Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control
Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control
Bo Meng1,2 and Xiaohong Wang2
1College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
Correspondence should be addressed to Bo Meng; nc.ude.tsuds@2290bm
Received 17 May 2018; Accepted 10 July 2018; Published 18 July 2018
Academic Editor: Xue-Jun Xie
Copyright © 2018 Bo Meng and Xiaohong Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Adaptive synchronization for a class of uncertain delayed fractional-order Hopfield neural networks (FOHNNs) with external disturbances is addressed in this paper. For the unknown parameters and external disturbances of the delayed FOHNNs, some adaptive estimations are designed. Firstly, a fractional-order switched sliding surface is proposed for the delayed FOHNNs. Then, according to the fractional-order extension of the Lyapunov stability criterion, a fractional-order sliding mode controller is constructed to guarantee that the synchronization error of the two uncertain delayed FOHNNs converges to an arbitrary small region of the origin. Finally, a numerical example of two-dimensional uncertain delayed FOHNNs is given to verify the effectiveness of the proposed method.
1. Introduction
The research of neural networks (NNs) is quite extensive, reflecting the characteristics of multidisciplinary technology. NNs have many successful applications in the fields of associative memories and image processing. Recently, the discussion on NNs has become a hot topic [1–3]. Guo et al. [4] studied the exponential stability analysis for complex-valued memristor-based bidirectional associative memory (BAM) NNs with time delays. Lv et al. [5] used NNs to discuss the adaptive tracking control for a class of uncertain nonlinear systems. Li et al. [6] studied Hopf bifurcation analysis of complex-valued neural networks model.
Fractional calculus (FC) has a long history. As early as 1695, the concept of fractional differential was mentioned in Leibnitz’s letter to L’Hospital. For a long time, FC continues to grow. Podlubny’s book [7] systematically introduced the concepts and properties of FC. Bai et al. (see [8–13], and the references therein) studied the existence and uniqueness of solutions for fractional differential equations (FDE). Wang et al. [14–16] studied the numerical analysis of FDE. In recent years, fractional-order systems (FOS) have attracted wide attentions. The control problems of all kinds of FOS were studied recently [17–21]. Many researchers focused on fractional-order neural networks (FONNs) [22–27]. Cao et al. [28] investigated the existence and uniqueness of the nontrivial solution of NNs and the uniform stability of the FONNs.
The researches on the stability of NNs, FOS, and stochastic systems have attracted the attention of a large number of researchers, and many achievements have been made [29–41]. The sliding mode control (SMC) is a very popular strategy for a general class of nonlinear uncertain systems, with a very large frame of applications fields. Due to the use of the discontinuous function, its main features are the robustness of closed-loop system and the finite-time convergence. Utkin et al. [42] studied the minimum possible value of control based on adaptation SMC methodology. Efe. et al. [43] discussed the fractional fuzzy adaptive SMC. Aghababa [44] designed a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems. The synchronization problems of FOHNNs have captured more and more researchers’ attention [45–48]. Xi et al. [22] have discussed SMC for uncertain FOHNNs. Liu et al. [24] have researched adaptive synchronization of a class of FONNs. It is well known that time delay is unavoidable due to finite switching speeds of the amplifiers, and it may cause oscillations or instability of dynamic systems. Wang et al. [26] have discussed the stability analysis of FOHNNs with time delay.
However, to the best of our knowledge, there are few attentions to adaptive synchronization for a class of uncertain delayed FOHNNs subject to external disturbances. The SMC technology was used to solve the above problems in the paper. The rest of this paper is organized as follows: some necessary definitions and lemmas are given in Section 2. The main works including the introduction of fractional-order network model, the fractional-order switched sliding mode surface (SMS), the design of adaptive synchronization controller, and stability analysis are included in Section 3. Section 4 presents a simulation example. Finally, the paper is concluded i (...truncated)