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Journal of Control Science and Engineering
Volume 2011 (2011), Article ID 938749, 9 pages
http://dx.doi.org/10.1155/2011/938749
Research Article

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

College of Computer, Dongguan University of Technology, Dongguan, Guangdong 523808, China

Received 7 December 2010; Accepted 22 July 2011

Academic Editor: Onur Toker

Copyright © 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.

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 ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),(1) where 𝑥(𝑡)𝑅𝑛 is the state vector, and 𝑢(𝑡)𝑅𝑝 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 𝜏sc between the sensor and the controller and computational delay 𝜏𝑐 in the controller, which is shown in Figure 1, the following control law is employed for the system (1) 𝑢𝑡+=𝐾𝑝𝑥𝑡𝜏sc𝑘𝜏𝑐𝑘,𝑡𝑘+𝜏sc𝑘+𝜏𝑐𝑘,,𝑘𝑁,(2) where is the sampling period, 𝑁 is the set of nonnegative integers, and 𝐾 is the controller gain to be determined. Second, substituting (2) into (1) yields the closed-loop system by considering the communication delay 𝜏ca between the controller and the actuator ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾𝑝𝑥(𝑘),𝑡𝑘+𝜏𝑘,(𝑘+1)+𝜏𝑘+1,𝑘𝑁,(3) where the time-delay 𝜏𝑘=𝜏sc𝑘+𝜏𝑐𝑘+𝜏ca𝑘>0 denotes the time from the instant 𝑘 when sensor nodes sample sensor data from a plant to the instant when actuators transfer data to the plant. Since 𝑥(𝑘)=𝑥(𝑡(𝑡𝑘)), defining 𝜏(𝑡)=𝑡𝑘, 𝑡[𝑘+𝜏𝑘,(𝑘+1)+𝜏𝑘+1),forall𝑘𝑁, rewrite (3) as ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾𝑝𝑥(𝑡𝜏(𝑡)),(4) where 𝜏(𝑡) is piecewise linear with derivative .𝜏(𝑡)=1 for 𝑡𝑘+𝜏𝑘, and 𝜏(𝑡) is discontinuous at the points 𝑡=𝑘+𝜏𝑘,forall𝑘𝑁. It is clear that 𝜏𝑘𝜏(𝑡)+𝜏𝑘+1 for 𝑡[𝑘+𝜏𝑘,(𝑘+1)+𝜏𝑘+1), for  all 𝑘𝑁.

938749.fig.001
Figure 1: A networked control system.

Let 𝜏1=min𝑘𝑁𝜏𝑘>0 and 𝜏2=max𝑘𝑁{+𝜏𝑘+1}. Then we have 0<𝜏1𝜏(𝑡)𝜏2.(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 𝜏1>0 is essential for NCSs.

To cover the routine case of 0𝜏(𝑡)𝜏2, we consider 𝜏(𝑡) as a uniformly continuous time-varying function satisfying 0𝜏1𝜏(𝑡)𝜏2.

Throughout this paper, we will analyze the following two scenarios of the time-varying delay 𝜏(𝑡).

Case 1. 𝜏(𝑡) is a differentiable function satisfying 0𝜏1𝜏(𝑡)𝜏2,|||.𝜏(𝑡)|||𝜇,𝑡0.(6)

Case 2. 𝜏(𝑡) is a continuous function satisfying 0𝜏1𝜏(𝑡)𝜏2,𝑡0,(7) where 𝜏1 and 𝜏2 are the lower and upper delay bounds, respectively, and 𝜏1,𝜏2 and 𝜇 are constants.

Remark 1. When 𝜏1=0, the interval delay becomes routine delay. When 𝜇 is zero, that is, 𝜏1=𝜏2, 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. 𝑅𝑛 denotes the 𝑛-dimensional Euclidean space, 𝑅𝑛×𝑚 is the set of 𝑛×𝑚 real matrices, 𝐼 is the identity matrix of appropriate dimensions, the notation 𝑋>0 (resp.,  𝑋0), for 𝑋𝑅𝑛×𝑛 means that the matrix 𝑋 is real positive definite (resp., positive semidefinite). For an arbitrary matrix 𝐵 and two symmetric matrices 𝐴 and 𝐶, 𝐴𝐵𝐶 denotes a symmetric matrix, where denotes a block matrix entry implied by symmetry.

