Synchronization of chaotic Liouvillian systems: An application to ...
Applied Mathematics and Computation 219 (2013) 10934–10944
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Synchronization of chaotic Liouvillian systems: An application to Chua’s oscillator Rafael Martínez-Guerra a, Dulce M.G. Corona-Fortunio a, Juan L. Mata-Machuca a,b,⇑ a
Departamento de Control Automático, Cinvestav-IPN, A.P. 14-740, Av. IPN 2508, 07360 D.F., Mexico Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Academia de Mecatrónica, Av. IPN 2580, 07340 D.F., Mexico b
a r t i c l e
i n f o
Keywords: Chaotic Liouvillian systems Chua’s circuit Exponential polynomial observer Numerical simulations
a b s t r a c t In this paper we deal with the synchronization of chaotic oscillators with Liouvillian properties (chaotic Liouvillian system) based on nonlinear observer design. The strategy consists of proposing a polynomial observer (slave system) which tends to follow exponentially the chaotic oscillator (master system). The proposed technique is applied in the synchronization of Chua’s circuit using Matlab-SimulinkÒ and MatlabÒ-LMI programs. Simulation results are used to visualize and illustrate the effectiveness of Chua’s oscillator in synchronization. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Since Pecora and Carroll’s observation on the possibility of synchronizing two chaotic systems [1] (so-called drive-response configuration), several synchronization schemes have been developed [2–5]. Synchronization can be classified into mutual synchronization (or bidirectional coupling) [6] and master–slave synchronization (or unidirectional coupling) [1,7]. In the past years, synchronization of chaotic systems problem has received a great deal of attention among scientists in many fields [8–11]. As it is well known, the study of the synchronization problem for nonlinear systems has been very important from the nonlinear science point of view, in particular, the applications to biology, medicine, cryptography, secure data transmission and so on [12,13]. In general, synchronization research has been focused on the following areas: nonlinear observers [2,14–17], nonlinear control [18], feedback controllers [5], nonlinear backstepping control [19], time delayed systems [12,20], directional and bidirectional linear coupling [21], adaptive control [9], adaptive observers [22,23], sliding mode observers [13,24], active control [25], among others. 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 first 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 finite 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. ⇑ Corresponding author at. Departamento de Control Automático, Cinvestav-IPN, A.P. 14-740, Av. IPN 2508, 07360 D.F., Mexico E-mail addresses: [email protected] (R. Martínez-Guerra), [email protected] (D.M.G. Corona-Fortunio), [email protected] (J.L. Mata-Machuca). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.056
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Chua’s circuit is a nonlinear electronic chaotic oscillator. This circuit is easily constructed [29] and has been employed in a variety of applications [30], e.g. communication systems [31]. The chaotic associative memory architecture proposed in [32] the use of a network of Chua’s oscillators coupled via piecewise linear conductances. In this paper the Chua’s oscillator is viewed as a chaotic system with some Liouvillian properties [26,28], refered 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 exploited as an observability criterion, that is to say, by this property we can know whether a variable can be reconstructed with the measurable output. This paper is organized as follows: In Section 2 we give some definitions about differential-algebraic approach and Liouvillian systems. In Section 3 we treated the synchronization problem and its solution by means of an exponential polynomial observer. In Section 4 we presented the synchronization of the Chua’s circuit [33] and we show some numerical simulations. Finally, in Section 5 we close the paper with some concluding remarks.
