Exponential Stability of Nonlinear Stochastic ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
493
Exponential Stability of Nonlinear Stochastic Systems with Time-delay Wei Qian School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Shaohua Wang School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China
Email: [email protected] Juan Liu Department of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Abstract—This note studies the exponential stability of nonlinear stochastic systems with time delay. Firstly, a more general Lyapunov-Krasovskii functional is constructed, and based on Ito calculus rules for stochastic systems, a novel delay-dependent sufficient condition for exponential stability in mean square is derived in terms of linear matrix inequalities, and it is proved in theory that the obtained stability condition is less conservative. Then, the proposed method is extended to nonlinear time-delay systems and retarded time-delay systems and the corresponding stability criteria are obtained. Finally, numerical examples are given to illustrate that the obtained results in this paper are less conservative than some existing ones. Index Terms—Stochastic systems, time delay, nonlinear uncertainties, exponential stability, linear matrix inequality (LMI)
I. INTRODUCTION Time delay and model uncertainty is commonly encountered in control systems. It has been shown that the existence of them often causes poor performance and even instability of systems. Therefore, robust stability of uncertain time-delay systems has attracted a great deal of attention over the years. In deterministic context, different methods, such as discretized Lyapunov functional[1], model transformation[2, 3], integral inequality[4, 5], free matrices[6, 7] and special Lyapunov Krasovskii functional[8, 9], have been developed to reduce the conservatism of the corresponding results and many delay-dependent stability criteria have been reported in the literature. When the environmental disturbances are taken into account, the systems under consideration become stochastic systems with time delay and uncertainties. Recently, considerable attention has been devoted to the study of stochastic systems due to the fact that stochastic modeling has played an important role in science and industry. Many fundamental results for
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deterministic systems have been extended to stochastic systems. In the aspect of robust stability of stochastic systems, in [10], the robust exponential stability of stochastic differential delay equations were discussed based on Lyapunov’s direct method. In [11], in terms of linear matrix inequality, a less conservative robust stability condition was obtained for stochastic delay systems with nonlinear uncertainties. By using a descriptor model transformation of the system and by applying Moon’s inequality, exponential stability of uncertain stochastic systems with multiple delays was studied in [12], and a delay-dependent result was proposed. By introducing some relaxation matrices, the exponential stability of stochastic systems with timevarying delay, nonlinearity, and markovian switching was investigated in [13]. In [14], based on the generalized Finsler lemma, by using model transformations, crossterm bounding techniques or additional matrix variables is avoided, and thus a simple and less conservative criterion was obtained. Motivated by above discussion, this paper is concerned with the robust exponential stability in mean square for stochastic systems with time delay and nonlinear uncertainties. Firstly, based on Ito calculus rules, a more general Lyapunov-Krasovskii functional is constructed and a novel delay-dependent stability criterion is obtained in terms of linear matrix inequalities. Secondly, a simplified Lyapunov-Krasovskii functional is considered and the corresponding stability condition is obtained, which illustrate the importance of the some LyapunovKrasovskii functional terms in reducing the conservatism of the stability result. Then the proposed method is used to deal with the stability of deterministic time-delay system. In the last, several examples are given to illustrate the effectiveness of the proposed method in this paper. Notations: In this paper, C [− τ ,0]; ℜ n denotes the
(
)
family of continuous functions from [− τ ,0] to ℜ n . For
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symmetric matrices A and B, λmax ( A) and λmin ( A) denote the largest and smallest eigenvalue of A, respectively; the notation A ≥ B (respectively, A > B ) means that the matrix A − B is positive semi-definite (respectively, positive definite). ⋅ denotes the Euclidean norm for vector or the spectral norm of matrices. Moreover, let (Ω, F , {Ft }t ≥ 0 , P ) be a complete probability
space with a filtration
{Ft }t ≥0 satisfying the usual E{} ⋅ denotes the mathematical
conditions. The notation expectation with respect to some probability measure P . L2F 0 [− τ ,0]; ℜ n denotes the family of all F0-measurable
(
(
[
'
α ⎛α ⎞ ⎛α ⎞ ⎜ w(σ )dσ ⎟ M ⎜ w(σ )dσ ⎟ ≤ (α − β ) w' (σ )Mw(σ )dσ ⎜ ⎟ ⎜ ⎟ β ⎝β ⎠ ⎝β ⎠
∫
∫
∫
)
)
C [− τ ,0]; ℜ n valued random variables ξ ={ξ(θ) : −τ ≤θ ≤0}
such that sup E ξ (θ ) < ∞ . In the symmetric block 2
−τ ≤θ ≤ 0
matrices, ∗ denotes a term that is induced by symmetry. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions.
III. MAIN RESULT In this section, by constructing a Lyapunov-Krasovskii functional with complete form, the delay-dependent stability criterion is obtained, and then some corollaries are deduced according the main method proposed in this paper. Firstly, let consider a LKF as V (x(t ), t ) =
II. PROBLEM FORMULAITON
+ g (t , x(t ), x(t − τ ))dw(t )
(1)
x(t ) = ϕ (t ), t ∈ [− τ ,0]
where x(t ) ∈ ℜ n is the state, A and A1 are known constant matrices, τ is the constant delay. w(t ) is the one dimensional standard Brownian motion and satisfies: Ε{dw(t )} = 0
}
Ε dw(t ) = dt 2
f : ℜ + × ℜ × ℜ → ℜ and g : ℜ + × ℜ × ℜ → ℜ denote the nonlinear uncertainties satisfying n
n
n
n
n
f (t , x(t ), x(t − τ )) ≤ G1 x(t ) + G2 x(t − τ )
[
]
Trace g (t , x(t ), x(t − τ ))g (t , x(t ), x(t − τ )) T
≤ G3 x(t ) + G4 x(t − τ ) 2
where G1 , G2 , G3 , G4
(
are
2
constant
)
matrices.
n×m
(3) Initial
(4) 1 [E (V (x(t + Δ ), t + Δ )) − V (x(t ), t )] Δ
The aim of this note is to develop a delay-dependent exponential stability condition in the mean square for system (1). For this purpose, the following definition and lemma are introduced. Definition 1. System (1) is said to be exponentially stable in mean square if there exists a positive constant λ such that
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(5)
V1 (x(t ), t ) = x T (t )Px(t )
(6)
V2 (x(t ), t ) =
(7)
V3 (x(t )) =
∫
t
t −τ
x T (s )R1 x(s )ds
∫ ∫ x (u )R x(u )duds t
t
T
2
t −τ s
V4 (x(t ), t ) = 2 x T (t )Q1 V5 (x(t ), t ) =
∫
0
−τ
ds
∫
0
−τ
∫ x(s )ds t
t −τ
(8) (9)
x T (t + s )Q2 x(t + θ )dθ
(10)
∫ (s − t + τ )x(s )ds
