Anti-control of chaos of single time scale brushless dc motors and chaos synchronization of different order systems

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Anti-control of chaos of single time scale brushless dc

motors and chaos synchronization of diﬀerent order systems

Zheng-Ming Ge

*

_{, Ching-Ming Chang, Yen-Sheng Chen}

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Accepted 4April 2005

Abstract

Anti-control of chaos of single time scale brushless dc motors (BLDCM) and chaos synchronization of diﬀerent order systems are studied in this paper. By addition of an external nonlinear term, we can obtain anti-control of chaos. Then, by addition of the coupling terms, by the use of Lyapunov stability theorem and by the linearization of the error dynamics, chaos synchronization between a third-order BLDCM and a second-order Duﬃng system are presented. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Chaos is undesirable in most engineering applications. Many researchers have devoted themselves to find new ways to suppress and control chaos more efficiently. However, chaos is desirable under certain circumstances. Chaotic phe-nomena are quite useful in many applications such as fluid mixing[1], human brain[2], and heart beat regulation[3], etc. Therefore, making a regular dynamical system chaotic, or preserving chaos of a chaotic dynamical system, is mean-ingful and worth to be investigated.

Chaos synchronization has been applied in many ﬁelds such as secure communication[4,5], chemical and biological systems[6,7]and others[8–16]etc. A lot of researchers have studied synchronization between two identical chaotic sys-tems. But, seldom researchers study synchronization of diﬀerent order chaotic syssys-tems. This motivates us to investigate this absorbing and challenging research topic.

The theme of this paper is brushless dc motor. The major advantage of BLDCM is the elimination of the physical contact between the brushes and the commutators. BLDCM has been widely applied in direct-drive applications such as robotics [17], aerospace[18], etc. In this paper, we investigate chaos anti-control of BLDCM and chaos synchro-nization of diﬀerent order systems. In order to verify periodic and chaotic phenomena of investigated systems, several numerical techniques such as time history, phase portrait, bifurcation diagram and Lyapunov exponents are employed.

This paper is organized as follows. Section 2 contains the dynamic characteristics of BLDCM[19–22]. First, the sys-tem model is described. Second, the syssys-tem equations are transformed to a compact form. Finally, the numerical results

*

Corresponding author. Tel.: +886 3571 2121; fax: +886 3572 0634. E-mail address:[email protected](Z.-M. Ge).

Chaos, Solitons and Fractals 27 (2006) 1298–1315

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of periodic and chaotic phenomena are presented. In Section 3, one method is investigated to achieve anti-control of chaos: the addition of a nonlinear term [23]. Chaos synchronization of different order systems [24] is discussed in Section 4. Two different chaotic dynamical systems, Duffing system and BLDCM, are applied in this section. Three methods are investigated to achieve chaos synchronization: the addition of the coupling terms, the use of Lyapunov stability theorem and the linearization of the error dynamics[25]. Finally, the conclusions of the whole paper are briefly stated.

2. Regular and chaotic dynamics of brushless dc motor

BLDCM is an electromechanical system[19–21]. By using an aﬃne transformation and a single time scale transfor-mation[22], its governing equations can be transformed into a dimensionless form as following:

d d^t^x1¼ ^vq ^x1 ^x2^x3þ q^x3 d d^t^x2¼ ^vd d^x2þ ^x1^x3 d d^t^x3¼ rð^x1 ^x3Þ þ g^x1^x2 bTL ð2:1Þ

where q = 60, ^vq¼ 0.168; ^vd¼ 20.66, d = 0.875, g = 0.26, bTL¼ 0.53 and the initial condition is ^x1ð0Þ ¼ ^x2ð0Þ ¼

^x3ð0Þ ¼ 0.01.

Fig. 1. (a) Phase portrait. (b) Bifurcation diagram. (c) Lyapunov exponents for BLDCM.

