Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 409819, 9 pages
http://dx.doi.org/10.1155/2014/409819
Research Article
Stability Criteria for Singular Stochastic Hybrid Systems with
Mode-Dependent Time-Varying Delay
Ming Zhao, Yueying Wang, Pingfang Zhou, and Dengping Duan
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Correspondence should be addressed to Ming Zhao; [email protected]
Received 31 October 2013; Revised 19 January 2014; Accepted 14 February 2014; Published 20 March 2014
Academic Editor: Daoyi Xu
Copyright © 2014 Ming Zhao 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.
This paper provides a delay-dependent criterion for a class of singular stochastic hybrid systems with mode-dependent time-varying
delay. In order to reduce conservatism, a new Lyapunov-Krasovskii functional is constructed by decomposing the delay interval into
multiple subintervals. Based on the new functional, a stability criterion is derived in terms of strict linear matrix inequality (LMI),
which guarantees that the considered system is regular, impulse-free, and mean-square exponentially stable. Numerical examples
are presented to illustrate the effectiveness of proposed method.
1. Introduction
Over the past years, a lot of attention has been devoted
to singular systems, which are also referred to as descriptor systems, generalized state-space systems, differentialalgebraic systems, or semistate systems, because they can
better describe physical systems than regular ones. Since
the introduction of the first model for a jump system by
Krasovskii and Lidskiı̆ [1], the Markovian jump systems have
become more popular in the area of control and operation
research due to the fact that they are more powerful and
appropriate to model varieties of systems mainly those with
abrupt changes in their structure [2–6]. Consequently, singular hybrid systems or say singular systems with Markovian
switching have been extensively studied in recent years. For
more details on this class of systems, we refer the readers to
see [7–10] and the references therein.
As is well known, environmental noise exists and cannot
be neglected in many dynamical systems [11]. The systems
with noise are called stochastic system. Up to date, some
results have been reported on stochastic systems with Markovian switching [12–16]. It should be pointed out that the
problems of stability and control for singular stochastic
systems with Markovian switching or say singular stochastic
hybrid systems are more complicated than that for regular
ones, because the singular systems usually have three types
of modes, namely, finite-dynamic modes, impulsive modes,
and nondynamic modes, while the latter two do not appear in
state-space systems. That is the reason why only a few results
have been reported on this class of systems in the literature.
In [17], the stochastic stability and stochastic stabilization
for a class of nonlinear singular stochastic hybrid systems
were investigated. The problem of sliding mode control for
singular stochastic hybrid systems was investigated in [18].
In [19], under an assumption, authors study the problem of
mean-square stability for singular stochastic systems with
Markovian switching. However, no time delay was considered in [17–19], which often causes instability and poor
performance of the systems [20]. In addition, the stability
conditions in [17–19] are difficult to calculate since the
equality constraints are used, which are often fragile and
usually not met perfectly. To the author’s best knowledge,
the problem of delay-dependent mean-square exponential
stability for singular stochastic hybrid systems with modedependent time-varying delay has not been fully investigated,
which remains important and challenging.
This paper will focus on the derivation of delaydependent mean-square exponential stability for a class of
singular stochastic hybrid systems with mode-dependent
time-varying delay. Inspired by the discretized Lyapunov
functional method proposed by Gu [21], a new LyapunovKrasovskii functional is constructed to reduce conservatism.
Based on the functional, a delay-dependent stability criterion
is proposed, which guarantees that the system is regular,
2
Abstract and Applied Analysis
impulse-free, and mean-square exponentially stable. The
criterion is formulated in terms of strict LMI, which can be
easily solved by standard software. Finally, two examples are
given to illustrate the effectiveness of proposed method.
proposed method can also be easily extended to systems with
multiple and distributed delays.
The following definition and lemma will be used.
Notation. R𝑛 denotes the 𝑛-dimensional Euclidean space;
R𝑚×𝑛 is the set of all 𝑚 × 𝑛 real matrices. 𝐶𝑛,𝑑2 =
𝐶([−𝑑2 , 0], R𝑛 ) denotes the Banach space of continuous
vector functions mapping the interval [−𝑑2 , 0] into R𝑛 with
norm ‖𝜑(𝑡)‖𝑑2 = sup−𝑑2≤𝑠≤0 ‖𝜑(𝑠)‖. (Ω, F, P) is a probability
space, Ω is the sample space, F is the 𝜎-algebra of subsets
of the sample space, and P is the probability measure on F.
E{⋅} denotes the expectation operator with respect to some
probability measure P. The superscript “∗” denotes the term
that is induced by symmetry.
Definition 2 (see [22]). (1) System (2) is said to be regular and
impulse-free for any time delay 𝑑𝑖 (𝑡) satisfying (3), if the pairs
(𝐸, 𝐴 𝑖 ) and (𝐸, 𝐴 𝑖 +𝐴 𝑑𝑖 ) are regular and impulse-free for each
𝑖 ∈ S.
(2) System (2) is said to be mean-square exponentially
stable, if there exist scalars 𝛼 > 0 and 𝛽 > 0 such that
𝐸{‖𝑥(𝑡)‖2 } ≤ 𝛼𝑒−𝛽𝑡 ‖𝜙(𝑡)‖2𝑑2 , 𝑡 > 0.
(3) System (2) is said to be mean-square exponentially
admissible, if it is regular, impulse-free, and mean-square
exponentially stable.
Lemma 3 (see [23]). Suppose that a positive continuous
function 𝑓(𝑡) satisfies
2. Problem Formulation and Preliminaries
Let {𝑟𝑡 , 𝑡 ≥ 0} be a continuous-time Markov process with a
right continuous trajectory taking values in a finite set S =
{1, 2, . . . , 𝑠} with transition probability matrix Λ = {𝜋𝑖𝑗 } given
by
P [𝑟𝑡+Δ𝑡 = 𝑗 | 𝑟𝑡 = 𝑖] = {
𝜋𝑖𝑗 Δ + 𝑜 (Δ)
1 + 𝜋𝑖𝑗 Δ + 𝑜 (Δ)
if 𝑗 ≠ 𝑖
if 𝑗 = 𝑖,
(1)
where limΔ → 0 𝑜(Δ)/Δ = 0, 𝜋𝑖𝑗 > 0, 𝑗 ≠ 𝑖, and 𝜋𝑖𝑖 = − ∑𝑗 ≠ 𝑖 𝜋𝑖𝑗
for each 𝑖 ∈ S.
Fix a probability space (Ω, F, P) and consider the singular
stochastic system with Markovian switching as follows:
E {𝑓 (𝑡)} ≤ 𝜆 1 E { sup 𝑓 (𝑡)} + 𝜆 2 𝑒−𝜀𝑡 ,
𝑡−𝑑≤𝑠≤𝑡
where 𝜀 > 0, 0 < 𝜆 1 < 1, 0 < 𝜆 1 𝑒𝜀𝛼 < 1, 𝜆 2 > 0, and 𝛼 > 0;
then
𝜆 2 𝑒−𝜀𝑡
󵄩
󵄩
.