(11)
t
t −τ
(2)
LV (x(t ), t ) Δ →0
i
where
V6 (x(t ), t ) = 2 x T (t )S1
function ϕ (⋅) ∈ L2F 0 [− τ ,0]; ℜ n . For a given function V , its infinitesimal operator L is defined as
= lim+
9
∑V (x(t ), t ) i =1
Consider the following stochastic systems with time delay and nonlinear uncertainties: dx(t ) = [Ax(t ) + A1 x(t − τ ) + f (t , x(t ), x(t − τ ))]dt
{
]
1 2 lim sup log Ε x(t ) ≤ −λ t t →∞ Lemma 1. For any scalars α , β with α > β , symmetric positive-definite constant matrix M , and vector-valued function w : [β , α ] → ℜ n such that the integrations concerned are well-defined, then
T
t t V7 (x(t ), t ) = 2⎛⎜ ∫ x(s )ds ⎞⎟ S 2 ∫ (s − t + τ )x(s )ds (12) − τ −τ t t ⎝ ⎠
V8 (x(t ), t ) =
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds t
t
T
3
t −τ
V9 (x(t ), t ) = ∫
t
t −τ
(13)
t −τ
(s − t + τ )3 x T (s )S 4 x(s )ds
(14)
In order to reduce the conservatism of the main result, we give the following proposition. Proposition 1. If there exist matrices P = P T , R1 = R1T > 0, R2 = R2T > 0, Q1 , Q2 = Q2T ∈ ℜ n×n , S1 , S 2 , S 3 = S 3T > 0, S 4 = S 4T > 0 satisfying the following LMI: Q1 S1 ⎤ ⎡P (15) Ω = ⎢⎢ ∗ τ −1 R1 + Q2 S 2 ⎥⎥ > 0 ⎢⎣ ∗ S 3 ⎥⎦ ∗
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then the Lyapunov-Krasovskii functional (5) is positive definite. Proof . Here we let: y1 (t ,τ ) := ∫ x(s )ds , t
t −τ
y 2 (t ,τ ) := ∫
t
t −τ
(s − t + τ )x(s )ds
χ T (t ) = [x(t ) y1 (t ,τ ) y 2 (t ,τ )]T then we can get:
V4 (x(t ), t ) = 2 x T (t )Q1 y1 (t ,τ )
[ [
by (3) and (13), we get: LV1 (x (t ), t ) ≤ 2 x T (t )PAx (t ) + 2 x T (t )PA1 x(t − τ )
T
t t V2 (x(t ), t ) ≥ τ −1 ⎡ ∫ x(s )ds ⎤ R1 ⎡ ∫ x(s )ds ⎤ ⎢⎣ t −τ ⎥⎦ ⎢⎣ t −τ ⎥⎦ = τ −1 y T (t ,τ )R1 y (t ,τ ) so we can conclude:
+ 2 x T (t )Pf (t , x (t ), x (t − τ ))
[
Noting (8) and (14), we know that if the LMI (15) holds, the Lyapunov-Krasovskii functional (5) is positive definite. This completes the proof. From the prove of the proposition 1, we can see that the constraint on functional parameter Q2 is relaxed, which is helpful to reduce the conservatism of the main results. In the following, we give the main stability condition of system (1). Theorem 1. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n ,
S1 , S 2 ,
S 3 = S 3T > 0, S 4 = S 4T > 0 satisfying Proposition 1, such that the following LMIs hold:
Π 15 ⎤ T A1 S1 − S 2 ⎥⎥ ⎥ < 0 (16) S1 ⎥ − S3 ⎥ − 3τ −1 S 4 ⎥⎦
P ≤ ρI
where Π 11 = PA + AT P + R1 + τR2 + Q1 + Q1T + τS1 + τS1T + τ 3 S 4 + εG1T G1 + ρG3T G3 Π 14 = AT Q1 + Q2 − S1 + τS 2T Π 15 = AT S1 + S 2 + τS 3
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T
]
T 3
+ 2 x (t )PA1 x (t − τ ) T
+ V6 (x(t ), t ) + V7 (x(t ), t ) + V8 (x(t ), t ) = χ (t )Ωχ (t )
Π 44 ∗
[
= x (t ) PA + A P + ρG G3 x (t ) T
T
− εI ∗ ∗
]
+ ρ x T (t )G3T G3 x(t ) + x T (t − τ )G4T G4 x (t − τ )
V1 (x(t ), t ) + V2 (x(t ), t ) + V4 (x(t ), t ) + V5 (x(t ), t )
Π 22 ∗ ∗ ∗
]
+ Trace g T (t , x (t ), x (t − τ ))Pg (t , x (t ), x (t − τ ))
V8 (x(t ), t ) = y 2T (t ,τ )S 3 y 2 (t ,τ ) by lemma 1, we also have:
Π 14 T A1 Q1 − Q2 Q1
T
+ 2 x T (t )Pf (t , x (t ), x (t − τ ))
V7 (x(t ), t ) = 2 y1T (t ,τ )S 2 y 2 (t ,τ )
P 0
]
= 2 x (t )PAx (t ) + 2 x (t )PA1 x (t − τ ) T
V6 (x(t ), t ) = 2 x T (t )S1 y 2 (t ,τ )
PA1 − Q1
Π 44 = −τ −1 R2 − S 2 − S 2T Proof . Choosing the Lyapunov-Krasovskii functional (5), by Ito calculus rules, the infinitesimal operator LV (x(t ), t ) of the stochastic process {xt , t ≥ 0} along the trajectory of system (1) are given by 1 LV1 (x (t ), t ) = lim+ [E (V1 (x (t + Δ ), t + Δ )) − V1 (x (t ), t )] Δ→0 Δ = 2 x T (t )P [ Ax (t ) + A1 x (t − τ ) + f (t , x (t ), x (t − τ ))] + Trace g T (t , x (t ), x (t − τ ))Pg (t , x (t ), x (t − τ ))
V5 (x(t ), t ) = y1T (t ,τ )Q2 y1 (t ,τ )
⎡Π 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎣
Π 22 = ρG4T G4 + εG2T G2 − R1
(17)
+ 2 x T (t )Pf (t , x (t ), x (t − τ )) + x T (t − τ )ρG4T G4 x(t − τ )
(18) LV2 (x(t ), t ) = x T (t )R1 x(t ) − x T (t − τ )R1 x(t − τ )
(19)
by lemma 2, we can get: LV3 (x(t ), t ) = τx T (t )R2 x(t ) − ≤ τx T (t )R2 x(t ) − τ −1
∫
t
t −τ
∫
t
t −τ
x T (s )R2 x(s )ds
(20)
x T (s )dsR2
∫ x(s )ds t
t −τ
1 [E (V4 (x(t + Δ ), t + Δ )) − V4 (x(t ), t )] Δ →0 Δ = 2 x T (t )Q1 x(t ) − 2 x T (t )Q1 x(t − τ )
LV4 (x (t ), t ) = lim+
+ 2 x T (t )AT Q1 ∫ x(s )ds t
t −τ
+ 2 x (t − τ )A Q1 ∫ x(s )ds T
+2f
T
T 1
t
t −τ
(t , x(t ), x(t − τ ))Q1 ∫t −τ x(s )ds t
(21) LV5 (x(t ), t ) = 2[x(t ) − x(t − τ )] Q2 T
= 2 x T (t )Q2 − 2 xT
∫ x(s )ds t
t −τ
∫ x(s )ds (t − τ )Q ∫ x(s )ds t
t −τ
t
2
t −τ
(22)
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1 [E (V6 (x(t + Δ ), t + Δ )) − V6 (x(t ), t )] Δ
LV6 (x(t ), t ) = lim+ Δ →0
= 2 x (t )S1[τx(t ) − T
+ 2 x T (t )AT + 2x
T
∫ x(s )ds]
It also can be observed:
V1 (x(t ), t ) = x T (t )Px(t ) ≤ λmax ( P) x(t )
t −τ
t
t
t −τ
t
t −τ
− 2xT
∫
+ 2τ −2
t
2
t
t
t −τ
LV8 (x(t ), t ) = 2τx (t )S 3 ∫ T
∫
t −τ
t
t −τ
(24)
x(s )ds
T
5
T
t
t −τ
(s − t + τ )x(s )ds
(25)
(s − t + τ ) ≤ τ 3 x T (t )S 4 x(t ) t −τ
−
3
τ
x (s )S 4 x(s )ds
4
t −τ
t
3
0
x(s )ds
⎞ ∫ x(s )ds ⎟⎠ t
T
S2
t −τ
t
∫ (s − t + τ )x(s )ds t
t −τ
t
T
2
t −τ
t −τ
t
T
t −τ
max
V8 (x(t ), t ) =
2
T 2
0
3
t
t
T
3
t −τ
t
9
3
0
3
LV (x(t ), t ) ≤ − μ x(t )
2
2
−τ
3
(29)
T
4
t −τ
0
3
T
t −τ
T
2
t
(28)
⎤ ∫t −τ (s − t +τ )x(s)ds⎥
2
−τ
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds ≤ τ ∫ (s − t + τ ) x (s )S x(s )ds ≤ τ λ (S )∫ x(t + θ ) dθ V (x(t ), t ) = ∫ (s − t + τ ) x (s )S x(s )ds ≤ τ λ (S )∫ x(t + θ ) dθ max
t
2
−τ
t −τ
4
2
−τ
so we can deduce that EV (x(0 ),0) ≤ (α + τβ ) sup E ξ (θ )
t −τ ⎦ ⎣ Therefore, we know that LMI (16) is equivalent to Π < 0 . Moreover, it is easy to see that there exists a scalar μ > 0 such that:
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1
2
3
where
∫
−τ
t
max
t
2
2
t −τ
η T (t , s )Πη (t , s )ds
η T (t, s ) = ⎡⎢ x(t ) x(t −τ ) f
2
t
LV (x(t ), t ) = ∑ LVi (x(t ), t ) t −τ
T
∫ x (s )dsS ∫ x(s )ds + ∫ (s − t + τ )x (s )ds ∫ (s − t + τ )x(s )ds ≤ (τλ (S S ) + τ )∫ x(t + θ ) dθ
9
∫
0
−τ
T 1 1
≤
t −τ
i =1
2
t −τ
V7 (x(t ), t ) = 2⎛⎜ ⎝
combining (18)- (26), we have:
τ
0
T
max
noting (2), we can calculate that, for any scalar ε > 0 , we have: εx T (t )G1T G1 x(t ) + εx T (t − τ )G2T G2 x(t − τ ) (27) − εf T (t ) f (t ) ≥ 0
≤
t −τ
0
−τ
−τ
t
(26)
t
t −τ
2
t −τ
t
T
t
T
∫ (s − t + τ )x (s )ds ∫ (s − t + τ )x(s )ds ≤ λ (S S ) x(t ) + τ ∫ x(t + θ ) dθ
T
t −τ
1
1
T 1
t
T
≤ x T (t )S1 S1T x T (t )
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds t
1
T
LV9 (x(t ), t ) = τ 3 x T (t )S 4 x(t ) − 3∫
2
−τ
t −τ
T 1
6
+
2
2
max
by lemma 2, we can get: t
2
0
(s − t + τ )x(s )ds
t −τ
T
t
1
max
− 2∫ x (s )dsS 3 ∫ t
t
T
x T (s )dsS 2 x(t ) t
∫ x (u )R x(u )duds ≤ τλ (R )∫ x(t + θ ) dθ V (x(t ), t ) = 2 x (t )Q ∫ x(s )ds ≤ x (t )Q Q x (t ) + ∫ x (s )ds ∫ x(s )ds ≤ λ (Q Q ) x(t ) + τ ∫ x(t + θ ) dθ V (x(t ), t ) = ∫ ds ∫ x (t + s )Q x(t + θ )dθ ≤ τλ (Q )∫ x(t + θ ) dθ V ( x(t ), t ) = 2 x (t )S ∫ (s − t + τ )x(s )ds t
4
t −τ
x T (s )dsS 2
V3 (x(t )) = ∫
max
t
t
2
0
t −τ
t −τ
x(t + θ ) dθ
0
−τ
t −τ s
