Synchronization of chaotic Liouvillian systems

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

Synchronization of chaotic Liouvillian systems Dulce M.G. Corona-Fortunio, Rafael. Martínez–Guerra, Juan L. Mata–Machuca Department of Automatic Control, CINVESTAV-IPN, México D.F., México E-mail: {dcorona, rguerra, jamata }@ctrl.cinvestav.mx

Abstract— In this paper we deal with the synchronization of the Chua oscillator, which is considered as a chaotic Liouvillian system. The synchronization problem is treated as an observation problem. The results of this work are based on a differential algebraic approach, which are used in order to determine observability with the measurements from the system, this strategy consists of proposing a polynomial observer (slave system) which tends to follow exponentially the chaotic oscillator (master system). Index Terms— Synchronization, chaotic Liouvillian systems, exponential polynomial observer, Chua's oscillator.

I. I NTRODUCTION In the past years, synchronization of chaotic systems problem has received a great deal of attention among scientists in many elds [1]-[3], in particular, secure communications, biological systems, chemical reactions, etc., [6], [9], [10].In general, synchronization research has been focused on the following areas: nonlinear observers [4], [11], feedback controllers [12], [13], nonlinear backstepping control [14], time delayed systems [15], [16], directional and bidirectional linear coupling [17], [18], adaptive control [19], [20], adaptive observers [21], [22], sliding mode observers [23], impulsive control [24], active control [25], among others. Since Pecora and Carroll's observation on the possibility of synchronizing two chaotic systems [1] (so-called driveresponse con guration), several synchronization schemes have been developed [4]-[6]. Synchronization can be classi ed into mutual synchronization (or bidirectional coupling) [7] and master-slave synchronization (or unidirectional coupling) [1], [8]. There are many methods to solve the synchronization problem since the control theory perspective. This work considers the master-slave synchronization problem via an exponential polynomial observer (EPO) based on differential an algebraic techniques [26]-[28]. Differential and algebraic concepts allow us to establish an algebraic observability condition, and therefore they provide a rst step for the construction of an algebraic observer. An observable system in this sense can be regarded as a system whose state variables can be expressed in terms of the input and output variables and a nite number of their time derivatives. Thus, chaos synchronization problem can be posed as an observer design procedure, where the coupling signal is viewed as output and the slave system is regarded as observer. The main characteristic is that the coupling signal is unidirectional, that is, the signal is transmitted from the master system (Chua's circuit) to the slave system (EPO), the slave is requested to recover the unknown state trajectories of the master. The strategy consists of proposing an EPO which exponentially reconstructs the unknown states of Chua's system.

IEEE Catalog Number: CFP11827-ART ISBN: 978-1-4577-1013-1 978-1-4577-1013-1/11/$26.00 ©2011 IEEE

127

In this paper the Chua's oscillator is viewed as a chaotic system with some Liouvillian properties [28], [26], refereed as Chaotic Liouvillian Oscillator (CLO). The Liouvillian character of the system (if a variable can be obtained by the adjunction of integrals or exponentials of integrals) is used as an observability criterion, that is to say, by this property we can know whether a variable can be reconstructed with the measurable output. The present paper is organized as follows: in section II we give some de nitions about differential-algebraic approach and Liouvillian systems. In section III is treated the synchronization problem and its solution by means of a polynomial observer.In section IV presents the numerical results applied to the oscillator chua. Finally, section V presents the conclusions of this work. II. D EFINITIONS Some basic de nitions are introduced in this section. De nition 1: (Algebraic Observability Condition–AOC). Let us consider a nonlinear dynamical system with input u, output y, and state vector x = (x1 ; x2 ; :::; xn )T : A state variable xi 2 R is said to be algebraically observable if it is algebraic over R hu; yi1 ,that is to say, xi satis es a differential algebraic polynomial in terms of fu; yg and some of their time derivatives, i.e., :

:

Pi (xi ; u; u; :::; y; y; :::) = 0;

i 2 f1; 2; :::; ng

(1)

with coef cients in R hu; yi Example 1: Considering the following nonlinear system x_ 1 = u + x2 x1 x_ 2 = x1 x_ 3 = x2 + ax1 If we de ne y = x2 , then x1 = y_ x2 = y x_ 3 = y + ay_

(2)

(3)

The previous system is not algebraically observable since x3 cannot be expressed as a differential algebraic polynomial in terms of fu; yg. For that reason, we present the next de nition. De nition 2 (Liouvillian system): A dynamical system is said to be Liouvillian if the elements (for example, state variables or parameters) can be obtained by an adjunction of integrals or exponentials of integrals of elements of R. 1 Rhu; yi denotes the differential eld generated by the eld R, the input u, the measurable output y, and the time derivatives of u and y.


