Novel discontinuous control for exponential synchronization of memristive recurrent neural networks with heterogeneous time-varying delays

Abstract

This paper investigates the exponential synchronization problem of memristive recurrent neural networks (MRNNs) with heterogeneous time-varying delays (HTVDs). First, a novel discontinuous feedback control is designed, in which a tunable scalar is introduced. The tunable scalar makes the controller more flexible in reducing the upper bound of the control gain. Based on this control scheme, the double integral term can be successfully used to construct the LKF. Second, New method for tackling memristive synaptic weights and new estimation technique are presented. Third, based on the LKF and estimation technique, synchronization criterion is derived. In comparison with existing results, the established criterion is less conservatism thanks to the double integral term of the LKF. Finally, numerical simulations are presented to validate the effectiveness and advantages of the proposed results.

Introduction

Leon Chua originally postulated memristor in 1971, which is associated with the fundamental properties of electrical circuits, namely, current, voltage, charge and magnetic flux [1]. While relationships for basic circuit elements such as resistor, inductor and capacitor were known, Chua established the relationship between electric charge and magnetic flux $(d\phi =Mdq)$ by conjecturing the existence of memristor, as shown in Fig. 1 [1]. About 40 years later, the practical memristor device based on Ti$O_2$ thin films was realized by scientists at Hewlett-Packard Laboratories who published their findings in [2], [3]. The memristor is a two terminal element with variable resistance called memristance. The value of memristance depends on the magnitude and polarity of the voltage applied to it and the length of the time that the voltage has been applied. When the voltage is turned off, the memristor remembers its most recent value until it is turned on next time. Therefore, the memristor can memorize its past dynamic history. Such memory features make the memristor as a promising candidate for next-generation computers, powerful brain-like neural computers, and neuromorphic computing systems [4]. As shown in [3], the memristor exhibits pinched-hysteresis loops, which can be observed in Fig. 2 [3]. Due to this feature, many researchers use memristors to design new models of neural networks to simulate the human brain. Recently, memristive recurrent neural networks (MRNNs) have received much attention because of their wide applications in variety of areas, such as pattern recognition, signal processing, optimization, and associative memories [5], [6], [7], [8], [9], [10], [11], [12], [13].

Since Pecora and Carroll firstly proposed a driving-response scheme for synchronization of coupled chaotic systems, synchronization has gained growing research interests due to its wide potential applications such as secure communication, biological informatics, and so on [14], [15], [16], [17]. From the control strategy point of view, the control is an effective synchronization method for systems, and many control approaches have been proposed such as adaptive control [7], feedback control [18], ${\mathcal{H}}_{\infty }$ control [19], and sampled-data control [20]. Due to the special physical properties and wide applications of memrisor, the synchronization control problem has been extended to MRNNs. Hence, the investigation of synchronization control for MRNNs is of great importance.

Meanwhile, due to the finite switching speeds of amplifiers and traffic congestions in signal transmission processes, time delays unavoidably exist in a variety of systems [21], [22], [23]. Therefore, the synchronization control for MRNNs with delays has become a hot research topic [6], [7], [8], [9], [11], [12], [13], [24], [25], [26], [27], [28], [29], [30]. The delays in [6], [12], [13], [30] are constants or neuron-independent variables. However, in very large-scale integrated (VLSI) circuits, the delays in MRNNs are always heterogeneous, which are depend on the neurons. In this case, most of the synchronization criteria for MRNNs with constant or neuron-independent delays may be not available anymore. It is known that time delay has a great effect on synchronization of MRNNs. Hence, it is germane to consider HTVDs in MRNNs.

