Stabilizability of complex complex-valued memristive neural networks using non-fragile sampled-data control

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Abstract

This paper investigates the stability and stabilizability of complex-valued memristive neural networks (CVMNNs) with random time-varying delays via non-fragile sampled-data control. Taking the influence of gain fluctuations into account, a non-fragile sampled-data controller is designed for CVMNNs. Compared with the existing control schemes, the one here is more applicable and can effectively save the communication resources. The assumption on activation functions of CVMNNs is relaxed by only needing the complex-valued activation functions satisfying the Lipschitz condition. By constructing a suitable Lyapunov–Krasovskii functional (LKF), new stability and stabilizability criteria are derived for CVMNNs. Different from the existing results with the maximum absolute values of memristive connection weights, our ones are based on the average values of the maximum and minimum of the memristive connection weights. Finally, numerical simulations are given to validate the effectiveness of the theoretical results.

Introduction

Memristor was firstly postulated by Chua in 1971 [1]. It is the fourth fundamental circuit element (others are resistor, inductor and capacitor) to describe the relationship between magnetic flux and electric charge. About 40 years later, based on TiO2 thin films, researchers in HP Labs built the first physical memristor device [2], [3]. Since then, memristor and memristor-based circuits have provoked substantial attention. Memristor is a two terminal element with changeable resistance. It not only captures the properties of resistors but has its own distinctive memory property. Therefore, memristor is considered to be a striking candidate to imitate biological synapses. Recently, based on memristor, a new type of network models called memristive neural networks (MNNs) have garnered increasing interests of researchers [4], [5], [6], [7], [8], [9], [10]. The dynamic behaviors of MNNs have been broadly investigated due to their wide-ranging engineering applications such as signal processing, brain emulation, pattern recognition, and secure communication [11], [12], [13].

As an expansion of real-valued neural networks (RVNNs), complex-valued neural networks (CVNNs) have complex-valued states, activation functions, and connection weights. In comparison with RVNNs, CVNNs have more plentiful properties, which make CVNNs have wider applications. For example, XOR and detection of symmetry problems [14], [15] cannot be solved by RVNNs but can be easily solved by CVNNs. Nowadays, real-valued memristive neural networks (RVMNNs) have been extended to complex-valued memristive neural networks (CVMNNs). The dynamic behaviors of CVMNNs have gained much attention because of their extensive applications such as remote sensing, filtering, electromagnetic wave imaging, speech synthesis, and wireless communications [16], [17], [18], [19].

Recently, many interesting works of CVMNNs with time delays are available in the literature [20], [21], [22], [23], [24], [25], [26], [27]. For example, in [20], by Lyapunov method and inequality techniques, the anti-synchronization problem has been considered for CVMNNs with time delays. In [21], the stabilizability and instabilizability have been studied for CVMNNs with time delays via a nonlinear state feedback controller. In [25], by Lyapunov function and M-matrix method, stability problem has been investigated for CVMNNs. In [26], based on differential inclusion and set-valued map theories, dissipativity analysis has been considered for CVMNNs with time delays. It is noted that all the works in [20], [21], [22], [23], [24], [25], [26], [27] are investigated under the framework of using the maximum absolute values of memristive connection weights, which overlooked the information of other memristive connection weights. The existing results in [20], [21], [22], [23], [24], [25], [26], [27] may be conservative to some extent. Moreover, in [20], [21], [22], [23], [25], the assumptions on the activation functions are strict since both real and imaginary parts of activation functions are required to satisfy the Lipschitz conditions. Thus, it is meaningful to make full use of the information of memristive connection weights and relax the assumption on activation functions for CVMNNs with time delays.

In the meantime, due to the signal transmission congestions and finite switching speeds of amplifiers, time delays are ubiquitous in dynamical systems [28], [29], [30]. However, time delays may bring about oscillation and instability to exasperate the performance of systems. Therefore, the control of CVMNNs with time delays is of great importance. Up to now, some effective control schemes have been proposed for stability of CVMNNs including state feedback control [20], [21], [22] and adaptive control [23], [24]. All the forenamed control methods need continuously feed back signals to CVMNNs, which constitute the communication loads all the time. But in practice, the communication resources are often limited. Therefore, it is of importance in both theory and application to reduce the utilization of communication bandwidth. Sampled-data control, in which only the sampling information is transmitted to the controller, can effectively save the communication resources. Compared with forenamed continuous feedback control approaches, sampled-data control has more wonderful features such as high reliability and efficiency, easy installation and maintenance, and low energy and cost consumption. Thus, sampled-data control has attracted increasing interests of numerous researchers [31], [32], [33], [34], [35], [36], [37]. However, few works have considered sampled-data control for stabilizability of CVMNNs with time delays.

In implementation, due the limitations of system equipments and influence of environment, fluctuations unavoidably exist in the control process of CVMNNs. However, the existence of fluctuations may result in instability of CVMNNs. Hence, it is significant to design non-fragile controllers for CVMNNs. Meanwhile, owing to the influence of uncertain factors, random time delays are universal, which belong to two intervals in a probabilistic way [38]. However, to our best knowledge, non-fragile sampled-data controllers and random time-varying delays have not been considered for CVMNNs.

