Elsevier

Neural Networks

Volume 75, March 2016, Pages 162-172
Neural Networks

Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control

https://doi.org/10.1016/j.neunet.2015.12.003Get rights and content

Abstract

The problem of exponential stabilization and synchronization for fuzzy model of memristive neural networks (MNNs) is investigated by using periodically intermittent control in this paper. Based on the knowledge of memristor and recurrent neural network, the model of MNNs is formulated. Some novel and useful stabilization criteria and synchronization conditions are then derived by using the Lyapunov functional and differential inequality techniques. It is worth noting that the methods used in this paper are also applied to fuzzy model for complex networks and general neural networks. Numerical simulations are also provided to verify the effectiveness of theoretical results.

Introduction

Memristor was postulated by Chua as the fourth basic circuit element in 1971 (Chua, 1971), and realized by Williams’s group in 2008 (Strukov, Snider, Stewart, et al., 2008). As a new circuit element, the memristor shares many properties of resistors and shares the same unit of measurement (ohm), and remembers information just as the neurons in human have. Because of this feature, memristors have been proposed to work as synaptic weights to build the models of neural networks to emulate the human brain, that is, memristor-based recurrent neural networks. In recent years, the memristor-based recurrent neural networks have been extensively investigated and successfully applied to signal processing, image processing, pattern classification, quadratic optimization, associative memory and so on (Chandrasekar and Rakkiyappan, 2015, Chandrasekar et al., 2014, Li et al., 2011, Wu and Zeng, 2012a, Wu et al., 2011, Zhang and Shen, 2014). As we know, the memristor-based recurrent neural networks can remember its past dynamical history, store a continuous set of states (Wu et al., 2011). It will open up new possibilities in the understanding of neural process using memory devices, and as an important step forward to reproduce complex learning, adaptive and spontaneous behavior with electronic neural networks. Due to the merits of memristor-based recurrent neural networks, its dynamic analysis has attracted many researchers’ attention (Wen et al., 2015, Wu and Zeng, 2013, Wu et al., 2011, Zhang et al., 2015). Zhang and Li studied the synchronization of memristor-based coupling recurrent neural networks with time-varying delays and impulses in Zhang et al. (2015). Wu, Zeng, Zhu et al. studied the exponential synchronization of memristor-based recurrent neural networks with time delays in Wu et al. (2011); Wu and Zeng studied the anti-synchronization of memristive recurrent neural networks in Wu and Zeng (2013); Wen studied the circuit design and the exponential stabilization of memristive neural networks in Wen et al. (2015). It is easy to see that those works discussed the stabilization or synchronization of normal model of memristive neural networks rather than fuzzy memristive neural networks. However, to the best of authors’ knowledge, there are few works that have been done to analyze the exponential stabilization and synchronization for fuzzy model of memristive neural networks.

In addition, much efforts have been devoted to the control and synchronization of neural networks due to its potential practical applications (Ali, 2015, Ali, 2014, Ali, Arik, & Saravanakumar, 2015, Boccaletti, Latora, Moreno, et al., 2006, Chandrasekar, Rakkiyappan, & Cao, 2015, Li, 2010, Li & Rakkiyappan, 2013b, Li, Ding, & Zhu, 2010, Stamova, Stamov, & Li, 2014, Strogatz, 2001, Suykens & Osipov, 2008). Meanwhile, many control approaches have been proposed to stabilize chaotic networks and nonlinear systems such as adaptive control (Zhu, Zhang, Fei, et al., 2009), impulsive control (Li et al., 2005, Li et al., 2015, Li and Rakkiyappan, 2013a, Li and Song, 2013, Li et al., 2014, Liu et al., 2013, Wen et al., 2015), intermittent control (Cai et al., 2011, Li, Feng et al., 2007, Li, Liao et al., 2007, Xia and Cao, 2009) and so on. In general, in order to stabilize a nonlinear system, it is natural to address the feedback stabilization problem, regardless of the different feedback mechanism (Fang and Chow, 2005, Wada et al., 2004, Yue et al., 2004). Recently, discontinuous control techniques such as impulsive control (Yang, 2001) and piecewise feedback control (Li, Liao, & Yang, 2006) have been attracted much attention. In this paper, we address the stabilization problem of nonlinear systems using another discontinuous feedback control, i.e., periodically intermittent control. Intermittent control, which was introduced to control nonlinear dynamical systems in Żochowski (2000), has been used for a variety of purposes such as manufacturing, transportation, communication, and signal processing in practice. Generally, compared with the continuous control methods, intermittent control is a straightforward engineering approach to control and synchronize the chaotic systems. In communications, in order to compensate the lost signal and enable received signal at the terminal to achieve the desired result or requirement, the external control signal is added as long as the strength of the system signal is below the required level (Hu, Yu, Jiang, et al., 2010a). And then, the external control can be considering the lower cost (Hu, Yu, Jiang, et al., 2010b). In view of those advantages, lots of works (Cai et al., 2011, Hu et al., 2010a, Hu et al., 2010b, Huang and Li, 2010, Li, Feng et al., 2007, Li, Liao et al., 2007, Xia and Cao, 2009, Zhang et al., 2011, Żochowski, 2000) have been obtained in recent years, which indicated the periodically intermittent control is more economical and efficient.

