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Mean square robust stability of stochastic switched discrete-time systems with convex polytopic uncertainties

Journal of Inequalities and Applications20122012:135

https://doi.org/10.1186/1029-242X-2012-135

Received: 28 April 2012

Accepted: 12 June 2012

Published: 12 June 2012

Abstract

This article is concerned with mean square robust stability of stochastic switched discrete time-delay systems with convex polytopic uncertainties. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the mean square robust stability for the stochastic switched system with convex polytopic uncertainties is designed via linear matrix inequalities. Numerical examples are included to illustrate the effectiveness of the results.

Keywords

switching designconvex polytopic uncertaintiesstochastic switched discrete systemmean square robust stabilityLyapunov functionlinear matrix inequality.

1 Introduction

Since the time delay is frequently viewed as a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, networked control systems, etc., the study of delay systems has received much attention and various topics have been discussed over the past years.

A switched system is a hybrid dynamical system consisting of a finite number of subsystems and a logical rule that manages switching between these subsystems. Switched systems have drawn a great deal of attention in recent years, see [119] and references therein. The motivation for studying switched systems comes partly from the fact that switched systems and switched multi-controller systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft and traffic control, and many other fields. On the other hand, time-delay phenomena are very common in practical systems. A switched system with time-delay individual subsystems is called a switched time-delay system; in particular, when the subsystems are linear, it is then called a switched time-delay linear system. During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [48]. The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functionals and linear matrix inequlity (LMI) approach for constructing a common Lyapunov function [810]. Although many important results have been obtained for switched linear continuous-time systems, there are few results concerning the stability of switched linear discrete systems with time-varying delays. It was shown in [5, 7, 11, 12] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems has been studied in [10], but the result was limited to constant delays. In [11, 12], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the average dwell time scheme.

This article studies mean square robust stability problem for stochastic switched linear discrete systems with convex polytopic uncertainties with interval time-varying delays. Specifically, our goal is to develop a constructive way to design switching rule to mean square robustly stable the system. By using improved Lyapunov-Krasovskii functionals combined with LMIs technique, we propose new criteria for the mean square robust stability of the system. Compared to the existing results, our result has its own advantages. First, the time delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, the delay function is bounded but not restricted to zero. Second, the approach allows us to design the switching rule for mean square robust stability in terms of LMIs, which can be solvable by utilizing Matlab's LMI Control Toolbox available in the literature to date.

The article is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Switching rule for the mean square robust stability is presented in Section 3. Numerical example is provided to illustrate the theoretical results in Section 4, and the conclusions are drawn in Section 5.

2 Preliminaries

The following notations will be used throughout this article. R+ denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the scalar product of two vectors 〈x, y〉 or x T y; Rn×rdenotes the space of all matrices of (n × r) - dimension. A T denotes the transpose of A; a matrix A is symmetric if A = A T , I is the identity matrix of appropriate dimention.

Matrix A is semi-positive definite (A ≥ 0) if 〈Ax, x〉 ≥ 0, for all x R n ; A is positive definite (A > 0) if 〈Ax, x〉 > 0 for all x ≠ 0; AB means A - B ≥ 0. λ(A) denotes the set of all eigenvalues of A; λmin(A) = min{Reλ: λ λ(A)}.