⎤ ∫ x(s )ds ⎥⎦
∫ (s − t + τ )x(s )ds (t − τ )S ∫ (s − t + τ )x(s )ds
t −τ
∫
∫t −τ (s − t + τ )x(s )ds t
x T (s )dsS 2 ⎡τx(t ) − ⎢⎣
= 2 x (t )S 2 T
≤ λmax (R1 )∫
t −τ
(23)
∫
2
V2 (x(t ), t ) = ∫ x T (s )R1 x(s )ds
t −τ
t
LV7 (x (t ), t ) = 2[x(t ) − x (t − τ )] S 2
(30)
0
t −τ
t
T
+2
t
EV (x(t ), t ) = EV (x(0),0) + E ∫ LV (x(s ), s )ds
t
∫ (s − t + τ )x(s )ds (t − τ )A ∫ (s − t + τ )x(s )ds ∫ (s − t + τ )x(s )ds T 1
+2f T
by Dynkin formula, we have:
2
−τ ≤θ ≤ 0
EV (x(t ), t ) ≥ λmin (P ) x(t )
where
2
(31)
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α = λmax (P ) + λmax (Q1Q1T ) + λmax (S1 S1T )
⎡Ξ11 ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢⎣ ∗
β = 1 + 2τ 3 + λmax (R1 ) + τλmax (R2 ) + τλmax (Q2 )
(
)
+ τλmax S 2 S 2T + τ 3 λmax (S 3 ) + +τ 3 λmax (S 4 )
PA1 − Q1 ρG G4 + εG2T G2 − R1 ∗ ∗ T 4
By Gronwell Inequality, we can get: E x(t ) ≤ 2
(α + τβ ) sup E ξ (θ ) 2 exp⎛⎜ − μ t ⎞⎟ ⎜ λ (P ) ⎟ λmin (P ) −τ ≤θ ≤ 0 ⎝ min ⎠
(32)
∫ (s − t + τ )x(s )ds
t
3
T
LV (x(t ), t ) ≤ − μ x(t )
the conservatism of the stability criterion. In the following section, we would proof it in theory. Choosing a simplified Lyapunov-Krasovskii functional as:
t
EV (x(t ), t ) = EV (x(0 ),0 ) + E ∫ LV (x(s ), s )ds 0
by Lyapunov-Krasovskii functional (33), we also can deduce that EV (x(0 ),0) ≤ (α + τβ ) sup E ξ (θ )
where
α = λmax (P ) + λmax (Q1Q1T ) , β = 1 + λmax (R1 ) + τλmax (R2 ) + τλmax (Q2 ) .
5
so by Gronwell Inequality, we also can get:
= x (t )Px(t ) + ∫ x (s )R1 x(s )ds t
T
T
t −τ
t
T
0
2
−τ
0
t
T
t −τ s
1
t −τ
T
−τ
2
(33) then we can get the proposition 2 as following: Proposition 2. If there exist matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying the following LMIs: ⎡P R1 > 0 , R2 > 0 , ⎢ T ⎣Q1
Q1
⎤
τ −1 R1 + Q2 ⎥⎦
>0
(34)
then the Lyapunov-Krasovskii functional (33) is positive definite. According to the proof of Theorem 1, by Ito calculus rules, the infinitesimal operator LV (x(t ), t ) of the stochastic process {xt , t ≥ 0} along the trajectory of system (1) can be calculated, and we can get the corresponding stability condition. Theorem 2. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices P = PT , R1 = R1T , R2 = R2T , Q1, Q2 = Q2T ∈ ℜ n×n
satisfying Proposition 2, such that the following LMIs hold: © 2013 ACADEMY PUBLISHER
E x(t ) ≤ 2
∫ x (u )R x(u )duds + 2 x (t )Q ∫ x(s )ds + ∫ ds ∫ x (t + s )Q x(t + θ )dθ +∫
t
2
−τ ≤θ ≤ 0
V (x(t ), t ) = ∑ Vi (x(t ), t ) i =1
2
Therefore, by Dynkin formula, we have
are very important to induce
4
t −τ
(36)
where Ξ11 = PA + AT P + R1 + τR2 + Q1 + Q1T + εG1T G1 + ρG3T G3 We know that LMI (35) is equivalent to Ξ < 0 . Moreover, it is easy to see that there exists a scalar μ > 0 such that
t
t −τ
in η (t, s ) and the Lyapunov-Krasovskii functional term
∫ (s − t + τ ) x (s )S x(s )ds
P ⎤ ⎥ 0 ⎥ < 0 (35) Q1T ⎥ ⎥ − εI ⎥⎦
P ≤ ρI
This completes the proof. Remark 1. LV (x(t ), t ) is calculated based on Ito calculus rules and the independent increments property of Brownian motion, which is different from the calculus rules in deterministic systems. It is just the difference that makes the difficulty in generalizing the stability analysis methods for time-delay systems, which is useful in reducing the conservatism of the stability result, to the stochastic case. In Theorem 1, the introduction of
AT Q1 + Q2 A1T Q1 − Q2 − τ −1 R2 ∗
(α + τβ ) sup E ξ (θ ) 2 exp⎛⎜ − μ t ⎞⎟ ⎜ λ (P ) ⎟ λmin (P ) −τ ≤θ ≤ 0 ⎠ ⎝ min (37)
which illustrate the stochastic system (1) is exponentially stable in the mean square. Remark 2. Compare Theorem 2 with Theorem 1, it is easy to see that Theorem 2 is the special case of Theorem t 1. Let η T (t , s ) = ⎡⎢ x T (t ) x T (t − τ ) ∫ x T (s )ds f T ⎤⎥ t −τ ⎦ ⎣ and S1 = S 2 = S 3 = S 4 = 0 , we can get Theorem 2 from Theorem 1. So, we can know that the conservatism of Theorem 1 is less conservative than Theorem 2. From (20), we can see that the integral inequality is used in order to get the main result, which cause conservatism of the result. Here we choose another method to avoid the integral inequality. Choosing the Lyapunov-Krasovskii functional (33) and calculated the V& (x(t ), t ) as following:
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x& (t ) = Ax(t ) + A1 x(t − τ ) + f (t,x(t ),x(t − τ ))
LV (x(t ), t ) = 2 x T (t )P[Ax(t ) + A1 x (t − τ ) + f (t , x (t ), x (t − τ ))]
[
]
x(t ) = φ(t ), t ∈ [− τ,0 ]
+ Trace g T (t , x(t ), x(t − τ ))Pg (t , x(t ), x(t − τ ))
+ x T (t )R1 x(t ) − x T (t − τ )R1 x(t − τ )
(38)
Using the Lyapunov-Krasovskii functional (5) and method in Theorem 1, we can get Corollary 1: + τx T (t )R2 x(t ) − ∫ x T (s )R2 x(s )ds Corollary 1. For any given τ > 0 , the system (38) is t −τ asymptotically stable. If there exist positive a scalar + 2 x T (t )Q1 x(t ) − 2 x T (t )Q1 x(t − τ ) ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , S1 , S 2 , t t + 2 x T (t )AT Q1 ∫ x(s )ds + 2 x T (t − τ )A1T Q1 ∫ x (s )ds t −τ t −τ S 3 = S 3T > 0, S 4 = S 4T > 0 Q2 = Q2T ∈ ℜ n× n satisfying t Proposition 1, such that the following LMI holds: + 2 f T (t , x (t ), x (t − τ ))Q1 x (s )ds t
∫
t −τ
+ 2 x T (t )Q2 ∫ x (s )ds t
t −τ
− 2 x T (t − τ )Q2 ∫ x (s )ds t
t −τ
⎡ A + A P + R1 + τR2 + Q1 + Q = x T (t )⎢ T T ⎢⎣+ ρG3 G3 + εG1 G1 + x T (t )[PA1 − Q1 ]x (t − τ ) T
[
T 1
⎤ ⎥ x(t ) ⎥⎦
]
+ x (t − τ ) ρG G4 + εG G2 − R1 x(t − τ ) T
T 4
T 2
+ 2 x (t )Pf (t , x(t ), x(t − τ ))
[
[
]∫
t
t −τ
+ 2 x T (t − τ ) A1T Q1 − Q2
x(s )ds
]∫
t
t −τ
x(s )ds
+ 2 f T (t , x(t ), x(t − τ ))Q1 ∫ x(s )ds t
t −τ
− ∫ x T (s )R2 x(s )ds t
t −τ
so we can get LV (x(t ), t ) = τ −1 ∫ γ T (t , s )Θγ (t , s )ds t
t −τ
and the following stability condition: Theorem 3. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices
(1,4) A1T Q1 − Q2
− εI
Q1
∗
(4,4)
∗
∗
(1,5) ⎤ A1T S1 − S 2 ⎥⎥ ⎥ 0 , the stochastic system (38) is asymptotically stable. If there exists a scalar ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying Proposition 2, such that the following LMI holds: ⎡(1,1)' ' PA1 − Q1 τAT Q1 + τQ2 P ⎤ ⎢ ⎥ εG2T G2 − R1 τA1T Q1 − τQ2 0 ⎥ ⎢ ∗ 0, S 4 = S 4T > 0 matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying Proposition 1, such that the following LMIs hold:
⎡(1,1)' ' ' PA1 − Q1 ⎢ ∗ − R1 ⎢ ⎢ ∗ ∗ ⎢ ∗ ⎣ ∗
(1,3)' ' '
(1,4)' ' '
A Q1 − Q2 T 1
(3,3)' ' ' ∗
⎤ A S1 − S 2 ⎥⎥ < 0 (39) − S3 ⎥ ⎥ − 3τ −1 S 4 ⎦ T 1
where (1,1)' ' ' = PA + AT P + R1 + τR2 + Q1 + Q1T + τS1 + τS1T + τ 3 S 4
(1,3)' ' ' = AT Q1 + Q2 − S1 + τS 2T (1,4)' ' ' = AT S1 + S 2 + τS 3 (3,3)' ' ' = −τ −1 R2 − S 2 − S 2T
Using the Lyapunov-Krasovskii functional (33) and method in Theorem 2, we can get Corollary 4: Corollary 4. For any given τ , the linear time-delay system (39) is asymptotically stable, if there exist R1 = R1T , Q1 , Q2 = Q2T ∈ ℜ n×n , matrices P = P T , R2 = R2T satisfying Proposition 1, such that the following LMI holds: ⎡(1,1)' ' ' ' PA1 − Q1 τAT Q1 + τQ2 ⎤ ⎢ ⎥ − R1 τ A1T Q1 − τQ2 ⎥ < 0 ⎢ ∗ ⎢ ∗ ⎥ ∗ − τR2 ⎣ ⎦
