#### Abstract

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.

#### 1. Introduction

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 [1]. The most significant advantage of fractional calculus is that it provides a powerful instrument of describing memory and hereditary properties of different substances [2]. In particular, fractional differential equations, as the basic theory for fractional-order control [3], have become a powerful tool in describing the dynamics of complex systems and gained great development very recently [4–6].

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 [7], there has emerged great interest in this novel and promising topic. On one hand, more and more fractional nonlinear systems which exhibit chaos have been discovered, and their chaotic behaviors have been studied with numerical simulations, such as the fractional-order Chua circuit [8], the fractional-order Van der Pol oscillator [9–11], the fractional-order Lorenz system [12, 13], the fractional-order Chen system [14–16], the fractional-order Lü system [17], the fractional-order Liu system [18], the fractional-order Rössler system [19, 20], the fractional-order Arneodo system [21], the fractional-order Lotka-Volterra system [22, 23], the fractional-order financial system [24, 25], and the discrete fractional logistic map [26]. On the other hand, control and synchronization of fractional-order dynamical systems have been attracting growing investigations. Linear-state feedback control approach has been designed in [14, 27–35], nonlinear feedback control in [36–40], fractional PID control in [41, 42], and open-plus-closed-loop control in [43]. To tackle with modeling inaccuracies and external noises which are unavoidable in the real-world application, fractional-order sliding mode control methodology has been established in [44–51].

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 2 presents some basic definitions and theorems about fractional calculus and fractional-order dynamical system. Section 3 describes the general form of fractional-order unified chaotic system and presents our main objective in this paper. Section 4 proposes the sliding mode control design for synchronization of nominal systems and adaptive sliding mode control design for the uncertain system. Numerical simulations are presented to show the effectiveness of the proposed schemes in Section 5. Finally, this paper is concluded in Section 6.

#### 2. Preliminaries

*Definition 1. *The most important function used in fractional calculus is Euler’s Gamma function, which is defined as

*Definition 2. *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 , where and are the bounds of the operation and . The continuous integrodifferential operator is defined as

The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [2, 23, 52, 53].

*Definition 3. *The Grünwald-Letnikov derivative definition of order is described as

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

*Definition 4. *The Riemann-Liouville derivative definition of order is described as

However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain , , and so forth. Unfortunately, the Riemann-Liouville approach fails to meet this practical need. It is M. Caputo who solved this conflict.

*Definition 5. *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 instead of in this paper.

Lemma 6 (see [22]). *Consider the following commensurate fractional-order dynamics system:
**
where and . The equilibrium points of system (9) are calculated by solving the following equation:
**
These points are locally asymptotically stable if all eigenvalues of the Jacobian matrix evaluated at the equilibrium points satisfy
*

Lemma 7 (see [22]). *Consider the following -dimensional linear fractional-order dynamics system:
**
where all ’s are rational numbers between and . Assume be the lowest common multiple of the denominators ’s of ’s, where , , and , for . Define
**Then, the zero solution of system (12) is globally asymptotically stable in the Lyapunov sense if all roots of the equation satisfy .*

Lemma 8 (see [53]). *Assume that there exists a scalar function of the state with continuous first-order derivative such that the following are given: *(i)* is positive definite, *(ii)* is negative definite, *(iii)* as . **Then, the equilibrium at the origin is globally asymptotically stable. *

#### 3. Problem Formulation

In [54], Lü et al. have considered a kind of chaotic systems and pointed out that these systems can be described in a unified form as follows: where , , and are state variables and is the system parameter. Lü et al. [54] call system (14) a unified chaotic system because it is chaotic for any . When , system (14) is called a the generalized Lorenz chaotic system. When , it is called the Lü chaotic system. And it is called the generalized Chen chaotic system when .

The fractional-order unified chaotic system has been firstly introduced and studied in [55] and reads as where is the fractional order.

System (15) is considered as the drive (master) system and the response (slave) system is a controlled system as follows:

Let us define the state errors between the response system (16) and the drive system (15) as , , and .

By subtracting (15) from (16), one can get the following error dynamical system: Our main objective in this paper is to investigate the synchronization issue for the fractional-order unified chaotic system (15). It is clear that the synchronization of systems (15) and (16) is equivalent to the stabilization of the error dynamical system (17).

#### 4. Synchronization Design

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 (15) and the response system (16). Two major steps are involved in the sliding mode control design: firstly, constructing an appropriate sliding surface on which the desired system dynamics is stable and, secondly, developing a suitable control law such that the sliding condition is attained.