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In addition, BLDCM is an autonomous system. It means that the period of the system is not explicitly known, so different choice of Poincare´ section would lead to different bifurcation diagrams. In the sections below, adding control inputs changes the dynamics of the system, thus we have to modify the choice of Poincare´ section. Modifying Poincare´ section, we obtain almost the same bifurcation diagram. The only difference is the shift in ^x3 axis. Therefore, we just

present the original bifurcation diagram.

The phase portrait, bifurcation diagram, and Lyapunov exponents are shown inFig. 1. It can be observed that the motion is period 1 for r = 4.05, period 2 for r = 4.15, and period 4 for r = 4.21. For r = 4.55, the motion is chaotic.

3. Anti-control of chaos

In order to preserve chaotic phenomena of BLDCM, a nonlinear term xjxj is added[23]. 3.1. Adding one term of xjxj

First, we add an external nonlinear input k1^x1j^x1j to the ﬁrst equation of(2.1). When k1> 0, the process of choice and

the numerical results are shown inFig. 2, it is quite clear that the chaotic phenomenon is not increased for k1= 0.052.

Fig. 2. (a) Bifurcation diagram of ^x3for k1= 0.01–0.07. (b) Bifurcation diagram of ^x3for k1= 0.052. (c) Lyapunov exponents for

k1= 0.052.

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When k1< 0, the process of choice and the numerical results are shown inFig. 3, it is clear that the chaotic phenomenon

is increased for k1=0.2 by comparison ofFigs. 1(c) and3(c).

Second, we add an external nonlinear input k2^x2j^x2j to the second equation of(2.1). When k2> 0, the process of

choice and the numerical results are shown in Fig. 4, it is clear that the chaotic phenomenon is increased for k2= 0.0051. When k2< 0, the process of choice and the numerical results are shown inFig. 5, it is clear that the chaotic

phenomenon is not increased for k2=0.0011.

Third, we add an external nonlinear input k3^x3j^x3j to the third equation of(2.1). When k3> 0, the process of choice

and the numerical results are shown inFig. 6, it is clear that the chaotic phenomenon is not increased for k3= 0.001.

When k3< 0, the process of choice and the numerical results are shown inFig. 7, it is clear that the chaotic phenomenon

is increased for k3=0.6.

From above numerical results, we can get some comments. First, when we choose positive value of k1, k2, k3, only

the choice of k2is successful. On the other hand, when we choose negative values of k1, k2, k3, only the choice of k2

fails. The eﬀect of negative k3is better than that of negative k1, and eﬀect of negative k1is better than that of positive

k2.

Fig. 3. (a) Bifurcation diagram of ^x3for k1=0.49 to 0.01. (b) Bifurcation diagram of ^x3for k1=0.2. (c) Lyapunov exponents for

k1=0.2.

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3.2. Adding two terms of xjxj

First, we choose two positive values of k1and k2, the process of choice and the numerical results are shown inFig. 8,

it is clear that the chaotic phenomenon is increased. We also investigate two negative values of k1and k2, the process of

choice and the numerical results are shown inFig. 9, it is clear that the chaotic phenomenon is also increased. Second, we choose two positive values of k1and k3, the process of choice and the numerical results are shown in Fig. 10, it is clear that the chaotic phenomenon is not increased. We also investigate two negative values of k1and k3,

the process of choice and the numerical results are shown in Fig. 11, it is clear that the chaotic phenomenon is increased.

Third, we choose two positive values of k2and k3, the process of choice and the numerical results are shown inFig. 12, it is clear that the chaotic phenomenon is increased. We also investigate two negative values of k2and k3, the process

of choice and the numerical results are shown inFig. 13, it is clear that the chaotic phenomenon is increased. From above numerical results, we can get some comments. When we choose two positive values of k1and k3, the

result fails. For the other choices, the results are all successful. The eﬀects of two negative values of k1, k3and two

neg-ative values of k2, k3are the best.

k2= 0.0051.

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From the results in this section, we ﬁnd that the necessary condition for success of increase of chaos is that there exists at least one successful addition of nonlinear term.

