Multiple integral inequalities and stability analysis of time delay systems

Highlights

• New multiple integral inequalities are derived.

• A set of sufficient LMI stability conditions for time delay systems are derived.

• The LMI conditions are arranged into a bidirectional hierarchy.

Abstract

This paper is devoted to stability analysis of continuous-time delay systems based on a set of Lyapunov–Krasovskii functionals. New multiple integral inequalities are derived that involve the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel–Legendre inequalities of Seuret and Gouaisbaut (2015) and the Wirtinger-based multiple-integral inequalities of Park et al. (2015) and Lee et al. (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.

Introduction

Time delays are present in many physical, industrial and engineering systems. The delays may cause instability or poor performance of systems, therefore much attention has been devoted to obtain tractable stability criteria of systems with time delay during the past few decades (see e.g. the monographs  [1], [2], [3], some recent papers  [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references therein). Several approaches have been elaborated and successfully applied for the stability analysis of time delay systems (see the references above for excellent overviews).

Lyapunov method is one of the most fruitful fields in the stability analysis of time delay systems. On the one hand, more and more involved Lyapunov–Krasovskii functionals (LKF) have been introduced during the past decades. On the other hand, much effort has been devoted to derive more and more tight inequalities (Jensen’s inequality and different forms of Wirtinger’s inequality  [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [18], [19], etc.) for the estimation of quadratic single, double and multiple integral terms in the derivative of the LKF. Simultaneously, augmented state vectors are introduced in part as a consequence of the improved estimations, in part on an ad hoc basis. The effectiveness of different methods is mainly compared using some numerical examples. Recently, the authors of [4], [13] have introduced a very appealing idea of the hierarchy of LMI conditions offering a rigorous theoretical basis for comparison of stability LMI conditions. Based on Legendre polynomials, they proposed a generic set of single integral inequalities opening the way to the derivation of a set of stability conditions forming a hierarchy of LMIs. A further possibility for the derivation of improved stability conditions have been proposed by  [5], [6] using multiple integral quadratic terms in the LKF, together with Wirtinger-based multiple integral inequalities. Naturally the question arises: how these two lines of investigations are related to each other, and how sufficient stability conditions can be derived unifying the approaches of using multiple integral quadratic terms in the LKF and refined estimations of these integral terms.

The aim of the present work is to answer these questions. On the one hand, multiple integral inequalities based on orthogonal hypergeometric polynomials will be derived that extend the results of   [4], [13]to multiple integrals and improve the estimations of   [5], [6]. On the other hand, a multi-parametric set of LMI conditions will be constructed, and it will be shown that a two parametric subset forms a bidirectional hierarchy of LMIs.

Analogous results have been presented for discrete-time systems in  [20].

The paper is organized as follows. In Section  2 it is shown, how the quadratic terms of the derivative of the LKF can be estimated by Bessel-type inequalities. It is also proven that these estimations relevantly improve a recently published result. A sufficient condition of asymptotic stability is presented in the form of an LMI in Section  3. The hierarchy of LMI conditions is established then in Section  4. Some benchmark numerical examples are shown in Section  5, the results of which are compared to earlier ones known from the literature. Finally, the conclusions will be drawn.

The notations applied in the paper are very standard, therefore we mention only a few of them. Symbol $A\otimes B$ denotes the Kronecker-product of matrices $A,B$, while $S_n$ and $S_n^{+}$ are the set of symmetric and positive definite symmetric matrices of size $n\times n$, respectively.