Chaos synchronization and chaotization of complex chaotic systems in series form by optimal control

Chaos synchronization and chaotization of complex chaotic systems

in series form by optimal control

Zheng-Ming Ge

a,*

, Cheng-Hsiung Yang

b

a

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC b

Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, 43 Section 4, Keelung Road, Taipei, 106, Taiwan, ROC

a r t i c l e

i n f o

Article history:

Accepted 23 February 2009

Communicated by Prof. Ji-Huan He

a b s t r a c t

By the method of quadratic optimum control, a quadratic optimal regulator is used for syn-chronizing two complex chaotic systems in series form. By this method the least error with less control energy is achieved, and the optimization on both energy and error is realized synthetically. The simulation results of two Quantum-CNN chaos systems in series form prove the effectiveness of this method. Finally, chaotization of the system is given by opti-mal control.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Chaos synchronization has been widely investigated and many effective methods have been presented recently. Thus, as a key technique of secret communication, chaos synchronization has become a very important goal. Since Pecora and Corrall discovered the synchronization of chaotic systems[1–5], many synchronization methods have been developed[6–9]. For chaos synchronization of practical engineering systems, the control cost must be taken into account. Optimal control method is preferable in such cases[10–13].

In this paper, a quadratic optimal regulator is used for chaos synchronization. In practical system, it is difficult to obtain the precise mathematical model, so in practical applications the investigators would like to employ simple and efficient controllers. Therefore, how to design a simple controller with limited information of a chaotic system is still an open problem[20-26].

As numerical example, recently developed Quantum Cellular Neural Network (Quantum-CNN) chaotic oscillator in series form is used. Quantum-CNN oscillator equations are derived from a Schrödinger equation taking account of quantum dots cellular automata structures to which in the last decade a wide interest has been devoted, with particular attention towards quantum computing[19].

Furthermore, chaotization is studied. Chaotization aims at creating or enhancing the system complexity. Chaotization of Quantum-CNN system is accomplished by an optimal control method.

This paper is organized as follows. In Section2, a linearly coupled chaos synchronization scheme by optimum control is given. In Section3, numerical results of the synchronization of two Quantum-CNN oscillator systems by unidirectional and by mutual linear coupling are presented, respectively. In Section4, chaotization of Quantum-CNN chaotic system and simulation results are obtained. Finally, conclusions are given in Section5.

2. Linearly coupled chaos synchronization scheme by optimum control

The optimum control is defined as a method by which the specified performance index of a system has optimum value when the desired control assignment is fulfilled.

0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.026

* Corresponding author. Fax: +886 3 5720634. E-mail address:[email protected](Z.-M. Ge).

Contents lists available atScienceDirect

Chaos, Solitons and Fractals

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c h a o s

The state equation of a linear system is

_xðtÞ ¼ AxðtÞ þ BuðtÞ; ð1Þ

where x(t) is an n-dimensional state variable of the system, A is an n  n dimensional constant matrix and B is an appropriate n  r dimensional constant matrix. The matrix [A B] is controllable entirely and u(t) is an r-dimensional control input of the system. Assuming that u(t) has no restriction and u(0) = 0, the performance index is

J ¼ Z 1

0

ðxT_{Qx þ u}T_{RuÞdt: ð2Þ}

In Eq.(2), Q is an n  n dimensional positive semidefinite real symmetric constant matrix; R is an r  r dimensional po-sitive definite real symmetric constant matrix. The choice of the weighting matrix Q or R is based on eclectic considerations which can enhance the control performance and reduce the control energy consumption. The aim of the optimum control is to get u(t) = Kx(t) and then make the performance index Eq.(2)to be minimum, where Kalman gain K is an r  n dimensional matrix.

So the design of the optimum control system is simplified to get the elements of matrix K. By stability theory, the opti-mization of the quadratic performance index indicated by Eq.(2)can be solved.

The feedback gain matrix K of the quadratic optimal regulator is obtained as follows[29]: K ¼ R1_BT_S:

ð3Þ The matrix S in Eq.(3)is a positive definite matrix and must satisfy the following Riccati equation[9]:

AT_{S þ SA  SBR}1_BT_{S þ Q ¼ 0:}

ð4Þ Then the following nonlinear chaotic system is considered:

_xðtÞ ¼ AxðtÞ þ Fðt; xÞ þ Bu1ðtÞ; ð5Þ

where A is an n  n dimensional constant matrix, x = (x1, x2, . . . , xn) 2 Rn is the state variable of the system,

F(x) = (F1, F2, . . . , Fn)Tis the nonlinear terms of the chaotic system and u1(t) = ka(y(t)  x(t)) is an r-dimensional control input

where kais a constant vector. The second chaotic system is

_yðtÞ ¼ AyðtÞ þ Fðt; yÞ þ Bu2ðtÞ; ð6Þ

where B is an appropriate constant matrix, u2(t) = ks(x(t)  y(t)) is an r-dimensional control input where ksis also a constant

vector.

