Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 524271, 22 pages
doi:10.1155/2012/524271
Research Article
Filtering-Based Fault Detection for Stochastic
Markovian Jump System with Distributed
Time-Varying Delays and Mixed Modes
Yucai Ding,1 Hong Zhu,1 Shouming Zhong,2
Yuping Zhang,1 and Jianwei Xia3
1
School of Automation Engineering, University of Electronic Science and Technology of China,
Chengdu 611731, China
2
School of Mathematical Sciences, University of Electronic Science and Technology of China,
Chengdu 611731, China
3
School of Mathematics Science, Liaocheng University, Liaocheng 252000, China
Correspondence should be addressed to Yucai Ding, dycyer@163.com
Received 10 May 2012; Revised 10 November 2012; Accepted 13 November 2012
Academic Editor: Jong Hae Kim
Copyright q 2012 Yucai Ding et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The problem of fault detection for stochastic Markovian jump system is considered. The system
under consideration involves discrete and distributed time-varying delays, Itô-type stochastic
disturbance, and different system and delay modes. The aim of this paper is to design a fault
detection filter such that the fault detection system is stochastically stable and satisfies a prescribed
H∞ disturbance attenuation level. By using a novel Lyapunov functional, a mix-mode-dependent
sufficient condition is formulated in terms of linear matrix inequalities. A numerical example is
given to illustrate the effectiveness of the proposed main results.
1. Introduction
Fault detection received considerable attention over the past decades because of the increasing demand for higher performance, safety, and reliability standards. In recently, many
effective methods have been developed for fault detection. To the best of the authors’
knowledge, the published results can be categorized into three approaches. The first category
is the filter- or observer-based approaches, where filters are used to generate residual signals
to detect and estimate the fault, for example, 1–9. In the fault detection scheme based
on filter or observer, a fault cannot only be detected but also be approximated, and the
2
Journal of Applied Mathematics
fault estimate can be further used in fault-tolerant control. The second category is the
statistic approach, where the Bayesian theory and likelihood method are used to evaluate the
fault signals 10. The third category is the geometric approach. By utilizing the geometric
framework, a set of residuals is generated such that each residual is affected by one fault
and is partially decoupled from others 11. In the framework of fault detection, faults are
detected by setting a predefined threshold on residual signals. Once the value of residual
evaluation function excesses the predefined threshold, an alarm of faults is generated. For
example, by using Luenberger type observers, the authors of 6, 12 present an explicit
expression of the filters for the fault such that both asymptotic stability and a prescribed level
of disturbance attenuation are satisfied for all admissible nonlinear perturbations; by using
the measured output probability density functions PDFs, the authors of 13, 14 construct a
stable filter-based residual generator.
Markovian jump systems MJSs are a special class of switched systems. The state
vector of such system has two components xt and rt. The first one is in general referred
to as the state, and the second one is regarded as the mode. In its operation, the jump system
will switch from one mode to another in a random way, based on a Markovian chain with
finite state space. These systems are very common in economic systems, communication
systems, robot manipulator systems and circuit systems, and so forth. Time delay is an
inherent characteristic of many physical systems, which occurs due to signal transmission,
inevitable defects of control equipment, and so on. The systems with or without time delays
are convergent when time delays are close to zero. otherwise, they may be divergent. In
other words, time delays, either constant or time varying, can degrade the performance of
systems designed without considering the delays and can even destabilize the systems. Due
to their extensive practical applications, considerable attention has been devoted to MJSs, see,
for example, 15–18 for stability, 18–25 for control, and 25–32 for state estimation. More
recently, the methods of fault estimation and fault detection have been extended successfully
to MJSs 33–40. From the published results, the delay mode is assumed to be the same
as the system matrices mode. However, the assumption cannot always be satisfied in real
applications. In some practical systems, variations of delay usually depend on phenomena
which may not cause abrupt changes in other system parameters. For instance, in networked
control systems, the randomness of delay is a result of communication network issues, but
the process itself may contain separate sources of randomness which means that the system
matrices mode may be different with the delay mode 41. Therefore, it is important and
necessary to pay attention to the study of Markovian jump systems with different system
and delay modes. Furthermore, it appears that general results pertaining to fault detection
for stochastic MJSs with discrete and distributed time delays, Itô-type stochastic disturbance
and different system and delay modes are few and restricted, despite its practical importance,
mainly due to the mathematical difficulties in dealing with such mixed modes. Research in
this area should be interesting yet challenging as it involves the combination of two different
jumping modes, which has motivated this paper.
This paper deals with the problem of fault detection for stochastic MJSs with discrete
and distributed time-varying delays, Itô-type stochastic disturbance, and different system
and delay modes. By using a novel mix-mode-dependent Lyapunov functional, a new
sufficient condition on stochastic stability with an H∞ performance is derived in terms of
linear matrix inequalities LMIs. Based on this, the existence condition of the fault detection
filter which guarantees stochastic stability and the H∞ performance of the corresponding
augmented system is presented. A numerical example is provided to show the effectiveness
of the proposed results.