2. System Descriptions and Preliminaries

Let 𝐵=𝐵𝐾𝑝 in (4), we have the following linear system with interval time-varying delay ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏(𝑡)),𝑥(𝑡)=𝜙(𝑡),𝑡𝜏2,𝜏1,(8) where 𝐴 and 𝐵 are the constant matrices with appropriate dimensions, 𝜙(𝑡) 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: 𝑉𝑡,𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝜏(𝑡)𝑥𝑇(𝜃)𝑍𝑥(𝜃)𝑑𝜃+𝑁𝑖=1𝑡(𝑖1)𝛿𝑡𝑖𝛿𝑥𝑇(𝜃)𝑄𝑖𝑥(𝜃)𝑑𝜃+𝐾𝑖=1𝑡𝜏1(𝑖1)𝜌𝑡𝜏1𝑖𝜌𝑥𝑇(𝜃)𝐺𝑖𝑥(𝜃)𝑑𝜃+𝑁𝑖=1(𝑖1)𝛿𝑖𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝛿𝑅𝑖̇𝑥(𝑣)𝑑𝑣𝑑𝜃+𝐾𝑖=1𝜏1(𝑖1)𝜌𝜏1𝑖𝜌𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝜌𝐻𝑖̇𝑥(𝑣)𝑑𝑣𝑑𝜃+𝜏1𝜏2𝑡𝑡+𝜃̇𝑥𝑇(𝑣)(𝛾𝑊)̇𝑥(𝑣)𝑑𝑣𝑑𝜃,(9) where 𝑃>0,  𝑍0,  𝑄𝑖>0,  𝑅𝑖>0(𝑖=1,,𝑁),  𝐺𝑖>0,  𝐻𝑖>0(𝑖=1,,𝐾),  𝑊>0 of appropriate dimensions, and 𝜏1=𝑁×𝛿,  𝜏2𝜏1=𝐾×𝜌,  𝛾=𝜏2𝜏1,  𝑁,  𝐾 are positive integer numbers of divisions of intervals [𝜏1,0] and [𝜏2,𝜏1], respectively, and 𝛿, 𝜌 are the length of each division. When 𝜏1=0 and 𝜏1=𝜏2, we assume that 𝑁=0 and 𝐾=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 𝑈𝑅𝑛×𝑛,𝑈=𝑈𝑇>0, scalar 𝜎>0, and vector function ̇𝑥[𝜎,0]𝑅𝑛 such that the following integration is well defined, then it holds that 𝜎𝑡𝑡𝜎̇𝑥𝑇(𝑤)𝑈̇𝑥(𝑤)𝑑𝑤𝑥(𝑡)𝑥(𝑡𝜎)𝑇𝑈𝑈𝑈×𝑥(𝑡)𝑥(𝑡𝜎).(10)
Applying the Lemma 1 yields the following new integral inequality for cross-product term.