Consider a stochastic switched linear discrete systems with convex polytopic uncertainties with interval time-varying delay of the form
x ( k + 1 ) = A γ ( ζ ) x ( k ) + B γ ( ζ ) x ( k - d ( k ) ) + σ γ ( x ( k ) , x ( k - d ( k ) ) , k ) ω ( k ) , k = 0 , 1 , 2 , x ( k ) = v k , k = - d 2 , - d 2 + 1 , , 0 ,
(2.1)
where x(k) R n is the state, γ ( . ) : R n N : = { 1 , 2 , , N } is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, γ(x(k)) = i implies that the system realization is chosen as the i th system, i = 1, 2,..., N. It is seen that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(k) hits predefined boundaries. A i , B i , i = 1, 2,..., N are given constant matrices. The system matrices are subjected to uncertainties and belong to the polytope Ω given by
Ω = { [ A i , B i ] ( ζ ) : = j = 1 N ζ j [ A i j , B i j ] , j = 1 N ζ j = 1 , ζ j 0 } ,
(2.2)
where A ij , B ij , i, j = 1, 2,..., N, are given constant matrices with appropriate dimensions. ω(k) is a scalar Wiener process (Brownian Motion) on Ω , F , P with
E { ω ( k ) } = 0 , E { ω 2 ( k ) } = 1 , E { ω ( i ) ω ( j ) } = 0 ( i j ) ,
(2.3)
and σ i : R n × R n × RR n , i = 1, 2,..., N is the continuous function, and is assumed to satisfy that
σ i T ( x ( k ) , x ( k - d ( k ) ) , k ) σ i ( x ( k ) , x ( k - d ( k ) ) , k ) ρ i 1 x T ( k ) x ( k ) + ρ i 2 x T ( k - d ( k ) ) x ( k - d ( k ) , x ( k ) , x ( k - d ( k ) R n ,
(2.4)

where ρi 1> 0 and ρi 2> 0, i = 1, 2,..., N are khown constant scalars. For simplicity, we denote σ i (x(k), x(k- d(k)), k) by σ i , respectively.

The time-varying function d(k) satisfies the following condition:
0 < d 1 d ( k ) d 2 , k = 0 , 1 , 2 , .

Remark 2.1. It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero.

Definition 2.1. The stochastic switched linear discrete systems with convex polytopic uncertainties (2.1) is robustly stable in the mean square if there exist a positive definite scalar function V (k, x(k): R+ × R n R and a switching function γ (·) such that
E { Δ V ( k , x ( k ) ) } = E { V ( k + 1 , x ( k + 1 ) ) - V ( k , x ( k ) ) } < 0 ,

along any trajectory of solution of the system (2.1) for all uncertainties which satisfy (2.2).

Definition 2.2. The system of matrices {J i }, i = 1, 2,..., N, is said to be strictly complete if for every x R n \{0} there is i {1, 2,..., N} such that x T J i x < 0.

It is easy to see that the system {J i } is strictly complete if and only if
i = 1 N α i = R n \ { 0 } ,
where
α i = { x R n : x T J i x < 0 } , i = 1 , 2 , , N .
Proposition 2.1. [11]The system {J i }, i = 1, 2,..., N, is strictly complete if there exist δ i 0 , i = 1 , 2 , , N , i = 1 N δ i > 0 such that
i = 1 N δ i J i < 0 .

If N = 2 then the above condition is also necessary for the strict completeness.

Proposition 2.2. For real numbers ζ j ≥ 0, j = 1,2,...,N, j = 1 N ζ j = 1 the following inequality hold
( N - 1 ) j = 1 N ζ j 2 - 2 j = 1 N - 1 l = j + 1 N ζ j ζ l 0 .
Proof. The proof is followed from the completing the square:
( N - 1 ) j = 1 N ζ j 2 - 2 j = 1 N - 1 l = j + 1 N ζ j ζ l = j = 1 N - 1 l = j + 1 N ( ζ j - ζ l ) 2 0 .

