Journal of Control Science and Engineering

Volume 2012 (2012), Article ID 517157, 12 pages

doi:10.1155/2012/517157

Research Article

State Feedback Stabilization for Neutral-Type Neural Networks with Time-Varying Discrete and Unbounded Distributed Delays

Yantao Wang, Xue Lin, and Xian Zhang

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received 19 December 2011; Revised 27 March 2012; Accepted 21 April 2012

Academic Editor: Onur Toker

Copyright © 2012 Yantao Wang 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.

The problem of stabilization for a class of neutral-type neural networks with discrete and unbounded distributed delays is investigated. By introducing an appropriate Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative, delay-range-dependent and rate-dependent stabilization criteria are presented in the form of LMIs with nonlinear constraints. In order to solve the nonlinear problem, a cone complementarity linearization (CCL) algorithm is offered. In addition, several numerical examples are provided to illustrate the applicability of the proposed approach.

1. Introduction

In 1943, McCulloch and Pitts [1] proposed the concept of artificial neurons and proved that a single neuron can perform logic functions. This is usually viewed as the beginning of era of artificial neural networks, which is called neural networks for short. Recently, neural networks have received considerable attention due to their wide applications in solving some optimization problems, pattern recognition, image processing, and signal processing. Since time delay is frequently a source of instability and/or oscillation of many practical systems, several approaches have been proposed for analysis and synthesis of delayed neural networks (see [2–8] and the references therein).

A neutral-type time-delay system contains delays in the state and its derivatives. Such system can be found in population ecology, lossless transmission lines, heat exchangers, and so forth. Because of its wider applications, the stability for the class of neutral-type systems has received considerable attention in the last several decades (see, e.g., [9–13]). Corresponding to the class of neutral-type time-delay systems, we will also get a class of neutral-type neural networks. Stability criteria for neutral-type neural networks have been proposed in [3, 14–21] by constructing appropriate Lyapunov-Krasovskii functional and applying the linear matrix inequality (LMI) approach. Recently, the stability conditions for the neutral-type neural networks with discrete and unbounded distributed delays were provided in [17–21]. Specifically, Rakkiyappan and Balasubramaniam [21] studied the global asymptotic stability for a class of neutral-type neural networks with two unbounded distributed delays by the so-called LMI technique. Lu [22] designed a state feedback controller stabilizing the neutral-type neural networks with discrete and bounded distributed delays. However, there are no results about the stabilization for neutral-type neural networks with unbounded distributed delay(s), this motivates our research.

This paper deals with the stabilization problem of a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays. New sufficient conditions for the existence of the state feedback controller are proposed by introducing a Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative. Since these sufficient conditions are presented in the form of LMIs with nonlinear constraints, the so-called CCL technique is employed to deal with the nonlinear constraints. This allows us to obtain a state feedback gain by LMI Control Toolbox of MATLAB. Several numerical examples are given to illustrate the effectiveness of the proposed approach.

2. Problem Formulation

Consider the class of neutral-type neural networks with time-varying discrete and unbounded distributed delays described by the integro differential equation:

where is the state variable of the th neuron, is a constant, , , , , and are connection weight coefficients of the neurons, is the neuron activation function, is the external bias vector element, is the external control input vector, denotes the initial condition, and are time-varying delays satisfying

and is a real value nonnegative continuous function on and satisfies

When for all , the global asymptotic stability of the system (1) has been considered in [21]. Here, the control input item in (1) is borrowed from [22].

Throughout this paper, we assume that the neuron activation functions () satisfy the following condition:

where () are known positive constants.

Let be an equilibrium point of the system (1). Set ,   . Then the system (1) can be written as

where , , , , , , , , , with , and . From (5), we have , , for all .

The main purpose of this paper is to develop a delay-range-dependent and rate-dependent condition for the existence of a state feedback controller

which stabilizes the neutral-type neural network (6) with time-varying discrete and unbounded distributed delays.

A delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained based on a Lyapunov-Krasovskii functional approach accompanied with a CCL technique, a delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained. Set , . Assume that there exists a scalar such that and . This guarantees that the so-called Lyapunov-Krasovskii functional theory can be applied to the considered stabilization problem.

3. Main Results

In this section, we will investigate a delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6). This requires the following several lemmas.

