Journal of Control Science and Engineering

Journal of Control Science and Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 938749 | https://doi.org/10.1155/2011/938749

Boren Li, "Delay-Dependent Stability Criteria for Systems with Interval Time-Varying Delay", Journal of Control Science and Engineering, vol. 2011, Article ID 938749, 9 pages, 2011. https://doi.org/10.1155/2011/938749

Delay-Dependent Stability Criteria for Systems with Interval Time-Varying Delay

Academic Editor: Onur Toker
Received07 Dec 2010
Accepted22 Jul 2011
Published11 Oct 2011

Abstract

This paper is concerned with robust stability of uncertain linear systems with interval time-varying delay. The time-varying delay is assumed to belong to an interval, which means that the derivative of the time-varying delay has an upper bound or a restriction. On other occasions, if we do not take restriction on the derivative of the time-varying delay into consideration, it allows the delay to be a fast time-varying function. The uncertainty under consideration includes a polytopic-type uncertainty and a linear fractional norm-bounded uncertainty. In order to obtain much less conservative results, a new Lyapunov-Krasovskii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to derive some new stability criteria. Numerical examples are given to demonstrate the effectiveness of the proposed stability criteria.

1. Introduction

Time delays are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, economy, and other areas. During the last two decades, the problem of stability of linear time-delay systems has been the subject of considerable research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to Gu et al. [1] and the references therein.

With the development of networked control technology, increasing attention has been paid to the study of stability analysis and controller design of networked control systems (NCSs) due to their low cost, simple installation and maintenance, and high reliability. For the NCSs, the sampling data and controller signals are transmitted through a network. As a result, it leads to a network-induced delay in a networked control closed-loop system. The existence of such kind of delay in a network-based control loop may induce instability or poor performance of NCSs. As pointed out by Yue et al. [2], NCSs are typical systems with interval time-varying delay. In fact, we consider the following system controlled through a network ฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๎‚๐ต๐‘ข(๐‘ก),(1) where ๐‘ฅ(๐‘ก)โˆˆ๐‘…๐‘› is the state vector, and ๐‘ข(๐‘ก)โˆˆ๐‘…๐‘ is the input vector. In the presence of the control network, which is shown in Figure 1, data transfer between the controller and the remote system, for example, sensors and actuators in a distributed control system will induce network delay in addition to the controller proceeding delay. First, since there exists the communication delay ๐œsc between the sensor and the controller and computational delay ๐œ๐‘ in the controller, which is shown in Figure 1, the following control law is employed for the system (1) ๐‘ข๎€ท๐‘ก+๎€ธ=๐พ๐‘๐‘ฅ๎€ท๐‘กโˆ’๐œsc๐‘˜โˆ’๐œ๐‘๐‘˜๎€ธ,๐‘กโˆˆ๎€ฝ๐‘˜โ„Ž+๐œsc๐‘˜+๐œ๐‘๐‘˜,๎€พ,โˆ€๐‘˜โˆˆ๐‘,(2) where โ„Ž is the sampling period, ๐‘ is the set of nonnegative integers, and ๐พ is the controller gain to be determined. Second, substituting (2) into (1) yields the closed-loop system by considering the communication delay ๐œca between the controller and the actuator ฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๎‚๐ต๐พ๐‘๐‘ฅ(๐‘˜โ„Ž),๐‘กโˆˆ๎€บ๐‘˜โ„Ž+๐œ๐‘˜,(๐‘˜+1)โ„Ž+๐œ๐‘˜+1๎€ธ,โˆ€๐‘˜โˆˆ๐‘,(3) where the time-delay ๐œ๐‘˜=๐œsc๐‘˜+๐œ๐‘๐‘˜+๐œca๐‘˜>0 denotes the time from the instant ๐‘˜โ„Ž when sensor nodes sample sensor data from a plant to the instant when actuators transfer data to the plant. Since ๐‘ฅ(๐‘˜โ„Ž)=๐‘ฅ(๐‘กโˆ’(๐‘กโˆ’๐‘˜โ„Ž)), defining ๐œ(๐‘ก)=๐‘กโˆ’๐‘˜โ„Ž, ๐‘กโˆˆ[๐‘˜โ„Ž+๐œ๐‘˜,(๐‘˜+1)โ„Ž+๐œ๐‘˜+1),forall๐‘˜โˆˆ๐‘, rewrite (3) as ฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๎‚๐ต๐พ๐‘๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก)),(4) where ๐œ(๐‘ก) is piecewise linear with derivative .๐œ(๐‘ก)=1 for ๐‘กโ‰ ๐‘˜โ„Ž+๐œ๐‘˜, and ๐œ(๐‘ก) is discontinuous at the points ๐‘ก=๐‘˜โ„Ž+๐œ๐‘˜,forall๐‘˜โˆˆ๐‘. It is clear that ๐œ๐‘˜โ‰ค๐œ(๐‘ก)โ‰คโ„Ž+๐œ๐‘˜+1 for ๐‘กโˆˆ[๐‘˜โ„Ž+๐œ๐‘˜,(๐‘˜+1)โ„Ž+๐œ๐‘˜+1), forโ€‰โ€‰all ๐‘˜โˆˆ๐‘.

Let ๐œ1โˆถ=min๐‘˜โˆˆ๐‘๐œ๐‘˜>0 and ๐œ2โˆถ=max๐‘˜โˆˆ๐‘{โ„Ž+๐œ๐‘˜+1}. Then we have 0<๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2.(5)

The system (3) is equivalent to the linear system (4) with interval time-varying delay described by (5). It should be pointed out that ๐œ1>0 is essential for NCSs.

To cover the routine case of 0โ‰ค๐œ(๐‘ก)โ‰ค๐œ2, we consider ๐œ(๐‘ก) as a uniformly continuous time-varying function satisfying 0โ‰ค๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2.

Throughout this paper, we will analyze the following two scenarios of the time-varying delay ๐œ(๐‘ก).

Case 1. ๐œ(๐‘ก) is a differentiable function satisfying 0โ‰ค๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2,|||.๐œ(๐‘ก)|||โ‰ค๐œ‡,โˆ€๐‘กโ‰ฅ0.(6)

Case 2. ๐œ(๐‘ก) is a continuous function satisfying 0โ‰ค๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2,โˆ€๐‘กโ‰ฅ0,(7) where ๐œ1 and ๐œ2 are the lower and upper delay bounds, respectively, and ๐œ1,๐œ2 and ๐œ‡ are constants.

