Exponential state estimation for Markovian jumping neural networks with mixed time-varying delays and discontinuous acti
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ORIGINAL ARTICLE
Exponential state estimation for Markovian jumping neural networks with mixed time-varying delays and discontinuous activation functions Huaiqin Wu1 • Leifei Wang1 • Yu Wang1 • Peifeng Niu2 • Bolin Fang3
Received: 17 February 2015 / Accepted: 14 October 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract This paper is concerned with the exponential state estimation issue for Markovian jumping neural networks with mixed time-varying delays and discontinuous activation functions. By introducing triple-integral terms and quadruple integrals term in Lyapunov–Krasovskii functional, the obtained Lyapunov matrices are distinct for different system modes. Based on the nonsmooth analysis theory and by applying stochastic analysis techniques, the full-order state estimator is designed to ensure that the corresponding error system is exponentially stable in mean square. The desired mode-dependent and delay-dependent estimator can be achieved by solving a set of linear matrix inequalities. Finally, two simulation examples are given to illustrate the validity of the theoretical results. Keywords Neural networks State estimation Discontinuous neuron activations Markovian jumping parameters Mixed time-varying delays
1 Introduction In the past few decades, neural networks (NNs), which usually show stability, periodic oscillation or chaotic behavior, have been a subject of intense research activities & Huaiqin Wu [email protected] 1
School of Science, Yanshan University, 066001 Qinhuangdao, China
2
School of Electrical Engineering, Yanshan University, 066001 Qinhuangdao, China
3
School of Information Science and Engineering, Yanshan University, 066001 Qinhuangdao, China
due to their wide applications in different areas such as image processing, pattern recognition, associative memory and combinational optimization, see [1]. In hardware implementation of NNs, time delay is inevitably encountered, and is usually discrete and distributed due to the finite switching speed of amplifiers. It is known that time delay is often the main cause for instability and poor performance of NNs. Therefore, the dynamics of NNs with time-delays have been an attractive and important subject of research, and a variety of interesting results have been reported, see [2–9, 15–31] and references therein. Recently, in [7], Forti et al. firstly discussed the global convergence of NNs with discontinuous activation functions. Subsequently, dynamical behaviors of NNs with delays and discontinuous activation functions have received considerable attention. It should be pointed out that most of existing papers concerning NNs with discontinuous activations focused on the stability of of periodic and almost periodic solutions [8–12] or even equilibrium points [13–16]. In the design of NNs, it is noted that the structures of real-world network systems are always variable and the information latching phenomenon often appears since the sudden external disturbances and environment changes. When solving these problems, the finite-s
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