Lemma 2 (see [4]). For any constant matrix 𝑈𝑅𝑛×𝑛,𝑈=𝑈𝑇>0, scalars 𝜎1𝜏(𝑡)𝜎2 and vector function ̇𝑥[𝜎2,𝜎1]𝑅𝑛 such that the following integration is well defined, then it holds that 𝜎2𝜎1𝑡𝜎1𝑡𝜎2̇𝑥𝑇(𝑤)𝑈̇𝑥(𝑤)𝑑𝑤𝜂𝑇(𝑡)Ω𝜂(𝑡),(11) where 𝜂(𝑡)=𝑥𝑡𝜎1𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜎2,Ω=𝑈𝑈02𝑈𝑈𝑈.(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 0𝜏1𝜏2 and 𝜇, the nominal system (8) satisfying (6) is asymptotically stable if there exit real symmetric matrices 𝑃>0,  𝑍0,  𝑄𝑖>0,  𝑅𝑖>0(𝑖=1,,𝑁),  𝐺𝑖>0,  𝐻𝑖>0(𝑖=1,,𝐾),  𝑊>0, and any matrix 𝑆=[𝑆𝑇1𝑆𝑇2𝑆𝑇𝑁+𝐾+3]𝑇 of appropriate dimensions such that the following LMI holds: Ξ=Ξ1+𝑆𝐴𝑑+𝑆𝐴𝑑𝑇<0,(13) where Ξ1=Ξ110𝑅10000000𝑃Ξ22000𝑊000𝑊0Φ1𝑅20000000Φ20000000Φ𝑁1𝑅𝑁00000Φ𝑁𝐻10000Ψ1𝐻2000Ψ2000Ψ𝐾1𝐻𝐾0Ψ𝐾0Υ,(14)Ξ11=𝑍+𝑄1𝑅1,Ξ22=(1𝜇)𝑍2𝑊,𝐴𝑑=𝐴𝐵000𝑁+𝐾𝐼,Φ𝑖=𝑄𝑖+1𝑄𝑖𝑅𝑖𝑅𝑖+1(𝑖=1,2,,𝑁1),Φ𝑁=𝑄𝑁𝑅𝑁𝑊+𝐺1𝐻1,Ψ𝑖=𝐺𝑖+1𝐺𝑖𝐻𝑖𝐻𝑖+1(𝑖=1,2,,𝐾1),Ψ𝐾=𝐺𝐾𝐻𝐾𝑊,Υ=𝑁𝑖=1𝛿2𝑅𝑖+𝐾𝑖=1𝜌2𝐻𝑖+𝛾2𝑊.(15)