Journal of Applied Mathematics
3
Notation. Throughout this paper, Rn denotes the n dimensional Euclidean space. λmax Q and
λmin Q denote, respectively, the maximal and minimal eigenvalue of matrix Q. E{·} refers to
the expectation operator with respect to some probability measure P. We use diag{·, ·, ·} as a
block-diagonal matrix. A > 0 < 0 means that A is a symmetric positive negative definite
matrix, A−1 denotes the inverse of matrix A. AT denotes the transpose of matrix A, and I is
the identity matrix with compatible dimension.
2. System Description and Definitions
Consider the following stochastic MJS with mode-dependent time-varying delays:
dxt Art xt A1 rt xt − τt, st A2 rt t
t−τt,st xsds
B0 rt ut B1 rt νt B2 rt ft dt G1 rt xtdωt,
dyt Crt xt C1 rt xt − τt, st C2 rt t
t−τt,st xsds
2.1
D1 rt νt D2 rt ft dt G2 rt xtdωt,
xt φt,
t ∈ −τ, 0,
where xt ∈ Rn is the state vector; ut is the exogenous disturbance input which belongs to
L2 0 ∞; νt the unknown input; ft is the fault to be detected; yt ∈ Rp is the measured
output; ωt is a zero-mean one-dimensional Wiener process satisfying E{ωt} 0 and
E{ω2 t} t; φt is a compatible vector-valued initial function defined on −τ, 0; Art ,
A1 rt , A2 rt , B0 rt , B1 rt , B2 rt , G1 rt , Crt , C1 rt , C2 rt , D1 rt , D2 rt , and G2 rt are real constant matrices with appropriate dimensions. τt, st is the mode-dependent timevarying delay. {rt , t 0} and {st , t 0} are continuous-time Markovian processes with right
continuous trajectories and taking values in finite sets S1 {1, 2, . . . , N}, S2 {1, 2, . . . , M}
with the transition probability matrices Π πil , i, l ∈ S1 and Λ λjk , j, k ∈ S2 ,
respectively, given by
Pr{rtΔ l | rt i} Pr stΔ k | st j πil Δ oΔ,
l
/ i,
1 πii Δ oΔ,
l i,
λjk Δ oΔ,
1 λjj Δ oΔ,
2.2
k
/ j,
k j,
where Δ > 0 and limΔ → 0 oΔ/Δ 0; πil 0 for i /
l is the transition rate from mode i at
π
; λjk 0 for j /
k is the transition rate
time t to mode l at time t Δ and πii − N
il
l1,l /
i
4
Journal of Applied Mathematics
from mode j at time t to mode k at time t Δ and λjj − M
k1,k /
j λjk . The processes rt and st
are assumed to be independent throughout this paper. For simplicity, a matrix Rrt will be
denoted by Ri . For example, Art is denoted by Ai , A1 rt is denoted by A1i , i ∈ S1 , τst , t
is denoted by τj t, j ∈ S2 , and so on. When the mode is in st j, the mode-dependent
time-varying delay satisfies
0 < τj t τj τ,
τ̇j t μj ,
2.3
where τ max{τj }.
Remark 2.1. In this work, we have assumed that the delay mode is different from the system
mode. This is more powerful and desirable in modeling of real systems, because the reason
for jump in delay value may not be the same as that for jump in other system parameters.
Remark 2.2. The generalized stochastic system 2.1 is quite general since it considers noise
perturbations, discrete, and distributed time-varying delays and Markovian jump processes
with different modes. To the best of our knowledge, the generalized stochastic system 2.1
has never been considered in the previous literature.
Remark 2.3. A fault detection system consists of a residual generator and an evaluation stage,
including an evaluation function and a threshold. Therefore, the fault detection problem to
be addressed in this paper can be stated as the following two steps. The first step is to design
a suitable filter to reduce the effect of disturbances on residual signals and to enhance the
influence of faults. The second step is to determine the residual evaluation function and an
appropriate threshold.
In this study, the following full-order fault detection filter is considered:
dxf t Afij xf tdt Bfij dyt,
rt Cfij xf t,
xf 0 0,
2.4
i ∈ S1 , j ∈ S2 ,
where xf t is the filter state vector. rt is its output which is sensitive to faults. Afij Bfij Cfij are appropriately dimensioned filter matrices to be determined.
To improve the sensitiveness of residual to fault, we add a weighting matrix function
into the fault ft, that is, Fω s WsFs, where Fs and Fω s denote, respectively, the
Laplace transforms of ft and fω t. One state-space realization of Fω s WsFs can be
ẋω t Aω xω t Bω ft,
fω t Cω xω t,
xω 0 0.
2.5
Journal of Applied Mathematics
5
Denoting re t rt − fω t, then the overall dynamic system can be governed by the
following augmented system:
dξt Aij ξt A1ij Kξ t − τj t A2ij K
t
t−τj t
ξsds B ij wt dt
Gij Kξtdωt, re t Cij ξt, ξt φt,
2.6
∀t ∈ −τ, 0,
T
T T
φT t 0T 0T , and
where ξt xT t xfT t xω t , wt uT t νT t f T t , φt
⎤
0
0
Ai
Aij ⎣Bfij Ci Afij 0 ⎦,
0
0 Aω
⎤
⎡
B2i
B0i B1i
Bij ⎣ 0 Bfij D1i Bfij D2i ⎦,
0
0
Bω
⎡
⎤
A1i
A1ij ⎣Bfij C1i ⎦,
0
⎤
⎡
G1i
Gij ⎣Bfij G2i ⎦,
0
K I 0 0 .
