Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 238487, 8 pages

http://dx.doi.org/10.1155/2014/238487

## Simplified Stability Criteria for Delayed Neutral Systems

^{1}School of Mathematical Science, Heilongjiang University, Harbin 150080, China^{2}College of Computer and Information Engineering, Heilongjiang University of Science and Technology, Harbin 150022, China

Received 4 March 2014; Accepted 1 April 2014; Published 13 May 2014

Academic Editor: Yuxin Zhao

Copyright © 2014 Xinghua Zhang 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.

#### Abstract

For a class of linear time-invariant neutral systems with neutral and discrete constant delays, several existing asymptotic stability criteria in the form of linear matrix inequalities (LMIs) are simplified by using matrix analysis techniques. Compared with the original stability criteria, the simplified ones include fewer LMI variables, which can obviously reduce computational complexity. Simultaneously, it is theoretically shown that the simplified stability criteria and original ones are equivalent; that is, they have the same conservativeness. Finally, a numerical example is employed to verify the theoretic results investigated in this paper.

#### 1. Introduction

Some practical systems, such as population ecology, neural networks, heat exchangers, and robots in contract with rigid environments, have been modeled by functional differential equations of neutral type (e.g., [1–6]). Since stability analysis is the primary task of analyzing and synthesizing a system, many researchers have paid more and more attention to establish stability criteria for delayed neutral systems (see [7, 8] and the references therein).

For a class of linear time-invariant neutral systems with neutral and discrete constant delays, both neutral- and discrete-delay-dependent stability criteria have been investigated in [9–27]. Many approaches have been provided for obtaining these stability criteria. Here we mention only some critical and representative approaches. Model transformation approach first transforms the neutral system with discrete delay into the one with distributed delays, and then delay-dependent stability criteria can be obtained by constructing a Lyapunov-Krasovskii functional [9]. However, model transformation approach introduces some additional dynamics, which leads still to conservative results [28]. To overcome this shortage, the so-called descriptor model transformation approach is introduced in [15, 24], which transforms the original system to an equivalent descriptor form. The descriptor model transformation approach will not introduce additional dynamics in the sense defined in [28]. Although the results obtained from descriptor model transformation approach may be less conservative than some existing ones, they can be improved by employing some other approaches (e.g., free weighting matrix approach [21, 22] and augmented Lyapunov-Krasovskii functional approach [14, 23, 27]) to get a larger bound for discrete delay.

By employing free weighting matrices and the Leibniz-Newton formula, He et al. [21] presented a neutral- and discrete-delay-dependent stability criterion for neutral systems with neutral and discrete constant delays, which reduces the conservativeness of methods involving a fixed model transformation. For the same system models, Qian et al. [27] established neutral- and discrete-delay-dependent stability criteria by constructing a Lyapunov-Krasovskii functional with additional functional parameters and employing free weighting matrices. To the best of our knowledge, the large-scale employment of functional parameters and/or free weighting matrices can cause complex stability criteria, and thereby the computational complexity increases obviously. Therefore, in order to reduce computational complexity, it is necessary to reduce the number of functional parameters and free weighting matrices. This motivates the present study.

The aim of this paper is to simplify several stability criteria proposed by He et al. [21] and Qian et al. [27]. The main contributions of the paper are as follows. (i) In the premise of not increasing conservativeness, a pair of stability criteria proposed in [27, Theorem 1 and Corollary 1] are simplified by using matrix analysis techniques; (ii) one of the pair of simplified stability criteria is theoretically presented to be less conservative than [26, Corollary 1], and another is theoretically proven to be equivalent to [21, Theorem 1]; and (iii) by one numerical example, our theoretic results are verified to be more effective than many existing results.

#### 2. Problem Statement and Preliminary Results

Consider a class of delayed neutral systems described as

where is the -dimensional state vector, , , and are known real constant matrices with appropriate dimensions, , is an -valued continuous initial function defined on , and positive scalars and represent the neutral and discrete delays, respectively.

Define as the set of continuous valued function on the interval , and let be a segment of system trajectory defined as

Define an operator by . The definitions on stability of operator and systems (1a) and (1b) can be seen in [29].

