A new asymptotic polynomial observer to synchronization problem

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

A #ew Asymptotic Polynomial Observer to Synchronization Problem J. L. Mata1, R. Martínez-Guerra1, R. Aguilar2 1

2

Department of Automatic Control, CINVESTAV-IPN, Mexico D.F., Mexico Department of Biotechnology and Bioengineering, CINVESTAV-IPN, Mexico D.F., Mexico Phone (55) 57473733 Fax (55) 57473982 E-mail: [email protected]

Abstract –– In this paper, we consider the synchronization problem via nonlinear observer design. A new asymptotic polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Lorenz and Rössler systems) by numerical simulation.

synchronization occurs which means that the difference of master and slave state vectors converges to zero for t → ∞ . The problem of observer design naturally arises in a system approach, as soon as one needs unmeasured internal information from external measurements. In general indeed, it is clear that one cannot use as many sensors as signals of interest characterizing the system behavior for technological constraints, cost reasons, and so on, especially since such signals can come in a quite large number, and they can be of various types: they typically include parameters, timevarying signals characterizing the system (state variables), and unmeasured external disturbances. The design of observers for nonlinear systems is a challenging problem (even for accurately known systems) that has received a considerable amount of attention. Since the observers developed by Kalman [9] and Luenberger [10] several years ago for linear systems, different state observation techniques have been proposed to handle the systems nonlinearities. A first category of techniques consists in applying linear algorithms to the system linearized around the estimated trajectory. These are known as the extended Kalman and Luenberger observers. Alternatively, the nonlinear dynamics are split into a linear part and a nonlinear one. The observer gains are then chosen large enough so that the linear part dominates the nonlinear one. Such observers are known as high-gain observers [11], [12]. In a third approach the nonlinear system is transformed into a linear one by an appropriate change of coordinates [13]. The estimate is computed in these new coordinates and the original coordinates are recovered through the inverse transformation. In most approaches, nonlinear coordinate transformations are employed to transform the nonlinear system into a block triangular observer canonical form. Then, high gain [14] or sliding mode observers [15], [16] can be designed. In this paper the synchronization scheme is proposed for a class of Lipschitz nonlinear systems. Many problems in engineering and other applications are globally Lipschitz for instance the sinusoidal terms in robotics. Nonlinearities which are square or cubic in nature are not globally Lipschitz, however, they are locally so, moreover when such functions occur in physical systems, they frequently have a saturation in their growth rate, making them globally Lipschitz functions [17]. Thus, this class of systems covered by this note is fairly general. References [17], [18], [19] established existence conditions of the full-order observers for Lipschitz nonlinear systems. The main purpose in this work is to extend these results by showing that the stability

Keywords –– #onlinear systems, Riccati equation, state observers, synchronization.

I. INTRODUCTION The growing interest in synchronization was probably caused in 1990 by the paper by Pecora and Carrol [1] where, among others, secure communication as a potential application has been indicated. In the last years, synchronization of chaotic systems problem has received a great deal of attention among scientist in many fields, for instance in [2], [3]. It is well known that study of the synchronization problem for nonlinear systems has been very important for nonlinear science, in particular the applications to biology, medicine, cryptography, secure data transmission and so on. In general, the synchronization research has been focused onto two areas. The first one relates with the employ of state observers, where the main applications lies on the synchronization of nonlinear oscillators [4], [5], [6], [7]. The second one is the use of control laws, which allows achieve the synchronization with different structure and order between nonlinear oscillators [2], [8]. A particular interest is the connection between the observers for nonlinear systems and chaos synchronization, which is also known as master- slave configuration [1]. Thus, chaos synchronization problem can be regarded as observer design procedure, where the coupling signal is viewed as output and the slave system is the observer. The purpose of this paper is to address the synchronization problem from a control theory perspective. More specifically, the paper addresses the synchronization problem from the perspective of nonlinear observer design. The phenomenon of synchronization also occurs for unidirectionally coupled systems and in this case the driven system (slave or receiver system) may be viewed as a nonlinear observer of the driving system (master or transmitter system). In this configuration, the two coupled systems are (almost) identical and therefore identical

