Predictive synchronization of chaotic satellites systems

Expert Systems with Applications 38 (2011) 9041–9045

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Short Communication

Predictive synchronization of chaotic satellites systems Djaouida Sadaoui a,⇑, Abdelkrim Boukabou b, Nadjim Merabtine a, Malek Benslama a a b

LET Laboratory, Electronics Department, Mentouri University, 25000 Constantine, Algeria Electronics Department, Faculty of Engineer Sciences, Jijel University, 18000 Jijel, Algeria

a r t i c l e

i n f o

Keywords: Chaotic satellite attitude Predictive control Synchronization

a b s t r a c t This paper presents a systematic design procedure to synchronize two identical chaotic satellites systems based on a predictive control. This method is developed on the basis of delayed feedback control of continuous-time chaotic systems combines with the prediction-based method of discrete-time chaotic systems. Moreover, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed method. Simulation results are presented to verify the effectiveness of the proposed scheme. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Chaotic synchronization is the important field of chaotic science researching (Lam & Seneviratne, 2008). Since the pioneering work of Pecora and Carroll (1990), much attention has been paid to the chaos synchronization in non-linear systems (Huang, Cheng, Yan, & Robust, 2009). Consideration was given to their application in various branches of engineering such as mechanical systems, chemical and processing industries, information systems, electrical and electronic systems, communication systems, and spacecraft. To date, various control approaches were reported to realize the chaotic synchronization, such as nonlinear control (Huang, Feng, & Wang, 2004), active control (Al-Sawalha, 2009; Xiaobing Zhou, Chen, & Hui, 2009), fuzzy control (Lam & Seneviratne, 2008; Lam, Ling, Iu, & Ling, 2008), impulsive control (Yang & Chua, 1997), adaptive control (Chen, Lee b, & Yang, 2007; Lina & Yan, 2009), phase control (Volkovskii, 1997) and finite-time control (Wang, Han, Xie, & Zhang, 2009). In synchronization theory, we define a Master (drive) system, which is the dominant system, and a bounded set of Slave (response) systems. The synchronization problem consists of creating either physical interconnections or control feedback loops, which forces the outputs of the slave systems to conform with those of the Master. As space technology progresses, the need for improved satellite systems by better understanding of satellite dynamics has continuously kept attention (Kuang, Leung, & Tan, 2003). Recently, non-linear dynamics, especially the chaotic attitude dynamics of a satellite have attracted the attention of many scientists (Kuang, Tan, Arichandran, & Leung, 2001; Kuang & tan, 2000; Kong & Zhoul,

2006; Tsui & Jones, 2000). The control of the Slave satellite, on the other hand, is a synchronization problem. A reference trajectory for the Slave satellite will therefore also depend on the states of the Master satellite. For many applications of formations of satellites the objective will be to point measuring instruments in the same direction. Let therefore the reference trajectory for the Slave satellite be the measured attitude of the Master satellite. In this paper, the synchronization of chaotic satellites systems is handled based on the predictive approach. The control input is based on the difference between the T-time delayed state and the current state, where T denotes a period of the stabilized orbits (Ushio & Yamamoto, 1999). On the other hand, states controlled by the delayed feedback control will converge to the stabilized orbits since the approximations are not used in the feedback loop, which is an advantage of the delayed feedback control. It was shown that this method guarantees the stability of the obtained controlled system (Boukabou, Chebbah, & Mansouri, 2008; Ushio & Yamamoto, 1999). 2. Predictive controller design Definition. Consider the nonlinear system described by

_ xðtÞ ¼ f ðxðtÞÞ þ uðtÞ; n

where x 2 R is the state vector and u 2 R is the feedback controller. We assume that f is differentiable. Our aim is to design a feedback controller u(t) such that the trajectory of the system (1) converge to an unstable fixed point xf. The control input u(t) is determined by the difference between the predicted states and the current states:

uðtÞ ¼ Kðxp ðtÞ  xðtÞÞ; ⇑ Corresponding author. E-mail address: [email protected] (D. Sadaoui). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.01.117

ð1Þ n

ð2Þ

where K is a gain vector, xp(t) is the predicted future state of uncontrolled chaotic systems from the current state x(t).

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Using a one-step-ahead-prediction, the predictive control (2) becomes

_  xðtÞÞ: uðtÞ ¼ KðxðtÞ

ð3Þ

Near xf, we can use the linear approximation for the uncontrolled system by

_  xf Þ ¼ AðxðtÞ  xf Þ; ðxðtÞ

ð4Þ

where A 2 Rnn is the Jacobian matrix of f(x(t)) evaluated at the fixed points xf, which is defined as follows:

A ¼ Dx f ðxf Þ ¼

 _  _ tÞ @ xð  : @xðtÞxf

ð5Þ

The controlled system will be described by:

_ xðtÞ ¼ f ðxðtÞÞ þ KðAðxðtÞ  xf ÞÞ:

0

1:2 0

pffiffi 6 2

10

1 ð15Þ

This torques are chosen so as to force the satellite into chaotic motion (Tsui & Jones, 2000). 3.2. Synchronization problem formulation Consider the following two identical satellites attitudes systems, where the drive system and response system are denoted with x and y, respectively. 3.2.1. Drive system

ð7Þ

With

dxðtÞ ¼ xðtÞ  xf :

1

xx B C B C CB @ hy A ¼ @ 0 A@ xy A 0:35 0 pffiffiffi xz hz  6 0 0:4 hx

ð6Þ

Eq. (4) is rewritten in the form

_ dxðtÞ ¼ AdxðtÞ:

We take Ix = 3, Iy = 2 and Iz = 1 with the perturbing torques

0

ð8Þ

The controlled system is linearized around xf by

8  pffiffi > x1 ¼ rx x2 x3  1:2 x þ 2I6x x3 ; > Ix 1 > <  x2 ; x2 ¼ ry x1 x3 þ 0:35 Iy > > pffiffi >  : x ¼ r x x  6 x  0:4 x : 3 z 1 2 Iz 1 Iz 3

ð16Þ

The chaotic motion of system (16) is shown in Fig. 1(a)–(d)



_ dxðtÞ ¼ AdxðtÞ þ KðdxðtÞ  dxðtÞÞ ¼ AdxðtÞ þ KðAdxðtÞ  dxðtÞÞ ¼ ðA þ KðA  IÞÞdxðtÞ;

ð9Þ

where I 2 Rnn is the identity matrix. In order to apply the proposed predictive control strategy, we have to determine the gain vector K and the vicinity of the fixed point to adjust the next point so it falls on the fixed one. The feedback gain K is determined as follows (Boukabou et al., 2008):

jA þ KðA  IÞj < I:

rðtÞ ¼ jxðtÞ  xðt  1Þj:

ð11Þ

f ðxðtÞÞ þ uðtÞ if rðtÞ < e f ðxðtÞÞ

ð17Þ

otherwise;

3.2.3. Error Let us define the synchronization errors between the response system (17) and the drive system (16) as follows:

eðtÞ ¼ ½e1 ðtÞ e2 ðtÞ e3 ðtÞT

The controlled system will be described by:



8  pffiffi > y1 ¼ rx y2 y3  1:2 y þ 2I6x y3 ; > Ix 1 > <  y2 ; y2 ¼ ry y1 y3 þ 0:35 Iy > > > : y ¼ r y y  pffiffi6 x  0:4 y : z 1 2 3 Iz 1 Iz 3

ð10Þ

And the vicinity of the fixed point is given by:

_ xðtÞ ¼

3.2.2. Response system

¼ ½y1 ðtÞ  x1 ðtÞ y2 ðtÞ  x2 ðtÞ y3 ðtÞ  x3 ðtÞT :

ð12Þ

Our aim is to design a feedback controller u(t) such that the controlled system (17) asymptotically synchronizes the system (16) in the sense that

where e is a positive small real number. 3. System description and problem formulation

lim eðtÞ ! 0:

t!1

3.1. Satellite system with chaotic dynamics The rotational motion equation for a general rigid spacecraft acting under the influence of external torques is given by McDuffie and Shtessel (1997), MacKunis, Dupree, Bhasin, and Dixon (2008), Guan, Liu, and Liu (2005), Kong’2, Zhoul, and Zou, 22 Sadaoui, Merabtine, and Benslama (2006), Show, Juang, and Jan (2007) and Show et al. (2003)

_ ¼ XIx þ h þ u; Ix

ð13Þ

where I is the moment of inertia tensor, x is the angular velocity vector, u is the control torque, and h contains any external disturbance torques. The dynamical equations of a satellite are

8  > > I x ¼ xy xz ðIy  Iz Þ þ hx þ ux ; > < x x 

Iy xy ¼ xx xz ðIz  Ix Þ þ hy þ uy ; > > > :  Iz xz ¼ xx xy ðIx  Iy Þ þ hz þ uz ;

ð14Þ

where Ix, Iy and Iz are the principal moments of inertia, xx, xy and xz are the angular velocities of the satellite, ux, uy and uz are the three control torques; and hx, hy and hz are perturbing torques.

Then the dynamics of the error system is determined, directly from subtracting (17) from (16), as follows:

8  pffiffi > e1 ¼ rx ðy2 y3  x2 x3 Þ  1:2 ðy1  x1 Þ þ 2I6x ðy3  x3 Þ; > Ix > <  ðy2  x2 Þ; e2 ¼ ry ðy1 y3  x1 x3 Þ þ 0:35 Iy > > > : e ¼ r ðy y  x x Þ  pffiffi6 ðy  x Þ  0:4 ðy  x Þ: 3 z 1 2 1 2 1 3 1 3 Iz Iz

ð18Þ

It is directly obtained that

8 > < y2 y3  x2 x3 ¼ e2 e3 þ y2 e3 þ y3 e2 ; y1 y3  x1 x3 ¼ e1 e3 þ y1 e3 þ y3 e1 ; > : y1 y2  x1 x2 ¼ e1 e2 þ y1 e2 þ y2 e1 :