Define error vector e = x  y. From Eqs.(5) and (6), the error system is

_eðtÞ ¼ ½A  Bðksþ kaÞe þ Fðt; xÞ  Fðt; yÞ: ð7Þ

In current schemes of chaos synchronization, maximum values of states must be determined by simulation[15–18]. They are half analytic method but not pure analytic method. In[14]F(t, x)  F(t, y) nonlinear item is ignored. This is incorrect since there exist linear terms of e in F(t, x)  F(t, y), which cannot be ignored. In this paper, the series expansion analysis offers a correct method.

The series expansion form of Eq.(7)is

_e ¼ A þ MðxðtÞ; yðtÞÞ  Bðk½ sþ kaÞe þ HðxðtÞ; yðtÞ; eÞ; ð8Þ

where M(x(t), y(t))e + H(x(t), y(t), e) = F(t, x)  F(t, y). The elements of M(x(t), y(t)) depend on state vectors x, y, and all of them are bounded convergent infinite series of x, y. H(x(t), y(t), e) contains higher degree terms of e only.

If we choose appropriate kaand ksto make A + M(x(t), y(t))  B(ks+ ka) asymptotically stable, then by first approximation

theory, the zero solution e = 0 of Eq.(8)is asymptotically stable, i.e., systems(5) and (6)are synchronized.

Now we construct an optimal regulator, which is used to synchronize chaotic systems according to the theory of the qua-dratic optimal regulator, respectively, and the aim is to get the feedback gain matrices kaand ksof system(5)and of system (6), respectively. The steps to get matrices kaand ksare: (a) selecting an n  n dimensional positive semidefinite real

sym-metric constant matrix Q, an r  r dimensional positive definite real symsym-metric constant matrix R and a constant matrix B, with the constant matrix A we can get a Riccati equation as shown in Eq.(4). Then, we should solve this equation to get ma-trix S. If the positive definite mama-trix S exists, the mama-trix A + M(x(t), y(t))  B(ks+ ka) is asymptotically stable and the design of

control for the synchronization of two systems is successful. Otherwise we should reselect Q, R and B and calculate again. (b) Putting the matrix S in Eq.(3), we can get the gain matrices kaand ksof the regulators. After getting the matrices kaand ks

according to the above steps, we put ka, ksand the matrix B in Eqs.(5) and (6). Then we get two synchronized systems.

3. Numerical results of the synchronization of two Quantum-CNN oscillator systems by unidirectional and by mutual linear coupling

For a two-cell Quantum-CNN, the following differential equations are obtained[19]: _x1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q sin x2; _x2¼ 

x

1ðx1 x3Þ þ 2a1 xffiffiffiffiffiffiffiffi1 1x2 1 p cos x2; _x3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 3 q sin x4; _x4¼ 

x

2ðx3 x1Þ þ 2a2 xffiffiffiffiffiffiffiffi3 1x2 3 p cos x4; 8 > > > > > > > > < > > > > > > > > : ð9Þ

where x1and x3are polarizations, x2and x4are quantum phase displacements, a1and a2are proportional to the inter-dot

energy inside each cell and

x

1and

x

2are parameters that weigh effects on the cell of the difference of the polarization

of neighboring cells, like the cloning templates in traditional CNNs. Let a1= a2= 2.47,

x

1= 1,

x

2= 1, chaos is obtained for this

system[20,23,24].

