∈ ℝ
^{l}
and
f
∈ ℝ
^{m}
belonging
ℒ
_{2}
to are the disturbance and the sensor fault, respectively. Matrices Δ
A
, Δ
B
, and Δ
C
represent the parametric uncertainties that satisfy the following assumption:
Assumption 1
: Δ
A
, Δ
B
, and Δ
C
are the real valued matrix functions fulfilling
where ■ means no restriction on that entry,
F
_{1}
,
F
_{2}
,
E
_{1}
, and
E
_{2}
are known constant real matrices of appropriate dimensions and 𝝨 is timevarying real matrix satisfying 𝝨
^{T}
𝝨≼
I
Without loss of generality, assume the observability of (
A
,
C
). We introduce an
m
observer bank in which the
p
th member should detect all faults on all sensors excluding the
p
th one. Such a requirement is realized if the
p
th observer takes the following form
where
p
∈ ℐ
_{M}
:= {1, 2,…,
m
},
∈ ℝ
^{n}
is the estimated state;
ŷ
∈ ℝ
^{m}
is the observer output;
r
∈ ℝ
^{m−1}
is the residual; and
L
and
H
are the observer and the residual gains to be designed, respectively.
Let
e
:=
x
−
and
. The residual is then generated by the following augmented error dynamics
where
and
For excellent FDI, (2) is desired to be designed so that the effect of
to
r
is encouraged and the effect of
w
to
r
is attenuated. In connection with this, we recall the following performance measures:
Definition 1
: For a map from
to
r
in (3), the
H
_ performance is defined by
[9]
For a map from
w
to
r
, the
H
_{∞}
performance is defined by
Definition 2
: Let the evaluation function be
and the threshold function be
where
T_{W}
∈ ℝ
_{>0}
is the constant time window. In cases where it is necessary to display explicitly the association of
J
_{r}
and
J
_{th}
with the residual from the
p
th observer, we write
J
_{rp}
and
J
_{th}
_{p}
, respectively. Define an FDI logic as
Table 1
.
FDI logic
Problem 1
: Given the fault sensitivity level
β
∈ ℝ
_{>0}
and the disturbance attenuation level 𝛾 ∈ ℝ
_{>0}
, find
L
and
H
in (2) so as to satisfy

(C1) (2) is asymptotically stable whenf= 0 andw= 0;

(C2)_>βwhenw= 0 with the initial conditionx(0) =(0) = 0;

(C3)∞<γwhenf= 0 withx(0) =(0) = 0.
3. Main Results
We recall the following lemmas, before proceeding further.
Lemma 1
: Given any matrices
X
=
X^{T}
≻ 0 , and
M
and vectors
a
(·) and
b
(·) defined on an interval 𝛀 , the following inequality holds
[17]
:
Lemma 2
: For any compatible matrices
S
=
S^{T}
≺ 0,
F
, and
E
, the following equivalence holds:
for some
ε
∈ ℝ
_{>0}
, where 𝝨
^{T}
𝝨 ≼
I
.
Lemma 3
: For any compatible matrices
Q
,
R
,
, and
Ȓ
, the following inequality holds:
Proof:
It is readily proved as follows:
Now, we summarize the main result on Problem 1.
Theorem 1
((β, 𝛾)H_H_{∞} FDI):
Given
β
, 𝛾 ∈ ℝ
_{>0}
(2) has
(β, 𝛾)H_H_{∞}
FDI performance and is asymptotically stable, if there exist ,
P
=
P^{T}
:= blockdiag {
P
_{1}
,
P
_{2}
} ≻ 0,
Q
=
Q^{T}
≻0,
X
=
X^{T}
≻ 0,
Z
=
Z^{T}
≻ 0,
Ĥ
=
Ĥ^{T}
≻ 0,
,
W
, and
Y
, such that
where
Then the gains are given by (
L
,
H
) = (
P
_{2}
^{−1}
N
,
Ĥ
^{−1}
W
).
