Research Article Synchronization of the Fractional-Order ...downloads.hindawi.com/journals/jcse/2016/1236210.pdfJournal of Control Science and Engineering 0 20 0 50 0 50 100 20 40

Research ArticleSynchronization of the Fractional-Order BrushlessDC Motors Chaotic System

Shiyun Shen1 and Ping Zhou1,2

1Center of SystemTheory and Its Applications, Chongqing University of Posts and Telecommunications,Chongqing 400065, China2Key Laboratory of Network Control and Intelligent Instrument of Ministry of Education,Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Correspondence should be addressed to Ping Zhou; [email protected]

Received 3 April 2016; Accepted 20 July 2016

Academic Editor: M. Junaid Khan

Copyright © 2016 S. Shen and P. Zhou.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the extension of Lyapunov directmethod for nonlinear fractional-order systems, chaos synchronization for the fractional-order Brushless DCmotors (BLDCM) is discussed. A chaos synchronization scheme is suggested. Bymeans of Lyapunov candidatefunction, the theoretical proof of chaos synchronization is addressed. The numerical results show that the chaos synchronizationscheme is valid.

1. Introduction

The brushless direct-current motors (BLDCM) have manyadvantages over brushed direct-current motors [1–7], includ-ing more torque per watt, high torque per weight, longerlifetime, lower noise, lower electromagnetic interference, andhigh reliability. So, the BLDCM has been used diffuselyin industrial automation and manufacturing engineering,for example, computer hard drives and CD/DVD players,electric vehicles and hybrid vehicles, motion control, posi-tioning or actuation systems, and micro radio-controlledairplane. Recently, chaotic behavior in the brushless direct-current motors (BLDCM) has been reported [1–7], andmanyresearchers investigated chaos synchronization and chaoscontrol for the BLDCM chaotic system [1–7]. Chaos is usefulin many applications, for example, in image steganography[8, 9], authenticated encryption [10], and chaotic communi-cations [11]. Itmust be pointed out that chaos synchronizationand chaos control are usually a prerequisite in chaos applica-tion.

On the other hand, based on the integer-order BLDCMchaotic system reported by Ge et al., a fractional-orderBLDCMchaotic system [12] has been proposed by Zhou et al.

By the adaptive control, back stepping design, and Lyapunovstability theory, the authors [2, 3] proposed some schemesof chaos synchronization and chaos control for the integer-order BLDCM chaotic system. Based on the generalizedGronwall inequality, Zhou et al. [12] presented two chaoscontrol strategies for the fractional-order BLDCM chaoticsystem. To the best of our knowledge, there are seldom resultson chaos synchronization for the fractional-order BLDCMchaotic system.Motivated by the above discussions, we inves-tigate chaos synchronization for the fractional-order BLDCMchaotic system in this paper. Based on the extension of Lya-punov direct method for nonlinear fractional-order system[13, 14], a chaos synchronization scheme is proposed. By aLyapunov candidate function, the theoretical proof of chaossynchronization is provided. Simulation results demonstratethe effectiveness of the synchronization scheme in our paper.

The rest of this paper is as follows: Section 2 introducesthe fractional-order BLDCM chaotic system, and chaoticattractors are given. Chaos synchronization for the fractional-order BLDCM chaotic system is discussed in Section 3, andsimulation results are obtained. Finally, Section 4 concludesthe work.

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016, Article ID 1236210, 5 pageshttp://dx.doi.org/10.1155/2016/1236210

2 Journal of Control Science and Engineering

0200

500

50

100

20 40 60 80

0

20

−20

−20

−50

xd

xdxq xa

xa

Figure 1: A chaotic attractor in the fractional-order BLDCM system (1) for 𝑞 = 0.967.

0200

500

50

100

20 40 60 80

0

20

−20

−20−50

xd

xd

xqxa

xa

Figure 2: A chaotic attractor in the FO-BLDCM system (1) for 𝑞 = 0.975.

