Journal of Applied Mathematics

Volume 2011, Article ID 138912, 20 pages

http://dx.doi.org/10.1155/2011/138912

## Robust Stability Criteria for Uncertain Neutral Systems with Interval Nondifferentiable Time-Varying Delay and Nonlinear Perturbations

^{1}Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand^{2}Center of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 7 July 2011; Accepted 13 September 2011

Academic Editor: James Buchanan

Copyright © 2011 W. Weera and P. Niamsup. 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

We study the robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.

#### 1. Introduction

It is well known that the existence of time delay in a system may cause instability and oscillations. Example, of time-delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [7–16]. The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small.

In many practical systems, models of system are described by neutral differential equations, in which the models depend on the delays of state and state derivatives. Heat exchanges, distributed networks containing lossless transmission lines and population ecology are examples of neutral systems because of its wider application. Therefore, several researchers have studied neutral systems and provided sufficient conditions to guarantee the asymptotic stability of neutral time delay systems, see [5, 9, 11–14, 16, 17] and references cited therein.

Well-known nonlinearities, as the delays, may cause instability and poor performance of practical systems, which have driven many researchers to study the problem of nonlinear perturbed systems with state delays during the recent years [5, 7, 9, 18]. In [18], the delay-dependent robust stability for linear time-varying systems with nonlinear perturbations is given, by using the Newton-Leibniz formula which has been taken into account instead of applying an integral inequality. In [7], a model transformation technique is used to deal with the stability of system with time varying for delays and nonlinear perturbations. In [9], based on a descriptor model transformation combined with a matrix decomposition approach, the robust stability of uncertain systems with time varying discrete delay is studied by applying an integral inequality. However, these model transformations often introduce additional dynamics which leads to relatively conservative results. In [5], the neutral delay and the discrete delay are all time-varying, while the derivative of discrete delay is less than 1 which limits its bigger application. In most studies the time-varying delays are required to be differentiable [1–5, 7, 9, 11–14, 16, 18]. Therefore their methods have a conservatism which can be improved upon. However, in most cases, these conditions are difficult to satisfy. From these reasons, the conditions are interesting to study, but there are fewer results for removing restriction to the derivative of interval time-varying delays. Therefore, in this paper we will employ some new techniques so that the above conditions can be removed.

In this paper, the problem of delay-dependent criterion for asymptotic stability for uncertain neutral system is studied with interval time-varying delay and time-varying nonlinear perturbations simultaneously. The restriction to the derivative of the interval time-varying delays is removed, which means that a fast interval time-varying delay is allowed. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. The new stability condition is much less conservative and is more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.

#### 2. Problem Formulation and Preliminaries

The following notations will be used in this paper: denotes the set of all real nonnegative numbers; denotes the -dimensional space and the vector norm ; denotes the space of all matrices of dimensions. denotes the transpose of matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; , ; denotes the set of all valued continuous functions on ; Matrix is called semipositive definite if , for all ; is positive definite () if , for all ; means . The symmetric term in a matrix is denoted by .

Consider the following neutral system with time-varying delay: where is the state vector, is a neutral delay, is a time-varying continuous function which satisfies where , , , are constants and ; the initial condition function denotes a continuous vector-valued initial function of , and are unknown nonlinear perturbations satisfying , and where and are positive real numbers.

The uncertain matrices , , , and satisfy where , , , are constant matrices with appropriate dimension, and , , , and are unknown real matrices of appropriate dimension representing the systems time-varying parameter uncertainties which satisfy where , , , , , , , and are known real constant matrices of appropriate dimension. is unknown time-varying matrix satisfying For simplicity, we denote , , by , , respectively.

Let and . Then can be expressed as where Obviously, . For this case, is a function belonging to the interval , where can be taken as the range of variation of the time-varying delay . Using the fact that system (2.1) can be rewritten as

Lemma 2.1 (see [17]). *There exists a symmetric matrix such that
**
if and only if
*

Lemma 2.2 (see [3]). *For any constant symmetric matrix , , , , and any differentiable vector function , we have
*

Lemma 2.3 (see [19]). *Given matrices , , , and with appropriate dimensions. Then
**
for all satisfying , if and only if there exists an such that
*

Proposition 2.4 (Cauchy inequality). * For any symmetric positive definite matrix and , we have
*

#### 3. Main Results

Now we present a new delay-dependent condition for the asymptotic stability of system (2.1).

*Assumption 3.1. *All the eigenvalues of matrix C are inside the unit circle.

First, we study the problem of stability for nominal system of (2.10) with , , , and .