2. Definitions We start with some basic definitions for the understanding of Liouvillian systems, the following definitions are presented. Definition 1 (Algebraic Observability Condition – AOC). Let us consider a nonlinear dynamical system with input u, output y, and state vector x ¼ ðx1 ; x2 ; . . . ; xn ÞT . A state variable xi 2 R is said to be algebraically observable if it is algebraic over Rhu; yi,1 that is to say, xi satisfies a differential algebraic polynomial in terms of fu; yg and some of their time derivatives, i.e.,
_ . . . ; y; y; _ . . .Þ ¼ 0; Pi ðxi ; u; u;
i 2 f1; 2; . . . ; ng;
ð1Þ
with coefficients in Rhu; yi. Example 1. Consider the following nonlinear system
x_ 1 ¼ x2 þ x23 ; x_ 2 ¼ x3 ; x_ 3 ¼ u;
ð2Þ
If we define y ¼ x2 , then
x2 ¼ y; _ x3 ¼ y;
ð3Þ
x_ 1 ¼ y þ y_ 2 ; The above system is not algebraically observable since x1 cannot be expressed as a differential algebraic polynomial in terms of fu; yg. Motivated by this fact, we present the next definition. Definition 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.
Example 2. We consider the nonlinear system as in Example 1. From (3) we can observe that, although x1 does not satisfy the AOC we can obtain it by means of the integral
x1 ¼
Z
ðy þ y_ 2 Þ;
Therefore the nonlinear system (2) is Liouvillian. Example 3. Consider the following nonlinear second order system that models a prey–predator situation,
x_ 1 ¼ x1 x2 x1 þ k; x_ 2 ¼ x1 x2 x2 ; y ¼ x1 ; so, we have: 1
Rhu; yi denotes the differential field generated by the field R, the input u, the measurable output y, and the time derivatives of u and y.
ð4Þ
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y_ ¼ yx2 y þ k; x_ 2 ¼ yx2 x2 ;
ð5Þ
we can solve directly (5), and obtain:
y_ ¼ yx2 y þ k; Z x2 ¼ exp ðy þ 1Þdt ; we obtain the parameter k to be:
Z k ¼ y_ exp ðy þ 1Þdt þ y and we say that system (4) is Liouvillian. For further information we recommend to see [26,28]. 3. Problem formulation and main result Let us consider the following chaotic Liouvillian system,
x_ ¼ Ax þ wðxÞ þ uðxÞ þ fðuÞ;
ð6Þ
y ¼ Cx;
where x 2 Rn , is the state vector2; u 2 Rl , is the input vector, l 6 n; y 2 R is the measured output; fðÞ : Rl ! Rn is an input dependent vector function; A 2 Rnn and C 2 R1n are constants; and wðÞ : Rn ! Rn , uðÞ : Rn ! Rn are state dependent nonlinear vector functions. We restrict each wi ðÞ to be nondecreasing, that is, for all a; b 2 R; a > b, it satisfies the following monotone sector condition
06
wi ðaÞ wi ðbÞ ; ab
i ¼ 1; . . . ; n:
ð7Þ
In the same manner, we restrict each ui ðÞ to be nonincreasing, that is, for all a; b 2 R; a > b, it satisfies
ui ðaÞ ui ðbÞ ab
6 0;
i ¼ 1; . . . ; n:
ð8Þ
To show the relation between the observers for nonlinear systems and chaos synchronization we give the observer’s definition. Definition 3 (Exponential Observer). An exponential observer for (6) is a system with state ^ x such that
kx ^xk 6 j expðntÞ; where j and n are positive constants. In the context of master–slave synchronization, x can be considered as the state variable of the master system, and ^ x can be viewed as the state variable of the slave system. Hence, the master–slave synchronization problem can be solved by designing an observer for (6). In what follows, we will solve the synchronization problem by using a exponential polynomial observer based upon the Lyapunov method [34]. To this end, we first 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 (6) is assumed to be a chaotic Liouvillian system, then by Definition 2 all states of (6) can be reconstructed. In this sense, we will propose an observer scheme. The observer structure. The observer for system (6) has the next form
b x þ wðb x Þ þ uðb x Þ þ fðuÞ þ x_ ¼ Ab
m X 2i1 Ki y Cb ; x i¼1
where ^ x 2 Rn , and K i 2 Rn , for 1 6 i 6 m. 2
Mathematically, chaotic systems are characterized by local instability and global boundedness of the trajectories, i.e. kxðtÞk is bounded for all t P 0.
