Elsevier

Neural Networks

Volume 110, February 2019, Pages 55-65
Neural Networks

Event-triggered impulsive control on quasi-synchronization of memristive neural networks with time-varying delays

https://doi.org/10.1016/j.neunet.2018.09.014Get rights and content

Abstract

This paper discusses the quasi-synchronization of memristive neural networks (MNNs) with time-varying delays via event-triggered impulsive and state feedback control approaches. The choice of different initial conditions may lead to the unexpected parameter mismatch in virtue of the state-dependent parameters of MNNs. Thus, the accurate synchronization error level and the exponential convergence rate are derived in view of the comparison principle of impulsive systems and the variable parameter formula. A co-design procedure that can be easily implemented is presented to make the synchronization error converge to a predetermined level. Then, no zeno-behavior is proved to exist in the controlled system with the proposed event-triggered condition. In addition, a self-triggered scheme is proposed to prevent continuous communication happening between the drive system and the response system. Finally, a numerical example is given to illustrate the availability of the proposed control scheme.

Introduction

The memristor concept was originally proposed by Chua (1971). As a contraction of memory and resistor, it was predicted as the fourth circuit element. By the end of 2008, scientists at Hewlett–Packard Laboratories found the material with memory properties (Strukov, Snider, Stewart, & Williams, 2008). This important discovery has given impetus to the development of computer and brain-like neural computers (Merrikh-Bayat & Shouraki, 2011). Because of the memristor’s memory characteristics, it has been used as synaptic weights to study artificial neural networks Liu et al., 2016, Pershin and Di Ventra, 2010, Wu and Zeng, 2012, Yang et al., 2018. Nevertheless, the connection weights of MNNs are changed according to the state dynamics, the choice of different initial conditions may lead to the unexpected parameter mismatch. Namely, the connection weights cannot be identical all the time Ding and Wang, 2017, Ding et al., 2017, Yang et al., 2017, Yang and Ho, 2016. Therefore, the traditional control programs and analytical techniques fail to ensure the synchronization of MNNs.

Synchronization can find many potential applications, such as the secure communications Sun et al., 2013, Yang and Chua, 1997, information science, and image encryption. With a growing interest in MNNs, many studies are investigating the synchronization problem of MNNs with delays Abdurahman and Jiang, 2016, Abdurahman et al., 2015, Mathiyalagan et al., 2016, Wang and Shen, 2015a, Wu et al., 2011, Zhang and Shen, 2013, Zhang et al., 2018. MNNs are likely to generate complex dynamic behaviors even strange chaotic attractors because of connection weights and time delays Sheng et al., 2017, Wang et al., 2014, Xiao and Zeng, 2017, Xiao and Zeng, 2018, Zhang and Shen, 2015. Thus, it is obliged to research the synchronization of time delayed MNNs. Meanwhile, various control schemes have been developed, which include state/output feedback control Wang and Shen, 2015b, Wu and Zeng, 2017, adaptive control Li and Cao, 2015, Yang et al., 2017, intermittent control Zhang and Shen, 2014, Zhang and Shen, 2015, impulsive control Lu et al., 2010, Tang, Park, Wang et al., 2018, Yang and Xu, 2005, Yang and Xu, 2007, Zhang et al., 2017, and time/event-triggered control Behera et al., 2015, Wen et al., 2016, Wen, Huang et al., 2017, Wen, Zeng et al., 2017, Yue et al., 2013, to research the synchronization and stability problems.

Because of the various applications in many realistic networks, impulsive control has attracted considerable attention. If the state of nodes is subject to instantaneous change at certain impulsive instants (Lu, Ho, Cao, & Kurths, 2011), the convergence rate of the systems may become faster or slower, or even not convergent. As a result, it is obliged to research the function of impulsive control on the synchronization and stability. At present, many existing works study the effect of time-triggered impulsive control for neural networks. But the working time of impulsive controller cannot be changed as long as the controller is designed. That is to say, the impulsive instants are predesigned, which makes impulsive instants stationary Duan et al., 2017, Tang, Park, and Feng, 2018, Zhang et al., 2017. In addition, delayed MNNs were investigated by impulsive control in Duan et al. (2017), where the impulsive instants could not be changeless as long as the controller was designed. To guarantee the performance of the controlled system, the time-triggered mechanism could inevitably lead to the frequent changes in the state of the actuator, resulting in unnecessary energy consumption. Namely, under the time-triggered impulsive control, high frequency of state switching might cause equipment and actuator attrition, which is not necessary for achieving the control objective.

To achieve the expected performance, the event-triggered mechanism has shown strong advantages in lessening controller update times Behera et al., 2015, Lu et al., 2015, Wen et al., 2016, Wen, Huang et al., 2017, Wen, Zeng et al., 2017, Zhu et al., 2018. The update instants of the controller are calculated by some event-triggered conditions. Compared with the periodic or aperiodic time-triggered mechanism, it has the advantage of less information exchange. This control method can ameliorate the utilization of the limited bandwidth resource. In Lu et al. (2015), the synchronization of linearly coupled systems has been researched by the proposed event-triggered coupling configurations. It is used the piecewise constant linear coupling state term as a controller to lessen the continuous sampling between network nodes. In Zhu et al. (2018), the exponential stabilization problem has been discussed for continuous time dynamic systems by means of the event-based impulsive control approaches. It should be stressed that the scheme in Zhu et al. (2018) is restricted to the systems without time delays and the state-dependent parameter mismatch is not taken into account in the synchronization of MNNs, which has much room for improvement.

