Abstract
A second-order sliding mode (SOSM) controller is proposed to synchronize a class of incommensurate fractional-order chaotic systems with model uncertainties and external disturbances. Based on the chattering free SOSM control scheme, it can be rigorously proved that the dynamics of the synchronization error is globally asymptotically stable by using the Lyapunov stability theorem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed controller design approach.
1. Introduction
For the last few decades, the study of fractional-order control systems has attracted increasing interest (see, e.g., [1–7] and the references therein), where the system equations were described by the so-called fractional derivatives and integrals (for the introduction to this theory see [1, 8]). Because fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes, then the advantages of using the fractional order model are that we have more degrees of freedom in the model and that a “memory” is included in the model. The modeling of dynamical systems by using the means of the fractional calculus has been reported in many engineering areas such as signal processing [9], electromagnetism [10], mechanics [11–14], image processing [15], bioengineering [16], automatic control [17, 18], and robotics [19, 20]. Among the existed literatures on the dynamics of fractional-order differential systems, it has been demonstrated that some fractional-order systems can behave chaotically or hyperchaotically [21–25].
Due to the existence of chaos in real practical systems and many potential applications in physics and engineering, the study of synchronizing/stabilizing chaotic/hyperchaotic systems has attracted considerable interests in the past decades [26–32]. Several methods have been proposed to achieve chaos synchronization. One of the methods is based on the sliding mode control (SMC) approach [27, 30, 31, 33–36]. The main feature of the SMC is to switch the control law to drive the states of the system from the initial states onto some predefined sliding surface in a finite time. The system on the sliding surface has desired properties such as stability, disturbance rejection capability, and tracking ability [34].
In general, the traditional sliding mode control is of the first order. And there exists an inevitable drawback when applying such standard SMC, that is the so-called chattering phenomenon, namely, the occurrence of undesirable high-frequency vibrations of the system variables which are caused by the discontinuous high-frequency nature of first-order sliding mode control signals. In order to improve the control accuracy and reduce the undesired chattering effect by removing the controller discontinuity while keeping similar properties of robustness analogous as those featured by the conventional first-order sliding mode approach, the second- (and higher) order sliding mode control method is proposed [37–40]. However, to the authors’ knowledge, there are few researches on the fractional-order system using the SOSM control approach so far.
Motivated by the above discussions, this article considers the robust synchronization problem for a class of uncertain incommensurate fractional-order chaotic systems raised by Aghababa in [41]. A chattering free SOSM controller is presented in the presence of model uncertainties and external disturbance.
The structure of this paper is as follows: Section 2 recalls some preliminaries on fractional calculus and gives the statement of the problem considered in this paper. Section 3 provides the SOSM controller together with the respective Lyapunov-based stability analysis. Section 4 illustrates some simulation results. Finally, a conclusion is drawn in Section 5.
2. Preliminaries and Problem Statement
2.1. Basic Definitions of Fractional Calculus
There are many ways to define the fractional integral and derivative. Two definitions, Riemann-Liouville definition and Caputo definition, are generally used in recent literatures.
Definition 1 (see [1]). The th-order Riemann-Liouville fractional integration of function is given by where is the Gamma function and is the initial time.
Definition 2 (see [1]). Letting , , the Riemann-Liouville fractional derivative of order of function is defined as follows:
Definition 3 (see [1]). The Caputo fractional derivative of order of a continuous function is defined as follows: where is the smallest integer number, larger than .
Lemma 4 (see [42]). Consider the system where is continuous on an open neighborhood . Suppose there exists a continuous differential positive-definite function , real numbers , , such that Then, the origin of system (4) is a locally finite-time stable equilibrium, and the settling time, depending on the initial state , satisfies . In addition, if and is also radially unbounded, then the origin is a globally finite-time stable equilibrium of system (4).
Lemma 5 (see [43]). Consider a vector signal . Let . If there exists such that , , then .
2.2. Problem Statement
Consider the following -dimensional uncertain incommensurate fractional-order chaotic/hyperchaotic slave system: where , , is the order of the system, is the state vector, , is a given nonlinear function of and , , and , denote unknown mode uncertain and external disturbances of the system, respectively, and is the control input.
Suppose the master system can be described as where is the state vector, , , is a given nonlinear function of and , , , and , , denote unknown mode uncertain and external disturbances of the system, respectively.
We define the chaos synchronization problem as follows: design an appropriate controller , , for the slave system (6) such that its state trajectories track the state trajectories of the master system (7) asymptotically.
Remark 6. If , , then systems (6) and (7) are called commensurate fractional-order chaotic systems. The finite-time synchronization between (6) and (7) with the same fractional orders has been addressed in [43] by using a discontinuous terminal sliding mode control method; this paper considers an SOSM controller design for synchronizing the incommensurate fractional-order system.
Assumption 7. The uncertainty terms , , the external disturbances and , , are derivable, and the bounds of their derivatives are known positive constants , , , and :
Remark 8. In order to design a chattering free second-order sliding mode controller, the smoothness hypotheses of the uncertainty and external disturbances terms are required as in Assumption 7, which is not necessary with the first-order sliding mode control approach. Indeed, this can be seen as a standard assumption when using second-order sliding mode technique [43].
Define the synchronization error as
Consequently, the synchronization error dynamics is obtained as follows:
The control task is to design a chattering free second-order sliding mode controller , , such that the synchronization error system (10) can be stabilized to zero as time goes to infinity.
Remark 9. It is clear that if , then the synchronization problem is transformed to the stabilization problem of the fractional-order uncertain chaotic system (6).
3. Main Results
To design a sliding mode controller, there are two steps. Firstly, a sliding surface should be constructed that represents a desired system dynamics. Secondly, a switching control law should be developed such that a sliding mode exists on every point of the sliding surface, and any states outside the surface are driven to reach the surface in a finite time [44].