Two Quantum-CNN chaotic systems using the unidirectional linear coupling can be written as _x1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q sin x2; _x2¼ 

x

1ðx1 x3Þ þ 2a1 xffiffiffiffiffiffiffiffi1 1x2 1 p cos x2; _x3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 3 q sin x4; _x4¼ 

x

2ðx3 x1Þ þ 2a2 xffiffiffiffiffiffiffiffi3 1x2 3 p cos x4 8 > > > > > > > > < > > > > > > > > : ð10Þ and _y1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  y2 1 q sin y2þ k1ðx1 y1Þ; _y2¼ 

x

1ðy1 y3Þ þ 2a1 yffiffiffiffiffiffiffiffi1 1y2 1 p cos y2þ k2ðx2 y2Þ; _y3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  y2 3 q sin y4þ k3ðx3 y3Þ; _y4¼ 

x

2ðy3 y1Þ þ 2a2 yffiffiffiffiffiffiffiffi3 1y2 3 p cos y4þ k4ðx4 y4Þ: 8 > > > > > > > > < > > > > > > > > : ð11Þ

The initial values for these linearly coupled Quantum-CNN systems are taken as x1(0) = 0.8, x2(0) = 0.77, x3(0) = 0.72,

x4(0) = 0.57, y1(0) = 0.2, y2(0) = 0.41, y3(0) = 0.25 and y4(0) =  0.81.

Expand the right hand sides of Eqs.(10) and (11)into power series: _x1¼ 2a1 12x 2 1x2þ121x 2 1x3218x 4 1x2þ x216x 3 2þ1201 x 5 2þ      ; _x2¼ 

x

1ðx1 x3Þ þ 2a1 x11₂x1x22þ241x1x 4 2þ12x 3 114x 3 1x22þ58x 5 1þ      ; _x3¼ 2a2 1₂x23x4þ₁₂1x23x3418x 4 3x4þ x41₆x34þ1201 x 5 4þ      ; _x4¼ 

x

2ðx3 x1Þ þ 2a2 x31₂x3x24þ241x3x 4 4þ12x 3 314x 3 3x24þ58x 5 3þ      8 > > > < > > > : ð12Þ and _y1¼ 2a1 12y21y2þ121y21y3218y41y2þ y261y32þ1201 y52þ      þ k1ðx1 y1Þ; _y2¼ 

x

1ðy1 y3Þ þ 2a1 y112y1y22þ 1 24y1y42þ 1 2y 3 1 1 4y 3 1y 2 2þ 5 8y 5 1þ      þ k2ðx2 y2Þ; _y3¼ 2a2 12y 2 3y4þ121y 2 3y 3 4 1 8y 4 3y4þ y416y 3 4þ 1 120y 5 4þ      þ k3ðx3 y3Þ; _y4¼ 

x

2ðy3 y1Þ þ 2a2 y312y3y24þ241y3y44þ12y 3 314y 3 3y24þ58y 5 3þ      þ k4ðx4 y4Þ: 8 > > > < > > > : ð13Þ

From Eqs.(12) and (13), the error dynamics is:

_e ¼ ½A þ MðxðtÞ; yðtÞÞ  Bkse þ Hðx; y; eÞ; ð14Þ

where e = (y1 x1, y2 x2, y3 x3, y4 x4)Tand MðxðtÞ; yðtÞÞ ¼ M11 2a1þ M21 0 0 2a1þ M12 M22 0 0 0 0 M33 2a2þ M43 0 0 2a2þ M34 M44 0 B B B @ 1 C C C A in which M11¼ a1 2x1y2 1 6x1y 3 2þ 1 4ðx1y 2 1y2þ 3x21y1y2Þ þ        

The infinite power series of the first element of M, i.e., M11is 2x1y2 1 6x1y 3 2þ 1 4ðx1y 2 1y2þ 3x21y1y2Þ þ    ð15Þ

It is well-known[28]that a necessary and sufficient condition for the convergence of the infinite series u1þ u2þ    þ unþ   

is that for any previously assigned positive

e

there exists an N such that, for any n > N and for positive p, unþ1þ unþ2þ    unþp



  <

e

: ð16Þ

FromFig. 1, we know that xi

j j < 1; j j < 1yi ði ¼ 1; 2; 3; 4Þ: ð17Þ

Therefore, M11and series contained in other elements of M(x(t), y(t)) are convergent series and they have bounded sums.

We can get the optimum gain ks= [k1, k2, k3, k4]Tby the method of constructing a quadratic optimal regulator. With

A ¼ 0 0 0 0 

x

1 0

x

1 0 0 0 0 0

x

2 0 

x

2 0 2 6 6 6 4 3 7 7 7 5 we choose B ¼ 0 0 0 1½ T; R ¼ 1½ ; Q ¼ 1 0 0 0 0 2 0 2 0 0 1 0 0 2 0 2 2 6 6 6 4 3 7 7 7 5: ð18Þ

After solving the corresponding Riccati equation, we get the gain matrix ks= [k1, k2, k3, k4]T= [0, 1, 0, 1]T.

From the simulation results ofFig. 1, it is shown that master system and slave system reach the synchronization state after they are controlled by the quadratic optimal regulator. It is noticed that the synchronization effect is good.