3 Main results

Let us set
x k = sup s [ - d 2 , 0 ] x ( k + s ) , W i j j = ( d 2 - d 1 ) Q j - P j + 2 ρ i 1 I S j - S j A i j - S j B i j S j T - A i j T S j T P j + S j + S j T - S j B i j - B i j T S j T - B i j T S j T - Q j + 2 ρ i 2 I , W i j l = ( d 2 - d 1 ) Q j - P j + 2 ρ i 1 I S j - S j A i l - S j B i l S j T - A i l T S j T P j + S j + S j T - S j B i l - B i l T S j T - B i l T S j T - Q j + 2 ρ i 2 I , W i l j = ( d 2 - d 1 ) Q l - P l + 2 ρ i 1 I S l - S l A i j - S l B i j S l T - A i j T S l T P l + S l + S l T - S l B i j - B i j T S l T - B i j T S l T - Q l + 2 ρ i 2 I , P ( ζ ) = j = 1 N ζ j P j , Q ( ζ ) = j = 1 N ζ j Q j , S ( ζ ) = j = 1 N ζ j S j , λ 1 = λ min ( P ( ζ ) ) , λ 2 = λ max ( P ( ζ ) ) , R = R 0 0 0 0 0 0 0 0 , J i j j : = Q j - S j A i j - A i j T S j T , J i j l = Q j - S j A i l - A i l T S j T , J i l j : = Q l - S l A i j - A i j T S l T , α i j j = { x R n : x T J i j j x < 0 , } , i = 1 , 2 , , N , j = 1 , 2 , , N , α i j l = { x R n : x T J i j l x < 0 , } , i = 1 , 2 , , N , j = 1 , 2 , , N - 1 ; l = j + 1 , , N , α i j l = { x R n : x T J i l j x < 0 , } , i = 1 , 2 , , N , j = 1 , 2 , , N - 1 ; l = j + 1 , , N , α ¯ 1 j j = α 1 j j , α ¯ i j j = α i j j \ i = 1 i - 1 α ¯ i j j , i = 2 , 3 , , N , j = 1 , 2 , , N α ¯ 1 j l = α 1 j l , α ¯ i j l = α i j l \ i = 1 i - 1 α ¯ i j l , i = 2 , 3 , , N , j = 1 , 2 , , N - 1 ; l = j + 1 , , N , α ¯ 1 l j = α 1 l j , α ¯ i l j = α i l j \ i = 1 i - 1 α ¯ i l j , i = 2 , 3 , , N , j = 1 , 2 , , N - 1 ; l = j + 1 , , N .
(3.1)

The main result of this article is summarized in the following theorem.

Theorem 3.1. The stochastic switched system with convex polytopic uncertainties (2.1) is robustly stable in the mean square if there exist symmetric matrices P j > 0, Q j > 0, R ≥ 0, j = 1, 2..., N and matrix S j , j = 1, 2..., N satisfying the following conditions

(i) δ i ≥ 0, i = 1 N δ i > 0 : i = 1 N δ i J i j j < 0 , and J ijj + < 0, i = 1,2,...,N, j = 1,2,...,N.

(ii) δ i ≥ 0, i = 1 N δ i > 0 : i = 1 N [ δ i J i j l + δ i J i l j ] < 0 , and J i j l + J i l j - 2 N - 1 R < 0 , i = 1,2,...,N, j = 1,2,..., N - 1; l = j + 1,...,N.

(iii) W ijj + < 0, i = 1, 2,...,N, j = 1, 2,...,N.

(iv) W i j l + W i l j - 2 N - 1 R < 0 , i = 1, 2,...,N, j = 1, 2,...,N - 1; l = j+1,...,N.

The switching rule is chosen as γ(x(k)) = i, whenever x ( k ) α ¯ i j l .

