This note studies an interesting phenomenon for stability conditions of discrete-time systems with time-varying delay. The underlying reason behind this phenomenon is revealed, and thereafter some conclusions are drawn: (i) Stability conditions of discrete-time systems with time-varying delay are generally divided into two types: those obtained by summation inequalities with free-matrix variables and those obtained by the combination of summation inequalities without free-matrix variables and the reciprocally convex lemma; (ii) The conservatism between the two types of stability conditions can not be theoretically compared. To clearly demonstrate this interesting phenomenon and meanwhile, to further verify these conclusions, several bounded real lemmas are obtained via different bounding-inequality methods and applied to a numerical example.

Introduction

Since time delay is often encountered in the real world such as network and mechanical engineering [1], [2], the stability analysis for discrete-time systems with time-varying delay has attracted considerable attention. Until now, many remarkable results have been reported in the literature [1]–[۳۹] such as the free-matrix-weighting technique [3], the delaypartitioning method [4], the bounding-equality method [1], [5]–[۷]. Compared with other techniques, the boundinginequality method, owing to its effectiveness and straightforwardness, has been widely used in the Lyapunov–Krasovskii (L–K) functional method. As far as the authors know, three bounding-inequality methods have been established to estimate the summation term δ(k). Owing to the conservatism of Jensen inequality [1], the Wirtinger-based summation inequality (WBSI) was proposed [8]–[۱۰]. Combined with the reciprocally convex lemma (RCL) [7], a more relaxed stability condition was obtained in [9]. This method is called WBSI + RCL method. Thereafter, a free-matrix-based summation inequality (FMBSI) was proposed by introducing some free matrix variables [11]. Based on FMBSI, a new stability condition was obtained in [11]. This method is called FMBSI method. Recently, an improved summation inequality was proposed in [12] by considering both δ۱(k) and δ۲(k) together. In fact, this inequality can be directly obtained by combining WBSI and the improved RCL (see (5) below).