E {𝑓 (𝑡)} ≤ 𝑒−𝜀𝑡 󵄩󵄩󵄩𝑓 (𝑠)󵄩󵄩󵄩𝑑 +
1 − 𝜆 1 𝑒𝜀𝛼
1
2𝑥𝑇 𝑦 ≤ 𝛾𝑥𝑇 𝑃−1 𝑥 + 𝑦𝑇 𝑃𝑦.
𝛾
+ [𝐶 (𝑟𝑡 ) 𝑥 (𝑡) + 𝐷 (𝑟𝑡 ) (𝑥 − 𝑑 (𝑡, 𝑟𝑡 ))] 𝑑𝜛 (𝑡) (2)
∀𝑡 ∈ [−𝑑, 𝑡] ,
𝑛
where 𝑥(𝑡) ∈ R is the state vector, 𝜛(𝑡) is a scalar Brownian
motion defined on a probability space. 𝐸 ∈ R𝑛×𝑛 may be
singular; we assume that rank 𝐸 = 𝑟 ≤ 𝑛. 𝐴(𝑟𝑡 ), 𝐴 𝑑 (𝑟𝑡 ),
𝐶(𝑟𝑡 ), and 𝐷(𝑟𝑡 ) are real constant matrices of appropriate
dimensions.
For notational simplicity, in the sequel, for each possible
𝑟𝑡 ∈ 𝑖, 𝑖 ∈ S, a matrix 𝑀(𝑟𝑡 ) will be denoted by 𝑀𝑖 ; for example,
𝐴(𝑟𝑡 ) is denoted by 𝐴 𝑖 , 𝐴 𝑑 (𝑟𝑡 ) is denoted by 𝐴 𝑑𝑖 , and so on.
The mode-dependent time-varying delay 𝑑𝑖 (𝑡) is a timevarying continuous function that satisfies
(3)
𝑑 ̇ (𝑡) ≤ 𝜇 < 1, ∀𝑖 ∈ S,
0 ≤ 𝑑 (𝑡) ≤ 𝑑 ,
𝑖
𝑖
𝑖
𝑖
where 𝑑𝑖 and 𝜇𝑖 are constants; 𝜙(𝑡) ∈ [−𝑑, 0] is the initial
condition of continuous state with 𝑑 = max{𝑑𝑖 , 𝑖 ∈ S}.
The following preliminary assumption is made for system
(2).
Assumption 1. For singular stochastic hybrid system (2), the
following condition is satisfied:
rank (𝐸) = rank [𝐸 𝐶𝑖 𝐷𝑖 ] ,
𝑖 ∈ S.
(4)
The objective of this paper is to present a criterion for
mean-square exponential stability of system (2). It should
be pointed out that, for simplicity only, we do not consider
uncertainties and disturbance input in our models. The
(6)
Lemma 4 (see [24]). For any vectors 𝑥, 𝑦 ∈ R𝑛 , scalar 𝛾 > 0,
and matrix 𝑃 > 0 with appropriate dimension, the following
inequality is always satisfied:
𝐸𝑑𝑥 (𝑡) = [𝐴 (𝑟𝑡 ) 𝑥 (𝑡) + 𝐴 𝑑 (𝑟𝑡 ) (𝑥 − 𝑑 (𝑡, 𝑟𝑡 ))] 𝑑𝑡
𝑥 (𝑡) = 𝜙 (𝑡) ,
(5)
(7)
3. Stability of Singular Stochastic
Hybrid Systems
In this section, we study the mean-square exponential stability of singular hybrid system (2).
Theorem 5. For any delays 𝑑𝑖 (𝑡) satisfying (3), the singular
stochastic hybrid system (2) is mean-square exponentially
admissible for 𝛿 > 0, if there exist symmetric matrices 𝑃𝑖 > 0,
𝑄𝑘𝑖 > 0, 𝑈𝑘 > 0, 𝑉𝑘 > 0, 𝑄𝑘 > 0, 𝑊𝑖 > 0, and 𝑊 > 0
and matrices 𝑆, 𝑀𝑘𝑙 , 𝑘 = 1, 2, . . . 𝑁, 𝑙 = 0, 1, . . . 𝑁 + 1, with
appropriate dimensions such that for each 𝑖 ∈ S,
̂ 𝑖12
̂ 𝑖13
Ξ
Ξ
Ω𝑖
𝑁
[
[ ∗ −𝛿 ∑ 𝑈
0
[
𝑘
[
𝑘=1
Θ𝑖 = [
[∗
∗
−𝑁𝑖
[
[∗
∗
∗
∗
∗
[∗
𝑠
∑𝜋𝑖𝑗 𝑄𝑘𝑗 ≤ 𝑄𝑘 ,
̂ 𝑖14 Ξ
̂ 𝑖15
Ξ
0
0
̂ 𝑖44
Ξ
∗
]
0 ]
]
]
] < 0,
0 ]
]
0 ]
̂ 𝑖55 ]
Ξ
𝑘 = 1, 2, . . . 𝑁,
𝑗=1
(8)
(9)
𝑠
∑𝜋𝑖𝑗 𝑊𝑗 ≤ 𝑊,
𝑗=1
(10)
Abstract and Applied Analysis
3
where
𝑇
𝑇
𝑇
𝑇
𝑇
𝑇
𝑀𝑘1
⋅ ⋅ ⋅ 𝑀𝑘(𝑁−1)
𝑀𝑘𝑁
𝑀𝑘(𝑁+1)
] ,
𝑀𝑘 = [𝑀𝑘0
𝑁
𝑁𝑖 = 𝐸𝑇 𝑃𝑖 𝐸 + 𝛿 ∑ 𝑉𝑘 ,
𝑘=1
𝑁
𝑁
𝑘=1
𝑘=1
𝑇
̂ 𝑖12 = [𝛿 ∑ 𝑈𝑘 𝐴 𝑖 0 ⋅ ⋅ ⋅ 0 𝛿 ∑ 𝑈𝑘 𝐴 𝑑𝑖 ] ,
Ξ
̂ 𝑖13 = [𝑁𝑇 𝐶𝑖 0 ⋅ ⋅ ⋅ 0 𝑁𝑇 𝐷𝑖 ]𝑇 ,
Ξ
𝑖
𝑖
̂ 𝑖14 = Ξ
̂ 𝑖15 = [𝑀1 𝑀2 ⋅ ⋅ ⋅ 𝑀𝑁] ,
Ξ
(11)
̂ 𝑖44 = − diag (𝑉1 , 𝑉2 , . . . , 𝑉𝑁) ,
Ξ
̂ 𝑖55 = − diag (𝑈1 , 𝑈2 , . . . , 𝑈𝑁) ,
Ξ
Ξ𝑖11 −𝑀10 𝐸 + 𝑀20 𝐸 −𝑀20 𝐸 + 𝑀30 𝐸
[∗
Ξ𝑖22
−𝑀21 𝐸 + 𝑀31 𝐸
[
[∗
∗
Ξ𝑖33
[
[ ..
