Exponential synchronization of coupled fuzzy neural networks with disturbances and mixed time-delays☆
Introduction
Recently, neural networks have been extensively applied in many areas, such as pattern recognition, image processing, secure communication, automatic control and associative memory, etc. [1], [2], [3], [4], [5], [6]. These applications greatly depend on the dynamical behaviors of networks. It is worth noting that realistic systems are often affected by some stochastic disturbances which may give rise to instability and poor performance. Considerable efforts [7], [8], [9], [10], [12], [13], [14], [15], [20], [24], [25], [27], [36] have been made to explore the mechanism of random noises.
In addition, it was pointed out in [5] that the combination of a set of neural networks could achieve higher level information processing. It is also an important issue to understand how the collective behaviors emerge from coupled systems [10]. As a typical collective behaviors, synchronization has gained increasing research interests [9], [10], [11], [12], [15], [16], [17], [18], [19], [23], [24], [25], [26], [28], [29], [31], [32], [33]. Taking constant, discrete-delay and distributed-delay couplings into consideration, the authors of [11] derived the global synchronization criteria for coupled neural networks. In [12], this result was extended to a discrete-time model with stochastic nonlinearities and mixed time-delays.
Generally, the exact modeling of practical systems is not always available. In traditional neural networks, present knowledge cannot be fully utilized to improve the network structure and speed up knowledge acquiring [20]. In this case, pre-assuming a certain model for dynamical system may lead to unpractical results. Nowadays, it has been wildly recognized that fuzzy logic theory is an efficient approach to deal with such problems. Since fuzzy logics were translated into cellular neural network [21], fuzzy cellular neural network (FCNN) has received widespread attentions [5], [20], [21], [22], [23], [24], [25], [26], [27]. Furthermore, it was proved in [25] that fuzziness could bring about more complexities. Therefore, studying the dynamical behaviors of fuzzy systems has both theoretical and practical significance.
However, the fuzzy logics and network synchronizability defined as the ability that system structure guarantees synchronization [19] are rarely considered in coupled stochastic neural networks with mixed time-delays. All of these motivate our work. In this paper, we study the exponential synchronization problem of coupled fuzzy neural networks (CFNNs) with disturbances and mixed time-delays. Time-delays are mixed in the sense that both discrete delay and distributed delay are involved. The former is employed to describe the propagation delay among neurons, and the latter is utilized to characterize the spatial nature of neural networks [9]. We aim to develop some sufficient criteria that guarantee the mean square exponential synchronization of CFNNs, and study the effects of network structure on network synchronizability.
This paper is organized as follows. In Section 2, the model is introduced. Sufficient conditions that ensure the mean square exponential synchronization of CFNNs are stated in Section 3. In Section 4, two numerical examples are given, and conclusion is made in Section 5.
Notations: and denote the n-dimensional Euclidean space and the set of m×n real matrices, respectively. For vector . For real symmetric matrix () means is negative definite (positive definite). We let denote the family of continuous functions from to with norm . Let be a complete probability space. Denote by the family of all random variables satisfying where stands for the mathematical expectation operator. and are separately the maximal and minimum eigenvalue of square matrix . denotes the determinant of . is a diagonal matrix with diagonal elements . Denote by In the n×n identity matrix. represents the Kronecker product of matrices.
Section snippets
Problem formulations and preliminaries
The fuzzy cellular neural network with Markovian switching parameters is given bywhere and are, respectively, the state variables and external inputs
Main results
In this section, some criteria are established in terms of linear matrix inequalities (LMIs) that guarantee the mean square exponential synchronization of CFNNs.
In order to prove that the CFNNs (2) (or (3)) is exponentially synchronized in the mean square, we need to ensure the exponential mean square stability of (4).
For simplifying representation, we denote , , , , ,
Numerical examples
In this section, two numerical examples are given to verify our results obtained in Section 3.
Consider the CFNNs (2) (or (3)) with Markovian switching between two modes , in which , , ,
Conclusion
In this paper, the network synchronizability of CFNNs with stochastic noises and mixed time-delays is discussed under a mild topology-based assumption. It is showed that coupling strength and spanning tree play a key role in synchronization process. LMI-based conditions are established to guarantee the exponential synchronization in the mean square sense for CFNNs.
Based on the properties of energy, passivity theory was proposed to analyze the input and output in neural networks [36]. We will
Guan Wang received his M.S. degree in Systems Engineering from Huazhong University of Science and Technology, Wuhan, China, in 2009. He is currently working toward his Ph.D. degree at the Department of Control Science and Engineering, Huazhong University of Science and Technology. His current research interests include neural networks and complex systems.
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Guan Wang received his M.S. degree in Systems Engineering from Huazhong University of Science and Technology, Wuhan, China, in 2009. He is currently working toward his Ph.D. degree at the Department of Control Science and Engineering, Huazhong University of Science and Technology. His current research interests include neural networks and complex systems.
Quan Yin received the B.S., M.S. degrees in Industrial Electrical Automation, and Ph.D. degree in Control Theory and Control Engineering from Huazhong University of Science and Technology, Wuhan, China, in 1990, 1995 and 2001, respectively. He was a Post-Doctoral Fellow with the Huazhong University of Science and Technology from July 2001 to June 2003. He is currently a Associate Professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He is the author of more than 30 research papers. His current research interests include intelligent control and embedded control system.
Yi Shen received the Ph.D. degree from Huazhong University of Science and Technology, Wuhan, China, in 1998. He was a Postdoctoral Research Fellow at the Huazhong University of Science and Technology during 1999–2001. He joined the Department of Control Science and Engineering, Huazhong University of Science and Technology, and became an Associated Professor in 2001 and was promoted to Professor in 2005. From 2006 to 2007 he was a Postdoctoral Research Fellow at the Department of Mechanical and Automation Engineering, the Chinese University of Hong Kong, Hong Kong. He has authored over 80 research papers. His main research interests lie in the fields of stochastic control and neural networks.