Elsevier

Neural Networks

Volume 121, January 2020, Pages 329-338
Neural Networks

A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks

https://doi.org/10.1016/j.neunet.2019.09.032Get rights and content

Abstract

This paper addresses the global stabilization of complex-valued neural networks (CVNNs) via event-triggered control. First, a waiting-time-based event-triggered scheme is designed to reduce the data transmission rate. Therein, an exponential decay term is introduced into the predefined threshold function, which may postpone the triggering instant of the necessary data and therefore reduce the frequency of data transmission. Then, with the help of the input delay approach, a time-dependent piecewise-defined Lyapunov–Krasovskii functional is constructed for closed-loop system to formulate a less conservative stability criterion. In addition, by resorting to matrix transformation, the co-design method for both the feedback gains and the trigger parameters is derived. Finally, a numerical example is given to illustrate the feasibility and superiority of the proposed event-triggered scheme and the obtained theoretical results.

Introduction

During the past few decades, neural networks have attracted considerable attention due to their wide applications in filtering, combinatorial optimization, pattern recognition, and some other areas (Cha and Kassam, 1995, Ding et al., 2017, Liu et al., 2016, Nitta, 2003, Nitta, 2004, Qian et al., 2017, Tanaka and Aihara, 2009, Zhang and Xia, 2015, Zhang et al., 2015). Complex-valued neural networks (CVNNs) are a kind of artificial neural network, whose states, activation functions and connection weights are all complex-valued vectors or matrices (Hirose, 2013). CVNNs can process complex-valued information and therefore have the advantage of dealing with the issues that cannot be settled by real-valued neural networks. However, delays are inevitably encountered in circuit implementation of neural networks and usually are time-varying because of the finite switching speed of amplifiers, which may cause oscillation, chaos or even instability of CVNNs (Dong et al., 2015, Li et al., 2018, Wang et al., 2017). Hence, it is very necessary and important to investigate the stability of delayed CVNNs. As a consequence, quantities of meaningful works have been developed in Chen and Song, 2013, Hu and Wang, 2012, Hu and Wang, 2015, Liang et al., 2016, Liu and Chen, 2015, Rakkiyappan et al., 2015, Song et al., 2016a, Song et al., 2016b, Velmurugan et al., 2017, Zhang et al., 2019, Zhang et al., 2014 and Zhou and Song (2013).

Stabilization problem is strongly linked to stability problem. Stabilization is often required in many practical applications to achieve satisfactory system performance (Huang, Huang, Chen, & Qian, 2013). Stabilization of CVNNs with time-varying delay has become a popular topic not only in theory but in practice. Moreover, different control methods have been presented such as sliding control (Zhang, Wang, & Lin, 2016), impulse control (Yu, Lu, Qiu, & Kurths, 2019), intermittent control (Wan, Jian, & Mei, 2018), and adaptive control (Hu & Zeng, 2017). However, it should be pointed out that the aforementioned control methods are all point-to-point control. A prerequisite for point-to-point control is that the controllers must receive the signals from sensors in a continuous way. Obviously, it is impossible and sometimes unnecessary in practice. In modern industrial control systems, the plants, the sensors, the controllers and the actuators are usually connected over a shared communication network, which is called networked control system (NCS). The two most common networked control systems are sampled-data control systems and event-triggered (ET) control systems. Compared to traditional point-to-point control system, NCS is distributed and therefore more advanced. It has advantages of robustness, low cost, and easy maintenance, etc. (Li et al., 2016, Yue et al., 2005). However, NCS is limited in the communication capacity, bandwidth, and power source. Thus, saving network resources becomes the most urgent problem and it has stimulated increasing interest from researchers all over the world.

