Some sufficient conditions, which are valid for stability check of fractional-order nonlinear systems, are given in this paper. Based on these results, the synchronization of two fractional-order chaotic systems is investigated. A novel fractional-order sliding surface, which is composed of a synchronization error and its fractional-order integral, is introduced. The asymptotical stability of the synchronization error dynamical system can be guaranteed by the proposed fractional-order sliding mode controller. Finally, two numerical examples are given to show the feasibility of the proposed methods.
In the past two decades, synchronization of chaotic systems (CSs) has received more and more attention, and a lot of interesting works have been done, which have potential application values in secret communications, signal processing, and complex systems [
It is well known that sliding mode control (SMC) is a very effective control method to cope with system uncertainties and external disturbances [
In control theory, stability analysis is an essential aspect. With respect to fractional-order linear systems, the stability condition was firstly investigated in [
In this section, we will give some properties of fractional calculus. The
The Laplace transform of Caputo fractional derivative is given as [
The Mittag-Leffler function is given by
Let
Let
Let
Consider a class of fractional-order systems described by
If
It follows from (
Using (
Thus, according to (
Noting that the Laplace transform of a Mittag-Leffler function is
It follows from Lemma
Consequently, for large enough time
It should be pointed out that Theorem
The equilibrium point of system (
It should be mentioned that Assumptions
Consider system (
Suppose that
It follows from (
After some straightforward manipulators, one has
Using Lemma
The master and slave FOCSs are defined, respectively, as
Define the synchronization error
To meet the synchronization object, let us construct the following fractional-order sliding mode surface:
Now, we can give the following results.
Consider the master FOCS (
It follows from (
Substituting (
Noting that
In this section, two examples will be given to show the effectiveness of the proposed method.
The fractional-order Duffing system is described by [
The Jacobian matrix of system (
It is easy to know that system (
Under the initial conditions
Phase attractor of FOCS (
According to (
In the simulation, the initial condition for the slave FOCS is
The simulation results are presented in Figures
Synchronization between
Synchronization between
Time response of synchronization errors
Time response of control inputs
Let us consider the following fractional-order chaotic neural networks expressed by [
Suppose that
Dynamical behavior of system (
It is easy to know in the master chaotic system (
The initial condition of the slave FOCS is
The simulation results are given in Figures
Simulation results in (a) synchronization between
Time response of control inputs
In this paper, two stability criteria for fractional-order nonlinear systems are given. Based on these theorems, the synchronization of two identical FOCSs is addressed. A fractional-order sliding surface, which contains a fractional-order integral of the synchronization errors, is given. The proposed controller can guarantee the asymptotical stability of the closed-loop systems. However, in the controller design, we need to know the exact value of the Lipchitz constant. How to reduce this condition is one of our future research directions.
The author does not have a direct financial relation with any commercial identity mentioned in this paper that might lead to conflicts of interest.
This work is supported by the National Natural Science Foundation of China (Grant no. 11302184) and the Young and Middle-Aged Teacher Education and Science Research Foundation of Fujian Province of China (Grant no. JAT170423).