Proof:
Denote
and
. Define the following positive definite function
V
(
z
(
t
−
α
),
α
∈ [0,
d_{u}
]) :=
V
_{1}
+
V
_{2}
+
V
_{3}
where
Using the basic calculus fact
(3) is rewritten as
Letting
and
b
(
α
) :=
Pz
(
α
) , and utilizing Lemma 1, we further compute
Replacing
Y
=
X M P
and assuming
yield
From (9), if
= 0, it holds
Integrating the Hamilton—Jacobi—Bellman (H—J—B) inequality
from 0 to ∞, the following relation holds
Indeed (10) is necessity for (6) because for all (
z
,
z
(
t
−
d
),
w
) ∈ ℝ
^{2n}
＼ {0} × ℝ
^{2n}
＼ {0} × ℝ
^{l}
where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (2) ((C1)). The proof of (7) ⇒ (C2) is along a similar line to the proof of (6) ⇒ (C3). Next, letting
w
= 0 in (9) allows one to manipulate
Considering the following H—J—B inequality
it is easy to see (12) is a sufficiency for
Using (11), we know for all (
z
,
z
(
t
−
d
),
)∈ ℝ
^{2n}
＼{0}×ℝ
^{2n}
＼{0}×ℝ
^{m−1}
, that
where we have used Schur complement, the congruence transformation, and Lemmas 2 and 3. ■
Algorithm 1
Iterative algorithm
Remark 1:
The matrix inequalities in Theorem 1 are not linear due to the terms
ĤĤ
and −
PZ
^{−1}
P
. It provokes a nonconvex feasibility problem that is generally difficult to solve. One may simply attempt to find a feasible solution set through a tractable convex optimization algorithm after recovering their convexity. For instance, we could set
Ĥ
=
I and
P
=
Z
, which however, may produce a quite conservative design result.
In what follows, we alternatively solve the problem to obtain a better result based on the cone complementary linearization technique
[12]
. Introduce
V
and
S
such that
ĤĤ
≽
V
and
PZ
^{−1}
P
≽
S
or
Let (6)
'
and (7)
'
be (6) and (7) with the respective replacements of
ĤĤ
and −
PZ
^{−1}
P
by
V
and
S
. Then we know that (6)
'
,(7)
'
,(8)⇒(6)−(8). Defining new variables (
M, K, U, J, R
) := (
V
^{−1}
,
Ĥ
^{−1}
,
S
^{−1}
,
P
^{−1}
,
Z
^{−1}
), (13) is represented as
Then, the problem concerned above is converted to the following nonlinear minimization one involving LMIs:
Algorithm 1 summarizes an iterative LMI approach to MP 1.
4. Paralleling to DiscreteTime Case
In this section, we discuss the FDI problem in the discretetime domain. Consider the following discretetime uncertain timedelay LTI system:
where
w_{k}
,
f_{k}
∈
l
_{2}
, and
d_{k}
∈ [0,
d_{u}
] ⊂
The
m
 observer bank whose
p
th observer is in the form of
is desired to be designed to possess the (
β
, 𝛾 )
H
_/
H
_{∞}
performance in the
l
_{2}
norm sense. The augmented error dynamics is written as
where
and
.
Lemma 4:
For any compatible matrices
X
=
X^{T}
≻ 0,
Y
,
N
,
Z
=
Z^{T}
≻ 0, following inequality is satisfied
[12]
:
where
Theorem 2:
Given
β
, 𝛾 ∈ ℝ
_{>0}
, (17) has (
β
, 𝛾)
H
_/
H
_{∞}
FDI performance and is asymptotically stable, if there exist
P
:=blockdiag{
P
_{1}
,
P
_{2}
} =
P^{T}
≻ 0,
Q
=
Q^{T}
≻ 0,
X
=
X^{T}
≻ 0,
Ĥ
=
Ĥ^{T}
≻ 0,
Z
=
Z^{T}
≻ 0,
,
W
, and
Y
, such that
and (19), where
Then the gains are given by
.