⎡
⎤
A2i
⎣Bfij C2i ⎦,
0
⎡
A2ij
Cij 0 Cfij −Cω ,
2.7
For simplicity, let
ϕt Aij ξt A1ij Kξ t − τj t A2ij K
t
t−τj t
ξsds Bij wt,
2.8
gt Gij Kξt.
Now, the problem of fault detection is transformed into an H∞ filtering problem
for system 2.1, which is described as follows: given a prescribed level of disturbance
attenuation γ, determine a series of filter matrices Afij , Bfij , and Cfij i ∈ S1 , j ∈ S2 such
that the augmented system 2.6 is stochastically stable.
After designing a fault detection filter, the remaining important task is to evaluate the
generated residual. One of widely adopted methods is to choose a residual evaluation function and a threshold. In this paper, residual evaluation function fr and a threshold Jth are
selected as
fr t0 T
t0
T
r trtdt,
Jth sup
E
t T
0
νt∈L2 ,ft0
t0
T
r trtdt ,
2.9
where t0 , t0 T is the finite-time window, T denotes the limited length, and t0 denotes the
initial evaluation time. The occurrence of fault can be detected by comparing fr and Jth,
according to the following logic:
fr > Jth ⇒ Faults ⇒ Alarm,
fr < Jth ⇒ No Faults.
2.10
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Journal of Applied Mathematics
The following lemma and definitions are introduced, which will be used in the proof
of the main results.
Lemma 2.4 see 42. For any matrix M > 0, scalar γ > 0, vector function ω : 0, γ → Rn such
that the integrations concerned are well defined, the following inequality holds:
γ
γ
γ
ωT sds M
ωsds γ
ωT sMωsds.
0
0
2.11
0
Definition 2.5. The filtering error system 2.6 with wt 0 is said to be stochastically stable,
the
if, for every system mode rt , every time-delay mode st and all finite initial state φt,
following relation holds: limt → ∞ E{|ξt|2 } 0.
Definition 2.6. Given a scalar γ > 0, the filtering error system 2.6 is said to be stochastically
stable with an H∞ performance γ, if, for every system mode rt and every time-delay mode
st , the filtering error system 2.6 with wt 0 is stochastically stable, and, under zero initial
condition, it satisfies re 2 γw2 for any nonzero wt ∈ L2 0, ∞.
It should be pointed out that the joint process ξt, rt , st is not Markovian. In order to
cast our model into the frame work for a Markovion system, let us define a new Markovion
process: ξt s ξt s, −τ s 0, and then {ξt , rt , st , t 0} is Markovian process with the
r0 , s0 .
initial state φ·,
Let CRn ×Rn ×R ×S1 ×S2 denote the family of all nonnegative functions V ξ, ξt , t, i, j
n
on R × Rn × R × S1 × S2 , which are continuously twice differentiable in ξ and differentiable
in t. If V ∈ CRn × Rn × R × S1 × S2 , then, along the trajectory of system 2.6, we define an
operator L· from Rn × Rn × R × S1 × S2 to R by
LV ξ, ξt , t, i, j Vt ξ, ξt , t, i, j Vξ ξ, ξt , t, i, j ϕt πil V ξ, ξt , t, l, j
l∈S1
k∈S2
1
λjk V ξ, ξt , t, i, k trace g T tVξξ ξ, ξt , t, i, j gt ,
2
2.12
where
∂V ξ, ξt , t, i, j
Vt ξ, ξt , t, i, j ,
∂t
∂V ξ, ξt , t, i, j
∂V ξ, ξt , t, i, j
,
Vξ ξ, ξt , t, i, j ,...,
∂ξ1
∂ξn
2 ∂ V ξ, ξt , t, i, j
Vξξ ξ, ξt , t, i, j .
∂ξi ξj
2.13
2.14
2.15
Journal of Applied Mathematics
7
3. Main Results
In this section, we first propose a delay-dependent sufficient condition for stochastic stability
with the H∞ performance of filtering error system 2.6. Now, define a stochastic Lyapunov
functional candidate for systems 2.6 as
6
V ξ, ξt , t, i, j Vn ξ, ξt , t, i, j ,
3.1
n1
where
V1 ξ, ξt , t, i, j ξT tP rt , st ξt,
t
ξT sK T Q1 rt , st Kξsds
V2 ξ, ξt , t, i, j t−τt,st V3 ξ, ξt , t, i, j t
0 t
V4 ξ, ξt , t, i, j −τ
tθ
0 t
ξT sK T Q2 rt , st Kξsds,
t−τt,st t−τst t
t
V5 ξ, ξt , t, i, j −τ
tθ
θ
−θ
0
T
ξ sK ds R1 rt , st t
θ
Kξsds dθ,
3.2
ϕT sK T ZKϕsds dθ,
ξT sK T R2 Kξsds dθ,
τ 0 t
V6 ξ, ξt , t, i, j T
ts
ξT αK T R3 Kξαdα ds dθ.