Example 2: We consider the nonlinear system (2). From (3) we can observe that, although x3 does not satisfy the AOC we can obtain it by means of the integral Z x3 = (y + ay) _ then the nonlinear system (2) is Liouvillian.

III. P ROBLEM FORMULATION AND MAIN RESULT If we consider the following chaotic Liouvillian system x_ = Ax + (x) + (u) y = Cx

(4)

De nition 3: (Exponential Observer). An exponential observer for (4) is a system with state x ^ such that x ^k

k exp(

t)

(6)

where k and are positive constants. In this work we will use the master-slave synchronization, in this type of synchronization x can be considered as the state variable of the master system, and x ^ can be considered as the state variable of the slave system. the master-slave synchronization problem can be solved by designing an observer for (4) We will solve the synchronization problem by using a exponential polynomial observer based upon the Lyapunov method [35]. To this end, we rst compute the dynamics of the synchronization error (difference between the master and the slave systems). Next, by means of a simple quadratic Lyapunov function, we prove the exponential convergence. System (4) is assumed to be a chaotic Liouvillian system, then by de nition 2 all states of (4) can be reconstructed. In this sense, we will propose the following observation scheme. The observer structure. The observer for system (4) has the next form m X x b = A^ x + (b x) + (u) + Ki (y C x b)2i 1 (7) i=1

n

n

where x b 2 R ;and Ki 2 R , for 1

i

m:

Remark 1: The meaning of m can be understood as follows. As it is well known, an Extended Luenberger observer can be


Observer Convergence Analysis. In order to prove the observer convergence, we analyze the observer error which is de ned as e = x x b. From eqs. (4) and (7), the dynamics of the state estimation error is given by m X e_ = (A K1 C)e + (e) Ki (y C x b)2i 1 (8) i=1

where x 2 Rn , is the state vector; u 2 Rl , is the input vector, l n; y 2 R is the measured output; ( ) : Rl ! Rn is an input dependent vector function; A 2 Rnxn and C 2 R1xn are constants; and ( ) : Rn ! R is an state dependent nonlinear vector function. We restrict each ( ) to be nondecreasing, that is, for all a; b 2 R, a > b, it satis es the following monotone sector condition i (a) i (b) ; i = 1; :::; n (5) 0 a b The following observer's de nition shows the relation between the observers for nonlinear systems and chaos synchronization.

kx

seen as a rst order Taylor series around the observed state, therefore to improve the estimation performance high order terms are included in the observer structure. In other words, the rate of convergence can be increased by injecting additional terms with increasing powers of the output error.

128

where (e) = (x) (b x): It follows from (5) that each component of (e) satis es i (ei )

0

; 8ei = 6 0 (9) ei this implies a relationship between (e) and e as follows, eT (e) =

n X

ei i (ei ) =

i=1

n X

e2i

i=1

i (ei )

ei

and using (9) we have the following condition 0

eT (e)

(10)

Properties (9) and (10) will allow us prove that the state estimation error e(t) decays exponentially. We have the main result. Proposition 1: Consider the chaotic Liouvillian system (4) and the observer (7). If there exists a matrix P = P T > 0,and scalars " > 0, "1 > 0 satisfying the linear matrix inequality (LMI) (A

K1 C)T P + P (A P "1 I

K1 C) + "I

P

"1 I 0

0 (11)

and min (Mi

+ MiT )

0; i = 2; :::; m

(12)

with Mi := P Ki C:Then, there exist positive constants k and such that, for all t 0 ke(t)k k exp( t) q " C ke(0)k, = , B = with k = B 2C max (P )

min (P ),

and C =

Proof: We proposed the following Lyapunov function candidate V = eT P e. From (8), the time derivative of V is V_

= eT [(A K1 C)P + P (A m X 2 (Ce)2i 2 eT Mi e

K1 C)]e + 2eT P (e)

i=2

= eT [(A K1 C)P + P (A K1 C)]e + 2eT P (e) m X (Ce)2i 2 eT Mi + MiT e i=2


Using KVL and KCL, the equations that describe the nonlinear dynamics of Chua's circuit are as follows:

From (11) and (12) we have V_

"eT e

2"1 eT (e)

by properties (9) and (10) we have 2

V_

" kek

(13) 2

We write the Lyapunov function as V Rayleigh-Ritz inequality we have that 2

B kek

2

= kekp , then by 2

kekp

dvc1 = G(vc2 vc1 ) g(vc1 ) (16) dt dvc2 c2 = G(vc1 vc2 ) + iL dt diL L = vc2 dt 1 and g( ) is a nonincreasing function de ned where G = R by: c1

C kek

(14)

where B = min (P ) and C = max (P ) 2 R+ (because P is positive de nite). By using (14) we obtain the following upper bound of (13) " 2 kekp (15) V_ C 2

Taking the time derivative of V = kekp and replacing in inequality (15), we obtain

1 g(vc1 ) = m0 vc1 + (m1 2

m0 ) fjvc1 + Bp j

jvc1

Bp jg (17) This function is shown graphically in Fig. 2 the slopes in the inner and outer regions are m0 and m1 respectively; Bp denote the breakpoints. The nonlinear resistor NR is termed voltage-controlled because the current in the element is a function of the voltage across its terminals.

" kekp 2C

d kekp dt Finally, the result follows with ke(t)k q C where k = B ke(0)k, and

k exp( =

t)

" 2C :

IV. N UMERICAL RESULTS Chua's circuit has become a paradigm for learning, understanding and studying nonlinear dynamics and chaos [29]. In recent years, a lot of modi ed Chua's circuits were developed [30], [31]; the applications of Chua's circuit were investigated well, especially in utilizing synchronization of Chua's circuit to realize secure communication and utilizing chaos in Chua's circuit to describe practical chaotic systems.

Fig. 2:Three-segment piecewise-linear v-i characteristic of the resistor in Chua`s circuit. The outer regions have slopes m0 , the inner region has slope m1 . There are two breakpoints at Bp .

In the rst reported study of this circuit, Matsumoto [33] showed by computer simulation that the system possesses a strange attractor called the Double Scroll. Experimental conrmation of the presence of this atractor was made shortly afterwards in [34]. If we consider the measured output y = vc2 from equations of (16) we obtain:

The circuit shown in Fig 1 contains three linear energystorage elements (an inductor, and two capacitors), a linear resistor, and a single nonlinear resistor NR .

vc1 vc2 iL

1 C2 y_ + y + G LG = y Z 1 ydt = L =

Z

ydt

(18)

From (18), the Chua's system (16) is Liouvillian. This implies that unknown variables vc1 and iL can be reconstructed with the selected output y = vc2 . Chua's system (16) is of the form (4) with (u) = 0, 3 2 g(x1 ) 3 2 G G 0 C1 C1 C1 G 1 5 5 (x) = 4 A = 4 CG2 0 C2 C2 1 0 0 0 L C

0

1

0

x=

vc1

vc2

iL

T

Since g(x1 ) is nonincreasing and C1 is a positive constant, then 1 (x) = g(x1 )=C1 is nondecreasing as in (5) and

Fig. 1: Chua's circuit


=

129


C2

C2 1 L

0 2 3 m k1;i X 4 k2;i 5 + i=1 k3;i

C2

0

0

7

6

5

4

3

V

2 (x) = 3 (x) = 0 are nondecreasing as in (5) and Chua's system is Liouvillian, so that, we proceed with the observer design. If we apply (9) to the system (16) we get the following 3 2 g(vc ) 3 2 G G 1 0 C1 C1 C1 G 1 5 5 x ^+4 x ^ = 4 G 0

2

0

1

0

e

1

0

2i 1

Master System Slave System -1

0

0.001

0.002

+

G ^2 C1 x 3

g(^ x1 ) C1

+ k1;1 e1;2

+k1;2 (e1;2 ) + ::: + k1;m (e1;2 )2m x ^2 = x ^3 =

3

1

0.006

0.007

0.008

0.009

0.01

x 10

-3

1

G ^2 + C12 + k2;1 e1;2 C2 x +k2;2 (e1;2 )3 + ::: + k2;m (e1;2 )2m 1 G ^1 C2 x

0

(19)

-1

1 ^3 Lx

+ k3;1 e1;2 +k3;2 (e1;2 )3 + ::: + k3;m (e1;2 )2m

0.005 time (seconds)

2

A

G ^1 C1 x

0.004

Fig. 4: Synchronization of x1 .

Hence, the state observer is rewritten as, x ^1 =

0.003

-2 -3 -4

1

-5

Figure 3 shows the general diagram of the synchronization of Chua's circuit (18) and the exponential observer (19) in masterslave con guration.

-6 -7

Master System Slave System 0

0.001

0.002

0.003

0.004


0.006

0.007

0.008

0.009

0.01

Fig. 5: Synchronization of x3 .

