and Lam [1] studied the delay-independent robust H
1
control. Chen et al. [4] investigated robust guaranteed control for uncer-
tain MJS with mode-dependent time delays by applying descriptor transformation method and moon’s inequality [21].Xuet
al. [35] addressed the problem of robust H
1
filtering for Markovian jump linear systems with parameter uncertainties and
mode-dependent time-varying delays. Wu et al. [33] applied slack variables method to develop the delay-dependent stability
and stabilization conditions.
In practice, input delay is always inevitable because the sampling data and controller signal are transmitted through a
network. So, the problem of controller design for linear systems with input delay has been paid a lot of consideration,
and some new controller design approaches have been reported respectively. Delay feedback approach is proposed in
[38], memoryless state feedback controllers are developed in [14,40]. However, there is few results on MJS with state and
input delay in the literatures. More recently, in [25], the authors considered delay-independent H
1
control problem for a
class of uncertain Markovian jump system with multiple delays in the state and in the input. Due to delay-dependent sta-
bility and stabilization conditions are less conservative than delay-independent ones because of using the information on the
size of delays, especially when time delays are small. To the best of the authors’ knowledge, however, the problem of delay-
dependent stabilization results for MJS with state and input delays has not been fully investigated and is very challenging.
This motivates us to do this study.
This paper considers the problem of delay-dependent stability analysis and controller synthesis for MJS with both state
and input delays. The delays are assumed to be constant and unknown, but their upper bounds are known. First, by con-
structing a new Lyapunov–Krasovskii functional and introducing some appropriate slack matrices, new robust stochastic sta-
bility conditions are presented in the form of LMIs. Second, based on the stability condition, memoryless state feedback
controllers are designed such that the closed-loop system is robustly stochastically stable. Finally, some numerical examples
are given to illustrate the effectiveness, and also give the comparison with the existing results.
Notation.Throughout this paper, for real symmetric matrices X and Y, the notation X P Y(respectively X > Y) means that
the X Y is positive semi-definite (respectively, positive definite); I is an identity matrix with appropriate dimension; The
superscript ‘‘T” represents the transpose; If A is symmetric matrix, k
max
and k
min
denote the largest and smallest eigenvalue of
A respectively; L
2
½0; 1Þ is the space of square-integrable vector functions over ½0; 1Þ; jjrefers the Euclidean vector norm;
kk
2
stands for the usual L
2
½0; 1Þ norm; ð
X
; F; PÞ is a probability space with
X
the sample and F the algebra of subsets of the
sample space; Efg stands for the expectation operator with respect to the given probability measure P;‘‘” denotes a block
that is readily inferred by symmetry; Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Problem statement
Consider the following MJS with state and input delays, which is defined on a complete probability space ð
X
; F; PÞ:
_
xtðÞ¼ Ar
t
ðÞþ
D
Ar
t
; tðÞ½xtðÞþA
s
r
t
ðÞþ
D
A
s
r
t
; tðÞ½xt
s
ðÞ
þ Br
t
ðÞþ
D
Br
t
; tðÞ½utðÞþ B
s
r
t
ðÞþ
D
B
s
r
t
; tðÞ½ut
s
ðÞ; ð1Þ
xtðÞ¼w tðÞ; utðÞ¼/ tðÞ; r 0ðÞ¼r
0
; t 2
s
; 0½; ð2Þ
where xðtÞ2R
n
is state vector and uðtÞ2R
m
denotes control input. fr
t
; t P 0g is a homogeneous finite-state Markovian pro-
cess with right continuous trajectories, which takes value in a finite-state space S ¼f1; 2; ...; sg with generator
P
¼f
p
ij
g; i; j 2 S, and has the mode transition probabilities
Pr r
tþ
D
t
¼ jjr
t
¼ iðÞ¼
p
ij
D
t þ o
D
tðÞ; i – j;
1 þ
p
ii
D
t þ o
D
tðÞ; i ¼ j;
where
D
t > 0, and lim
D
t!0
oð
D
tÞ
D
t
¼ 0.
p
ij
P 0ði; j 2 S; i – jÞ denote the switching rate from mode i to mode j, and
p
ii
¼
P
s
j¼1;j–i
p
ij
for all i 2 S. Aðr
t
Þ; A
s
ðr
t
Þ; Bðr
t
Þ and B
s
ðr
t
Þ are constant matrices with appropriate dimension.
D
Aðr
t
; tÞ;
D
A
s
ðr
t
; tÞ;
D
Bðr
t
; tÞ and
D
B
s
ðr
t
; tÞ are the unknown matrices which denote time-varying parametric uncertainties
and are assumed to belong to certain bounded compact sets.
s
> 0 is an unknown constant time delay, and the scalar
s
> 0 is an upper bound of
s
. For notational simplicity, which r
t
¼ i; i 2 S, the matrices Aðr
t
Þ; A
s
ðr
t
Þ; Bðr
t
Þ; B
s
ðr
t
Þ;
D
Aðr
t
; tÞ;
D
A
s
ðr
t
; tÞ;
D
Bðr
t
; tÞ and
D
B
s
ðr
t
; tÞ will be represented by A
i
; A
s
i
; B
i
; B
s
i
;
D
A
i
ðtÞ;
D
A
s
i
ðtÞ;
D
B
i
ðtÞ and
D
B
s
i
ðtÞ, respectively. In this
paper, the parameter uncertainties are assumed to be of the following form:
D
A
i
tðÞ
D
A
s
i
tðÞ
D
B
i
tðÞ
D
B
s
i
tðÞ½¼H
i
F
i
tðÞE
1i
E
2i
E
3i
E
4i
½; ð3Þ
where H
i
; E
1i
; E
2i
; E
3i
and E
4i
are known constant matrices with appropriate dimensions, and F
i
ðtÞði 2 SÞ are unknown matrix
functions with the property
F
T
i
tðÞF
i
tðÞ6 I: ð4Þ
To state our main results, the following definitions and lemmas are first introduced, which are essential for the proof in
the sequel.
Definition 1. The MJS (1) with uðtÞ¼0 is said to be robustly stochastically stable if for all finite wðtÞ defined on ½
s
; 0 and
initial mode r
0
, there exists a finite number
N
ðwðÞ;
s
; r
0
Þ > 0, such that
2852 B. Chen et al. / Information Sciences 179 (2009) 2851–2860