Predictive feedback control and synchronization of

Applied Mathematics and Computation 247 (2014) 235–243

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Predictive feedback control and synchronization of hyperchaotic systems D. Sadaoui a, A. Boukabou b,⇑, S. Hadef b a b

Department of Electronics, Hadj Lakhdar University Batna, Batna 05000, Algeria Department of Electronics, Jijel University, Ouled Aissa, Jijel 18000, Algeria

a r t i c l e

i n f o

Keywords: Hyperchaotic systems Predictive control Synchronization

a b s t r a c t The paper deals with the problem of control and synchronization of hyperchaotic systems. The proposed control method is based on the predictive principle which employ an instantaneous control input to guarantee the convergence of the chaotic trajectory towards an unstable equilibrium point. The synchronization is constructed around the drive-response principle. Numerical simulations on some hyperchaotic systems are given to demonstrate the main results. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction The growing interest in control and synchronization of chaotic systems was probably caused by the paper [1,2]. The two mentioned papers have formed an impulse for much research along these lines. Many possible applications of chaos control and synchronization methods have been discussed by computer simulation and realized in laboratory condition [3–14]. As chaos theory progresses, many new chaotic systems have been proposed, specially hyperchaotic systems [15–23]. A hyperchaotic system is usually characterized as a chaotic system with more than one positive Lyapunov exponent, implying that the dynamics expand in more than one direction; giving rise to more complex chaotic dynamics. Moreover, a hyperchaotic system has the following characteristics: (i) the minimal dimension of the phase space of the system is at least four. (ii) the number of terms in coupled equations is at least two, of which at least one has a nonlinear function. The aim of this paper is to apply both control and synchronization to hyperchaotic systems. This is done by extending the predictive chaos control method presented in [10,11] and applied for the control of the n-scroll Chua circuit [12], the control of multiscroll Chen system [13], and developed for the synchronization of chaotic satellites systems [14]. Under this framework, Section 2 deals with the details of the predictive control and synchronization principles. In Section 3, we present the numerical simulation studies for the control and synchronization of hyperchaotic Lorenz and Chen systems. Conclusion is given in Section 4. 2. Hyperchaotic predictive control and synchronization principles Consider the two non linear systems

X_ 1 ¼ f ðX 1 ; uÞ; X_ 2 ¼ f ðX 2 ; X 1 Þ; ⇑ Corresponding author. http://dx.doi.org/10.1016/j.amc.2014.09.016 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

ð1Þ ð2Þ

236

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243

where f : RN ! RN ; g : RN  RN ! RN are continuous and differentiable, X 1 ; X 2 2 RN are the state variables and u represents the feedback controller. The system given by Eq. (1) will be called the drive system and the system given by Eq. (2) will be called the response system. 2.1. Predictive control principle The objective of the proposed hyperchaotic predictive control method is to design a controller, added to system (1) in a feedback configuration in order to drive the drive system out of the chaotic attractor and then converges to an unstable equilibrium point. The hyperchaotic system (1) is under predictive law given by

X_ 1 ¼ f ðX 1 Þ þ u;

ð3Þ

where the control input u is determined by the difference between the predicted state X p and the current state X 1 as follows

  u ¼ K Xp  X1 ;

ð4Þ

where gain K to be determined. Let E be an unstable equilibrium point of system (1). Then, we have

X_ 1 ¼ 0 ) E ¼ f ðEÞ:

ð5Þ

Near E, we can use the approximation for the uncontrolled system by

dX_ 1 ¼ AdX 1 ;

ð6Þ

where dX_ 1 ¼ X_ 1  E; dX 1 ¼ X 1  E and A 2 RN  RN is the Jacobian matrix of f ðX 1 Þ evaluated at the unstable equilibrium point E, which is defined as follows

A¼

 @ X_ 1   : @X 1 

ð7Þ

E

Using a one-step ahead prediction of the uncontrolled state variable X 1 , i.e. X p ¼ X_ 1 , then the control law (4) becomes

  u ¼ K X_ 1  X 1 ¼ K ðf ðX 1 Þ  X 1 Þ

ð8Þ

and the controlled system is described as follows

X_ 1 ¼ f ðX 1 Þ þ K ðf ðX 1 Þ  X 1 Þ:

ð9Þ

Thus, the controlled drive system is linearized around E by

dX_ 1 ¼ AdX 1 þ K ðAdX 1  dX 1 Þ ¼ ðA þ K ðA  IÞÞdX 1 ;