However, in implementation of memristive neural networks, time delays in particular time-varying delays are unavoidably encountered in the signal transmission among the neurons due to the finite speed of switching and transmitting signals, which may result in oscillatory behavior or network instability. Meanwhile, the stabilization problem of memristive neural networks especially the memristive neural networks with time-varying delays have been discussed by many researchers. For instance, Zhang et al. investigated the exponential stabilization of memristor-based chaotic neural networks with time-varying delays via intermittent control in Zhang and Shen (2014); Wen et al. analyzed the exponential stability about memristor-based recurrent neural networks with time-varying delays in Wen, Zeng, and Huang (2012); Wu studied the exponential stabilization of memristive neural networks with time-varying delays in Wu and Zeng (2012b). However, the problem of synchronization has also attracted increased attention from scientists and engineers due to its wide-scope potential applications in various scientific fields (Wen et al., 2014, Wu and Zeng, 2013, Wu et al., 2011, Zhang et al., 2015). In those literatures, there exists a common requirement to regulate the behavior of large ensembles of interacting units. Hence, investigating the stabilization and synchronization problem of memristive neural networks with time-varying delays is also important.

Motivated by the above analysis, the aim of this paper is to discuss the exponential stabilization and synchronization for fuzzy memristive neural networks by using intermittent control. Compared with the previous works (Hu et al., 2010b, Huang and Li, 2010, Zhang et al., 2011), it is noted that the modeling process and nonlinear terms covered in normal model of memristive neural network are complicated and copious. However, the fuzzy model of memristive neural network used in this paper just need two sub-systems for modeling and only require two sets of control gains for stabilizing and synchronizing. Hence, it is vital to analyze the exponential stabilization and synchronization for fuzzy model of memristive neural network. The obtained conditions of our results are new and less conservative. Furthermore, simulations are given to illustrate the effectiveness of the proposed method. The main contributions of this paper can be listed as follows: (1) The fuzzy memristive neural networks are employed to give a new method to analyze the complicated MNNs with only two subsystems; (2) The intermittent control was applied to stabilize and synchronize chaotic systems and neural networks with or without constant delay in earlier works. However, the fuzzy memristive neural network stability criteria and synchronization conditions were firstly proposed based on the approaches of periodically intermittent control in this paper, which can be applied to fuzzy chaotic systems or other memristor-based chaotic systems. (3) Our results are applicable to model and stabilize the high-dimensional nonlinear systems, specially high-order neural networks due to their excellent approximation capabilities. (4) The analysis methods used in this paper are also applied to discuss the dynamic behaviors of nonautonomous systems with variable moments of impulses. We will investigate the applications of our results in the future works.

This paper is organized as follows: In the following section, the theoretical model for fuzzy memristive neural network, some definitions and lemmas are presented. In Section  3, the exponential stabilization of the fuzzy memristive neural network is analyzed, and the simulation results of periodically intermittent control are given. Exponential synchronization of two fuzzy memristive neural networks is investigated in Section  4. Finally, the conclusion is given in Section  5.

Section snippets

Problem statement and preliminaries

In this section, we shall formulate the considered problem and present some preliminaries including the fuzzy memristive neural network and some necessary definitions and lemmas.

