Advances in Mathematical Physics

Volume 2017 (2017), Article ID 5201634, 9 pages

https://doi.org/10.1155/2017/5201634

## Robust Output Synchronization of Arrays of Chaotic Sprott Circuits

Facultad de Ingeniería, Universidad Autónoma de Baja California, Blvd. Benito Juárez s/n, Mexicali, BC, Mexico

Correspondence should be addressed to David I. Rosas Almeida

Received 31 January 2017; Accepted 25 July 2017; Published 23 August 2017

Academic Editor: Giampaolo Cristadoro

Copyright © 2017 Ernesto V. Gonzalez Solis and David I. Rosas Almeida. 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 article presents a technique for synchronizing arrays of a class of chaotic systems known as Sprott circuits. This technique can be applied to different topologies and is robust to parametric uncertainties caused by tolerances in the electronic components. The design of coupling signals is based on the definition of a set of functionals which depend on the errors between the outputs of the nodes and the errors between the output of a reference system and the outputs of the nodes. When there are no parametric uncertainties, we establish a criterion to design the coupling signals using only one state variable of each system. When the parametric uncertainties are present, we add a robust observer and a low pass filter to estimate the perturbation terms, which are subsequently compensated through the coupling signals, resulting in a robust closed loop system. The performance of the synchronization technique is illustrated by real-time simulations.

#### 1. Introduction

Synchronization is a dynamical behavior that two or more systems exhibit when a correlated motion between them is established [1]. This phenomenon appears very often in nature, but it can also be forcibly induced by introducing some coupling elements or input signals in a convenient way. This last case is often denoted as controlled synchronization because it becomes a control objective [2].

Controlled synchronization can be achieved in dynamical systems on chaotic regime [3–8]. However, in practice, chaos synchronization is far from being straightforward because different aspects, as the presence of external disturbances and parameter uncertainties, affect significantly the ability of systems to synchronize. Therefore, robustness is a very desirable property of a synchronization approach.

Chaos synchronization has important applications, for example, in private communication systems; see, for example, [4, 5, 9], where simplicity is always a desirable characteristic to consider in a practical implementation. As shown by Sprott [10], a practical chaos generator can be constructed with a simple electronic circuit with nonlinear terms as piecewise linear functions.

Recently, some synchronization techniques for Sprott circuits and other kinds of systems have been proposed for the master/slave scheme and for arrays or networks of them. Some important papers are [5, 9, 11–14].

A proposal to synchronize Sprott circuits and their application in secure communication are presented in [4]. Here a proportional-integral-derivative (PID) control scheme is used to solve the synchronization problem of identical chaotic systems and without disturbances under the master/slave configuration. To verify the system performance, basic electronic components are used to implement the proposed chaotic secure communication system. Something similar is presented in [5], but here, the information is incorporated as a disturbance, which is recovered through the equivalent control produced by a variable structure control.

A robust algorithm to synchronize Sprott circuits, under the master/slave configuration, using sliding mode control is presented in [14]. The objective is to obtain identical synchronization between the master and slave systems in spite of existence of external disturbances. An important condition is that both systems must have the same nonlinear function. The coupling signal contains a term to linearize the error dynamics and add a discontinuous part that gives robustness to the closed loop system. With the same objective and conditions, a technique to synchronize two Sprott circuits in finite time is presented in [15]. An important characteristic in this work is that the slave system incorporates two control inputs. These are designed based on variable structure control and depend on all of states of the systems. As in [14], the master and the slave systems must have the same nonlinear function.

In [16], a technique to synchronize arrays of dynamical systems is presented. The arrays are formed by uncertain nonlinear second-order systems where only the generalized position is available. The synchronization technique can be applied to many array topologies where the connections can be unidirectional or bidirectional with different weights; this produces a connection matrix that it is not necessarily symmetric. The design of the coupling signals is based on a robust discontinuous controller and on an exact deriver that estimates the velocity of each node. This is an interesting proposal but it is not applicable to Sprott circuits.

In this paper, we present a technique, based on [16], for synchronizing arrays of a class of chaotic systems known as Sprott circuits. This technique can be applied to different topologies and is robust to parametric uncertainties caused by tolerances in the electronic components. The design of coupling signals is based on the definition of a set of functionals, which depend on the errors between the outputs of the nodes and on the errors between the output of a reference system and the outputs of the nodes; when there are no parametric uncertainties, we establish a criterion to design the coupling signals using only one state variable of each system. When the parametric uncertainties are present, we add a robust observer and a low pass filter to estimate the perturbation terms, which are subsequently compensated through the coupling signals, resulting in a robust closed loop system. The performance of the synchronization technique is illustrated by experimental results.

The organization of the paper is as follows. Section 2 includes some preliminary definitions and the statement of the synchronization objective. In Section 3, the synchronization technique is described. Here, a sufficient condition on synchronizability is established. Also, in this section, a methodology to design the coupling signals is presented. In Section 4, the performance of the synchronization technique is shown through a real-time simulation where an array formed by four nodes and a reference system is synchronized. Finally, in Section 5, some conclusions are presented.

#### 2. Preliminary Definitions and Problem Statement

Consider nonlinear dynamical systems, known as Sprott circuits [10], modeled byfor , where is one of the following functions: where is the output system; , , and are nominal values proposed in [10], for which system (1) presents chaotic behavior; , , and are parametric variations due to tolerances in electronic components; and are the coupling signals to be designed for each node.

A state space representation of system (1), taking the state variables as , , and , iswhere is a disturbance term due to parametric variations; it is unknown but bounded. Also, consider a reference system given bySystem (4) is defined in the same way of system (3), but in this case a coupling signal is not present.

Let us consider functionals . Here are the sets of all output functions.

*Definition 1. *One says that outputs of the systems are synchronized with respect to the functionals if

*Definition 2. *One says that outputs of the systems with initial conditions , are asymptotically synchronized with respect to the functionals if

*Definition 3. *One says that outputs of the systems with initial conditions , are approximately synchronized with respect to the functionals if there is an such that

Finally, we define the concept of connection graph. A connection graph is a figure that describes the presence or absence of couplings among reference system (4) and nodes (3) in an array of systems. Figure 1 shows an example of a connection graph with three nodes , orange spheres, and a reference system , green sphere. Here solid lines denote connections among nodes and dashed lines connect the reference system with nodes. Both kinds of couplings have a particular direction represented by an arrow. An arrow from to indicates that the th node has access to the output of th node. It is important to notice that couplings between nodes may be bidirectional; however, couplings from the reference system to nodes are unidirectional.