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

Abstract

This paper is concerned with the stability analysis problems of discrete-time systems with time-varying delays using summation inequalities. In the literature focusing on the Lyapunov-Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel-Legendre inequalities constructed with arbitrary-order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete-time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities.Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete-time counterparts of the Bessel-Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete-time systems with time-varying delays. KEYWORDS discrete orthogonal polynomials, discrete-time system with delays, LMI, Lyapunov-Krasovskii stability theorem, stability analysis, summation inequality Int J Robust Nonlinear Control. 2019;29:473-491.wileyonlinelibrary.com/journal/rnc

An improved exponential stability analysis method for discrete‐time systems with a time‐varying delay

“…Stability analysis problems of discrete‐time systems with a time‐varying delay are basic problems in control theory 1 . Thus, much attention has been paid to how to deal with the problems 2‐6 . Existing research indicates that the Lyapunov–Krasovskii functional method is an effective tool to obtain stability criteria of delay systems 7 .…”

Enhanced Stability Criteria for Discrete-time Systems with Time-varying Delay

“…In terms of focusing on estimating the upper bound of the difference of the LKF, a lot of tight inequality techniques can be found in [25, 29-31, 33, 35-39], such as, estimating the upper bound on the difference of the LKF via the free-weighting matrix approach without ignoring some important terms [25], new discrete inequalities for single summation and double summation [30,40], discrete Wirtinger-based inequality approach [35,36], etc. It is worth emphasizing that Nthorder discrete Legendre polynomials-based inequality and Bessel summation inequality are proposed in [38,41], respectively, where N-dependent stability criteria are given. The results of [38,41] show that the conservativeness de-creases with the increase of N. Two general free-matrixbased summation inequalities are proposed in [36,42] based on second-order Bessel summation inequality, respectively.…”

“…It is worth emphasizing that Nthorder discrete Legendre polynomials-based inequality and Bessel summation inequality are proposed in [38,41], respectively, where N-dependent stability criteria are given. The results of [38,41] show that the conservativeness de-creases with the increase of N. Two general free-matrixbased summation inequalities are proposed in [36,42] based on second-order Bessel summation inequality, respectively. However, the upper bounds of the summations with respect to the time delay intervals [h 1 , h(k)] and [h(k), h 2 ] are always estimated separately based on the Legendre polynomials-based inequality, Bessel summation inequality above or the general free-matrix-based summation inequality.…”

Gain‐scheduled control for discrete‐time non‐linear parameter‐varying systems with time‐varying delays

“…For instance, in [31], an auxiliary function based inequality has been introduced, in [38], a free-matrix-based summation inequality has been employed, and, in [39], an improvement in the method of [35] has been presented. Several strategies have been reported for stability analysis and stabilisation problems for linear time-delay systems [32,[40][41][42][43][44][45][46][47][48], uncertain linear time-delay systems [49][50][51][52], N-TS, and TS fuzzy model with time-delays [10,[53][54][55][56][57][58][59][60][61].…”