Proof. Taking the time derivative of 𝑉(𝑡,𝑥𝑡) with respect to 𝑡 along the trajectory of (8) yields ̇𝑉𝑡,𝑥𝑡2𝑥𝑇(𝑡)𝑃̇𝑥(𝑡)+𝑥𝑇(𝑡)𝑍𝑥(𝑡)(1𝜇)𝑥𝑇(𝑡𝜏(𝑡))𝑍𝑥(𝑡𝜏(𝑡))+𝑁𝑖=1𝑥𝑇(𝑡(i1)𝛿)𝑄𝑖𝑥(𝑡(𝑖1)𝛿)𝑁𝑖=1𝑥𝑇(𝑡𝑖𝛿)𝑄𝑖𝑥(𝑡𝑖𝛿)+𝐾𝑖=1𝑥𝑇𝑡𝜏1(𝑖1)𝜌𝐺𝑖𝑥𝑡𝜏1(𝑖1)𝜌𝐾𝑖=1𝑥𝑇𝑡𝜏1𝑖𝜌𝐺𝑖𝑥𝑡𝜏1𝑖𝜌+̇𝑥𝑇(𝑡)𝑁𝑖=1𝛿2𝑅𝑖+𝐾𝑖=1𝜌2𝐻𝑖+𝛾2𝑊̇𝑥(𝑡)𝑁𝑖=1𝑡(𝑖1)𝛿𝑡𝑖𝛿̇𝑥𝑇(𝜃)𝛿𝑅𝑖̇𝑥(𝜃)𝑑𝜃𝐾𝑖=1𝑡𝜏1(𝑖1)𝜌𝑡𝜏1𝑖𝜌̇𝑥𝑇(𝜃)𝜌𝐻𝑖̇𝑥(𝜃)𝑑𝜃𝑡𝜏1𝑡𝜏2̇𝑥𝑇(𝜃)(𝛾𝑊)̇𝑥(𝜃)𝑑𝜃.(16)
Applying the Lemma 1, the following inequalities hold: 𝑁𝑖=1𝑡(𝑖1)𝛿𝑡𝑖𝛿̇𝑥𝑇(𝜃)𝛿𝑅𝑖̇𝑥(𝜃)𝑑𝜃𝑁𝑖=1𝑥(𝑡(𝑖1)𝛿)𝑥(𝑡𝑖𝛿)𝑇𝑅𝑖𝑅𝑖𝑅𝑖𝑥(𝑡(𝑖1)𝛿)𝑥(𝑡𝑖𝛿),𝐾𝑖=1𝑡𝜏1(𝑖1)𝜌𝑡𝜏1𝑖𝜌̇𝑥𝑇(𝜃)𝜌𝐻𝑖̇𝑥(𝜃)𝑑𝜃𝐾𝑖=1𝑥(𝑡𝜏1(𝑖1)𝜌)𝑥𝑡𝜏1𝑖𝜌𝑇𝐻𝑖𝐻𝑖𝐻𝑖×𝑥𝑡𝜏1(𝑖1)𝜌𝑥𝑡𝜏1𝑖𝜌.(17)
Applying the Lemma 2, the following inequality holds: 𝑡𝜏1𝑡𝜏2̇𝑥𝑇(𝜃)(𝛾𝑊)̇𝑥(𝜃)𝑑𝜃𝑥𝑡𝜏1𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜏2𝑇𝑊𝑊02𝑊𝑊𝑊×𝑥𝑡𝜏1𝑥(𝑡𝜏(𝑡))𝑥𝑡𝜏2.(18)
For any matrix 𝑆=[𝑆𝑇1𝑆𝑇2𝑆𝑇𝑁+𝐾+3]𝑇 of appropriate dimensions, the following equality holds: 2𝜉𝑇(𝑡)𝑆[𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏(𝑡))̇𝑥(𝑡)]=0,(19) where is defined in the top of next page, 𝜉𝑇(𝑡)=𝑥𝑇(𝑡)𝑥𝑇(𝑡𝜏(𝑡))𝜉𝑇1(𝑡)𝜉𝑇2(𝑡)̇𝑥(𝑡),𝜉𝑇1(𝑡)=𝑥𝑇(𝑡𝛿)𝑥𝑇(𝑡2𝛿)𝑥𝑇(𝑡(𝑁1)𝛿)𝑥𝑇𝑡𝜏1,𝜉𝑇2(𝑡)=𝑥𝑇𝑡𝜏1𝜌𝑥𝑇𝑡𝜏12𝜌𝑥𝑇𝑡𝜏1(𝐾1)𝜌𝑥𝑇𝑡𝜏2.(20)
Considering (16)–(19) together, we have ̇𝑉(𝑡,𝑥𝑡)𝜉𝑇(𝑡)Ξ𝜉(𝑡). If (13) is satisfied, then ̇𝑉(𝑡,𝑥𝑡)𝜆𝑥𝑇(𝑡)𝑥(𝑡) for some scalar 𝜆>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 𝑍0 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 0𝜏1𝜏2, the nominal system (8) satisfying (7) is asymptotically stable, if there exit real symmetric matrices 𝑃>0, 𝑄𝑖>0,  𝑅𝑖>0(𝑖=1,,𝑁), 𝐺𝑖>0,  𝐻𝑖>0(𝑖=1,,𝐾), 𝑊>0 and any matrix 𝑆 of appropriate dimensions such that the LMI (13) holds with 𝑍0.

Remark 2. For system (8) with the routine delay case described by 𝜏1=0 and 𝜏2>0, that is, 𝑁=0,𝐾>0, the corresponding Lyapunov-Krasovskii functional reduces to 𝑉𝑡,x𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑡𝑡𝜏(𝑡)𝑥𝑇(𝜃)𝑍𝑥(𝜃)𝑑𝜃+𝐾𝑖=1𝑡(𝑖1)𝜌𝑡𝑖𝜌𝑥𝑇(𝜃)𝐺𝑖𝑥(𝜃)𝑑𝜃+𝐾𝑖=1(𝑖1)𝜌𝑖𝜌𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝜌𝐻𝑖̇𝑥(𝑣)𝑑𝑣𝑑𝜃+0𝜏2𝑡𝑡+𝜃̇𝑥𝑇(𝑣)(𝛾𝑊)̇𝑥(𝑣)𝑑𝑣𝑑𝜃.(21)
And for system (8) with the constant delay case described by 𝜏1=𝜏(𝑡)=𝜏2>0, that is, 𝑁>0,𝐾=0, the corresponding Lyapunov-Krasovskii functional reduces to 𝑉𝑡,𝑥𝑡=𝑥𝑇(𝑡)𝑃𝑥(𝑡)+𝑁𝑖=1𝑡(𝑖1)𝛿𝑡𝑖𝛿𝑥𝑇(𝜃)𝑄𝑖𝑥(𝜃)𝑑𝜃+𝑁𝑖=1(𝑖1)𝛿𝑖𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝛿𝑅𝑖̇𝑥(𝑣)𝑑𝑣𝑑𝜃.(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 𝐴 and 𝐵 in (8) can be expressed as [𝐴𝐵]=𝑟𝑗=1𝜁𝑗𝐴𝑗𝐵𝑗,(23) where 𝑟𝑗=1𝜁𝑗=1,0𝜁𝑗1. Based on Theorem 1, we can easily obtain the following result.

