Research Article | Open Access

Wei Qian, Tao Li, Juan Liu, "Improving Delay-Range-Dependent Stability Condition for Systems with Interval Time-Varying Delay", *Mathematical Problems in Engineering*, vol. 2013, Article ID 680272, 6 pages, 2013. https://doi.org/10.1155/2013/680272

# Improving Delay-Range-Dependent Stability Condition for Systems with Interval Time-Varying Delay

**Academic Editor:**Bo-Chao Zheng

#### Abstract

This paper discusses the delay-range-dependent stability for systems with interval time-varying delay. Through defining the new Lyapunov-Krasovskii functional and estimating the derivative of the LKF by introducing new vectors, using free matrices and reciprocally convex approach, the new delay-range-dependent stability conditions are obtained. Two well-known examples are given to illustrate the less conservatism of the proposed theoretical results.

#### 1. Introduction

It is well known that time-varying delays are frequently encountered in many practical control systems, and they are usually regarded as a source of instability and poor performance. So the stability issue of time delay systems has received considerable attention [1–16]. In the last years, in order to further reduce conservatism of the stability results, some methods were developed, such as the delay-fraction approach in [1, 2], free weighting matrices method in [3, 4], the convex analysis method in [5, 6], the reciprocally convex approach in [7], LKF constructing method with matrices that depend on the time delays [8], LKF constructing method with triple-integral terms in [9], LKF constructing method with quadruple-integral terms in [10], and simple LKF having quadratic terms multiplied by a higher degree scalar function [11, 12]. These methods reduced the conservatism of the stability results. But when delay is interval time-varying, the information of the delay derivative is not full used, which causes the conservatism of the stability results.

Motivated by recent methods, in this paper, we further discuss the stability of linear systems with interval time-varying delay. Firstly, a novel LKF is introduced. Then, by introducing new vectors, using free matrices and reciprocally convex approach, the derivative of LKF is estimated less conservatively, and as a result, the stability criterion is obtained in terms of LMI. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Throughout the note, the used notations are standard. denotes the -dimensional Euclidean space, is a set of real matrix, is the transpose of , means symmetric positive (negative) definite matrix, and in the matrix denotes the symmetric element; is the identity matrix of appropriate dimensions, . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

#### 2. Problem Formulations

Consider the following time-delay system: where is the state vector, the initial condition is a continuously differentiable vector-valued function, are known real constant matrices, and is the time-varying delay satisfying where , , and are constants.

To obtain the main results, the following lemmas are needed.

Lemma 1 (see [9]). *For any symmetric matrix , scalars such that the integration is well defined; then
*

Lemma 2 (see [7]). *Let have positive values in an open subset of . Then, the reciprocally convex combination of over satisfies
*

Lemma 3 (see [16]). *The symmetric appropriately dimensional matrices , if hold; then there exists a matrix of appropriate dimension such that the following LMI holds:
*

#### 3. Main Results

In this section, the stability of system (1) is investigated. Through constructing a novel LKF and estimating the derivative of it, new stability condition is provided.

Firstly, define the following vector as and then construct the L-K functional as follows: where and matrices , .

In order to make the proof process clear, the following notations are used:and block entry matrices (e.g., ).

Then taking the time derivatives of along the trajectory of system (1) yields where .

To reduce the conservatism of the main result, by using the Jesen inequality and Lemma 2 with introducing appropriate matrices , we can calculate as where by using Lemma 2 with matrix , one can get and by using Lemma 2 with matrix , one can get

To the time-derivative of , it can be calculated as where by using Lemma 1, we can get

Similar to the method in (12)-(13), by introducing matrix and using Lemma 2, we can estimate the following:

In the last, in order to obtain the stability result base on LMI, we can deal with as follows: where Also, for appropriate matrices , one can have

Therefore, combining (10)–(19), we can obtain where As we know, if holds, then , which means that system (1) is asymptotically stable. So, by Lemma 3, there exists a matrix of appropriate dimension such that the following LMI holds:

So, we give the main theorem of this paper as follows.

Theorem 4. *The time-delayed system (1) having contrains (2) is asymptotically stable, if there exist matrices , matrix , and appropriate matrices such that the following LMIs hold:
**
where , , , and are defined in (18) and (21).*

*Proof. *From above, one can see that is a convex combination of matrix as
So (22) can be handled nonconservatively by the two corresponding boundary LMIs:
where , . This completes the proof.

*Remark 5. *From (17) and (22), it can be seen that, in order to obtain the less conservative stability condition in terms of LMI, , are used, but free matrices are also introduced, which increases the computational burden; this is a disadvantage of the proposed method.

*Remark 6. *When , , let , , and using the convex combination again with , , , , the corresponding stability criterion can also be obtained.

#### 4. Numerical Examples

In this section, the effectiveness of the obtained results in this paper is shown by the two well-known numerical examples.

*Example 1. *Consider the linear system (1) with

The purpose is to compare the admissible upper bounds given lower bound and .

For , , in [14], the result is , in [17], the result is , and our results are .

For , , in [14], the result is , in [17], the result is , and our results are .

For , , in [14], the result is , in [17], the result is , and our results are .

For various , the maximum upper bounds on delay by different methods are also listed in Table 1. It can be seen that the result obtained in this paper is less conservative.

*Example 2. *Consider the linear system (1) with

The purpose is to compare the admissible upper bounds which guarantee the asymptotic stability of the above system for given lower bound and . This example is used in many recent papers, such as [6, 9], For , Table 2 lists the comparison of our results with some recent ones, and it is easy to see that the results obtained in this paper are less conservative.

#### 5. Conclusions

In this note, the stability of interval time-varying delay systems has been discussed. Through constructing a novel LKF and using some new analysis methods, the delay-range-dependent stability criteria were derived. Compared with some previous stability conditions, the obtained main results in this paper have less conservatism. In the end, numerical examples were given to show the superiority of the obtained criteria and their improvements over the existing results.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the National Nature Science Foundation under Grants 61104119, the Science and Technology Innovation Talents Project of Henan University under Grant 13HASTIT044, the Doctoral Foundation from Henan Polytechnic University under Grants B2010-50, and the Young Core Instructor Foundation from the Department of Education of Henan Province under Grant 2011GGJS-054.

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#### Copyright

Copyright © 2013 Wei Qian et al. 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.