Motivated by the above discussions, by designing a non-fragile sampled-data controller, the stability and stabilizability problems are studied for CVMNNs with random time-varying delays. The main contributions of this paper are summarized below.

  • 1)

    A non-fragile sampled-data controller is designed for CVMNNs for the first time. Compared with some existing control schemes, our one is more applicable and can effectively save the communication resources.

  • 2)

    Less conservative criteria are established for CVMNNs since the average values of the maximum and minimum of the memristive connection weights are used instead of the maximum absolute values of memristive connection weights.

  • 3)

    The assumption on activation functions of CVMNNs is relaxed. Different from the existing assumptions with both real and imaginary parts of activation functions satisfying the Lipschitz conditions, only the complex-valued activation functions are required to satisfy the Lipschitz condition.

  • 4)

    A random time-varying delay is considered for CVMNNs, which belongs to two intervals in a probabilistic way.

Notations: R and C represent, respectively, the set of real numbers and the set of complex numbers. Rn and Cn denote the n-dimensional Euclidean space and n-dimensional complex space, respectively. Rn×m and Cn×m are the set of all n×m real matrices and the set of all n×m complex matrices, respectively. In, 0n, and 0n,m stand for n×n identity matrix, n×n, and n×m zero matrices, respectively. For real symmetric matrices X and Y, the notation X>Y means that the matrix XY is positive definite. Sym{X}=X+XT. diag{} means a block-diagonal matrix. The superscript T represents matrix transposition. The symmetric term in a matrix is denoted by *. λmin(·) is the minimum eigenvalue of a real symmetric matrix. · denotes the Euclidean vector norm. E{·} represents the mathematical expectation.

Section snippets

Problem description and preliminaries

Replacing resistors by memristors in circuit and according to the Kirchhoff’s current law, a model of MNNs with a random time-varying delay is described as:Ckψ˙k(t)=[j=1n(Nfkj+Ngkj)+1Rk]ψk(t)+j=1nsgnkjNfkjfj(ψj(t))+j=1nsgnkjNgkjfj(ψj(tτ(t)))+u¯k(t),t0,k=1,2,,n,where ψk(t) means the voltage of the capacitor Ck; fj(·) is the activation function; u¯k(t) is the control input; Rk is the resistor; Nfkj and Ngkj denote, respectively, the memductances of memristors Mfkj and Mgkj, in which Mfkj

Main results

In this section, the stability and stabilizability of CVMNN (3) are studied via non-fragile sampled-data control mechanism (22). We first derive sufficient conditions for CVMNN (3) to be globally asymptotically stable. Then, a design method for the non-fragile sampled-data controller (22) is proposed.

LetW̲1=[W̲1RW¯1IW̲1IW̲1R],W¯1=[W¯1RW̲1IW¯1IW¯1R],W̲2=[W̲2RW¯2IW̲2IW̲2R],W¯2=[W¯2RW̲2IW¯2IW¯2R],W˜1=W̲1+W¯12,W˜2=W̲2+W¯22,W1*R=W¯1RW̲1R2=(w¯1kjRw̲1kjR2)n×n,W1*I=W¯1IW̲1I2=(w¯1kjIw̲1kjI2)n×n,W

Illustrative example

In this section, the effectiveness of the obtained theoretical results is illustrated by the following example.

Consider a 2-D CVMNN as follows:{ϑ˙1(t)=1.5ϑ1(t)+j=12(w11jR(σ1(t))+iw11jI(ς1(t)))fj(ϑj(t))+j=1n(w21jR(σ1(t))+iw21jI(ς1(t)))fj(ϑj(tτ(t)))+u1(t),ϑ˙2(t)=1.5ϑ2(t)+j=12(w12jR(σ2(t))+iw12jI(ς2(t)))fj(ϑj(t))+j=1n(w22jR(σ2(t))+iw22jI(ς2(t)))fj(ϑj(tτ(t)))+u2(t),where the activation functions fj(ϑj(t))=tanh(σj(t))+itanh(ςj(t)), j=1,2, andw111R(σ1(t))={1.8,|σ1(t)|1.50.9,|σ1(t)|>1.5,w112R

Conclusion

In this note, we have investigated, by designing a non-fragile sampled-data control mechanism, the stability and stabilizability problems of CVMNNs with random time-varying delays. In comparison with the existing results of CVMNNs with time-varying delays, our work has four superiorities. First, the non-fragile sampled-data controller is more applicable and can effectively save the communication resources, since the influence of fluctuations is considered and only the sampling information is

Declaration of Competing Interest

The authors declare that they have no conflict of interest to this work.

Acknowlgedgments

The work was supported by Sichuan Science and Technology Program (2019YJ0382) and National Natural Science Foundation of China (No. 62003229). Also, the work of J.H. Park was supported by the National Research Foundation of KoreaNRF) grant funded bythe Korea government (Ministry of Science and ICT) (No. 2019R1A5A808029011).

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