Consider a class of memristive neural networks as follows (Li, Liao et al., 2007, Wen et al., 2014): ẋi(t)=di(xi(t))xi(t)+j=1naijfj(xj(t))+j=1nbijfj(xj(tτj(t)))+si, where aij=signijCiRfij,bij=signijCiRgij,si=IiCi,di(xi(t))=iCi[j=1n(iRfij+iRgij)+Wi(xi(t))]={d1i,xi(t)0,d2i,xi(t)>0. and fj is the activation

Main results

In this section, we will address the exponential stability problem of the system (4) by means of the aforementioned lemmas. The main results are stated as follows.

Theorem 3.1

Suppose that there exists a continuous and positive-definite function V:RnR+ satisfying the following conditions:

  • (i)

    there exist positive constants c1,r such thatc1yrV(y)φ(y)where φ:R+R+ is a continuous and strictly monotonic increasing function with φ(0)=0.

  • (ii)

    there exist positive constants ai,bi(i=1,2) such that the derivative of V

Periodically intermittent synchronization

In this section, we study exponential synchronization of the addressed fuzzy model of memristive neural networks using periodically intermittent control. In order to deal with synchronization, we need to design control input for a response system so that the response system achieves synchronization with the drive system, provided that the two systems start from different initial conditions. The drive system is given by system (3), suppose the response system is represented as ỹ(t)=l=12ΠlDlỹ(

Conclusions

In this paper, intermittent control technique is generalized to study the exponential stabilization and synchronization problem for a class of fuzzy models of memristive neural network by using analysis technique. By adding intermittent controllers, the general exponential stability criterion and synchronization conditions, together with its simplified versions have been obtained. Evidently, our results are novel and easily verified. Compared with corresponding previous works, our results are

References (52)

  • S.P. Wen et al.

    Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays

    Neurocomputing

    (2012)
  • A. Wu et al.

    Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays

    Neural Networks

    (2012)
  • A. Wu et al.

    Anti-synchronization control of a class of memristive recurrent neural networks

    Communications in Nonlinear Science and Numerical Simulation

    (2013)
  • A. Wu et al.

    Exponential synchronization of memristor-based recurrent neural networks with time delays

    Neurocomputing

    (2011)
  • A. Wu et al.

    Dynamic behaviors of a class of memristor-based Hopfield networks

    Physics Letters A

    (2011)
  • W. Zhang et al.

    Weak synchronization of chaotic neural networks with parameter mismatch via periodically intermittent control

    Applied Mathematical Modelling

    (2011)
  • G.D. Zhang et al.

    Exponential stabilization of memristor-based chaotic neural networks with time-varying delays via intermittent control

    Neural Networks

    (2014)
  • G.D. Zhang et al.

    Global exponential stability of a class of memristor-based recurrent neural networks with time-varying delays

    Neurocomputing

    (2012)
  • Q. Zhu et al.

    Adaptive neural control for a class of output feedback time delay nonlinear systems

    Neurocomputing

    (2009)
  • M. Żochowski

    Intermittent dynamical control

    Physica D: Nonlinear Phenomena

    (2000)
  • M.S. Ali

    Stability analysis of Markovian jumping stochastic Cohen Grossberg neural networks with discrete and distributed time varying delays

    Chinese Physics B

    (2014)
  • M.S. Ali

    Stability of Markovian jumping recurrent neural networks with discrete and distributed time-varying delays

    Neurocomputing

    (2015)
  • M.S. Ali et al.

    Delay-dependent stability criteria of uncertain Markovian jump neural networks with discrete interval and distributed time-varying delays

    Neurocomputing

    (2015)
  • D. Bainov et al.

    Integral inequalities and applications

    (1992)
  • S. Cai et al.

    Exponential synchronization of chaotic systems with time-varying delays and parameter mismatches via intermittent control

    Chaos: An Interdisciplinary Journal of Nonlinear Science

    (2011)
  • A. Chandrasekar et al.

    Impulsive controller design for exponential synchronization of delayed stochastic memristor-based recurrent neural networks

    Neurocomputing

    (2015)
  • Cited by (148)

    View all citing articles on Scopus

    This publication was made possible by NPRP grant NPRP 4-1162-1-181 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 Natural Science Foundation of China (Grant No. 61374078) and Natural Science Foundation Project of Chongqing CSTC (Grant No. cstc2014jcyjA40014).

    View full text