By Itô’s formula, we obtain the stochastic differential as
6
dV L Vn ξ, ξt , t, i, j dt 2ξT tPij gtdωt,
3.3
n1
where L is the weak infinitesimal generator of the random process {ξt , rt , st } along the system
2.6.
Using the operator 2.12, we have
T
LV1 ξ, ξt , t, i, j 2ξ tPij Aij ξt A1ij Kξ t − τj t A2ij K
T
ξ t
l∈S1
πil Plj k∈S2
t
t−τj t
ξsds Bij wt
λjk Pik ξt g T tPij gt.
3.4
8
Journal of Applied Mathematics
The derivative of the first term in V2 ξ, ξt , t, i, j is given as follows:
L
t
t−τt,st ξT sK T Q1 rt , st Kξsds
1
lim E
Δ→0 Δ
tΔ
tΔ−τstΔ , tΔ
−
T
t−τj t
t
T
t−τj t
t
ξ sK
ξT sK T Q1 rtΔ , stΔ Kξsds
T
T
ξ sK Q1ij Kξsds
Kξsds
πil Q1lj
l∈S1
t
λjk
ξT sK T Q1ik Kξsds
t−τk t
k∈S2
ξT tK T Q1ij Kξt − 1 − τ̇j t ξT t − τj t K T Q1ij Kξ t − τj t
ξT tK T Q1ij Kξt − 1 − μj ξT t − τj t K T Q1ij Kξ t − τj t
T
t−τj t
λjj
t
ξ sK
T
t−τj t
πil Q1lj
Kξsds
ξT sK T Q1ij Kξsds
⎛
t−τ
l∈S1
t
t
ξT sK T ⎝
⎞
λjk Q1ik ⎠Kξsds.
k
/j
Following a similar method of 3.5, it is easy to obtain
L
t
t−τst ξT sK T Q2 rt , st Kξsds
ξT tK T Q2ij Kξt − ξT t − τj K T Q2ij Kξ t − τj
t
T
t−τj
k∈S2
ξ sK
λjk
T
πil Q2lj
Kξsds
l∈S1
t
t−τk
ξT sK T Q2ik Kξsds
3.5
Journal of Applied Mathematics
9
ξT tK T Q2ij Kξt − ξT t − τj K T Q2ij Kξ t − τj
t
T
T
ξ sK
πil Q2lj Kξsds
t−τj
λjj
l∈S1
t
t−τj
ξT sK T Q2ij Kξsds
⎛
t
ξT sK T ⎝
t−τ
LV3 ξ, ξt , t, i, j
θ
t
t−τj t
Kξsds dθ ξT sK T dsR1ij
πil
λjk
t
k∈S2
t
t−τk t
θ
t
t−τj t
t
t
t−τj t
l∈S1
λjk Q2ik ⎠Kξsds,
− 1 − τ̇j t
×
⎞
k/
j
t
θ
t
Kξsds 2
T
T
ξ sK ds R1lj
t−τj t
t
θ
ξT tK T R1ij
Kξsds dθ
ξT sK T R1ik Kξsds dθ.
3.6
Using Lemma 2.4 and considering 2.3, we have
LV3 ξ, ξt , t, i, j
t
t
− 1 − μj
t
t−τj t
t−τj t
θ
− 1 − μj
t
t−τj t
×
T
t
t
ξ tK
t−τj t
t
t−τj t
t − θ
T
θ
ξ sK ds R1ij
t
t−τj t
πil
l/
i
t−τk t
k/
j
T
ξT sK T R1ij Kξsds dθ t
λjk
T
t
θ
t
t−τj t
t − θ
t
θ
1 2
τj R1ij Kξt
2
ξT sK T R1lj Kξsds dθ
ξT sK T R1ik Kξsds dθ
T
ξT tK T
Kξsds
T
ξ sK ds R1ij
t
t−τj t
Kξsds
1 2
τ R1ij Kξt
2 j
T
T
ξ sK R1ij Kξsds dθ t − θdθ ds l/
i
k/
j
λjk
t
t−τk t
πil
t
t−τj t
ξT sK T R1lj Kξs
ξT sK T R1ik Kξs
t
t−τk t
t − θdθ ds
10
Journal of Applied Mathematics
t
t
T
T
T
T 1 2
τ R1ij Kξt
− 1 − μj
ξ sK ds R1ij
Kξsds ξ tK
2 j
t−τj t
t−τj t
t
t
t−τj t
1 ξ sK R1ij Kξsds dθ τj2 πil
2 l / i
θ
T
1 2
τ λjk
2 k / j k
T
t
t−τ
t
t−τ
ξT sK T R1lj Kξsds
ξT sK T R1ik Kξsds.
3.7
Moreover,
T
T
LV4 ξ, ξt , t, i, j τϕ tK ZKϕt −
LV5 ξ, ξt , t, i, j τξT tK T R2 Kξt −
t
t−τ
t
t−τ
1
LV6 ξ, ξt , t, i, j τ 2 ξT tK T R3 Kξt −
2
ϕT sK T ZKϕsds,
ξT sK T R2 Kξsds,
t t
t−τ
θ
3.8
ξT sK T R3 Kξsds dθ.