IEEE Catalog Number:CFP09827 ISBN: 978-1-4244-4689-6 Library of Congress: 2009904789 978-14244-4689-6/09/$25.00 ©2009 IEEE

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

conditions given in [19] also guarantee the existence of a full-order observer with a high-order correction term. The main contribution of this paper consists in the solution of the synchronization problem via an asymptotic polynomial observer. The obtained state space estimation error is shown to be bounded, and this bound depends on observer’s gain and a Lipchitz constant. This communication presents some fundamental insights into polynomial observer design for the class of Lipschitz nonlinear systems, that it means, that any autonomous nonlinear system of the form x& = f (x, u ) can be regarded as Lipschitz continuous system with respect to x , with a Lipchitz constant L . The intention of choosing examples as Lorenz and Rössler system is to clarify the proposed methodology. However, it is worth to mention that this technique can be applied to almost any chaotic synchronization problem. In what follows, an asymptotic polynomial observer is proposed as well as an easy numerical design is given. Numerical results show its satisfactory performance. Finally we close this paper with some concluding remarks.

B. Observer design

Proposition 1. For any initial conditions, the following nonlinear dynamic system is a full order state observer of the system (1b) m

•

xˆ = Axˆ + Ψ (xˆ , u ) +

∑K

i

[y − C xˆ ]2 i −1

(2)

i =1

In proposition 1 the following assumptions should be considered A1. K1 can be chosen such as the following Algebraic Riccati Equation (ARE) has a symmetric positive-definite

solution P ∈ ℜ n × n for some ε > 0

( A − K1C )T P + P( A − K1C ) + L2 PP + I + ε I = 0 A2. λmin (P Ki C ) ≥ 0 , i ∈ {2, 3, K , m}

In

[

xˆ ∈ ℜ n ,

(2),

K i = k1,i

k 2, i

(3) (4)

K k n ,i

]T ∈ ℜ n ,

i ∈ {1, 2, K , m} . II. ASYMPTOTIC POLYNOMIAL OBSERVER

C. Stability analysis A. Problem Statement Consider the following nonlinear system:

x& = f (x, u ) y = Cx where

x ∈ ℜn

(1a)

is the vector of the state variables;

f (o ) : ℜ × ℜ → ℜ , (m ≤ n) is a nonlinear smooth vector function and Lipschitz in x and uniformly bounded n

m

To prove that system (2) is an observer for (1b) we analyze the estimation error (difference between real states and their estimated) employing Lyapunov stability theory. Let us define the estimation error as e := x − xˆ . The corresponding dynamic of the estimation error is m

e& = A e −

n

in u, y ∈ ℜ is the vector of measured states. Any nonlinear system of the form (1a) can be expressed in the form (1b) as long as f (x, u ) is differentiable with respect to x . x& = A x + Ψ (x, u )

(1b)

i

(C e )2 i −1 ]+ [Ψ (x, u ) − Ψ (xˆ, u )]

(5)

i =1

Equation (5) is written alternatively as m

e& = ( A − K 1C )e −

∑ [K

i

(C e )2 i −1 ]+ [Ψ (x, u ) − Ψ (xˆ, u )]

(6)

i=2

Consider the Lyapunov function candidate V = eT Pe , where 0 < P = P T and satisfies (3). Its derivative is

y = C x , x0 = x(t 0 ) In this paper, we always assume that the pair ( A, C ) is observable. In system (1b), Ψ (x, u ) is a nonlinear vector function which satisfies the Lipschitz condition with a Lipschitz constant L , i.e,

Ψ ( x, u ) − Ψ (xˆ , u ) ≤ L x − xˆ

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∑ [K

V& = e&T Pe + eT Pe&

[

]

= e T ( A − K 1C )T P + P ( A − K 1C ) e m

−2

∑ [ (C e )

]

e PK i Ce + 2e T P[Ψ (x, u ) − Ψ (xˆ, u )] (7)

2 i −2 T

i=2

In [17] is presented the next inequality as a lemma which is useful for this proof,

