Delay-dependent robust stability and stabilization of uncertain memristive delay neural networks
Introduction
Motivated by many systems in science and humanities [1], [2], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], memristive neural networks which consist of bio-inspired neuron oscillatory circuits with nanoscale memristors have attracted extensive interest in both modeling studies and neurobiological research in the past few years due to their feasibility to achieve the large connectively and highly parallel processing power of biological systems [3], [4], [16], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. Since memristive neural networks were introduced by Hu and Wang [3], many researchers have paid attention to the new neural networks model. Research shows that memristive neural networks have many important applications in the fields of pattern recognition, signal processing, optimization and associative memories, especially since they can remember their past dynamical history, store a continuous set of states, and be plastic according to the pre-synaptic and post-synaptic neuronal activity. With more and more successful research results about memristive neural networks being published, we find that it will help us build a brain-like machine to implement the synapses of biological systems. However, the existing memristive neural networks which many researchers had constructed have been found to be computationally restrictive. In these circumstances, the applicability of these memristive neural networks in this area only has limited success.
Recently, we note that many researchers have turned their attention to the dynamical analysis of memristive neural networks [3], [16], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. Furthermore, an interesting issue is to investigate the dynamic behavior of memristor-based recurrent neural networks, an ideal model for the case where the memristor-based circuit networks exhibit complex switching phenomena. Hu and Wang [3] considered the global asymptotic stability of memristor-based recurrent neural networks. Wu et al. [32], [33] considered the synchronization control of a class of memristor-based neural networks. Wen [35], [36], [37], [38], [39], [40], [41] presented dynamic behaviors of a class of memristor-based recurrent neural networks with time-varying delay. Zhang et al. [16] presented some sufficient conditions for exponential stability for memristor-based recurrent neural networks. However, most of the studies overlook the problem that the noise may destroy the stability of the memristive neural networks. So we should further study the memristive neural networks, i.e. conduct robust stability analysis. To the best of authors׳ knowledge, there are few results in the open literature dealing with the delay-dependent asymptotical stability and robust control design for the memristive neural networks. More importantly, further studies of the dynamic analysis of memristive neural networks will make us build a brain-like model more precisely.
We also note that noise and some other external disturbances are often the sources of instability and they may destabilize stable neural networks. Generally, these disturbances could easily cause the neural network to be unstable. Although various stability properties of neural networks have been analyzed extensively in recent years [8], [9], [28], [29], [30], [31],43,44] the robustness of the uncertain memristive neural networks is rarely investigated directly. Motivated by the aforementioned discussion, this paper is concerned with the problem of delay-dependent robust stability analysis and robust control design for uncertain memristive neural networks with delay state and norm-bounded parameter uncertainty. We consider the case of a single constant time-delay. The focal point of this paper is on developing methods for robust stability analysis and robust stabilization based on linear matrix inequalities [28] and depend on the size of time-delay.
The paper is organized as follows: In the next section, the problems investigated in this paper are formulated and some preliminaries are presented. In Section 3, the delay-dependent robust stability and robust control design are derived. Several numerical examples and simulations including comparison analysis of conservatism are presented in Section 4. Finally,some conclusions are drawn in Section 5.
Section snippets
Problem formulation and preliminaries
Consider uncertain memristive time-delay neural networks described by the following systems:where denotes the state of memristive neuron, and are feedback activation functions; ui(t) denotes the control inputs and τj is time delay satisfying , where is a real constant. are
Main results
In this sequel, we shall present our main results concerning delay-dependent robust stability analysis and robust stabilization.Before we present the results, the following assumption which is useful for the main results proof procedure is presented firstly.
(A): Robust stability analysis: Assumption 3.1 The matrix A+B has all eigenvalues in the open left-plane. In connection with (7), we introduce the matrix function
Simulation examples
In this section, two example are given to illustrate our results. Simulation results show that the obtained conclusions are valid. Example 1 We study numerically the following system, by applying Theory 1 presented in the previous sectionswhere
Conclusion
Memristive neural network is a ground-breaking concept that is helping us understand the behavior of many physical, biological, social,and technical systems. It is significant to study memristive neural networks, which can approximately simulate synapses so that they behave like the real thing about the memristor minds to some extent. The problems of robust stability analysis and robust stabilization of uncertain memristive neural networks have been addressed.We considered systems described by
Acknowledgments
This publication was made possible by NPRP grant #NPRP-4-1162-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work was also supported by National Natural Science Foundation of China (grant no:61374078), the Graduate Innovation Foundation of Chongqing University Grand NO.CDJXS12 18 00 05.
Xin Wang got his B.Sc. degree from Hunan University of Arts and Science, Changde, China, in 2010. Now he is a Ph.D. candidate with the Department of Computer Science at Chongqing University, Chongqing, China. His research interests include neural network, memristor, and impulsive system theory.
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Xin Wang got his B.Sc. degree from Hunan University of Arts and Science, Changde, China, in 2010. Now he is a Ph.D. candidate with the Department of Computer Science at Chongqing University, Chongqing, China. His research interests include neural network, memristor, and impulsive system theory.
Chuandong Li received his B.S. degree in Applied Mathematics from Sichuan University, Chengdu, China, in 1992, and M.S. degree in operational research and control theory and Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2001 and in 2005, respectively. He has been a Professor at the College of Computer Science, Chongqing University, Chongqing 400044, China, since 2007, and been IEEE Senior member since 2010. From November 2006 to November 2008, he serves as a research fellow in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China. He has published more than 100 journal papers. His current research interest covers computational intelligence, neural networks, memristive systems, chaos control and synchronization, and impulsive dynamical systems.
Tingwen Huang obtained his B.S. degree from Southwest Normal University in 1990, M.S. degree from Sichuan University in 1993 and Ph.D. from Texas A&M University in 2002. Since he graduated at Texas A&M University, he has been working in the Mathematics Department of Texas A&M University as a Visiting Assistant Professor. In 2003, he worked at Texas A&M University at Qatar until recently. He is now an associate professor of Mathematics. His research fields include neural networks, chaos and its applications, etc. He has published about 30 journal papers on neural networks and nonlinear dynamics.