Remark 1. When ๐œ1=0, the interval delay becomes routine delay. When ๐œ‡ is zero, that is, ๐œ1=๐œ2, the time-varying delay becomes constant delay, Case 1 is a special case of the Case 2. We may obtain a less conservative result using Case 1 than that using Case 2. However, if the time-varying delay is not differentiable, or information about derivative of time-varying delay is absent, only Case 2 can be used to handle the situation.

Notation 1. ๐‘…๐‘› denotes the ๐‘›-dimensional Euclidean space, ๐‘…๐‘›ร—๐‘š is the set of ๐‘›ร—๐‘š real matrices, ๐ผ is the identity matrix of appropriate dimensions, the notation ๐‘‹>0 (resp.,โ€‰โ€‰๐‘‹โ‰ฅ0), for ๐‘‹โˆˆ๐‘…๐‘›ร—๐‘› means that the matrix ๐‘‹ is real positive definite (resp., positive semidefinite). For an arbitrary matrix ๐ต and two symmetric matrices ๐ด and ๐ถ, ๎€บ๐ด๐ตโˆ—๐ถ๎€ป denotes a symmetric matrix, where โˆ— denotes a block matrix entry implied by symmetry.

2. System Descriptions and Preliminaries

Let ๐ต=๎‚๐ต๐พ๐‘ in (4), we have the following linear system with interval time-varying delay ฬ‡๐‘ฅ(๐‘ก)=๐ด๐‘ฅ(๐‘ก)+๐ต๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก)),๐‘ฅ(๐‘ก)=๐œ™(๐‘ก),๐‘กโˆˆ๎€บโˆ’๐œ2,โˆ’๐œ1๎€ป,(8) where ๐ด and ๐ต are the constant matrices with appropriate dimensions, ๐œ™(๐‘ก) is the initial condition of the system.

In this paper, we will study stability criteria of system described by (8) satisfying (6) or (7), employing the following new Lyapunov-Krasovskii functional: ๐‘‰๎€ท๐‘ก,๐‘ฅ๐‘ก๎€ธ=๐‘ฅ๐‘‡(๐‘ก)๐‘ƒ๐‘ฅ(๐‘ก)+๎€œ๐‘ก๐‘กโˆ’๐œ(๐‘ก)๐‘ฅ๐‘‡(๐œƒ)๐‘๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ๐‘กโˆ’๐‘–๐›ฟ๐‘ฅ๐‘‡(๐œƒ)๐‘„๐‘–๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐พ๎“๐‘–=1๎€œ๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๐‘ฅ๐‘‡(๐œƒ)๐บ๐‘–๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘๎“๐‘–=1๎€œโˆ’(๐‘–โˆ’1)๐›ฟโˆ’๐‘–๐›ฟ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๎€ท๐›ฟ๐‘…๐‘–๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ+๐พ๎“๐‘–=1๎€œโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒโˆ’๐œ1โˆ’๐‘–๐œŒ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๎€ท๐œŒ๐ป๐‘–๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ+๎€œโˆ’๐œ1โˆ’๐œ2๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)(๐›พ๐‘Š)ฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ,(9) where ๐‘ƒ>0,โ€‰โ€‰๐‘โ‰ฅ0,โ€‰โ€‰๐‘„๐‘–>0,โ€‰โ€‰๐‘…๐‘–>0(๐‘–=1,โ€ฆ,๐‘),โ€‰โ€‰๐บ๐‘–>0,โ€‰โ€‰๐ป๐‘–>0(๐‘–=1,โ€ฆ,๐พ),โ€‰โ€‰๐‘Š>0 of appropriate dimensions, and ๐œ1=๐‘ร—๐›ฟ,โ€‰โ€‰๐œ2โˆ’๐œ1=๐พร—๐œŒ,โ€‰โ€‰๐›พ=๐œ2โˆ’๐œ1,โ€‰โ€‰๐‘,โ€‰โ€‰๐พ are positive integer numbers of divisions of intervals [โˆ’๐œ1,0] and [โˆ’๐œ2,โˆ’๐œ1], respectively, and ๐›ฟ, ๐œŒ are the length of each division. When ๐œ1=0 and ๐œ1=๐œ2, we assume that ๐‘=0 and ๐พ=0, respectively. Moreover, we will consider a poly-topic uncertainty and a linear fractional norm-bounded uncertainty which includes a routine norm-bounded uncertainty as a special case. Some numerical examples will be given to show the improvement over some previous results.

For any delay satisfying Case 1 or Case 2, our objective of this study is to develop new stability criteria which guarantee that system (8) is asymptotically stable and the system (8) subject to some uncertainties is robustly stable. For this purpose, the following lemmas are introduced.

Lemma 1 (see [3]). For any constant matrix ๐‘ˆโˆˆ๐‘…๐‘›ร—๐‘›,๐‘ˆ=๐‘ˆ๐‘‡>0, scalar ๐œŽ>0, and vector function ฬ‡๐‘ฅโˆถ[โˆ’๐œŽ,0]โ†’๐‘…๐‘› such that the following integration is well defined, then it holds that โˆ’๐œŽ๎€œ๐‘ก๐‘กโˆ’๐œŽฬ‡๐‘ฅ๐‘‡(๐‘ค)๐‘ˆฬ‡๐‘ฅ(๐‘ค)๐‘‘๐‘คโ‰คโŽกโŽขโŽขโŽฃ๐‘ฅ(๐‘ก)๐‘ฅ(๐‘กโˆ’๐œŽ)โŽคโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽฃโˆ’๐‘ˆ๐‘ˆโˆ—โˆ’๐‘ˆโŽคโŽฅโŽฅโŽฆร—โŽกโŽขโŽขโŽฃ๐‘ฅ(๐‘ก)๐‘ฅ(๐‘กโˆ’๐œŽ)โŽคโŽฅโŽฅโŽฆ.(10)
Applying the Lemma 1 yields the following new integral inequality for cross-product term.

