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Journal of Applied Mathematics
Volume 2012, Article ID 689820, 13 pages
http://dx.doi.org/10.1155/2012/689820
Research Article

Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components

Tiejun Li1,2 and Junkang Tian1,2,3

1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China
2School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China
3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 27 November 2011; Revised 26 December 2011; Accepted 27 December 2011

Academic Editor: Chong Lin

Copyright © 2012 Tiejun Li and Junkang Tian. 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 delay-dependent stability for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

1. Introduction

Robust stability of dynamic interval systems covering interval matrices and interval polynomials has attracted considerable attention over last decades. Reference [1] presents some necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. It is well known that time delay frequently occurs in many industrial and engineering systems, such as manufacturing systems, telecommunication, and economic systems, and is a major cause of instability and poor performance. Over the past decades, much efforts have been invested in the stability analysis of time-delay systems [216]. Reference [2] deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete time systems with state delay. Reference [3] deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed linear dynamic systems. However, almost all the reported results mentioned above on time-delay systems are based on the following basic mathematical model:̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴𝑑𝑥(𝑡𝑑(𝑡)),(1.1) where 𝑑(𝑡) is a time delay in the state 𝑥(𝑡), which is often assumed to be either constant or time-varying satisfying certain conditions, for example,0𝑑(𝑡)̇𝑑<,𝑑(𝑡)𝜏<.(1.2)

Sometimes in practical situations, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. Thus, in recent papers [15, 16], a new model for time-delay systems with multiple additive time-varying delay components has been proposed: ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴𝑑𝑥𝑡𝑛𝑖=1𝑑𝑖(,𝑡)(1.3)0𝑑𝑖(𝑡)𝑑𝑖̇𝑑<,𝑖(𝑡)𝜏𝑖<.(1.4) To make the stability analysis simpler, we proceed by considering the system (1.3) with two additive delay components as follows: ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴𝑑𝑥𝑡𝑑1(𝑡)𝑑2,[].(𝑡)𝑥(𝑡)=𝜙(𝑡),𝑡𝑑,0(1.5) Here, 𝑥(𝑡)𝑛 is the state vector; 𝑑1(𝑡) and 𝑑2(𝑡) represent the two delay components in the state, and we denote 𝑑(𝑡)=𝑑1(𝑡)+𝑑2(𝑡); 𝐴, 𝐴𝑑 are system matrices with appropriate dimensions. It is assumed that0𝑑1(𝑡)𝑑1̇𝑑<,1(𝑡)𝜏1<,0𝑑2(𝑡)𝑑2̇𝑑<,2(𝑡)𝜏2<,(1.6) and 𝑑=𝑑1+𝑑2, 𝜏=𝜏1+𝜏2. 𝜙(𝑡) is the initial condition on the segment [𝑑,0].

The purpose of our paper is to derive new stability conditions under which system (1.5) is asymptotically stable for all delays 𝑑1(𝑡) and 𝑑2(𝑡) satisfying (1.6). One possible approach to check the stability of this system is to simply combine 𝑑1(𝑡) and 𝑑2(𝑡) into one delay 𝑑(𝑡) with0𝑑(𝑡)𝑑1+𝑑2̇<,𝑑(𝑡)𝜏1+𝜏2<.(1.7) Then, the system (1.5) becomes ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴𝑑𝑥[].𝑥(𝑡𝑑(𝑡)),(𝑡)=𝜙(𝑡),𝑡𝑑,0(1.8) By using some existing stability conditions, the stability of system (1.8) can be readily checked. As discussed in [15, 16], however, since this approach does not make full use of the information on 𝑑1(𝑡) and 𝑑2(𝑡), it would be inevitably conservative for some situations. Recently, some new delay-dependent stability criteria have been obtained for system (1.5) in [15, 16], by making full use of the information on 𝑑1(𝑡) and 𝑑2(𝑡). However, the stability result is conservative because of overly bounding some integrals appearing in the derivative of the Lyapunov functional. On the one hand, the integral 𝑡𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 in [15] was enlarged as 𝑡𝑡𝑑1(𝑡)̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠, with 𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 discarded. On the other hand, some integrals were estimated conservatively. Take 𝑡𝑡𝑑1(𝑡)̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 as an example, by introducing0=2𝜁𝑇𝑆𝑥(𝑡)𝑥𝑡𝑑1(𝑡)𝑡𝑡𝑑1(𝑡)̇𝑥(𝑠)𝑑𝑠(1.9) with an appropriate vector 𝜁(𝑡) and a matrix 𝑆, respectively, it was estimated as2𝜁𝑇(𝑡)𝑆𝑥(𝑡)𝑥𝑡𝑑1(𝑡)+𝜁𝑇(𝑡)𝑑1𝑆𝑍11𝑆𝑇𝜁(𝑡)(1.10) with 𝑑1(𝑡)𝑆𝑍11𝑆𝑇 enlarged as 𝑑1𝑆𝑍11𝑆𝑇.

