Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

TL;DR

This paper introduces new multiple summation inequalities for discrete-time delay systems, providing a hierarchy of LMIs that improve stability analysis accuracy and reduce conservatism.

Contribution

It develops novel multiple summation inequalities involving Jensen's and Wirtinger's inequalities, forming a hierarchy of LMIs for better stability analysis of delay systems.

Findings

New inequalities improve stability bounds

Hierarchy of LMIs allows method comparison

Numerical examples confirm efficiency

Abstract

This paper is devoted to stability analysis of discrete-time delay systems based on a set of Lyapunov-Krasovskii functionals. New multiple summation inequalities are derived that involve the famous discrete Jensen's and Wirtinger's inequalities, as well as the recently presented inequalities for single and double summation in P.T. Nam, H. Trinh, P.N. Pathirana, Discrete inequalities based on a multiple auxiliary functions and their applications to stability analysis of time-delay systems, Journal of the Franklin Institute, (2015). The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method.