This paper deals with the synchronization of a class of fractional order chaotic systems with unknown parameters and external disturbance. Based on the Lyapunov stability theory, a fractional order sliding mode is constructed and a controller is proposed to realize chaos synchronization. The presented method not only realizes the synchronization of the considered chaotic systems but also enhances the robustness of sliding mode synchronization. Finally, some simulation results demonstrate the effectiveness and robustness of the proposed method.

Fractional calculus is as old as conventional calculus and with more than 300 years’ history, but its application to physics and engineering is in recent years. It has been found that many systems can be described by fractional order differential equations, for example, in interdisciplinary fields, such as viscoelastic [

Since the pioneering work of Pecora and Carroll [

Motivated by the aforementioned analysis, in this paper, we construct a robust synchronization of a class of uncertain fractional chaotic systems via adaptive sliding model control. Based on the designed fractional order integral type sliding surface, an adaptation algorithm is proposed to realize the synchronization of fractional order chaotic systems with unknown parameters, even the fractional order master and slave chaotic system with external disturbance. The numerical simulations show the effectiveness of the proposed method. This paper is organized as follows. In Section

Although fractional calculus is very important in modern science, it has no uniform definition up till now. There are many fractional calculus definitions and among them Riemann-Liouville and Caputo definitions are more important than others. As the constant’s fractional derivative is zero and its Laplace translation has the traditional initial value, the Caputo definition is used in this paper:

Consider a class of fractional order chaotic systems with unknown parameters [

Note that many fractional order chaotic systems belong to the class characterized by (3) in [

Let system (

The aim in this paper is that, for different initial conditions of systems (

It is assumed that the external disturbances are norm-bounded; that is,

The error between the driver system (

The first step is to select an appropriate sliding mode surface with the desired behavior:

To ensure the existence of the sliding motion, a discontinuous control law is proposed as

The fractional order error system is changed into the following formation:

The following update laws are defined to tackle the uncertainties, external disturbances, and unknown parameters:

If the controller is selected as (

Selecting a positive definite function as a Lyapunov function candidate

Introducing the sliding motion (

Substituting

Assorting to the update laws (

In this section, two numerical simulations are presented to show the efficiency of the proposed method.

Consider the fractional order Chen system [

Regarding (

The discontinuous control law corresponding to (

2.89-order fractional Chen system.

Synchronization errors of the drive and slave of fractional Chen systems.

Time response of the update parameters

Time response of the update parameters

Time response of the update parameter

Consider the fractional order Lorenz system [

2.97-order fractional Lorenz system.

The discontinuous control law corresponding to (

Synchronization errors of the drive and slave of fractional Lorenz systems.

Time response of the update parameters

Time response of the update parameters

Time response of the update parameter

In this paper, a robust adaptive sliding mode controller has been designed to synchronize a class of uncertain fractional chaotic systems with unknown parameters. Based on the Lyapunov stability theory, the designed closed-loop system is stable and the proposed robust adaptive controller can realize chaotic systems’ synchronization. Finally, two numerical examples have been shown to demonstrate the effectiveness of the proposed scheme.

The authors declared that they have no conflict of interests regarding the publication of this work. The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work.

This research was supported by the National Nature Science Foundation of Hebei Province under Project no. F2016502025, the Fundamental Research Funds for the Central Universities under Project no. 9161015007, and the National Nature Science Foundation of China under Project no. 61403137.