Proof:
Define the positive definite function
V
_{k}
:=
V
_{1k}
+
V
_{2k}
+
V
_{3k}
+
V
_{4k}
, where
and
where Δ
z_{h}
:=
z_{h}
_{+1}
−
z_{h}
. Since
, one rewrite (18) as
By assigning
a
:=
z_{k}
,
b
:= Δ
z_{h}
and
in Lemma 4, we obtain
Moreover, we have
From the above relations, Δ
V_{k}
is majorized by
In case of
is written as
Summing up the discretetime H—J—B inequality
along (18) from 0 to ∞ yields
Inequality (21) guarantees this 𝛾
H
_{∞}
performance in the
l
_{2}
norm sense, because for all (
z_{k}
,
z_{k}
−
d_{k}
,
w_{k}
) ∈ ℝ
^{2n}
＼ {0} × ℝ
^{2n}
＼ {0} × ℝ
^{l}
one can derive
where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (17). Finally, it is not difficult to prove that (21) is the sufficiency for the
β

H
performance in the
l
_{2}
norm sense when one considers (22) with
w_{k}
= 0 to induce the following inductions
where we have used Schur complement, the congruence transformation, Lemma 2, and Lemma 3 with the assignments
,
R
:=
H
,
,
therein.
Though the design condition in Theorem 2 is casted in a nonlinear form, it can be efficiently solved by convex optimization algorithms in a manner similar to Algorithm 1 through the following conversion.
5. Example
Consider the statespace data for (1) borrowed from
[15]
and
d_{u}
= 1. We further suppose by Assumption 1 that Δ
A
, Δ
B
, and Δ
C
and are decomposed as
Since
y
∈ ℝ
^{3}
, a threeobserver bank is employed. With a slight abuse of notations, in the sequel, the subscript
p
∈ { 1, 2, 3 } denotes each observer in the bank. First, we attempt to find the gains by fixing
Ĥ_{p}
=
I
and
P_{p}
=
Z_{p}
in Theorem 1, which convexifies the inequalities. However, as is concerned, it turns out to fail in finding any feasible solutions, even though we do our best in adjusting
β_{p}
’s,
𝛾_{p}
’s, and
ε_{p}
’s. On the other hand, for the given (
β
_{1}
,
β
_{2}
,
β
_{3}
)=(1.1, 1.2, 1.6), and
𝛾_{p}
= 1, and ,
ε_{p}
= 0.1,
p
∈ { 1, 2, 3 }, we succeed in finding the following gains by Algorithm 1 in 8 iterations for the first and the second observer, respectively, and 16 iterations for the third one:
For simulation, we introduce a delay
d
∈ ℝ
_{≽0}
and a disturbance
w
∈
ℒ
_{2}
randomly varying within [0,
d_{u}
] and (−0.5, 0.5) , respectively. A fault
f
∈
ℒ
_{2}
forms the transient pattern as
for 50 s . Set
Tw_{p}
= 3 , then the threshold is generally calculated as
under the zeroinitial condition for both the system and the observer bank.
Three cases are simulated to investigate the validity of the designed bank.
i) In the faultfreedisturbanceactivated case, the threshold is computed as
J
_{th}
_{p}
= 0.5
𝛾_{p}
= 0.5,
p
∈ {1, 2, 3 }. As shown in
Fig.1
,
J
_{r}
_{p}
,
p
∈ { 1, 2, 3 }, does not exceed
J
_{th}
_{p}
for all
t
∈ [0.50], from which one judge based on
Table 1
that there does not occur any fault, as it really is. The evaluation
=(0.8691,0.7275,0.9093)≺(𝛾
_{1}
,𝛾
_{2}
,𝛾
_{3}
)=(1, 1, 1) confirms that the design goal — the 𝛾
H
_{∞}
performance — is achieved.
