This paper deals with robust synchronization of the fractional-order unified chaotic systems. Firstly, control design for synchronization of nominal systems is proposed via fractional sliding mode technique. Then, systematic uncertainties and external disturbances are considered in the fractional-order unified chaotic systems, and adaptive sliding mode control is designed for the synchronization issue. Finally, numerical simulations are carried out to verify the effectiveness of the two proposed control techniques.

Even though the theory of fractional calculus dates back to the end of the 17th century, the subject only really came to life over the last few decades [

One of the most important areas of application is the fractional-order chaotic systems, which have wide potential applications in engineering. Since Hartley et al. firstly discovered chaotic phenomenon in fractional dynamics systems [

In this paper, we investigate robust synchronization of the fractional-order unified chaotic systems. We firstly propose controllers to synchronize the nominal systems via fractional sliding mode technique. Secondly, we consider systematic uncertainties and external disturbances in the fractional-order unified chaotic systems and establish adaptive sliding mode control for synchronization of the uncertain systems.

The rest of this paper is organized as follows. Section

The most important function used in fractional calculus is Euler’s Gamma function, which is defined as

Another important function is a two-parameter function of the Mittag-Leffler type defined as

Fractional calculus is a generalization of integration and differentiation to noninteger-order fundamental operator

The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [

The Grünwald-Letnikov derivative definition of order

For binomial coefficients calculation, we can use the relation between Euler’s Gamma function and factorial defined as

The Riemann-Liouville derivative definition of order

However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain

The Caputo definition of fractional derivative can be written as

In the following, we use the Caputo approach to describe the fractional chaotic systems and the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional-order derivative as

Consider the following commensurate fractional-order dynamics system:

Consider the following

Then, the zero solution of system (

Assume that there exists a scalar function

In [

The fractional-order unified chaotic system has been firstly introduced and studied in [

System (

Let us define the state errors between the response system (

By subtracting (

In the following, the sliding mode control technique, which can maintain low sensitivity to unmodeled dynamics and external disturbances, is applied to establish an effective control law to guarantee the synchronization of the drive system (

In this subsection, let us firstly consider a simple case: the nominal fractional-order unified chaotic system; that is, the system contains no systematic uncertainties or external disturbances. The design procedure is elaborated in the rest part of this subsection.

In order to achieve the stability of system (

As long as system (

By using Lemmas

Choose the first control Lyapunov function

Choose the second control Lyapunov function

Equation (

Choose the third control Lyapunov function

With this choice, (

Finally, we gather the above three control functions as

In terms of Lemma

In this subsection, we will proceed to study the synchronization of the fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances which can be hardly ignored in the real-world application. It is assumed that systematic uncertainties

The uncertain fractional-order unified chaotic system can be described as

The adaptive sliding mode design of system (

Consider the first control Lyapunov function

Choose the second control Lyapunov function

Choose the third Lyapunov function

Finally, we gather the above three control functions as

By using Lemma

In [

The chaotic behaviors are presented in Figures

Chaotic trajectories with

Chaotic trajectories with

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed sliding model control in Section

When

Synchronization of determined fractional-order unified chaotic system with

Synchronization errors states with

Sliding surfaces states with

Synchronization of determined fractional-order unified chaotic system with

Synchronization errors states with

Sliding surfaces states with

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive sliding model control in Section

When

Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances with

Synchronization errors states in the presence of systematic uncertainties and external disturbances with

Sliding surfaces states in the presence of systematic uncertainties and external disturbances with

Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances with

Synchronization errors states in the presence of systematic uncertainties and external disturbances with

Sliding surfaces states in the presence of systematic uncertainties and external disturbances with

This work is concerned with robust synchronization of the fractional-order unified chaotic system. The sliding mode control technique was applied to propose the control design of nominal system and adaptive sliding mode control scheme was designed to develop the control laws and adaptive laws for uncertain system with systematic uncertainties and external disturbances whose bounds are unknown. Numerical simulations were presented to demonstrate the effectiveness of the two kinds of techniques.