where (1,1)' ' ' ' = PA + AT P + R1 + τR2 + Q1 + Q1T IV. NUMERICAL EXAMPLE In this section, several examples are given to show that Theorem 1 is less conservative than Theorem 2, and they are less conservative than some existing results. Example 1. Consider the stochastic delay system (1) with 0⎤ ⎡− 2 0 ⎤ ⎡ −1 , A1 = ⎢ A=⎢ ⎥ ⎥ ⎣ 1 − 1⎦ ⎣− 0.5 − 1⎦ ΔA(t ) ≤ 0.1 , ΔA1 (t ) ≤ 0.1
[ ]
Trace g T g ≤ 0.1 x(t ) + 0.1 x(t − τ ) 2
2
The comparing results are listed in Table 1. It can be seen that the delay-dependent exponential stability criteria in this note is less conservative in the sense of the computed maximum delay bound. Example 2. Consider the time-delay system (38) with 0 ⎤ ⎡− 2 ⎡− 1 0 ⎤ A=⎢ ⎥ , A1 = ⎢− 1 − 1⎥ 0 − 0 . 9 ⎣ ⎦ ⎣ ⎦ For this deterministic system, we compared the results in Table 2. The results have demonstrated that the delay-
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dependent stability condition in this note is less conservative. TABLE I. COMPARISON OF STABILITY CONDITIONS IN EXAMPLE 1 Methods Mao [6] Yue and Won [7] Chen and Guan [8] Theorem 2 Theorem 1
Maximu τ allowed 0.175 0.8635 1.1997 2.0859 2.8375
TABLE II. COMPARISON OF STABILITY CONDITIONS IN EXAMPLE 2 Methods Moon et al. [5] Fridman and Shaked [2] Xu and Lam [11] Corollary 2 Corollary 1
Maximu τ allowed 4.3588 4.47 4.4721 4.4721 4.564
V. CONCLUSION The exponential stability for stochastic delay systems with nonlinear uncertainties is investigated. Based on Ito calculus rules, a general type of Lyapunov-Krasovskii functional is introduced, and a novel stability condition is obtained in terms of linear matrix inequalities. Numerical examples are given to show that the criteria perform much better than existing stability results. ACKNOWLEDGMENT This work is supported by National Nature Science Foundation under Grant 61104119 and 61104079, the Doctoral Foundation and the Young Core Instructor Foundation from Henan Polytechnic University under Grant B2010-50 and 649166, the Young Core Instructor Foundation in Higher School of Henan Province under grant 2011GGJS-054. REFERENCES [1] Q. L. Han, “A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems, “Automatica, vol. 44, pp. 272- 277, 2008. [2] E. Fridman, and U. Shaked, “A descriptor system approach to H control of linear time-delay systems, “IEEE Trans. Automat. Control, vol. 47, pp. 253-270, 2002. [3] Y. He, Q. G. Wang, C. Lin, and M. Wu, “Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems, “ Int. J. Robust Nonlinear Control, vol. 15, pp. 923–933, 2005. [4] P. Park, “A Delay-Dependent Stability Criterion for Systems with Uncertain Time-Invariant Delays, “IEEE Trans. on Automatic Control, Vol. 44, pp. 876-877, 1999. [5] Y.S. Moon, P. Park, and W. H. Kwon, “Delay-dependent robust stabilization of uncertain state-delayed systems, “Int. J. Control, vol. 74, pp. 1447-1457, 2001. [6] W. Qian, J. Liu, Y. X. Sun, and S. M. Fei, “A less conservative robust stability criteria for uncertain neutral systems with mixed delays, “Mathematics and Computers in Simulation, vol. 80, pp. 1007–1017, 2010. [7] W. Qian, S. Cong, Y. X. Sun and S. M. Fei, “Novel robust stability criteria for uncertain systems with time-varying
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[9]
[10]
[11]
[12]
[13]
[14]
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delay, “Applied Mathematics and Computation, vol. 215, pp. 866-872, 2009. M. N. A. Parlakci, “Extensively augmented Lyapunov functional approach for the stability of neutral time-delay systems, “ IET Control Theory Appl., Vol. 2, No. 5, pp. 431-436, 2008. W. Qian, T. Li, S. Cong, and S. M. Fei, “Improved stability analysis on delayed neural networks with linear fractional uncertainties, “ Applied Mathematics and Computation, vol. 217, pp. 3596-3606, 2010. X. Mao, “Robustness of exponential stability of stochastic differential delay equations, “IEEE Trans. Automat. Control, vol. 41, pp. 442-447, 1996. D. Yue and S. Won, “Delay-dependent robust stability of stochastic systems with the time delay and nonlinear uncertainties, “IEE Electron. Lett, vol. 37, pp. 992-993, 2001. W. Chen, Z. Guan and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach, “Systems Control Lett., vol. 54, pp. 547-555, 2005. D. Yue, and Q. Han, “Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and markovian switching, “IEEE Trans. Automat. Control, vol. 50, pp. 217-222, 2005. Y. Chen, W. X. Zheng, and A. Xue, “A new result on stability analysis for stochastic neutral systems” Automatica, vol. 46, pp. 2100-2104, 2010.
Wei Qian was born in China in 1978. He received his B.Sc. degree in Mechanical Design and Manufacturing Technology from Zhengzhou University in 1999, M.Sc. degree in Control Theory and Application from Southeast University in 2005 and Ph.D. degree in Control Science and Engineering from Zhejiang University in 2009, respectively. From 1999 to 2002, he served as a teacher at Department of Basic Courses, Jiaozuo, China. From 2002 to 2005, he studied for his M.Sc. degree at Department of Automation of Southeast University, Nanjing, China. From 2006 to 2009, he studied for
© 2013 ACADEMY PUBLISHER
his Ph.D. degree at State Key Lab of Industrial Control Technology, Zhejiang University, Hangzhou, China. Since 2009, he served as a teacher at Department of Electrical Engineering and Automation at Henan Polytechnic University, Jiaozuo, China. He has published 30 papers in international journals and conferences. His research interests include stability analysis, robust control of time-delay systems, network control systems and stochastic systems.
Juan Liu was born in China in 1976. She received her B.Sc. degree from Henan Normal University in 1999, and M.Sc. degree from Southwest Jiaotong University in 2005, all in Mathematics. From 1999 to 2002, she served as a teacher at Department of Basic Courses, Jiaozuo, China. From 2002 to 2005, she studied at Department of Mathematics of Southwest University, Chendu, China. Since 2002, she served as a teacher at Department of Mathematics and Information Science of Henan Polytechnic University. She has published 15 papers in international journals and conferences. Her research interests include robust control and fuzzy control of systems with time-delay.
Shaohua Wang was born in China in 1963. He received his B.Sc. degree from Jiaozuo Institute of Technology and M.Sc. degree from China University of Mining and Technology, in 1987 and 1995, respectively, all in Electrical Engineering. From 1987 to 1993, he served as a teacher at Department of Electrical Engineering at Jiaozuo Institute of Technology, Jiaozuo, China. From 1993 to 1995, he studied for his M.Sc. degree at Department of Electrical Engineering at China University of Mining and Technology, Beijing, China. Since 1995, he served as a teacher at Department of Electrical Engineering and Automation at Henan Polytechnic University, Jiaozuo, China. He has published 15 papers in in international journals and conferences. His research interests include date fusion, robust control and electrical engineering.