As mentioned in Introduction, neutral- and discrete-delay-dependent stability criteria for systems (1a) and (1b) have been investigated by He et al. [21] and Qian et al. [27]. In this paper, we will simplify the stability criteria [21, Theorem 1] and [27, Theorem 1 and Corollary 1] in the premise of not increasing conservativeness of stability criteria.

In order to present conveniently our main results, the following lemmas are required.

Lemma 1 (Schur complement lemma [30]). *Given constant matrices , , and of appropriate dimensions, where and , then
**
if and only if
*

*Lemma 2 (see [31]). Given a real symmetric matrix and a pair of real matrices and , the following LMI problem
is solvable with respect to decision variable if and only if
where and are matrices whose columns form a basis of the right null spaces of and , respectively.*

*3. Several Stability Criteria in the Literature*

*In this section we introduce several existing stability criteria for systems (1a) and (1b), which will be useful to present conveniently the main results of this paper in the next section.*

*Proposition 3 (see [21, Theorem 1]). For given scalars and , systems (1a) and (1b) are asymptotically stable, if the operator is stable and there exist real matrices , (), , , (), , and () such that the following LMIs are feasible:
where
*

*Proposition 4 (see [27, Theorem 1]). For given scalars and , systems (1a) and (1b) are asymptotically stable, if the operator is stable and there exist real matrices , , , , , , , , , , , , (), and () such that the following LMIs are feasible:
where
*

*Proposition 5 (see [27, Corollary 1]). When , for a given scalar , systems (1a) and (1b) are asymptotically stable, if the operator is stable and there exist real matrices , , , , , , , , , and () such that the following LMIs are feasible:
*

where

*Remark 6. *Proposition 3 simplifies the notations in [21, Theorem 1].

*Remark 7. *Propositions 4 and 5 correct some slips of the pen in [27, Theorem 1 and Corollary 1]. After [27, Theorem 1] is amended as in Remark 7, [27, Theorem 2] will be correct.

*4. Simplified Stability Criteria*

*4. Simplified Stability Criteria**In this section we will simplify the stability criteria introduced in the previous section. Firstly, the stability criterion presented in Proposition 4 can be simplified by the following theorem.*

*Theorem 8. For given scalars and , systems (1a) and (1b) are asymptotically stable, if the operator is stable and one of the following cases, (i)–(iv), holds. (i)There exist real matrices , , , , , , , , , , , , (), and () such that the LMIs (10)–(15) hold.(ii)There exist real matrices , , , , , , , , , , , , (), and () such that the LMIs (11)–(15) and the following LMI (20) hold:
(iii)There exist real matrices , , , , , , , , and () such that the LMIs (11), (12), and (13) and the following LMI (21) hold:
(iv)There exist real matrices , , , , , , , , and () such that the LMIs (11), (12), (13), and (20) and the following LMI (22) hold:
where
*

*Proof. *Due to Proposition 4, it suffices to show that the conditions (i)–(iv) are equivalent.

(i)(ii) Since the eigenvalues of a matrix are continuous functions of its elements, it follows from (i) that there exists a sufficiently small positive scalar such that

Therefore, the LMIs (11)–(15) and (20) are feasible.

(ii)(i) The proof is very easy, and hence it is omitted.

(ii)(iii) By Lemma 1, it follows from (20) and (15) that

This, together with (14) and Lemma 1, implies that (21) holds.

(iii)(ii) It follows from (21) and Lemma 1 that (20) holds and
where
and hence (ii) holds.

(iii)(iv) Let

Then LMI (21) can be written as
where the matrix is obtained from the matrix on the left of (21) by deleting all parts containing one of , , and (e.g., the th subblock of equals , which is obtained from by deleting ). Choose

Noting that , one can derive from (12) and Lemma 2 that the LMI (29) (i.e., (21)) is feasible if and only if LMIs (22) and (20) are feasible.

The proof is completed.

