Elsevier

Applied Mathematics and Computation

Volume 311, 15 October 2017, Pages 164-177
Applied Mathematics and Computation

Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities

https://doi.org/10.1016/j.amc.2017.05.004Get rights and content

Abstract

This paper is devoted to stability analysis of continuous-time delay systems with interval time-varying delays having known bounds on the delay derivatives. A parameterized family of Lyapunov–Krasovskii functionals involving multiple integral terms is introduced, and novel multiple integral inequalities are utilized to derive sufficient stability condition for systems with time-varying delays. The efficiency of the proposed method is illustrated by numerical examples.

Introduction

It is well-known that time-delays are frequently present in various physical, industrial and engineering systems. The delays may cause poor performance or even instability of systems, therefore much attention has been devoted to obtain tractable stability criteria for systems with time delay during the past few decades (see e.g., the monographs [1], [2], some recent papers [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and the references therein).

Let us consider the following system with time-varying delay: x˙(t)=Ax(t)+Adx(tτ(t)),x(t)=φ(t),t[τ¯,0],where x(t)Rnx is the state, and A,AdRnx×nx are given matrices, the time-varying delay τ is supposed to be a differentiable function satisfying conditions 0τ̲τ(t)τ¯,μ̲τ˙(t)μ¯,τ̲<τ¯with known bounds τ̲,τ¯,μ̲,μ¯, and function φ gives the initial condition.

Several approaches have been elaborated and successfully applied for the stability analysis of time delay systems (see the references above for excellent overviews). A major issue in this respect is to develop delay-dependent stability conditions, which provide as wide delay bounding interval guaranteeing stability as possible. Many important results on asymptotic stability of time-delay systems have been established using the Lyapunov–Krasovskii functional (LKF) approach.

In order to enhance the effectiveness, more and more involved Lyapunov–Krasovskii functionals have been introduced during the past decades involving single/multiple integral terms with respect to quadratic functions. 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], [22], [28], etc.) for the estimation of single, double and multiple quadratic integral terms in the derivative of the LKF. Lately, the so-called Bessel–Legendre inequality has been developed in [12] to deal with single integral terms of quadratic functions, and a wide range of multiple integral inequalities has been derived in [9], [10], [11]. Very recently, general integral inequalities that encompass all the above-mentioned inequalities are parallel presented in [16], [17] based on orthogonal polynomials in different Euclidean spaces with integral inner products.

Utilizing these results, sufficient stability conditions for linear time delay systems partly with constant, partly with time varying delays are derived in the form of linear matrix inequalities (LMIs). We note that a great part of recent works investigates time-delay systems with bounded delays (i.e., when τ̲=0) and with different conditions on the delay derivative.

The aim of the present work is to investigate the stability problem of linear systems with interval time-varying delays (i.e., when τ >0) having known bounds of the delay derivative via the application of the inequalities proposed in [17].

It has to be emphasized that the time-variance of the delay causes some new problems compared to the case of constant delay: (1) new terms and different augmented variables have to be considered in the LKF for efficiency; (2) the presence of time-varying delays implies the non-convexity of the obtained matrix inequalities with respect to the length of the intervals, which necessitates special handling.

The contribution of the paper can be summarized as follows:

  • 1.

    A parameterized family of LKFs is introduced involving multiple integral terms and augmented state variables that are chosen in accordance with the applied estimations. The possible choice of the parameters gives great flexibility to find a compromise between the computational complexity and the reduction of conservatism.

  • 2.

    Relaxed sufficient LMI stability conditions for systems with interval time-varying and bounded time-varying delays are derived.

  • 3.

    The derivation is based on the multiple integral inequalities published in [17].

  • 4.

    It will be shown that the proposed condition yields less conservative results with substantially less number of decision variables than several others known from the literature.

The paper is organized as follows. Some preliminary results will be recalled in Section 2. The main result will be derived in Section 3, which starts by introducing a new Lyapunov–Krasovskii functional in Section 3.1. Next, the analysis problem will be investigated in Section 3.2. In Section 4, some numerical examples illustrate the effectiveness of the results. Finally, the conclusion will be drawn.

Standard notations are used. As usual, P > 0 ( ≥ 0) denotes the positive (semi-)definiteness of P. Symbols Sn (S+n) denote the set of symmetric (and positive definite) matrices of size n × n. For any ARn×n, symbol He(A) is defined as He(A)=A+AT. For the sake of brevity, asterisks replace the blocks in hypermatrices that are inferred readily by symmetry. If n=0 and/or m=0, then vector xRn and matrix CRn×m are considered to be empty. The Euclidean vector norm in Rn is ‖·‖. We also denote xt(θ)=x(t+θ),x˙t(θ)=x˙(t+θ),(θ[τ¯,0]).

Section snippets

Preliminaries

This paper aims to derive sufficient stability conditions in LMI form based on some multiple integral inequalities recently published in [17]. For convenience of the readers, we evoke these estimations in the following lemmas.

Lemma 1

[17]

LetWS+n¯, let E be a Euclidean space with the scalar product ⟨.,.⟩, and letπiE,(i=0,1,) form an orthogonal system. If ν ≥ 0 is a given integer, then for anyfEn¯, the following inequality holds JW(f)=f,Wfj=0ν1πj2wjTWwj,wherewj=f,πj, and the scalar product is

Lyapunov–Krasovskii functional

The stability of system (1) can be analysed via different Lyapunov–Krasovskii functionals (LKFs). In order to define the LKF, let mi, νjk,Mij be nonnegative integers chosen according to Table 1.

Let N1=(1+2M21+M34)nx, and let the extended state variable x˜(t)RN1 be introduced as x˜(t)=col{x(t),X˜(1)(t),X˜(2)(t),X˜(4)(t)},where X˜(i)(t)RM21nx,(i=1,2),X˜(4)(t)RM34nx,X˜(1)(t)=0, if τ(t)=τ̲,X˜(2)(t)=0, if τ(t)=τ¯,X˜j(1)(t)=tτ(t)tτ̲P0j(s(tτ(t))τ(t)τ̲)x(s)ds,ifτ̲<τ(t),X˜j(2)(t)=tτ¯tτ(t)P0j(

Numerical examples

In this section, we apply the proposed method to examples that have been used in the literature to compare the results. Consider system (1) with coefficient matrices listed in Table 2, where τ̲*,τ¯* denote the analytical delay bounds of constant delays. The computations have been performed by using YALMIP [31] together with MATLAB.

Case τ >0. First, we consider Example 1, which is one of the most frequently used numerical examples, it is considered in numerous papers, to mention but a few, in

Conclusions

This paper has been devoted to the stability problem for continuous-time delay systems with interval time-varying delays having known bounds on the delay derivatives. A parameterized family of Lyapunov–Krasovskii functionals involving multiple integral terms has been introduced, and novel multiple integral inequalities proposed by Gyurkovics and Takács [17] have been utilized to derive sufficient stability conditions in the form of LMIs. As a corollary, sufficient stability condition has been

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