493
Exponential Stability of Nonlinear Stochastic Systems with Time-delay Wei Qian School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Shaohua Wang School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China
Email: [email protected] Juan Liu Department of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China Email: [email protected]
Abstract—This note studies the exponential stability of nonlinear stochastic systems with time delay. Firstly, a more general Lyapunov-Krasovskii functional is constructed, and based on Ito calculus rules for stochastic systems, a novel delay-dependent sufficient condition for exponential stability in mean square is derived in terms of linear matrix inequalities, and it is proved in theory that the obtained stability condition is less conservative. Then, the proposed method is extended to nonlinear time-delay systems and retarded time-delay systems and the corresponding stability criteria are obtained. Finally, numerical examples are given to illustrate that the obtained results in this paper are less conservative than some existing ones. Index Terms—Stochastic systems, time delay, nonlinear uncertainties, exponential stability, linear matrix inequality (LMI)
I. INTRODUCTION Time delay and model uncertainty is commonly encountered in control systems. It has been shown that the existence of them often causes poor performance and even instability of systems. Therefore, robust stability of uncertain time-delay systems has attracted a great deal of attention over the years. In deterministic context, different methods, such as discretized Lyapunov functional[1], model transformation[2, 3], integral inequality[4, 5], free matrices[6, 7] and special Lyapunov Krasovskii functional[8, 9], have been developed to reduce the conservatism of the corresponding results and many delay-dependent stability criteria have been reported in the literature. When the environmental disturbances are taken into account, the systems under consideration become stochastic systems with time delay and uncertainties. Recently, considerable attention has been devoted to the study of stochastic systems due to the fact that stochastic modeling has played an important role in science and industry. Many fundamental results for
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.2.493-500
deterministic systems have been extended to stochastic systems. In the aspect of robust stability of stochastic systems, in [10], the robust exponential stability of stochastic differential delay equations were discussed based on Lyapunov’s direct method. In [11], in terms of linear matrix inequality, a less conservative robust stability condition was obtained for stochastic delay systems with nonlinear uncertainties. By using a descriptor model transformation of the system and by applying Moon’s inequality, exponential stability of uncertain stochastic systems with multiple delays was studied in [12], and a delay-dependent result was proposed. By introducing some relaxation matrices, the exponential stability of stochastic systems with timevarying delay, nonlinearity, and markovian switching was investigated in [13]. In [14], based on the generalized Finsler lemma, by using model transformations, crossterm bounding techniques or additional matrix variables is avoided, and thus a simple and less conservative criterion was obtained. Motivated by above discussion, this paper is concerned with the robust exponential stability in mean square for stochastic systems with time delay and nonlinear uncertainties. Firstly, based on Ito calculus rules, a more general Lyapunov-Krasovskii functional is constructed and a novel delay-dependent stability criterion is obtained in terms of linear matrix inequalities. Secondly, a simplified Lyapunov-Krasovskii functional is considered and the corresponding stability condition is obtained, which illustrate the importance of the some LyapunovKrasovskii functional terms in reducing the conservatism of the stability result. Then the proposed method is used to deal with the stability of deterministic time-delay system. In the last, several examples are given to illustrate the effectiveness of the proposed method in this paper. Notations: In this paper, C [− τ ,0]; ℜ n denotes the
(
)
family of continuous functions from [− τ ,0] to ℜ n . For
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symmetric matrices A and B, λmax ( A) and λmin ( A) denote the largest and smallest eigenvalue of A, respectively; the notation A ≥ B (respectively, A > B ) means that the matrix A − B is positive semi-definite (respectively, positive definite). ⋅ denotes the Euclidean norm for vector or the spectral norm of matrices. Moreover, let (Ω, F , {Ft }t ≥ 0 , P ) be a complete probability
space with a filtration
{Ft }t ≥0 satisfying the usual E{} ⋅ denotes the mathematical
conditions. The notation expectation with respect to some probability measure P . L2F 0 [− τ ,0]; ℜ n denotes the family of all F0-measurable
(
(
[
'
α ⎛α ⎞ ⎛α ⎞ ⎜ w(σ )dσ ⎟ M ⎜ w(σ )dσ ⎟ ≤ (α − β ) w' (σ )Mw(σ )dσ ⎜ ⎟ ⎜ ⎟ β ⎝β ⎠ ⎝β ⎠
∫
∫
∫
)
)
C [− τ ,0]; ℜ n valued random variables ξ ={ξ(θ) : −τ ≤θ ≤0}
such that sup E ξ (θ ) < ∞ . In the symmetric block 2
−τ ≤θ ≤ 0
matrices, ∗ denotes a term that is induced by symmetry. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions.
III. MAIN RESULT In this section, by constructing a Lyapunov-Krasovskii functional with complete form, the delay-dependent stability criterion is obtained, and then some corollaries are deduced according the main method proposed in this paper. Firstly, let consider a LKF as V (x(t ), t ) =
II. PROBLEM FORMULAITON
+ g (t , x(t ), x(t − τ ))dw(t )
(1)
x(t ) = ϕ (t ), t ∈ [− τ ,0]
where x(t ) ∈ ℜ n is the state, A and A1 are known constant matrices, τ is the constant delay. w(t ) is the one dimensional standard Brownian motion and satisfies: Ε{dw(t )} = 0
}
Ε dw(t ) = dt 2
f : ℜ + × ℜ × ℜ → ℜ and g : ℜ + × ℜ × ℜ → ℜ denote the nonlinear uncertainties satisfying n
n
n
n
n
f (t , x(t ), x(t − τ )) ≤ G1 x(t ) + G2 x(t − τ )
[
]
Trace g (t , x(t ), x(t − τ ))g (t , x(t ), x(t − τ )) T
≤ G3 x(t ) + G4 x(t − τ ) 2
where G1 , G2 , G3 , G4
(
are
2
constant
)
matrices.
n×m
(3) Initial
(4) 1 [E (V (x(t + Δ ), t + Δ )) − V (x(t ), t )] Δ
The aim of this note is to develop a delay-dependent exponential stability condition in the mean square for system (1). For this purpose, the following definition and lemma are introduced. Definition 1. System (1) is said to be exponentially stable in mean square if there exists a positive constant λ such that
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(5)
V1 (x(t ), t ) = x T (t )Px(t )
(6)
V2 (x(t ), t ) =
(7)
V3 (x(t )) =
∫
t
t −τ
x T (s )R1 x(s )ds
∫ ∫ x (u )R x(u )duds t
t
T
2
t −τ s
V4 (x(t ), t ) = 2 x T (t )Q1 V5 (x(t ), t ) =
∫
0
−τ
ds
∫
0
−τ
∫ x(s )ds t
t −τ
(8) (9)
x T (t + s )Q2 x(t + θ )dθ
(10)
∫ (s − t + τ )x(s )ds
(11)
t
t −τ
(2)
LV (x(t ), t ) Δ →0
i
where
V6 (x(t ), t ) = 2 x T (t )S1
function ϕ (⋅) ∈ L2F 0 [− τ ,0]; ℜ n . For a given function V , its infinitesimal operator L is defined as
= lim+
9
∑V (x(t ), t ) i =1
Consider the following stochastic systems with time delay and nonlinear uncertainties: dx(t ) = [Ax(t ) + A1 x(t − τ ) + f (t , x(t ), x(t − τ ))]dt
{
]
1 2 lim sup log Ε x(t ) ≤ −λ t t →∞ Lemma 1. For any scalars α , β with α > β , symmetric positive-definite constant matrix M , and vector-valued function w : [β , α ] → ℜ n such that the integrations concerned are well-defined, then
T
t t V7 (x(t ), t ) = 2⎛⎜ ∫ x(s )ds ⎞⎟ S 2 ∫ (s − t + τ )x(s )ds (12) − τ −τ t t ⎝ ⎠
V8 (x(t ), t ) =
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds t