ð10Þ

where I represents the identity matrix. The stability is guaranteed in the local neighborhood of the unstable equilibrium E if all real parts of eigenvalues of ðA þ K ðA  IÞÞ are negative. In general, any periodic orbit embedded in the hyper chaotic system is hyperbolic so that it satisfies detðI  AÞ – 0. Thus, the predictive feedback control is more applicable than the delay feedback control [10]. Moreover, the predictive control demonstrates its efficiency for the control of continuous chaotic systems by only controlling one of the states variables. The feedback gain K is determined as follows

jA þ K ðA  IÞj < I:

ð11Þ

Stabilization can be achieved when the drive system shadows neighborhoods of unstable equilibrium point. This is done by verifying the following test

rðt Þ ¼ jX 1 ðtÞ  X 1 ðt  1Þj < e:

ð12Þ

The controlled drive system will be described by

X_ 1 ¼



f ðX 1 Þ þ K ðf ðX 1 Þ  X 1 Þ; if r ðtÞ < e; f ðX 1 Þ;

otherwise

for small positive constant e. 2.2. Synchronization principle Let X 1 ðt; X 1 ð0ÞÞ and X 2 ðt; X 2 ð0ÞÞ be solutions to the drive system (1) and to the response system (2), respectively.

ð13Þ

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243

237

In this framework, complete synchronization is defined as the identity between the trajectories of the response system X 2 and of one replica X 02 of it X_ 02 ¼ gðX 1 ; X 02 Þ for the same chaotic driving system X 1 . If the solutions X 1 ðt; X 1 ð0ÞÞ and X 2 ðt; X 2 ð0ÞÞ satisfy

lim kX 1 ðt; X 1 ð0ÞÞ  X 2 ðt; X 2 ð0ÞÞk ¼ 0:

t!1

ð14Þ

Then, the drive system and the response system are said synchronized. In other words, the response system forgets its initial conditions, though evolving on a chaotic attractor. This kind of synchronization can be achieved and provide that all the Lyapunov exponents of the response system under the action of the driver (the conditional Lyapunov exponents) are negative [2]. This implies that the response system is asymptotically stable. 3. Numerical results 3.1. Predictive control and synchronization of hyperchaotic Lorenz system The hyperchaotic Lorenz system is a differential system with a chaotic behavior for some values of parameters, described by [15]

8 x_ ¼ rðy  xÞ; > > > < y_ ¼ ðr  zÞx  y þ w; > z_ ¼ xy  bz; > > : _ ¼ cx; w

ð15Þ

where the state variable w represents the extra variable and c is a new parameter. All the other variables and parameters come from the original Lorenz model. For parameters values r ¼ 10; b ¼ 8=3; r ¼ 30 and c ¼ 10, and starting from the initial conditions ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð3; 2; 0:5; 13Þ, the hyperchaotic Lorenz system is shown in Fig. 1(a)–(d). The hyperchaotic Lorenz system admits an equilibrium point at the origin E ¼ ð0; 0; 0; 0Þ. The eigenvalues of the jacobian matrix at the equilibrium are 0:28; 0:24; 0:00 and 14:24. Thus, the equilibrium point E is an unstable saddle-node point. In order to control the hyperchaotic Lorenz drive system to the unstable equilibrium point E, we have to determine the correction which will be applied to the current states of the hyperchaotic drive system. For this purpose, we determine the control input defined by Eq. (8). The predictive control is applied to the only state variable y1 as follows

8 x_ 1 ¼ rðy1  x1 Þ; > > > < y_ 1 ¼ ðr  z1 Þx1  y1 þ w1 þ u; > z > _ 1 ¼ x1 y1  bz1 ; > : _ 1 ¼ cx1 w

ð16Þ

u ¼ Kðy_ 1  y1 Þ ¼ Kððr  z1 Þx1  2y1 þ w1 Þ:

ð17Þ

with

Thus, the controlled hyperchaotic drive system will be given by

8 x_ 1 ¼ rðy1  x1 Þ; > > > < y_ 1 ¼ ðr  z1 Þx1  y1 þ w1 þ Kððr  z1 Þx1  2y1 þ w1 Þ; > z > _ 1 ¼ x1 y1  bz1 ; > : _ 1 ¼ cx1 : w

ð18Þ

By linearization around the state y1 , we obtain

dy_ 1 ¼ ð1  2K Þdy1 :

ð19Þ

Thus, gain K must satisfies the inequality

j1  2K j < 1:

ð20Þ

This implies that

1 < K < 0:

ð21Þ

And the vicinity of the equilibrium point is given by

rðt Þ ¼ jy1 ðt Þ  y1 ðt  1Þj < 0:01:

ð22Þ

The control input is applied for t > 8. From Fig. 2, it is possible to see that, after a transitory phase, the hyperchaotic Lorenz system is quickly stabilized around its unstable equilibrium point E with extremely small applied force.