The problem of delay-dependent stability criterion for continuous systems with two additive time-varying delay components has been considered in this paper. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to indicate less conservatism of the stability results.

Definition 1.1. Let Φ1,Φ2,,Φ𝑁𝑚𝑛 be a given finite number of functions such that they have positive values in an open subset 𝐷 of 𝑚. Then, a reciprocally convex combination of these functions over 𝐷 is a function of the form 1𝛼1Φ1+1𝛼2Φ21++𝛼𝑁Φ𝑁𝐷𝑅𝑛,(1.11) where the real numbers 𝛼𝑖 satisfy 𝛼𝑖>0 and 𝑖𝛼𝑖=1.

The following Lemma 1.2 suggests a lower bound for a reciprocally convex combination of scalar positive functions Φ𝑖=𝑓𝑖.

Lemma 1.2 (See [10]). Let 𝑓1,𝑓2,,𝑓𝑁𝑚 have positive values in an open subset 𝐷 of 𝑚. Then, the reciprocally convex combination of 𝑓𝑖 over 𝐷 satisfies min𝛼𝑖𝛼𝑖>0,𝑖𝛼𝑖=1𝑖1𝛼𝑖𝑓𝑖(𝑡)=𝑖𝑓𝑖(𝑡)+max𝑔𝑖,𝑗(𝑡)𝑖𝑗𝑔𝑖,𝑗(𝑡)(1.12) subject to 𝑔𝑖,𝑗𝑅𝑚𝑅,𝑔𝑗,𝑖(𝑡)Δ𝑔𝑖,𝑗𝑓(𝑡),𝑖(𝑡)𝑔𝑖,𝑗𝑔(𝑡)𝑖,𝑗(𝑡)𝑓𝑗(𝑡)0.(1.13) In the following, we present a new stability criterion by a convex polyhedron method and Lemma 1.2.