In this paper, as a choice, we propose an integral type sliding surface as follows:
According to sliding mode control (SMC) method, when in the sliding mode, the switching surface and its derivative must satisfy the following conditions: from which, one can obtain the so-called equivalent control and then derive the sliding mode controller.
But, in this paper, different from the traditional sliding mode control, the SOSM controller to be designed will drive all the states of sliding variables to zero in a finite time; then, by using Lemma 5, one has
Next, we will give the main results.
Theorem 10. Under Assumption 7, consider the uncertain fractional-order chaotic synchronization error system (10) and the sliding surface (11) and take the following SOSM control law: where , and is the sign function; denote the design parameters satisfying thatwhere , . Then the closed-loop system of (10) is globally and asymptotically stable.
Proof. According to Definition 2, we have
Substituting (14) into the previous equation, it yields
where . From Assumption 7, one has
which implies that
Letting , then system (17) can be rewritten as
Selecting a Lyapunov function for system (20),
which can also be written as a quadratic form , where
It is obvious that is continuous but is not differentiable at ; it is positive and radially unbounded if ; that is,
where is the Euclidean norm of and and are the minimal eigenvalue and the largest eigenvalue of matrix , respectively. Taking the time derivative of along system (20), we have
By a simple derivation, we have
The previous formula can be simplified as
Therefore, we can rewrite as
where
. Next, we will prove that the matrixes and are positive definite.
For all , let
Then, by simple calculations and from the first inequality of (15), we have
which implies that .
As for , by direct calculation, three positive eigenvalues of it can be obtained:
which means .
Noting that , according to (19) and (27), one has
By using the second inequality of (15) and formula (31) results
Hence, from (32), one has
Because
we have
Therefore, from (34), we have
By Lemma 4 it follows easily that and therefore , globally converge to zero in a finite time. According to the sliding surface dynamics (11) and Lemma 5, we obtain as .
This completes the proof of Theorem 10.
Remark 11. It is difficult to obtain all the possible solutions of nonlinear inequalities (15). However, in the process of selecting parameters, we observe that Therefore, if , we may present a set of feasible solutions of design parameters , , and step by step. First, choosing , satisfies and . Next, select such that . Thus it is an easy task to get a group of appropriate design parameters in this way.
4. Simulations
A useful approximate numerical technique for solving the fractional differential equations has been developed by many researchers; see, for example, Diethelm et al. [45], which is the generalization of the Adams-Bashforth-Moulton algorithm.
Based on the numerical algorithms of fractional-order systems, we consider the simulation example for the synchronization problem of the uncertain fractional-order hyperchaotic Lorenz system as the slave system and the fractional-order hyperchaotic Chen system as the master system [41].
Firstly, we describe the hyperchaos phenomenon in the fractional-order hyperchaotic Chen system and the fractional-order hyperchaotic Lorenz system, respectively. For simplicity, we consider the following systems: where we take the fractional orders , , , and . Assume the initial conditions are and . By using the numerical algorithm similar to [46], the time step is s. Figures 1 and 2 show the hyperchaotic phenomenon.


Next, we consider the synchronization simulations of these two uncertain fractional-order hyperchaotic systems; the first is hyperchaotic Lorenz system:
The second is hyperchaotic Chen system:
The uncertainty terms of systems (40) and (41) are selected as follows:
As pointed out in [47, 48], to ensure the existence of chaos for the hyperchaotic Lorenz and Chen systems, the initial conditions of the slave and master systems are chosen as , , , , , , , and , respectively. By Remark 11, choose parameters , , and .
According to (40) and (41), the synchronization error dynamics is described as with the initial conditions being , , , and . Substituting the second-order sliding mode controller (14) into (43), we can obtain the closed-loop error system.
By using the numerical algorithm [45], with the sampling interval being s, next we present the simulation result to show the convergence of , .
From Figure 3, we observe that all the states of synchronization error system (43) converge to zero driven by the second-order sliding mode controller, which implies that the control approach is valid to address the robust synchronization problem for the uncertain hyperchaotic systems.

Remark 12. As given by Aghababa in [41], the uncertainty terms of systems (40) and (41) are chosen as bounded periodic function containing sine and cosine forms. Of course, other uncertain cases satisfying (8) can also be selected as the simulate examples.
Remark 13. As in [41], in this section, the fractional-order hyperchaotic Lorenz system and the fractional-order hyperchaotic Chen system are selected as slave system and the master system, respectively. In fact, there are many other fractional-order chaotic systems that can be considered; here we cannot discuss each case for lack of space.
Remark 14. In this section, we adopt the traditional numerical algorithm [45] for fractional-order system, as for the other method [46] with MATLAB implementation that can also be applied in our simulation section.
Remark 15. For the chaotic fractional systems, the orders should be lower than 3, as for the hyperchaotic systems in our paper, even though the systems are of order >3, but, as pointed out in [47, 48], the existence of chaos can be guaranteed just as shown in Figures 1 and 2.
5. Conclusion
A second-order sliding mode controller is proposed in this article in order to address the synchronization problem for a class of uncertain fractional-order chaotic systems. The stability analysis is given based on the Lyapunov theorem; a simple numerical example is adopted to show the effectiveness of our control approach.
Acknowledgments
This paper was supported by the National Basic Research Program of China (973 Project, 2010CB832702), the National Science Funds for Distinguished Young Scholars (11125208), the 111 Project (B12032), the R&D Special Fund for Public Welfare Industry (Hydrodynamics Project, 201101014), the Natural Science Foundation of China (11202066), the China Postdoctoral Science Foundation (2013M531263), and the Fundamental Research Funds for the Central Universities (2013B10114).