Up to now, many interesting results concerning synchronization control of MRNNs with time delays have been published in [6], [7], [8], [9], [11], [12], [13], [27], [29], [30], [31], [32], [33], [34], [35]. In [11], by discontinuous control scheme, the global exponential synchronization problem has been studied for MRNNs with HTVDs. In [12], asymptotic synchronization of MRNNs with time-varying delays has been considered. However, in [11], [12], constrained by their controllers, the double integral term ${\int }_{-\tau }^0{\int }_{t+\theta }^t{\dot{e}}^T\left(s\right)S\dot{e}\left(s\right)dsd\theta$ cannot be used to construct the LKF, even though it is very effective for conservatism reducing. On the other hand, since the MRNNs have the feature of initial-value-sensitivity and parameters of MRNNs are state-dependent, the parameters of driver and response MRNNs are uncertain and may be mismatched. In [27], [32], [33], [34], the following assumption is necessary to solve the problem of parameter mismatch,

$$\begin{array}{cc}& co\{{\check{a}}_{ij},{\widehat{a}}_{ij}\}{f}_j\left(y_j\left(t\right)\right)-co\{{\check{a}}_{ij},{\widehat{a}}_{ij}\}{f}_j\left(x_j\left(t\right)\right)\subseteq co\{{\check{a}}_{ij},{\widehat{a}}_{ij}\}(f_j\left(y_j\left(t\right)\right)-f_j\left(x_j\left(t\right)\right)),\\ & co\{{\check{b}}_{ij},{\widehat{b}}_{ij}\}{f}_j\left(y_j\left(t\right)\right)-co\{{\check{b}}_{ij},{\widehat{b}}_{ij}\}{f}_j\left(x_j\left(t\right)\right)\subseteq co\{{\check{b}}_{ij},{\widehat{b}}_{ij}\}(f_j\left(y_j\left(t\right)\right)-f_j\left(x_j\left(t\right)\right)).\end{array}$$

However, the assumption has been proved not always to be correct in [28]. To avoid this assumption, further investigations on the parameter mismatch problem are proposed in [7], [29], where the activation functions are zero at switching points. But this assumption is still strict. Further, most of the existing results in [7], [9], [29], [30], [31], [35] on synchronization control of MRNNs are studied under the framework of using the maximum absolute values of memristive synaptic weights. Thus, the obtained synchronization criteria in these papers may be conservative. However, to our best knowledge, these problems have not been successfully solved for the synchronization problem of MRNNs with HTVDs.

Motivated by the above discussions, in this paper, the global exponential synchronization problem of MRNNs with HTVDs is investigated via discontinuous control law. The main contributions are summarized below.

(1) A novel discontinuous feedback control scheme is designed. Based on this controller, the double integral term can be successfully utilized to reduce control cost and conservatism. Moreover, due to the discontinuous part of the controller, the parameter mismatch problem of MRNNs can be solved by relaxing some strict assumptions. In addition, a tunable scalar is introduced, which can be adjusted to reduce the upper bound of the control gain. Thus, compared with existing controllers in [7], [11], [12], [27], [29], [32], [33], [34], the controller here is more effective and flexible than those in the existing papers.

(2) A new method for tackling memristive synaptic weights is proposed. Based on the new method in Lemma 1, the state-dependent parameters of MRNNs can be transformed into traditional neural networks with uncertain parameters. Compared with existing method used in [7], [9], [29], [30], [31], [35], this method is much less conservative, since the average values of the maximum and minimum of memristive synaptic weights is adopted, instead of the maximum absolute values of memristive synaptic weights.

(3) A new estimation technique is the first time to be proposed for MRNNs with HTVDs in Lemma 3, By this estimation technique, the derivative of the double integral term can be easily manipulated.

The remainder of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are introduced. In Section 3, new exponential synchronization criteria for MRNNs with HTVDs are proposed by designing discontinuous controller. In Section 4, numerical examples are given to demonstrate the effectiveness and the benefit of the obtained results. Finally, the conclusion is drawn in Section 5.

Notations: Throughout this paper, the superscripts T stands for matrix transposition. ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space, ${\mathbb{R}}^{n\times m}$ is the set of all n × m real matrices. I_n, 0_n, and 0_n,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 $X-Y$ is positive define. The symmetric term in a matrix is denoted by *, $\text{diag}\{\dots \}$ stands for a block-diagonal matrix, and $\text{Sym}\left\{X\right\}=X+X^T$. λ_min( · ) and λ_max( · ) denote the minimum and maximum eigenvalue of a real symmetric matrix, respectively. ‖ · ‖ stands for the Euclidean vector norm.