Lemma 1 (Jensen inequality [23]). Given a real symmetry positive-definite matrix , a pair of scalars and satisfying . If a vector-valued function is derivable on , then

Lemma 2 (Schur complementary lemma [24]). For a given matrix with and , the following conditions are equivalent.

(i). (ii) and .(iii) and .

Lemma 3 (Cauchy-Schwarz inequality [25]). If the functions and are integral on , then

Based on the above three lemmas, the following delay-range-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6) can be investigated by constructing an appropriate Lyapunov-Krasovskii functional and applying the CCL technique.

Theorem 4. For given scalars , , , , , and such that and , the system (6) subject to (3) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices , , , , , a matrix ,  and positive scalars and ,  such that

where

Furthermore, when the LMIs (10)–(12) with the constraint (13) are feasible, a desired state feedback gain is given by .

Proof. Choose a Lyapunov-Krasovskii functional as follows:

with

Set . By some derivation, we have

where

which implies that the functional is a legitimate due to [26]. Next we deal with the derivatives of .

Set

When the controller (7) is applied to the system (6), the resultant closed-loop system is obtained as follows:

Due to Lemma 1, (3), the derivatives of () along the trajectory of (21) are

By Lemma 3 and (4), we can obtain the derivatives of and as follows:

For the functions , , using , we get that

where . Noting that , we have . The combination of (22)–(28) gives

where

By Lemma 2 and (10)–(13), we can get that

where

Thus it is easy to see that , which, together with (7) and [26, Theorem  3.1.6], implies that the system (6) is asymptotically stabilizable. The proof is completed.

Remark 5. A delay-range-dependent and rate-dependent condition under which the system (6) is asymptotically stabilizable is investigated in Theorem 4. In [7, 8], the stability of a class of the neural networks with unbounded distributed delays has been studied; however, the neutral term was not considered in their models. In [18], only a unbounded distributed delay is taken into account. But in this paper the neutral-type neural networks with two unbounded distributed delays are studied. Therefore, Theorem 4 in this paper can be viewed as an extension of the corresponding results in [7, 8, 18].

Remark 6. In order to apply [26, Theorem 3.1.6] to conclude that the system (6) is asymptotically stabilizable, we denote a new norm on a space of functions.

Noting that the stabilization criterion proposed in Theorem 4 includes the nonlinear constraint (13), we cannot obtain the desired gain by using the LMI Controller Toolbox of MATLAB. In order to deal with the nonlinear constraint (13), we offer the following CCL algorithm. The algorithm is available to determine the maximum value of for given , , , , and and the corresponding gain .

Algorithm 7 (CCL algorithm). Step 1. Choose a sufficiently small initial , such that there exists a feasible solution to (10)–(12) and

Set .

Step 2. Find a feasible set of , , , , , , , , , and satisfying (10)–(12), (33). Set .

Step 3. Solve the following LMI problem for the variables , , , , , , , , , , and :

where

Set , , , , , , , , and .

Step 4. If the LMI.

is feasible for the variables , , , and and the matrices and obtained in Step 3, then set and stop. If the LMI (36) is infeasible within a specified number of iterations, then stop; otherwise, set and go to Step 3.

Remark 8. The idea of the above algorithm is taken from [27]. Compared with the CCL algorithms proposed in [28, 29], the merit of the above algorithm is that more freedoms are provided to test the iteration stop conditions (see [27, Remark  4] for details).

Next, we consider the following neutral-type neural network (37), which is a special case of (6). A stability criterion for the neural network (37) has been presented in [21]. The following corollary can be immediately obtained from Theorem 4, which gives a stabilization criterion for the neutral-type neural network (37):

where

Corollary 9. For given scalars and such that , the neutral-type neural network (37) subject to (38) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices , and , a matrix and positive scalars and , such that

where

Furthermore, when the LMIs (39) with the constraint (40) are feasible, a desired state feedback gain is given by .

Proof. If we use the following and instead of and in the proof of Theorem 4, respectively, then the proof can be easily completed:

with

The following system model is a practical partial element equivalent circuit (PEEC) that is described in [30, Figure 1] by Bellen et al.:

where is a positive scalar representing the system delay.

It should be emphasized that the matrix in (44) is not necessarily diagonal, which is different from the one in (6). However, one can easily find that the proof of Theorem 4 is always available whether the matrix is diagonal or not. For this reason, we can derive the following Corollaries 10 and 12 by a method similar to obtaining Theorem 4.

