Simplified Stability Criteria for Delayed Neutral Systems

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.


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][2][3][4][5][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][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][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 delaydependent 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.

Problem Statement and Preliminary Results
Consider a class of delayed neutral systems described as where () is the -dimensional state vector, , Define an operator D : ([−, 0], R  ) → R  by D  = () − ( − ).The definitions on stability of operator D and systems (1a) and (1b) can be seen in [29].
As mentioned in Introduction, neutral-and discretedelay-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 Ω 1 , Ω 2 , and Ω 3 of appropriate dimensions, where if and only if Lemma 2 (see [31]).Given a real symmetric matrix Ψ and a pair of real matrices  and , the following LMI problem is solvable with respect to decision variable  if and only if where N  and N  are matrices whose columns form a basis of the right null spaces of  and , respectively.

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. where Proposition 4 (see [27,Theorem 1]).For given scalars  > 0 and  > 0, systems (1a) and (1b) are asymptotically stable, if the operator D is stable and there exist real  ×  matrices , 5), and   (1 ≤  <  ≤ 5) such that the following LMIs are feasible: where Proposition 5 (see [27,Corollary 1]).When  = , for a given scalar  > 0, systems (1a) and (1b) are asymptotically stable, if the operator D is stable and there exist real × matrices and   ( = 1, 2, 4, 5) such that the following LMIs are feasible: where

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  > 0 and  > 0, systems (1a) and (1b) are asymptotically stable, if the operator D is stable and one of the following cases, (i)-(iv), holds.
(ii)⇒(i) The proof is very easy, and hence it is omitted.
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  > 0, systems (1a) and (1b) are asymptotically stable, if the operator D is stable, and one of the following cases, (i)-(ii), holds.(ii) There exist real  ×  matrices and  1 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.
Proof.The "Only If " Part.It follows from ( 7) and ( 8 and hence Noting that one can conclude from (36) that Let Then LMI (7) holds, and Furthermore, LMI (8) is derived from (39).The proof is completed.
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.

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.

Table 1 :
The maximum allowable bounds of  = .