Case II. The synchronization by mutual linear coupling.

Two Quantum-CNN systems with mutual linear coupling are given: _x1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q sin x2þ k11ðy1 x1Þ; _x2¼ 

x

1ðx1 x3Þ þ 2a1 ffiffiffiffiffiffiffiffix1 1x2 1 p cos x2þ k12ðy2 x2Þ; _x3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 3 q sin x4þ k13ðy3 x3Þ; _x4¼ 

x

2ðx3 x1Þ þ 2a2 ffiffiffiffiffiffiffiffix3 1x2 3 p cos x4þ k14ðy4 x4Þ 8 > > > > > > > > < > > > > > > > > : ð19Þ and _y1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  y2 1 q sin y2þ k21ðx1 y1Þ; _y2¼ 

x

1ðy1 y3Þ þ 2a1 yffiffiffiffiffiffiffiffi1 1y2 1 p cos y2þ k22ðx2 y2Þ; _y3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  y2 3 q sin y4þ k23ðx3 y3Þ; _y4¼ 

x

2ðy3 y1Þ þ 2a2 yffiffiffiffiffiffiffiffi3 1y2 3 p cos y4þ k24ðx4 y4Þ: 8 > > > > > > > > < > > > > > > > > : ð20Þ

Expand the right hand sides of Eqs.(19) and (20)into power series: _x1¼ 2a1 12x 2 1x2þ121x 2 1x 3 2 1 8x 4 1x2þ x216x 3 2þ 1 120x 5 2þ      þ k11ðy1 x1Þ; _x2¼ 

x

1ðx1 x3Þ þ 2a1 x112x1x22þ 1 24x1x42þ 1 2x 3 1 1 4x 3 1x 2 2þ 5 8x 5 1þ      þ k12ðy2 x2Þ; _x3¼ 2a2 12x 2 3x4þ121x 2 3x3418x 4 3x4þ x416x 3 4þ1201 x 5 4þ      þ k13ðy3 x3Þ; _x4¼ 

x

2ðx3 x1Þ þ 2a2 x312x3x24þ241x3x44þ12x 3 314x 3 3x24þ58x 5 3þ      þ k14ðy4 x4Þ 8 > > > < > > > : ð21Þ and _y1¼ 2a1 1₂y21y2þ121y 2 1y3218y 4 1y2þ y216y 3 2þ1201 y 5 2þ      þ k21ðx1 y1Þ; _y2¼ 

x

1ðy1 y3Þ þ 2a1 y112y1y22þ241y1y42þ12y 3 114y 3 1y22þ58y 5 1þ      þ k22ðx2 y2Þ; _y3¼ 2a2 1₂y23y4þ121y 2 3y3418y 4 3y4þ y416y 3 4þ1201 y 5 4þ      þ k23ðx3 y3Þ; _y4¼ 

x

2ðy3 y1Þ þ 2a2 y312y3y24þ241y3y44þ12y3314y33y24þ58y53þ      þ k24ðx4 y4Þ: 8 > > > < > > > : ð22Þ

From Eqs.(21) and (22), the error dynamics is:

_e ¼ ½A þ MðxðtÞ; yðtÞ  Bðksþ kaÞÞe þ Hðx; y; eÞ; ð23Þ

where e = (y1 x1, y2 x2, y3 x3, y4 x4)Tand MðxðtÞ; yðtÞÞ ¼ M11 2a1þ M21 0 0 2a1þ M12 M22 0 0 0 0 M33 2a2þ M43 0 0 2a2þ M34 M44 0 B B B @ 1 C C C A x2 y2

(b)

0 5 10 15 20 25 30 35 40 45 50 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x1 y1

Time (sec)

x

1

,

y

1

(a)

x3 y3

(c)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 e1 e2 e3 e4

e

1

,

e

2

,

e

3

,

e

4

(e)

x4 y4

(d)

0 5 10 15 20 25 30 35 40 45 50 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (sec)

x

2

,

y

2 0 5 10 15 20 25 30 35 40 45 50 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (sec)

x

3

,

y

3 0 5 10 15 20 25 30 35 40 45 50 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (sec)

x

1

,

y

1 0 5 10 15 20 25 30 35 40 45 50

Time (sec)

in which M11¼ a1 2x1y2 1 6x1y 3 2þ 1 4ðx1y 2 1y2þ 3x21y1y2Þ þ        

Similar to Case I, fromFig. 2, jxij < 1, jyij < 1 (i = 1, 2, 3, 4), the infinite power series elements of M(x(t), y(t)) are all convergent

and have bounded sums[27,28].