4. Chaos synchronization of diﬀerent order systems

We discuss chaos synchronization of two diﬀerent order systems [24]in this section. These two systems are the autonomous third-order BLDCM system and the nonautonomous second-order Duﬃng system. Three methods are applied: the addition of the coupling terms, the Lyapunov stability theorem, and the linearization of the error dynamics

[25]. BLDCM is described by _x1¼ Vq x1 x2x3þ px3 _x2¼ Vd Bx2þ x1x3 _x3¼ aðx1 x3Þ þ h; x1x2 T3 ð4:1Þ

Fig. 5. (a) Bifurcation diagram of ^x3for k2=0.02 to 0.0. (b) Bifurcation diagram of ^x3for k2=0.0011. (c) Lyapunov exponents for

k2=0.0011.

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where Vq= 0.168, p = 60, Vd= 20.66, B = 0.875, a = 4.55, h = 0.26, T3= 0.53, and the initial condition is

x1(0) = x2(0) = x3(0) = 0.01.

Duﬃng system is described by _y1¼ y2

_y2¼ y1 y 3

1 dy2þ a cos xt

ð4:2Þ

where d = 0.15, a = 0.3, x = 1.0, and the initial condition is y1(0) = y2(0) = 0.

4.1. Chaos synchronization of coupled diﬀerent order chaotic systems

First, we choose BLDCM as the master system and Duﬃng system as the slave system. For leading (y1, y2) to

(x1, x2), we add two coupling terms, k1(x1 y1) and k2(x2 y2), to the ﬁrst and second equation of (4.2),

respectively.

k3= 0.001.

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We use the random optimization method [26] to find the critical coupling strength. If the critical coupling strength does exist, the coupling strength should converge to some constant value, and the difference U should be zero. Define U by U¼ Z T 0.9T jx yj2dt ð4:3Þ

where x = [x1 x2]T, y = [y1 y2]T, and T is the simulation time.

In numerical simulation, the larger the coupling strength, the better the synchronization is. The diﬀerence U can be rather small but not zero, this means that chaos synchronization of diﬀerent order systems can be practically achieved. The numerical results are shown inFigs. 14and15.

Second, we choose Duﬃng system as the master system and BLDCM as the slave system. For leading (x1, x2) to

(y1, y2), we add two coupling terms, k1(y1 x1) and k2(y2 x2), to the ﬁrst and second equation of(4.1), respectively. Fig. 7. (a) Bifurcation diagram of ^x3for k3=1.5 to 0.05. (b) Bifurcation diagram of ^x3for k3=0.6. (c) Lyapunov exponents for

k3=0.6.

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The numerical results are shown inFigs. 16and17. Chaos synchronization of diﬀerent order systems can be practically achieved.

Up to now, the coupling methods are both uni-directional. The bi-directional coupling method is studied now. We add two coupling terms, k1(y1 x1) and k2(y2 x2), to the ﬁrst and second equation of(4.1). We also add two coupling

terms, k1(x1 y1) and k2(x2 y2), to the ﬁrst and second equation of(4.2). The numerical results are shown inFigs. 18

and19. Chaos synchronization of diﬀerent order systems can be practically achieved.

4.2. Chaos synchronization of diﬀerent order systems by Lyapunov stability theorem

BLDCM is chosen as the master system and Duﬃng system is chosen as the slave system. For leading (y1, y2) to

(x1, x2), we add u1and u2on the ﬁrst and second equation of(4.2), respectively.

_y₁¼ y2þ u1

_y2¼ y1 y31 dy2þ a cos xt þ u2

ð4:4Þ

Fig. 8. (a) Bifurcation diagram of ^x3 for k1= 0.052, k2= 0.001–0.005. (b) Bifurcation diagram of ^x3 for k1= 0.052, k2= 0.003.

(c) Lyapunov exponents for k1= 0.052, k2= 0.003.

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Subtracting Eq.(4.4)from the ﬁrst two equations of(4.2), we can obtain the error dynamics _e1¼ Vq x1 x2x3þ px3 y2 u1 _e2¼ Vd Bx2þ x1x3 y1þ y 3 1þ dy2 a cos xt u2 ð4:5Þ where e1= x1 y1, e2= x2 y2.

