LMIs conditions to robust pinning synchronization of uncertain fractional-order neural networks with discontinuous activ
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LMIs conditions to robust pinning synchronization of uncertain fractional-order neural networks with discontinuous activations Xinxin Zhang1 • Yunpeng Ma1 Published online: 24 September 2020 Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper deals with the robust pinning synchronization issue of uncertain fractional-order neural networks with discontinuous activations (FNNDAs) by means of the linear matrix inequalities (LMIs). In this paper, a class of FNNDAs model is presented. Moreover, an appropriate pinning controller is designed to ensure the error dynamical system gets robust Mittag–Leffler stability via Lyapunov function approach, non-smooth analysis theory and inequality analysis technique. In addition, the robust pinning synchronization conditions of FNNDAs drive system and FNNDAs response system are obtained in terms of the LMIs. Finally, a typical numerical simulation is provided to show the effectiveness of the obtained results. Keywords Fractional-order neural networks Discontinuous activations Robust pinning synchronization Lyapunov function Linear matrix inequality (LMI)
1 Introduction Compared with integer order models, a model established with fractional differential equations provides a more accurate instrument for the description of memory and hereditary properties of various processes. Thus, it is well known that the dynamical behaviors of fractional-order systems have drawn much attention during the past few decades and many associated results have been investigated (Chen et al. 2012; Liu et al. 2012; Wei et al. 2016). Moreover, the research about fractional-order neural networks (FNNs) has become more and more significant because of the wide application into various areas (Hilfer 2000; Koeller 1984; Jiang and Xu 2010). It is known to all that synchronization is very important and interesting phenomenon of dynamical behaviors of FNNs. Synchronization refers to that dynamic behaviors of
Communicated by A. Di Nola. & Yunpeng Ma [email protected] Xinxin Zhang [email protected] 1
School of Information Engineering, Tianjin University of Commerce, Tianjin, China
two FNNs systems can gradually reach the same time and space state with the growth of time. Recently, lots of literature related to the synchronization of FNNs have been put forward. Based on the fractional Lyapunov stability criterion, the paper (Liu et al. 2018) considered synchronization between two FNNs with full/under-actuation using fractional-order sliding mode control. It was point out that some new sufficient conditions are derived to ensure the finite-time synchronization of FNNs by using Laplace transform, the generalized Gronwall’s inequality, Mittag– Leffler functions and linear feedback control technique (Velmurugan and Rakkiyappan 2016). It was noted that projective synchronization of nonidentical FNNs is studied based on sliding mode controller (Wu et al. 2017; Ding and Shen 2016). In addition, the global Mittag–Leffle
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