Complexity

Volume 2019, Article ID 4706491, 10 pages

https://doi.org/10.1155/2019/4706491

## Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

^{1}Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Azcapotzalco, Instituto Politécnico Nacional, Ciudad de México 02250, Mexico^{2}Área de Mecatrónica, Centro de Innovación y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, Ciudad de México 07700, Mexico^{3}Departamento de Ingeniería Electrónica, Tecnológico Nacional de México, CENIDET, Cuernavaca 62490, Mexico

Correspondence should be addressed to J. Humberto Pérez-Cruz; xm.npi@zerephj

Received 28 January 2019; Revised 3 June 2019; Accepted 23 June 2019; Published 25 July 2019

Academic Editor: Mohammed Chadli

Copyright © 2019 J. Humberto Pérez-Cruz et al. 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

In this paper, a new linear feedback controller for synchronization of two identical chaotic systems in a master-slave configuration is presented. This controller requires knowing a priori Lipschitz constant of the nonlinear function of the chaotic system on its attractor. The controller development is based on an algebraic Riccati equation. If the gain matrix and the matrices of Riccati equation are selected in such a way that a unique positive definite solution is obtained for this equation, then, with respect to previous works, a stronger result can be guaranteed here: the exponential convergence to zero of the synchronization error. Additionally, the nonideal case is also studied, that is, when unmodeled dynamics and/or disturbances are present in both master system and slave system. On this new condition, the synchronization error does not converge to zero anymore. However, it is still possible to guarantee the exponential convergence to a bounded zone. Numerical simulation confirms the satisfactory performance of the suggested approach.

#### 1. Introduction

The problem of the unidirectional synchronization of chaotic systems consists of finding an appropriate control law such that when this is applied to a system with coupled inputs called “slave” or “response,” such system follows the dynamics of an autonomous chaotic system called “master” or “drive” [1–11]. This proper control action is necessary because, without it, two identical autonomous chaotic systems could never be synchronized due to their high sensitivity to initial conditions [12–18].

During the last three decades, several strategies have been proposed to solve the synchronization problem when the structure and the parameters of both chaotic systems are known. One of these approaches is active control [19–31]. In this approach, the controller is selected based on the synchronization error dynamics in such a way that the nonlinearities are compensated and the dynamics equations are decoupled [32]. Another approach is nonlinear control [33–40]. In this technique, given a Lyapunov function candidate , the control law is selected considering that the first derivative of must be compelled to be negative definite [41]. Hence, the asymptotic convergence to zero of the synchronization error can be guaranteed. The aforementioned strategies present two drawbacks: (a) their practical implementation can be difficult, particularly for analog systems; (b) the controller can be expensive with respect to the consumption of energy. These problems can be avoided if a linear feedback controller is used [42–45]. This kind of controller is formed by the product of a gain matrix and the negative synchronization error, , that is, In [46], Wang et al. proposed an approach for synchronization of Chen system using linear feedback control. Given a proper Lyapunov function candidate, they could express the first time derivative of this function as a quadratic form of the synchronization error absolute value. Thus, by taking into account the conditions on which this quadratic form is negative definite, they could guarantee the asymptotic convergence to zero. This result was obtained by using only a unique control input. Based on this technique, the synchronization of the unified chaotic system and Lü system was presented in [47] and [48], respectively. It is important to mention that both Chen system and unified system are formed by five linear terms and two cross-product terms. Lü system is formed by four linear terms and two cross-product terms. Both cross-product terms are exactly the same expression in the three cases. By using Wang’s technique, Yassen synchronized a four-scroll chaotic system formed by three linear terms and three cross-product terms [49]. However, to be able to guarantee the asymptotic convergence, Yassen had to use three control inputs. Likewise, Liu chaotic system was synchronized in [50] with three control inputs and using the same procedure but applied from a more general perspective. Based on the same technique but with a unique control input, hyperchaotic Lorenz–Stenflo system formed by eight linear terms and two cross-product terms, and Lü hyperchaotic system formed by seven linear terms and two cross-product terms were synchronized in [51] and [52], respectively. In spite of the aforementioned successful applications, this technique has two main disadvantages: (a) the design procedure must be particularized to each system; (b) the technique is valid only for a very constrained class of systems. On the other hand, by using linearization and Lyapunov’s direct methods and linear feedback control, a sufficient synchronization criterion was presented in [42] for the case of generalized Lorenz systems. A wider class (but still restrictive) of systems were considered in [44] for synchronization based on linear feedback control. In all the aforementioned works, only the asymptotic convergence of the synchronization error can be guaranteed. To overcome this situation and the limitations of Wang's technique, in this paper, a new linear feedback controller based on a matrix Riccati equation is presented. This procedure requires knowing a priori Lipschitz constant of nonlinear function of the chaotic system on its attractor. If the gain matrix and the matrices of the Riccati equation are selected in such a way that this equation has a unique positive definite solution, then a stronger result can here be established: the exponential convergence to zero of the synchronization error. The proposed strategy can be applied to a very wide class of systems. However, for simplicity, the attention is focused on a system recently reported by Xian et al in [53]. Despite the similarity between this system and the system synchronized by Yassen in [49] (the main difference is only a unique cross-product term), Wang’s technique is not able to handle this new system.

Throughout this paper, given a vector denotes the Euclidean norm of , that is, ; given a matrix and denote the minimum and maximum eigenvalues of , respectively; denotes the two-norm of , that is, .

#### 2. System Description

The system proposed by Xian [53] is a new generalized third-order Lü chaotic system formed by three linear terms and four cross-product terms. The system can be described as where , and are the system states and , and are constant parameters. This system shows chaotic behavior for the values and the initial condition , and . In Figure 1, the corresponding attractor is shown. The time series for the states , and during the first twenty seconds of numerical simulation are plotted in Figure 2.