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

2eT P[Ψ (x, u ) − Ψ (xˆ, u )] ≤ L2 eT PP e + eT e

The system (10) may be written in the form given by (1b),

From Rayleigh inequality [20], and taking into account A2, we have

(

)

− e T PK i Ce ≤ −λ min PK i C e

2

where i ∈ {2, 3, K , m} . Equation (7) leads to

[ ] − 2 ∑ [ (C e ) λ (PK C )] e + L e PP e + e = e [( A − K C ) P + P( A − K C ) + L PP + I ]e − 2 ∑ [ (C e ) λ (PK C )] e 2

2 i −2

min

2 T

T

i

T

e

2

1

1

m

2

2 i −2

min

−1 a 0

0  −1    0  , C = [1 0 0] 0  , Ψ= b + x1 x 3  − c 

the corresponding pair ( A, C ) is observable. By choosing the observer’s gains appropriately, the observer given by (2) may achieve local synchronization. According to proposition 1, we get the following equations system (slave system) as the observer,

i =2 T

 0 A =  1  0

x3 ]T ,

x2

It can easily be shown that with the selection of y = x1 ,

V& ≤ e T ( A − K 1C )T P + P ( A − K 1C ) e m

where x = [x1

i

(8)

i =2

From A2, the second term in the right hand side of (8) always will be positive or zero, then

x&ˆ1 = − (xˆ 2 + xˆ 3 ) + x&ˆ 2 = xˆ1 + a xˆ 2 +

m

∑k

1,i

i =1 m

∑k

2, i

[y − C xˆ ]2 i −1

[y − C xˆ ]2 i − 1

(11)

i =1

[

]

V& ≤ e T ( A − K1C )T P + P( A − K 1C ) + L2 PP + I e

(9)

x&ˆ 3 = b + xˆ 3 (xˆ1 − c ) +

m

∑k

3,i

[y − C xˆ ]2 i −1

i =1

According to A1, since

ε > 0 , it is clear that

( A − K1C )T P + P( A − K1C ) + L2 PP + I < 0 .

Hence, V& < 0 . This implies that system (2) is an observer for system (1b) and the corresponding dynamic of the estimation error (5) is asymptotically stable. III. APLICATION TO SYNCHRONIZATION OF CHAOTIC SYSTEMS To illustrate our methodology, we give two applications to chaotic systems. In fact, these are applications to the socalled Rössler system [21] which presents a chaotic behavior and exhibits the simplest possible strange attractor, and the well known Lorenz chaotic system [22]. A. Example 1: Rössler system We consider the popular nonlinear Rössler system, which is described by x&1 = −( x2 + x3 ) x& 2 = x1 + ax2

(10)

y = x1 It is well known that in a large neighborhood of { a = b =0.2, c =5} this system has a chaotic behavior. Remark 1. It is not difficult to prove that above system is Lipschitz.

68

k 2, i

k 3,i

]T ∈ ℜ 3 ,

i ∈ {1, 2, K , m} and

a , b, c > 0 . Now, to illustrate the effectiveness of the proposed approach, some numerical simulations are presented. The design of the full-order observer presented in this paper is based on the solution of the Riccati equation which can be obtained by using the Matlab function ARE. We have chosen the values for the Rössler system (10) and the observer (11) as a = b = 0.2 , c = 5 , and the observer’s gains have been taken as K1 = [ 5 − 5 5]T and

K 2 = [10 10 10]T , K 3 = [10 10 10]T . All simulations results in this paper were carried out with the help of Matlab 7.1 Software with Simulink 6.3 as the toolbox. In this work, the performance index of the corresponding synchronization process was calculated as [12] J (t ) =

x&3 = b + x3 (x1 − c )

IEEE Catalog Number:CFP09827 ISBN: 978-1-4244-4689-6 Library of Congress: 2009904789 978-14244-4689-6/09/$25.00 ©2009 IEEE

[

where, K i = k1,i

1 t + 0.001

t

∫ e(t )

2 dτ Q0

(12)

0

where e(t ) denotes the estimation error and Q0 = I . Fig. 1 shows the convergence of the estimated states (slave system) to the real states (master system), without any noise in the system output. The initial conditions are x1 = −0.5 , x2 = 0.5 , x3 = 4 , xˆ1 = −4 , xˆ 2 = 3 , xˆ3 = −4 . It should be noted that the rate of convergence is faster for m =3.