Theorem 2. For some given scalars 0𝜏1𝜏2 and 𝜇, the system described by (8) satisfying (6) subject to polytopic uncertainty (23) is robustly stable, if there exit real symmetric matrices 𝑃𝑗>0,𝑍𝑗0,𝑄𝑖𝑗>0,𝑅𝑖𝑗>0(𝑖=1,,𝑁), 𝐺𝑖𝑗>0,𝐻𝑖𝑗>0(𝑖=1,,𝐾), 𝑊𝑗>0, and any matrix 𝑆 of appropriate dimensions such that the following LMIs hold for 𝑗=1,2,,𝑟: Ξ(𝑗)=Ξ(𝑗)1+𝑆𝐴(𝑗)𝑑+𝑆𝐴(𝑗)𝑑𝑇<0,(24) where Ξ(𝑗)1=Ξ(𝑗)110𝑅1𝑗0000000𝑃𝑗Ξ(𝑗)22000𝑊𝑗000𝑊𝑗0Φ(𝑗)1𝑅2𝑗0000000Φ(𝑗)20000000Φ(𝑗)𝑁1𝑅𝑁𝑗00000Φ(𝑗)𝑁𝐻1𝑗0000Ψ(𝑗)1𝐻2𝑗000Ψ(𝑗)2000Ψ(𝑗)𝐾1𝐻𝐾𝑗0Ψ(𝑗)𝐾0Υ(𝑗),(25)𝑆 is defined in Theorem 1, and Ξ(𝑗)11=𝑍𝑗+𝑄1𝑗𝑅1𝑗,Ξ(𝑗)22=(1𝜇)𝑍𝑗2𝑊𝑗,𝐴(𝑗)𝑑=𝐴𝑗𝐵𝑗000𝑁+𝐾𝐼,Φ(𝑗)𝑖=𝑄𝑖+1,𝑗𝑄𝑖𝑗𝑅𝑖𝑗𝑅𝑖+1,𝑗(𝑖=1,2,,𝑁1),Φ(𝑗)𝑁=𝑄𝑁𝑗𝑅𝑁𝑗𝑊𝑗+𝐺1𝑗𝐻1𝑗,Ψ(𝑗)𝑖=𝐺𝑖+1,𝑗𝐺𝑖𝑗𝐻𝑖𝑗𝐻𝑖+1,𝑗(𝑖=1,2,,𝐾1),Ψ(𝑗)𝐾=𝐺𝐾𝑗𝐻𝐾𝑗𝑊𝑗,Υ(𝑗)=𝑁𝑖=1𝛿2𝑅𝑖𝑗+𝐾𝑖=1𝜌2𝐻𝑖𝑗+𝛾2𝑊𝑗.(26)

