Sliding Mode Control of the Fractional-Order Unified Chaotic System

and Applied Analysis 3 Lemma8 (see [53]). Assume that there exists a scalar function V of the state xwith continuous first-order derivative such that the following are given: (i) V(x) is positive definite, (ii) ?̇?(x) is negative definite, (iii) V(x) → ∞ as ‖x‖ → ∞. 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: ?̇? 1 = (25α + 10) (x 2 − x 1 ) , ?̇? 2 = (28 − 35α) x 1 − x 1 x 3 + (29α − 1) x 2 , ?̇? 3 = x 1 x 2 − (8 + α) x 3


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][5][6].
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 fractionalorder 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 2 Abstract and Applied Analysis 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.

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  ∈ R. 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 0    in this paper.
Lemma 6 (see [22]).Consider the following commensurate fractional-order dynamics system: where 0 <  ≤ 1 and  ∈ R  .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 0 and 1. Assume  be the lowest common multiple of the denominators   's of   's, where   = V  /  , (  , V  ) = 1, and   , V  ∈  + , for  = 1, 2, . . ., .Define Then, the zero solution of system (12) is globally asymptotically stable in the Lyapunov sense if all roots of the equation det(Δ()) = 0 satisfy | arg()| > /2.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: Then, the equilibrium at the origin is globally asymptotically stable.

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  1 ,  2 , and  3 are state variables and  ∈ [0, 1] is the system parameter.Lü et al. [54] call system ( 14) a unified chaotic system because it is chaotic for any  ∈ [0, 1].When  ∈ [0, 0.8), system ( 14) is called a the generalized Lorenz chaotic system.When  = 0.8, it is called the Lü chaotic system.And it is called the generalized Chen chaotic system when  ∈ (0.8, 1].
The fractional-order unified chaotic system has been firstly introduced and studied in [55] and reads as where  1 ,  2 ,  3 ∈ (0, 1] 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 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).

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.

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.

Sliding Surfaces Design.
In order to achieve the stability of system (17), three sliding surfaces  1 ,  2 , and  3 are introduced as the time derivative of which becomes As long as system (17) operates on the sliding surfaces, it satisfies   = 0 and ṡ  = 0,  = 1, 2, 3, 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.

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  1 > 0 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  2 > 0. 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  3 > 0.
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  > 0 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).

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 Δ 1 , Δ 2 , and Δ 3 and external disturbances  1 ,  2 , and  3 are all bounded; that is, |Δ  | <   and |  ()| <   , where   and   are unknown positive constants,  = 1, 2, 3. Let us denote that ρ is an estimate of   , while θ 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 ρ and θ ,  = 1, 2, 3.
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  3 is Step 4. Finally, we gather the above three control functions as It is clear from (48), (55), and (61) that There exists some  > 0 such that The resulting derivative of V 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).

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 2.76.

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  1 = 0.93,  2 = 0.94, and  3 = 0.95.The initial conditions of the drive system (15) and the response system ( 16) are chosen as ( 1 (0),  2 (0),  3 (0)) = (−1, −2, 1) and ( 1 (0),  2 (0),  3 (0)) = (4, −4, 4), respectively.Parameters in (18) are chosen as  1 =  2 =  3 = 4. Gains of the control inputs in ( 24), (30), and (35) are chosen as When  = 0.5 and  = 1, 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.Figure 3: Synchronization of determined fractional-order unified chaotic system with  = 0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).Figure 6: Synchronization of determined fractional-order unified chaotic system with  = 1 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).Figure 9: Synchronization of fractional-order unified chaotic system in the presence of systematic uncertainties and external disturbances with  = 0.5 (blue line represents the trajectories of the drive system, while red line represents the trajectories of the response system).

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.

2 Figure 4 : 20 Figure 5 :
Figure 4: Synchronization errors states with  = 0.5 (red line represents the first error state  1 , blue line represents the second error state  2 , and black line represents the third error state  3 ).

1 Figure 7 :
Figure 7: Synchronization errors states with  = 1 (red line represents the first error state  1 , blue line represents the second error state  2 , and black line represents the third error state  3 ).

20 Figure 8 :
Figure 8: Sliding surfaces states with  = 1 (red line represents the first sliding surface state  1 , blue line represents the second sliding surface state  2 , and black line represents the third sliding surface state  3 ).

2 Figure 10 :
Figure 10: Synchronization errors states in the presence of systematic uncertainties and external disturbances with  = 0.5 (red line represents the first error state  1 , blue line represents the second error state  2 , and black line represents the third error state  3 ).

5 Figure 11 :Figure 12 :
Figure 11: Sliding surfaces states in the presence of systematic uncertainties and external disturbances with  = 0.5 (red line represents the first sliding surface state  1 , blue line represents the second sliding surface state  2 , and black line represents the third sliding surface state  3 ).

Figure 13 : 10 Figure 14 :
Figure 13: Synchronization errors states in the presence of systematic uncertainties and external disturbances with  = 1 (red line represents the first error state  1 , blue line represents the second error state  2 , and black line represents the third error state  3 ).