Theorem 3.2. *Under Assumption 3.1, nominal system of (2.10) with time-varying delay satisfying (2.2) is asymptotically stable if there exist positive definite matrices , , , , , , matrices , , , , of appropriate dimension and such that
**
where
*

*Proof. *We prove that Theorem 3.2 is true for three cases, namely, ; ; .*Case 1 (). *Choose a Lyapunov-Krasovskii functional candidate as
where , , , , , and are positive definite matrices. Taking the derivative of with respect to along the trajectory of (2.10) yields
since
Based on Lemma 2.2, we obtain
and from the following equalities:
where , , and , , are some matrices of appropriate dimension. Next, from (4.5), for any scalars and , we obtain
By adding the terms on left of (3.7)–(3.10) to , we may express as
where
, , , , , , are some parameter matrices of appropriate dimensions. From (3.11) if and , then for some . Pre and postmultiplying both sides of by , we get that
By Schur complement lemma, this implies . In light of Lemma 2.1, (3.1) holds if and only if and (3.13) simultaneously hold. Then (3.1) holds if and only if there exists a symmetric matrix and (3.13) simultaneously hold. Therefore, nominal system of (2.10) is asymptotically stable.*Case 2 (). * For this case, we choose a Lyapunov-Krasovskii functional candidate as
where , , , and positive definite matrices are the same as those in .*Case 3 (). *For this case, we choose the Lyapunov-Krasovskii functional candidate as
where , , , , , and are positive definite matrices and are the same as those in .

By similar arguments used in proof of Theorem 3.2, we conclude that the nominal system of (2.10) is robustly asymptotically stable. The proof is complete.

Based on Theorem 3.2, we can perform the robust stability analysis for system (2.10) with uncertainties (2.5) and (2.6).

Theorem 3.3. *Under Assumption 3.1, system (2.10) with time-varying delay satisfying (2.2) and uncertainties (2.5) and (2.6) is asymptotically stable if there exist positive definite matrices , , , , , and , , , , of appropriate dimension and scalars , such that
**
where
*

*Proof. *We choose Lyapunov-Krasovskii functional as in Theorem 3.2, we may proof this Theorem by using a similar arguments as in the proof of Theorem 3.2. By replacing , , , and in (3.11) with , , and , respectively. For Case 1
Applying Lemmas 2.3. and 2.4., the following estimations hold:
Therefore, from (3.18)–(3.28), it follows that
where
Applying Schur complement lemma, the inequalities and are equivalent to and , respectively. Therefore, system (2.10) is robust asymptotically stable if the condition (3.16) holds.

By using arguments similar to the proof of Case 1 for Case 2 and Case 3, we may conclude that the close-loop system (2.10) is robust asymptotically stable.

*Remark 3.4. *In this paper, the restriction that the state delay is differentiable is not required, which allows state delay to be fast time varying. Meanwhile, this restriction is required in some existing results, see [1–5, 7, 9, 11–14, 16, 18].

*Remark 3.5. *In the proof of Theorem 3.3, we need negative definiteness of matrices , and simultaneously. In order to do so, we need to have certain diagonal terms of matrices , and being negative. This leads to the splitting of the term as which is one possibility to achieve such goal.

#### 4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of our theoretical results.

*Example 4.1. *Consider the following uncertain neutral system with time-varying delay and nonlinear uncertainties which is studied in [1, 2]:
where
It is assumed that the nonlinear uncertainties satisfy
Applying Theorem 3.3, the maximum allowable value of is given in Table 1 when and in Table 2 for . The results obtained in [1, 2] may not be used for the case when . Moreover, the differentiability of the time delay is not required in Theorem 3.3. Tables 1 and 2 show that our results significantly improve the results of [1, 2].

Moreover, it should be pointed out that if we let and , then from Theorem 3.3, the solutions of LMI (3.16) are given as follows:

Figure 1 shows the trajectories of solutions and of the system (4.1) with time-varying delay , , , for all , , and . Since the time-delay is not differentiable, the stability criterion in [1, 2] cannot be applied to this case because it is only applicable to the system with the differentiable delay.

*Example 4.2. *Consider the following uncertain neutral system with time-varying delay in [3, 4]:
where
where , and , are unknown parameter satisfying , , , and .

*Case 1. *For , , the maximum values of are listed in Table 3 for by applying criteria in [3, 4] and in this paper. We see that the maximum allowable bounds for obtained from Theorem 3.3 are much better than that obtained in [3, 4].

*Case 2. *For , , the maximum value obtained form Theorem 3.3 is listed in Table 4. In [3, 4] the neutral delay is constant, then its stability criterion cannot be applied to systems with time-varying neutral delay. Furthermore, the stability criterion in [5, 6] cannot be applied to this case because Theorem 3.3 does not have restriction on the derivative of time-varying delay. It is obvious that the obtained results are significantly better than those in [3–6].

Figure 2 shows the trajectories of solutions and of the system (4.5) with time-varying delay , , , for all .

#### 5. Conclusions

In this paper, we have investigated the delay-dependent robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. Based on Lyapunov-krasovskii theory, new delay-dependent sufficient conditions for robust stability have been derived in terms of LMIs. The interval time-varying delay function is not required to be differentiable, which allows time-delay function to be a fast time-varying function. Numerical examples are given to illustrate the effectiveness of the theoretic results which show that our results are much less conservative than some existing results in the literature.

#### Acknowledgments

Financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0355/2552) to W. Weera and P. Niamsup is acknowledged. The first author is also supported by the Graduate School, Chiang Mai University, and the Center of Excellence in Mathematics, Thailand. The second author is also supported by the Center of Excellence in Mathematics, Thailand and Commission for Higher Education, Thailand.

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