IMPROVED DELAY-RANGE-DEPENDENT STABILITY CRITERIA FOR DISCRETE-TIME LINEAR SYSTEMS WITH INTERVAL TIME-VARYING DELAY AND NONLINEAR PERTURBATIONS

Abstract: In this paper, we study the problem of stability analysis for discrete-time linear system with interval time-varying delay and nonlinear perturbations. By constructing a new Lyapunov-Krasovskii functional with triple summation terms, mixed model transformation, Jensen-type summation inequality and utilization of zero equation, new delay-range-dependent asymptotic stability criteria are obtained and formulated in terms of linear matrix inequalities (LMIs). Moreover, we obtain new delay-range-dependent asymptotic stability criteria of discrete-time linear system with interval time-varying delay. Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.


Introduction
Time-delay systems exist in many fields such as electric systems, chemical processes systems, networked control systems, telecommunication systems and economical systems.During the past two decades, the problems of robust stability and stabilization analysis for dynamic systems with time delays have been widely investigated by many researchers [1]- [26].It is well known that nonlinearities, as time delays, may cause instability and poor performance of practical systems.Commonly, stability criteria for dynamic systems with time delay are generally divided into two classes: a delay-independent one and a delay-dependent one.The delay-independent stability criteria tend to be more conservative, especially for a small size delay, such criteria do not give any information on the size of delay.On the other hand, delay-dependent stability criteria are concerned with the size of delay and usually provide a maximal delay size.Recently, a special type of time delay in practical engineering systems, that is interval time-varying delay, is investigated.The characteristic of interval time-varying delay is that time delay can vary in an interval in which the lower bound of delay is not restricted to zero.The typical examples of systems with interval time-varying delay are networked control systems, chemical process and flight systems [10].
Discrete-time systems with state delay have strong background in engineering applications, among which network based control has been well recognized to be a typical example.If the delay is constant in discrete systems, one can transform a delayed system in to a delay-free one by using state augmentation techniques.However, when the delay is large, the augmented system will become much complex and thus difficult to analyze and synthesize [5].Hence, researchers have focussed on the delay-range-dependent stability and stabilization problems of discrete-time systems with interval time-varying delay and many existing results mainly focus on discrete-time linear delay systems [1]- [12], [14]- [15], [17]- [23], [25]- [26].However, most real systems hold nonlinear dynamics.Therefore, researchers have been investigated the delay-range-dependent stability criteria for discrete-time linear systems with interval time-varying delay and nonlinear perturbations [13], [16], [24].However, the existing results of delay-range-dependent stability criteria do not take into account the presence of nonlinear perturbations uncertainties in the discrete-time delay systems by model transformation.
This paper will focus on the delay-range-dependent stability analysis for discrete-time linear system with interval time-varying delay and nonlinear perturbations.By using the combination of mixed model transformation, Jensen-type summation inequality, utilization of zero equation and new Lyapunov-Krasovskii functional, new delay-range-dependent asymptotic stability criteria are obtained and formulated in terms of LMIs.Then, we can obtain new delayrange-dependent asymptotic stability criteria of discrete-time linear systems with interval time-varying delay.Finally, numerical examples will be given to show the effectiveness of the obtained results.
We introduce some notations, definition and lemmas that will be used throughout the paper.Z + denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the vector norm • ; x denotes the Euclidean vector norm of x ∈ R n ; R n×r denotes the set of n × r real constant matrices; A T denotes the transpose of the matrix A; A is symmetric if A = A T ; I denotes the identity matrix

Problem Formulation and Preliminaries
Consider the discrete-time linear system with interval time-varying delay and nonlinear perturbations of the form where k ∈ Z + , x(k) ∈ R n is the state variable and φ(s) is a initial value at s.A and B ∈ R n×n are known real constant matrices.f (k, x(k)) and g(k, x(k − h(k))) are the nonlinear perturbations with respect to current state x(k) and discrete delay state x(k − h(k)), respectively, and are bounded in magnitude: where α and β are given positive real constants.In addition, we assume that the time-varying delay h(k) is upper and lower bounded.It satisfies the following assumption of the form 0 where h 1 and h 2 are known positive real constants.Rewrite the system (1) in the following system: By utilizing the following zero equation, we have where J ∈ R n×n will be chosen to guarantee the asymptotic stability of the system (1)- (2).By (7), system ( 5)-( 6) can be represented by the form Definition 2.1.
[18] The system (1)-( 2) is said to be asymptotically stable if there exists a positive definite function along any trajectory of solution for system (1)-( 2).

Lemma 2.2. [6] [Schur complement lemma] Given constant symmetric matrices X, Y and Z of appropriate dimensions with
For any constant matrix W ∈ R n×n , W = W T > 0, two integers r M and r m satisfying r M ≥ r m and vector function x : [r m , r M ] → R n , the following inequality holds: where δ = r M − r m + 1.
[19] Let M ∈ R n×n be a positive-definite matrix, X i ∈ R n , i = 1, 2, ....If the sums concerned are well defined, then We introduce the following notations for later use. where

Delay-Range-Dependent Stability Criteria
Theorem 3.1.The system (1)-( 2) is asymptotically stable, if there exist positive definite symmetric matrices P, Q, R, S, T, U, V, W, X, Y, Z, any appropriate dimensional matrices L w , C j , D j , E j , G j , M, N, K, L, J, w = 1, 2, 3, j = 1, 2, . . ., 13 and positive real constants ǫ 1 and ǫ 2 such that the following symmetric linear matrix inequalities hold: Proof.Consider the following Lyapunov-Krasovskii function for system (8)-( 9) of the form where Evaluating the forward deference of V (k), it is defined as Let us define for i = 1, 2, ..., 6, Taking the forward deference of y(i) ∆V By Lemma 2.3, the increments of V 3 (k) and V 4 (k) are easily computed as Taking the forward deference of V 5 (k) yields By ( 12) and ( 13), it is easy to see that From ( 22) and ( 23), we can obtain and we have Therefore, we conclude that We take the forward difference of V 6 (k) as Furthermore, it follows from Lemma 2.4 that y T (j)Hy(j) Therefore, we obtain It is obvious that The following equations are true for any matrices with appropriate dimensions: that our results in Theorem 3.1 are much less conservative than in [16] and [24].

Conclusion
In this paper, the problem of asymptotic stability analysis for discrete-time linear system with interval time-varying delay and nonlinear perturbations has been presented.The method combining augmented Lyapunov-Krasovskii functional, mixed model transformation, Jensen-type summation inequality and utilization of zero equation have been studied.New delay-range-dependent asymptotic stability criteria have been obtained and formulated in terms of LMIs.By comparing the proposed results with the results available in the existing literature, it is shown that the derived criteria are less conservative.
LMI Toolbox in MATLAB (with accuracy 0.01) for Corollary 3.2 to system (39)-(40) with (47), the maximum upper bounds h 2 for asymptotic stability of Example 4.2 is listed in the comparison in Table

Table 1 :
Upper bounds of time delay h 2 for different conditions for Example 4.1.

Table 2 :
Upper bounds of time delay h 2 for different conditions for Example 4.2.