Corollary 10. For a given scalar , the neutral-type system (44) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices , ,  , , , , and and a matrix , such that

where

Furthermore, when the LMIs (46) and (47) with the nonlinear constraint (48) are feasible, a desired state feedback gain is given by .

Remark 11. Similar to Algorithm 7, one can solve the stabilization criteria proposed in Corollaries 9 and 10.

Corollary 12. For a given scalar , the unforced system of (44) is asymptotically stable if there exist real symmetry positive-definite matrices and such that

where

and is defined as in Corollary 10.

4. Numerical Examples

In this section, we will illustrate our approach by several numerical examples.

Example 13. Consider the neutral-type neural network (6), where

Solving the LMIs (10)–(12) with nonlinear constraint (13) by Algorithm 7 and the LMI Control Toolbox of MATLAB, we can get a desired state feedback gain stabilizing the system as follows:

The state response curves of the closed-loop system are given in Figures 1 and 2, where the initial functions in Figures 1 and 2 are, respectively, chosen as , and , .

(a)

(b)

Figure 1: State responses of the closed-loop system for , .

(a)

(b)

Figure 2: State responses of the closed-loop system for , .

Example 14. Consider the system (37) with the following parameter matrices:

When by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.1] is not feasible, so it cannot guarantee that the unforced system is stable. Now we check the stabilization condition proposed in Corollary 9 of this paper and find that it is feasible. So, by Corollary 9 the system is asymptotically stabilizable by state feedback controller, and a desired state feedback gain is given by

and, further, the state response curves of the closed-loop system are given in Figure 3.

When , by the LMI Control Toolbox of MATLAB, it can be verified that the LMI in [21, Theorem 3.2] is not feasible, which cannot guarantee the unforced system is stable. However, the stabilization condition proposed in Corollary 9 can be satisfied. Hence, by Corollary 9, the neutral-type neural network is asymptotically stabilizable via state feedback controller, and a desired state feedback controller gain is given by

and, further, the state response curves of the closed-loop system are given in Figure 4.

(a)

(b)

Figure 3: State responses of the closed-loop system when .

(a)

(b)

Figure 4: State responses of the closed-loop system when .

Example 15. Consider the PEEC system (44) with the following parameter matrices:

Applying Corollaries 12 and 10, the maximum delay bounds for the various are presented in Table 1, which clearly shows the effectiveness of our approach. Through Table 1, we can get that the stability criterion proposed in this paper is less conservative than the ones in [31–33].

The following example is provided for applying the approach proposed in this paper to a realistic problem which is motivated by the small PEEC model in [30, Figure 1].

Table 1: Allowable upper bound of for different .

[31, Theorem 3.1]

[32, Theorem  1]

[33, Theorem  1]

Corollary  12  in this paper

Corollary  10  in this paper

Example 16. Consider the PEEC system (44) with

When , Bellen et al. [30] gave sufficient conditions for the asymptotic stability of the zero solution to (44) by utilizing a suitable reformulation of the system.

By applying Algorithm 7 and the LMI Control Toolbox of MATLAB to solve LMIs (46) and (47) with the nonlinear constraint (48), a required state feedback gain is obtained as follows:

The simulation results for the PEEC system are shown by Figures 5 and 6 with the initial value . Figure 5 represents the state responses of the PEEC system when , and Figure 6 represents the state responses of the resultant closed-loop system. From Figures 5 and 6, we can see that both the open-loop and closed-loop systems are stable, however, the speed of convergence towards the null point in the closed-loop system is faster than one in the open-loop system.

Figure 5: State responses of the open-loop system.

Figure 6: State responses of the closed-loop system.

5. Conclusions

In this paper, the delay-range-dependent and rate-dependent stabilization criteria for a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays are established. The criteria are derived by constructing an appropriate Lyapunov-Krasovskii functional and using certain matrix technique. A CCL algorithm is developed to obtain the state feedback gain . Numerical examples are provided to show that our results are more suitable than some existing ones, which illustrate the merits of the proposed approach.

Acknowledgments

This work is supported by the fund of Heilongjiang Education Committee under Grant no. 12521429 and the fund of Heilongjiang University Innovation Team Support Plan under Grant no. Hdtd2010-03. The authors thank the anonymous referees for their helpful comments and suggestions that improved greatly this paper.

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