238

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243 30

50

20

40

10 30 z

y

0

20 −10 10

−20

−30 −20

−15

−10

−5

0 x

5

10

15

0 −20

20

40

−5

0 x

5

10

15

20

0

y

−20 0 50

30 z

10

15

20

5

10

15

20

5

10

15

20

5

10 Times(s)

15

20

z

20

5

0

−50 0 50

0 0 80 w

10

0 −60

−10

20 x

50

−15

−20

w

20

60

0

−80 0

Fig. 1. The hyperchaotic Lorenz system. (a) Phase plane xy.(b) Phase planexz. (c) Phase plane wz. (d) Time response of variable states.

control on

x

20 0

y

−20 0 50

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

0

z

−50 0 50

w

0 0 80 0

u

−80 0 0.05 0

−0.05 0

Fig. 2. Time response of the controlled drive system.

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243

239

Once controlled drive system is obtained, we construct a response system which exhibits a generalized kind of synchronization motion with the driver by making a simple nonlinear transformation among the response variable x2 . Thus, the response system will be given by

8 x_ 2 ¼ rðy1  x2 Þ; > > > < y_ 2 ¼ ðr  z2 Þx2  y2 þ w2 ; > z_ 2 ¼ x2 y2  bz2 ; > > : _ 2 ¼ cx2 : w

ð23Þ

Let us define the synchronization errors between the response system (23) and the drive system (18) as follows

2

x1  x2

3

6e 7 6 y  y 6 27 6 1 2 6 7¼6 4 e 3 5 4 z1  z2

7 7 7: 5

e1

e4

3

2

ð24Þ

w1  w2

And we obtain the error dynamics

8 e_ 1 > > > < e_ 2 > > e_ 3 > : e_ 4

¼ re1 ; ¼ re1  x1 z1 þ x2 z2  e2 þ e4 ; ¼ x1 y1  x2 y2  be3 ;

ð25Þ

¼ ce1 :

In order to synchronize between the drive and response systems, then error dynamics must converge to zero, that is

8 e_ 1 ¼ re1 ¼ 0; > > > < e_ ¼ re  x z þ x z  e þ e ¼ 0; 2 1 1 1 2 2 2 4 > e_ 3 ¼ x1 y1  x2 y2  be3 ¼ 0; > > : e_ 4 ¼ ce1 ¼ 0; 8 r e1 ¼ 0; > > > < re1  x1 z1 þ x1 z2  x1 z2 þ x2 z2  e2 þ e4 ¼ 0; ) > x1 y1  x1 y2 þ x1 y2  x2 y2  be3 ¼ 0; > > : ce1 ¼ 0; 8 > re1 ¼ 0; > > < ðr  z Þe  e  x e þ e ¼ 0; 2 1 2 1 3 4 ) > y e þ x e  be ¼ 0; 1 1 2 3 > 2 > : ce1 ¼ 0:

ð26Þ

Because the parameters r; r; b; c and the state variables x1 ; y2 ; z2 are different from zero, it follows that the error variables ðe1 ; e2 ; e3 ; e4 Þ asymptotically converge to ð0; 0; 0; 0Þ. In other words, the response system asymptotically synchronizes with the drive system no matter how they are initialized. The results of simulation are shown in Figs. 3 and 4 with response system starting from ðx2 ð0Þ; y2 ð0Þ; z2 ð0Þ; w2 ð0ÞÞ ¼ ð8; 4; 12; 35Þ. The synchronization process is switched on for time t > 5. 3.2. Predictive control and synchronization of hyperchaotic Chen system The hyperchaotic Chen system is described as follows [16]

8 x_ ¼ aðy  xÞ; > > > < y_ ¼ ðd  zÞx þ cy  w; > z_ ¼ xy  bz; > > : _ ¼ x þ k: w

ð27Þ

When a ¼ 36; b ¼ 3; c ¼ 28; d ¼ 16 and k ¼ 0:3, the system shows complex hyperchaotic behavior. The phase portraits of the hyperchaotic Chen system are shown in Fig. 5(a)–(d) when starting from the initial conditions ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð3; 7; 2; 0:5Þ. This hyperchaotic system has an unstable equilibrium point given by 2

E¼

k; k;

3

k k ;  kðc þ dÞ b b

!T ¼ ð0:3; 0:3; 0:03; 3:591ÞT :

The eigenvalues of the jacobian matrix at the equilibrium E are 17:13; 0:00; 2:98 and 25:14.