Proof. Construct the following parameter-dependent Lyapunov-Krasovskii functional: 𝑉𝑢𝑡,𝑥𝑡=𝑟𝑗=1𝑥𝑇(𝑡)𝜁𝑗𝑃𝑗𝑥(𝑡)+𝑟𝑗=1𝑡𝑡𝜏(𝑡)𝑥𝑇(𝜃)𝜁𝑗𝑍𝑗𝑥(𝜃)𝑑𝜃+𝑟𝑗=1𝑁𝑖=1𝑡(𝑖1)𝛿𝑡𝑖𝛿𝑥𝑇(𝜃)𝜁𝑗𝑄𝑖𝑗𝑥(𝜃)𝑑𝜃+𝑟𝑗=1𝐾𝑖=1𝑡𝜏1(𝑖1)𝜌𝑡𝜏1𝑖𝜌𝑥𝑇(𝜃)𝜁𝑗𝐺𝑖𝑗𝑥(𝜃)𝑑𝜃+𝑟𝑗=1𝑁𝑖=1(i1)𝛿𝑖𝛿𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝜁𝑗𝛿𝑅𝑖𝑗̇𝑥(𝑣)𝑑𝑣𝑑𝜃+𝑟𝑗=1𝐾𝑖=1𝜏1(𝑖1)𝜌𝜏1𝑖𝜌𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝜁𝑗𝜌𝐻𝑖𝑗̇𝑥(𝑣)𝑑𝑣𝑑𝜃+𝑟𝑗=1𝜏1𝜏2𝑡𝑡+𝜃̇𝑥𝑇(𝑣)𝜁𝑗𝛾𝑊𝑗̇𝑥(𝑣)𝑑𝑣𝑑𝜃,(27) where 𝑃𝑗>0,  𝑍𝑗0,  𝑄𝑖𝑗>0,  𝑅𝑖𝑗>0(𝑖=1,,𝑁),  𝐺𝑖𝑗>0,  𝐻𝑖𝑗>0(𝑖=1,,𝐾),  𝑊𝑗>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: ̇𝑉𝑢𝑡,𝑥𝑡𝑟𝑗=1𝜉𝑇(𝑡)𝜁𝑗Ξ(𝑗)𝜉(𝑡),(28) where 𝜉(𝑡) is defined in the formula (19), then ̇𝑉𝑢(𝑡,𝑥𝑡)𝜆𝑥𝑇(𝑡)𝑥(𝑡) for some scalar 𝜆>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 0𝜏1𝜏2, the system described by (8) satisfying (7) subject to polytopic uncertainty (23) is robustly stable if there exit real symmetric matrices 𝑃𝑗>0,  𝑄𝑖𝑗>0,  𝑅𝑖𝑗>0(𝑖=1,,𝑁),  𝐺𝑖𝑗>0,  𝐻𝑖𝑗>0(𝑖=1,,𝐾),  𝑊𝑗>0, and any matrix 𝑆 of appropriate dimensions such that the LMIs (24) hold with 𝑍𝑗0 for 𝑗=1,2,,𝑟.

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 𝑟 different LMIs, which are distinguished with Proposition 7 in Jiang and Han [6], in which used fixed Lyapunov matrices for 𝑟 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 𝐴 and 𝐵 have parameter perturbations as Δ𝐴(𝑡) and Δ𝐵(𝑡), which are in the form of [Δ𝐴(𝑡)Δ𝐵(𝑡)]=𝐷(𝐼𝐹(𝑡)𝑀)1𝐹(𝑡)𝐸1𝐸2,(29) where 𝐷,𝑀,𝐸1, and 𝐸2 are known real constant matrices of appropriate dimensions, and 𝑀𝑇𝑀<𝐼; 𝐹(𝑡) is an unknown matrix function with Lesbesgue measurable elements satisfying 𝐹𝑇(𝑡)𝐹(𝑡)𝐼.

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

Theorem 3. For some given scalars 0𝜏1𝜏2 and 𝜇, 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 𝑃>0,  𝑍0, 𝑄𝑖>0,  𝑅𝑖>0(𝑖=1,,𝑁), 𝐺𝑖>0,  𝐻𝑖>0(𝑖=1,,𝐾), 𝑊>0, any matrix 𝑆 of appropriate dimensions, and a scalar 𝜀>0 such that the following LMI holds: Ξ𝑆𝐷𝜀Γ𝜀𝐼𝜀𝑀𝑇𝜀𝐼<0,(30) where Γ𝑇=𝐸1𝐸2000𝑁+𝐾+1,  Ξ is defined in the formula (13), and 𝑆 is defined in Theorem 1.