.
..
..
Ω𝑖1 = [ .
.
[
[∗
∗
∗
[
[∗
∗
∗
∗
∗
[∗
⋅ ⋅ ⋅ Ξ𝑖1𝑁
⋅ ⋅ ⋅ Ξ𝑖2𝑁
⋅ ⋅ ⋅ Ξ3𝑁
.
d ..
−𝑀𝑁0
−𝑀𝑁1
−𝑀𝑁𝑁
..
.
⋅ ⋅ ⋅ Ξ𝑁𝑁 −𝑀𝑁(𝑁−1)
⋅ ⋅ ⋅ ∗ Ξ(𝑁+1)(𝑁+1)
⋅⋅⋅ ∗
∗
Ξ𝑖1(𝑁+2)
Ξ𝑖2(𝑁+2)
Ξ𝑖3(𝑁+2)
..
.
]
]
]
]
]
]
]
Ξ𝑖𝑁(𝑁+2) ]
]
Ξ(𝑁+1)(𝑁+2) ]
Ξ(𝑁+2)(𝑁+2) ]
𝑇
𝑇
+ 𝐸𝑇 𝑀(𝑁+1)(𝑁+1)
,
Ξ𝑖(𝑁+1)(𝑁+2) = −𝐸𝑇 𝑀𝑁(𝑁+1)
with
Ξ(𝑁+2)(𝑁+2) = − (1 − 𝜇𝑖 ) 𝑊𝑖 ,
Ξ𝑖11 =
𝐴𝑇𝑖 (𝑃𝑖 𝐸
𝑠
𝑇
𝑇
𝑇
+ 𝑅𝑆 ) + (𝐸 𝑃𝑖 + 𝑆𝑅 ) 𝐴 𝑖
(12)
and 𝑅 ∈ R𝑛×(𝑛−𝑟) is any matrix with full rank and satisfying
𝐸𝑇 𝑅 = 0.
+ ∑𝜋𝑖𝑗 𝐸𝑇 𝑃𝑗 𝐸
𝑗=1
𝑁
𝑇
+𝑄1𝑖 + 𝑊𝑖 + 𝛿 ∑ 𝑄𝑘 + 𝑑𝑊 + 𝑀10 𝐸 + 𝐸𝑇 𝑀10
,
𝑘=1
Ξ𝑖1𝑁 = −𝑀(𝑁−1)0 𝐸 + 𝑀𝑁0 𝐸,
𝑇
Ξ𝑖1(𝑁+2) = (𝐸𝑇 𝑃𝑖 + 𝑆𝑅𝑇 ) 𝐴 𝑑𝑖 + 𝐸𝑇 𝑀1(𝑁+1)
,
Ξ𝑖22 = 𝑄2𝑖 − 𝑄1𝑖 − 𝑀11 𝐸 − 𝐸
𝑇
𝑇
𝑀11
+ 𝑀21 𝐸 + 𝐸
𝑇
𝑇
𝑀21
,
Ξ𝑖2𝑁 = −𝑀(𝑁−1)1 𝐸 + 𝑀𝑁1 𝐸,
𝑇
𝑇
Ξ𝑖2(𝑁+2) = −𝐸𝑇 𝑀1(𝑁+1)
+ 𝐸𝑇 𝑀2(𝑁+1)
,
𝑇
𝑇
+ 𝑀32 𝐸 + 𝐸𝑇 𝑀32
,
Ξ𝑖33 = 𝑄3𝑖 − 𝑄2𝑖 − 𝑀22 𝐸 − 𝐸𝑇 𝑀22
Ξ3𝑁 = −𝑀(𝑁−1)2 𝐸 + 𝑀𝑁2 𝐸,
𝑇
Ξ𝑁𝑁 = 𝑄𝑁𝑖 − 𝑄(𝑁−1)𝑖 − 𝑀(𝑁−1)(𝑁−1) 𝐸 − 𝐸𝑇 𝑀(𝑁−1)(𝑁−1)
𝑇
+𝑀𝑁(𝑁−1) 𝐸 + 𝐸𝑇 𝑀𝑁(𝑁−1)
,
𝑇
𝑇
+ 𝐸𝑇 𝑀𝑁(𝑁+1)
,
Ξ𝑖𝑁(𝑁+2) = −𝐸𝑇 𝑀(𝑁−1)(𝑁+1)
𝑇
Ξ(𝑁+1)(𝑁+1) = −𝑄𝑁𝑖 − 𝑀𝑁𝑁𝐸 − 𝐸𝑇 𝑀𝑁𝑁
,
Proof. We first show that the system (2) is regular and
impulse-free.
Since rank 𝐸 = 𝑟 ≤ 𝑛, there exist nonsingular matrices 𝐺
and 𝐻 such that
𝐼 0
𝐸̂ = 𝐺𝐸𝐻 = [ 𝑟 ] .
0 0
(13)
Then, 𝑅 can be parameterized as 𝑅 = 𝐺𝑇 [ Φ0 ], where Φ ∈
is any nonsingular matrix.
R
Define
(𝑛−𝑟)×(𝑛−𝑟)
̂𝑖 = 𝐺𝐴 𝑖 𝐻 = [𝐴 𝑖11 𝐴 𝑖12 ] ,
𝐴
𝐴 𝑖21 𝐴 𝑖22
−𝑇
−1
𝐺 𝑃𝑖 𝐺
𝑃 𝑃
= [ 𝑖11 𝑖12 ] ,
𝑃𝑖21 𝑃𝑖22
𝑆
𝐻 𝑆 = [ 11 ] .
