Stability analysis of discrete-time systems with time-varying delays: generalized zero equalities approach

Abstract

This paper suggests a generalized zero equality lemma for summations, which leads to making a new Lyapunov-Krasovskii functional with more state terms in the summands and thus applying various zero equalities for deriving stability criteria of discrete-time systems with interval time-varying delays. Also, using a discrete-time counter part of Wirtinger-based integral inequality, Jensen inequality, and a lower bound lemma for reciprocal convexity, the forward difference of the Lyapunov-Krasovskii functional is bounded by the combinations of various state terms including not only summation terms but also their interval-normalized versions, which contributes to making the criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds.

Polynomials‐based summation inequalities and their applications to discrete‐time systems with time‐varying delays

“…Considering the facts (18), (20), the choice of p r .i/ as Q p r .i/ .r D 0; 1/ and .m D 1/ in Lemma 7 provides the following summation inequality.…”

“…Because mathematical techniques deriving lower bounds of summation quadratic functions are crucial to derive less conservative stability criteria, inequality with slack matrices [9,10] and Jensen inequality [7,[11][12][13][14][15] have been utilized. Further, with the summation inequalities, a lower bound lemma for reciprocal convexity [14,16], zero equality approaches [14,17,18], and a delay-partitioning method [19,20] have been used to reduce the conservatism of stability criteria for discrete-time systems with time-varying delays. Among the methods, the Jensen inequality has been frequently used because of its compactness.…”

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

“…Time delay is a common phenomenon in many dynamic systems such as chemical systems, biological systems, mechanical engineering systems, and networked control systems . Since such phenomenon often causes control performance degradation or even system instability, there have been considerable efforts to solve stability analysis problems for time‐delay systems before implementing control strategies . Stability analysis for delayed discrete‐time systems can be classified into two types of problems depending on delay properties: constant time delays and time‐varying delays.…”

“…In the case of constant time delays, it has been noted that necessary and sufficient stability criteria can be obtained using a system augmentation method at a price of large computational burden. However, in the case of time‐varying delays, the stability analysis of delayed discrete‐time systems is still an open problem, and thus, various numerical methods such as reciprocally convex combination lemma, summation inequalities, and free‐weighting matrices approach have been proposed to obtain less conservative stability criteria . In the presence of time‐varying delays in systems, the Lyapunov‐Krasovskii (L‐K) approach is one of major methods to derive numerically tractable optimization problems, which are usually formulated in terms of linear matrix inequalities (LMIs), for stability analysis of time‐delay systems.…”

“…In the literature using the L‐K approach, the Jensen integral/summation inequalities have been frequently utilized in stability analysis for delayed continuous/discrete‐time systems to derive numerically tractable lower bounds of single integral/summation quadratic functions. In the field of stability analysis for delayed continuous‐time systems, it has been noted that a more precise bound of an integral quadratic function leads to a less conservative stability condition based on appropriate L‐K functionals .…”