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

Rössler

10

0

1

5

x

1

x

5

-5

15 10 Time(seconds)

5

(a)

(a)

Master m=1 m=2 m=3

0

15 10 Time(seconds)

5

(b)

Master m=1 m=2 m=3

-10 0

20

15 10 Time(seconds)

5

Rössler

10

5

-5 0

0

15

x

3

x

20

0

(c)

Rössler

Master m=1 m=2 m=3

Rössler

15

10

20

15 10 Time(seconds)

5

-5

3

-10 0

-10 0

5

-5

(b)

0

10

x

2

x

20 Rössler

10

5

Master m=1 m=2 m=3

-5

2

-10 0

Rössler

10

Master m=1 m=2 m=3

Master m=1 m=2 m=3

5

0

15 10 Time(seconds)

5

-5 0

20

Fig. 1. Synchronization between system (10) and its observer (11),

x1 and xˆ1 ; (b) signals x 3 and xˆ 3 .

without any noise in the system output, (a) signals

x2

and

xˆ 2 ; (c) signals

Fig. 2. Synchronization between system (10) and its observer (11), with

and

Perf ormance i ndex

(a)

Perf ormance i ndex

Let us consider the Lorenz chaotic system described by the following set of differential equations,

(b)

(13)

and

xˆ1 ; (b) signals x 2

xˆ 2 ; (c) signals x 3 and xˆ 3 . Rössler

m=1 m=2 m=3

1.5

1

0.5

0 0

5

15 10 Time(seconds)

20

Rössler

2

B. Example 2: Lorenz system

m=1 m=2 m=3

1.5

1

0.5

0 0

5

15 10 Time(seconds)

20

Fig. 3. Quadratic estimation error, (a) without any noise in the system output; (b) with white noise in the system output.

y = x1

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x1

2

x3 = 4 , xˆ1 = −4 , xˆ 2 = 3 , xˆ3 = −4 . We can see that synchronization is possible, i.e., the estimated states tend to the real states. In Fig. 3 is illustrated the performance index given by (12) for the corresponding synchronization process. It should be noted that the quadratic estimation error (performance index) is bounded on average and has a tendency to decrease. Clearly, we can see that the proposed observer is robust against noisy measurements.

x& 3 = x1 x 2 − β x 3

20

15 10 Time(seconds)

white noise in the system output, (a) signals

Now, the effect of noise in the measurements is analyzed. In Fig. 2 are presented the numerical results when a noise is added in the system output (white noise with σ = 0.1 , ±10% around the current value of the system output). The initial conditions are x1 = −0.5 , x2 = 0.5 ,

x&1 = α (x 2 − x1 ) x& 2 = ρ x1 − x 2 − x1 x 3

5

(c)

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

with positive parameters ( α , ρ , β > 0 ) system (13) exhibits chaotic behaviour.

Lorenz

25

Master m=1 m=2 m=3

20 15

1

Remark 2. It is not difficult to prove that system (13) is Lipschitz.

x

10 5 0 -5

The system (13) may be written in the form given by (1b), where x = [x1

x2

− α A =  ρ  0

-10

x3 ]T ,

-15 0

0  0   − 1 0  , Ψ =  − x1 x3  , C = [1 0 0]  x1 x2  0 − β 

α

4

5

Master m=1 m=2 m=3

20

2

x

15 10 5 0 -5 -10 -15 0

1

(b)

3 2 Time(seconds)

4

5

Lorenz 50

Master m=1 m=2 m=3

40

3

30

k1,i [ y − C xˆ ]2 i − 1

x

∑

3

Lorenz 30

one may obtain a stable matrix ( A − K1C ) and use the observer given by (2) for synchronization of chaos. According to proposition 1, we get the following equations system (slave system) as the observer, m

2

Time(seconds)