ð19Þ

By Eq. (19), system (18) can be rewritten in the following form:

8   pffiffi 6 > e1 ¼  1:2 e þ r y e þ r y þ e3  rx e2 e3 ; > 1 x 2 x 3 2 I 2Ix > x > <  0:35 e2 ¼ ry y3 e1 þ Iy e2 þ ry y1 e3  ry e1 e3 ; > >  pffiffi > > : e3 ¼ rz y  6 e1 þ rz y e2  0:4 e3  rz e1 e2 : 2 1 Iz Iz

ð20Þ

In order to control the system to the unstable equilibrium point [0 0 0]T, we have to determine the correction which will be applied

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a

c

Trajectories of the drive system

6 5

4 4 3

0

x2

x3

2

-2

2

-4 6

1 4 2 0

-2

-2 -4

0 -1 -4

x2

b

4

2

0

d

2

2

1

1

x3

x3

3

0

0

-1

-2

-2

-3

-3

-2

-1

0

1

1

2

3

4

0

-1

-3

-1

4

3

-4

-2

x1

4

-4

-3

x1

2

3

4

-4 -1

0

1

2

x1

3

4

5

6

x2

Fig. 1. (a) Trajectories of the drive system. (b) Trajectories projected on the x1–x2 plane. (c) Trajectories projected on the x1–x3 plane. (d) Trajectories projected on the x2–x3 plane.

to the current state of the chaotic system. For this purpose, we determine the control input u(t) defined by Eq. (3).

8 0 >   <  0:35 uðtÞ ¼ K ry y3 e1 þ Iy e2 þ ry y1 e3  ry e1 e3  e2 : > : 0

ð21Þ

The controlled system is given by:

8   pffiffi > > e1 ¼  1:2 e1 þ rx y3 e2 þ rx y2 þ 2I6x e3  rx e2 e3 ; > I x > > >  > > < e2 ¼ ry y3 e1 þ 0:35 e2 þ ry y1 e3  ry e1 e3 Iy    0:35 > > > þK ry y3 e1 þ Iy e2 þ ry y1 e3  ry e1 e3  e2 ; > >   > > > e ¼ r y  pffiffi6 e þ r y e  0:4 e  r e e : : 3

z 2

Iz

1

z 1 2

Iz

3

@ e_ 2 ðtÞ de2 ðtÞ; @e2 ðtÞ   0:35 0:35 þK  1 de2 ðtÞ: de_ 2 ðtÞ ¼ Iy Iy

z 1 2

ð23Þ ð24Þ

Thus, the controlled system is described by

8 > < Kððry y3 e1 þ 0:175e2 þ ry y1 e3  ry e1 e3 Þ  e2 Þ if je2 ðtÞ  e2 ðt  1Þjhe u¼ > : 0 otherwise:

ð28Þ

In this section, the fourth-order Runge–Kutta integration method is used to solve the system of differential equations. In the simulation process the initial states of the drive system are (x1(0), x2(0), x3(0))textsuperscriptT = (3, 4.1, 2)T and those of the response are (y1(0), y2(0), y3(0))T = (5, 3, 1)T. The time responses of state variables of the satellite system are shown in Fig. 2 and the time responses of synchronization error are shown in Fig. 3. From the simulation results, it shows that the time responses of synchronization error under the proposed predictive control converge quickly to zero; which means that perfect synchronization responses can be achieved. These results showed the efficiency of the control strategy.

ð25Þ 5. Conclusion

This implies that:

1hKh1:424:

ð27Þ

4. Simulation results

Then, k must satisfy the inequality

   0:35  0:35    I þ K I  1 h1: y y

rðtÞ ¼ je2 ðtÞ  e2 ðt  1Þj:

ð22Þ

Its linearized system around the fixed point is given by:

de_ 2 ðtÞ ¼

And the vicinity of the fixed point is given by:

ð26Þ

In this paper, a feedback predictive control has been proposed to synchronize two chaotic satellites systems coupled in a drive/re-

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3

6 x1 y1

5

e1

2.5 2 1.5

e1

x1/y1

4 3

1 0.5

2

0 1

-0.5

0 -1

-1 0

10

20

30

40

50

60

70

80

90

30

40

50

60

70

80

90

100

100

e2

4 x2 y2

5

3 2

e2

4 3

x2/y2

20

5

6

1

2

0

1

-1

0

-2

-1

-3 0

10

20

30

40

50

60

70

80

e3

-1

-1

-2

-2

-3

-3

-4 40

50

50

60

70

80

90

60

70

80

90

100

e3

0

0

30

40

1

1

20

30

2

x3 y3

10

20

3

2

0

10

90 100

3

-4

0

t

t

x3/y3

10

t

t

-2

0

100

t

0

10

20

30

40

50

60

70

80

90

100

t Fig. 3. Synchronization errors of the satellites systems.

Fig. 2. State variables of the satellites drive and response systems.

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