In recent years, event-triggered (ET) control, as one of the most popular networked control method, has drawn considerable attention, which is first introduced to deal with the stabilization issue in Tabuada (2007). In fact, ET mechanism provides an effective way to determine when the sampling action is triggered. To be specific, there is an event generator, which is positioned between the sensor and the controller. The sampling is executed and the controller is updated only when the trigger function is greater than a predefined threshold function. Obviously, the incorporation of the ET mechanism into the NCS can guarantee that only really necessary data will be sent to the controller. However, it is generally known that continuous event-trigger may cause the Zeno phenomenon (i.e., there are an infinite number of events within a finite time interval). It makes continuous ET control inapplicable in the real-world systems. To solve this problem, a novel ET scheme with triggering algorithm tk+1=min{(k+j)h|jN,[x((k+j)h)x(kh)]TΩ[x((k+j)h)x(kh)]ϵxT((k+j)h)Ωx((k+j)h)}was proposed in Yue, Tian, and Han (2013). Therein, periodic sampling is executed first and then the triggering condition is checked. Hence, the above mechanism determines only some of the sampled states that violate the triggering condition will be sent to the controller. Thus, the interval between two transmitted data is at least h and Zeno phenomenon can be avoided. In addition, a waiting-time-based event-triggered (WET) scheme with triggering algorithm tk+1=min{ttk+h|(x(t)x(tk))TΩ(x(t)x(tk))>ϵxT(t)Ωx(t)} was presented in Fei, Guan, and Gao (2018) and Selivanov and Fridman (2016). Therein, a waiting time is designated artificially. That is to say, when the measurement x(tk) has been sent at instant tk, the sensor waits for h seconds and then starts to check the triggering condition continuously. The next measurement x(tk+1) will be sent to the controller at the instant tk+1 when the event-trigger condition is violated. Due to the insertion of the waiting-time h, it can be seen that the interval between two successively triggered events is at least h, which can avoid the Zeno phenomenon well.

Although the existing WET schemes in Fei et al. (2018) and Selivanov and Fridman (2016) have many remarkable advantages, it still needs further improvement in the sense of cutting the triggering times down further to save the limited communication resources from the following two aspects. (i) A larger threshold usually means a lower triggering frequency. Hence, how to design a novel WET scheme with larger threshold function to further reduce the triggering times while still preserving the stability of the closed-loop system is a problem worthy of consideration. (ii) The waiting-time in Fei et al. (2018) and Selivanov and Fridman (2016) is designated to be fixed. Actually, an unfixed waiting-time is flexible and therefore preferable if taking the intermittent sensor breakdowns into account. Besides, it should be mentioned that there is no literature concentrated on stabilization of CVNNs in the framework of ET control. Hence, the first objective of this paper is to design a WET scheme to achieve the stabilization of CVNNs.

On the other hand, the stability analysis of NCS is a challenging problem. Usually, input delay approach, as one of the most important methods, is often used together with the time-dependent Lyapunov functionals to the stability analysis of NCS. It should be noted that the most frequently used Lyapunov functionals for traditional point-to-point control systems are time-independent. However, NCS is essentially a kind of hybrid system since zero-order hold (ZOH) is often used to generate the control signal. Hence, time-dependent Lyapunov functionals can not only better reflect the inherent characteristics of hybrid systems, but also benefit to obtaining less conservative stability criteria. In fact, many excellent results concerning the stability analysis of NCS have been proposed. To be specific, based on input delay approach, linear system under sampled-data control was converted into a system with input delay, and a time-dependent discontinuous Lyapunov functional was constructed for stability analysis, which does not grow after the sampling instances (Fridman, 2010). Based on free-matrix-based integral inequality and time-dependent discontinuous Lyapunov functions, stability analysis of sampled-data control systems was carried out in Lee and Park (2017). However, for CVNNs under ET control, the construction of Lyapunov functionals is more challenging. Hence, the second objective of this paper is to construct an appropriate Lyapunov functional exclusively for CVNNs to address the stabilization of CVNNs under ET control.

Motivated by above discussions, this paper investigates the globally asymptotical stabilization of delayed CVNNs in the framework of ET control. The main contributions of this paper include: (i) an exponential-decay-dependent WET scheme with unfixed waiting-time, named EWET scheme, is proposed by introducing an exponential decay term into the threshold function. The proposed scheme can enlarge the interval between two successively triggered events and therefore can further reduce the triggering times; (ii) a time-dependent piecewise-defined Lyapunov functional is constructed and some less conservative stabilization criteria are obtained. The requirement on V̇(t)0 in each inter-event interval [tk,tk+1) is relaxed; (iii) a co-design method for both the feedback gains and the trigger parameters is proposed, which provides a systematic way for the design of controller and triggering mechanism.

The rest of this paper is organized as follows. In Section 2, the EWET scheme is proposed, system description is introduced and some assumptions on activation functions are made. In Section 3, two stabilization criteria are derived to co-design the controller and the triggering scheme. In Section 4, numerical comparisons of different triggering schemes are carried out to demonstrate the superiority of the obtained theoretical results. Finally, some conclusions are drawn in Section 5.