2. The Fractional-Order BLDCMChaotic System

Recently, a fractional-order BLDCM chaotic system wasreported by Zhou et al. [12], and this chaotic system can bedescribed as follows:

𝐶

0𝐷𝑞

𝑡𝑥𝑑 = −𝜎𝑥𝑑 + 𝑥𝑞𝑥𝑎,

𝐶

0𝐷𝑞

𝑡𝑥𝑞= −𝑥𝑞+ 𝛽𝑥𝑎− 𝑥𝑑𝑥𝑎,

𝐶

0𝐷𝑞

𝑡𝑥𝑎= 𝛾 (𝑥

𝑞− 𝑥𝑎) ,

(1)

where 0.96 < 𝑞 ≤ 1 is the fractional order and 𝐶0𝐷𝑞

𝑡𝑥𝑖=

Γ−1(1 − 𝑞) ∫

𝑡

0(𝑡 − 𝜏)

−𝑞𝑑𝑥𝑖(𝜏) (𝑖 = 𝑑, 𝑞, 𝑎). 𝑥

𝑑is the direct

axis current of the motor, 𝑥𝑞is quadrature axis current of the

motor, and 𝑥𝑎is the angular velocity of the motor.Themotor

parameters are chosen as 𝜎 = 0.875, 𝛽 = 55, and 𝛾 = 4.The authors obtained the maximum Lyapunov exponent onvarying 𝑞 in [12], and system (1) exhibits chaotic behavior if0.96 < 𝑞 ≤ 1. Now, we can choose 𝑞 = 0.967 and 𝑞 =

0.975 and obtain themaximumLyapunov exponent as 0.7767and 0.8954, respectively. The positive maximum Lyapunovexponent implies that fractional-order BLDCM system (1) ischaotic under 𝑞 = 0.967 and 𝑞 = 0.975, and the chaoticattractors are shown as Figures 1 and 2, respectively.

3. Synchronization for the Fractional-OrderBLDCM Chaotic System

In this section, chaos synchronization for the fractional-orderBLDCM chaotic system (1) is considered. First, we recallsome results for the Caputo derivative.

Lemma 1 (see [15]). For absolutely continuous functions 𝑎(𝑡)and 𝑏(𝑡), one can obtain the following equality:

𝐶

0𝐷𝑞

𝑡[𝑎 (𝑡) 𝑏 (𝑡)] − 𝑎 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑏 (𝑡) − 𝑏 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑎 (𝑡)

=−𝑞

Γ (1 − 𝑞)

⋅ ∫

𝑡

0

𝑑𝜏

(𝑡 − 𝜏)1−𝑞

[∫

𝜏

0

𝑑𝑎 (𝜔)

(𝑡 − 𝜔)1−𝑞

∫

𝜏

0

𝑑𝑏 (𝜌)

(𝑡 − 𝜌)1−𝑞

] ,

(2)

where 0 < 𝑞 < 1.

According to Lemma 1, for absolutely continuous func-tion 𝑎(𝑡), the following equality can be obtained:

𝐶

0𝐷𝑞

𝑡[𝑎 (𝑡)]

2− 2𝑎 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑎 (𝑡)

=−𝑞

Γ (1 − 𝑞)∫

𝑡

0

𝑑𝜏

(𝑡 − 𝜏)1−𝑞

[∫

𝜏

0

𝑑𝑎 (𝜔)

(𝑡 − 𝜔)1−𝑞

]

2

.

(3)

Due to (1/(𝑡 − 𝜏)1−𝑞)[∫𝜏0(𝑑𝑎(𝜔)/(𝑡 − 𝜔)

1−𝑞)]2≥ 0 for 𝜏 ∈

[0, 𝑡], according to (3), one has the following result:

𝐶

0𝐷𝑞

𝑡[𝑎 (𝑡)]

2− 2𝑎 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑎 (𝑡) ≤ 0. (4)

Now, choosing the fractional-order BLDCM chaotic sys-tem (1) as drive system, we have the following main result.

Journal of Control Science and Engineering 3

Theorem 2. Let the response fractional-order BLDCM systembe

(

𝐶

0𝐷𝑞

𝑡𝑦𝑑

𝐶

0𝐷𝑞

𝑡𝑦𝑞

𝐶

0𝐷𝑞

𝑡𝑦𝑎

) =(

−𝜎𝑦𝑑+ 𝑦𝑞𝑦𝑎

−𝑦𝑞 + 𝛽𝑦𝑎 − 𝑦𝑑𝑦𝑎

𝛾 (𝑦𝑞− 𝑦𝑎)

) + Φ, (5)

where 0.96 < 𝑞 ≤ 1, Φ = (

−𝑥𝑞

𝑥𝑑+𝑘−𝛽−𝛾

0

) (𝑦𝑎 − 𝑥𝑎) is a feedbackcontroller, and 𝑘 is a real number. Choosing 𝑘 = 2𝑘

𝑞𝑘𝑎, 𝑘𝑞, 𝑘𝑎

are real numbers. If |𝑘𝑞| < 1 and |𝑘

𝑎| < √𝛾, then chaos syn-

chronization between the response fractional-order BLDCMsystem (5) and the drive fractional-order BLDCM chaoticsystem (1) can be reached.