We define
T
ηj t ξ t ξ
T
t − τj t K
T
ξ
T
t − τj K
T
T
ϕ tK
T
t
T
T
t−τj t
T
ξ sK ds
.
3.9
The following equations are true for any matrices L, M, N, and Y with appropriate
dimensions:
0
2ηjT tL
0
2ηjT tM
0
2ηjT tN
Kξt − Kξ t − τj t −
t
t−τj t
Kξ t − τj t − Kξ t − τj −
K ϕsds
−
t−τj t
t−τj
t
t−τj t
K ϕsds
−
K g sdωs ,
t−τj t
t−τj
−Kϕt KAij ξt KA1ij Kξ t − τj t KA2ij K
K g sdωs ,
t
t−τj t
ξsds KB 1ij νt ,
3.10
0 τj ηjT tYηj t −
t−τj t
t−τj
ηjT tYηj tds −
t
t−τj t
ηjT tYηj tds,
3.11
Journal of Applied Mathematics
11
where
T
L LT1 K LT2 LT3 LT4 LT5 ,
T
M M1T K M2T M3T M4T M5T ,
T
N 0 0 0 NT 0 ,
⎡ T
⎤
K Y11 K K T Y12 K T Y13 K T Y14 K T Y15
⎢ ∗
Y22
Y23
Y24
Y25 ⎥
⎢
⎥
⎢
⎥
Y⎢ ∗
∗
Y33
Y34
Y35 ⎥.
⎢
⎥
⎣ ∗
∗
∗
Y44
Y45 ⎦
∗
∗
∗
∗
Y55
3.12
Considering 3.4–3.11, we obtain that
LV ξ, ξt , i, j
T ηj t
Σij Φ1ij ηj t
wt
wt
∗
0
−
−
⎡
t
T
t−τj t
t
t−τ
t
t−τj
ξT sK T ⎣
πil Q1lj τj2
2
T
ξ sK
T
⎤
πii Q1ij λjj Q1ij ⎦Kξsds
R1lj
λjk Q1ik Q2ik k∈S2 ,k /
j
τk2
2
R1ik
⎤
− R2 ⎦Kξsds
πil Q2lj λjj Q2ij
Kξsds
l∈S1
θ
t
t−τj t
ξT sK T R1ij − R3 Kξsds dθ
ηj t
Kϕs
t−τj t t−τj
l∈S1 ,l /
i
⎡
t t
t−τ
ξ sK
T⎣
T Y L ηj t
ds
∗ Z Kϕs
T ηj t
Y M ηj t
ds ft,
Kϕs
∗ Z Kϕs
3.13
12
Journal of Applied Mathematics
where
2ξT tPij gtdωt − 2ηT tL
ft
j
⎡
Σ11 Σ12 Σ13
⎢∗ Σ Σ
⎢
22
23
⎢
∗ Σ33
Σij ⎢ ∗
⎢
⎣∗
∗
∗
∗
∗
∗
Σ14
Σ24
Σ34
Σ44
∗
t
t−τj t
t−τj t
t−τj
Kgsdωs,
⎤
Σ15
Σ25 ⎥
⎥
⎥
Σ35 ⎥,
⎥
Σ45 ⎦
Σ55
T
T
Σ11 Pij Aij Aij Pij K T Gij Pij Gij K Σ12
Kgsdωs −
2ηjT tM
πil Plj l∈S1
λjk Pik
k∈S2
1
1
K T Q1ij Q2ij τR2 τ 2 R3 τj2 R1ij L1 LT1 τj Y11 K,
2
2
"
#
Pij A1ij K T −L1 LT2 M1 τj Y12 ,
"
#
Σ13 K T LT3 − M1 τj Y13 ,
"
#
T
Σ14 K T LT4 τj Y14 Aij K T N T ,
"
#
Σ15 Pij A2ij K T LT5 τj Y15 ,
Σ22 − 1 − μj Q1ij − L2 − LT2 M2 M2T τj Y22 ,
Σ23 −LT3 − M2 M3T τj Y23 ,
T
Σ24 −LT4 M4T A1ij K T N T τj Y24 ,
Σ25 −LT5 M5T τj Y25 ,
Σ33 −Q2ij − M3 − M3T τj Y33 ,
Σ34 −M4T τj Y34 ,
Σ35 −M5T τj Y35 ,
Σ44 τZ − N − N T τj Y44 ,
Σ45 NKA2ij τj Y45 ,
Σ55 − 1 − μj R1ij τj Y55 ,
T
T
T
Φ1ij Bij Pij 0 0 Bij K T N T 0 .
3.14
Therefore, we have the following result for the H∞ performance analysis.
