Stability Analysis for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Delays

Abstract

This paper is concerned with the stability problem of a class of discrete-time stochastic fuzzy neural networks with mixed delays. New Lyapunov-Krasovskii functions are proposed and free weight matrices are introduced. The novel sufficient conditions for the stability of discrete-time stochastic fuzzy neural networks with mixed delays are established in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate the effectiveness and benefits of the proposed method.

1. Introduction

Such applications of neural networks heavily depend on the dynamical behaviors of the networks. Because neural network systems and their various generalizations have strong self-adaptive ability, learning ability, robustness, and fault tolerance, they have been extensively applied to intelligent signal processing, speech recognition, data mining, robot control, earthquake exploration, and so on.

As is well-known, the existence-uniqueness and stability constitute a hot topic in the field of dynamics systems (e.g., see [1–23] and references therein). Just as dynamics systems, the main task of the neural networks designers is to ensure the stability of the equilibrium point of the designed system. At the same time, time-delay and stochastic disturbance are often considered as two main resources to affect the work performance of different neural networks systems. So in theory research, in order to keep the reliability of systems, the researchers have taken stability theory, Lyapunov functional technique, line matrix inequality and free weight matrix method, time-delay decomposition methods to guarantee stability and low conservatism of neural networks systems. In recent many decades, the stability and passivity problem for stochastic neural networks with different time-delays were investigated widely, and many important results were reported (see, e.g., [24–37] and references therein).

Furthermore, most physical systems in the real world are nonlinear ones; fuzzy model has been widely used as an efficient method to deal with nonlinear systems because the controlled plant does not need exact mathematical model. When establishing dynamical systems, fuzzy systems in the form of Takagi-Sugeno (T-S) model, being described by a set of IF-THEN rules and nonlinear membership functions, have been extensively adapted. By a T-S fuzzy model, a nonlinear system can be described as a weighted sum of some simple linear subsystems. Recently, it has been used as an efficient tool when approximating the complex nonlinear systems. During the past many years, many results on the research of fuzzy neural networks were reported [38–43]. In [43], the authors studied the stability of fuzzy Markovian jump neural networks. In [42], by utilizing the inequality technic and free-weighting matrices, constructing novel Lyapunov functions, some new results about the robust exponential stability issue of the fuzzy uncertain neural networks were obtained, but adding on the free weight matrices methods has increased computation load and the conservatism of system. Also, a lot of literature published new research results about stability analysis of fuzzy neural networks with different time-delays [38, 39] and many new ideas were introduced. In [39], mean square exponential stability of the fuzzy neural networks with mixed delays was studied, but time-delay was only the constant one, which is not practical in the real applications. In particular, when implementing the delayed continuous-time neural networks for computer simulation, it becomes essential to formulate discrete-time neural network that is an analogue of the continuous-time delayed neural network.

Motivated by the above discussion, in this paper we investigate the stability problem for a class of discrete-time stochastic fuzzy neural networks system with mixed time-varying delays. The mixed delays consist of both discrete and distributed delays. One of the main contributions in this paper is that our obtained results reduces the conservatism than the existing achievement. In order to reduce the conservatism, many inequalities technic are utilized and new Lyapunov-Krasovskii functions are proposed. Novel stability criteria are derived in terms of LMIs. Finally, numerical examples are given to show the new established results are less conservative than the existing ones.

Throughout this paper, denotes the -dimensional Euclidean space, and is the set of real matrices. is the identity matrix. denotes Euclidean norm for vectors. is a complete probability space with a filtration satisfying the usual conditions. stands for the transpose of the matrix . For symmetric matrices and , the notation (respectively, ) means that the is positive definite (respectively, positive semidefinite). denotes a block that is readily inferred by symmetry. stands for the mathematical expectation operator with respect to the given probability measure .

2. Problem Description

Consider the following discrete stochastic fuzzy neural networks with mixed delays:

Rule : IF is and is and and is , THENwhere are the fuzzy sets, is the state vector of neural networks, and are the neural activation functions, are positive diagonal matrices representing the self-feedback term with , is the connection weighting matrix, are the time-delay connection matrices, r is the number of IF-THEN rules, is the time-varying delay and satisfies , and is a scalar Wiener process (Brownian motion) on withThen, the final model of discrete-time stochastic fuzzy neural networks is described aswhere =, =, and is the grade of membership of in . Suppose , , and for all . Therefor, for and for all .

Assumption 1. The function is Borel measurable and is locally Lipschitz continuous, satisfying the following assumption:

There exist two positive constants , such that

Assumption 2. For any , i=1,2… n,and, for presentation convenience, we defineThe following Lemmas will be essential in the proof of main results.

Lemma 3 (see [44]). For any constant positive-definite matrix , , two positive scalars , such that the sums concerned are well defined, then

Lemma 4 (see [45]). Suppose that , , are constant matrices of appropriate dimensions, ; thenholds, if the following inequalities hold simultaneously

Lemma 5 (see [46]). For the symmetric appropriate dimensional matrices , , constant matrix , the following two statements are equivalent:
(i)  ,
(ii) There exists a matrix of appropriate dimension such that

Lemma 6 (see [46]). Let : have positive values in an open subset D of . Then, the reciprocally convex combination of over D satisfies subject to

Lemma 7 (see [47]). Let be a positive semidefinite matrix, , and . If the series concerned are convergent, the following inequality holds:

3. Main Results

Theorem 8. System (4) is said to be asymptotically stable, if there exist symmetric positive matrices , , , , , , , , , diagonal matrices , , , , , matrices , , , of appropriate dimensions, and a scalar such that the following LMIs hold:whereand

Proof. Choose the following Lyapunov functionalBy defining , calculating the difference of along the trajectory of neural networks (4), and taking the mathematical expectation, we getBy Jensen’s inequality ([48]), it is observedand from convex reciprocal Lemma 6, we getwhere , . If there exists matrix such that holds, thenwhich leads toBy utilizing Lemma 3, we can getwhereBy Lemma 6 again, we have the following.By Lemma 3 again, we havewhereAlso, from Lemma 7 we can get thatSince , we haveThen for any matrix , we haveFrom Assumption 2, we havewhich lead towhere denotes the unit column vector having the element 1 on its rth row and zeros elsewhere. Let , , , thenSimilarly, we can getAlso, from Assumption 1, we get thatCombining (28)-(53), we havewhereFrom LMI (15), we can get that for . Then by Lemma 4, the terms guaranteewhich makes the following inequality true:By Lemma 5 and (58), we can get thatThen, there must exist a positive scalar such thatBy Lyapunov stability theory, the neural network systems is globally asymptotically stable; this completes the proof.

Remark 9. If we have the following discrete-time fuzzy neural network system without the distributed delays:for the neural network system (61), we have the following stability results.

Corollary 10. System (61) is said to be exponentially stable, if there exist symmetric positive matrices , , , , , , , , , diagonal matrices , , , , matrices , , , , and scalar such that the following LMIs hold:where

Proof. The proof is similar to that of Theorem 8 and hence is omitted here.

Remark 11. If we have the following deterministic neural network system without the distributed delays: