Fractional-Order Sliding Mode Synchronization for Fractional-Order Chaotic Systems

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.


Introduction
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 [1][2][3][4][5][6][7][8][9].Recently, control and synchronization of fractional-order chaotic systems (FOCSs), which can be seen as a generalization of the integer-order CSs, have been studied extensively.A lot of controllers have been implemented such as active control [10], feedback control [11], sliding mode control [12,13], adaptive control, [14,15], and adaptive fuzzy control [8,9,16].
It is well known that sliding mode control (SMC) is a very effective control method to cope with system uncertainties and external disturbances [17][18][19][20][21][22][23][24][25][26][27].Consequently, it has been used to synchronize FOCSs.For example, a novel FOCS and its SMC have been studied in [28]; SMC of a 3D FOCS using a fractional-order switching type controller is investigated in [29].Using a hierarchical fuzzy neural network, [30] proposed a new adaptive SMC method for the synchronization of uncertain FOCSs.On the other hand, it is well known that, in stability analysis of nonlinear systems, quadratic Lyapunov functions are most commonly used.However, [31,32] show that it is not realistic to use quadratic Lyapunov functions in the stability analysis of fractionalorder nonlinear systems due to the complicated infinite series produced by differentiating the squared Lyapunov function with fractional order.It should be mentioned that, in most aforementioned works, the stability analysis is given based on fractional Lyapunov methods.How to establish some stability analysis methods according to the model of FOCSs is a meaningful work.
In control theory, stability analysis is an essential aspect.With respect to fractional-order linear systems, the stability condition was firstly investigated in [33].Then, using LMI, some sufficient conditions are given in [34].The related results on the stability analysis of fractional-order nonlinear systems can be seen in [35][36][37][38][39][40][41] and the references therein.It should be pointed out that the stability criterion for fractional-order nonlinear systems requires further study.Thus, proposing some new stability criterion for FOCSs is necessary.In this paper, we will give two sufficient conditions for the stability of a class of FOCSs.Based on these theorems, a fractional-order SMC will be given.The contributions of this paper are concluded as follows: (1) two sufficient conditions are proposed to check the stability of the fractionalorder nonlinear system and (2) a novel fractional-order SMC is given, and the stability of the closed-loop system is proven rigorously.

Preliminaries
In this section, we will give some properties of fractional calculus.The th fractional-order integral is expressed as [42] The Caputo fractional derivative is given by where  is the fractional order satisfying  − 1 ≤  < .

Some Sufficient Conditions for the Stability Analysis of
Fractional-Order Systems.Consider a class of fractionalorder systems described by or equivalently where  = 1, 2, . . ., , 0 <  < 1, and then there exist two positive constants  0 and  such that for all  >  0 .
Proof.It follows from (11) that Using (5), one solves (15) as Thus, according to (13), one has Noting that the Laplace transform of a Mittag-Leffler function is then one has where Consequently, for large enough time , one has lim where  = max 1≤≤ {  /  }.This ends the proof of Theorem 4.
It should be pointed out that Theorem 4 can only drive   () to a small region of zero.To discuss the asymptotic stability, one needs the following assumptions.
Assumption 5.The equilibrium point of system ( 11) is the origin.Assumption 6. (()) is a Lipshitz continuous function; that is, the following inequality holds: where  > 0 is a Lipshitz constant.
Remark 7. It should be mentioned that Assumptions 5 and 6 are reasonable.In fact, every equilibrium point of system (11) can be moved to the origin by some linear transformations.In many FOCSs, the nonlinear functions are smooth and Lipshitz continuous, for example, fractional-order Lorenz system, fractional-order Chen system, fractional-order Lü system, fractional-order financial system, and fractional Volta system [45].
After some straightforward manipulators, one has Solving (25) yields According to Assumption 6 and Lemma 1, one can find a constant  > 0 such that Using Lemma 2, one has Noting that (/)‖‖ < , where  = ‖‖ = max 1≤≤ −   , then according to (28) one has which completes the proof.

Synchronization Controller Design. The master and slave
FOCSs are defined, respectively, as where (), ζ() ∈ R  are the state vectors of the master FOCS and slave FOCS, respectively, , ,  ∈ R × are three constant matrices,  is a positive definite control gain matrix, and () ∈ R  represents the control input.Define the synchronization error () = () − ζ().The objective of this section is to design a proper control input () such that () converges to zero eventually.To proceed, let us give the following assumption first.Assumption 9. ℏ is a Lipshitz continuous function; that is, the following inequality holds: where  0 > 0 is a constant.
To meet the synchronization object, let us construct the following fractional-order sliding mode surface: where Λ,  ∈ R × are two design matrices.Then, it follows from ( 30), (31), and ( 33 Consequently, let D  () = 0; the control input can be given as Now, we can give the following results.
Theorem 10.Consider the master FOCS (30) and the slave FOCS (31) under Assumption 9. Suppose that the sliding surface is given by (33) and the control input is designed as (35).If the design matrices satisfy  − Λ < 0 and  0 ‖ − Λ‖ ≤ , where  is the smallest eigenvalue of Λ − , then one can conclude that the synchronization error converges to the origin asymptotically.

Simulation Results
In this section, two examples will be given to show the effectiveness of the proposed method.
The simulation results are presented in Figures 2-5.The results where the state variables of the slave FOCS track the master system's states are presented in Figures 2 and 3.The time response of the synchronization errors is depicted in Figure 4. From these pictures, we can see that the synchronization controller works well, and the synchronization  errors converge to the origin fast.From (35), we know that the synchronization control input is a continuous function.The smoothness of the control input is given in Figure 5, from which we can see that the proposed controller has small fluctuation.
The simulation results are given in Figures 7 and 8. Just like the results in Figures 2-5, we know that good synchronization performance has been obtained.

Conclusion
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 fractionalorder 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.