Journal of Applied Mathematics

Volume 2012, Article ID 689820, 13 pages

http://dx.doi.org/10.1155/2012/689820

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

^{1}State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China^{2}School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China^{3}School 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 [2–16]. 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: where is a time delay in the state , which is often assumed to be either constant or time-varying satisfying certain conditions, for example,

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: To make the stability analysis simpler, we proceed by considering the system (1.3) with two additive delay components as follows: Here, is the state vector; and represent the two delay components in the state, and we denote ; , are system matrices with appropriate dimensions. It is assumed that and , . is the initial condition on the segment .

The purpose of our paper is to derive new stability conditions under which system (1.5) is asymptotically stable for all delays and satisfying (1.6). One possible approach to check the stability of this system is to simply combine and into one delay with Then, the system (1.5) becomes 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 and , 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 and . 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 in [15] was enlarged as , with discarded. On the other hand, some integrals were estimated conservatively. Take as an example, by introducing with an appropriate vector and a matrix , respectively, it was estimated as with enlarged as .

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 be a given finite number of functions such that they have positive values in an open subset of . Then, a reciprocally convex combination of these functions over is a function of the form
where the real numbers satisfy and .

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

Lemma 1.2 (See [10]). * Let have positive values in an open subset of . Then, the reciprocally convex combination of over satisfies
**
subject to
**
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 and satisfying (1.6) is asymptotically stable if there exist symmetric positive definite matrices,,,,,,, ,, and any matrices,,,,,, with appropriate dimensions, such that the following LMIs hold:
**
where
*

*Proof. *Construct a new Lyapunov functional candidate as

*Remark 2.2. *Our paper fully uses the information about ,, and , but [15, 16] only use the information about and , when constructing the Lyapunov functional . 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 along the trajectory of system (1.5) is given by
The is upper bounded by
where the inequality in (2.12) comes from the Jensen inequality lemma, and that of (2.13) from Lemma 1.2 as
where
, . Note that when or , one can obtain or , respectively. So the relation (2.13) also holds.

By the Jensen inequality lemma, it is easy to obtain
where

From the Leibniz-Newton formula, the following equations are true for any matrices ,, ,,, with appropriate dimensions
Hence, according to (2.8)–(2.19), we can obtain
where
If , then there exists a scalar , such that
The leads for to and leads for to , where
It is easy to see that results from , where we deleted the zero row and the zero column. Define
The LMI (2.23) and (2.24) imply (2.22) because
and is convex in .

LMI (2.23) leads for to LMI (2.2) and for to LMI (2.3). It is easy to see that results from , where we deleted the zero row and the zero column. The LMI (2.2) and (2.3) imply (2.23) because
and is convex in , where
and are defined in Theorem 2.1.

Similarly, the LMI (2.4) and (2.5) imply (2.24) because
and is convex in , where
and are defined in Theorem 2.1. According to the above analysis, one can conclude that the system (1.5) with delays and 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 is negative definite in the rectangle , , 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
which results in a convex combination on . 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 in the proof of Theorem 2.1 is less conservative due to the convex polyhedron method is employed. More specifically, is retained, while is divided into and . When the two integrals together with others are handled by using free weighting matrix method, instead of enlarging some term as . The convex polyhedron method is employed to verify the negative definiteness of . 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
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 and 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:

Our purpose is to calculate the upper bound of delay , or of delay , 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.

*Example 3.2. *Consider system (1.5) with the following parameters:
We assume condition 1: , ; condition 2: , , and under the two cases above, respectively. Table 2 lists the corresponding upper bounds of for given . This numerical illustrates the effectiveness of the derived results.

#### 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.

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