Lemma 2 (see [4]). For any constant matrix ๐‘ˆโˆˆ๐‘…๐‘›ร—๐‘›,๐‘ˆ=๐‘ˆ๐‘‡>0, scalars ๐œŽ1โ‰ค๐œ(๐‘ก)โ‰ค๐œŽ2 and vector function ฬ‡๐‘ฅโˆถ[โˆ’๐œŽ2,โˆ’๐œŽ1]โ†’๐‘…๐‘› such that the following integration is well defined, then it holds that โˆ’๎€ท๐œŽ2โˆ’๐œŽ1๎€ธ๎€œ๐‘กโˆ’๐œŽ1๐‘กโˆ’๐œŽ2ฬ‡๐‘ฅ๐‘‡(๐‘ค)๐‘ˆฬ‡๐‘ฅ(๐‘ค)๐‘‘๐‘คโ‰ค๐œ‚๐‘‡(๐‘ก)ฮฉ๐œ‚(๐‘ก),(11) where ๐œ‚(๐‘ก)=โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ๎€ท๐‘กโˆ’๐œŽ1๎€ธ๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))๐‘ฅ๎€ท๐‘กโˆ’๐œŽ2๎€ธโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ,ฮฉ=โŽกโŽขโŽขโŽขโŽขโŽฃโˆ’๐‘ˆ๐‘ˆ0โˆ—โˆ’2๐‘ˆ๐‘ˆโˆ—โˆ—โˆ’๐‘ˆโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.(12)

3. New Stability Criteria

We first consider asymptotic stability for the nominal system (8). Employing Lyapunov-Krasovskii functional (9), we have the following result.

Theorem 1. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2 and ๐œ‡, the nominal system (8) satisfying (6) is asymptotically stable if there exit real symmetric matrices ๐‘ƒ>0,โ€‰โ€‰๐‘โ‰ฅ0,โ€‰โ€‰๐‘„๐‘–>0,โ€‰โ€‰๐‘…๐‘–>0(๐‘–=1,โ€ฆ,๐‘),โ€‰โ€‰๐บ๐‘–>0,โ€‰โ€‰๐ป๐‘–>0(๐‘–=1,โ€ฆ,๐พ),โ€‰โ€‰๐‘Š>0, and any matrix ๐‘†=[๐‘†๐‘‡1๐‘†๐‘‡2โ‹ฏ๐‘†๐‘‡๐‘+๐พ+3]๐‘‡ of appropriate dimensions such that the following LMI holds: ฮž=ฮž1+๐‘†๐ด๐‘‘+๎€ท๐‘†๐ด๐‘‘๎€ธ๐‘‡<0,(13) where ฮž1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃฮž110๐‘…10โ‹ฏ0000โ‹ฏ00๐‘ƒโˆ—ฮž2200โ‹ฏ0๐‘Š00โ‹ฏ0๐‘Š0โˆ—โˆ—ฮฆ1๐‘…2โ‹ฏ0000โ‹ฏ000โˆ—โˆ—โˆ—ฮฆ2โ‹ฏ0000โ‹ฏ000โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโˆ—โˆ—โˆ—โˆ—โ‹ฏฮฆ๐‘โˆ’1๐‘…๐‘00โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—ฮฆ๐‘๐ป10โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—ฮจ1๐ป2โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—ฮจ2โ‹ฏ000โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฏโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏฮจ๐พโˆ’1๐ป๐พ0โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—ฮจ๐พ0โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—ฮฅโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,(14)ฮž11=๐‘+๐‘„1โˆ’๐‘…1,ฮž22=โˆ’(1โˆ’๐œ‡)๐‘โˆ’2๐‘Š,๐ด๐‘‘=โŽกโŽขโŽฃ๐ด๐ต00โ‹ฏ0๎„ฟ๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…Œ๐‘+๐พโˆ’๐ผโŽคโŽฅโŽฆ,ฮฆ๐‘–=๐‘„๐‘–+1โˆ’๐‘„๐‘–โˆ’๐‘…๐‘–โˆ’๐‘…๐‘–+1(๐‘–=1,2,โ€ฆ,๐‘โˆ’1),ฮฆ๐‘=โˆ’๐‘„๐‘โˆ’๐‘…๐‘โˆ’๐‘Š+๐บ1โˆ’๐ป1,ฮจ๐‘–=๐บ๐‘–+1โˆ’๐บ๐‘–โˆ’๐ป๐‘–โˆ’๐ป๐‘–+1(๐‘–=1,2,โ€ฆ,๐พโˆ’1),ฮจ๐พ=โˆ’๐บ๐พโˆ’๐ป๐พโˆ’๐‘Š,ฮฅ=๐‘๎“๐‘–=1๐›ฟ2๐‘…๐‘–+๐พ๎“๐‘–=1๐œŒ2๐ป๐‘–+๐›พ2๐‘Š.(15)