Residual evaluations when f = 0 but w ∈ ℒ_{2} ＼ {0}: J_{r}_{p} (solid) J_{th}_{p} and (dashed).
ii) Next, we simulate the faultactivateddisturbancefree case. In this case, the threshold is modified to
Fig. 2
reveals the comparison
Residual evaluations when f ∈ ℒ_{2} ＼ {0} but : 0: J_{rp} (solid) and J_{thp} (dashed).
By the FDI logic in
Table 1
, the observer bank successfully declares that
f
_{1}
,
f
_{2}
,
f
_{3}
are isolated for
t
∈ (5.97, 16.54),
t
∈ (21.73, 30.70),
t
∈ (36.54, 45.96), respectively, with a retard up to 2 s from the faultarising time. The evaluated values
=(1.4049, 1.5931, 2.1226) are greater than (
β
_{1}
,
β
_{2}
,
β
_{3}
)=(1.1, 1.2, 1.6), entrywisely, which verifies the
β

H
_ performance of the observer bank.
iii) Now, we consider the faultdisturbanceactivated case. The fault and the disturbance data applied in the previous runs are activated at the same time. Analyzing the result in
Fig. 3
with the threshold
Residual evaluation when f ≠ 0 and w ≠ 0: J_{rp}(solid) and J_{thp} (dashed).
According to the FDI logic in
Table 1
, we recognize that each sensor malfunctions for
t
∈ (6.40, 16.10),
t
∈ (22.06, 30.23),
t
∈ (36.97, 45.48), in regular order. This means that the observer bank designed by the proposed method is robust against the disturbance, the uncertainties, and the time delay so that any misleading alarm is not issued, as should be expected.
iv) To highlight the benefit of the proposed method, we simulate a comparable scheme, Theorem 1 in
[18]
that does not consider the timedelay nor uncertainties. By this technique we compute
and apply them to the faultdisturbanceactivated case. As shown in
Fig. 4
, the compared only isolates
f
_{1}
for
t
∈ [4.87, 19.89) For the rest of the simulationrun time, any isolation is not declared but detection is kept alarmed although there exist no fault for
t
∈ [30.35) and
t
∈ [45,50), which less accurate than ours.
Residual evaluation by Theorem 1 in [18] when f ≠ 0 and w ≠ 0: J_{rp}(solid) and J_{thp} (dashed).
6. Conclusions
In this paper, we presented the
H
_/
H
_{∞}
FDI observer bank design techniques for uncertain timedelay LTI systems for both continuous and discretetime settings. Design conditions are developed in the format of LMIs using the cone complementary linearization algorithm. Simulation results convincingly demonstrated the effectiveness of the developed methodology.
Acknowledgements
This work was supported by INHA UNIVERSITY Research Grant.
BIO
Sung Chul Jee received B.S . and M.S. degrees from the Department of Electronic Engineering, Inha University, Incheon, Korea, in 2009 and 2011, respectively. Now, He is currently pursuing a Ph.D. degree at the same university. His research interests include fuzzy control systems, fault detection and isolation, and their applications.
Ho Jae Lee received B.S., M.S., and Ph. D. degrees from the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 1998, 2000, and 2004, respectively. In 2005, he was a Visiting Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. Since 2006, he has been with the School of Electronic Engineering, Inha University, Incheon, Korea, where he is currently an Assistant Professor. His research interests include fuzzy control systems, hybrid dynamical systems, largescale systems, and digital redesign.
Do Wan Kim received the B.S., M.S., and Ph.D. degrees from the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2002, 2004, and 2007, respectively. He was with the Engineering Research Institute, Yonsei University, and a Visiting Scholar with the Department of Mechanical Engineering, University of California, Berkeley. In 2009, he was a Research Professor with the Department of Electrical and Electronic Engineering, Yonsei University. Since 2010, he has served on the faculty in the Department of Electrical Engineering, Hanbat National University, Daejeon, Korea. His current research interests include analysis and synthesis of nonlinear sampleddata control systems.
Bai L.
,
Tian Z.
,
Shi S.