t
T
3
t −τ
V9 (x(t ), t ) = ∫
t
t −τ
(13)
t −τ
(s − t + τ )3 x T (s )S 4 x(s )ds
(14)
In order to reduce the conservatism of the main result, we give the following proposition. Proposition 1. If there exist matrices P = P T , R1 = R1T > 0, R2 = R2T > 0, Q1 , Q2 = Q2T ∈ ℜ n×n , S1 , S 2 , S 3 = S 3T > 0, S 4 = S 4T > 0 satisfying the following LMI: Q1 S1 ⎤ ⎡P (15) Ω = ⎢⎢ ∗ τ −1 R1 + Q2 S 2 ⎥⎥ > 0 ⎢⎣ ∗ S 3 ⎥⎦ ∗
JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
495
then the Lyapunov-Krasovskii functional (5) is positive definite. Proof . Here we let: y1 (t ,τ ) := ∫ x(s )ds , t
t −τ
y 2 (t ,τ ) := ∫
t
t −τ
(s − t + τ )x(s )ds
χ T (t ) = [x(t ) y1 (t ,τ ) y 2 (t ,τ )]T then we can get:
V4 (x(t ), t ) = 2 x T (t )Q1 y1 (t ,τ )
[ [
by (3) and (13), we get: LV1 (x (t ), t ) ≤ 2 x T (t )PAx (t ) + 2 x T (t )PA1 x(t − τ )
T
t t V2 (x(t ), t ) ≥ τ −1 ⎡ ∫ x(s )ds ⎤ R1 ⎡ ∫ x(s )ds ⎤ ⎢⎣ t −τ ⎥⎦ ⎢⎣ t −τ ⎥⎦ = τ −1 y T (t ,τ )R1 y (t ,τ ) so we can conclude:
+ 2 x T (t )Pf (t , x (t ), x (t − τ ))
[
Noting (8) and (14), we know that if the LMI (15) holds, the Lyapunov-Krasovskii functional (5) is positive definite. This completes the proof. From the prove of the proposition 1, we can see that the constraint on functional parameter Q2 is relaxed, which is helpful to reduce the conservatism of the main results. In the following, we give the main stability condition of system (1). Theorem 1. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n ,
S1 , S 2 ,
S 3 = S 3T > 0, S 4 = S 4T > 0 satisfying Proposition 1, such that the following LMIs hold:
Π 15 ⎤ T A1 S1 − S 2 ⎥⎥ ⎥ < 0 (16) S1 ⎥ − S3 ⎥ − 3τ −1 S 4 ⎥⎦
P ≤ ρI
where Π 11 = PA + AT P + R1 + τR2 + Q1 + Q1T + τS1 + τS1T + τ 3 S 4 + εG1T G1 + ρG3T G3 Π 14 = AT Q1 + Q2 − S1 + τS 2T Π 15 = AT S1 + S 2 + τS 3
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T
]
T 3
+ 2 x (t )PA1 x (t − τ ) T
+ V6 (x(t ), t ) + V7 (x(t ), t ) + V8 (x(t ), t ) = χ (t )Ωχ (t )
Π 44 ∗
[
= x (t ) PA + A P + ρG G3 x (t ) T
T
− εI ∗ ∗
]
+ ρ x T (t )G3T G3 x(t ) + x T (t − τ )G4T G4 x (t − τ )
V1 (x(t ), t ) + V2 (x(t ), t ) + V4 (x(t ), t ) + V5 (x(t ), t )
Π 22 ∗ ∗ ∗
]
+ Trace g T (t , x (t ), x (t − τ ))Pg (t , x (t ), x (t − τ ))
V8 (x(t ), t ) = y 2T (t ,τ )S 3 y 2 (t ,τ ) by lemma 1, we also have:
Π 14 T A1 Q1 − Q2 Q1
T
+ 2 x T (t )Pf (t , x (t ), x (t − τ ))
V7 (x(t ), t ) = 2 y1T (t ,τ )S 2 y 2 (t ,τ )
P 0
]
= 2 x (t )PAx (t ) + 2 x (t )PA1 x (t − τ ) T
V6 (x(t ), t ) = 2 x T (t )S1 y 2 (t ,τ )
PA1 − Q1
Π 44 = −τ −1 R2 − S 2 − S 2T Proof . Choosing the Lyapunov-Krasovskii functional (5), by Ito calculus rules, the infinitesimal operator LV (x(t ), t ) of the stochastic process {xt , t ≥ 0} along the trajectory of system (1) are given by 1 LV1 (x (t ), t ) = lim+ [E (V1 (x (t + Δ ), t + Δ )) − V1 (x (t ), t )] Δ→0 Δ = 2 x T (t )P [ Ax (t ) + A1 x (t − τ ) + f (t , x (t ), x (t − τ ))] + Trace g T (t , x (t ), x (t − τ ))Pg (t , x (t ), x (t − τ ))
V5 (x(t ), t ) = y1T (t ,τ )Q2 y1 (t ,τ )
⎡Π 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎣
Π 22 = ρG4T G4 + εG2T G2 − R1
(17)
+ 2 x T (t )Pf (t , x (t ), x (t − τ )) + x T (t − τ )ρG4T G4 x(t − τ )
(18) LV2 (x(t ), t ) = x T (t )R1 x(t ) − x T (t − τ )R1 x(t − τ )
(19)
by lemma 2, we can get: LV3 (x(t ), t ) = τx T (t )R2 x(t ) − ≤ τx T (t )R2 x(t ) − τ −1
∫
t
t −τ
∫
t
t −τ
x T (s )R2 x(s )ds
(20)
x T (s )dsR2
∫ x(s )ds t
t −τ
1 [E (V4 (x(t + Δ ), t + Δ )) − V4 (x(t ), t )] Δ →0 Δ = 2 x T (t )Q1 x(t ) − 2 x T (t )Q1 x(t − τ )
LV4 (x (t ), t ) = lim+
+ 2 x T (t )AT Q1 ∫ x(s )ds t
t −τ
+ 2 x (t − τ )A Q1 ∫ x(s )ds T
+2f
T
T 1
t
t −τ
(t , x(t ), x(t − τ ))Q1 ∫t −τ x(s )ds t
(21) LV5 (x(t ), t ) = 2[x(t ) − x(t − τ )] Q2 T
= 2 x T (t )Q2 − 2 xT
∫ x(s )ds t
t −τ
∫ x(s )ds (t − τ )Q ∫ x(s )ds t
t −τ
t
2
t −τ
(22)
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JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
1 [E (V6 (x(t + Δ ), t + Δ )) − V6 (x(t ), t )] Δ
LV6 (x(t ), t ) = lim+ Δ →0
= 2 x (t )S1[τx(t ) − T
+ 2 x T (t )AT + 2x
T
∫ x(s )ds]
It also can be observed:
V1 (x(t ), t ) = x T (t )Px(t ) ≤ λmax ( P) x(t )
t −τ
t
t
t −τ
t
t −τ
− 2xT
∫
+ 2τ −2
t
2
t
t
t −τ
LV8 (x(t ), t ) = 2τx (t )S 3 ∫ T
∫
t −τ
t
t −τ
(24)
x(s )ds
T
5
T
t
t −τ
(s − t + τ )x(s )ds
(25)
(s − t + τ ) ≤ τ 3 x T (t )S 4 x(t ) t −τ
−
3
τ
x (s )S 4 x(s )ds
4
t −τ
t
3
0
x(s )ds
⎞ ∫ x(s )ds ⎟⎠ t
T
S2
t −τ
t
∫ (s − t + τ )x(s )ds t
t −τ
t
T
2
t −τ
t −τ
t
T
t −τ
max
V8 (x(t ), t ) =
2
T 2
0
3
t
t
T
3
t −τ
t
9
3
0
3
LV (x(t ), t ) ≤ − μ x(t )
2
2
−τ
3
(29)
T
4
t −τ
0
3
T
t −τ
T
2
t
(28)
⎤ ∫t −τ (s − t +τ )x(s)ds⎥
2
−τ
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds ≤ τ ∫ (s − t + τ ) x (s )S x(s )ds ≤ τ λ (S )∫ x(t + θ ) dθ V (x(t ), t ) = ∫ (s − t + τ ) x (s )S x(s )ds ≤ τ λ (S )∫ x(t + θ ) dθ max
t
2
−τ
t −τ
4
2
−τ
so we can deduce that EV (x(0 ),0) ≤ (α + τβ ) sup E ξ (θ )
t −τ ⎦ ⎣ Therefore, we know that LMI (16) is equivalent to Π < 0 . Moreover, it is easy to see that there exists a scalar μ > 0 such that:
© 2013 ACADEMY PUBLISHER
1
2
3
where
∫
−τ
t
max
t
2
2
t −τ
η T (t , s )Πη (t , s )ds
η T (t, s ) = ⎡⎢ x(t ) x(t −τ ) f
2
t
LV (x(t ), t ) = ∑ LVi (x(t ), t ) t −τ
T
∫ x (s )dsS ∫ x(s )ds + ∫ (s − t + τ )x (s )ds ∫ (s − t + τ )x(s )ds ≤ (τλ (S S ) + τ )∫ x(t + θ ) dθ
9
∫
0
−τ
T 1 1
≤
t −τ
i =1
2
t −τ
V7 (x(t ), t ) = 2⎛⎜ ⎝
combining (18)- (26), we have:
τ
0
T
max
noting (2), we can calculate that, for any scalar ε > 0 , we have: εx T (t )G1T G1 x(t ) + εx T (t − τ )G2T G2 x(t − τ ) (27) − εf T (t ) f (t ) ≥ 0
≤
t −τ
0
−τ
−τ
t
(26)
t
t −τ
2
t −τ
t
T
t
T
∫ (s − t + τ )x (s )ds ∫ (s − t + τ )x(s )ds ≤ λ (S S ) x(t ) + τ ∫ x(t + θ ) dθ
T
t −τ
1
1
T 1
t
T
≤ x T (t )S1 S1T x T (t )
∫ (s − t + τ )x (s )dsS ∫ (s − t + τ )x(s )ds t
1
T
LV9 (x(t ), t ) = τ 3 x T (t )S 4 x(t ) − 3∫
2
−τ
t −τ
T 1
6
+
2
2
max
by lemma 2, we can get: t
2
0
(s − t + τ )x(s )ds
t −τ
T
t
1
max
− 2∫ x (s )dsS 3 ∫ t
t
T
x T (s )dsS 2 x(t ) t
∫ x (u )R x(u )duds ≤ τλ (R )∫ x(t + θ ) dθ V (x(t ), t ) = 2 x (t )Q ∫ x(s )ds ≤ x (t )Q Q x (t ) + ∫ x (s )ds ∫ x(s )ds ≤ λ (Q Q ) x(t ) + τ ∫ x(t + θ ) dθ V (x(t ), t ) = ∫ ds ∫ x (t + s )Q x(t + θ )dθ ≤ τλ (Q )∫ x(t + θ ) dθ V ( x(t ), t ) = 2 x (t )S ∫ (s − t + τ )x(s )ds t
4
t −τ
x T (s )dsS 2
V3 (x(t )) = ∫
max
t
t
2
0
t −τ
t −τ
x(t + θ ) dθ
0
−τ
t −τ s
⎤ ∫ x(s )ds ⎥⎦
∫ (s − t + τ )x(s )ds (t − τ )S ∫ (s − t + τ )x(s )ds
t −τ
∫
∫t −τ (s − t + τ )x(s )ds t
x T (s )dsS 2 ⎡τx(t ) − ⎢⎣
= 2 x (t )S 2 T
≤ λmax (R1 )∫
t −τ
(23)
∫
2
V2 (x(t ), t ) = ∫ x T (s )R1 x(s )ds
t −τ
t
LV7 (x (t ), t ) = 2[x(t ) − x (t − τ )] S 2
(30)
0
t −τ
t
T
+2
t
EV (x(t ), t ) = EV (x(0),0) + E ∫ LV (x(s ), s )ds
t
∫ (s − t + τ )x(s )ds (t − τ )A ∫ (s − t + τ )x(s )ds ∫ (s − t + τ )x(s )ds T 1
+2f T
by Dynkin formula, we have:
2
−τ ≤θ ≤ 0
EV (x(t ), t ) ≥ λmin (P ) x(t )
where
2
(31)
JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
497
α = λmax (P ) + λmax (Q1Q1T ) + λmax (S1 S1T )
⎡Ξ11 ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎢⎣ ∗
β = 1 + 2τ 3 + λmax (R1 ) + τλmax (R2 ) + τλmax (Q2 )
(
)
+ τλmax S 2 S 2T + τ 3 λmax (S 3 ) + +τ 3 λmax (S 4 )
PA1 − Q1 ρG G4 + εG2T G2 − R1 ∗ ∗ T 4
By Gronwell Inequality, we can get: E x(t ) ≤ 2
(α + τβ ) sup E ξ (θ ) 2 exp⎛⎜ − μ t ⎞⎟ ⎜ λ (P ) ⎟ λmin (P ) −τ ≤θ ≤ 0 ⎝ min ⎠
(32)
∫ (s − t + τ )x(s )ds
t
3
T
LV (x(t ), t ) ≤ − μ x(t )
the conservatism of the stability criterion. In the following section, we would proof it in theory. Choosing a simplified Lyapunov-Krasovskii functional as:
t
EV (x(t ), t ) = EV (x(0 ),0 ) + E ∫ LV (x(s ), s )ds 0
by Lyapunov-Krasovskii functional (33), we also can deduce that EV (x(0 ),0) ≤ (α + τβ ) sup E ξ (θ )
where
α = λmax (P ) + λmax (Q1Q1T ) , β = 1 + λmax (R1 ) + τλmax (R2 ) + τλmax (Q2 ) .