The performance of the proposed observer is evaluated by means of the relative error which in this case is de ned as ei =

;

i = 1; 3:

Figs. 6-7 illustrate the corresponding relative errors, it should be noted that e1 = 0 and e3 = 0 for t > 0:006s, as was expected. Finally.

and K2 is chosen such that (12) is satis ed, then we obtain 2 3 2 3 k1;2 7:1573 K2 = 4 k2;2 5 = 4 14:1040 5 : k3;2 0:0268

0.5 0.45 0.4 0.35 Relative error x1

Numerical simulations for the synchronization of Chua's system are carried out in order to show the performance of the exponential observer. The parameter values considered in the numerical simulations correspond to chaotic behavior [32] and these are: C1 = 10 nF, C2 = 100 nF, R = 1:8 kA, L = 18 mH, m0 = 0:409 mS, m1 = 0:756 mS and Bp = 1:08 V. The Matlab-Simulinkr program uses the Dormand–Prince integration algorithm, with the integration step set to 1 10 5 . We x m = 2 in the observer (19). The LMI (11) is feasible with " = 0:001 and F1 = 0:001, a solution is 2 3 0:0008 0:0006 0:1021 0:0005 0:0805 5 P = 4 0:0006 15:0959 2 0:1021 3 2 0:0805 3 k1;1 1:5 K1 = 4 k2;1 5 = 4 0:5 5 ; k3;1 45

0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.005

0.01


0.02

0.025

0.03

g. 6: Relative error e1 0.8 0.7 0.6 Relative error x3

Fig. 3: Synchronization diagram.

0.5 0.4 0.3 0.2 0.1 0 0

0.005

0.01


0.02

0.025

0.03

g. 7: Relative error e3

Figs. 4-5 show the obtained results for the initial conditions x1 = 0:5, x2 = 1, x3 = 0, x ^1 = 1; x ^2 = 0:5 and x ^3 = 0:002:


jxi x ^i j jxi j

130

Fig. 9 presents the synchronization in a phase diagram, where clearly is observed the chaotic behavior of the Chua's circuit.


-3

x 10

x1: 1 x2: 0.5 x3: 0.002

2 x1: -0.5 x2: 1 x3: 0

Master System Slave System

x

3

0

-2

-4 2 1 0 -1 -2

-1

0

1

2

3

4

5

x

Fig. 8: Phase diagram

V. C ONCLUSIONS The synchronization problem of chaotic Liouvillian systems has been treated by using differential and algebraic techniques. We proposed a polynomial observer, and by means of properties of nondecreasing functions, linear matrix inequalities and with the help of the Lyapunov method we proved that the estimation error exponentially converges to zero. This observer has been used as a slave system whose states are synchronized with the chaotic system (Chua's circuit). A reduced set of measurable state variables were needed to achieve the synchronization with this approach. R EFERENCES [1] L.M. Pecora, T.L. Carroll, "Synchronization in chaotic systems", Physical Review Letters 64, 821–824, 1990. [2] O. Morgul, & E. Solak, "Observer based synchronization of chaotic systems", Phys. Rev. E 54, 4803– 4811, 1996. [3] U. Parlitz, L. O. Chua, L. Kocarev, K. S. Halle, & A. Shang, “Transmission of digital signals by chaotic synchronization", Int. J. Bifurcation and Chaos 2, 973–977, 1992. [4] H. Nijmeijer, I.M.Y. Mareels, "An observer looks at synchronization", IEEE Trans. Circuits Syst. I 44, 882–890, 1997. [5] M. Feki, "Observer-based exact synchronization of ideal and mismatched chaotic systems", Phys. Lett. A 309, 53–60, 2003. [6] A. Fradkov, "Cybernetical physics: from control of chaos to quantum control, Springer", Berlin, 2007. [7] Y. Ushio, "Synthesis of synchronized chaotic systems based on observers", Int. J. Bifurcat. Chaos 9, 541–546, (1999). [8] T.L. Carroll, L.M. Pecora, "Synchronizing chaotic circuits", IEEE Trans. Circuits Syst. I 38, 453–456, 1991. [9] R. Martínez-Guerra, and J. J. Rincón Pasaye, "Synchronization and anti-synchronization of chaotic systems:A differential and algebraic approach", Chaos Solitons and Fractals, vol. 28, pp. 511–517, 2009. [10] M. Chen, D. Zhou and Y. Shang, "A sliding mode observer based secure communication scheme", Chaos, Solitons Fractals, vol. 25, pp. 573-578, 2005. [11] E. Cherrier, M. Boutayeb, J. Ragot, "Observers-based synchronization and input recovery for a class of nonlinear chaotic models", IEEE Trans. Circuits Syst. I 53, 1977–1988, 2006. [12] Y. Ushio, "Synthesis of synchronized chaotic systems based on observers", Int. J. Bifurcat. Chaos 9, 541–546, 1999. [13] F. Wang, C. Liu, "A new criterion for chaos and hyperchaos synchronization using linear feedback control", Phys. Lett. A 360, 274–278, 2006. [14] C. Wang, S. Ge, "Adaptive backstepping control of uncertain Lorenz system", Int. J. Bifurcat. Chaos 11, 1115–1119, 2001.


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