Proof. Taking the time derivative of ๐‘‰(๐‘ก,๐‘ฅ๐‘ก) with respect to ๐‘ก along the trajectory of (8) yields ฬ‡โ€Œ๐‘‰๎€ท๐‘ก,๐‘ฅ๐‘ก๎€ธโ‰ค2๐‘ฅ๐‘‡(๐‘ก)๐‘ƒฬ‡๐‘ฅ(๐‘ก)+๐‘ฅ๐‘‡(๐‘ก)๐‘๐‘ฅ(๐‘ก)โˆ’(1โˆ’๐œ‡)๐‘ฅ๐‘‡(๐‘กโˆ’๐œ(๐‘ก))๐‘๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))+๐‘๎“๐‘–=1๐‘ฅ๐‘‡(๐‘กโˆ’(iโˆ’1)๐›ฟ)๐‘„๐‘–๐‘ฅ(๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ)โˆ’๐‘๎“๐‘–=1๐‘ฅ๐‘‡(๐‘กโˆ’๐‘–๐›ฟ)๐‘„๐‘–๐‘ฅ(๐‘กโˆ’๐‘–๐›ฟ)+๐พ๎“๐‘–=1๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๎€ธ๐บ๐‘–๐‘ฅ๎€ท๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๎€ธโˆ’๐พ๎“๐‘–=1๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๎€ธ๐บ๐‘–๐‘ฅ๎€ท๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๎€ธ+ฬ‡๐‘ฅ๐‘‡(๐‘ก)โŽ›โŽœโŽ๐‘๎“๐‘–=1๐›ฟ2๐‘…๐‘–+๐พ๎“๐‘–=1๐œŒ2๐ป๐‘–+๐›พ2๐‘ŠโŽžโŽŸโŽ ฬ‡๐‘ฅ(๐‘ก)โˆ’๐‘๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ๐‘กโˆ’๐‘–๐›ฟฬ‡๐‘ฅ๐‘‡(๐œƒ)๎€ท๐›ฟ๐‘…๐‘–๎€ธฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒโˆ’๐พ๎“๐‘–=1๎€œ๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒฬ‡๐‘ฅ๐‘‡(๐œƒ)๎€ท๐œŒ๐ป๐‘–๎€ธฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒโˆ’๎€œ๐‘กโˆ’๐œ1๐‘กโˆ’๐œ2ฬ‡๐‘ฅ๐‘‡(๐œƒ)(๐›พ๐‘Š)ฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒ.(16)
Applying the Lemma 1, the following inequalities hold: โˆ’๐‘๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ๐‘กโˆ’๐‘–๐›ฟฬ‡๐‘ฅ๐‘‡(๐œƒ)๎€ท๐›ฟ๐‘…๐‘–๎€ธฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒโ‰ค๐‘๎“๐‘–=1โŽกโŽขโŽขโŽฃ๐‘ฅ(๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ)๐‘ฅ(๐‘กโˆ’๐‘–๐›ฟ)โŽคโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽฃโˆ’๐‘…๐‘–๐‘…๐‘–โˆ—โˆ’๐‘…๐‘–โŽคโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽฃ๐‘ฅ(๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ)๐‘ฅ(๐‘กโˆ’๐‘–๐›ฟ)โŽคโŽฅโŽฅโŽฆ,โˆ’๐พ๎“๐‘–=1๎€œ๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒฬ‡๐‘ฅ๐‘‡(๐œƒ)๎€ท๐œŒ๐ป๐‘–๎€ธฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒโ‰ค๐พ๎“๐‘–=1โŽกโŽขโŽขโŽฃ๐‘ฅ(๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ)๐‘ฅ๎€ท๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๎€ธโŽคโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽฃโˆ’๐ป๐‘–๐ป๐‘–โˆ—โˆ’๐ป๐‘–โŽคโŽฅโŽฅโŽฆร—โŽกโŽขโŽขโŽฃ๐‘ฅ๎€ท๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๎€ธ๐‘ฅ๎€ท๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๎€ธโŽคโŽฅโŽฅโŽฆ.(17)
Applying the Lemma 2, the following inequality holds: โˆ’๎€œ๐‘กโˆ’๐œ1๐‘กโˆ’๐œ2ฬ‡๐‘ฅ๐‘‡(๐œƒ)(๐›พ๐‘Š)ฬ‡๐‘ฅ(๐œƒ)๐‘‘๐œƒโ‰คโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ๎€ท๐‘กโˆ’๐œ1๎€ธ๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))๐‘ฅ๎€ท๐‘กโˆ’๐œ2๎€ธโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽขโŽขโŽฃโˆ’๐‘Š๐‘Š0โˆ—โˆ’2๐‘Š๐‘Šโˆ—โˆ—โˆ’๐‘ŠโŽคโŽฅโŽฅโŽฅโŽฅโŽฆร—โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ๎€ท๐‘กโˆ’๐œ1๎€ธ๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))๐‘ฅ๎€ท๐‘กโˆ’๐œ2๎€ธโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.(18)
For any matrix ๐‘†=[๐‘†๐‘‡1๐‘†๐‘‡2โ‹ฏ๐‘†๐‘‡๐‘+๐พ+3]๐‘‡ of appropriate dimensions, the following equality holds: 2๐œ‰๐‘‡(๐‘ก)๐‘†[๐ด๐‘ฅ(๐‘ก)+๐ต๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))โˆ’ฬ‡๐‘ฅ(๐‘ก)]=0,(19) where is defined in the top of next page, ๐œ‰๐‘‡(๐‘ก)=๎€บ๐‘ฅ๐‘‡(๐‘ก)๐‘ฅ๐‘‡(๐‘กโˆ’๐œ(๐‘ก))๐œ‰๐‘‡1(๐‘ก)๐œ‰๐‘‡2(๐‘ก)ฬ‡๐‘ฅ(๐‘ก)๎€ป,๐œ‰๐‘‡1(๐‘ก)=๎€บ๐‘ฅ๐‘‡(๐‘กโˆ’๐›ฟ)๐‘ฅ๐‘‡(๐‘กโˆ’2๐›ฟ)โ‹ฏ๐‘ฅ๐‘‡(๐‘กโˆ’(๐‘โˆ’1)๐›ฟ)๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1๎€ธ๎€ป,๐œ‰๐‘‡2(๐‘ก)=๎€บ๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1โˆ’๐œŒ๎€ธ๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1โˆ’2๐œŒ๎€ธโ‹ฏ๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ1โˆ’(๐พโˆ’1)๐œŒ๎€ธ๐‘ฅ๐‘‡๎€ท๐‘กโˆ’๐œ2๎€ธ๎€ป.(20)
Considering (16)โ€“(19) together, we have ฬ‡โ€Œ๐‘‰(๐‘ก,๐‘ฅ๐‘ก)โ‰ค๐œ‰๐‘‡(๐‘ก)ฮž๐œ‰(๐‘ก). If (13) is satisfied, then ฬ‡โ€Œ๐‘‰(๐‘ก,๐‘ฅ๐‘ก)โ‰คโˆ’๐œ†๐‘ฅ๐‘‡(๐‘ก)๐‘ฅ(๐‘ก) for some scalar ๐œ†>0, from which we conclude that the nominal system (8) is asymptotically stable. This completes the proof.

When the restriction on the derivative of the interval time-varying delay is removed, that is, choosing ๐‘โ‰ก0 in Theorem 1, we can obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 1. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2, the nominal system (8) satisfying (7) is asymptotically stable, if there exit real symmetric matrices ๐‘ƒ>0, ๐‘„๐‘–>0,โ€‰โ€‰๐‘…๐‘–>0(๐‘–=1,โ€ฆ,๐‘), ๐บ๐‘–>0,โ€‰โ€‰๐ป๐‘–>0(๐‘–=1,โ€ฆ,๐พ), ๐‘Š>0 and any matrix ๐‘† of appropriate dimensions such that the LMI (13) holds with ๐‘โ‰ก0.