The self-triggered mechanism had been proposed to generate time series instead of the event-triggered mechanism Behera et al., 2015, Wen, Huang et al., 2017. The main difference is that the next triggered instant is calculated by the last sampled state information, hence the continuous state information is not needed, which lessens the load and the network communication.

From what has been discussed above, few works focus on the quasi-synchronization of time-varying delayed MNNs with state-dependent parameter mismatch, which is discussed via the designed event-triggered impulsive and state feedback control approaches in this paper. The main contributions are summarized as follows.

  • (i)

    The quasi-synchronization is investigated for time-varying delayed MNNs with state-dependent parameter mismatch via the proposed event-triggered impulsive and state feedback control approaches. The accurate synchronization error level and the exponential convergence rate are derived in view of the comparison principle of impulsive systems and the variable parameter formula, respectively.

  • (ii)

    The impulsive and state feedback instants are determined by the proposed event-triggered condition. It is worth noting that the event-triggered instants determine the update time of the state feedback control input and the working time of the impulsive controller. A co-design procedure that can be easily implemented is presented to make the synchronization error converge to a predetermined level.

  • (iii)

    The event-triggered rules are proved to perform well and can exclude the zeno-behavior. In addition, a self-triggered scheme is proposed to prevent continuous communication happening between the drive system and the response system, and to refrain from continuous monitoring the triggered condition.

This paper is structured as follows: The drive-response MNNs model, definitions, and important lemmas are given in Section 2. The proposed control scheme is required to achieve the quasi-synchronization of MNNs with parameter mismatch and a self-triggered mechanism is further developed in Section 3. A numerical simulation is carried out in Section 4. The conclusions are reached in Section 5.

Notations: N={1,2,}. R denotes the real number set. Rn denotes n-dimensional Euclidean space. x=(xTx)12 is the Euclidean norm of x=(x1,x2,,xn)T Rn. For a given vector or matrix H=(hij)n×m, |H|=(|hij|)n×m. P<0(P>0) denotes a negative (positive) definite matrix. λmin(P), λmax(P), and PT denote the minimum eigenvalue, the maximum eigenvalue, and the transpose of matrix P, respectively. The n-order identity matrix is depicted as In. diag{} stands for a diagonal matrix. C([τ,0],Rn) denotes the Banach space of all continuous functions mapping [τ,0] into Rn. PC(n) denotes the set of piecewise right continuous function χ:[τ,+)Rn. For χ:RR, denote χ(t+)=lims0+χ(t+s), χ(t)=lims0χ(t+s), and the upper-right Dini derivative D+χ(t)=lims0+supχ(t+s)χ(t)s.

Section snippets

Preliminaries

Consider MNNs with time-varying delays described by: ẋl(t)=dlxl(t)+m=1nalm(xl(t))fm(xm(t))+m=1nblm(xl(t))×gm(xm(tτm(t))),where t0, l=1,2,,n; alm(xl(t))=álm,|xl(t)|Ϝl,àlm,|xl(t)|>Ϝl,blm(xl(t))=b́lm,|xl(t)|Ϝl,b̀lm,|xl(t)|>Ϝl,the switching jumps Ϝl>0; xl(t) is the state variable of the l-th neuron; τm(t) is the time-varying delay which satisfies 0τm(t)τ (τ is a constant); dl>0álmàlm, b́lm, and b̀lm are constant numbers; alm(xl(t)) and blm(xl(t)) denote the feedback connection

Main results

In this section, the event-triggered impulsive and state feedback control approaches will be used to study the quasi-synchronization of system (4) and system (5) with time-varying delays.

For convenience, we denote A=(ālm)n×n, B=(b̄lm)n×n, F=diag{f1,f2,,fn}, G=diag{g1,g2,,gn}, Γ1=ÃW, Γ2=B̃W, Ã=(|álmàlm|)n×n, B̃=(|b́lmb̀lm|)n×n, W=(ω1,ω2,,ωn)T, W=(ω1,ω2,,ωn)T.

Theorem 1

Under Assumption 1 , 2 and the control law (7) , if there exist constants ρ2ρ1>0, αi>0(i=1,2,3,4), α̃>0, ϖ>0

Numerical example

In this section, one example is provided to validate the availability of the previous results.

The following MNNs with time-varying delays is considered as the drive system ẋl(t)=dlxl(t)+m=12alm(xl(t))fm(xm(t))+m=12blm(xl(t))×gm(xm(tτm(t))),t0,l=1,2, where d1=1,d2=1,a11(x1(t))=1.75,a22(x2(t))=2.85,b11(x1(t))=1.6,b22(x2(t))=2.38,a12(x1(t))=2.9,|x1(t)|Ϝ,2.8,|x1(t)|>Ϝ,a21(x2(t))=2.9,|x2(t)|Ϝ,2.8,|x2(t)|>Ϝ,b12(x1(t))=0.08,|x1(t)|Ϝ,0.11,|x1(t)|>Ϝ,b21(x2(t))=0.11,|x2(t)|Ϝ,0.1,|x2(t)|

Conclusions

This paper has been studied the quasi-synchronization of MNNs with and without time-varying delays via the designed event-triggered impulsive and state feedback controllers. The influence of parameter mismatch on synchronization has been estimated. The accurate synchronization error level and the exponential convergence rate have been derived in view of the comparison principle of impulsive systems and the variable parameter formula, respectively. A co-design procedure that can be easily

Acknowledgments

The work was supported by the Natural Science Foundation of China under Grants 61673188, 61761130081 and 61821003, the National Key Research and Development Program of China under Grant 2016YFB0800402, the Foundation for Innovative Research Groups of Hubei Province of China under Grant 2017CFA005 and the Fundamental Research Funds for the Central Universities of HUST , China under Grant 2018KFYXKJC051.

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