2. Main Results

Theorem 2.1. System (1.5) with delays 𝑑1(𝑡) and 𝑑2(𝑡) satisfying (1.6) is asymptotically stable if there exist symmetric positive definite matrices𝑃,𝑄1,𝑄2,𝑄3,𝑄4,𝑄5,𝑄6, 𝑍,𝑍1,𝑍2 and any matrices𝑆12,𝑁,𝑀,𝐿,𝑆,𝑃1,𝑃2 with appropriate dimensions, such that the following LMIs hold: 𝑍𝑆12𝑍0,(2.1)𝐸13=𝐸𝑑1𝑁𝑑2𝐿𝑑1𝑍10𝑑2𝑍2<0,(2.2)𝐸14=𝐸𝑑1𝑁𝑑2𝑆𝑑1𝑍10𝑑2𝑍2<0,(2.3)𝐸23=𝐸𝑑1𝑀𝑑2𝐿𝑑1𝑍10𝑑2𝑍2<0,(2.4)𝐸24=𝐸𝑑1𝑀𝑑2𝑆𝑑1𝑍10𝑑2𝑍2<0,(2.5) where EE=11E12𝑆𝑇120000E18E22E230000𝐴𝑇𝑑𝑃𝑇2E3300000E440000𝑄4000E6600𝑄60E88++𝑁+𝐿00𝑀𝑁𝑀𝑆𝐿𝑆0𝑁+𝐿00𝑀𝑁𝑀𝑆𝐿𝑆0𝑇,𝐸11=𝑄1+𝑄2+𝑄3+𝑄4+𝑄5+𝑄6𝑍+𝑃1𝐴+𝐴𝑇𝑃𝑇1,𝐸12=𝑆𝑇12+𝑍+𝑃1𝐴𝑑,𝐸18=𝑃𝑃1+𝐴𝑇𝑃𝑇2,𝐸22=(1𝜏)𝑄12𝑍+𝑆12+𝑆𝑇12,𝐸23=𝑆𝑇12𝐸+𝑍,33=𝑄2𝑍,𝐸44=1𝜏1𝑄3,𝐸66=1𝜏2𝑄5,𝐸88=𝑑2𝑍+𝑑1𝑍1+𝑑2𝑍2𝑃2𝑃𝑇2.(2.6)

Proof. Construct a new Lyapunov functional candidate as 𝑉(𝑥(𝑡))=𝑉1(𝑥(𝑡))+𝑉2(𝑥(𝑡))+𝑉3(𝑥(𝑡))+𝑉4𝑉(𝑥(𝑡)),1(𝑥(𝑡))=𝑥𝑇𝑉(𝑡)𝑃𝑥(𝑡),2(𝑥(𝑡))=𝑡𝑡𝑑(𝑡)𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑1(𝑡)𝑥𝑇(𝑠)𝑄3+𝑥(𝑠)𝑑𝑠𝑡𝑡𝑑1𝑥𝑇(𝑠)𝑄4𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑2(𝑡)𝑥𝑇(𝑠)𝑄5𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑2𝑥𝑇(𝑠)𝑄6𝑥𝑉(𝑠)𝑑𝑠,3(𝑥(𝑡))=𝑑0𝑑𝑡𝑡+𝜃̇𝑥𝑇𝑉(𝑠)𝑍̇𝑥(𝑠)𝑑𝑠𝑑𝜃,4(𝑥(𝑡))=0𝑑1𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠𝑑𝜃+0𝑑2𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑑𝜃.(2.7)