Remark 2. For system (8) with the routine delay case described by ๐œ1=0 and ๐œ2>0, that is, ๐‘=0,๐พ>0, the corresponding Lyapunov-Krasovskii functional reduces to ๐‘‰๎€ท๐‘ก,x๐‘ก๎€ธ=๐‘ฅ๐‘‡(๐‘ก)๐‘ƒ๐‘ฅ(๐‘ก)+๎€œ๐‘ก๐‘กโˆ’๐œ(๐‘ก)๐‘ฅ๐‘‡(๐œƒ)๐‘๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐พ๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐œŒ๐‘กโˆ’๐‘–๐œŒ๐‘ฅ๐‘‡(๐œƒ)๐บ๐‘–๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐พ๎“๐‘–=1๎€œโˆ’(๐‘–โˆ’1)๐œŒโˆ’๐‘–๐œŒ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๎€ท๐œŒ๐ป๐‘–๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ+๎€œ0โˆ’๐œ2๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)(๐›พ๐‘Š)ฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ.(21)
And for system (8) with the constant delay case described by ๐œ1=๐œ(๐‘ก)=๐œ2>0, that is, ๐‘>0,๐พ=0, the corresponding Lyapunov-Krasovskii functional reduces to ๐‘‰๎€ท๐‘ก,๐‘ฅ๐‘ก๎€ธ=๐‘ฅ๐‘‡(๐‘ก)๐‘ƒ๐‘ฅ(๐‘ก)+๐‘๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ๐‘กโˆ’๐‘–๐›ฟ๐‘ฅ๐‘‡(๐œƒ)๐‘„๐‘–๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘๎“๐‘–=1๎€œโˆ’(๐‘–โˆ’1)๐›ฟโˆ’๐‘–๐›ฟ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๎€ท๐›ฟ๐‘…๐‘–๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ.(22)

Similar to the proof of the Theorem 1, one can easily derive a less conservative results than some exiting ones, respectively, which will be shown through numerical examples in the next section. For the sake of simplicity, the results are omitted.

In what follows, we consider robust stability of the system (8) satisfying (6) or (7) subject to a poly-topic uncertainty and a linear fractional norm-bounded uncertainty. For the poly-topic uncertainty, that is, the matrices ๐ด and ๐ต in (8) can be expressed as [๐ด๐ต]=๐‘Ÿ๎“๐‘—=1๐œ๐‘—๎€บ๐ด๐‘—๐ต๐‘—๎€ป,(23) where โˆ‘๐‘Ÿ๐‘—=1๐œ๐‘—=1,0โ‰ค๐œ๐‘—โ‰ค1. Based on Theorem 1, we can easily obtain the following result.

Theorem 2. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2 and ๐œ‡, the system described by (8) satisfying (6) subject to polytopic uncertainty (23) is robustly stable, if there exit real symmetric matrices ๐‘ƒ๐‘—>0,๐‘๐‘—โ‰ฅ0,๐‘„๐‘–๐‘—>0,๐‘…๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐‘), ๐บ๐‘–๐‘—>0,๐ป๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐พ), ๐‘Š๐‘—>0, and any matrix ๐‘† of appropriate dimensions such that the following LMIs hold for ๐‘—=1,2,โ€ฆ,๐‘Ÿ: ฮž(๐‘—)=ฮž(๐‘—)1+๐‘†๐ด(๐‘—)๐‘‘+๎‚€๐‘†๐ด(๐‘—)๐‘‘๎‚๐‘‡<0,(24) where ฮž(๐‘—)1=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃฮž(๐‘—)110๐‘…1๐‘—0โ‹ฏ0000โ‹ฏ00๐‘ƒ๐‘—โˆ—ฮž(๐‘—)2200โ‹ฏ0๐‘Š๐‘—00โ‹ฏ0๐‘Š๐‘—0โˆ—โˆ—ฮฆ(๐‘—)1๐‘…2๐‘—โ‹ฏ0000โ‹ฏ000โˆ—โˆ—โˆ—ฮฆ(๐‘—)2โ‹ฏ0000โ‹ฏ000โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโˆ—โˆ—โˆ—โˆ—โ‹ฏฮฆ(๐‘—)๐‘โˆ’1๐‘…๐‘๐‘—00โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—ฮฆ(๐‘—)๐‘๐ป1๐‘—0โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—ฮจ(๐‘—)1๐ป2๐‘—โ‹ฏ000โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—ฮจ(๐‘—)2โ‹ฏ000โ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฏโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏฮจ(๐‘—)๐พโˆ’1๐ป๐พ๐‘—0โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—ฮจ(๐‘—)๐พ0โˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—โˆ—โˆ—โ‹ฏโˆ—โˆ—ฮฅ(๐‘—)โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,(25)๐‘† is defined in Theorem 1, and ฮž(๐‘—)11=๐‘๐‘—+๐‘„1๐‘—โˆ’๐‘…1๐‘—,ฮž(๐‘—)22=โˆ’(1โˆ’๐œ‡)๐‘๐‘—โˆ’2๐‘Š๐‘—,๐ด(๐‘—)๐‘‘=โŽกโŽขโŽฃ๐ด๐‘—๐ต๐‘—00โ‹ฏ0๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘+๐พโˆ’๐ผโŽคโŽฅโŽฆ,ฮฆ(๐‘—)๐‘–=๐‘„๐‘–+1,๐‘—โˆ’๐‘„๐‘–๐‘—โˆ’๐‘…๐‘–๐‘—โˆ’๐‘…๐‘–+1,๐‘—(๐‘–=1,2,โ€ฆ,๐‘โˆ’1),ฮฆ(๐‘—)๐‘=โˆ’๐‘„๐‘๐‘—โˆ’๐‘…๐‘๐‘—โˆ’๐‘Š๐‘—+๐บ1๐‘—โˆ’๐ป1๐‘—,ฮจ(๐‘—)๐‘–=๐บ๐‘–+1,๐‘—โˆ’๐บ๐‘–๐‘—โˆ’๐ป๐‘–๐‘—โˆ’๐ป๐‘–+1,๐‘—(๐‘–=1,2,โ€ฆ,๐พโˆ’1),ฮจ(๐‘—)๐พ=โˆ’๐บ๐พ๐‘—โˆ’๐ป๐พ๐‘—โˆ’๐‘Š๐‘—,ฮฅ(๐‘—)=๐‘๎“๐‘–=1๐›ฟ2๐‘…๐‘–๐‘—+๐พ๎“๐‘–=1๐œŒ2๐ป๐‘–๐‘—+๐›พ2๐‘Š๐‘—.(26)