Proof. Let synchronization errors be 𝑒𝑑(𝑡) = 𝑦

𝑑− 𝑥𝑑, 𝑒𝑞(𝑡) =

𝑦𝑞− 𝑥𝑞, and 𝑒

𝑎(𝑡) = 𝑦

𝑎− 𝑥𝑎. Thus, the error system between

the response system (5) and the drive system (1) can be shownas follows:

𝐶

0𝐷𝑞

𝑡𝑒𝑑 (𝑡) = −𝜎𝑒𝑑 (𝑡) + 𝑥𝑎 (𝑡) 𝑒𝑞 (𝑡) ,

𝐶

0𝐷𝑞

𝑡𝑒𝑞 (𝑡) = −𝑒𝑞 (𝑡) − 𝑥𝑎 (𝑡) 𝑒𝑑 (𝑡) + (𝑘 − 𝛾) 𝑒𝑎 (𝑡) ,

𝐶

0𝐷𝑞

𝑡𝑒𝑎 (𝑡) = 𝛾 [𝑒𝑞 (𝑡) − 𝑒𝑎 (𝑡)] .

(6)

Now, we choose one positive definite Lyapunov functionas follows:

𝐿 (𝑡) = 0.5 [𝑒2

𝑑(𝑡) + 𝑒

2

𝑞(𝑡) + 𝑒

2

𝑎(𝑡)] . (7)

Thus, the fractional derivative of the Lyapunov function𝐿(𝑡) is𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) = 0.5 [

𝐶

0𝐷𝑞

𝑡𝑒2

𝑑(𝑡) +𝐶

0𝐷𝑞

𝑡𝑒2

𝑞(𝑡) +𝐶

0𝐷𝑞

𝑡𝑒2

𝑎(𝑡)] . (8)

By inequality (4), according to (8), one has the followinginequality:

𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) ≤ 𝑒𝑑 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑒𝑑 (𝑡) + 𝑒𝑞 (𝑡)

𝐶

0𝐷𝑞

𝑡𝑒𝑞 (𝑡)

+ 𝑒𝑎 (𝑡)𝐶

0𝐷𝑞

𝑡𝑒𝑎 (𝑡) .

(9)

By the error dynamical system (6), inequality (9) can bechanged as follows:𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) ≤ −𝜎𝑒

2

𝑑(𝑡) − 𝑒

2

𝑞(𝑡) − 𝛾𝑒

2

𝑎(𝑡) + 𝑘𝑒𝑞 (𝑡) 𝑒𝑎 (𝑡) . (10)

Using 𝑘 = 2𝑘𝑞𝑘𝑎, inequality (10) can be changed as

follows:𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) ≤ − [𝑘𝑞𝑒𝑞 (𝑡) + 𝑘𝑎𝑒𝑎 (𝑡)]

2

− 𝜎𝑒2

𝑑(𝑡)

− (1 − 𝑘2

𝑞) 𝑒2

𝑞(𝑡) − (𝛾 − 𝑘

2

𝑎) 𝑒2

𝑎(𝑡) .

(11)

Due to |𝑘𝑞| < 1 and |𝑘

𝑎| < √𝛾, thus, it can be obtained

that𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) ≤ 0, ∀𝑒 (𝑡) , (12)

𝐶

0𝐷𝑞

𝑡𝐿 (𝑡) < 0, ∀𝑒 (𝑡) ̸= 0, (13)

where 𝑒(𝑡) = (𝑒𝑑(𝑡), 𝑒𝑞(𝑡), 𝑒𝑎(𝑡)).

eq(t)

ea(t)

ed(t)

−20

−10

0

10

,eq(t),e a

(t)

e d(t)

2 4 6 8 100t

Figure 3: Evolution of the synchronization errors 𝑒𝑑(𝑡), 𝑒𝑞(𝑡), and

𝑒𝑎(𝑡).

t

−60

−40

−20

0

20

Con

troller

2 6 8 104

(xd − 62)(ya − xa)

−xq(ya − xa)

Figure 4: The time series for controller Φ.