Notations

Throughout this paper, R denotes the real space and Rn represents the n-dimensional Euclidean space. Rn×n and n×n denote the set of n×n real or complex matrices. For matrix ARn×n, AT and A1 denote the transpose and the inverse of matrix A. diag{l1,l2,,ln} represents a diagonal matrix. Ca,b,n denotes the family of continuous functions from a,b to n. Sym(X) represents the expression X+XT. For matrix ARn×n,A>0 (or A<0) means A is a symmetric positive definite (or symmetric negative definite) matrix. denotes the symmetric elements of a symmetric matrix. col{} denotes a column vector.

Section snippets

System description

Consider a class of CVNNs with time-varying delay, which is represented by ż(t)=Dz(t)+Af(z(t))+Bg(z(tτ(t)))+u(z(t)),t>0,z(t)=ϕ(t),t[τ,0],where z(t)=(z1(t),z2(t),,zn(t))Tn and zi(t) denotes the state of the ith neuron; D=diag{d1,d2,,dn}Rn×n>0 represents the neuron self-inhibition; A=(aij)n×n, B=(bij)n×nn×n are the synaptic connection weight matrices; τ(t) is the time-varying delay satisfying 0τ(t)τM and τ̇(t)<a<1; f(z(t)), g(z(tτ(t))) are the activation functions for the neurons

Main results

In this section, some sufficient conditions are derived to guarantee the globally asymptotical stability of closed-loop system (5).

Theorem 1

For given positive scalars h,σ,ϵ, and feedback gains KR,KI, closed-loop system (5) is globally asymptotically stable, if there exist matricesP>0,Q>0,U>0,V>0,R1>0,Q1>0,Ω1>0,Ω2>0, diagonal matrices Λi>0(i=1,2,,4) and matrices X,X1,Y,Y1,Y2,Y3,T1,T2,W1,W2, andPi(i=2,,7), such that for any hk(0,h], the following constraints are satisfied P+hkSym(X2)0hk(X+X1)0Q+hkSym(

Numerical example

In this section, a numerical example is given to testify the effectiveness and show the advantage of the obtained theoretical results.

Example

Consider a 3-D CVNN with system matrices: D=diag{0.8,0.8,0.8}, A=2.21.7i10.1i0.1+0.1i1.3+1.8i0.70.2i0.5+0.2i1.41.9i0.8+0.2i0.30.2i,B=1.2+1.5i0.7+0.2i0.3+0.4i1.3+1.7i0.60.3i0.4+0.5i1.2+1.6i0.9+0.1i0.3+0.2i, the activation functions are chosen as f(z(t))=g(z(t))=0.2tanh(x(t))+i0.2tanh(y(t)), and the time-varying delay is set as τ(t)=et1+et. The

Conclusions

In this paper, global stabilization of CVNNs with time-varying delay is addressed for the first time in the framework ofET control. An EWET scheme is employed to reduce the data transmission rate as well as avoid the Zeno phenomenon. In particular, an exponential decay term is introduced into the predefined threshold function, which provides the possibility of deferring the next update and reducing the triggering times. A time-dependent piecewise-defined Lyapunov–Krasovskii functional is

References (39)

  • QianX. et al.

    Efficient construction of sparse radial basis function neural networks using L1-regularization

    Neural Networks

    (2017)
  • RakkiyappanR. et al.

    Multiple μ-stability analysis of complex-valued neural networks with unbounded time-varying delays

    Neurocomputing

    (2015)
  • SongQ. et al.

    Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays

    Neural Networks

    (2016)
  • SongQ. et al.

    Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects

    Neural Networks

    (2016)
  • VelmuruganG. et al.

    Dissipativity and stability analysis of fraction-order complex-valued neural networks with time delay

    Neural Networks

    (2017)
  • YueD. et al.

    Network-based robust H control of systems with uncertainty

    Automatica

    (2005)
  • ZhangS. et al.

    A complex-valued neural dynamical optimization approach and its stability analysis

    Neural Networks

    (2015)
  • ChaI. et al.

    Channel equalization using adaptive complex radial basis function networks

    IEEE Journal on Selected Areas in Communications

    (1995)
  • DongT. et al.

    Stability and Hopf bifurcation of a complex-valued neural network with two time delays

    Nonlinear Dynamics

    (2015)
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    This work was supported by the National Natural Science Foundation of China (Nos. 61573008, 61773004, 61973199) and the SDUST, China graduate innovation project (No.SDKDYC190367).

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