Proof. Consider the following Lyapunov-Krasovskii functional for any i th system (2.1)
V ( k ) = V 1 ( k ) + V 2 ( k ) + V 3 ( k ) ,
where
V 1 ( k ) = x T ( k ) P ( ζ ) x ( k ) , V 2 ( k ) = i = k - d ( k ) k - 1 x T ( i ) Q ( ζ ) x ( i ) , V 3 ( k ) = j = - d 2 + 2 - d 1 + 1 l = k + j k - 1 x T ( l ) Q ( ζ ) x ( l ) .
We can verify that
λ 1 x ( k ) 2 V ( k ) λ 2 x k 2 .
(3.2)
Let us set ξ(k) = [x T (k) x T (k + 1) x T (k - d(k)) ω T (k)], and
H ( ζ ) = 0 0 0 0 0 P ( ζ ) 0 0 0 0 0 0 0 0 0 0 , G ( ζ ) = P ( ζ ) 0 0 0 I I 0 0 0 0 I 0 0 0 0 I .
Then, the difference of V1(k) along the solution of the system (2.1) and taking the mathematical expectation, we obtained
E { Δ V 1 ( k ) } = E { x T ( k + 1 ) P ( ζ ) x ( k + 1 ) - x T ( k ) P ( ζ ) x ( k ) } = E ξ T ( k ) H ( ζ ) ξ ( k ) - 2 ξ T ( k ) G T ( ζ ) 0 . 5 x ( k ) 0 0 0 .
(3.3)
because of
E { ξ T ( k ) H ( ζ ) ξ ( k ) } = E { x ( k + 1 ) P ( ζ ) x ( k + 1 ) } , E 2 ξ T ( k ) G T ( ζ ) 0 . 5 x ( k ) 0 0 0 = E { x T ( k ) P ( ζ ) x ( k ) } .
Using the expression of system (2.1)
0 = - S ( ζ ) x ( k + 1 ) + S ( ζ ) A i ( ζ ) x ( k ) + S ( ζ ) B i ( ζ ) x ( k - d ( k ) ) + S ( ζ ) σ i ω ( k ) , 0 = - σ i T x ( k + 1 ) + σ i T A i ( ζ ) x ( k ) + σ i T B i ( ζ ) x ( k - d ( k ) ) + σ i T σ i ω ( k ) ,
we have
E - 2 ξ T ( k ) G T ( ζ ) × 0 . 5 x ( k ) - S ( ζ ) x ( k + 1 ) + S ( ζ ) A i ( ζ ) x ( k ) + S ( ζ ) B i ( ζ ) x ( k - d ( k ) ) + S ( ζ ) σ i ω ( k ) 0 - σ i T x ( k + 1 ) + σ i T A i ( ζ ) x ( k ) + σ i T B i ( ζ ) x ( k - d ( k ) ) + σ i T σ i ω ( k ) ξ ( k ) = E - ξ T ( k ) G T ( ζ ) 0 . 5 I 0 0 0 S ( ζ ) A i ( ζ ) - S ( ζ ) S ( ζ ) B i ( ζ ) S ( ζ ) σ i 0 0 0 0 σ i T A i ( ζ ) - σ i T σ i T B i ( ζ ) σ i T σ i ξ T ( k ) - E ξ T ( k ) 0 . 5 I 0 0 0 S ( ζ ) A i ( ζ ) - S ( ζ ) S ( ζ ) B i ( ζ ) S ( ζ ) σ i 0 0 0 0 σ i T A i ( ζ ) - σ i T σ i T B i ( ζ ) σ i T σ i T G ( ζ ) ξ ( k ) .
Therefore, from (3.3) it follows that
E { Δ V 1 ( k ) } = E { x T ( k ) [ - P ( ζ ) - S ( ζ ) A i ( ζ ) - A i T ( ζ ) S T ( ζ ) ] x ( k ) } + E { 2 x T ( k ) [ S ( ζ ) - S ( ζ ) A i ( ζ ) ] x ( k + 1 ) } + E { 2 x T ( k ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { 2 x T ( k ) [ - S ( ζ ) σ i - σ i T A i ( ζ ) ] ω ( k ) } + E { x ( k + 1 ) [ P ( ζ ) + S ( ζ ) + S T ( ζ ) ] x ( k + 1 ) } + E { 2 x ( k + 1 ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { 2 x ( k + 1 ) [ σ i T - S ( ζ ) σ i ] ω ( k ) } + E { 2 x T ( k - d ( k ) ) [ - σ i T B i ( ζ ) ] ω ( k ) } + E { ω T ( k ) [ - 2 σ i T σ i ] ω ( k ) } .
By asumption (2.3), we have