𝑆21
𝑇
(14)
Premultiplying and postmultiplying Ξ𝑖11 < 0 by 𝐻𝑇 and
𝐻, respectively, we have
𝑇
𝐴𝑇𝑖22 Φ𝑆21
+ 𝑆21 Φ𝑇 𝐴 𝑖22 < 0
(15)
and thus 𝐴 𝑖22 is nonsingular. Otherwise, supposing 𝐴 𝑖22
is singular, there must exist a nonzero vector 𝜁 ∈ R𝑛−𝑟
4
Abstract and Applied Analysis
which ensures that 𝐴 𝑖22 𝜁 = 0. Then, we can conclude that
𝑇
𝜁𝑇 (𝐴𝑇𝑖22 Φ𝑆21
+ 𝑆21 Φ𝑇 𝐴 𝑖22 )𝜁 = 0, and this contradicts (15). So
̂𝐴
̂𝑖 ) is regular and
𝐴 𝑖22 is nonsingular, and thus the pair (𝐸,
̂𝑖 ),
impulse-free for each 𝑖 ∈ S. Since det(𝑠𝐸 − 𝐴 𝑖 ) = det(𝑠𝐸̂ − 𝐴
we can easily see that the pair (𝐸, 𝐴 𝑖 ) is regular and impulsefree. On the other hand, premultiplying and postmultiplying
Ω𝑖1 < 0 by [𝐼 𝐼 ⋅ ⋅ ⋅ 𝐼] and [𝐼 𝐼 ⋅ ⋅ ⋅ 𝐼]𝑇 , respectively, we
have
𝑇
𝑠
+ 𝑥𝑇 (𝑡) [ ∑𝜋𝑖𝑗 𝐸𝑇 𝑃𝑗 𝐸] 𝑥 (𝑡)
]
[𝑗=1
+ 𝑔𝑇 (𝑡) 𝐸𝑇 𝑃𝑖 𝐸𝑔 (𝑡)
𝑁
+ ∑ 𝑥𝑇 (𝑡 − (𝑘 − 1) 𝛿) 𝑄𝑘𝑖 𝑥 (𝑡 − (𝑘 − 1) 𝛿)
𝑘=1
𝑁
− ∑ 𝑥𝑇 (𝑡 − 𝑘𝛿) 𝑄𝑘𝑖 𝑥 (𝑡 − 𝑘𝛿)
𝑇
(𝐴 𝑖 + 𝐴 𝑑𝑖 ) (𝑃𝑖 𝐸 + 𝑅𝑆 )
+ (𝐸𝑇 𝑃𝑖 + 𝑆𝑅𝑇 ) (𝐴 𝑖 + 𝐴 𝑑𝑖 )
𝑠
𝑘=1
𝑁
(16)
+∑∫
𝑘=1 𝑡−𝑘𝛿
+ ∑𝜋𝑖𝑗 𝐸𝑇 𝑃𝑗 𝐸 < 0.
− (1 − 𝜇𝑖 ) 𝑥𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊𝑖 𝑥 (𝑡 − 𝑑𝑖 (𝑡))
𝑠
𝑡−(𝑘−1)𝛿
+∑∫
𝑘=1 𝑡−𝑘𝛿
𝑡
+∫
𝑡−𝑑𝑖 (𝑡)
−(𝑘−1)𝛿
+∑∫
𝑁
𝑘=1
𝑘=1 −𝑘𝛿
𝑡+𝛽
−(𝑘−1)𝛿
+∑∫
𝑘=1 −𝑘𝛿
𝑁
∫
𝑡
𝑡+𝛽
−(𝑘−1)𝛿
+∑∫
𝑘=1 −𝑘𝛿
∫
𝑡
𝑡+𝛽
0
+∫ ∫
𝑁
−∑∫
𝑡
−𝑑 𝑡+𝛽
𝑁
𝑘=1
𝑁
−∑∫
(17)
𝑔𝑇 (𝛼) 𝑉𝑘 𝑔 (𝛼) 𝑑𝛼
𝑁
𝑔 (𝛼) 𝑉𝑘 𝑔 (𝛼) 𝑑𝛼 𝑑𝛽
+ 𝛿 ∑ 𝑥𝑇 (𝑡) 𝑄𝑘 𝑥 (𝑡)
𝑥𝑇 (𝛼) 𝑄𝑘 𝑥 (𝛼) 𝑑𝛼 𝑑𝛽
−∑∫
𝑘=1
𝑁
+ 2𝑥𝑇 (𝑡) 𝐸𝑇 𝑃 (𝑟𝑡 ) 𝐸𝑔 (𝑡) 𝑑𝜛 (𝑡) ,
(18)
−∫
𝑥𝑇 (𝛼) 𝑊𝑥 (𝛼) 𝑑𝛼.
(20)
From (19), for any appropriately dimensioned matrix 𝑀𝑘 ,
𝑘 = 1, 2, . . . 𝑁, we have
𝑁
2 ∑ 𝜁𝑇 (𝑡) 𝑀𝑘
(19)
𝑘=1
× [𝐸𝑥 (𝑡 − (𝑘 − 1) 𝛿) − 𝐸𝑥 (𝑡 − 𝑘𝛿)
−∫
≤ 2𝑥 (𝑡) (𝐸 𝑃𝑖 + 𝑆𝑅 ) 𝑓 (𝑡)
𝑡
𝑡−𝑑
where
𝐿𝑉 (𝑥𝑡 , 𝑖, 𝑡)
𝑥𝑇 (𝛼) 𝑄𝑘 𝑥 (𝛼) 𝑑𝛼
+ 𝑑𝑥𝑇 (𝑡) 𝑊𝑥 (𝑡)
By Itô’s Lemma, we have
𝑑𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) = 𝐿𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) 𝑑𝑡
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑥𝑇 (𝛼) 𝑊𝑥 (𝛼) 𝑑𝛼 𝑑𝛽,
𝑔 (𝑡) = [𝐶 (𝑟𝑡 ) 𝑥 (𝑡) + 𝐷 (𝑟𝑡 ) (𝑡 − 𝑑𝑖 (𝑡))] .
𝑇
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑇
𝑓 (𝑡) = [𝐴 (𝑟𝑡 ) 𝑥 (𝑡) + 𝐴 𝑑 (𝑟𝑡 ) (𝑡 − 𝑑𝑖 (𝑡))] ,
𝑇
𝑓𝑇 (𝛼) 𝑈𝑘 𝑓 (𝛼) 𝑑𝛼
+ 𝛿 ∑ 𝑔𝑇 (𝑡) 𝑉𝑘 𝑔 (𝑡)
where 𝛿 = 𝜏/𝑁 and 𝑁 is the number of divisions of the
interval [−𝑑, 0],
𝑇
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑇
𝑓 (𝛼) 𝑈𝑘 𝑓 (𝛼) 𝑑𝛼 𝑑𝛽
𝑥𝑇 (𝛼) 𝑊𝑗 𝑥 (𝛼) 𝑑𝛼
+ 𝛿 ∑ 𝑓𝑇 (𝑡) 𝑈𝑘 𝑓 (𝑡)
𝑥𝑇 (𝛼) 𝑊 (𝑟𝑡 ) 𝑥 (𝛼) 𝑑𝛼
∫
𝑡−𝑑𝑗 (𝑡)
𝑗=1
𝑥𝑇 (𝛼) 𝑄𝑘 (𝑟𝑡 ) 𝑥 (𝛼) 𝑑𝛼
𝑡
𝑡
+ ∑𝜋𝑖𝑗 ∫
𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) = 𝑥𝑇 (𝑡) (𝐸𝑇 𝑃 (𝑟𝑡 ) + 𝑆𝑅𝑇 ) 𝐸𝑥 (𝑡)
𝑁
𝑗=1
+ 𝑥 (𝑡) 𝑊𝑖 𝑥 (𝑡)
This implies that the pair (𝐸, 𝐴 𝑖 + 𝐴 𝑑𝑖 ) is regular and
impulse-free for each 𝑖 ∈ S according to Theorem 10.1 of [25].
Then, by Definition 2, system (2) is regular and impulse part.