25

With the selection of y = x1 , the corresponding pair ( A, C ) is detectable, hence, by an appropriate choice of K1,

xˆ&1 = α (xˆ 2 − xˆ1 ) +

1

(a)

20

i =1

xˆ& 2 = ρ xˆ1 − xˆ 2 − xˆ1 xˆ 3 +

10

m

∑k

2, i

[y − C xˆ ]2 i −1

(14)

0

i =1

xˆ& 3 = xˆ1 xˆ 2 − β xˆ 3 +

m

∑k

-10 0

(c)

[y − C xˆ ]

2 i −1

3,i

[

k 2,i

k 3,i

]T ∈ ℜ 3 ,

i ∈ {1, 2, K , m} and

α, ρ, β > 0 . We show some simulations for the Lorenz system. The parameter values for system (13) and its observer (14) are taken as α = 10, ρ = 28, β = 8 / 3 . The observer’s gains have been fixed as K1 = [10 10 10]T , K 2 = [10 10 10]T , and

K 3 = [ 5 5 5]T . Fig. 4 shows the convergence of the estimated states (slave system) to the real states (master system), without any noise in the system output. The initial conditions are x1 = 1 , x 2 = 0 , x3 = −5 , xˆ1 = 4 , xˆ 2 = −5 , xˆ 3 = 8 . It should be noted that the rate of convergence is faster for m =3. To analyze the effect of noise in the measurements a noise is added in the system output as in example 1. In Fig. 5 are presented the numerical results when a noise is added in the system output (white noise with σ = 0.1 , ±10% around the current value of the system output). The initial conditions are x1 = 1 , x 2 = 0 , x3 = −5 , xˆ1 = 4 , xˆ 2 = −5 , xˆ 3 = 8 . We can see that synchronization is still possible, i.e., the estimated states tend to the real states.

IEEE Catalog Number:CFP09827 ISBN: 978-1-4244-4689-6 Library of Congress: 2009904789 978-14244-4689-6/09/$25.00 ©2009 IEEE

3 2 Time(seconds)

4

5

Fig. 4. Synchronization between system (13) and its observer (14), without any noise in the system output, (a) signals x1 and xˆ1 ; (b) signals

i =1

where, K i = k1,i

1

70

x 2 and xˆ 2 ; (c) signals x 3 and xˆ 3 . In Fig. 6 is illustrated the performance index given by (12) for the corresponding synchronization process, without any noise system output and with noise in the system output (white noise with σ = 0.1 , ±10% around the current value of the measured output). We can see that the quadratic estimation error (performance index) is bounded on average and has a tendency to decrease. Clearly, we can see that the proposed observer is robust against noisy measurements.

IV. CONCLUSION In this paper, we have designed a new asymptotic polynomial observer (high order polynomial type) for a class of nonlinear oscillators to attack the synchronization problem. Also, we have proven the asymptotic stability of the resulting state estimation error and by means of simple algebraic manipulations we construct the observer (slave system). Finally, we have presented some simulations to illustrate the effectiveness of the suggested approach, which shows some robustness properties against noisy measurements.

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2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2009) (Formerly known as ICEEE) Toluca, México. November 10-13, 2009

Lorenz

25

3.5

1

x

10 5 0 -5 -10

(a)

-15 0

1

m=1 m=2 m=3

3 Perf or mance i ndex

15

Lorenz

4

Master m=1 m=2 m=3

20

3 2 Time(seconds)

4

5

2.5 2 1.5 1

(a)

0.5 0 0

Lorenz

50

30 Master m=1 m=2 m=3

25 20

Perf ormance i ndex

2

x

3

5 0 -5 -10 1

3 2 Time(seconds)

4

5

2.5 2 1.5 1

Lorenz 50

(b)

