Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 8529053, 13 pages
https://doi.org/10.1155/2019/8529053

Research Article

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

YaJun Li¹ and Quanxin Zhu²

¹Department of Electronics and Information Engineering, Shunde Polytechnic, Foshan, 528300, China
²MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, China

Correspondence should be addressed to Quanxin Zhu; zqx22@126.com

Received 6 November 2018; Accepted 30 January 2019; Published 14 February 2019

Academic Editor: Alessandro De Luca

Copyright © 2019 YaJun Li and Quanxin Zhu. 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.

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 , THEN

where 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 with

Then, the final model of discrete-time stochastic fuzzy neural networks is described as

where =, =, 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 define

The 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, ; then

holds, 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:

where

and

Proof. Choose the following Lyapunov functional

By defining , calculating the difference of along the trajectory of neural networks (4), and taking the mathematical expectation, we get

By Jensen’s inequality ([48]), it is observed

and from convex reciprocal Lemma 6, we get

where , . If there exists matrix such that holds, then

which leads to

By utilizing Lemma 3, we can get

where

By Lemma 6 again, we have the following.

By Lemma 3 again, we have

where

Also, from Lemma 7 we can get that

Since , we have

Then for any matrix , we have

From Assumption 2, we have

which lead to

where denotes the unit column vector having the element 1 on its rth row and zeros elsewhere. Let , , , then

Similarly, we can get

Also, from Assumption 1, we get that

Combining (28)-(53), we have

where

From LMI (15), we can get that for . Then by Lemma 4, the terms guarantee

which makes the following inequality true:

By Lemma 5 and (58), we can get that

Then, there must exist a positive scalar such that

By 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:

for the neural network system (67), Corollary 12 can be obtained.

Corollary 12. The neural network (67) is said to be exponentially stable if there exist symmetric positive matrices , , , , , , , , , diagonal matrices , , , , matrices , , , , and a scalar such that the following LMIs hold:

where

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

Remark 13. If we have the following fuzzy neural network system with mixed delays:

for the neural network system (73), Corollary 14 can be obtained.

Corollary 14. System (73) is said to be exponentially stable if there exist symmetric positive matrices , , , , , , , , , diagonal matrices , , , , matrices , , , of appropriate dimensions, and a scalar such that the following LMIs hold:

where

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

Remark 15. In [39], the stability problem for the discrete-time stochastic fuzzy neural networks with constant time-delay is studied. Compared with the stochastic fuzzy neural network system in [39], system (73) in this paper has taken the time-varying delay into consideration. Time-varying delays exist in most practical applications and the constant delay is a special case of it. From this point of view, it is more practical to take the time-varying delays into consideration when studying the fuzzy neural networks.

4. Numerical Examples

Example 1. Consider the neural networks (61) with

The activation functions are described by and . Then from Assumption 2 we can get and we have =diag, =diag, =diag, =diag. Let , by solving the LMIs in Corollary 10, the allowable upper bounds of the time-delay for different are shown in Table 1.

Table 1: Maximum for given in Corollary 10.

2
4
6
8
10

[42]
5
7
9
11
13

Corollary 10
8
10
12
14
16

When the initial condition is with , and the fuzzy membership functions are taken as

the simulation results of neural networks (61) are plotted in Figure 1. It is clearly observed from the simulation result that the neural networks system (61) is globally asymptotically stable.

Figure 1: State trajectory of the neural network (61).

Example 2. Consider the neural networks (67) with

The activation functions are described by . Then from Assumption 2 we can get that and =diag, =diag, =diag, =diag. Let , by solving the LMIs (68)-(71) in Corollary 12, we can obtain the allowable upper bounds of the time-delay for different in Table 2.

Table 2: Calculated maximum for given for Corollary 12.

2
4
6
8
10

[34]
11
12
13
14
16

Corollary 12
16
17
18
19
21

When the initial condition is with , the simulation results of neural networks are plotted in Figure 2. It is clearly observed from the simulation results that the neural networks system (67) is globally asymptotically stable.

Figure 2: State trajectory of the neural network (67).

Example 3. Consider the neural networks (4) with

The activation functions are described by and . Then from Assumption 2 we can get that and , then we have =diag, =diag, =diag, =diag and =diag, =diag, . Let , by solving the LMIs in Theorem 8, the allowable upper bounds of the time-delay for different are listed in Table 3.

Table 3: Calculated maximum for given for Theorem 8.

2
4
6
8
10

Theorem 8
4
6
8
10
12

When the initial condition is set by with and the fuzzy membership functions are taken as

the state trajectory of the neural network system (4) is plotted in Figure 3. It is clearly shown from the simulation result that the neural networks system (4) is asymptotically stable.

Figure 3: State trajectory of the neural network (4).

5. Conclusions

The stability problem for a class of discrete-time stochastic fuzzy neural networks system with mixed time-delays has been investigated. With the purpose of reducing the conservatism and obtaining larger time-delay bounds, some new Lyapunov-Krasovskii functions and free-weighting matrices are introduced. The sufficient condition which makes the studied system asymptotically stable has been established in terms of LMIs. Numerical examples and simulations are given to illustrate the effectiveness of our presented method compared with the existing result. It should be mentioned that the proposed method in this paper can be extended to the passivity analysis, robustness research, and filter design of fuzzy neural network systems.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors have made the same contribution. All authors read and approved the final manuscript.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (61773217, 61374080), the Natural Science Foundation of Jiangsu Province (BK20161552), the Construct Program of the Key Discipline in Hunan Province, and the Nature Science Fund of Guang Dong Province of China (No. 2015A030310336 and No. 2017A030310640).

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