In the following, we will prove the mean-square exponential stability of system (2). Define a new process {(𝑥𝑡 , 𝑟𝑡 ), 𝑡 ≥
0}, by 𝑥𝑡 (𝑠) = 𝑥(𝑡 + 𝑠), −2𝑑 ≤ 𝑠 ≤ 0; then {(𝑥𝑡 , 𝑟𝑡 ), 𝑡 ≥
0} is a Markov process with initial state (𝜙(⋅), 𝑟0 ). Choose a
Lyapunov function candidate as follows:
𝑁
𝑠
𝑥𝑇 (𝛼) ( ∑𝜋𝑖𝑗 𝑄𝑘𝑗 ) 𝑥 (𝛼) 𝑑𝛼
𝑇
𝑗=1
𝑁
𝑡−(𝑘−1)𝛿
𝑡−(𝑘−1)𝛿
𝑡−𝑘𝛿
𝑓 (𝛼) 𝑑𝛼 − ∫
𝑡−(𝑘−1)𝛿
𝑡−𝑘𝛿
𝑔 (𝛼) 𝑑𝜛 (𝛼)] = 0,
(21)
Abstract and Applied Analysis
5
where
𝑇
𝜁 (𝑡) = [𝑥𝑇 (𝑡) 𝑥𝑇 (𝑡 − 𝛿) 𝑥𝑇 (𝑡 − 2𝛿) ⋅ ⋅ ⋅ 𝑥𝑇 (𝑡 − (𝑁 − 1)𝛿) 𝑥𝑇 (𝑡 − 𝑁𝛿) 𝑥𝑇 (𝑡 − 𝑑𝑖 (𝑡))] .
(22)
𝑁
From Lemma 4, we obtain
+ 𝛿 ∑ 𝜁𝑇 (𝑡) 𝑀𝑘 𝑈𝑘−1 𝑀𝑘𝑇 𝜁 (𝑡)
𝑘=1
𝑁
2𝜁𝑇 (𝑡) 𝑀𝑘 ∫
𝑡−(𝑘−1)𝛿
𝑡−𝑘𝛿
+ (∫
−∑∫
𝑡−(𝑘−1)𝛿
𝑡−(𝑘−1)𝛿
𝑔 (𝛼) 𝑑𝜛 (𝛼)) 𝑉𝑘 (∫
𝑡−𝑘𝛿
𝑘=1 𝑡−𝑘𝛿
𝑔 (𝛼) 𝑑𝜛 (𝛼) ≤ 𝜁𝑇 (𝑡) 𝑀𝑘 𝑉𝑘−1 𝑀𝑘𝑇 𝜁 (𝑡)
𝑇
𝑡−𝑘𝛿
𝑡−(𝑘−1)𝛿
𝑁
−∑∫
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑔 (𝛼) 𝑑𝜛 (𝛼)) .
𝑔𝑇 (𝛼) 𝑉𝑘 𝑔 (𝛼) 𝑑𝛼
[𝜁𝑇 (𝑡) 𝑀𝑘 + 𝑓𝑇 (𝛼) 𝑈𝑘 ]
𝑇
× 𝑈𝑘−1 [𝜁𝑇 (𝑡) 𝑀𝑘 + 𝑓𝑇 (𝛼) 𝑈𝑘 ] 𝑑𝛼
(23)
𝑁
+ ∑ (∫
𝑡−𝑘𝛿
𝑘=1
Thus, it follows from (20)–(23) that
𝑡−(𝑘−1)𝛿
𝑇
𝑔 (𝛼) 𝑑𝜛 (𝛼))
𝑡−(𝑘−1)𝛿
× 𝑉𝑘 (∫
𝑡−𝑘𝛿
𝐿𝑉 (𝑥𝑡 , 𝑖, 𝑡)
𝑁
𝑇
𝑇
+∑∫
𝑇
≤ 2𝑥 (𝑡) (𝐸 𝑃𝑖 + 𝑆𝑅 ) 𝑓 (𝑡)
𝑘=1 𝑡−𝑘𝛿
𝑠
𝑠
𝑥𝑇 (𝛼) 𝑊𝑗 𝑥 (𝛼) 𝑑𝛼
𝑡−𝑑𝑗 (𝑡)
𝑗=1
−∑∫
𝑇
+ ∑ 𝑥 (𝑡 − (𝑘 − 1) 𝛿) 𝑄𝑘𝑖 𝑥 (𝑡 − (𝑘 − 1) 𝛿)
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑘=1
𝑁
𝑗=1
𝑡
𝑁
𝑁
𝑠
𝑥𝑇 (𝛼) ( ∑ 𝜋𝑖𝑗 𝑄𝑘𝑗 ) 𝑥 (𝛼) 𝑑𝛼
+ ∑𝜋𝑖𝑗 ∫
+ 𝑥 (𝑡) [∑ 𝜋𝑖𝑗 𝐸𝑇 𝑃𝑗 𝐸] 𝑥 (𝑡) + 𝑔𝑇 (𝑡) 𝐸𝑇 𝑃𝑖 𝐸𝑔 (𝑡)
]
[𝑗=1
𝑇
𝑡−(𝑘−1)𝛿
−∫
𝑇
− ∑ 𝑥 (𝑡 − 𝑘𝛿) 𝑄𝑘𝑖 𝑥 (𝑡 − 𝑘𝛿)
𝑔 (𝛼) 𝑑𝜛 (𝛼))
𝑥𝑇 (𝛼) 𝑄𝑘 𝑥 (𝛼) 𝑑𝛼
𝑡
𝑡−𝑑
𝑥𝑇 (𝛼) 𝑊𝑥 (𝛼) 𝑑𝛼.
𝑘=1
(24)
𝑇
+ 𝑥 (𝑡) 𝑊𝑖 𝑥 (𝑡)
Noting 𝜋𝑖𝑗 > 0 for 𝑗 ≠ 𝑖 and 𝜋𝑖𝑖 < 0, then we have
− (1 − 𝜇𝑖 ) 𝑥𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊𝑖 𝑥 (𝑡 − 𝑑𝑖 (𝑡))
𝑁
𝑁
𝑘=1
𝑘=1
+ 𝛿 ∑ 𝑓𝑇 (𝑡) 𝑈𝑘 𝑓 (𝑡) + 𝛿 ∑ 𝑔𝑇 (𝑡) 𝑉𝑘 𝑔 (𝑡)
𝑠
∑ 𝜋𝑖𝑗 ∫
𝑗=1
𝑘=1
𝑇
+ 2 ∑ 𝜁 (𝑡) 𝑀𝑘 [𝐸𝑥 (𝑡 − (𝑘 − 1) 𝛿) − 𝐸𝑥 (𝑡 − 𝑘𝛿)]
𝑘=1
𝑁
𝑡−(𝑘−1)𝛿
− 2 ∑ 𝜁𝑇 (𝑡) 𝑀𝑘 ∫
𝑘=1
𝑁
(𝑡) 𝑀𝑘 𝑉𝑘−1 𝑀𝑘𝑇 𝜁 (𝑡)
(25)
𝑠
𝑇
𝑥 (𝛼) ( ∑ 𝜋𝑖𝑗 𝑊𝑗 ) 𝑥 (𝛼) 𝑑𝛼.
𝑗=1,𝑗 ≠ 𝑖
Note
E { ∑ (∫
𝑘=1
𝑡−(𝑘−1)𝛿
𝑡−𝑘𝛿
× 𝑉𝑘 (∫
𝑇
𝑔 (𝛼) 𝑑𝜛 (𝛼))
𝑡−(𝑘−1)𝛿
𝑡−𝑘𝛿
𝑁
𝑇
+ ∑𝜁
𝑘=1
𝑡−𝑘𝛿
𝑓 (𝛼) 𝑑𝛼
𝑡
𝑡−𝑑
𝑁
𝑁
𝑥𝑇 (𝛼) 𝑊𝑗 𝑥 (𝛼) 𝑑𝛼
𝑡−𝑑𝑗 (𝑡)
≤∫
𝑁
+ 𝛿 ∑ 𝑥𝑇 (𝑡) 𝑄𝑘 𝑥 (𝑡) + 𝑑𝑥𝑇 (𝑡) 𝑊𝑥 (𝑡)
𝑡
= E {∑ ∫
𝑡−(𝑘−1)𝛿
𝑘=1 𝑡−𝑘𝛿
𝑔 (𝛼) 𝑑𝜛 (𝛼)) }
𝑔𝑇 (𝛼) 𝑉𝑘 𝑔 (𝛼) 𝑑𝛼} .
(26)
6
Abstract and Applied Analysis
Then, for each 𝑖 ∈ S, system (2) is equivalent to
From (9)-(10) and (24)–(26), we have
E𝐿𝑉 (𝑥𝑡 , 𝑖, 𝑡) ≤ E {𝜁𝑇 (𝑡) [Ω𝑖1 + Ω𝑖2 ] 𝜁 (𝑡)} ,
(27)
̂𝑖11 𝜁1 (𝑡) + 𝐴 𝑖𝑑1 𝜁1 (𝑡 − 𝑑𝑖 (𝑡))
𝑑𝜁1 (𝑡) = [𝐴
+𝐴 𝑖𝑑2 𝜁2 (𝑡 − 𝑑𝑖 (𝑡)) ] 𝑑𝑡
where
+ [𝐶𝑖11 𝜁1 (𝑡) + 𝐶𝑖12 𝜁2 (𝑡)
𝑁
Ω𝑖2 = 𝛿 ∑ Γ1𝑖𝑇 𝑈𝑘 Γ1𝑖 + Γ2𝑖𝑇 𝐸𝑇 𝑃𝑖 𝐸Γ2𝑖
𝑘=1
+
+𝐷𝑖11 𝜁1 (𝑡 − 𝑑𝑖 (𝑡))
𝑁
𝛿 ∑ Γ2𝑖𝑇 𝑉𝑘 Γ2𝑖
𝑘=1
(28)
𝑁
𝑁
𝑘=1
𝑘=1
+𝐷𝑖12 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))] 𝑑𝜛 (𝑡) ,
̂𝑖21 𝜁1 (𝑡) + 𝐴 𝑖𝑑3 𝜁1 (𝑡 − 𝑑𝑖 (𝑡))
−𝜁2 (𝑡) = 𝐴
+ ∑ 𝑀𝑘 𝑉𝑘−1 𝑀𝑘𝑇 + 𝛿 ∑ 𝑀𝑘 𝑈𝑘−1 𝑀𝑘𝑇
+𝐴 𝑖𝑑4 𝜁2 (𝑡 − 𝑑𝑖 (𝑡)) ,
𝜁 (𝑡) = 𝜓 (𝑡) = 𝐻−1 𝜙 (𝑡) ,
with
Γ1𝑖 = [𝐴 𝑖 0 0 ⋅ ⋅ ⋅ 0 𝐴 𝑑𝑖 ] ,
(29)
Γ2𝑖 = [𝐶𝑖 0 0 ⋅ ⋅ ⋅ 0 𝐷𝑖 ] .
To prove the mean-square exponential stability, we define
a new function as
where 𝜀 > 0. By Dynkin’s formula [26], we get that for each
𝑖 ∈ S,
(30)
𝑡
+ E {∫ 𝑒𝜀𝑠 [𝜀𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) − 𝜌‖𝑥 (𝑠)‖2 ] 𝑑𝑠} .
0
Since 𝐴 𝑖22 is nonsingular for each 𝑖 ∈ S, we set 𝐺 =
𝐼𝑟 𝐴 𝑖12 𝐴−1
𝑖22
0 𝐴−1
𝑖22
] 𝐺.
It is easy to see that
̂𝑖11 =
where 𝐴
𝐼𝑟 0
],
0 0
𝐺
𝑃̂
𝑃𝑖 𝐻 = [ ̂𝑖11
𝑃𝑖21
𝐻−1 𝐶𝑖 𝐻 = [
𝑃̂𝑖12
],
𝑃̂𝑖22
𝐶𝑖11 𝐶𝑖12
],
0
0
𝜁 (𝑡) = [
=
(31)
𝐴−1
𝑖22 𝐴 𝑖21 . Define
𝐴
𝐴
𝐺𝐴 𝑑𝑖 𝐻 = [ 𝑖𝑑1 𝑖𝑑2 ] ,
𝐴 𝑖𝑑3 𝐴 𝑖𝑑4
𝑃𝑖 = 𝑃𝑖 𝐸 + 𝑅𝑆𝑇 ,
−𝑇
̂
𝐴
0
𝐺𝐴 𝑖 𝐻 = [ ̂𝑖11 ] ,
𝐴 𝑖21 𝐼
̂
𝐴 𝑖11 −𝐴 𝑖12 𝐴−1
𝑖22 𝐴 𝑖21 and 𝐴 𝑖21
𝐻𝑇 𝑊𝑖 𝐻 = [
(34)
E {𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡)} ≤ E {𝑉 (𝑥0 , 𝑟0 , 0)}
E𝐿𝑉 (𝑥𝑡 , 𝑖, 𝑡) ≤ −𝜌E‖𝑥 (𝑡)‖2 .
𝐺𝐸𝐻 = [
𝑡 ∈ [−𝑑2 , 0] .
𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) = 𝑒𝜀𝑡 𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡) ,
From (8) and using Schur complement lemma, it is easy
to see that there exists a scalar 𝜌 > 0 such that for each 𝑖 ∈ S,
[
(33)
(35)
By using the similar method of [27], it can be seen from
(17), (34), and (35) that, if 𝜀 is chosen small enough, a constant
𝜅 > 0 can be found such that
󵄩2
󵄩
min {𝜆 min (𝑃̂𝑖11 )} E {󵄩󵄩󵄩𝜁1 (𝑡)󵄩󵄩󵄩 }
𝑖∈S
≤ E {𝑥𝑇 (𝑡) 𝐸𝑇 𝑃𝑖 𝑥 (𝑡)}
(36)
󵄩2
󵄩
≤ E {𝑉 (𝑥𝑡 , 𝑟𝑡 , 𝑡)} ≤ 𝜅𝑒−𝜀𝑡 󵄩󵄩󵄩𝜙 (𝑡)󵄩󵄩󵄩𝑑 .