2007
“Robust fault detection for a class of nonlinear timedelay systems,”
Journal of the Franklin Institute
344
(6)
873 
888
DOI : 10.1016/j.jfranklin.2006.11.004
Ding S. X.
2008
Modelbased Fault Diagnosis Techniques
Springer
Tsai J. S.H.
,
Wei C.L.
,
Guo S.M.
,
Shieh L. S.
,
Liu C. R.
2008
“Epbased adaptive tracker with observer and fault estimator for nonlinear timevarying sampleddata systems against actuator failures,”
Journal of the Franklin Institute
345
(5)
508 
535
DOI : 10.1016/j.jfranklin.2008.02.003
Persis C. D.
,
Isidori A.
2001
“A geometric approach to nonlinear fault detection and isolation,”
IEEE Transactions on Automatic Control
46
(6)
853 
865
Caliskan F.
,
Genc I.
2008
“A robust fault detection and isolation method in load frequency control loops,”
IEEE Transactions on Power Systems
23
(4)
1756 
1767
DOI : 10.1109/TPWRS.2008.2004831
Liu J.
,
Wang J. L.
,
Yang G.H.
2005
“An LMI approach to minimum sensitivity analysis with application to fault detection,”
Automatica
41
(11)
1995 
2004
DOI : 10.1016/j.automatica.2005.06.005
Guo J.
,
Huang X.
,
Cui Y.
2009
“Design and analysis of robust fault detection filter using LMI tools,”
Computers & Mathematics with Applications
57
(1112)
1743 
1747
DOI : 10.1016/j.camwa.2008.10.032
Jee S. C.
,
Lee H. J.
,
Joo Y. H.
2012
“ℋ_ℋ∞ sensor fault detection observer design for nonlinear systems in Takagi—Sugeno’s form,”
Nonlinear Dynamics
67
(4)
2343 
2351
DOI : 10.1007/s1107101101486
Paviglianiti G.
,
Pierri F.
,
Caccavale F.
,
Mattei M.
2010
“Robust fault detection and isolation for proprioceptive sensors of robot manipulators,”
Mechatronics
20
(1)
162 
170
DOI : 10.1016/j.mechatronics.2009.09.003
Moon Y. S.
,
Park P.
,
Kwon W. H.
,
Lee Y. S.
2001
“Delaydependent robust stabilization of uncertain statedelayed systems,”
International Journal of Control
74
(14)
1447 
1455
DOI : 10.1080/00207170110067116
Casavola A.
,
Famularo D.
,
Franze G.
2005
“A robust deconvolution scheme for fault detection and isolation of uncertain linear systems: An LMI approach,”
Automatica
41
(8)
1463 
1472
DOI : 10.1016/j.automatica.2005.03.019
Huang D.
,
Nguang S. K.
2010
“Robust fault estimator design for uncertain networked control systems with random time delays: An ILMI approach,”
Information Sciences
180
(3)
465 
480
DOI : 10.1016/j.ins.2009.10.002
Bai L.
,
Tian Z.
,
Shi S.
2006
“Design of H robust fault detection filter for linear uncertain timedelay systems,”
ISA Transactions
45
(4)
491 
502
DOI : 10.1016/S00190578(07)602274
Karimi H.
,
Zapateiro M.
,
Luo N.
2010
“A linear matrix inequality approach to robust fault detection filter design of linear systems with mixed timevarying delays and nonlinear perturbations,”
Journal of the Franklin Institute
347
(6)
957 
973
DOI : 10.1016/j.jfranklin.2010.03.004
Park P.
1999
“A delaydependent stability criterion for systems with uncertain timeinvariant delays,”
IEEE Transactions on Automatic Control
44
(4)
876 
877
DOI : 10.1109/9.754838
Jee S. C.
,
Lee H. J.
,
Joo Y. H.
2012
“H_H∞ sensor fault detection and isolation in linear timeinvariant systems,”
International Journal of Control Automation and Systems
10
(4)
841 
848
DOI : 10.1007/s1255501204225