Remark 2.2. Our paper fully uses the information about 𝑑(𝑡),𝑑1(𝑡), and 𝑑2(𝑡), but [15, 16] only use the information about 𝑑1(𝑡) and 𝑑2(𝑡), when constructing the Lyapunov functional 𝑉(𝑥(𝑡)). So the Lyapunov functional in our paper is more general than that in [15, 16], and the stability criteria in our paper may be more applicable.
The time derivative of 𝑉(𝑥(𝑡)) along the trajectory of system (1.5) is given by ̇𝑉1(𝑥(𝑡))=2𝑥𝑇̇𝑉(𝑡)𝑃̇𝑥(𝑡),(2.8)2(𝑥(𝑡))=𝑥𝑇𝑄(𝑡)1+𝑄2+𝑄3+𝑄4+𝑄5+𝑄6𝑥(𝑡)(1𝜏)𝑥𝑇(𝑡𝑑(𝑡))𝑄1𝑥(𝑡𝑑(𝑡))𝑥𝑇(𝑡𝑑)𝑄2𝑥(𝑡𝑑)1𝜏1𝑥𝑇𝑡𝑑1𝑄(𝑡)3𝑥𝑡𝑑1(𝑡)𝑥𝑇𝑡𝑑1𝑄4𝑥𝑡𝑑11𝜏2𝑥𝑇𝑡𝑑2𝑄(𝑡)5𝑥𝑡𝑑2(𝑡)𝑥𝑇𝑡𝑑2𝑄6𝑥𝑡𝑑2,̇𝑉(2.9)3(𝑥(𝑡))=𝑑2̇𝑥𝑇(𝑡)𝑍̇𝑥(𝑡)𝑑𝑡𝑑(𝑡)𝑡𝑑̇𝑥𝑇(𝑠)𝑍̇𝑥(𝑠)𝑑𝑠𝑑𝑡𝑡𝑑(𝑡)̇𝑥𝑇̇𝑉(𝑠)𝑍̇𝑥(𝑠)𝑑𝑠,(2.10)4(𝑥(𝑡))=̇𝑥𝑇𝑑(𝑡)1𝑍1+𝑑2𝑍2̇𝑥(𝑡)𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝑑1(𝑡)̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠𝑡𝑑2(𝑡)𝑡𝑑2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝑑2(𝑡)̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠.(2.11) The ̇𝑉3(𝑥(𝑡)) is upper bounded by ̇𝑉3(𝑥(𝑡))𝑑2̇𝑥𝑇𝑑(𝑡)𝑍̇𝑥(𝑡)𝜁𝑑𝑑(𝑡)𝑇𝑒(𝑡)2𝑒3𝑍𝑒2𝑒3𝑇𝑑𝜁(𝑡)𝑑𝜁(𝑡)𝑇𝑒(𝑡)1𝑒2𝑍𝑒1𝑒2𝑇𝜁(𝑡)(2.12)𝑑2̇𝑥𝑇(𝑡)𝑍̇𝑥(𝑡)𝜁𝑇𝑒(𝑡)𝑇2𝑒𝑇3𝑒𝑇1𝑒𝑇2𝑇𝑍𝑆12𝑆𝑇12𝑍𝑒𝑇2𝑒𝑇3𝑒𝑇1𝑒𝑇2𝜁(𝑡),(2.13) where the inequality in (2.12) comes from the Jensen inequality lemma, and that of (2.13) from Lemma 1.2 as 𝜁𝑇(𝑡)𝛽𝛼𝑒2𝑒3𝑇𝛼𝛽𝑒1𝑒2𝑇𝑇𝑍𝑆12𝑆𝑇12𝑍𝛽𝛼𝑒2𝑒3𝑇𝛼𝛽𝑒1𝑒2𝑇𝜁(𝑡)0,(2.14) where 𝜁𝑇𝑥(𝑡)=𝑇(𝑡)𝑥𝑇(𝑡𝑑(𝑡))𝑥𝑇𝑥(𝑡𝑑)𝑇𝑡𝑑1(𝑥𝑡)𝑇𝑡𝑑1𝑥𝑇𝑡𝑑2𝑥(𝑡)𝑇𝑡𝑑2̇𝑥𝑇,𝑒(𝑡)1=𝐼0000000𝑇,𝑒2=0𝐼000000𝑇,𝑒3=00𝐼00000𝑇,(2.15)𝛼=(𝑑𝑑(𝑡))/𝑑,  𝛽=𝑑(𝑡)/𝑑. Note that when 𝑑(𝑡)=𝑑 or 𝑑(𝑡)=0, one can obtain 𝜁𝑇(𝑡)(𝑒2𝑒3)=0 or 𝜁𝑇(𝑡)(𝑒1𝑒2)=0, respectively. So the relation (2.13) also holds.
By the Jensen inequality lemma, it is easy to obtain 𝑡𝑡𝑑1(𝑡)̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠𝑑1(𝑡)𝑈𝑇1𝑍1𝑈1,𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1𝑑̇𝑥(𝑠)𝑑𝑠1𝑑1𝑈(𝑡)𝑇2𝑍1𝑈2,𝑡𝑡𝑑2(𝑡)̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑑2(𝑡)𝑈𝑇3𝑍2𝑈3,𝑡𝑑2(𝑡)𝑡𝑑2̇𝑥𝑇(𝑠)𝑍2𝑑̇𝑥(𝑠)𝑑𝑠2𝑑2𝑈(𝑡)𝑇4𝑍2𝑈4,(2.16) where 𝑈1=1𝑑1(𝑡)𝑡𝑡𝑑1(𝑡)̇𝑥(𝑠)𝑑𝑠,𝑈2=1𝑑1𝑑1(𝑡)𝑡𝑑1(𝑡)𝑡𝑑1𝑈̇𝑥(𝑠)𝑑𝑠,3=1𝑑2(𝑡)𝑡𝑡𝑑2(𝑡)̇𝑥(𝑠)𝑑𝑠,𝑈4=1𝑑2𝑑2(𝑡)𝑡𝑑2(𝑡)𝑡𝑑2̇𝑥(𝑠)𝑑𝑠,(2.17)lim𝑑1(𝑡)01𝑑1(𝑡)𝑡𝑡𝑑1(𝑡)̇𝑥(𝑠)𝑑𝑠=̇𝑥(𝑡),lim𝑑1(𝑡)𝑑11𝑑1𝑑1(𝑡)𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥(𝑠)𝑑𝑠=̇𝑥𝑡𝑑1,lim𝑑2(𝑡)01𝑑2(𝑡)𝑡𝑡𝑑2(𝑡)̇𝑥(𝑠)𝑑𝑠=̇𝑥(𝑡),lim𝑑2(𝑡)𝑑21𝑑2𝑑2(𝑡)𝑡𝑑2(𝑡)𝑡𝑑2̇𝑥(𝑠)𝑑𝑠=̇𝑥𝑡𝑑2.(2.18)