Hence, for any 𝑡 > 0
𝑊11 𝑊12
],
∗ 𝑊22
𝐷
𝐷
𝐻−1 𝐷𝑖 𝐻 = [ 𝑖11 𝑖12 ] ,
0
0
𝜁1 (𝑡)
] = 𝐻−1 𝑥 (𝑡) .
𝜁2 (𝑡)
(32)
𝑇
From 𝐸𝑇 𝑃𝑖 = 𝑃𝑖 𝐸, we can deduce that 𝑃̂𝑖11 > 0 and 𝑃̂𝑖12 =
0 for each 𝑖 ∈ S.
󵄩2
󵄩
󵄩2
󵄩
E {󵄩󵄩󵄩𝜁1 (𝑡)󵄩󵄩󵄩 } ≤ 𝜂𝑒−𝜀𝑡 󵄩󵄩󵄩𝜙 (𝑡)󵄩󵄩󵄩𝑑 ,
(37)
−1
where 𝜂 = (min𝑖∈S {𝜆 min (𝑃̂𝑖11 )}) 𝜅. Define
̂𝑖21 𝜁1 (𝑡) + 𝐴 𝑖𝑑3 𝜁1 (𝑡 − 𝑑𝑖 (𝑡)) .
𝑒 (𝑡) = 𝐴
(38)
Then, from (37), there exists a constant 𝑚 > 0 such that
when 𝑡 > 0
󵄩2
󵄩
(39)
E {‖𝑒(𝑡)‖2 } ≤ 𝑚𝑒−𝜀𝑡 󵄩󵄩󵄩𝜙 (𝑡)󵄩󵄩󵄩𝑑 .
Define a function as follows:
𝐽 (𝑡) = 𝜁2𝑇 (𝑡) 𝑊22 𝜁2 (𝑡) − 𝜁2𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊22 𝜁2 (𝑡 − 𝑑𝑖 (𝑡)) .
(40)
Abstract and Applied Analysis
7
By premultiplying the second equation of (33) with
we get
𝑇
𝜁2𝑇 (𝑡)𝑃̂𝑖22
,
𝑇
𝑇
𝜁2 (𝑡) + 𝜁2𝑇 (𝑡) 𝑃̂𝑖22
𝐴 𝑖𝑑4 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
𝜁2𝑇 (𝑡) 𝑃̂𝑖22
+
𝑇
𝑒 (𝑡)
𝜁2𝑇 (𝑡) 𝑃̂𝑖22
= 0.
(41)
where 0 < 𝛿 < min{𝜀, 𝑑−1 ln 𝜏3 }, 𝑓(𝑡) = 𝜁2𝑇 (𝑡)𝑊22 𝜁2 (𝑡), and
𝜏4 = (𝜏3 𝜏1 )−1 𝑚 max𝑖∈S ‖𝑊22 ‖2 ‖𝜙(𝑡)‖2𝑑 .
Therefore, by Lemma 3, the above inequality yields that
󵄩2
󵄩
−1
−𝛿𝑡 󵄩
󵄩󵄩𝜁2 (𝑡)󵄩󵄩󵄩2
E󵄩󵄩󵄩𝜁2 (𝑡)󵄩󵄩󵄩 ≤ 𝜆−1
min (𝑊22 ) 𝜆 max (𝑊22 ) 𝑒
󵄩𝑑
󵄩
+
Adding (41) to (40) yields that
𝑇
𝐽 (𝑡) = 𝜁2𝑇 (𝑡) (𝑃̂𝑖22
+ 𝑃̂𝑖22 + 𝑊22 ) 𝜁2 (𝑡)
𝑇
𝑒 (𝑡)
− 𝜁2𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊22 𝜁2 (𝑡 − 𝑑𝑖 (𝑡)) + 2𝜁2𝑇 (𝑡) 𝑃̂𝑖22
𝜁2 (𝑡)
]
𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
𝑇
𝑇
𝜁2 (𝑡)
𝐴 𝑖𝑑4
𝑃̂𝑇 + 𝑃̂𝑖22 + 𝑊22 𝑃̂𝑖22
× [ 𝑖22
][
]
𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
∗
−𝑊22
𝑇
+ 𝜏1 𝜁2𝑇 (𝑡) 𝜁2 (𝑡) + 𝜏1−1 𝑒𝑇 (𝑡) 𝑃̂𝑖22 𝑃̂𝑖22
𝑒 (𝑡) ,
(42)
where 𝜏1 is any positive scalar. Premultiplying and postmultiplying Ω𝑖1 < 0 by diag(𝐻𝑇 ⋅ ⋅ ⋅ 𝐻𝑇 )and diag(𝐻 ⋅ ⋅ ⋅ 𝐻),
respectively, we get a constant 𝜏2 > 0 such that
𝑇
𝜏𝐼 0
𝐴 𝑖𝑑4
𝑃̂𝑇 + 𝑃̂𝑖22 + 𝑊22 𝑃̂𝑖22
] ≤ −[ 2
[ 𝑖22
].
0 0
∗
−𝑊22
(49)
which means that the system (2) is mean-square exponentially 10 stable. This completes the proof.
𝑇
+ 2𝜁2𝑇 (𝑡) 𝑃̂𝑖22
𝐴 𝑖𝑑4 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
≤[
−𝛿𝑡
𝜆−1
min (𝑊22 ) 𝜏4 𝑒
,
1 − 𝜏3−1 𝑒𝛿𝑑
(43)
Remark 6. In Theorem 5, a delay-dependent criterion for
the mean-square exponential admissibility of system (2) is
derived in terms of strict LMI, which can be easily solved
by standard software. In order to reduce conservatism, a
new Lyapunov-Krasovskii functional is constructed base
on the discretized Lyapunov functional method. The functional in (17) allows to take different weighing matrices
on different subintervals, which will yield less conservative
delay-dependent stability criterion. The conservatism will
be reduced with 𝑁 increasing and will be illustrated via
examples in the next section. The method can also be applied
to this class of systems with interval time-varying delays by
decomposing the lower bound of the time delay.
Remark 7. Specially, if there are no stochastic terms, the
system (2) becomes a singular Markovian jump system with
mode-dependent time-varying delay:
Since 𝜏1 can be chosen arbitrarily, 𝜏1 is chosen small enough
such that 𝜏2 − 𝜏1 > 0. Then, we can always find a scalar 𝜏3 > 1
such that
𝐸𝑥̇ (𝑡) = 𝐴 (𝑟𝑡 ) 𝑥 (𝑡) + 𝐴 𝑑 (𝑟𝑡 ) (𝑥 − 𝑑 (𝑡, 𝑟𝑡 ))
𝑊22 − (𝜏1 − 𝜏2 ) 𝐼 ≥ 𝜏3 𝑊22 .