Proof. Construct the following parameter-dependent Lyapunov-Krasovskii functional: ๐‘‰๐‘ข๎€ท๐‘ก,๐‘ฅ๐‘ก๎€ธ=๐‘Ÿ๎“๐‘—=1๐‘ฅ๐‘‡(๐‘ก)๐œ๐‘—๐‘ƒ๐‘—๐‘ฅ(๐‘ก)+๐‘Ÿ๎“๐‘—=1๎€œ๐‘ก๐‘กโˆ’๐œ(๐‘ก)๐‘ฅ๐‘‡(๐œƒ)๐œ๐‘—๐‘๐‘—๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘Ÿ๎“๐‘—=1๐‘๎“๐‘–=1๎€œ๐‘กโˆ’(๐‘–โˆ’1)๐›ฟ๐‘กโˆ’๐‘–๐›ฟ๐‘ฅ๐‘‡(๐œƒ)๐œ๐‘—๐‘„๐‘–๐‘—๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘Ÿ๎“๐‘—=1๐พ๎“๐‘–=1๎€œ๐‘กโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒ๐‘กโˆ’๐œ1โˆ’๐‘–๐œŒ๐‘ฅ๐‘‡(๐œƒ)๐œ๐‘—๐บ๐‘–๐‘—๐‘ฅ(๐œƒ)๐‘‘๐œƒ+๐‘Ÿ๎“๐‘—=1๐‘๎“๐‘–=1๎€œโˆ’(iโˆ’1)๐›ฟโˆ’๐‘–๐›ฟ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๐œ๐‘—๎€ท๐›ฟ๐‘…๐‘–๐‘—๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ+๐‘Ÿ๎“๐‘—=1๐พ๎“๐‘–=1๎€œโˆ’๐œ1โˆ’(๐‘–โˆ’1)๐œŒโˆ’๐œ1โˆ’๐‘–๐œŒ๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๐œ๐‘—๎€ท๐œŒ๐ป๐‘–๐‘—๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ+๐‘Ÿ๎“๐‘—=1๎€œโˆ’๐œ1โˆ’๐œ2๎€œ๐‘ก๐‘ก+๐œƒฬ‡๐‘ฅ๐‘‡(๐‘ฃ)๐œ๐‘—๎€ท๐›พ๐‘Š๐‘—๎€ธฬ‡๐‘ฅ(๐‘ฃ)๐‘‘๐‘ฃ๐‘‘๐œƒ,(27) where ๐‘ƒ๐‘—>0,โ€‰โ€‰๐‘๐‘—โ‰ฅ0,โ€‰โ€‰๐‘„๐‘–๐‘—>0,โ€‰โ€‰๐‘…๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐‘),โ€‰โ€‰๐บ๐‘–๐‘—>0,โ€‰โ€‰๐ป๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐พ),โ€‰โ€‰๐‘Š๐‘—>0 are unknown matrices of appropriate dimensions. Then the proof follows a linear similar to the proof of Theorem 1, we can obtain the following inequality: ฬ‡โ€Œ๐‘‰๐‘ข๎€ท๐‘ก,๐‘ฅ๐‘ก๎€ธโ‰ค๐‘Ÿ๎“๐‘—=1๐œ‰๐‘‡(๐‘ก)๐œ๐‘—ฮž(๐‘—)๐œ‰(๐‘ก),(28) where ๐œ‰(๐‘ก) is defined in the formula (19), then ฬ‡โ€Œ๐‘‰๐‘ข(๐‘ก,๐‘ฅ๐‘ก)โ‰คโˆ’๐œ†๐‘ฅ๐‘‡(๐‘ก)๐‘ฅ(๐‘ก) for some scalar ๐œ†>0, from which we conclude that the system described by (8) satisfying (6) subject to polytopic uncertainty (23) is robustly stable. This completes the proof.

Similar to Corollary 1, we can easily obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 2. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2, the system described by (8) satisfying (7) subject to polytopic uncertainty (23) is robustly stable if there exit real symmetric matrices ๐‘ƒ๐‘—>0,โ€‰โ€‰๐‘„๐‘–๐‘—>0,โ€‰โ€‰๐‘…๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐‘),โ€‰โ€‰๐บ๐‘–๐‘—>0,โ€‰โ€‰๐ป๐‘–๐‘—>0(๐‘–=1,โ€ฆ,๐พ),โ€‰โ€‰๐‘Š๐‘—>0, and any matrix ๐‘† of appropriate dimensions such that the LMIs (24) hold with ๐‘๐‘—โ‰ก0 for ๐‘—=1,2,โ€ฆ,๐‘Ÿ.

Remark 3. We succeed to separate the system matrices and Lyapunov matrices in Theorem 1, so we can easily use parameter-dependent Lyapunov-Krasovskii functional method. Different Lyapunov matrices are used in Theorem 2 for ๐‘Ÿ different LMIs, which are distinguished with Proposition 7 in Jiang and Han [6], in which used fixed Lyapunov matrices for ๐‘Ÿ different LMIs. In fact parameter-dependent Lyapunov-Krasovskii functional method can reduce stability criteria conservatism significantly. Numerical example will be given to show the improvement with Jiang and Han [6] in the next section.

Next we address the linear fractional norm-bounded uncertainty. Suppose that matrices ๐ด and ๐ต have parameter perturbations as ฮ”๐ด(๐‘ก) and ฮ”๐ต(๐‘ก), which are in the form of [ฮ”๐ด(๐‘ก)ฮ”๐ต(๐‘ก)]=๐ท(๐ผโˆ’๐น(๐‘ก)๐‘€)โˆ’1๐น(๐‘ก)๎€บ๐ธ1๐ธ2๎€ป,(29) where ๐ท,๐‘€,๐ธ1, and ๐ธ2 are known real constant matrices of appropriate dimensions, and ๐‘€๐‘‡๐‘€<๐ผ; ๐น(๐‘ก) is an unknown matrix function with Lesbesgue measurable elements satisfying ๐น๐‘‡(๐‘ก)๐น(๐‘ก)โ‰ค๐ผ.

For system (8) with uncertainty (29), we can establish the following result by considering Theorem 1 and applying ๐‘† procedure [9].

Theorem 3. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2 and ๐œ‡, the system described by (8) satisfying (6)subject to the linear fractional norm-bounded uncertainty (29) is robustly stable if there exit real symmetric matrices ๐‘ƒ>0,โ€‰โ€‰๐‘โ‰ฅ0, ๐‘„๐‘–>0,โ€‰โ€‰๐‘…๐‘–>0(๐‘–=1,โ€ฆ,๐‘), ๐บ๐‘–>0,โ€‰โ€‰๐ป๐‘–>0(๐‘–=1,โ€ฆ,๐พ), ๐‘Š>0, any matrix ๐‘† of appropriate dimensions, and a scalar ๐œ€>0 such that the following LMI holds: โŽกโŽขโŽขโŽขโŽขโŽฃฮž๐‘†๐ท๐œ€ฮ“โˆ—โˆ’๐œ€๐ผ๐œ€๐‘€๐‘‡โˆ—โˆ—โˆ’๐œ€๐ผโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ<0,(30) where ฮ“๐‘‡=๎‚ƒ๐ธ1๐ธ200โ‹ฏ0๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œ๐‘+๐พ+1๎‚„,โ€‰โ€‰ฮž is defined in the formula (13), and ๐‘† is defined in Theorem 1.