Journal of Applied Mathematics
13
Theorem 3.1. Given scalars τ, τj , and μj , the fault detection system 2.6 is stochastically stable with
an H∞ performance γ for any time delay τj t satisfying 2.3, if there exist matrices Pij > 0, Z > 0,
Q1ij > 0, Q2ij > 0, R1ij > 0, R2 > 0, R3 > 0, and matrices L, M, N, Y denoted in 3.10–3.11 such
that for each i ∈ S1 , j ∈ S2
⎡
⎤
ij Φ1ij Φ2ij Φ3ij
Σ
⎢
⎥
0 ⎥
⎢ ∗ −γ 2 I 0
⎢
⎥ < 0,
⎣∗
∗
−I
0 ⎦
∗
∗
∗ −Pij
πil Q1lj l∈S1 ,l /i
k∈S2 ,k /
j
τj2
2
R1lj
πii Q1ij λjj Q1ij < 0,
λjk Q1ik Q2ik 3.15
τk2
2
R1ik
− R2 < 0,
3.16
πil Q2lj λjj Q2ij < 0,
l∈S1
R1ij − R3 < 0,
Y L
Y M
0,
0,
∗ Z
∗ Z
3.17
where
⎡
T
Σ11 − K T Gij Pij Gij K Σ12 Σ13
⎢
⎢
∗
Σ22 Σ23
ij ⎢
Σ
⎢
∗
∗ Σ33
⎢
⎣
∗
∗
∗
∗
∗
∗
T
Φ2ij Cij 0 0 0 0 ,
Σ14
Σ24
Σ34
Σ44
∗
⎤
Σ15
⎥
Σ25 ⎥
⎥
,
Σ35 ⎥
⎥
⎦
Σ45
Σ55
3.18
T
Φ3ij Pij Gij K 0 0 0 0 .
Proof. Using Schur complement formula to 3.15, it can be seen that 3.15 is equivalent to
Σij Φ2ij ΦT2ij Φ1ij
< 0.
∗
−γ 2 I
3.19
Now, we show that the filtering error system 2.6 with wt 0 is stochastically stable. If
wt 0, from 3.13 and 3.16–3.17, we can obtain
%
$
E LV ξ, ξt , t, i, j E ξT tΣij ξt .
3.20
14
Journal of Applied Mathematics
Inequality 3.20 implies that Σij < 0. Thus, we have
LV ξ, ξt , t, i, j −α1 ξT tξt,
3.21
where α1 mini∈S1 ,j∈S2 {λmin −Σij } > 0. Therefore, for any T > 0, by Dynkin’s formula, we
have
T $
%
#
"
E ξT sξs ds α−1
1 V φ0, r0 , s0 ,
3.22
0
which means that limt → ∞ E{|ξt|2 } 0. Thus, the filtering error system 2.6 with wt 0 is
stochastically stable by Definition 2.5.
In the sequel, we will deal with the H∞ performance of the filtering error system 2.6.
Using 3.19 and H∞ performance, we have
% η tT Σ Φ ΦT Φ
$
ηj t
ij
2ij 2ij
1ij
j
T
2 T
< 0.
E LV ξ, ξt , t, i, j re tre t − γ w twt wt
∗
−γ 2 I wt
3.23
Noting that the zero initial condition, then it follows from 3.23 that
JH E
& ∞ '
reT tre t − γ 2 wT twt dt
0
E
& ∞ 0
'
reT tre t − γ 2 wT twt LV ξ, ξt , t, i, j dt
3.24
< 0.
Hence, if 3.15–3.17 hold, JH < 0 can be guaranteed. That is, re 2 γw2 for all
nonzero wt. Therefore, the filtering error system 2.6 is stochastically stable with the H∞
performance γ by Definition 2.6. This completes the proof.
Remark 3.2. Theorem 3.1 presents a new stochastic stability criterion by employing a novel
mixed mode-dependent Lyapunov functional. The Lyapunov functional in this paper uses
all information about rt , st , and τt, st . Also, the Lyapunov matrices P rt , st , Q1 rt , st ,
Q2 rt , st , and R1 rt , st depend on both the system mode rt and the delay mode st . Hence,
the Lyapunov functional in this paper is more general, and the condition on stability is
more applicable. In the most published papers about Markovian jump systems with mixed
time delays, the authors choose the mode-independent Lyapunov matrices which may lead
to some conservativeness, such as 15, 17, 23, 31, 33, 37–39, to name a few among many
important results in the literature. But, the selected mode-dependent Lyapunov matrices in
Journal of Applied Mathematics
15
this paper can reduce some conservativeness because they allow more freedom in choosing
feasible solutions of LMIs.
Remark 3.3. In Theorem 3.1, μj < 1 can be extended to a wider range μj < ∞ by dealing
(t
with the integral term λjj t−τj t ξT sK T Q1ij Kξsds in 3.5. Noting that λjj < 0, utilizing
Lemma 2.4, one has
λjj
t
λjj
ξ sK Q1ij Kξsds τj
t−τj t
T
T
t
t−τj
T
T
ξ sK dsQ1ij
t
t−τj t
Kξsds.
3.25
Further, deleting λjj Q1ij in 3.16 and adding λjj /τj Q1ij to Σ55 in 3.15, we can obtain a
more general stability condition.
Based on Theorem 3.1, the fault detection filter synthesis problem can be developed in
terms of LMIs for the system 2.1 with different system and delay modes.
