Brief PapersNew mean square exponential stability condition of stochastic fuzzy neural networks☆
Introduction
Over the past years, neural networks have been widely used in many practical applications such as signal processing, combinatorial optimization and image processing. Many research results have been proposed for neural networks, such as [1], [2], [3], [4], [5], [6], [7], [8], and the references therein. In real nervous systems, it is easy to see that stochastic disturbances are nearly inevitable and affect the stability of the neural networks. Hence, it implies that the stability analysis of stochastic neural networks has primary significance in the research of neural networks. Some related research results have been published in [9], [10], [11], [12], [13].
Fuzzy logic control scheme has been proposed as an effective method to deal with complex nonlinear systems [14], [15], [16]. In recent years, the fuzzy logic control method has been widely used in many practical applications such as chemical processes, automotive systems and robotics systems [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. Takagi–Sugeno (T–S) fuzzy system [27] is a well-known fuzzy system in model-based fuzzy control. The results on stability analysis and stabilization, control problem and filtering problem for T–S fuzzy systems were reported in [28], [29], [30], [31], [32], [33]. Recently, the T–S fuzzy control approach has been used in the study of stochastic fuzzy neural network [34], [35], [36], [37], [38]. In [35], the authors considered the mean square exponential stability problem of stochastic fuzzy neural networks with time-varying delays. However, it should be mentioned that the above research results are based on type-1 T–S stochastic fuzzy neural networks. By using the type-1 T–S fuzzy model, the stability problem cannot be studied if the nonlinear plants contain parameter uncertainties. Once nonlinear plants contain parameter uncertainties, it will result in the uncertainties of membership functions. To solve this problem, the authors in [39] proposed an IT2 fuzzy logic model, which can be utilized to investigate the nonlinear plants problem. Recently, the authors in [40], [41] have studied the problem of the IT2 fuzzy systems by using the upper and lower membership functions which can be used to deal with parameter uncertainties problem. However, it is mentioned that, so far, there are few results on stability analysis of IT2 stochastic fuzzy neural networks.
Motivated by the above discussion, this paper investigates the stability problem for IT2 stochastic fuzzy neural networks subject to parameter uncertainties. The parameter uncertainties, which can be obtained by the lower and upper membership functions, are bounded. By using stochastic analysis approach and Itô differential formula, a novel sufficient condition ensuring mean square exponential stability for stochastic fuzzy neural networks is achieved. Two examples are given to show the feasibility of the proposed scheme. The remaining parts of this paper are organized as follows. Section 2 introduces IT2 stochastic fuzzy neural networks. Section 3 proposes stability condition for IT2 stochastic fuzzy neural networks and Section 4 provides two illustrative examples to show the merits of the proposed results. Section 5 concludes this paper.
Notation: The superscript “T” denotes matrix transposition, and “−1” stands for matrix inverse. is the n-dimensional Euclidean space and the notation stands for a symmetric and positive definite (semi-definite) matrix. The symbol within a symmetric block matrix represents the symmetric terms, and denotes a block-diagonal matrix. is defined as for simplicity. If not explicitly stated, all matrices are assumed to be compatible dimensions for algebraic operations. stands for the Hilbert space of square integrable functions over means the expectation of the stochastic variable x.
Section snippets
Problem formulation
As discussed in [34], we consider the following IT2 stochastic fuzzy neural network.
Plant Rule η1i where denotes the premise variable and ηai is an IT2 fuzzy set, , . p is a positive integer. stands for the neural state, denotes the neuron activation function, and w(t) is a one-dimensional Brownian
Main results
In this section, the stability condition for the IT2 stochastic fuzzy neural network (2) is first proposed in Theorem 1. Based on the linear matrix inequality (LMI) approach, we can have the following theorem. For presentation convenience, we denote , . Theorem 1 The IT2 stochastic fuzzy neural network (2) is mean square exponentially stable, if there exist matrices , , , with appropriate
Simulation results
In this section, two examples are used to show the effectiveness and the merits of the proposed method. Example 1 Consider the IT2 stochastic fuzzy neural network (2) with the parameters as follows:
Take the activation functions as and . Therefore, it is easy to have , . Membership
Conclusions
In this paper, the stability problem has been considered for a class of IT2 stochastic fuzzy neural networks. Firstly, the IT2 stochastic fuzzy neural networks have been constructed. Sufficient condition has been given to ensure that the IT2 stochastic fuzzy neural network is mean square exponential stability. The existence condition of the stability can be obtained in terms of LMI. Two numerical examples have been given to show the merits of the proposed approach.
Xing Xing was born in Liaoning Province, China, in 1982. He obtained the M.S. degree in Computer Software and Theory in 2008, and the Ph.D. degree in Computer Applied Technology in 2013, from Dalian Maritime University, China. He was a visiting student at the Department of Computer Science, University of New Mexico, USA, from September 2009 to September 2011. Currently, he is an associate professor at the Department of Information Science and Technology, Bohai University, China. From February
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Xing Xing was born in Liaoning Province, China, in 1982. He obtained the M.S. degree in Computer Software and Theory in 2008, and the Ph.D. degree in Computer Applied Technology in 2013, from Dalian Maritime University, China. He was a visiting student at the Department of Computer Science, University of New Mexico, USA, from September 2009 to September 2011. Currently, he is an associate professor at the Department of Information Science and Technology, Bohai University, China. From February 2014, he carries out his postdoctoral research in the School of Astronautics, Harbin Institute of Technology. His research interests include data mining, machine learning, and neural networks.
Yingnan Pan received the B.S. degree in mathematics from Bohai University, Jinzhou, China, in 2012. He is studying for M.S. degree in applied mathematics in Bohai University, Jinzhou, China. His research interests include fuzzy control, robust control and their applications.
Qing Lu received the B.E. degree in automation from Hebei University of Technology, Tianjin, China in 2013, and is studying for M.E. degree in Control Theory and Control Engineering in Automation Research Institute, Bohai University, Jinzhou, China. She is a master degree candidate in the College of Engineering, Bohai University, Jinzhou, China. Her research interest includes fuzzy control and model predictive control.
Hongxia Cui received the B.Sc. degree in photogrammetry and remote sensing from Wuhan University, Hubei Province, China, in 1991, the M.E. degree in computer science from Shanxi University, Shanxi, China, in 2002 and the Ph.D. degree in photogrammetry and remote sensing from Wuhan University, Hubei Province, China, in 2006. She is currently a professor at the College of Computer Science and Technology, Bohai University. Her current research interests include image processing, computer vision, neural networks and their applications.