E { Δ V 1 ( k ) } = E { x T ( k ) [ - P ( ζ ) - S ( ζ ) A i ( ζ ) - A i T ( ζ ) S T ( ζ ) ] x ( k ) } + E { 2 x T ( k ) [ S ( ζ ) - S ( ζ ) A i ( ζ ) ] x ( k + 1 ) } + E { 2 x T ( k ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { x ( k + 1 ) [ P ( ζ ) + S ( ζ ) + S T ( ζ ) ] x ( k + 1 ) } + E { 2 x ( k + 1 ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { 2 [ - σ i T σ i ] } .
Applying asumption (2.4), the following estimations hold
E { - σ i T ( x ( k ) , x ( k - d ( k ) ) , k ) σ i ( x ( k ) , x ( k - d ( k ) ) , k ) } E { ρ i 1 x T ( k ) x ( k ) + ρ i 2 x T ( k - d ( k ) ) x ( k - d ( k ) } .
Therefore, we have
E { Δ V 1 ( k ) } = E { x T ( k ) [ - P ( ζ ) - S ( ζ ) A i ( ζ ) - A i T ( ζ ) S T ( ζ ) + 2 ρ i 1 I ] x ( k ) } + E { 2 x T ( k ) [ S ( ζ ) - S ( ζ ) A i ( ζ ) ] x ( k + 1 ) } + E { 2 x T ( k ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { x ( k + 1 ) [ P ( ζ ) + S ( ζ ) + S T ( ζ ) ] x ( k + 1 ) } + E { 2 x ( k + 1 ) [ - S ( ζ ) B i ( ζ ) ] x ( k - d ( k ) ) } + E { x T ( k - d ( k ) ) [ 2 ρ i 2 I ] x ( k - d ( k ) ) } .
(3.4)
The difference of V2(k) is given by
E { Δ V 2 ( k ) } = E i = k + 1 - d ( k + 1 ) k x T ( i ) Q ( ζ ) x ( i ) - i = k - d ( k ) k - 1 x T ( i ) Q ( ζ ) x ( i ) = E i = k + 1 - d ( k + 1 ) k - d 1 x T ( i ) Q ( ζ ) x ( i ) + x T ( k ) Q ( ζ ) x ( k ) - x T ( k - d ( k ) ) Q ( ζ ) x ( k - d ( k ) ) + E i = k + 1 - d 1 k - 1 x T ( i ) Q ( ζ ) x ( i ) - i = k + 1 - d ( k ) k - 1 x T ( i ) Q ( ζ ) x ( i ) .
(3.5)
Since d(k) ≥ d1 we have
E i = k + 1 - d 1 k - 1 x T ( i ) Q ( ζ ) x ( i ) - i = k + 1 - d ( k ) k - 1 x T ( i ) Q ( ζ ) x ( i ) 0 ,
and hence from (3.5) we have
E { Δ V 2 ( k ) } E { i = k + 1 - d ( k + 1 ) k - d 1 x T ( i ) Q ( ζ ) x ( i ) + x T ( k ) Q ( ζ ) x ( k ) - x T ( k - d ( k ) ) Q ( ζ ) x ( k - d ( k ) ) } .
(3.6)
The difference of V3(k) is given by
E { Δ V 3 ( k ) } = E { j = - d 2 + 2 - d 1 + 1 l = k + j + 1 k x T ( l ) Q ( ζ ) x ( l ) - j = - d 2 + 2 - d 1 + 1 l = k + j k - 1 x T ( l ) Q ( ζ ) x ( l ) } = E j = - d 2 + 2 - d 1 + 1 [ l = k + j k - 1 x T ( l ) Q ( ζ ) x ( l ) + x T ( k ) Q ( ζ ) ( ξ ) x ( k ) - E { l = k + j k - 1 x T ( l ) Q ( ζ ) x ( l ) - x T ( k + j - 1 ) Q ( ζ ) x ( k + j - 1 ) ] } = E j = - d 2 + 2 - d 1 + 1 [ x T ( k ) Q ( ζ ) x ( k ) - x T ( k + j - 1 ) Q ( ζ ) x ( k + j - 1 ) ] = E ( d 2 - d 1 ) x T ( k ) Q ( ζ ) x ( k ) - j = k + 1 - d 2 k - d 1 x T ( j ) Q ( ζ ) x ( j )
(3.7)
Since d(k) ≤ d2, and
E i = k = 1 - d ( k + 1 ) k - d 1 x T ( i ) Q ( ζ ) x ( i ) - i = k + 1 - d 2 k - d 1 x T ( i ) Q ( ζ ) x ( i ) 0 ,
we obtain from (3.6) and (3.7) that
E { Δ V 2 ( k ) + Δ V 3 ( k ) } E { ( d 2 - d 1 + 1 ) x T ( k ) Q ( ζ ) x ( k ) - x T ( k - d ( k ) ) Q ( ζ ) x ( k - d ( k ) ) } .
