Elsevier

Neural Networks

Volume 141, September 2021, Pages 261-269
Neural Networks

Saturated impulsive control for synchronization of coupled delayed neural networks

https://doi.org/10.1016/j.neunet.2021.04.012Get rights and content

Abstract

The paper focuses on the synchronization problem for a class of coupled neural networks with impulsive control, where the saturation structure of impulse action is fully considered. The coupled neural networks under consideration are subject to mixed delays including transmission delay and coupled delay. The sector condition in virtue of a new constraint of set inclusion is given for a addressed network, based on which a sufficient condition for exponential synchronization problem is obtained by replacing saturation nonlinearity with a dead-zone function. In the framework of saturated impulses, our results relying on the domain of attraction can still achieve the synchronization of coupled delayed neural networks. In addition, the estimating domain of attraction is proposed as large as possible by solving an optimization problem. Finally, a numerical simulation example is presented to demonstrate the effectiveness of the proposed results.

Introduction

In recent decades, there has been a substantial growth of research interest in the dynamics of neural networks comprising stability (Song, Long, Zhao, Liu, & Alsaadi, 2020), stabilization (Li, O’Regan, & Akca, 2015), state estimation (Ali, Usha, Orman, & Arik, 2019), periodic attractors (Duan, Huang, Guo, & Fang, 2017), robust stability (Song, Chen, Zhao, Liu, & Alsaadi, 2021), and synchronization (Yu, Cao, & Lu, 2008). A coupled neural network consisting of coupled nodes is a special neural network, whose complexity originates from not only the dynamic behavior of each node itself but also the complex structure produced by them interlaced together. Up to now, the synchronization problem for coupled neural networks has attracted a great deal of attention due to the wide applications in various branches of science and engineering including parallel image processing, automotive vehicle control, and secure communication, see Kuntimad and Ranganath (1999) and Kwon, Park, and Lee (2011). Therefore, it is important to investigate synchronous behaviors in a class of coupled neural networks. Synchronization has been studied from different aspects such as chaotic synchronization (Liu, 2018), outer synchronization (Tan, Zhou, Chu, & Li, 2020), and cluster synchronization (Wei, Zhou, & Chen, 2008). Besides, synchronizing a network to a desired trajectory is also an important issue, and it has been well formulated as leader-following synchronization by using different methods, such as intermittent control (Chen, Wang, Shen, & Dong, 2018), sampled-data control (Lee, Park, Lee, & Kwon, 2013), pinning control (He, Qian, & Cao, 2017), distributed control (Guan, Liu, Feng, & Wang, 2010), event-triggered control (Senan, Ali, Vadivel, & Arik, 2017), impulsive control (Lu et al., 2010, Yang et al., 2020, Zhang and Bao, 2020) and so on.

Especially, impulsive control is characterized by lower control cost, higher confidentiality, and stronger robustness. Hence, it has been widely used in currency supply control in the financial market (Sun, Qiao, & Wu, 2005), static multisynchronization (Lv, Li, Cao, & Perc, 2018), cooperative models (Li, Yang, & Huang, 2019), stability (Yang, Li, Xi, & Duan, 2018), and communication security (Yang & Chua, 1997). However, it is worth noting that most of the existing results on the design of impulsive controller (Li et al., 2019, Lu et al., 2010, Lv et al., 2018, Sun et al., 2005, Yang and Chua, 1997, Yang et al., 2020, Yang et al., 2018, Zhang and Bao, 2020) are based on the condition that there is no restriction on the impulsive strength. In other words, in order to achieve the desired control performance, we can design the impulsive strength artificially. In fact, from the applications point of view, it is actually very hard to achieve the design objective for every control input due to the fact that actuator saturation is prevalent in virtually all control systems. For example, the reduction in the speed of the car may be subject to saturation when a car goes through speed bumps at a very high speed, where speed bumps act as impulses. Therefore, impulses saturation can greatly affect the dynamical behaviors of systems, see Hu and Lin, 2001, Li and Lin, 2018 and Tarbouriech, Garcia, da Silva, and Queinnec (2011). If such limits are not treated carefully or even the relevant controllers do not take them into account, then adverse behaviors may be observed caused by saturation. Naturally, it is necessary to consider the influence of saturated impulses when studying the impulsive control of coupled neural networks.

Generally speaking, there are two methods to deal with actuator saturation. The first one takes into account the saturation nonlinearity at the outset of the control design based on a control law, where saturation nonlinearity is often expressed as the form of linear differential inclusion, see Hu et al., 2006, Li et al., 2020, Li and Lin, 2013 and Xiong, Zhang, and Ma (2019). Another one is called “anti-windup compensator”, which is to introduce additional feedbacks to keep the actuator within its proper limits. The significant advantage of this method is that the compensator works together with the existing linear controller and only acts when saturation occurs, see Cao et al., 2004, Da Silva and Tarbouriech, 2005, Ouyang et al., 2020, Zaccarian and Teel, 2011 and Zhu, Li, and Wang (2020). However, the synchronization problem on saturation structure of impulse action is tackled in only a few works, see Li et al., 2020, Ouyang et al., 2020 and Xiong et al. (2019). By using the first method, Li et al. (2020) and Xiong et al. (2019) only studied asymptotic synchronization with the variable saturated impulsive controller and hybrid controller consisting of saturated impulse and sampled data, respectively. In addition, the estimating domain of attraction was essentially excluded in both Li et al. (2020) and Xiong et al. (2019). In terms of anti-windup compensator, Ouyang et al. (2020) considered the impulsive synchronization of coupled delayed neural networks with actuator saturation. However, it is difficult to satisfy the impulsive condition in application because of the uncertainty of the assumption in Ouyang et al. (2020). Moreover, coupled delay among different nodes was ignored. Hence, we consider the synchronization problem for a class of coupled delayed neural networks with saturated impulsive control. Compared with the existing results, the contribution and novelty of this paper can be summarized as follows:

  • One may note that most of the existing results of impulsive synchronization, such as those in Lu et al., 2010, Lv et al., 2018, Yang et al., 2020 and Zhang and Bao (2020), always design an idealized threshold for the impulsive control at every impulse point to achieve a desired performance without considering saturation structure of impulse action. In this paper, to meet inherent limitation at the impulse points, we consider the synchronization control problem in which saturation structure produced by the maximum and minimum limits of the control inputs is taken into account.

  • For continuous linear system with input saturation, Da Silva and Tarbouriech (2005) proposed a set inclusion constraint for designing the anti-windup gains. Subsequently, it has been applied to a variety of continuous systems. Recently,Ouyang et al. (2020) and Zhu et al. (2020) have extended previous works on actuator saturation of continuous systems to impulsive systems. Especially, Ouyang et al. (2020) developed the impulsive synchronization problem by applying anti-windup technique. Compared with Ouyang et al. (2020), a new constraint of set inclusion is proposed, which allows the existence of both transmission delay and coupled delay.

  • Note that the estimating domain of attraction was ignored in both Li et al. (2020) and Xiong et al. (2019). As we know, when studying the systems involving saturation structure, the estimating domain of attraction is crucial to guarantee the stability. In this paper, a novel saturated impulsive controller is designed such that the estimating domain of attraction can be derived.

The outline of this paper is given as follows. Firstly, in Section 2, some matrices and polyhedral sets will be introduced to describe the sector condition. In Section 3, based on the sector condition, a sufficient condition of synchronization and the domain of attraction shall be obtained for coupled neural networks with mixed delay. In Section 4, a numerical simulation example is given to show the effectiveness of our new results. Finally, the paper is concluded in Section 5.

Notations. Let N={1,2,,N}, M={1,2,,n}, R denote the set of real numbers, R+ the set of nonnegative real numbers, Z+ the set of positive integer numbers, Rn and Rn×m the n-dimensional and n×m-dimensional real spaces equipped with the Euclidean norm , respectively. A>0 or A<0 denotes that the matrix A is a symmetric and positive definite or negative definite matrix. The notation λmax(A), λmin(A), A1, and AT mean the maximum eigenvalue, minimum eigenvalue, the inverse and the transpose of matrix A, respectively. denotes the symmetric block in one symmetric matrix, and I the identity matrix with appropriate dimensions. For any interval Y1R and any set Y2Rn, we put PC(Y1,Y2)={v:Y1Y2 is bounded and continuous everywhere except at finite number of points t, at which v(t),v(t+) exist and v(t+)=v(t)}. ab and ab are the maximum and minimum value of a and b, respectively. denotes the Kronecker-product. In addition, to present our main results, we need some special notation. Given a symmetric and positive definite matrix PRn×n and a positive constant κ, the ellipsoid ε(INP,κ){ϱRNn:ϱT(INP)ϱκ}.

Section snippets

Preliminaries

Consider an array of coupled neural networks, in which both current-state coupling and delayed coupling are taken into account simultaneously. The dynamic of the ith neural network with impulsive control is described by: xi̇(t)=Cxi(t)+Af(xi(t))+Bf(xi(tτ(t)))+J(t)+j=1NGijDxj(t)+j=1NGijD¯xj(tσ(t))+ui(t),where iN, xi(t)=[xi1(t),xi2(t),,xin(t)]TRn is the state vector of the ith network at time t; C=diag{c1,c2,,cn}>0 with ci>0 describes the rate with which the ith neuron will reset its

Main results

In this section, we shall design a class of impulsive controllers involving saturation structure, which are formalized in terms of LMIs and ADT technique. Firstly, a sufficient condition for exponential synchronization of coupled neural networks is established based on the set inclusion constraint.

Example

In this section, an example is given to show the validity of the proposed theoretical results.

Example 1

Consider coupled delayed neural network (1) and the isolated node described in He et al. (2017) with N=3, n=2, J=(0,0)T, τ=1, σ=1, fi(yi)=(|yi+1||yi1|)2, i=1,2, and matrices C, A, and B are given by: C=1001,A=1+π4200.11+π4,B=2π1.340.10.12π1.34.

Fig. 1 shows the phase plot of the isolated node with initial condition y(s)=(0.1,0.2)T, s[1,0].

The topology structure of the network is shown in Fig. 2

Conclusion

In this paper, a sufficient condition for the exponential synchronization problem of coupled neural networks, which takes into account two types of delays simultaneously. The designed saturated impulsive controller can be derived by solving LMIs, under which the estimating domain of attraction is presented as large as possible. Moreover, simulations are given to confirm that the effectiveness of the proposed results. It should be pointed out that our results can be only applied to the cases

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported by National Natural Science Foundation of China (61673247), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (2019KJI008). The paper has not been presented at any conference.

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