Choose a Lyapunov function of the form Vðe1; e2Þ ¼ 1 2ðe 2 1þ e 2 2Þ ð4:6Þ

its derivative along the solution of Eq.(4.5)is _

V ¼ e1ðVq e1 y1 x2x3þ px3 y2 u1Þ þ e2ðVd Be2 Bx2þ x1x3 y1þ x31þ dy2 a cos xt u2Þ ð4:7Þ Fig. 9. (a) Bifurcation diagram of ^x3for k1=0.2, k2=0.006 to 0.001. (b) Bifurcation diagram of ^x3for k1=0.2, k2=0.0015.

(c) Lyapunov exponents for k1=0.2, k2=0.0015.

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Choose

u1¼ Vq y1 x2x3þ px3 y2

u2¼ Vd Bx2þ x1x3 y1þ x 3

1þ dy2 a cos xt

Eq.(4.7)can be rewritten as _

V ¼ e2

1 Be

2

2<0

this means that chaos synchronization between diﬀerent order systems, Duﬃng system and BLDCM, can be achieved. The numerical results are shown inFig. 20.

4.3. Chaos synchronization of diﬀerent order systems by linearization of error dynamics

BLDCM is chosen as the master system and Duﬃng system is chosen as the slave system. For leading (y1, y2) to

(x1, x2), we add u1and u2on the ﬁrst and second equation of(4.2), respectively.

Fig. 10. (a) Bifurcation diagram of ^x3 for k1= 0.052, k3= 0.0–0.03. (b) Bifurcation diagram of ^x3 for k1= 0.052, k3= 0.001.

(c) Lyapunov exponents for k1= 0.052, k3= 0.001.

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_y1¼ y2þ u1 _y₂¼ y1 y 3 1 dy2þ a cos xt þ u2 ð4:8Þ Deﬁne e1¼ x1 y1 e2¼ x2 y2

Subtracting Eq.(4.8)from the ﬁrst two equations of(4.1), we can obtain the error dynamics _e1¼ e1þ e2þ Vq x2x3þ px3 y1 x2 u1

_e2¼ e1 ðB þ dÞe2þ Vdþ x1x3þ y31 a cos xt By2 x1þ dx2 u2

ð4:9Þ

Fig. 11. (a) Bifurcation diagram of ^x3 for k1=0.2, k3=0.4to 0.1. (b) Bifurcation diagram of ^x3 for k1=0.2, k3=0.3.

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To delete the nonlinear terms in Eq.(4.9) [25], we design u1and u2as

u1¼ Vq x2x3þ px3 y1 x2þ k11e1þ k12e2

u2¼ Vdþ x1x3þ y31 a cos xt By2 x1þ dx2þ k21e1þ k22e2

Eq.(4.9)can be rewritten as _e¼ Ae

where

A¼ 1 k11 1 k12 1 k21 ðB þ d þ k22Þ

" #

If each eigenvalue of A is negative, e would converge to zero. By designing k11= 0, k12= 0, k21= 2, and

k22=0.025, we can get two negative eigenvalues of A: 1 and 1. Thus u1and u2can be obtained, and e would

con-verge to zero. This means that chaos synchronization between diﬀerent order systems, Duﬃng system and BLDCM, can be achieved. The numerical results are shown inFig. 21.

Fig. 12. (a) Bifurcation diagram of ^x3for k2= 0.0051, k3= 0.01–0.09. (b) Bifurcation diagram of ^x3 for k2= 0.0051, k3= 0.05. (c)

Lyapunov exponents for k2= 0.0051, k3= 0.05.

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Fig. 13. (a) Bifurcation diagram of ^x3for k2=0.0011, k3=0.5 to 0.1. (b) Bifurcation diagram of ^x3for k2=0.0011, k3=0.3.

0 10 20 30 40 50 60 70 80 90 100 0 2000 4000 6000 step number k₁ k₂ 0 10 20 30 40 50 60 70 80 90 100 100 105 1010 step number U

Fig. 14. Time evolution of k1and k2by random optimization process.