Master m=1 m=2 m=3

40

3

x

100 Time(seconds)

150

200

[8] R. Femat, G. Solís-Perales, Robust synchronization of chaotic systems via feedback, Springer Verlag, 2008. [9] R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng., Series D, vol. 82, pp. 35-45, 1960. [10] D. Luenberger, “An introduction to observers,” IEE Trans. Aut. Control, vol. 16, pp. 596-602, 1971. [11] R. Aguilar, R. Martínez-Guerra, R. Maya-Yescas, “State Estimation for Partially Unknown Nonlinear Systems: A Class of Integral High Gain Observers,” IEE Proc. Control Theory App., vol. 150, pp. 240-4, 2003. [12] R. Martínez-Guerra, A. Poznyak, V. Díaz, “Robustness of highgain observers for closed-loop nonlinear systems: theoretical study and robotics control application,” Int. J. Systems Science, vol. 31, pp. 1519-1529, 2000. [13] H. Keller, “Non-linear observer design by transformation into a generalized observer canonical form,” Int. J. Control, vol. 46, pp. 1915-1930, 1987. [14] J. Gauthier, H. Hammouri, S. Othman, “A simple observer for nonlinear systems. Applications to bioreactors”. IEEE Trans. Aut. Control, vol. 37, pp. 875-880, 1992. [15] Y. Xiong, M. Saif, “Sliding mode observer for nonlinear uncertain systems,” IEEE Trans. Aut. Control, vol. 46, pp. 20122017, 2001. [16] R. Martinez-Guerra, W. Yu, “Chaotic synchronization and secure communication via sliding mode observer,” Int. J. Bif. Chaos, vol. 18, no. 1, pp. 235-243, 2008. [17] S. Raghavan, J. Hedrick, “Observer design for a class of nonlinear systems,” Int. J. Control, vol. 59, pp. 515-528, 1994. [18] F. E. Thau, “Observing the states of nonlinear dynamic systems,” Int. J. Control, vol. 17, no. 3, pp. 471-9, 1994. [19] R. Rajamani, “Observers for Lipschitz nonlinear systems,” IEEE Trans. Aut. Control, vol. 43, no.3, pp. 397-401, 1998. [20] R. Horn, C. Johnson, Matrix analysis, Cambridge University Press, New York, 1985, pp. 176. [21] O. Rössler, “An Equation for Continuous Chaos,” Phys. Lett., vol. 57 A, pp. 397-398, 1976. [22] E. Lorenz, “Deterministic nonperiodic flow,” J. Atm. Sci., vol. 20, pp. 130-141, 1963.

0 -10 0

50

Fig. 6. Quadratic estimation error, (a) without any noise in the system output; (b) with white noise in the system output.

20 10

(c)

0.5 0 0

30

200

m=1 m=2 m=3

3.5

10

-15 0

150

Lorenz

15

(b)

100 Time(seconds)

4

1

3 2 Time(seconds)

4

5

Fig. 5. Synchronization between system (13) and its observer (14), with

x1 and xˆ1 ; (b) signals x 2 and xˆ 2 ; (c) signals x 3 and xˆ 3 .

white noise in the system output, (a) signals

ACKNOWLEDGMENT J. L. Mata thanks the support of CONACYT for a postgraduate scholarship.

REFERENCES [1] L. Pecora, T. Caroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, pp. 821-4, 1990. [2] A. Fradkov, Cybernetical physics: from control of chaos to quantum control, Springer-Verlag, Berlin, 2007. [3] M. Chen, D. Zhou, Y. Shang, “A sliding mode observer based secure communication scheme,” Chaos, Solitons Fractals, vol. 25, pp. 573-8, 2005. [4] C. Hua, X. Guan, “Synchronization of chaotic systems based on PI observer design,” Phys. Lett. A, vol. 334, pp. 382-9, 2005. [5] R. Martínez-Guerra, J. Cruz, R. Gonzalez, R. Aguilar, “A new reduced-order Observer design for the synchronization of Lorenz systems,” Chaos, Solitons Fractals, vol. 28, pp. 511-7, 2006. [6] O. Morgül, E. Solak, “Observed based synchronization of chaotic systems,” Phys. Rev. E, vol. 54, pp. 4803–4811, 1996. [7] M. Feki, “Observer-based exact synchronization of ideal and mismatched chaotic systems,” Phys. Lett. A, vol. 309, pp. 53-60, 2003

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