New Results on Stability of Linear Discrete-Time Systems With Time-Varying Delay

This paper is concerned with the stability problem for linear discrete-time systems with a time-varying delay. Some relations between two general free-matrix-based summation inequalities are discussed. A novel Lyapunov–Krasovskii functional (LKF) is proposed by modifying the single- and double-summation LKF terms. As a result, a new stability condition is obtained by employing the general free-matrix-based summation inequalities reported recently. Numerical examples are given to show that the obtained stability condition is more relaxed than some of existing results.


I. INTRODUCTION
It is well known that time delay, as a natural phenomenon, widely exists in various practical systems such as networked control systems, fuzzy systems and neural networks [1]- [3]. Its existence usually brings oscillation, divergence, and even instability [1]. On the other hand, a proper introduction of small delay can make an unstable system become stable or improve dynamic performances [1]. Therefore, during past decades, linear discrete-time systems with a timevarying delay have attracted extensive attention from academic research [4]- [12] and a number of remarkable results have been reported in the literature [13]- [24].
Nowadays, the LKF method is a powerful tool to deal with the stability problem for linear delayed systems. To achieve a relaxed condition, choosing an appropriate LKF candidate is a key point, which is required to take more information of various state vectors into account. Another important point is to tightly estimate the forward difference of this candidate by employing advanced techniques such as free-weighting-matrix technique [25], summation inequalities [26]- [33], reciprocally convex combination lemmas (RCCLs) [34]- [36] and free-matrix-based (FMB) summation inequalities [37]- [39].
The associate editor coordinating the review of this manuscript and approving it for publication was Xiaojie Su. In recent years, instead of free-weighting-matrix technique, summation inequalities are commonly used to deal with summation terms arising in the forward differences of LKFs. For examples, the Wirtinger-based summation inequality is proposed in [26], [27] and [29], which produces a tighter bound than Jensen summation inequality. Later, another kind of summation inequality, called the freematrix-based summation inequality, is developed by introducing several free matrices [38]. As stated in [38], it covers the Wirtinger-based summation inequality as a special case. In the case that the FMB summation inequality is applied, RCCL is no longer required to further estimate summation terms. Very recently, a general free-matrix-based (GFMB) summation inequality is developed [40], which leads to a relaxed stability condition for linear discrete-time systems with a time-varying delay. However, the interest of the GFMB inequality is not fully utilized [40] when estimating summation terms. It is noted that to analyse the stability of discretetime neural networks with a time-varying delay [37], another GFMB inequality is proposed by providing more freedom to choose the undetermined augmented vectors. The relations between the two GFMB inequalities deserve to be discussed.
As stated before, constructing an appropriate LKF candidate plays a key role in achieving a relaxed stability condition.
To mention a few, to coordinate with the Wirtinger-based summation inequality, a new augmented LKF is proposed in [26], in which the quadratic augmented vector contains three vectors x(k), the state, h 1 and h 2 are, respectively, the lower and upper bounds of the discrete-time delay. This proposed LKF is later widely used in the next few years in the literature. Additionally, in order to cooperate with the use of double-summation inequalities, triple-summation LKF terms are included in the LKF proposed in [28]. It is worth pointing out that on the basis of the LKF proposed in [26], a double-summation LKF term [40] that is really helpful to reduce the conservatism of the obtained stability condition by employing a GFMB summation inequality. However, there is still some room to develop a new and appropriate LKF candidate. This motives this research.
In this paper, we further study the stability problem for linear discrete-time systems with a time-varying delay via the LKF method. First, some relations between the two GFMB summation inequalities recently proposed in [37] and [40] are discussed. It is pointed out the GFMB summation inequality proposed in [37] covers that proposed in [40] as a special case. Meanwhile, for convenience of applications, summation inequalities [37] are rewritten in a more compact form. Second, a novel LKF is constructed by modifying the single-and double-summation LKF terms so that more relations among different state vectors are taken into account. Finally, two numerical examples are given to show that the obtained stability condition produces more relaxed results than existing ones, especially one recently reported in [40].
Notations. Throughout this paper, Sym{X } denotes X + X T for any square real matrix X . R n denotes the n-dimensional Euclidean space and R n×m the set of all n × m real matrices. I and 0 denote the identity and zero matrices of appropriate dimension, respectively. S n + represents the set of symmetric positive-definite matrices of R n×n .