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0 20 40 60 80 100 120 140 160 180 200 -50 0 50 k₁=5105.007381,k₂=4434.127551 x1 -y1 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 100 150 x2 -y2 110 120 130 140 150 160 170 180 190 200 -0.05 0 0.05 x1 -y1 110 120 130 140 150 160 170 180 190 200 -0.4 -0.2 0 0.2 x2 -y2 t(sec)

Fig. 15. Time history of errors for k1= 5105.007381, and k2= 4434.127551.

0 10 20 30 40 50 60 70 80 90 100 0 2000 4000 6000 step number k 1 k 2 0 10 20 30 40 50 60 70 80 90 100 102 104 106 108 step number U

0 20 40 60 80 100 120 140 160 180 200 -50 0 50 100 k₁=4951.462167, k₂=5040.654891 y1 -x1 0 20 40 60 80 100 120 140 160 180 200 -150 -100 -50 0 50 y2 -x2 110 120 130 140 150 160 170 180 190 200 -0.02 0 0.02 0.04 y1 -x1 110 120 130 140 150 160 170 180 190 200 -4 -2 0 2 4 y2 -x2 t(sec)

Fig. 17. Time history of errors for k1= 4951.462167, k2= 5040.654891.

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0 10 20 30 40 50 60 70 80 90 100 0 2000 4000 6000 step number k₁ k₂ 0 10 20 30 40 50 60 70 80 90 100 100 105 1010 step number U

0 20 40 60 80 100 120 140 160 180 200 -50 0 50 k₁=5285.682274,k₂=4652.413356 x1 -y1 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 100 150 x2 -y2 110 120 130 140 150 160 170 180 190 200 -0.4 -0.2 0 0.2 x1 -y1 110 120 130 140 150 160 170 180 190 200 -2 0 2 x2 -y2 t(sec)

Fig. 19. Time history of errors for k1= 5285.682274, k2= 4652.413356.

0 20 40 60 80 100 120 140 160 180 200 -50

0 50

Lyapunov's stability theorem

x1 -y1 0 20 40 60 80 100 120 140 160 180 200 -50 0 50 100 150 x2 -y2 130 140 150 160 170 180 190 200 -1 0 1 x1 -y1 130 140 150 160 170 180 190 200 -1 0 1 x2 -y2 t(sec)

Fig. 20. Time history of errors.

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5. Conclusions

Brushless dc motor (BLDCM) is studied in this paper. It is an autonomous third-order electromechanical system. In order to verify periodic and chaotic phenomena of investigated systems, several numerical techniques such as time his-tory, phase portrait, bifurcation diagram, and Lyapunov exponents are employed.

The dynamic characteristics of BLDCM are discussed in Section 2. The system model is described, and the numerical results of periodic and chaotic phenomenon are presented.

In Section 3, the nonlinear term k^xj^xj added to achieve anti-control of chaotic BLDCM. When one nonlinear posi-tive term is added, only the choice of k2succeeds. However, whenone nonlinear negative term is added, only the choice

of k2fails. The eﬀect of negative k3is better than that of negative k1, and eﬀect of negative k1is better than that of

positive k2. When we add two nonlinear terms, the necessary condition for success of increase of chaos is that there

exists at least one successful addition of nonlinear term.

Three methods to achieve chaos synchronization of different order systems are investigated in Section 4. Two dif-ferent chaotic dynamical systems, Duffing system and BLDCM, are applied. First, the coupling terms are added. We study two kinds of uni-directional coupling methods and a bi-directional coupling method. The larger the coupling strength, the better the synchronization is. The difference can be rather small but not zero, this means that chaos syn-chronization of different order systems can be practically achieved. Second, Lyapunov stability theorem is used. Chaos synchronization between different order systems can be achieved. Third, linearization of the error dynamics is used. Chaos synchronization between different order systems can be achieved.

Acknowledgement

This research was supported by the National Science Council, Republic of China, under Grant Number NSC 92-2212-E-009-027.

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