From the Leibniz-Newton formula, the following equations are true for any matrices 𝑁,𝑀, 𝐿,𝑆,𝑃1,𝑃2 with appropriate dimensions 2𝜁𝑇𝑥(𝑡)𝑁(𝑡)𝑥𝑡𝑑1(𝑡)𝑑1(𝑡)𝑈1=0,2𝜁𝑇(𝑥𝑡)𝑀𝑡𝑑1(𝑡)𝑥𝑡𝑑1𝑑1𝑑1(𝑈𝑡)2=0,2𝜁𝑇𝑥(𝑡)𝐿(𝑡)𝑥𝑡𝑑2(𝑡)𝑑2(𝑡)𝑈3=0,2𝜁𝑇(𝑥𝑡)𝑆𝑡𝑑2(𝑡)𝑥𝑡𝑑2𝑑2𝑑2(𝑈𝑡)42𝑥=0,𝑇(𝑡)𝑃1+̇𝑥𝑇(𝑡)𝑃2̇𝑥(𝑡)+𝐴𝑥(𝑡)+𝐴𝑑𝑥(𝑡𝑑(𝑡))=0.(2.19) Hence, according to (2.8)–(2.19), we can obtain ̇𝑉(𝑥(𝑡))𝜉𝑇(𝑡)𝐸𝜉(𝑡),(2.20) where 𝜉𝑇𝜁(𝑡)=𝑇(𝑡)𝑈𝑇1𝑈𝑇2𝑈𝑇3𝑈𝑇4,𝐸=𝐸𝑑1𝑑(𝑡)𝑁1𝑑1(𝑡)𝑀𝑑2𝑑(𝑡)𝐿2𝑑2𝑆(𝑡)𝑑1(𝑡)𝑍1𝑑0001𝑑1𝑍(𝑡)100𝑑2(𝑡)𝑍20𝑑2𝑑2𝑍(𝑡)2.(2.21) If 𝐸<0, then there exists a scalar 𝜀>0, such that ̇𝑉(𝑥(𝑡))𝜉𝑇(𝑡)𝐸𝜉(𝑡)𝜀𝜉𝑇(𝑡)𝜉(𝑡)𝜀𝑥𝑇(𝑡)𝑥(𝑡)<0,𝑥(𝑡)0.(2.22) The 𝐸<0 leads for 𝑑1(𝑡)𝑑1 to 𝐸1<0 and leads for 𝑑1(𝑡)0 to 𝐸2<0, where 𝐸1=𝐸𝑑1𝑁𝑑2𝑑(𝑡)𝐿2𝑑2𝑆(𝑡)𝑑1𝑍100𝑑2(𝑡)𝑍20𝑑2𝑑2𝑍(𝑡)2𝐸<0,(2.23)2=𝐸𝑑1𝑀𝑑2𝑑(𝑡)𝐿2𝑑2𝑆(𝑡)𝑑1𝑍100𝑑2(𝑡)𝑍20𝑑2𝑑2𝑍(𝑡)2<0.(2.24) It is easy to see that 𝐸1 results from 𝐸|𝑑1(𝑡)=𝑑1, where we deleted the zero row and the zero column. Define 𝜉𝑇1𝜁(𝑡)=𝑇(𝑡)𝑈𝑇1𝑈𝑇3𝑈𝑇4,𝜉𝑇2𝜁(𝑡)=𝑇(𝑡)𝑈𝑇2𝑈𝑇3𝑈𝑇4,(2.25) The LMI (2.23) and (2.24) imply (2.22) because 𝑑1(𝑡)𝑑1𝜁𝑇1(𝑡)𝐸1𝜁1𝑑(𝑡)+1𝑑1(𝑡)𝑑1𝜁𝑇2(𝑡)𝐸2𝜁2(𝑡)=𝜉𝑇(𝑡)𝐸𝜉(𝑡)𝜀𝑥𝑇(𝑡)𝑥(𝑡)(2.26) and 𝐸 is convex in 𝑑1(𝑡)[0,𝑑1].
LMI (2.23) leads for 𝑑2(𝑡)𝑑2 to LMI (2.2) and for 𝑑2(𝑡)0 to LMI (2.3). It is easy to see that 𝐸13 results from 𝐸1|𝑑2(𝑡)=𝑑2, where we deleted the zero row and the zero column. The LMI (2.2) and (2.3) imply (2.23) because 𝑑2(𝑡)𝑑2𝜁𝑇13(𝑡)𝐸13𝜁13𝑑(𝑡)+2𝑑2(𝑡)𝑑2𝜁𝑇14(𝑡)𝐸14𝜁14(𝑡)=𝜉𝑇1(𝑡)𝐸1𝜉1(𝑡)<0(2.27) and 𝐸1is convex in 𝑑2(𝑡)[0,𝑑2], where 𝜉𝑇13𝜁(𝑡)=𝑇(𝑡)𝑈𝑇1𝑈𝑇3,𝜉𝑇14𝜁(𝑡)=𝑇(𝑡)𝑈𝑇1𝑈𝑇4(2.28)𝐸13and 𝐸14 are defined in Theorem 2.1.
Similarly, the LMI (2.4) and (2.5) imply (2.24) because 𝑑2(𝑡)𝑑2𝜁𝑇23(𝑡)𝐸23𝜁23𝑑(𝑡)+2𝑑2(𝑡)𝑑2𝜁𝑇24(𝑡)𝐸24𝜁24(𝑡)=𝜉𝑇2(𝑡)𝐸2𝜉2(𝑡)<0(2.29) and 𝐸2is convex in 𝑑2(𝑡)[0,𝑑2], where 𝜉𝑇23𝜁(𝑡)=𝑇(𝑡)𝑈𝑇2𝑈𝑇3,𝜉𝑇24𝜁(𝑡)=𝑇(𝑡)𝑈𝑇2𝑈𝑇4.(2.30)𝐸23and 𝐸24are defined in Theorem 2.1. According to the above analysis, one can conclude that the system (1.5) with delays 𝑑1(𝑡) and 𝑑2(𝑡) satisfying (1.6) is asymptotically stable if the LMIs (2.1)–(2.5) hold.
From the proof of Theorem 2.1, one can obtain that 𝐸 is negative definite in the rectangle 0𝑑1(𝑡)𝑑1, 0𝑑2(𝑡)𝑑2, only if it is negative definite at all vertices. We call this method as the convex polyhedron method.