II. PRELIMINARY AND USEFUL LEMMAS
Consider the following linear discrete-time system with a time-varying delay: where x(k) ∈ R n is the state vector; φ(k) is the initial condition; A, A d ∈ R n×n are constant matrices; h(k), abbreviated as h k , is a time-varying delay, satisfying where h 1 and h 2 are known integers satisfying h 1 < h 2 . This article aims to develop a stability condition that is of less conservatism, compared to some of existing results recently reported. Based on the new stability condition, larger maximum allowable upper bounds (MAUBs) should be obtained for different h 1 so that the considered system (1) is ensured to be stable with the delay h k varying within the interval [h 1 , h 2 ] as large as possible. To do so, a novel LKF will be developed and the recently-reported GFMB summation inequalities will be employed.
In the rest of this section, we will present some lemmas that are useful to obtain the main results. Before proceeding, two functions are first defined: s 1 (h) = h + 1 and Lemma 1 [37]: For matrices R ∈ S n + , N 0 , N 1 , N 2 with appropriate dimensions, and a vector function holds, where ω 0 , ω 1 and ω 2 are any vectors, and x a (i), (3) can be rewritten in a more compact form. So do summation inequalities developed in Lemma 2 proposed in [37].
Lemma 2: For matrices R ∈ S n + , N ∈ R 3n×pn and vector hold, where ω ∈ R pn is any vector, χ 0 , χ 1 and χ 2 are defined in Lemma 1, and Inequalities (4)-(7) are all induced from Inequality (3). Recently, a novel GFMB summation inequality is proposed in [40]. For convenience of comparison, it is recalled in the following lemma.
Lemma 3 [40]: For a given n × n-matrix R > 0, and three given nonnegative integers a, b, k, satisfying a < b ≤ k, a vector function x(·) ∈ R n , taking n × pn-matrices i (i = 1, 2) and a vector ζ ∈ R pn such that then, for any n × pn-matrices N i (i = 1, 2), the following inequality holds:
Remark 3: Compared to Theorem 2 proposed in [40], there exist three points in Theorem 1 that may lead to the reduction of conservatism. One is to include the LKF term in the LKF candidate. The second is to utilize Inequality (5) to estimate summation terms in (22), instead of Inequality (8), which could take more relations among the double-summation terms into account. The third is to construct the augmented vector η 1 (k) = col{x(k), x(i)}, instead of the state x(i), which could make the coupling information between the state x(k) and its single-summation terms considered.

Example 1: Consider system (1) with
To compare the conservatism of stability conditions, MAUBs for different h 1 are obtained by Theorem 1 proposed in this paper and some other conditions reported recently in the literature and listed in Table 1. From Table 1, it is seen that Theorem 1 produces more relaxed results than conditions proposed in [17], [26]- [28], [30], [34] and [38] and the same results as those proposed in [40]. This means that Theorem 1 is not more conservative than other conditions, including one proposed in [40]. For Example 2, MAUBs are obtained by Theorem 1 proposed in this paper and other conditions proposed in [26]- [28], [30], [34], [38] and [40]. It is seen from  [40]. This clearly shows that Theorem 1 is more relaxed than all of other conditions, although the number of decision variables is larger than those involved in other conditions. Generally speaking, less conservatism is usually achieved at the cost of more computational complexity. The proposed condition is no exception.

V. CONCLUSION
This article has investigated the stability problem for linear discrete-time systems with a time-varying delay. Some relations between two general free-matrix-based summation inequalities have been discussed and meanwhile, their compact versions have been presented. By modifying the single-and double-summation LKF terms, a novel Lyapunov-Krasovskii functional has been constructed. By employing newly-developed summation inequalities, a new stability condition has been obtained, which is of less conservatism compared to some existing results. The proposed method can be used to address the stability problem for other discrete-time delayed systems such as discrete-time Takagi-Sugeno fuzzy systems with timevarying delay [41], [42]. Especially, the idea of introducing free matrices to improve freedoms may be extended to other control fields like delayed conic-type nonlinear systems [14]- [16].