5
so by Gronwell Inequality, we also can get:
= x (t )Px(t ) + ∫ x (s )R1 x(s )ds t
T
T
t −τ
t
T
0
2
−τ
0
t
T
t −τ s
1
t −τ
T
−τ
2
(33) then we can get the proposition 2 as following: Proposition 2. If there exist matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying the following LMIs: ⎡P R1 > 0 , R2 > 0 , ⎢ T ⎣Q1
Q1
⎤
τ −1 R1 + Q2 ⎥⎦
>0
(34)
then the Lyapunov-Krasovskii functional (33) is positive definite. According to the proof of Theorem 1, by Ito calculus rules, the infinitesimal operator LV (x(t ), t ) of the stochastic process {xt , t ≥ 0} along the trajectory of system (1) can be calculated, and we can get the corresponding stability condition. Theorem 2. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices P = PT , R1 = R1T , R2 = R2T , Q1, Q2 = Q2T ∈ ℜ n×n
satisfying Proposition 2, such that the following LMIs hold: © 2013 ACADEMY PUBLISHER
E x(t ) ≤ 2
∫ x (u )R x(u )duds + 2 x (t )Q ∫ x(s )ds + ∫ ds ∫ x (t + s )Q x(t + θ )dθ +∫
t
2
−τ ≤θ ≤ 0
V (x(t ), t ) = ∑ Vi (x(t ), t ) i =1
2
Therefore, by Dynkin formula, we have
are very important to induce
4
t −τ
(36)
where Ξ11 = PA + AT P + R1 + τR2 + Q1 + Q1T + εG1T G1 + ρG3T G3 We know that LMI (35) is equivalent to Ξ < 0 . Moreover, it is easy to see that there exists a scalar μ > 0 such that
t
t −τ
in η (t, s ) and the Lyapunov-Krasovskii functional term
∫ (s − t + τ ) x (s )S x(s )ds
P ⎤ ⎥ 0 ⎥ < 0 (35) Q1T ⎥ ⎥ − εI ⎥⎦
P ≤ ρI
This completes the proof. Remark 1. LV (x(t ), t ) is calculated based on Ito calculus rules and the independent increments property of Brownian motion, which is different from the calculus rules in deterministic systems. It is just the difference that makes the difficulty in generalizing the stability analysis methods for time-delay systems, which is useful in reducing the conservatism of the stability result, to the stochastic case. In Theorem 1, the introduction of
AT Q1 + Q2 A1T Q1 − Q2 − τ −1 R2 ∗
(α + τβ ) sup E ξ (θ ) 2 exp⎛⎜ − μ t ⎞⎟ ⎜ λ (P ) ⎟ λmin (P ) −τ ≤θ ≤ 0 ⎠ ⎝ min (37)
which illustrate the stochastic system (1) is exponentially stable in the mean square. Remark 2. Compare Theorem 2 with Theorem 1, it is easy to see that Theorem 2 is the special case of Theorem t 1. Let η T (t , s ) = ⎡⎢ x T (t ) x T (t − τ ) ∫ x T (s )ds f T ⎤⎥ t −τ ⎦ ⎣ and S1 = S 2 = S 3 = S 4 = 0 , we can get Theorem 2 from Theorem 1. So, we can know that the conservatism of Theorem 1 is less conservative than Theorem 2. From (20), we can see that the integral inequality is used in order to get the main result, which cause conservatism of the result. Here we choose another method to avoid the integral inequality. Choosing the Lyapunov-Krasovskii functional (33) and calculated the V& (x(t ), t ) as following:
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x& (t ) = Ax(t ) + A1 x(t − τ ) + f (t,x(t ),x(t − τ ))
LV (x(t ), t ) = 2 x T (t )P[Ax(t ) + A1 x (t − τ ) + f (t , x (t ), x (t − τ ))]
[
]
x(t ) = φ(t ), t ∈ [− τ,0 ]
+ Trace g T (t , x(t ), x(t − τ ))Pg (t , x(t ), x(t − τ ))
+ x T (t )R1 x(t ) − x T (t − τ )R1 x(t − τ )
(38)
Using the Lyapunov-Krasovskii functional (5) and method in Theorem 1, we can get Corollary 1: + τx T (t )R2 x(t ) − ∫ x T (s )R2 x(s )ds Corollary 1. For any given τ > 0 , the system (38) is t −τ asymptotically stable. If there exist positive a scalar + 2 x T (t )Q1 x(t ) − 2 x T (t )Q1 x(t − τ ) ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , S1 , S 2 , t t + 2 x T (t )AT Q1 ∫ x(s )ds + 2 x T (t − τ )A1T Q1 ∫ x (s )ds t −τ t −τ S 3 = S 3T > 0, S 4 = S 4T > 0 Q2 = Q2T ∈ ℜ n× n satisfying t Proposition 1, such that the following LMI holds: + 2 f T (t , x (t ), x (t − τ ))Q1 x (s )ds t
∫
t −τ
+ 2 x T (t )Q2 ∫ x (s )ds t
t −τ
− 2 x T (t − τ )Q2 ∫ x (s )ds t
t −τ
⎡ A + A P + R1 + τR2 + Q1 + Q = x T (t )⎢ T T ⎢⎣+ ρG3 G3 + εG1 G1 + x T (t )[PA1 − Q1 ]x (t − τ ) T
[
T 1
⎤ ⎥ x(t ) ⎥⎦
]
+ x (t − τ ) ρG G4 + εG G2 − R1 x(t − τ ) T
T 4
T 2
+ 2 x (t )Pf (t , x(t ), x(t − τ ))
[
[
]∫
t
t −τ
+ 2 x T (t − τ ) A1T Q1 − Q2
x(s )ds
]∫
t
t −τ
x(s )ds
+ 2 f T (t , x(t ), x(t − τ ))Q1 ∫ x(s )ds t
t −τ
− ∫ x T (s )R2 x(s )ds t
t −τ
so we can get LV (x(t ), t ) = τ −1 ∫ γ T (t , s )Θγ (t , s )ds t
t −τ
and the following stability condition: Theorem 3. For any given τ > 0 , the stochastic system (1) is exponentially stable in the mean square. If there exist positive scalars ρ > 0 , ε > 0 and matrices
(1,4) A1T Q1 − Q2
− εI
Q1
∗
(4,4)
∗
∗
(1,5) ⎤ A1T S1 − S 2 ⎥⎥ ⎥ 0 , the stochastic system (38) is asymptotically stable. If there exists a scalar ε > 0 and matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying Proposition 2, such that the following LMI holds: ⎡(1,1)' ' PA1 − Q1 τAT Q1 + τQ2 P ⎤ ⎢ ⎥ εG2T G2 − R1 τA1T Q1 − τQ2 0 ⎥ ⎢ ∗ 0, S 4 = S 4T > 0 matrices P = P T , R1 = R1T , R2 = R2T , Q1 , Q2 = Q2T ∈ ℜ n×n satisfying Proposition 1, such that the following LMIs hold:
⎡(1,1)' ' ' PA1 − Q1 ⎢ ∗ − R1 ⎢ ⎢ ∗ ∗ ⎢ ∗ ⎣ ∗
(1,3)' ' '
(1,4)' ' '
A Q1 − Q2 T 1
(3,3)' ' ' ∗
⎤ A S1 − S 2 ⎥⎥ < 0 (39) − S3 ⎥ ⎥ − 3τ −1 S 4 ⎦ T 1
where (1,1)' ' ' = PA + AT P + R1 + τR2 + Q1 + Q1T + τS1 + τS1T + τ 3 S 4
(1,3)' ' ' = AT Q1 + Q2 − S1 + τS 2T (1,4)' ' ' = AT S1 + S 2 + τS 3 (3,3)' ' ' = −τ −1 R2 − S 2 − S 2T
Using the Lyapunov-Krasovskii functional (33) and method in Theorem 2, we can get Corollary 4: Corollary 4. For any given τ , the linear time-delay system (39) is asymptotically stable, if there exist R1 = R1T , Q1 , Q2 = Q2T ∈ ℜ n×n , matrices P = P T , R2 = R2T satisfying Proposition 1, such that the following LMI holds: ⎡(1,1)' ' ' ' PA1 − Q1 τAT Q1 + τQ2 ⎤ ⎢ ⎥ − R1 τ A1T Q1 − τQ2 ⎥ < 0 ⎢ ∗ ⎢ ∗ ⎥ ∗ − τR2 ⎣ ⎦
where (1,1)' ' ' ' = PA + AT P + R1 + τR2 + Q1 + Q1T IV. NUMERICAL EXAMPLE In this section, several examples are given to show that Theorem 1 is less conservative than Theorem 2, and they are less conservative than some existing results. Example 1. Consider the stochastic delay system (1) with 0⎤ ⎡− 2 0 ⎤ ⎡ −1 , A1 = ⎢ A=⎢ ⎥ ⎥ ⎣ 1 − 1⎦ ⎣− 0.5 − 1⎦ ΔA(t ) ≤ 0.1 , ΔA1 (t ) ≤ 0.1
[ ]
Trace g T g ≤ 0.1 x(t ) + 0.1 x(t − τ ) 2
2
The comparing results are listed in Table 1. It can be seen that the delay-dependent exponential stability criteria in this note is less conservative in the sense of the computed maximum delay bound. Example 2. Consider the time-delay system (38) with 0 ⎤ ⎡− 2 ⎡− 1 0 ⎤ A=⎢ ⎥ , A1 = ⎢− 1 − 1⎥ 0 − 0 . 9 ⎣ ⎦ ⎣ ⎦ For this deterministic system, we compared the results in Table 2. The results have demonstrated that the delay-