Proof. Replacing 𝐴 and 𝐵 in (13) with 𝐴+𝐷(𝐼𝐹(𝑡)𝑀)1𝐹(𝑡)𝐸1 and 𝐵+𝐷(𝐼𝐹(𝑡)𝑀)1𝐹(𝑡)𝐸2, respectively, and multiply both sides of the resulting matrix by vectors 𝑥𝑖 for 𝑖=1,2,,𝑁+𝐾+3. Next, define 𝑝=(𝐼𝐹(𝑡)𝑀)1𝐹(𝑡)𝐸1𝑥1,𝑞=(𝐼𝐹(𝑡)𝑀)1𝐹(𝑡)𝐸2𝑥2. Then, we have the following condition for the admissible uncertainty 𝐹(𝑡): 𝛽𝑇Ξ𝑆𝐷0𝛽<0,(31) where 𝛽𝑇=[𝑥𝑇1,𝑥𝑇2,,𝑥𝑇𝑁+𝐾+3,(𝑝+𝑞)𝑇]. It is easy to know that 𝑝 and 𝑞 can be rewritten as 𝑝=𝐹(𝑡)𝑀𝑝+𝐹(𝑡)𝐸1𝑥1 and 𝑞=𝐹(𝑡)𝑀𝑞+𝐹(𝑡)𝐸2𝑥2, since 𝐹𝑇(𝑡)𝐹(𝑡)𝐼, it is obvious that the following inequality holds: (𝑝+𝑞)𝑇(𝑝+𝑞)𝑥1𝑥2𝑝+𝑞𝑇𝐸𝑇1𝐸1𝐸𝑇1𝐸2𝐸𝑇1𝑀𝐸𝑇2𝐸2𝐸𝑇2𝑀𝑀𝑇𝑀𝑥1𝑥2𝑝+𝑞.(32) Applying 𝑆 procedure, both inequalities (31) and (32) are true if and only if there is a 𝜀>0, promising that the following condition holds: 𝛽𝑇Ξ+𝜀ΓΓ𝑇𝑆𝐷+𝜀Γ𝑀𝜀𝑀𝑇𝑀𝜀𝐼𝛽<0.(33) By Schur complement, for any 𝛽0, (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 0𝜏1𝜏2, 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 𝑃>0, 𝑄𝑖>0,  𝑅𝑖>0(𝑖=1,,𝑁), 𝐺𝑖>0,  𝐻𝑖>0(𝑖=1,,𝐾), 𝑊>0, any matrix 𝑆 of appropriate dimensions, and a scalar 𝜀>0 such that the LMI (30) holds with 𝑍0.

Remark 4. It is clear to see that if we set 𝑀=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: ̇𝑥(𝑡)=2000.9𝑥(𝑡)+1011𝑥(𝑡𝜏(𝑡)).(34) Table 1 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different 𝜏1, and those in He et al. [5], Jiang and Han [6], and Shao [7].(1)If 𝜏1=0, by using Corollary 1, we have 𝜏2=1.3454 for 𝑁=0 and any integer 𝐾2, which is larger than 𝜏2=1.34 in derived in He et al. [5] and close to 𝜏2=1.3453 in derived in Jiang and Han [6].(2)If 0<𝜏1𝜏(𝑡)𝜏2 and 𝜏1𝜏2. By using Corollary 1 for 𝑁=2,𝐾=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 𝑁=3,𝐾=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 𝜏2=4.6007 for 𝜏1=4.4697, not have 4.6009a.(4)For the constant delay 𝜏1=𝜏2, that is, 𝐾=0. By using Corollary 1, we have 𝜏max=4.4721 for 𝑁=1; 5.7175 for 𝑁=2; 5.9677 for 𝑁=3; 6.0983 for 𝑁=5; 6.1537 for 𝑁=10, which is close to analytical delay limit for stability 6.17258.

tab1
Table 1: MAUB of the time-varying delay for different 𝜇.

