Complexity

Volume 2018, Article ID 5431619, 8 pages

https://doi.org/10.1155/2018/5431619

## On Designing Feedback Controllers for Master-Slave Synchronization of Memristor-Based Chua’s Circuits

^{1}School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China^{2}Jiangxi E-commerce High Level Engineering Technology Research Centre, Jiangxi University of Finance and Economics, Nanchang 330013, China

Correspondence should be addressed to Ke Ding; nc.ude.efuxj@gnidek

Received 26 April 2018; Accepted 22 July 2018; Published 16 October 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Ke Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with designing feedback controllers for master-slave synchronization of two chaotic memristor-based Chua’s circuits. The memductance function of memristor-based Chua’s circuits is a bounded function with a bounded derivative which is more generalized than those piecewise constant-valued functions or quadratic functions in some existing papers. The main contributions are that one master-slave synchronization criterion is established for two chaotic memristor-based Chua’s circuits, and the feedback controller gain is easily obtained by solving a set of linear matrix inequalities. One numerical example is given to illustrate the effectiveness of the design method.

#### 1. Introduction

Since the memristor, a missing circuit element, was first introduced by Chua in 1971 [1] and was realized in 2008 [2], memristor-based Chua’s circuits have received some attention, see, for example, [3–5].

When some equipment of circuits in oscillators were replaced by memristors, complex and dynamical properties were revealed in the circuits. Chaotic attractors have been studied in memristor-based Chua’s circuits in which the memductance functions of memristors were characterized by a piecewise constant-valued function [3, 6, 7] or a quadratic function [4, 5, 8, 9]. It should be pointed out that the memductance function of memristor can be represented by a bounded function with a bounded derivative [2], which is more generalized than those piecewise constant-valued functions or quadratic functions in some existing papers [3, 4]. However, to the best of author’s knowledge, there is no result available in the existing published literature to study memristor-based Chua’s circuits with abovementioned memductance function, which is the first motivation of this paper.

Chaotic synchronization and chaos control have received much attention due to its theoretical importance and practical applications [10–33]. Due to the existence of memristors, the product of the memductance function and voltage can give rise to chaotical behaviors in circuits. Most research efforts [3–5, 7–9, 17–19] were made to chaotic behaviors of memristor-based circuits, rather than master-slave synchronization and chaos control for two memristor-based circuits. Zhang et al. [6] conducted stability analysis for a single circuit with a piecewise constant-valued memductance function, but they did not consider the synchronization problem of two circuits. In [15, 16], synchronization of memristor-based Chua’s circuits has been investigated, in which the memductance elements were piecewise linear functions. In addition, the memristor with a passive nonlinearity and a piecewise constant-valued memductance function is essential to generate the high signal-to-noise ratio which is not suitable for achieving the secure communication [5]. Therefore, the memristor with nonlinear memductance function which is suitable for secure communication should be worth studying. The memristor in which the memductance function is a bounded function with a bounded derivative can satisfy this criterion, but the mathematical model of corresponding circuit is a set of nonlinear differential equations as well as the corresponding error systems derived by the master-slave scheme. Thus, how to derive master-slave synchronization criteria for two memristor-based Chua’s circuits in which the memductance function is a bounded function with a bounded derivative and how to design a feedback controller matrix gain to achieve synchronization is the second motivation of this paper.

In this paper, we will deal with the problem of the controller design for master-slave synchronization of chaotic memristor-based Chua’s circuits. The master-slave scheme will be constructed by using an error state feedback control. We will derive one synchronization criterion. Based on the obtained synchronization criterion, we will give the sufficient conditions on the existence of an error state feedback controller. Moreover, we will obtain the controller gain. We will also use one numerical example to illustrate the effectiveness of the synchronization criterion and the design method.

*Notation 1. * denotes the -dimensional Euclidean space. is the set of all real matrices. For symmetric matrices and , the notation (respectively, ) means that matrix is positive definite (respectively, positive semidefinite). and are the maximum and minimum eigenvalues of the matrix , respectively.

#### 2. Memristor-Based Chua’s Circuits

The memristor in Chua’s circuits is a two-terminal element. The magnetic flux of memristor between the terminals is a function of the electric charge which passes through the device [1]. A flux-controlled memristor can be characterized by the incremental menductance function describing the flux-dependent rate of change of charge [2], i.e., . Therefore, the voltage across and the current through the memristor can be described as [3].

Figure 1 shows a smooth flux-controlled memristor-based Chua’s circuit, where and are the voltages across capacitors and , respectively; is the current through the inductances ; is a linear resistor; the Chua’s diode is replaced by a memristor. The mathematical model of the Chua’s circuit with the memristor can be described as
with the initial condition *.*