Remark 2.3. To avoid the emergence of the reciprocally convex combination in (2.12), similar to [9], the integral terms in (2.10) can be upper bounded by 𝑑𝑡𝑡𝑑̇𝑥𝑇([]𝑠)𝑍̇𝑥(𝑠)𝑑𝑠𝑥(𝑡𝜏(𝑡))𝑥(𝑡𝑑)𝑇𝑍[][]𝑥(𝑡𝜏(𝑡))𝑥(𝑡𝑑)𝑥(𝑡)𝑥(𝑡𝜏(𝑡))𝑇𝑍[][]𝑥(𝑡)𝑧(𝑡𝜏(𝑡))(1𝛾)𝑥(𝑡𝜏(𝑡))𝑥(𝑡𝑑)𝑇𝑍[][]𝑥(𝑡𝜏(𝑡))𝑥(𝑡𝑑)𝛾𝑥(𝑡)𝑥(𝑡𝜏(𝑡))𝑇𝑍[]𝑥(𝑡)𝑥(𝑡𝜏(𝑡))(2.31) which results in a convex combination on 𝛾. However, Theorem 2.1 directly handles the inversely weighted convex combination of quadratic terms of integral quantities by utilizing the result of Lemma 1.2, which achieves performance behavior identical to the approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on the Jensen inequality lemma.