(3.8)
Therefore, combining the inequalities (3.4), (3.8) gives
E { Δ V ( k ) } E { x T ( k ) J i ( ζ ) x ( k ) + ψ T ( k ) W i ( ζ ) ψ ( k ) } ,
(3.9)
where
ψ ( k ) = [ x T ( k ) x T ( k + 1 ) x T ( k - d ( k ) ) ] , W i ( ζ ) = ( d 2 - d 1 ) Q ( ζ ) - P ( ζ ) + 2 ρ i 1 I S ( ζ ) - S ( ζ ) A i ( ζ ) - S ( ζ ) B i ( ζ ) S T ( ζ ) - A i T ( ζ ) S T ( ζ ) P ( ζ ) + S ( ζ ) + S T ( ζ ) - S ( ζ ) B i ( ζ ) - B i T ( ζ ) S T ( ζ ) - B i T ( ζ ) S T ( ζ ) - Q ( ζ ) + 2 ρ i 2 I , J i ( ζ ) = Q ( ζ ) - S ( ζ ) A i ( ζ ) - A i T ( ζ ) S T ( ζ ) .
From the convex combination of the expression of P(ζ), Q(ζ), S1(ζ), A(ζ), B(ζ), we have
W i ( ζ ) = j = 1 N ζ j 2 ( d 2 - d 1 ) Q j - P j + 2 ρ i 1 I S j - S 1 j A i j - S j B i j S j T - A i j T S j T P j + S j + S j T - S j B i j - B i j T S j T - B i j T S j T - Q j + 2 ρ i 2 I + j = 1 N - 1 l = j + 1 N ζ j ζ l ( d 2 - d 1 ) Q j l - P j l + 2 ρ i 1 I S j l - ( S A i ) j l - ( S B i ) j l S j l T - ( S A i ) j l T P j l + S j l + S j l T - ( S B i ) j l - ( S B i ) j l T - ( S B i ) j l T - Q j l + 2 ρ i 2 I = j = 1 N ζ j 2 W i j j + j = 1 N - 1 l = j + 1 N ζ j ζ l [ W i j l + W i l j ] ,
J i ( ζ ) = j = 1 N ζ j 2 ( Q j - S j A i j - A i j T S j T ) + j = 1 N - 1 l = j + 1 N ζ j ζ l ( Q j l - ( S A i ) j l - ( S A i ) j l T = j = 1 N ζ j 2 J i j j + j = 1 N - 1 l = j + 1 N ζ j ζ l [ J i j l + J i l j ] ,
where
( S A i ) j l : = S j A i l + S l A i j , ( S B i ) j l : = S j B i l + S l B i j , Q j l = Q j + Q l , P j l = P j + P l , S j l = S j + S l .
Then the conditions (i)-(iv) give
W i ( ζ ) < - j = 1 N ζ j 2 R + 2 N - 1 j = 1 N - 1 l = j + 1 N ζ j ζ l R 0 , J i ( ζ ) < - j = 1 N ζ j 2 R + 2 N - 1 j = 1 N - 1 l = j + 1 N ζ j ζ l R 0 ,
because of Proposition 2.2:
( N - 1 ) j = 1 N ζ j 2 - 2 j = 1 N - 1 l = j + 1 N ζ j ζ l = j = 1 N - 1 l = j + 1 N ( ζ j - ζ l ) 2 0 .
Therefore, we finally obtain from (3.9) and the condition (iii), (iv) that
E { Δ V ( k ) } < E { x T ( k ) J i ( ζ ) x ( k ) } , i = 1 , 2 , . , N , k = 0 , 1 , 2 , .
We now apply the condition (i), (ii) and Proposition 2.1, the system J i (ζ) is strictly complete, and the sets α ijl and α ¯ i j l by (3.1) are well defined such that
i = 1 N α i j l = R n \ { 0 } ,
i = 1 N α ¯ i j l = R n \ { 0 } , α ¯ i j l α ¯ t j l = , i t .
Therefore, for any x(k) R n , k = 0, 1, 2,...., there exists i {1, 2,..., N} such that x ( k ) α ¯ i j l . By choosing switching rule as γ(x(k)) = i whenever x ( k ) α ¯ i j l , from the condition (i) and (ii) we have
E { Δ V ( k ) } E { x T ( k ) J i ( ζ ) x ( k ) } < 0 , k = 0 , 1 , 2 , ,