The optimum gains ka= [k11, k12, k13, k14]Tand ks= [k21, k22, k23, k24]Tcan be obtained by the method of constructing a

qua-dratic optimal regulator. With

x2 y2

(b)

x1 y1

(a)

x3 y3

(c)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 e1 e2 e3 e4

e

1

,

e

2

,

e

3

,

e

4

(e)

x4 y4

(d)

0 5 10 15 20 25 30 35 40 45 50 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (sec)

x

1

,

y

1 0 5 10 15 20 25 30 35 40 45 50 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (sec)

x

2

,

y

2 0 5 10 15 20 25 30 35 40 45 50 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Time (sec)

0 5 10 15 20

_{Time (sec)}

25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

Time (sec)

x

4

,

y

4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x

3

,

y

3 -1

A ¼ 0 0 0 0 

x

1 0

x

1 0 0 0 0 0

x

2 0 

x

2 0 2 6 6 6 4 3 7 7 7 5 we choose B ¼ 0 0 0 1½ T; R ¼ 1½ ; Q ¼ 1 0 0 0 0 2 0 2 0 0 1 0 0 2 0 2 2 6 6 6 4 3 7 7 7 5: ð24Þ

After solving the corresponding Riccati equation, we then get two gain matrices ka= [k11, k12, k13, k14]T= [0, 0.5, 0, 0.5]Tand

ks= [k21, k22, k23, k24]T= [0, 0.5, 0, 0.5]T.

From the simulation results ofFig. 2two systems reach the synchronization state after they are controlled by the qua-dratic optimal regulator. It is noticed that the synchronization effect is also satisfactory.

4. Chaotization of Quantum-CNN chaotic system scheme and simulation

Optimal control is a well-established engineering control strategy, and is useful for both linear and nonlinear system with linear or nonlinear controllers[3]. Now, we use a typical optimal control for the chaotization of Quantum-CNN system. Con-sider the system(9)with a controller u and define the Hamilton function:

Hðx1;x2;x3;x4;u; pÞ ¼ pTFðx1;x2;x3;x4;u; pÞ; pT

¼ ½p1p2p3p4;

ð25Þ where p is a Lagrange multiplier, called a co-state vector, F is the right hand side of Eq.(9). Following the variational principle of optimal control, we can obtain

p2 

x

1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q cos x2 0 B @ 1 C A þ p3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 3 q sin x4  þ p4 

x

2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 3 q cos x4 0 B @ 1 C A ¼ 0; ð26Þ p2 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q sin x2¼ 0: ð27Þ

This yield a non-trivial solution for (p2, p3, p4) if and only if

2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 1 q sin x2¼ 0: ð28Þ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10-4 -8 -6 -4 -2 0 2 4 6 8 10

kb

L

yapuno

v e

xponent.

x 10-5

It gives an optimal surface singularly in the state space. This type of control assumes values on the two allowable bound-aries(27) and (28)alternatively according to a switching surface. Locating system trajectories on the surface, a typical feed-back control in the form

u ¼ kbsgn 2affiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1  x2 1 q sin x2 2 6 4 3 7 5 ð29Þ

can be used. By adjusting the value of kbfrom zero initial value to kb= 1.6  104in the above controller with the signum

function sgn½

v

 ¼ 1 if

v

>0; 0 if

v

¼ 0; 1 if

v

<0 8 > < > : ð30Þ

the chaotic motion with one positive Lyapunov exponent can be controlled to chaotic motion with two positive Lyapunov exponents as shown by the simulation result inFig. 3.

5. Conclusions

Two chaotic Quantum-CNN systems are synchronized in two cases: unidirectional linear coupling by optimum control, mutual linear coupling by optimum control. The number of controllers for optimum control is less than that for synchroni-zation only by linear couplings. This results in lower cost. In chaos synchronisynchroni-zation cases, by a theorem of convergent series, we prove that all infinite power series elements of A + M(x(t), y(t))  B(ks+ ka) are convergent and have bounded sums. This

synchronization of chaos systems can be used to increase the security of communication. Next, the optimum control is used for chaotization, i.e., to enhance original chaotic state to more complex chaotic state. Numerical simulations are used to ver-ify the effectiveness of the proposed scheme.

Acknowledgments

This research was supported by the National Science Council, Republic of China, under Grant Number NSC 96-2221-E-009-145-MY3.

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