ð28Þ

240

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243 synchronization on

x ,x

2

20 1

0 5

10

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

y ,y

2

−20 0 50 1

0

z1, z2

−50 0 60

w1, w2

0 0 80 0

−80 0

Fig. 3. Time response of the synchronized drive-response hyperchaotic Lorenz systems.

e1

50 0

e

2

−50 0 50

e3

5

10

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

0

−50 0 50 0

−50 0 100 e4

synchronization on

0

−100 0

Fig. 4. Time response of the error variables.

The hyperchaotic Chen drive system is under predictive control of the form

8 x_ 1 ¼ aðy1  x1 Þ; 8_ > > > ¼ aðy  x Þ; x 1 1 1 > > > > > > y_ ¼ ðd  z1 Þx1 þ cy1  w1 > > < y_ 1 ¼ ðd  z1 Þx1 þ cy  w1 þ u; < 1 1 ) þKððd  z1 Þx1 þ cy1  w1  y1 Þ; > > > z_ 1 ¼ x1 y1  bz1 ; > > > > > _ z ¼ x1 y1  bz1 ; : > 1 > > _ 1 ¼ x1 þ k; w : _ 1 ¼ x1 þ k: w

ð29Þ

The system state variable y1 is linearized around E by

dy_ 1 ¼ ðc þ K ðc  1ÞÞdy1 :

ð30Þ

The predictive feedback control stabilizes the unstable equilibrium point if K satisfies the inequality:

jc þ K ðc  1Þj < 1:

ð31Þ

241

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243 30 35 15 25

z

y

0

15

−15

−30 −20

5

−10

0

x

10

−5 −20

20

−10

0

x

10

20

30 x

35

0

−30 0 30 y

25

z

z w −5 −12

−8

−4 w

0

4

15

20

5

10

15

20

5

10

15

20

5

10 Times (s)

15

20

0

−40 0 20

5

10

0

−30 0 40

15

5

0

−20 0

Fig. 5. The hyperchaotic Chen system. (a) Phase plane xy.(b) Phase plane xz. (c) Phase plane wz. (d) Time response of variable states.

control on

x

30 0

y

−30 0 30

z

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

0

−40 0 10 w

10

0

−30 0 40

0

−10 0 0.02 u

5

0

−0.02 0

Fig. 6. Time response of the controlled drive system.

242

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243

x1, x2

30

synchronization on

0

y1, y2

−30 0 30

5

10

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

0

1

z ,z

2

−30 0 40 0

w1, w2

−40 0 10 0

−10 0

Fig. 7. Time response of the synchronized drive-response hyperchaotic Chen systems.

e

1

30

synchronization on

0

e2

−30 0 30

3

e

15

20

5

10

15

20

5

10

15

20

5

10 Time (s)

15

20

0

−40 0 10 4

10

0

−30 0 40

e

5

0

−10 0

Fig. 8. Time response of the error variables.

In other hand, the response system is given by

8 x_ 2 ¼ aðy2  x2 Þ; > > > > > > < y_ 2 ¼ ðd  z2 Þx2 þ cy1  w2 ; > > z_ 2 ¼ x2 y2  bz2 ; > > > > : _ 2 ¼ x2 þ k: w

ð32Þ

The dynamic of the error variables will be given by

8 e_ 1 ¼ aðe2  e1 Þ; > > > > > > < e_ 2 ¼ de1  z1 x1 þ z2 x2  e4 ; > > e_ 3 ¼ x1 y1  x2 y2  be3 ; > > > > : e_ 4 ¼ e1 ;

ð33Þ

D. Sadaoui et al. / Applied Mathematics and Computation 247 (2014) 235–243

243

Demanding that all of the equations of system (33) are zero, we get the following

8 e_ 1 ¼ aðe2  e1 Þ ¼ 0; > > > < e_ ¼ de  x z þ x z  e ¼ 0; 2 1 1 1 2 2 4 > _ 3 ¼ x1 y1  x2 y2  be3 ¼ 0; e > > : e_ 4 ¼ e1 ¼ 0; 8 e1 ¼ e2 ; > > > < de  x z þ x z  x z þ x z  e ¼ 0; 1 1 1 1 2 1 2 2 2 4 ) > x y  x y þ x y  x y  be ¼ 0; 1 1 1 2 3 > 1 2 2 2 > : e1 ¼ 0; 8 8 > e1 ¼ e2 ; > e1 ¼ 0; > > > > < ðd  z Þe  x e  e ¼ 0; < e ¼ 0; 2 1 1 3 4 2 ) ) > > y e e þ x e  be ¼ 0; 1 1 2 3 3 ¼ 0; > > 2 > > : : e1 ¼ 0; e4 ¼ 0:

ð34Þ

Simulation results of the control and synchronization processes switched on for time t > 10 and time t > 5, respectively, and of the error variables, are given in Figs. 6–8 with ðx2 ð0Þ; y2 ð0Þ; z2 ð0Þ; w2 ð0ÞÞ ¼ ð0:1; 3; 0; 5Þ. 4. Conclusion We realize in this paper both the control and synchronization of hyperchaotic systems; specially the hyperchaotic Lorenz and Chen systems. The controllers designed by predictive control method are used to stabilize the systems trajectories on the unstable equilibrium points successfully. Moreover, the response systems are perfectly synchronized with the controlled drive systems. Numerical simulations showed the effectiveness of the proposed method. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. K. Pyragas, Continues control of chaos by self-controlling feedback, Phys. Lett. A 170 (1992) 421–428. L. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, U. Parlitz, Experimental demonstration of secure communications via chaotic synchronization, Int. J. Bifurcation Chaos 2 (1992) 709–713. K.M. Cuomo, A.V. Openheim, Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Lett. 71 (1993) 65–68. T. Ushio, S. Yamamoto, Prediction-based control of chaos, Phys. Lett. A264 (1999) 30–35. E. Alvarez, A. Fernandez, P. Garcia, J. Jimenez, A. Marcano, New approach to chaotic encryption, Phys. Lett. A 263 (1999) 373–375. G. Hu, Z. Feng, R. Meng, Chosen ciphertext attack on chaos communication based on chaotic synchronization, IEEE Trans. Circuits Syst. I 50 (2003) 275– 279. A. Boukabou, N. Mansouri, Controlling chaos in higher-order dynamical systems, Int. J. Bifurcation Chaos 14 (2004) 4019–4025. A. Boukabou, A. Chebbah, N. Mansouri, Predictive control of continuous chaotic systems, Int. J. Bifurcation Chaos 18 (2008) 587–592. A. Boukabou, N. Mekircha, Generalized chaos control and synchronization by nonlinear high-order approach, Math. Comput. Simul. 82 (2012) 2268– 2281. A. Boukabou, B. Sayoud, H. Boumaiza, N. Mansouri, Control of n-scroll Chua’s circuit, Int. J. Bifurcation Chaos 19 (2009) 3813–3822. S. Hadef, A. Boukabou, Control of multi-scroll Chen system, J. Franklin Inst. 351 (2014) 2728–2741. D. Sadaoui, A. Boukabou, N. Merabtine, M. Benslama, Predictive synchronization of chaotic satellites systems, Exp. Syst. Appl. 38 (2011) 9041–9045. R. Barboza, Dynamics of a hyperchaotic Lorenz system, Int. J. Bifurcation Chaos 17 (2007) 4285–4294. T. Gao, Z. Chen, Z. Yuan, G. Chen, A hyperchaos generated from Chen’s system, Int. J. Mod. Phys. C 17 (2006) 471–478. Z. Chen, Y. Yang, G. Qi, Z. Yuan, A novel hyperchaos-system only with one equilibrium, Phys. Lett. A 360 (2007) 696–701. X. Wang, M. Wang, A hyperchaos generated from Lorenz system, Physica A 387 (2008) 3751–3758. N. Yujun, W. Xingyuan, W. Mingjun, Z. Huaguang, A new hyperchaotic system and its circuit implementation, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3518–3524. S. Pang, Y. Liu, A new hyperchaotic system from the Lü system and its control, J. Comput. Appl. Math. 235 (2011) 2775–2789. X. Liu, X. Shen, H. Zhang, Multi-scroll chaotic and hyperchaotic attractors generated from Chen system, Int. J. Bifurcation Chaos 22 (2012) 1250033. L.J. Ontañón-García, E. Jiménez-Ló pez, E. Campos-Cantón, M. Basin, A family of hyperchaotic multi-scroll attractors in Rn , Appl. Math. Comput. 233 (2014) 522–533. A.M.A. El-Sayed, H.M. Nour, A. Elsaid, A.E. Matouk, A. Elsonbaty, Circuit realization, bifurcations, chaos and hyperchaos in a new 4D, Appl. Math. Comput. 239 (2014) 333–345.