*Clearly, the stability condition (iv) in Theorem 8 is more simpler than (i) in Theorem 8 (i.e., Proposition 4). By a process similar to investigating Theorem 8, one can easily obtain the following theorem which simplifies the stability criterion presented in Proposition 5.*

*Theorem 9. When , for a given scalar , systems (1a) and (1b) are asymptotically stable, if the operator is stable, and one of the following cases, (i)-(ii), holds.(i)There exist real matrices , , , , , , , , , and () such that the LMIs (17) and (18) hold.(ii)There exist real matrices , , , , , , , and such that (17) and the following LMI (31) hold:
where
*

*Remark 10. *Theorems 8 and 9 indicate that, without increasing conservativeness of stability criteria, the free weighting matrices required in Propositions 4 and 5 (i.e., [27, Theorem 1 and Corollary 1]) can be eliminated. This will obviously reduce the computational complexity, and hence the stability conditions (iv) of Theorem 8 and (ii) of Theorem 9 are more effective.

*Remark 11. *In the special case that , , , and ( is a sufficiently small positive number), Theorem 9 reduces to [26, Corollary 1]. So, by choosing suitable , , , and , Theorem 9 could reduce the conservativeness of [26, Corollary 1], by which the improvement over [26, Corollary 1] is theoretically demonstrated.

*If we choose , , , , and ( is a sufficiently small positive number) () in Theorem 8, the following corollary can be easily derived.*

*Corollary 12. For given scalars and , systems (1a) and (1b) are asymptotically stable, if the operator is stable and there exist real matrices , , , and () such that LMI (33) holds:
where
*

*Next we will show that the pair of stability criteria presented in Proposition 3 (i.e., [21, Theorem 1]) and Corollary 12 have the same conservativeness. However, the stability criterion presented in Corollary 12 is more simpler than the one in Proposition 3 (i.e., [21, Theorem 1]), which reduces greatly the computational complexity.*

*Theorem 13. For given scalars and , LMIs (7) and (8) are feasible if and only if the LMI (33) is feasible.*

*Proof. **The “Only If” Part.* It follows from (7) and (8) that
and hence

Noting that
one can conclude from (36) that
which becomes LMI (33) by letting , , and . That is, LMI (33) is feasible.*The “If” Part.* Since LMI (33) is feasible, there exist real matrices , , , and () such that LMI (33) is satisfied. Set , , and . Then it follows from (33) and Lemma 1 that

Let

Then LMI (7) holds, and

Furthermore, LMI (8) is derived from (39).

The proof is completed.

*5. Numerical Comparisons of Stability Criteria*

*5. Numerical Comparisons of Stability Criteria*

*In this section we will present one example to demonstrate effectiveness of the theoretic results described above.*

*Example 1 . *Consider the stability of systems (1a) and (1b) with

*Case 1 (). *The upper bounds on the delay under which robust stability of this system can be guaranteed using the methods in [9–27] and Theorem 9 of this paper are listed in Table 1. It is clear that our results are significantly better than those in [9–26] because our allowable maximum time delay is larger.

*Case 2 (). *For different values of , Table 2 lists the maximum allowable bounds on that guarantee the stability of the system. It can be seen that the maximum allowable bound on decreases as increases when is small but that remains almost unchanged when increases to certain extent.

*6. Conclusions*

*6. Conclusions*

*By using matrix analysis techniques, several stability criteria (i.e., [21, Theorem 1] and [27, Theorem 1 and Corollary 1]) for delayed neutral systems have been simplified. The numbers of LMI variables in [21, Theorem 1] and [27, Theorem 1 and Corollary 1] are reduced into the simplified ones, which can obviously reduce computational complexity. Furthermore, it is theoretically proven that the simplified stability criteria have the same conservativeness as the original ones. A numerical example is given to illustrate the theoretic results investigated in this paper.*

*Extending the idea of this paper to other system models, including singular delayed systems [32–35], stochastic systems [36], Markovian jump systems [37, 38], and genetic regulatory networks [39–41], is under consideration.*

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no commercial or associative interest that represents a conflict of interest in connection with the work submitted.*

*Acknowledgments*

*Acknowledgments*

*This work is supported by the National Natural Science Foundation of China under Grant no. 11371006, the fund of Heilongjiang Province Innovation Team Support Plan under Grant no. 2012TD007, the fund of Heilongjiang University Innovation Team Support Plan under Grant no. Hdtd2010-03, the Fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province, (Heilongjiang University), China, and the fund of Heilongjiang Education Committee. The authors thank the anonymous referees for their helpful comments and suggestions which improve greatly this paper.*

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