Remark 2.4. Compared to some existing ones, the estimation of ̇𝑉(𝑥(𝑡)) in the proof of Theorem 2.1 is less conservative due to the convex polyhedron method is employed. More specifically, 𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 is retained, while 𝑡𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 is divided into 𝑡𝑡𝑑1(𝑡)̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠 and 𝑡𝑑1(𝑡)𝑡𝑑1̇𝑥𝑇(𝑠)𝑍1̇𝑥(𝑠)𝑑𝑠. When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term 𝑑1(𝑡)𝑆𝑍11𝑆𝑇 as 𝑑1𝑆𝑍11𝑆𝑇. The convex polyhedron method is employed to verify the negative definiteness of ̇𝑉(𝑥(𝑡)). Therefore, Theorem 2.1 is expected to be less conservative than some results in the literature.

Remark 2.5. The case in which only two additive time-varying delay components appear in the state has been considered, and the idea in this paper can be easily extended to the system (1.3) with multiple additive delay components satisfying (1.4). Choose the Lyapunov functional as 𝑉(𝑥(𝑡))=𝑉1(𝑥(𝑡))+𝑉2(𝑥(𝑡))+𝑉3(𝑥(𝑡))+𝑉4𝑉(𝑥(𝑡)),1(𝑥(𝑡))=𝑥𝑇𝑉(𝑡)𝑃𝑥(𝑡),2(𝑥(𝑡))=𝑡𝑡𝑑(𝑡)𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡𝑑𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠+𝑛𝑖=1𝑡𝑡𝑑𝑖(𝑡)𝑥𝑇(𝑠)𝑄3𝑖+𝑥(𝑠)𝑑𝑠𝑛𝑖=1𝑡𝑡𝑑𝑖𝑥𝑇(𝑠)𝑄4𝑖𝑥𝑉(𝑠)𝑑𝑠,3(𝑥(𝑡))=𝑑0𝑑𝑡𝑡+𝜃̇𝑥𝑇𝑉(𝑠)𝑍̇𝑥(𝑠)𝑑𝑠𝑑𝜃,4(𝑥(𝑡))=𝑛𝑖=10𝑑𝑖𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑍𝑖̇𝑥(𝑠)𝑑𝑠𝑑𝜃.(2.32) Then, the corresponding stability result can be easily derived similar to the proof of Theorem 2.1. The result is omitted due to complicated notation.