According to the stability of fractional order systems [13,14], inequality (13) indicates that origin (𝑒

𝑑(𝑡), 𝑒𝑞(𝑡), 𝑒𝑎(𝑡)) =

(0, 0, 0)of error system (6) is asymptotically stable in the senseof Lyapunov.This result indicates that the chaos synchroniza-tion between the fractional-order BLDCM system (5) and thefractional-order BLDCM chaotic system (1) can be achieved.The proof is finished.

Remark 3. According toTheorem2, if the chaos synchroniza-tion between the response system (5) and the drive system (1)is reached, the controller Φ = (

−𝑥𝑞

𝑥𝑑+𝑘−𝛽−𝛾

0

) (𝑦𝑎 − 𝑥𝑎) coulddecrease to zero.

Next, simulation results are given. In numerical simula-tion, we set initial conditions as (𝑥

𝑑, 𝑥𝑞, 𝑥𝑎) = (10, 20, 30) and

(𝑦𝑑, 𝑦𝑞, 𝑦𝑎) = (20, 8, 10). The fractional-order 𝑞 is 0.975 innumerical simulation.

Case 1. Choosing 𝑘𝑞 = −0.8 and 𝑘𝑎 = 1.875, thus 𝑘 = −3

and Φ = (

−𝑥𝑞

𝑥𝑑−62

0

) (𝑦𝑎− 𝑥𝑎). According to Theorem 2, the

chaos synchronization between the response fractional-orderBLDCM system (5) and the drive fractional-order BLDCMchaotic system (1) can be achieved. Figure 3 depicts theevolution of synchronization errors. Figure 4 shows the timeseries for controllerΦ = (

−𝑥𝑞

𝑥𝑑−62

0

) (𝑦𝑎− 𝑥𝑎).

Case 2. Choosing 𝑘𝑞 = 𝑘𝑎 = 0, thus 𝑘 = 0 and Φ =

(

−𝑥𝑞

𝑥𝑑−59

0

) (𝑦𝑎− 𝑥𝑎). According to Theorem 2, the response

fractional-order BLDCM system (5) and the drive fractional-order BLDCM chaotic system (1) can be synchronized.

4 Journal of Control Science and Engineering

eq(t)

ea(t)

ed(t)

−20

−10

0

10

,eq(t),e a

(t)

e d(t)

2 4 6 8 100t


𝑒𝑎(𝑡).

t

2 6 8 104−40

−20

0

20

40

60

Con

troller (xd − 59)(ya − xa)

−xq(ya − xa)


eq(t)

ea(t)

ed(t)

−20

−10

0

10

,eq(t),e a

(t)

e d(t)

2 4 6 8 100t


𝑒𝑎(𝑡).

Figure 5 displays the evolution of synchronization errors. Fig-ure 6 shows the time series for controller Φ = (

−𝑥𝑞

𝑥𝑑−59

0

) (𝑦𝑎−

𝑥𝑎).

Case 3. Choosing 𝑘𝑞 = 0.9 and 𝑘𝑎 = 5/3, thus 𝑘 = 3

and Φ = (

−𝑥𝑞

𝑥𝑑−56

0

) (𝑦𝑎− 𝑥𝑎). According to Theorem 2, the

chaos synchronization between the response fractional-orderBLDCM system (5) and the drive fractional-order BLDCMchaotic system (1) can be arrived. Figure 7 displays theevolution of synchronization errors. Figure 8 shows the timeseries for controllerΦ = (

−𝑥𝑞

𝑥𝑑−56

0

) (𝑦𝑎− 𝑥𝑎).

According to Figures 3–8, the simulative results show theeffectiveness of the proposed theorem in our paper.

t

−100

−50

0

50

Con

troller

4 6 8 102

(xd − 56)(ya − xa)

−xq(ya − xa)


4. Conclusions

In this paper, the chaos synchronization for a fractional-orderBLDCMchaotic system is discussed. One feedback controlleris given. By the extension of Lyapunov direct method fornonlinear fractional-order system, a Lyapunov candidatefunction is established, and the theoretical proof of chaossynchronization is given. Finally, the numerical results aregiven, and it shows that the chaos synchronization scheme inour paper is effective. Up to now, to the best of our knowledge,there are no similar results on chaos synchronization of thefractional-order chaotic BLDCM system.

Competing Interests

The authors declare no competing interests.

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Research Article Synchronization of the Fractional-Order ...downloads.hindawi.com/journals/jcse/2016/1236210.pdfJournal of Control Science and Engineering 0 20 0 50 0 50 100 20 40