which, combining the condition (3.2), and Definition 2.1, the system (2.1) is exponentially stable in the mean square. The proof is complete.

Remark 3.1. Note that the results purposed in [46, 9, 12] for switching systems to be asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems studied in [9] was limited to constant delays. In [10], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the averaged well time scheme.

Remark 3.2. It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero but in [18, 19] were limited to constant delays and the lower bound of delay was restricted to zero.

4 Numerical examples

Example 4.1. Consider the stochastic switched discrete-time system with convex polytopic uncertainties (2.1) for N = 2, where the delay function d(k) is given by
d ( k ) = 1 + 28 s i n 2 k π 2 , k = 0 , 1 , 2 , .
and
( A 11 , B 11 ) = - 1 0 . 1 0 . 2 - 2 , - 1 0 . 2 0 . 1 - 3 , ( A 12 , B 12 ) = - 2 0 . 3 0 . 5 - 3 , - 3 0 . 01 0 . 24 - 1 . 8 ,
( A 21 , B 21 ) = - 0 . 1 0 . 01 0 . 02 - 0 . 2 , - 0 . 1 0 . 02 0 . 01 - 0 . 3 , ( A 22 , B 22 ) = - 0 . 2 0 . 03 0 . 05 - 0 . 3 , - 0 . 3 0 . 01 0 . 024 - 0 . 18 .
By LMI toolbox of Matlab, we find that the conditions (i) - (iv) of Theorem 3.1 are satisfied with δ1 = 0.01, δ2 = 0:01, ζ 1 = 0:5, ζ 2 = 0.5, ρ11 = 0.1, ρ12 = 0.1, ρ21 = 0.1, ρ22 = 0.1, d1 = 1, d2 = 29, and
P 1 = 28 . 0877 0 . 0742 0 . 0742 22 . 3782 , P 2 = 23 . 3129 0 . 0057 0 . 0057 29 . 0647 , Q 1 = 0 . 3324 0 . 0128 0 . 0128 0 . 0868 , Q 2 = 0 . 0160 0 . 0097 0 . 0097 0 . 1173 ,
S 1 = - 5 . 0578 - 0 . 1508 - 0 . 6586 - 3 . 1947 , S 2 = - 3 . 9473 - 0 . 2175 - 0 . 6347 - 2 . 9459 , R = 0 . 3270 - 0 . 0153 - 0 . 0153 0 . 4768 .
In this case, we have
( J 111 ( R , Q ) , J 211 ( R , Q ) ) = - 9 . 7229 0 . 1973 0 . 1973 - 12 . 5603 , - 0 . 6731 0 . 0313 0 . 0313 - 1 . 1780 ,
( J 122 ( R , Q ) , J 222 ( R , Q ) ) = - 15 . 5556 0 . 7448 0 . 7448 - 17 . 1775 , - 1 . 5412 0 . 0833 0 . 0833 - 1 . 6122 ,
( J 112 ( R , Q ) , J 212 ( R , Q ) ) = - 19 . 7481 1 . 3578 1 . 3578 - 18 . 6862 , - 1 . 6756 0 . 1473 0 . 1473 - 1 . 7905 ,
( J 121 ( R , Q ) , J 221 ( R , Q ) ) = - 7 . 7916 - 0 . 0761 - 0 . 0761 - 11 . 5395 , - 0 . 7648 0 . 0012 0 . 0012 - 1 . 0484 .
Moreover, the sum
δ 1 J 111 ( R , Q ) + δ 1 J 211 ( R , Q ) + δ 1 J 122 ( R , Q ) + δ 1 J 222 ( R , Q ) + δ 1 J 112 ( R , Q ) + δ 1 J 121 ( R , Q ) + δ 1 J 212 ( R , Q ) + δ 1 J 221 ( R , Q ) + δ 2 J 111 ( R , Q ) + δ 2 J 211 ( R , Q ) + δ 2 J 122 ( R , Q ) + δ 2 J 222 ( R , Q ) + δ 2 J 112 ( R , Q ) + δ 2 J 121 ( R , Q ) + δ 2 J 212 ( R , Q ) + δ 2 J 221 ( R , Q ) = - 1 . 1495 0 . 0497 0 . 0497 - 1 . 3119 ,