Remark 2.6. The stability condition presented in Theorem 2.1 is for the nominal system. However, it is easy to further extend Theorem 2.1 to uncertain systems, where the system matrices 𝐴 and 𝐴𝑑 contain parameter uncertainties either in norm-bounded or polytopic uncertain forms. The reason why we consider the simplest case is to make our idea more lucid and to avoid complicated notations.

3. Illustrative Example

Example 3.1. Consider system (1.5) with the following parameters: 𝐴=2000.9,𝐴𝑑=̇𝑑1011,assuming1̇𝑑(𝑡)0.1,2(𝑡)0.8.(3.1)
Our purpose is to calculate the upper bound 𝑑1 of delay 𝑑1(𝑡), or 𝑑2 of delay 𝑑2(𝑡), when the other is known, below which the system is asymptotically stable. By combining the two delay components together, some existing stability results can be applied to this system. The calculation results obtained by Theorem 2.1, in this paper, Theorem  1 in [6, 12, 15, 16], [14, Theorem  2] for different cases are listed in Table 1. It can be seen from the Table 1 that Theorem 2.1, in this paper, yields the least conservative stability test than other results.

tab1
Table 1: Calculated delay bounds for different cases.

Example 3.2. Consider system (1.5) with the following parameters: 𝐴=0.01.01.02.0,𝐴𝑑=0.00.01.01.0.(3.2) We assume condition 1: ̇𝑑1(𝑡)0.2, ̇𝑑2(𝑡)0.5; condition 2: ̇𝑑1(𝑡)0.2, ̇𝑑2(𝑡)0.3, and under the two cases above, respectively. Table 2 lists the corresponding upper bounds of 𝑑2 for given 𝑑1. This numerical illustrates the effectiveness of the derived results.

tab2
Table 2: Allowable upper bound of 𝑑2 for various 𝑑1.

4. Conclusions

This paper has investigated the stability problem for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors would like to thank the editors and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by research on the model and method of parameter identification in reservoir simulation under Grant PLN1121.

References

  1. W.-J. Mao and J. Chu, “Quadratic stability and stabilization of dynamic interval systems,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 1007–1012, 2003. View at Publisher · View at Google Scholar
  2. M. da la Sen, “Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 508–526, 2007. View at Publisher · View at Google Scholar
  3. S. Xu, J. Lam, and C. Yang, “Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay,” Systems & Control Letters, vol. 43, no. 2, pp. 77–84, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Y. He, Q.-G. Wang, C. Lin, and M. Wu, “Delay-range-dependent stability for systems with time-varying delay,” Automatica, vol. 43, no. 2, pp. 371–376, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. X. Jiang and Q.-L. Han, “On H control for linear systems with interval time-varying delay,” Automatica, vol. 41, no. 12, pp. 2099–2106, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X.-J. Jing, D.-L. Tan, and Y.-C. Wang, “An LMI approach to stability of systems with severe time-delay,” IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1192–1195, 2004. View at Publisher · View at Google Scholar
  7. E. Fridman, U. Shaked, and K. Liu, “New conditions for delay-derivative-dependent stability,” Automatica, vol. 45, no. 11, pp. 2723–2727, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. P. Park and J. W. Ko, “Stability and robust stability for systems with a time-varying delay,” Automatica, vol. 43, no. 10, pp. 1855–1858, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. 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 Zentralblatt MATH
  10. P. Park, J. W. Ko, and C. K. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. Park, “A delay-dependent stability criterion for systems with uncertain time-invariant delays,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 876–877, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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 · View at Zentralblatt MATH
  13. J.-H. Kim, “Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty,” IEEE Transactions on Automatic Control, vol. 46, no. 5, pp. 789–792, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. Wu, Y. He, J.-H. She, and G.-P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Lam, H. Gao, and C. Wang, “Stability analysis for continuous systems with two additive time-varying delay components,” Systems & Control Letters, vol. 56, no. 1, pp. 16–24, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. H. Gao, T. Chen, and J. Lam, “A new delay system approach to network-based control,” Automatica, vol. 44, no. 1, pp. 39–52, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH