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

Abstract

Summary This paper proposes a novel summation inequality, say a polynomials‐based summation inequality, which contains well‐known summation inequalities as special cases. By specially choosing slack matrices, polynomial functions, and an arbitrary vector, it reduces to Moon's inequality, a discrete‐time counterpart of Wirtinger‐based integral inequality, auxiliary function‐based summation inequalities employing the same‐order orthogonal polynomial functions. Thus, the proposed summation inequality is more general than other summation inequalities. Additionally, this paper derives the polynomials‐based summation inequality employing first‐order and second‐order orthogonal polynomial functions, which contributes to obtaining improved stability criteria for discrete‐time systems with time‐varying delays. Copyright © 2017 John Wiley & Sons, Ltd.

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.…”

“…The first one is to construct L‐K functionals and the second one is to derive sufficient conditions guaranteeing that the forward difference of an L‐K functional is negative. Since there has been growing tendency to construct L‐K functionals using more summation quadratic functions, bounding techniques such as summation inequalities have played essential roles to derive numerically tractable upper bounds of forward difference of L‐K functionals. Among the bounding techniques, this paper focuses on single summation inequalities extended from the Jensen summation inequality …”

“…Similarly, for stability analysis of delayed discrete‐time systems, there have been various efforts to reduce a bounding gap between the Jensen summation inequality and a summation quadratic function by using zero‐, first‐, second‐, and third‐order discrete orthogonal polynomials obtained from Gram‐Schmidt orthogonalization process . These efforts have resulted in the discrete‐time Wirtinger‐based inequality, auxiliary‐function‐based summation inequalities, and free‐weighting matrix‐based summation inequalities . Differently from the case of integral inequalities, however, the existing summation inequalities are hardly extensible to high‐order or arbitrary‐order cases since there have still not been general discrete orthogonal polynomials applicable to the deriving summation inequalities.…”

“…Differently from the case of integral inequalities, however, the existing summation inequalities are hardly extensible to high‐order or arbitrary‐order cases since there have still not been general discrete orthogonal polynomials applicable to the deriving summation inequalities. As described in a conversion lemma, discrete orthogonal polynomials given in terms of a binomial coefficient form or a rising factorial form can be directly used to derive summation inequalities. However, the existing discrete Legendre polynomial function is given in terms of falling factorials, and Gram‐Schmidt orthogonalization, which is an iterative process, does not provide a unique general form.…”