Theorem 3.4. Consider the system 2.1. Given scalars τ, τj , and μj , the fault detection system 2.6
is stochastically stable with an H∞ performance γ for any time delay τj t satisfying 2.3 if there
exist matrices Vij > 0, Wij > 0, Uij > 0, Z > 0, Q1ij > 0, Q2ij > 0, R1ij > 0, R2 > 0, R3 > 0, Afij ,
Bfij , Cfij and matrices L, M, N, Y denoted in 3.10–3.11 such that for each i ∈ S1 , j ∈ S2
⎡
Σij Φ1ij Φ2ij
⎢
⎢∗
⎢
⎢∗
⎣
∗
−γ 2 I 0
∗
−I
∗
πil Q1lj l∈S1 ,l /
i
Φ3ij
τj2
2
∗
0
0
⎥
⎥
⎥ < 0,
⎥
⎦
−Φ4ij
R1lj
πii Q1ij λjj Q1ij < 0,
λjk Q1ik Q2ik k∈S2 ,k /
j
⎤
τk2
2
R1ik
− R2 < 0,
πil Q2lj λjj Q2ij < 0,
l∈S1
R1ij − R3 < 0,
Y L
0,
∗ Z
Y M
0,
∗ Z
3.26
16
Journal of Applied Mathematics
where
⎤
⎡
⎡
Σ11 Σ12 0 Σ14 Σ15 Σ16 Σ17
Vij B0i Vij B1i
⎥
⎢∗ Σ
⎢
0
Σ
0
0
Σ
⎢
22
24
27 ⎥
Bfij D1i
⎢ 0
⎥
⎢
⎢ 0
⎥
⎢∗
∗
Σ
0
0
0
0
0
33
⎢
⎥
⎢
⎢
⎥,
Σij ⎢
Φ
⎢
Σ
Σ
Σ
Σ
∗
∗
∗
0
0
1ij
44
45
46
47
⎥
⎢
⎢
⎥
⎢
⎢ 0
0
∗
∗
∗ Σ55 Σ56 Σ57 ⎥
⎢∗
⎢
⎥
⎢
⎣
NB
NB
⎦
⎣∗
0i
0i
∗
∗
∗
∗ Σ66 Σ67
0
0
∗
∗
∗
∗
∗
∗ Σ77
⎡
⎡
⎤
⎤
T
0
GT1i VijT GT2i Bfij 0
⎢ T ⎥
⎢
⎥
⎢ Cfij ⎥
⎢ 0
0
0⎥
⎢
⎢
⎥
⎥
⎡
⎢−CT ⎥
⎢ 0
Vij 0
0
0⎥
⎢ ω⎥
⎢
⎥
⎢
⎥
⎥
⎣
,
,
Φ2ij ⎢
Φ
Φ
∗ Wij
3ij
4ij
0
0⎥
⎢ 0 ⎥
⎢ 0
⎢
⎢
⎥
⎥
∗ ∗
⎢ 0 ⎥
⎢ 0
0
0⎥
⎢
⎢
⎥
⎥
⎣ 0 ⎦
⎣ 0
0
0⎦
0
0
0
0
Σ11 Vij Ai ATi Vij πil Vlj λjk Vik Q1ij Q2ij τR2
l∈S1
⎤
Vij B2i
Bfij D2i ⎥
⎥
Uij Bω ⎥
⎥
⎥
0 ⎥,
⎥
0 ⎥
⎥
NB0i ⎦
0
⎤
0
0 ⎦,
Uij
k∈S2
3.27
1
1
τ 2 R3 τj2 R1ij L1 LT1 τj Y11 ,
2
2
T
Σ12 CiT Bfij ,
Σ14 Vij A1i − L1 M1 LT2 τj Y12 ,
Σ15 LT3 − M1 τj Y13 ,
Σ16 LT4 τj Y14 ATi N T ,
Σ17 Vij A2i LT5 τj Y15 ,
Σ24 Bfij C1i ,
T
Σ22 Afij Afij Σ27 Bfij C2i ,
πil Wlj l∈S1
Σ33 Uij Aω ATω Uij λjk Wik ,
k∈S2
πil Ulj l∈S1
λjk Uik ,
k∈S2
Σ44 − 1 − μj Q1ij − L2 − LT2 M2 M2T τj Y22 ,
Σ45 − LT3 − M2 M3T τj Y23 ,
Σ46 − LT4 M4T τj Y24 AT1i N T ,
Σ55 − Q2ij − M3 − M3T τj Y33 ,
Σ66 τZ − N T − N τj Y44 ,
Σ47 −LT5 M5T τj Y25 ,
Σ56 −M4T τj Y34 ,
Σ67 NA2i τj Y45 ,
Σ57 −M5T τj Y35 ,
Σ77 − 1 − μj R1ij τj Y55 .
In this case, the parameters of the desired fault detection filter can be chosen by
Afij Wij−1 Afij ,
Bfij Wij−1 Bfij ,
Cfij Cfij .
3.28
Journal of Applied Mathematics
17
Proof. For each rt i ∈ S1 , st j ∈ S2 , we define a matrix Pij > 0 by Pij diagVij Wij Uij .
Then, with the parameters in 3.28, it can be verified that, for each i ∈ S1 , j ∈ S2 , the
LMI 3.26 can be rewritten as 3.15. Then, we can obtain the results in Theorem 3.1. This
completes the proof.
Remark 3.5. Noting that the first diagonal element Σ11 in 3.15 includes Pij , Plj , and Pik .