is negative definite; i.e. the first entry in the first row and the first column -1:1495 < 0 is negative and the determinant of the matrix is positive. The sets α ijl , i, j, l = 1, 2, are given as
α 111 = { ( x 1 , x 2 ) : - 9 . 7229 x 1 2 + 0 . 3946 x 1 x 2 - 12 . 5603 x 2 2 < 0 } , α 211 = { ( x 1 , x 2 ) : 0 . 6731 x 1 2 - 0 . 0626 x 1 x 2 + 1 . 1780 x 2 2 > 0 } , α 122 = { ( x 1 , x 2 ) : - 15 . 5556 x 1 2 + 1 . 4896 x 1 x 2 - 17 . 1775 x 2 2 < 0 } , α 222 = { ( x 1 , x 2 ) : 1 . 5412 x 1 2 - 0 . 1666 x 1 x 2 + 1 . 6122 x 2 2 > 0 } , α 112 = { ( x 1 , x 2 ) : - 19 . 7481 x 1 2 + 2 . 7156 x 1 x 2 - 18 . 6862 x 2 2 < 0 } , α 212 = { ( x 1 , x 2 ) : 1 . 6756 x 1 2 - 0 . 2946 x 1 x 2 + 1 . 7905 x 2 2 > 0 } , α 121 = { ( x 1 , x 2 ) : - 7 . 7916 x 1 2 - 0 . 1522 x 1 x 2 - 11 . 5395 x 2 2 < 0 } , α 221 = { ( x 1 , x 2 ) : 0 . 7648 x 1 2 - 0 . 0024 x 1 x 2 + 1 . 0484 x 2 2 > 0 } .
Obviously, the union of these sets is equal to R2 \ {0}. The switching regions are defined as
α ¯ 111 = { ( x 1 , x 2 ) : - 9 . 7229 x 1 2 + 0 . 3946 x 1 x 2 - 12 . 5603 x 2 2 < 0 } , α ¯ 211 = α 211 \ α ¯ 111 , α ¯ 122 = { ( x 1 , x 2 ) : - 15 . 5556 x 1 2 + 1 . 4896 x 1 x 2 - 17 . 1775 x 2 2 < 0 } , α ¯ 222 = α 222 \ α ¯ 122 , α ¯ 112 = { ( x 1 , x 2 ) : - 19 . 7481 x 1 2 + 2 . 7156 x 1 x 2 - 18 . 6862 x 2 2 < 0 } , α ¯ 212 = α 212 \ α ¯ 112 , α ¯ 121 = { ( x 1 , x 2 ) : - 7 . 7916 x 1 2 - 0 . 1522 x 1 x 2 - 11 . 5395 x 2 2 < 0 } , α ¯ 221 = α 221 \ α ¯ 121 .

By Theorem 3.1 the system is robustly stable in the mean square and the switching rule is chosen as γ(x(k)) = i whenever x ( k ) α ¯ i j l .

5 Conclusion

This article has proposed a switching design for the mean square robust stability of stochastic switched linear discrete-time systems with convex polytopic uncertainties with interval time-varying delays. Based on the discrete Lyapunov functional, a switching rule for the mean square robust stability for the stochastic system with convex polytopic uncertainties is designed via linear matrix inequalities.

Declarations

Acknowledgements

This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the article.

Authors’ Affiliations

(1)
Major of Mathematics and Statistics, Faculty of Science, Maejo University, Chiangmai, Thailand

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© Rajchakit and Rajchakit; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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