Design of a nonlinear controller and its intelligent optimization for exponential synchronization of a new chaotic system

Abstract

This paper deals with the synchronization problem of two chaotic systems in a master-slave configuration. The two systems are based on a new chaotic system recently reported by Akgul and Pehlivan in [1]. The global and exponential convergence to zero of the synchronization error is guaranteed by means of a nonlinear controller designed from the Lyapunov stability theory. In order to find the appropriate parameters of this controller, the compromise between magnitude of control signal and convergence speed is quantified by using a quadratic performance index. Next, such index is minimized via four different intelligent optimization algorithms: differential evolution, brain storm, cuckoo search, and harmony search. These four algorithms are exhaustively tested and the results are systematically compared. Finally, the performance of the optimized nonlinear controller is showed graphically.

Introduction

Chaos theory has been considered as one of the three more important discoveries of the last century in physics, at the same level that relativity theory and quantum mechanics [2], [3], [4]. From pioneer work of Edward Lorenz in 1963 [5], chaos theory has given rise to an intense research activity not only for its undeniable intrinsic transcendence but by its applications in areas as diverse as economy [6], [7], biology [8], [9], [10], finance [11], [12], [13], [14], optics [15], [16], medicine [17], hydrology [18], secure communications [19], [20], [21], [22], [23], [24], ecology [25], chemistry [26], [27], [28], [29], mechanics [30], [31], etc.

A chaotic system is a nonlinear aperiodic oscillator with an extremely high sensitivity to initial conditions which causes the impossibility of carrying out accurate predictions about its long-term dynamic behavior. Nevertheless, one of the most intriguing properties of these systems is that despite their lack of predictability they can be controlled [32], that is, their dynamic behavior can be modified.

For chaotic systems control, two cases of interest can be distinguished: chaos suppression and chaos synchronization. In the first case, the objective is the extinction of the chaotic behavior by compelling the trajectories to converge to a periodic orbit or an equilibrium point. In the second one, the dynamic behavior of two o more subsystems must converge to a same unique chaotic behavior. Such subsystems can be coupled bidirectionally, that is, it exists a mutual influence between them, or unidirectionally. This last configuration is known as master-slave and is formed by two subsystems: a subsystem with coupled inputs called slave is compelled to follow the chaotic dynamics of an autonomous subsystem called master.

From the initial work of Pecora and Caroll [33], several proposals have been presented for synchronization of chaotic systems. One of simplest approaches is based on the use of what is known in technical literature as “linear state error feedback control” [34], [35], [36]. Although the implementation of this strategy is easy, its corresponding mathematical analysis to guarantee the synchronization error convergence to zero could be complicated. Other approach consists of using feedback linearization [37], [38]. By the appropriate control law, the nonlinearities are compensated and consequently the dynamics of the error synchronization is linearized. Once this objective has been achieved, any of the well-developed techniques for control of linear systems can be applied. Additionally, other proposals have been reported such as synchronization by filtrering [39] and synchronization by using the theory of passivity [40], [41]. The most of the aforementioned approaches consider the case when the states and the parameters of the chaotic system are well-known. When no all states are available, observer based synchronization has been developed in [42], [43], [44]. And, for the case of parametric uncertainty, several strategies of adaptive synchronization has been presented in [45], [46], [47], [48], [49]. Nonetheless, the control laws obtained by means of this last approach can be very complicated and their implementation could become difficult. For this reason, the context in which each strategy will be applied must be distinguished thoroughly. If the chaotic system is implemented by means of analog electronics [50], [51], [52] then certainly the parameters cannot be known exactly and consequently the adaptive approach could be justified. However, when the implementation is realized numerically by means of a high performance digital platform such as FPGA [53], [54], [55] or DSP [56], [57], then the assumption of a well knowledge about the parameters is reasonable and simpler approaches such as feedback linealization or nonlinear control can be utilized. The main idea behind of this last approach consists of, given a Lyapunov function candidate such as $V=e^{T}e/2$ where $e$ is the synchronization error, determining a control law such that ${\displaystyle \dot{V}}<0$. Thus, from Lyapunov stability theory, the asymptotic convergence of the synchronization error to zero can be guaranteed. In [58], a nonlinear controller was developed to achieve the synchronization of two chaotic Genesio systems. In [59], [60], [61], identical or inclusive different chaotic systems were synchronized by means of systematically designed nonlinear controllers.

In this paper, we propose to use a nonlinear controller for accomplishing the synchronization of a master-slave configuration based on a new chaotic system reported recently by Akgul and Pehlivan in [1]. The convergence to zero of the synchronization error is guaranteed by means of Lyapunov analysis. In order to find the appropriate parameters of the controller, the compromise between magnitude of control signal and convergence speed is quantified by using a quadratic performance index. Next, such index is minimized via four different intelligent optimization algorithms: differential evolution, brain storm, cuckoo search, and harmony search. With respect to previous works about chaos synchronization using nonlinear control [59], [60], [61], the main contributions of this paper are as follows: a) in the aforementioned works, the asymptotic synchronization was guaranteed. Here a stronger result is found, the exponential convergence to zero of the synchronization error, b) control gains are introduced to manipulate the convergence speed, c) four different algorithms of intelligent optimization are exhaustively tested for the tuning of such gains and the results are systematically compared. This is the first time, up to the best knowledge of the authors, that a nonlinear control strategy is proposed for the synchronization of the chaotic system discovered by Akgul and Pehlivan [1].