Example 2. Consider the following time-delay system: ̇𝑥(𝑡)=2+𝛿1001+𝛿2𝑥(𝑡)+1+𝜌1011+𝜌2𝑥(𝑡𝜏(𝑡)),(35) where |𝛿1|1.6, |𝛿2|0.05, |𝜌1|0.1, |𝜌2|0.3, and suppose that 𝜏(𝑡) 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 𝜏(𝑡) is a constant, applying Corollary 2 in this paper yields the result 𝜏2=1.8130 for 𝜏1=𝜏2, which is less conservative than the result 𝜏2=1 given in Fridman and Shaked [10], the result 𝜏2=1.02 given in Han et al. [11] and the result 𝜏2=1.3500 given in Jiang and Han [6]; (2)if 𝜏1=0, we have 𝜏2=1.0238, which is larger than 𝜏2=0.7692 in derived in Fridman and Shaked [10],𝜏2=0.8654 in derived in Jiang and Han [8] and 𝜏2=0.8748 in derived in Jiang and Han [6], respectively;(3)if 0<𝜏1𝜏(𝑡)𝜏2, 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.

tab2
Table 2: MAUB of the time varying delay for different 𝜏1.

Next, the uncertainty is handled as norm-bounded uncertainty, and we choose 𝐷,𝐸1 and 𝐸2 as𝐷=0.01000.04,𝐸1=160001.25,𝐸2=10007.5.(36)

Table 3 lists the maximum allowable upper bound (MAUB) of the time-varying delay for different 𝜏1, and those in Jiang and Han [6, 8].(1)If 𝜏1=0, we have 𝜏2=0.9442 for 𝑁=0 and any integer 𝐾2, which is larger than 𝜏2=0.8654 in derived in He et.al [5] and same as 𝜏2=0.9442 in derived in Jiang and Han [6].(2)If 0<𝜏1𝜏(𝑡)𝜏2 and 𝜏1𝜏2, By using Corollary 3, our results listed in the second last column for 𝑁=2,𝐾=2 and in the last column for 𝑁=3,𝐾=3, which is larger than those in derived in Jiang and Han [6].(3)If 𝜏(𝑡) is a constant, this is 𝜏1=𝜏2, we have 𝜏2=1.4368 for 𝑁=2,𝐾=0; 1.4810 for 𝑁=3,𝐾=0.

tab3
Table 3: MAUB of the time-varying delay for different 𝜇.

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 𝜏10 by increasing divisions of the intervals [𝜏1,0] or [𝜏2,𝜏1] 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 [𝜏1,0] and [𝜏2,𝜏1] into 𝑁 and 𝐾 divisions, respectively, to derive some new stability criteria. Numerical examples show that our criteria would be less conservative along with 𝑁,𝐾 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.

References

  1. K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Brikhauser, Boston, Mass, USA, 2003.
  2. D. Yue, Q. L. Han, and C. Peng, “State feedback controller design of networked control systems,” IEEE Transactions on Circuits and Systems II, vol. 51, no. 11, pp. 640–644, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. Q. L. Han, “Absolute stability of time-delaysy stems with sector-bounded nonlinearity,” Automatica, vol. 41, no. 12, pp. 2171–2176, 2005.
  4. C. Peng and Y. C. Tian, “Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay,” IET Control Theory and Applications, vol. 2, no. 9, pp. 752–761, 2008. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. He, Q. G. Wang, L. Xie, and C. Lin, “Further improvement of free-weighting matrices technique for systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 293–299, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. X. Jiang and Q. L. Han, “New stability criteria for linear systems with interval time-varying delay,” Automatica, vol. 44, no. 10, pp. 2680–2685, 2008. View at Publisher · View at Google Scholar · View at Scopus
  7. H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. X. Jiang and Q. L. Han, “Delay-dependent robust stability for uncertain linear systems with interval time-varying delay,” Automatica, vol. 42, no. 6, pp. 1059–1065, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Sytem and Control Theory, SIAM, Philadelphia, Pa, USA, 1994.
  10. E. Fridman and U. Shaked, “Delay-dependent stability and H control: constant and time-varying delays,” International Journal of Control, vol. 76, no. 1, pp. 48–60, 2003. View at Publisher · View at Google Scholar
  11. Q. L. Han, X. H. Yu, and K. Q. Gu, “Improved stability criteria and controller design for linear neutral systems,” IEEE Transactions on Automatic Control, vol. 49, no. 12, pp. 2281–2285, 2004.