Proof. Replacing ๐ด and ๐ต in (13) with ๐ด+๐ท(๐ผโˆ’๐น(๐‘ก)๐‘€)โˆ’1๐น(๐‘ก)๐ธ1 and ๐ต+๐ท(๐ผโˆ’๐น(๐‘ก)๐‘€)โˆ’1๐น(๐‘ก)๐ธ2, respectively, and multiply both sides of the resulting matrix by vectors ๐‘ฅ๐‘– for ๐‘–=1,2,โ€ฆ,๐‘+๐พ+3. Next, define ๐‘=(๐ผโˆ’๐น(๐‘ก)๐‘€)โˆ’1๐น(๐‘ก)๐ธ1๐‘ฅ1,๐‘ž=(๐ผโˆ’๐น(๐‘ก)๐‘€)โˆ’1๐น(๐‘ก)๐ธ2๐‘ฅ2. Then, we have the following condition for the admissible uncertainty ๐น(๐‘ก): ๐›ฝ๐‘‡โŽกโŽขโŽขโŽฃฮž๐‘†๐ทโˆ—0โŽคโŽฅโŽฅโŽฆ๐›ฝ<0,(31) where ๐›ฝ๐‘‡=[๐‘ฅ๐‘‡1,๐‘ฅ๐‘‡2,โ€ฆ,๐‘ฅ๐‘‡๐‘+๐พ+3,(๐‘+๐‘ž)๐‘‡]. It is easy to know that ๐‘ and ๐‘ž can be rewritten as ๐‘=๐น(๐‘ก)๐‘€๐‘+๐น(๐‘ก)๐ธ1๐‘ฅ1 and ๐‘ž=๐น(๐‘ก)๐‘€๐‘ž+๐น(๐‘ก)๐ธ2๐‘ฅ2, since ๐น๐‘‡(๐‘ก)๐น(๐‘ก)โ‰ค๐ผ, it is obvious that the following inequality holds: (๐‘+๐‘ž)๐‘‡(๐‘+๐‘ž)โ‰คโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ1๐‘ฅ2๐‘+๐‘žโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽขโŽขโŽฃ๐ธ๐‘‡1๐ธ1๐ธ๐‘‡1๐ธ2๐ธ๐‘‡1๐‘€โˆ—๐ธ๐‘‡2๐ธ2๐ธ๐‘‡2๐‘€โˆ—โˆ—๐‘€๐‘‡๐‘€โŽคโŽฅโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ฅ1๐‘ฅ2๐‘+๐‘žโŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.(32) Applying ๐‘† procedure, both inequalities (31) and (32) are true if and only if there is a ๐œ€>0, promising that the following condition holds: ๐›ฝ๐‘‡โŽกโŽขโŽขโŽฃฮž+๐œ€ฮ“ฮ“๐‘‡๐‘†๐ท+๐œ€ฮ“๐‘€โˆ—๐œ€๐‘€๐‘‡๐‘€โˆ’๐œ€๐ผโŽคโŽฅโŽฅโŽฆ๐›ฝ<0.(33) By Schur complement, for any ๐›ฝโ‰ 0, (33) is equivalent to (30). This completes the proof.

Similar to Corollary 1, we can easily obtain a delay variety rate-independent criterion for a delay that only satisfies (7).

Corollary 3. For some given scalars 0โ‰ค๐œ1โ‰ค๐œ2, the system described by (8) satisfying (7) subject to the linear fractional norm-bounded uncertainty (29) is robustly stable if there exit real symmetric matrices ๐‘ƒ>0, ๐‘„๐‘–>0,โ€‰โ€‰๐‘…๐‘–>0(๐‘–=1,โ€ฆ,๐‘), ๐บ๐‘–>0,โ€‰โ€‰๐ป๐‘–>0(๐‘–=1,โ€ฆ,๐พ), ๐‘Š>0, any matrix ๐‘† of appropriate dimensions, and a scalar ๐œ€>0 such that the LMI (30) holds with ๐‘โ‰ก0.

Remark 4. It is clear to see that if we set ๐‘€=0, the linear fractional norm-bounded uncertainty reduces to the routine norm-bounded uncertainty, and we can derive a corresponding results for the routine norm-bounded uncertainty from Theorem 3 and Corollary 3. For the sake of simplicity, the results are omitted.

4. Numerical Examples

In this section, two examples are given to show the effectiveness of the results derived in this paper.

Example 1. Consider the following time-delay system: ฬ‡๐‘ฅ(๐‘ก)=โŽกโŽขโŽขโŽฃโˆ’200โˆ’0.9โŽคโŽฅโŽฅโŽฆ๐‘ฅ(๐‘ก)+โŽกโŽขโŽขโŽฃโˆ’10โˆ’1โˆ’1โŽคโŽฅโŽฅโŽฆ๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก)).(34) Table 1 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different ๐œ1, and those in He et al. [5], Jiang and Han [6], and Shao [7].(1)If ๐œ1=0, by using Corollary 1, we have ๐œ2=1.3454 for ๐‘=0 and any integer ๐พโ‰ฅ2, which is larger than ๐œ2=1.34 in derived in He et al. [5] and close to ๐œ2=1.3453 in derived in Jiang and Han [6].(2)If 0<๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2 and ๐œ1โ‰ ๐œ2. By using Corollary 1 for ๐‘=2,๐พ=2, our results listed in the second last column is same as the results derived in Jiang and Han [6]. Moreover, by using Corollary 1 for ๐‘=3,๐พ=3, we have the results listed in the last column, which are larger than the results derived in He et al. [5] and Jiang and Han(2008).(3)By using Proposition 8 in Jiang and Han [6], we can have ๐œ2=4.6007 for ๐œ1=4.4697, not have 4.6009a.(4)For the constant delay ๐œ1=๐œ2, that is, ๐พ=0. By using Corollary 1, we have ๐œmax=4.4721 for ๐‘=1; 5.7175 for ๐‘=2; 5.9677 for ๐‘=3; 6.0983 for ๐‘=5; 6.1537 for ๐‘=10, which is close to analytical delay limit for stability 6.17258.


