Synchronization of the Fractional-Order Brushless DC Motors Chaotic System

Based on the extension of Lyapunov direct method for nonlinear fractional-order systems, chaos synchronization for the fractional-order Brushless DC motors (BLDCM) is discussed. A chaos synchronization scheme is suggested. By means of Lyapunov candidate function, the theoretical proof of chaos synchronization is addressed. The numerical results show that the chaos synchronization scheme is valid.

On the other hand, based on the integer-order BLDCM chaotic system reported by Ge et al., a fractional-order BLDCM chaotic system [12] has been proposed by Zhou et al.
By the adaptive control, back stepping design, and Lyapunov stability theory, the authors [2,3] proposed some schemes of chaos synchronization and chaos control for the integerorder BLDCM chaotic system.Based on the generalized Gronwall inequality, Zhou et al. [12] presented two chaos control strategies for the fractional-order BLDCM chaotic system.To the best of our knowledge, there are seldom results on chaos synchronization for the fractional-order BLDCM chaotic system.Motivated by the above discussions, we investigate chaos synchronization for the fractional-order BLDCM chaotic system in this paper.Based on the extension of Lyapunov direct method for nonlinear fractional-order system [13,14], a chaos synchronization scheme is proposed.By a Lyapunov candidate function, the theoretical proof of chaos synchronization is provided.Simulation results demonstrate the effectiveness of the synchronization scheme in our paper.
The rest of this paper is as follows: Section 2 introduces the fractional-order BLDCM chaotic system, and chaotic attractors are given.Chaos synchronization for the fractionalorder BLDCM chaotic system is discussed in Section 3, and simulation results are obtained.Finally, Section 4 concludes the work.

The Fractional-Order BLDCM Chaotic System
Recently, a fractional-order BLDCM chaotic system was reported by Zhou et al. [12], and this chaotic system can be described as follows: where 0.96 <  ≤ 1 is the fractional order and ).   is the direct axis current of the motor,   is quadrature axis current of the motor, and   is the angular velocity of the motor.The motor parameters are chosen as  = 0.875,  = 55, and  = 4.The authors obtained the maximum Lyapunov exponent on varying  in [12], and system (1) exhibits chaotic behavior if 0.96 <  ≤ 1.Now, we can choose  = 0.967 and  = 0.975 and obtain the maximum Lyapunov exponent as 0.7767 and 0.8954, respectively.The positive maximum Lyapunov exponent implies that fractional-order BLDCM system (1) is chaotic under  = 0.967 and  = 0.975, and the chaotic attractors are shown as Figures 1 and 2, respectively.

Synchronization for the Fractional-Order BLDCM Chaotic System
In this section, chaos synchronization for the fractional-order BLDCM chaotic system (1) is considered.First, we recall some results for the Caputo derivative.
According to Lemma 1, for absolutely continuous function (), the following equality can be obtained: according to (3), one has the following result: Now, choosing the fractional-order BLDCM chaotic system (1) as drive system, we have the following main result.
According to Figures 3-8, the simulative results show the effectiveness of the proposed theorem in our paper.

Conclusions
In this paper, the chaos synchronization for a fractional-order BLDCM chaotic system is discussed.One feedback controller is given.By the extension of Lyapunov direct method for nonlinear fractional-order system, a Lyapunov candidate function is established, and the theoretical proof of chaos synchronization is given.Finally, the numerical results are given, and it shows that the chaos synchronization scheme in our paper is effective.Up to now, to the best of our knowledge, there are no similar results on chaos synchronization of the fractional-order chaotic BLDCM system.

Figure 8 :
Figure 8: The time series for controller Φ.
and  is a real number.Choosing  = 2    ,   ,   are real numbers.If |  | < 1 and |  | < √ , then chaos synchronization between the response fractional-order BLDCM system (