Passivity and passification of memristive neural networks with leakage term and time-varying delays

Abstract

This paper investigates passivity and passification for memristive neural networks (MNNs) with both leakage and time-varying delays. MNNs are converted into traditional neural networks (NNs) by nonsmooth analysis, then sufficient conditions are derived to guarantee the passivity based on Lyapunov method. A novel Lyapunov–Krasovskii functional (LKF) is constructed without requiring all the symmetric matrices to be positive definite. The relaxed passivity criteria with less conservativeness or complexity are obtained in the form of linear matrix inequalities (LMIs), which can be verified easily by the LMI toolbox. Then, the passification controller is designed with the relaxed criteria to ensure that MNNs with both leakage and time-varying delays are passive. Finally, two pertinent examples are presented to show the effectiveness of the theoretical results.

Introduction

Memristor, as predicted by Chua [1] in 1971, is considered as the fourth passive circuit element, in addition to resistor, capacitor and inductor. Since discovering memristive behavior on nanodevices in Hewlett–Packard laboratories in 2008 [2], the researches on memristor and its widespread applications have been deeply developed. It is observed that the resistance of the memristor, also called memristance, can change according to the hostory of flowing current, which means that a lag will occur between the working and idle state and its subsequent effect. Variable resistance and the memory characteristics make memristor more suitable for application in neural networks than the conventional resistor. Hence the neural networks with memristors, also called memristive neural networks (MNNs), have been widely studied in image processing, video generation, biological systems and so on (see [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]).

As an important theoretical part in neural network field, dynamic analysis of MNNs has been investigated in many aspects such as stability [17], [18], passivity [19], [20], passification [21], [22], synchronization [23], [24], [25], [26], finite time control [27], [28], non-fragile control [29], [30], etc. The passivity theory, originating from circuit theory, provides a tool to analyze the stability of control systems. The passification analysis means to design a passive system with a feedback controller in order to reduce the parts that are ‘over passive’ or not passive in original systems [31]. Guo et al. [22] investigated the passivity and passification problem on MNNs for the first time. Zhang et al. [32] and Xiao et al. [33] reduced the conservativeness of passivity conclusions by combining various approaches. Passivity based control problems have also attracted a lot of attention in recent years (See [19], [20], [25]). At the same time, more research on passification of MNNs should also be investigated to reduce the conservativeness by combining new techniques, which has not been studied yet.

In the modeling of neural network architectures, varies delays in the neuron interconnection, which may have a negative impact on dynamical behaviors such as oscillation and instability, need to be considered to restore the original attributes of the physical models owing to the restricted speed of transmission and information processing. The leakage delay may exist in the negative feedback terms of systems due to the decay process of neurons, and will have a destabilizing influence on the dynamical behaviors of neural networks. In recent years, a few of results on the dynamic analysis of neural networks considering the constant leakage term or the time-varying leakage term have been derived (see [34], [35]). However, to the best of our knowledge, the passivity and passification problems of MNNs with both leakage and time-varying delays have not been solved yet.

One common method that is widely used for dynamic analysis of neural networks is Lyapunov method. By constructing the appropriate Lyapunov–Krasovskii functional (LKF) which can reasonably describe the states of the dynamic system, the stability of the equilibrium states can be easily analyzed. Whether LKF can fully and accurately describe the characteristics of the systems and how to reduce the conservativeness of derived conditions are two important issues of Lyapunov method in recent studies. The structure of LKF is mainly determined by the specific system models and is relatively fixed, while the conservatism of derived conditions depends on different processing methods. In previous researches, one way can reduce conservatism to a certain extent by adding multiple integral quadratic terms [36], [37] or decomposing the domains(the upper/lower limit of integration) [38] of integrations. Nevertheless, it is showed in [37] that the effectiveness of multiple integration is gradually decreasing, but the computational complexity are increasing. In this perspective, it may not be a good way to get more relaxed conditions. Another way to reduce conservativeness is finding the sharper bounds of integral terms from the derivatives of LKF. Free-weighting matrix (FWM) approaches and inequalities like Jensen’s inequality are used to deal with the integral terms. As the study moving on, improved methods have been widely used. Wirtinger-based integral inequality is widely used to obtain a sharper upper bound than the Jensen’s inequality. Improved FWM approaches are involved in [39]. For the multiple integral terms, the extended Wirtinger-based multiple integral inequality [40] and the extended auxiliary function-based inequalities [41] have been applied to theoretical researches. Furthermore, by adding the zero-value terms, such as He’s FWM techniques based on Newton–Leibniz formula [42] and Kwon’s FWM-based zero-value equality based on integration [43], more related terms will be obtained in the original integral terms from the derivatives of LKF, which will be helpful to reduce conservativeness [44].

Motivated by the above discussions, we investigate the passivity and passification of MNNs with both leakage and time-varying delays in this paper. By balancing the conservativeness and complexity, effective methods are proposed to obtain more relaxed passivity conditions. Based on the passivity theorem, we investigate the passification of MNNs with time-varying leakage term, which is studied for the first time. The main contributions of this paper are listed as follow:

• We construct a new LKF that can effectively relax the passivity conditions with little complexity increment, which can be easily transplanted to other researches. We construct an LKF without all the symmetric matrix to be positive definite. By combining Jensen’s inequality, Wirtinger-besed inequality, reciprocally convex combination and FWM-based techniques, the more sharper bounds of integral terms are obtained.

• In our paper, we analyze the effect of Wirtinger-based inequality and FWM-based zero-value equalities. In theory, we analyze the advantages of various methods and their effects on conservativeness and computational complexity, and propose a more balanced conditions in passivity and passification analysis. Meanwhile, numerical examples and comparison experiment are given to show that the combination of the two methods can greatly reduce the conservativeness with less amount of calculation.

• The passification problem of MNNs with both leakage term and time-varying delays is investigated for the first time. The process of matrix congruence transformation can be used in other passivation or stabilization studies. The feedback gain variation can be easily obtained by an LMI toolbox. The feedback control effect on different neurons can be changed artificially, which can be better applied to neural network systems.

The remainder of this paper is organized as follows: MNNs model and definitions of passivity and passification are formulated in Section 2. Section 3 presents the analyses of passivity and passification of MNNs. Then, two illustrative examples are presented in Section 4. Finally, concluding remarks and further work are included in Section 5.

Notations: Through this paper, $\mathbb{R}$ and ${\mathbb{R}}^{n\times m}$ represent the space of real number and the n × m dimensional Euclidean space, respectively. The notation ||·|| is used to denote Euclidean vector norm for a vector or for a matrix. $diag(a_1,a_2,\dots ,a_n)$ denotes a diagonal matrix, while sign(·) represents the sign function. * represents the symmetric block in symmetric matrix. A^T is the transpose of matrix A. A > 0 means that the matrix A is symmetric positive definite. Define Sym(A) as $A+A^{\mathrm{T}}$. I_n × n denotes the identity matrix in ${\mathbb{R}}^{n\times n}$. For $f:\mathbb{R}⟶\mathbb{R},$ denote $f\left(t^{-}\right)={\lim }_{s\to 0^{-}}f(t+s)$ and the upper-left Dini derivative $D^{-}f\left(t\right)={\lim }_{s\to 0^{-}}\sup \frac{f(t+s)-f\left(t\right)}{s}$.