Owing to the restrictions on the authors’ knowledge and the technique difficulties, Pij
is assumed to be diagonal matrices to obtain the parameters of the fault detection filter.
Although this assumption may cause some conservativeness, considering complete Pij results
in bilinear matrix inequalities and not LMIs which are more conservative.
4. A Numerical Example
In this section, a numerical example will be presented to show the validity of the main results
derived above.
Example 4.1. Let us consider the stochastic system 2.1 with the following system of matrices:
A1 B1 1 C1 1 G2 1 A2 B1 2 C1 2 G2 2 −10 0
−1 0.3
0.1
,
A1 1 ,
A2 1 0.5I,
B0 1 ,
0.6 −12
2 −1
0.2
0.7
0.6
C1 2 2.1 ,
,
B2 1 ,
G1 1 I,
0.1
0
1.5 0 ,
C2 1 0.1 0.1 ,
D1 1 0.1,
D2 1 0.2,
−0.5 −0.5 ,
−1 1.3
0.1
−12
1
B0 2 ,
A2 2 0.1I,
,
,
A1 2 0.7 −1.1
0.2
−2 −14.3
0.1
0.3
1 0
,
B2 2 ,
G1 2 ,
C2 2 2 ,
0.3
0
1 1
1.2 0.7 ,
C2 2 0.1 0.1 ,
D1 2 0.2, D2 2 0.2,
−0.5 −0.5 .
4.1
The transition probability matrix is considered as
−5 5
Π
,
3 −3
−0.6 0.6
Λ
.
0.5 −0.5
4.2
In this example, the weighting matrix Ws in Fω s WsFs is supposed to be Ws 5/s 5. Its state-space realization is given as 2.5 with Aω −5, Bω 5 and Cω 1. Also,
Journal of Applied Mathematics
System mode
18
2
1
5
10
15
20
Times (s)
25
30
Figure 1: System jumping mode.
we assume that τ1 0.6, τ2 0.4, μ1 0.4, μ2 0.3. For γ 2.0, by the Theorem 3.4 in this
paper, the filter matrices are obtained as
−3.1724 1.1747
0.6403
Af11 ,
Bf11 ,
−1.1416 −0.5011
−0.0123
Cf11 4.3588 −0.1614 ,
−2.3595 −0.9045
0.5199
Af12 ,
Bf12 ,
−4.1454 −3.4820
0.8936
Cf12 5.5396 3.8431 ,
−0.3648 0.0560
0.0161
Af21 ,
Bf21 ,
−0.1076 −2.8163
1.0228
Cf21 −0.0077 4.4208 ,
−3.4133 0.0210
0.6410
Af22 ,
Bf22 ,
0.0135 −0.5002
−0.0114
Cf22 4.8422 −0.0192 .
4.3
For simulation purposes, we assume the initial condition x0 0.6 − 0.6T . The time
delays are τ1 t 0.2 0.4 sint, τ2 t 0.1 0.3 cost. The control input ut is chosen
to be sinte−2t . The unknown input νt t ∈ 0 30 is assumed to be the band-limited
white noise. The fault signal ft is simulated as a square wave signal with unit amplitude
that occurred from the 10 s to 20 s. Figures 1–5 illustrate the simulation results. The possible
realizations of the Markovian jumping modes of system and delay are plotted in Figures
1 and 2, respectively, where the initial modes are assumed to be r0 1 and s0 1.
Figure 3 shows the unknown input νt. Figure 4 shows the residual signal. Figure 5 is the
Journal of Applied Mathematics
19
Delay mode
2
1
5
10
15
20
25
30
25
30
Times (s)
Figure 2: Delay jumping mode.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
5
10
15
20
Times (s)
Figure 3: The unknown input νt.
simulation results of the evaluation function fr. Under the above conditions, with a selected
threshold Jth 0.1486, the simulation of evaluation function fr with fault shows that
( 10.9 T
r trtdt 0.1492 > Jth. Thus, the appeared fault can be detected after 0.9 s. The
0
simulation results demonstrate that the designed fault detection filter is feasible and effective.
Remark 4.2. In this study, the fault signal ft is assumed to be a square wave signal that
occurred from the 10 s to 20 s. Figure 4 shows that the generated residual signal is sensitive
to the fault and possesses robustness to exogenous disturbance. Furthermore, if no less than
one fault appears in the systems, the designed filter is also effective to estimate fault.
20
Journal of Applied Mathematics
0.7
0.6
Generated residual r(t)
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
0
5
10
15
20
25
30
Times (s)
Figure 4: Generated residual signal rt.
2
Residual evaluation function f(r)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
Times (s)
Fault case
Fault-free case
Figure 5: Evolution of residual evaluation function fr.
5. Conclusion
The problem of fault detection for a class of stochastic MJS is investigated in this paper.
Different system mode and delay mode are considered in the model. By using the Lyapunov
functional, a mixed mode-dependent sufficient condition is developed to design the stable
filter. A numerical example demonstrates the effectiveness of the given method.
Journal of Applied Mathematics
21
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities
103.1.2E022050205 and the National Natural Science Foundation of China Grant no.
61004046.
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