##### 4.1. Synchronization of the Nominal 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.

###### 4.1.1. Sliding Surfaces Design

In order to achieve the stability of system (17), three sliding surfaces , , and are introduced as the time derivative of which becomes

As long as system (17) operates on the sliding surfaces, it satisfies and , , which yields the following sliding mode dynamics:

By using Lemmas 6 or 7, system (20) is asymptotically stable. As a result, the sliding mode surfaces (18) we have just constructed are appropriate for the control design.

###### 4.1.2. Control Laws Design

*Step 1. *Choose the first control Lyapunov function
Taking time derivative gives
Substituting the first state equation of (17) into (22), one has
Therefore, by designing the first control law as
where and
then, (23) becomes

*Step 2. *Choose the second control Lyapunov function
By taking its derivative with respect to time yields
Substituting the second state equation of (17) into (28), one has
Therefore, by designing the second control law as
where .

Equation (29) becomes

*Step 3. *Choose the third control Lyapunov function
Its time derivative is given by
Substituting the third state equation of (17) into (33), one has
We are, then, in the position to design the third control law as follows:
where .

With this choice, (34) can be rewritten as

*Step 4. *Finally, we gather the above three control functions as
It is clear from (26), (31), and (36) that
There exists some such that
where
The resulting derivative of is

In terms of Lemma 8, the Lyapunov function (37) provides the proof of globally asymptotical stability with the control laws (24), (30), and (35).

##### 4.2. Synchronization of the Uncertain System

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 , , and and external disturbances , , and are all bounded; that is, and , where and are unknown positive constants, . Let us denote that is an estimate of , while is an estimate of . Since and are unknown, our task in this subsection is fulfilled with an adaptive controller consisting of control laws and update laws to obtain and , .

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

The adaptive sliding mode design of system (42) consists of four steps which are elaborated as follows.

*Step 1. *Consider the first control Lyapunov function
the time derivative of which becomes
Substituting the first state equation of (17) into (44), one has
By designing the first control law and adaptive law as
then, (45) becomes

*Step 2. *Choose the second control Lyapunov function
whose derivative is
Substituting the second state equation of (17) into (50), one has
We choose the control law and the adaptive law
With this choice, (51) becomes

*Step 3. *Choose the third Lyapunov function
Taking time derivative gives
Substituting the third state equation of (17) into (57), one has
We choose the third control law and adaptive law
Then, the resulting derivative of is

*Step 4. *Finally, we gather the above three control functions as
It is clear from (48), (55), and (61) that
There exists some such that
The resulting derivative of is
where

By using Lemma 8, this Lyapunov function provides the proof of globally asymptotical stability with the control laws (46), (52), and (59) and adaptive laws (47), (53), and (60).

#### 5. Numerical Simulations

##### 5.1. Chaotic Behaviors of Fractional-Order Chaotic System

In [55], the authors have provided us with numerical methods of fractional calculus. In [56], the chaotic behaviors of the fractional-order unified system were numerically investigated, where it is found that the lowest order to exhibit chaos is .

The chaotic behaviors are presented in Figures 1 and 2 with fractional orders of , , and and the initial conditions of . The numerical algorithm is based on the following Grünwald-Letnikov’s definition: where is the simulation time, , for .

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##### 5.2. Simulations of Synchronization of the Nominal System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed sliding model control in Section 4.1. In the numerical simulations, the fractional orders are chosen as , , and . The initial conditions of the drive system (15) and the response system (16) are chosen as and , respectively. Parameters in (18) are chosen as . Gains of the control inputs in (24), (30), and (35) are chosen as .

When and , numerical simulations of synchronization of system (15) are presented in Figures 3, 4, 5, 6, 7, and 8 with control inputs (24), (30), and (35). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.

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##### 5.3. Simulations of Synchronization of the Uncertain System

In this subsection, numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive sliding model control in Section 4.2. In the numerical simulations, the fractional orders are always chosen as , , and . The initial conditions of the drive system (15) and the response system (16) are also chosen as and , respectively. Parameters in (18) are chosen as . Gains of the control laws (46), (52), and (59) are chosen as . Gains of the adaptive laws (47), (53), and (60) are chosen as . Systematic uncertainties and external disturbances are assumed to be , , , , , and .

When and , numerical simulations are presented in Figures 9, 10, 11, 12, 13, and 14 with control inputs (46), (52), and (59) and adaptive laws (47), (53), and (60). For interpretations of the references to colors in these figure legends, the reader is referred to the web version of this paper.

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#### 6. Conclusions

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.