© 2013 ACADEMY PUBLISHER
499
dependent stability condition in this note is less conservative. TABLE I. COMPARISON OF STABILITY CONDITIONS IN EXAMPLE 1 Methods Mao [6] Yue and Won [7] Chen and Guan [8] Theorem 2 Theorem 1
Maximu τ allowed 0.175 0.8635 1.1997 2.0859 2.8375
TABLE II. COMPARISON OF STABILITY CONDITIONS IN EXAMPLE 2 Methods Moon et al. [5] Fridman and Shaked [2] Xu and Lam [11] Corollary 2 Corollary 1
Maximu τ allowed 4.3588 4.47 4.4721 4.4721 4.564
V. CONCLUSION The exponential stability for stochastic delay systems with nonlinear uncertainties is investigated. Based on Ito calculus rules, a general type of Lyapunov-Krasovskii functional is introduced, and a novel stability condition is obtained in terms of linear matrix inequalities. Numerical examples are given to show that the criteria perform much better than existing stability results. ACKNOWLEDGMENT This work is supported by National Nature Science Foundation under Grant 61104119 and 61104079, the Doctoral Foundation and the Young Core Instructor Foundation from Henan Polytechnic University under Grant B2010-50 and 649166, the Young Core Instructor Foundation in Higher School of Henan Province under grant 2011GGJS-054. REFERENCES [1] Q. L. Han, “A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems, “Automatica, vol. 44, pp. 272- 277, 2008. [2] E. Fridman, and U. Shaked, “A descriptor system approach to H control of linear time-delay systems, “IEEE Trans. Automat. Control, vol. 47, pp. 253-270, 2002. [3] Y. He, Q. G. Wang, C. Lin, and M. Wu, “Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems, “ Int. J. Robust Nonlinear Control, vol. 15, pp. 923–933, 2005. [4] P. Park, “A Delay-Dependent Stability Criterion for Systems with Uncertain Time-Invariant Delays, “IEEE Trans. on Automatic Control, Vol. 44, pp. 876-877, 1999. [5] Y.S. Moon, P. Park, and W. H. Kwon, “Delay-dependent robust stabilization of uncertain state-delayed systems, “Int. J. Control, vol. 74, pp. 1447-1457, 2001. [6] W. Qian, J. Liu, Y. X. Sun, and S. M. Fei, “A less conservative robust stability criteria for uncertain neutral systems with mixed delays, “Mathematics and Computers in Simulation, vol. 80, pp. 1007–1017, 2010. [7] W. Qian, S. Cong, Y. X. Sun and S. M. Fei, “Novel robust stability criteria for uncertain systems with time-varying
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[8]
[9]
[10]
[11]
[12]
[13]
[14]
JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
delay, “Applied Mathematics and Computation, vol. 215, pp. 866-872, 2009. M. N. A. Parlakci, “Extensively augmented Lyapunov functional approach for the stability of neutral time-delay systems, “ IET Control Theory Appl., Vol. 2, No. 5, pp. 431-436, 2008. W. Qian, T. Li, S. Cong, and S. M. Fei, “Improved stability analysis on delayed neural networks with linear fractional uncertainties, “ Applied Mathematics and Computation, vol. 217, pp. 3596-3606, 2010. X. Mao, “Robustness of exponential stability of stochastic differential delay equations, “IEEE Trans. Automat. Control, vol. 41, pp. 442-447, 1996. D. Yue and S. Won, “Delay-dependent robust stability of stochastic systems with the time delay and nonlinear uncertainties, “IEE Electron. Lett, vol. 37, pp. 992-993, 2001. W. Chen, Z. Guan and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach, “Systems Control Lett., vol. 54, pp. 547-555, 2005. D. Yue, and Q. Han, “Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and markovian switching, “IEEE Trans. Automat. Control, vol. 50, pp. 217-222, 2005. Y. Chen, W. X. Zheng, and A. Xue, “A new result on stability analysis for stochastic neutral systems” Automatica, vol. 46, pp. 2100-2104, 2010.
Wei Qian was born in China in 1978. He received his B.Sc. degree in Mechanical Design and Manufacturing Technology from Zhengzhou University in 1999, M.Sc. degree in Control Theory and Application from Southeast University in 2005 and Ph.D. degree in Control Science and Engineering from Zhejiang University in 2009, respectively. From 1999 to 2002, he served as a teacher at Department of Basic Courses, Jiaozuo, China. From 2002 to 2005, he studied for his M.Sc. degree at Department of Automation of Southeast University, Nanjing, China. From 2006 to 2009, he studied for
© 2013 ACADEMY PUBLISHER
his Ph.D. degree at State Key Lab of Industrial Control Technology, Zhejiang University, Hangzhou, China. Since 2009, he served as a teacher at Department of Electrical Engineering and Automation at Henan Polytechnic University, Jiaozuo, China. He has published 30 papers in international journals and conferences. His research interests include stability analysis, robust control of time-delay systems, network control systems and stochastic systems.
Juan Liu was born in China in 1976. She received her B.Sc. degree from Henan Normal University in 1999, and M.Sc. degree from Southwest Jiaotong University in 2005, all in Mathematics. From 1999 to 2002, she served as a teacher at Department of Basic Courses, Jiaozuo, China. From 2002 to 2005, she studied at Department of Mathematics of Southwest University, Chendu, China. Since 2002, she served as a teacher at Department of Mathematics and Information Science of Henan Polytechnic University. She has published 15 papers in international journals and conferences. Her research interests include robust control and fuzzy control of systems with time-delay.
Shaohua Wang was born in China in 1963. He received his B.Sc. degree from Jiaozuo Institute of Technology and M.Sc. degree from China University of Mining and Technology, in 1987 and 1995, respectively, all in Electrical Engineering. From 1987 to 1993, he served as a teacher at Department of Electrical Engineering at Jiaozuo Institute of Technology, Jiaozuo, China. From 1993 to 1995, he studied for his M.Sc. degree at Department of Electrical Engineering at China University of Mining and Technology, Beijing, China. Since 1995, he served as a teacher at Department of Electrical Engineering and Automation at Henan Polytechnic University, Jiaozuo, China. He has published 15 papers in in international journals and conferences. His research interests include date fusion, robust control and electrical engineering.