Based on the proof of Theorem 5, we have the following
corollary.
(44)
From (40), (42), and (43), we get
𝜁2𝑇 (𝑡) 𝑊22 𝜁2 (𝑡) − 𝜁2𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊22 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
≤ −𝜏2 𝜁2𝑇 (𝑡) 𝜁2 (𝑡) + 𝜏1 𝜁2𝑇 (𝑡) 𝜁2 (𝑡)
(45)
𝑇
+ 𝜏1−1 𝑒𝑇 (𝑡) 𝑃̂𝑖22 𝑃̂𝑖22
𝑒 (𝑡) .
It is easy to see that the above inequality and (44) imply
𝜏3 𝜁2𝑇 (𝑡) 𝑊22 𝜁2 (𝑡) ≤ 𝜁2𝑇 (𝑡) (𝑊22 − (𝜏1 − 𝜏2 ) 𝐼) 𝜁2 (𝑡)
≤ 𝜁2𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊22 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
(46)
𝑇
𝑒 (𝑡) .
+ 𝜏1−1 𝑒𝑇 (𝑡) 𝑃̂𝑖22 𝑃̂𝑖22
−1 𝑇
𝑇
+ (𝜏3 𝜏1 ) 𝑒 (𝑡) 𝑃̂𝑖22 𝑃̂𝑖22
𝑒 (𝑡) .
𝑡−𝑑≤𝑠≤𝑡
(50)
Corollary 8. For any delays 𝑑𝑖 (𝑡) satisfying (3), the singular
Markovian jump system (50) is mean-square exponentially
admissible for 𝛿 > 0, if there exist matrices 𝑃𝑖 > 0, 𝑄𝑘𝑖 > 0,
𝑈𝑘 > 0, 𝑄𝑘 > 0, 𝑀𝑘𝑙 , 𝑘 = 1, 2, . . . 𝑁, 𝑙 = 0, 1, . . . 𝑁 + 1, 𝑊𝑖 > 0,
and 𝑊 > 0 and matrix 𝑆 with appropriate dimensions such
that for each 𝑖 ∈ S,
̂ 𝑖12
̂ 𝑖14
Ξ
Ξ
Ω𝑖
𝑁
]
[
]
[
Θ𝑖 = [ ∗ −𝛿 ∑ 𝑈𝑘 0 ] < 0
]
[
𝑘=1
̂ 𝑖55 ]
∗
Ξ
[∗
𝑠
𝑘 = 1, 2, . . . 𝑁
(51)
𝑗=1
(47)
𝑠
∑𝜋𝑖𝑗 𝑊𝑗 ≤ 𝑊.
𝑗=1
Then, from (39) and (47), we deduce that
E {𝑓 (𝑡)} ≤ 𝜏3−1 E { sup 𝑓 (𝑠)} + 𝜏4 𝑒−𝛿𝑡 ,
∀𝑡 ∈ [−𝑑, 𝑡] .
∑𝜋𝑖𝑗 𝑄𝑘𝑗 ≤ 𝑄𝑘 ,
Then, from above inequality, we have
𝜁2𝑇 (𝑡) 𝑊22 𝜁2 (𝑡) ≤ 𝜏3−1 𝜁2𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑊22 𝜁2 (𝑡 − 𝑑𝑖 (𝑡))
𝑥 (𝑡) = 𝜙 (𝑡) ,
(48)
Remark 9. Corollary 8 provides a new exponential stability criterion for continuous-time singular Markovian jump
systems with mode-dependent time delays, which has only
been studied by Balasubramiam et al. [28]. It should be
8
Abstract and Applied Analysis
pointed out that, even in the case 𝑁 = 1, our criterion
still has less conservatism than the results in [28], which
will be illustrated via a numerical example. The reduced
conservatism of Corollary 8 comes from the matrices 𝑄𝑘𝑖 and
𝑊𝑖 , which are selected to be mode dependent in our paper.
Example 1. Consider the system (50) with two modes and the
parameters as follows:
𝐴1 = [
−1.5 −2.6
],
−2.6 −1.5
−1.8 −2.7
𝐴2 = [
],
−2.7 −1.8
𝐴 𝑑1 = [
−1.2 0.7
],
0.7 −1.2
−1.3 0.6
𝐴 𝑑2 = [
],
0.6 −1.3
−5 5
],
4 −4
𝐸=[
1 0
],
0 0
(52)
0
𝑅 = [ ].
1
Example 2. Consider the following singular stochastic hybrid
system with mode-dependent time-varying delay:
𝐴 𝑑1 = [
0.3 0.1
],
0.9 0.4
Corollary 8
𝑁=1
Corollary 8
𝑁=2
Corollary 8
𝑁=3
2.1841
2.3936
2.4285
1.2741
−0.9 0.0
𝐴2 = [
],
0.0 −1.0
𝐴 𝑑2 = [
𝐶1 = [
0.5 0
],
0 0
𝐶2 = [
𝐷1 = [
0.1 0
],
0 0
𝐷2 = [
Λ= [
−1 1
],
2 −2
𝐸=[
−0.8 0
],
−0.5 −0.6
0.3 0
],
0 0
(53)
0.3 0
],
0 0
1 0
],
0 0
Theorem 5, 𝑁 = 1
Theorem 5, 𝑁 = 2
Theorem 5, 𝑁 = 3
0.3
7.836
8.163
8.311
𝜇
0.6
4.397
4.712
4.890
0.9
1.037
1.373
1.514
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
For given 𝜇1 = 0.4, 𝜇2 = 0.6, and 𝑑1 = 𝑑2 = 𝑑,
the comparison results of the maximum upper bounds 𝑑 of
the time delay are given in Table 1. From Table 1, we can see
that our results still have less conservatism than the results
obtained in [28] even with the case 𝑁 = 1.
−1.1 0.0
],
𝐴1 = [
0.0 −0.9
Balasubramaniam
et al. [28]
Table 2: Maximum allowed 𝑑2 via different 𝜇.
4. Numerical Examples
Λ= [
Table 1: Maximum allowed delay bound 𝑑 via different method.
0
𝑅 = [ ].
1
By Theorem 5, for given 𝑑1 = 0.1 and 𝜇1 = 𝜇2 = 𝜇,
the maximum allowable upper bound of 𝑑2 to guarantee the
system is mean-square exponentially stable for different 𝜇 is
listed in Table 2.
From Table 2, we can see that the conservatism will be
reduced with 𝑁 increasing.
5. Conclusions
In this paper, the problem of delay-dependent stability
problem for a class of singular hybrid systems with modedependent time delay has been investigated. Based on the
discretized Lyapunov functional method, a criterion which
guarantees the systems is regular, impulse-free, and meansquare exponentially stable has been derived in terms of strict
LMI. Two numerical examples have been given to show the
effectiveness of proposed method.
Acknowledgment
This study was supported by the National Natural Science
Foundation of China (Grant nos. 51205253 and 11272205).
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