๐œ 1 He et al. [5]Jiang and Han [6]Shao [7]This work for ๐‘ = 2 , ๐พ = 2 This work for ๐‘ = 3 , ๐พ = 3

01.341.3453 โ€”1.3454 (N = 0)1.3454 (N = 0)
11.741.80431.87371.80431.8111
22.432.52132.50492.52132.5339
33.223.33113.25913.33113.3475
44.074.18804.07444.18804.2072
4.46974.47 4 . 6 0 0 9 a โ€”4.60074.6210
5 โ€”5.0722โ€”5.07225.0937
5.71 โ€”5.7107โ€”5.71075.7337

Example 2. Consider the following time-delay system: ฬ‡๐‘ฅ(๐‘ก)=โŽกโŽขโŽขโŽฃโˆ’2+๐›ฟ100โˆ’1+๐›ฟ2โŽคโŽฅโŽฅโŽฆ๐‘ฅ(๐‘ก)+โŽกโŽขโŽขโŽฃโˆ’1+๐œŒ10โˆ’1โˆ’1+๐œŒ2โŽคโŽฅโŽฅโŽฆ๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก)),(35) where |๐›ฟ1|โ‰ค1.6, |๐›ฟ2|โ‰ค0.05, |๐œŒ1|โ‰ค0.1, |๐œŒ2|โ‰ค0.3, and suppose that ๐œ(๐‘ก) is continuous function. We handle the uncertainty as polytopic uncertainty and norm-bounded uncertainty, respectively. First, the uncertainty is handled as polytopic uncertainty. Then(1)if ๐œ(๐‘ก) is a constant, applying Corollary 2 in this paper yields the result ๐œ2=1.8130 for ๐œ1=๐œ2, which is less conservative than the result ๐œ2=1 given in Fridman and Shaked [10], the result ๐œ2=1.02 given in Han et al. [11] and the result ๐œ2=1.3500 given in Jiang and Han [6]; (2)if ๐œ1=0, we have ๐œ2=1.0238, which is larger than ๐œ2=0.7692 in derived in Fridman and Shaked [10],๐œ2=0.8654 in derived in Jiang and Han [8] and ๐œ2=0.8748 in derived in Jiang and Han [6], respectively;(3)if 0<๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2, Table 2 lists the maximum allowable upper bounds of the time-varying delay for different lower bounds the delay by Corollary 2. From Table 2, we can see that our results are larger than the previous ones.


๐œ 1 Jiang and Han [6] This work for ๐‘ = 2 , ๐พ = 2

00.87481.0238 ( ๐‘ = 0 )
0.50.99761.1887
1.01.19491.4511
1.20511.28701.5361
๐œ 2 = ๐œ 1 1.35001.8130 (K = 0)

Next, the uncertainty is handled as norm-bounded uncertainty, and we choose ๐ท,๐ธ1 and ๐ธ2 as๐ท=โŽกโŽขโŽขโŽฃ0.01000.04โŽคโŽฅโŽฅโŽฆ,๐ธ1=โŽกโŽขโŽขโŽฃ160001.25โŽคโŽฅโŽฅโŽฆ,๐ธ2=โŽกโŽขโŽขโŽฃ10007.5โŽคโŽฅโŽฅโŽฆ.(36)

Table 3 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different ๐œ1, and those in Jiang and Han [6, 8].(1)If ๐œ1=0, we have ๐œ2=0.9442 for ๐‘=0 and any integer ๐พโ‰ฅ2, which is larger than ๐œ2=0.8654 in derived in He et.al [5] and same as ๐œ2=0.9442 in derived in Jiang and Han [6].(2)If 0<๐œ1โ‰ค๐œ(๐‘ก)โ‰ค๐œ2 and ๐œ1โ‰ ๐œ2, By using Corollary 3, our results listed in the second last column for ๐‘=2,๐พ=2 and in the last column for ๐‘=3,๐พ=3, which is larger than those in derived in Jiang and Han [6].(3)If ๐œ(๐‘ก) is a constant, this is ๐œ1=๐œ2, we have ๐œ2=1.4368 for ๐‘=2,๐พ=0; 1.4810 for ๐‘=3,๐พ=0.


๐œ 1 Jiang and Han [8]Jiang and Han [6]This work for ๐‘ = 2 , ๐พ = 2 This work for ๐‘ = 3 , ๐พ = 3

0.00.86540.94420.9442 ( ๐‘ = 0 ) 0.9442 ( ๐‘ = 0 )
0.20.90970.97570.98590.9864
0.40.95751.02081.02801.0307
0.61.01041.07951.08291.0888
0.81.06921.15001.15061.1601
1.01.13361.23081.23021.2432
1.2051 โ€”1.32311.32231.3389
๐œ 1 = ๐œ 2 1.20511.43801.4368 ( ๐พ = 0 ) 1.4810 ( ๐พ = 0 )

It is clear that proposed stability criteria in this paper can significantly improve some exiting results in the literature.

Remark 5. From Tables 2 and 3, we can find that the conservatism using the criterion derived for the poly-topic uncertainty is less than using the criterion derived for norm-bounded uncertainty, which is distinguished to the result derived in Jiang and Han [6], and there are some reverse results, for there are some fixed Lyapunov matrices in Jiang and Han [6] for different LMIs. Moreover, the conservatism can be reduced for ๐œ1โ‰ 0 by increasing divisions of the intervals [โˆ’๐œ1,0] or [โˆ’๐œ2,โˆ’๐œ1] in this paper.

5. Conclusion

This note presents some new stability criteria for the interval time-varying delay system with a poly-topic uncertainty and a linear fractional norm-bounded uncertainty. We have proposed a new Lyapunov-Krasovskii functional, which is based on dividing intervals [โˆ’๐œ1,0] and [โˆ’๐œ2,โˆ’๐œ1] into ๐‘ and ๐พ divisions, respectively, to derive some new stability criteria. Numerical examples show that our criteria would be less conservative along with ๐‘,๐พ increased and demonstrate our criteria are less conservative than previous ones. Furthermore, based on time-delay system stability results, some future research can be focused on designing feedback controller, which can promise system robust stabilization and satisfy some system performance.

Acknowledgments

This work is supported by NSFC Project 11071087, NSFC-Guangdong Joint Foundation Key Project U0735003, and Guangdong Province Natural Science Foundation of China Project 06105413 and DGUT-NSYF Project 2010ZQ23.

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Copyright ยฉ 2011 Boren Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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