eu STABILITY ANALYSIS OF SWITCHED DISCRETE TIME-DELAY SYSTEMS WITH CONVEX POLYTOPIC UNCERTAINTIES

This article is concerned with robust stability of 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 robust stability for switched system with convex polytopic uncertainties is designed via linear matrix inequalities. AMS Subject Classification: 47N10, 93C55, 93D20, 94C10


Introduction
Many dynamical systems in real world involve variables that be always confined to the positive orthant, and such systems are generally termed as positive systems in the literature.In particular, positive linear systems have drawn considerable interest due to their numerous applications in the areas such as economics, biology, communications, etc.In the context of positive linear systems, stability analysis is a major concern and the so-called linear copositive Lyapunov function has been proved to be an efficient approach.Also, based on the linear copositive Lyapunov function approach, the analysis and synthesis problems of positive linear systems with time delay have been successfully tackled.
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.By using the Floquet theory, the author gives new sufficient conditions for stabilize in terms of a controllability rank condition for a linear time-invariant discrete system as well as of matrix inequalities in [1][2][3][4][5][6][7][8].During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [4,9,10,14].The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach for constructing a common Lyapunov function [11][12][13][14][15][16][17][18].Although many important results have been obtained for switched linear continuous-time systems, there are a few results concerning the stability of switched linear discrete systems with time-varying delays.It was shown in [11][12][13][14][15][16][17] 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 [19], but the result was limited to constant delays.In [20], 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 paper studies the robust stability problem for switched linear discrete systems with convex polytopic uncertainties with interval time-varying delays.Specifically, our goal is to develop a constructive way to design a switching rule to be robustly stable the system.By using improved Lyapunov-Krasovskii functional combined with the LMIs technique, we propose new criteria for the 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 robust stability in terms of LMIs, which can be solvable by utilizing Matlab's LMI Control Toolbox available in the literature to date.
The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results.The robust stability of switched discrete time-delay systems with convex polytopic uncertainties is presented in Section 3.

Preliminaries
The following notations will be used throughout this paper.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; R n×r denotes the space of all matrices of (n × r)− dimension.A T denotes the transpose of A; Consider a discrete systems with convex polytopic uncertainties with interval time-varying delay of the form 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 (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 polytopes Ω given by where A ij , B ij , i, j = 1, 2, ..., N, are given constant matrices with appropriate dimensions.The time-varying function d(k) satisfies the following condition: 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 1.The switched linear discrete systems with convex polytopic uncertainties (1) is robustly stable if there exist a positive definite scalar function V (k, x(k) : R + × R n → R and a switching function γ(.) such that along any trajectory of solution of the system (1) for all uncertainties which satisfy (2).Definition 2. The switched system (1) is robustly stable if there exists a switching function γ(.) such that the zero solution of the system is robustly stable.
It is easy to see that the system {J i } is strictly complete if and only if where Proposition 1. (see [21]) The system {J i }, i = 1, 2, . . ., N, is strictly complete if there exist If N = 2 then the above condition is also necessary for the strict completeness.

Main Results
Let us set Theorem 1.The switched system with convex polytopic uncertainties (1) is robustly stable if there exist symmetric matrices The switching rule is chosen as γ(x(k)) = i.
Proof.Consider the following Lyapunov-Krasovskii functional for any ith system (1) where We can verify that Then, the difference of V 1 (k) along the solution of the system is given by Therefore, from (6) it follows that where The difference of V 2 (k) is given by The difference of V 3 (k) is given by We obtain from ( 9) and ( 10) that Therefore, combining the inequalities ( 7), (11) gives where Therefore, we finally obtain from (12) and the condition (i), (ii) that which, combining the condition (5) and the Lyapunov stability theorem [21], concludes the proof of the theorem.

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