ð9Þ
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Remark 1. The meaning of m can be understood as follows. As it is well known, an Extended Luenberger observer can be seen as a first 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. Observer convergence analysis. In order to prove the observer convergence, we analyze the observer error which is defined as e ¼ x ^ x. From Eqs. (6) and (9), the dynamics of the state estimation error is given by
e_ ¼ ðA K 1 CÞe þ /ðeÞ þ qðeÞ
m X K i ðCeÞ2i1 ;
ð10Þ
i¼2
where /ðeÞ :¼ wðxÞ wð^ xÞ and qðeÞ :¼ uðxÞ uð^ xÞ. It follows from (7) that each component of /ðeÞ satisfies
06
/i ðei Þ ; ei
8ei – 0;
ð11Þ
which implies a relationship between /ðeÞ and e as follows,
eT /ðeÞ ¼
n X
ei /i ðei Þ ¼
i¼1
n X / ðei Þ e2i i ; ei i¼1
by using (11) we have the following condition
0 6 eT /ðeÞ:
ð12Þ
By a similar analysis, from (8) we have
eT qðeÞ 6 0:
ð13Þ
Properties (12) and (13) will allow us prove that the state estimation error eðtÞ decays exponentially. We have the main result. Proposition 1. Consider the chaotic Liouvillian system (6) and the observer (9). If there exists a matrix P ¼ P T > 0, and scalars e > 0; 1 > 0, 2 > 0 satisfying the linear matrix inequality (LMI)
2 6 4
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P þ 1 I
P 2 I
P þ 1 I
0
0
P 2 I
0
0
3 7 560
ð14Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m;
ð15Þ
with M i :¼ PK i C. Then, there exist positive constants j and n such that, for all t P 0,
keðtÞk 6 j expðntÞ; where
j¼
qffiffi b e akeð0Þk, n ¼ 2ba ¼ kmin ðPÞ, and b ¼ kmax ðPÞ.
Proof. We use the following Lyapunov function candidate V ¼ eT Pe. From (10), the time derivative of V is m h i X V_ ¼ eT ðA K 1 CÞT P þ PðA K 1 CÞ e þ 2eT P/ðeÞ þ 2eT PqðeÞ 2 ðCeÞ2i2 eT M i e i¼2
h
i
¼ eT ðA K 1 CÞT P þ PðA K 1 CÞ e þ 2eT P/ðeÞ þ 2eT PqðeÞ
m X
ðCeÞ2i2 eT ðMi þ MTi Þe
i¼2
and, in view of (14) and (15),
V_ 6 eeT e 21 eT /ðeÞ þ 22 eT qðeÞ: By properties (12) and (13) we have
V_ 6 ekek2 :
ð16Þ
We write the Lyapunov function as V ¼ 2
akek 6
kek2P
kek2P ,
then by Rayleigh–Ritz inequality we have that
2
ð17Þ
6 bkek ; þ
where a :¼ kmin ðPÞ, and b :¼ kmax ðPÞ 2 R (because P is positive definite).
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By using (17) we obtain the following upper bound of (16)
e V_ 6 kek2P : b
ð18Þ
Taking the time derivative of V ¼ kek2P and replacing in inequality (18), we obtain
d e kekP 6 kekP : dt 2b Finally, the result follows with
keðtÞk 6 j expðntÞ;
qffiffi e. where j ¼ abkeð0Þk, and n ¼ 2b
ð19Þ h
Corollary 1. Let us consider wðÞ 0. Then system (9) is an exponential observer of system (6) if there exists a matrix P ¼ P T > 0, and scalars e > 0; 2 > 0 satisfying
"
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P 2 I
P 2 I
0
# 60
ð20Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m;
ð21Þ
With j and n defined as in Proposition 1. Proof. The result is proven as in Proposition 1. h Corollary 2. Let us consider uðÞ 0. Then system (9) is an exponential observer of system (6) if there exists a matrix P ¼ P T > 0, and scalars e > 0; 1 > 0 satisfying the LMI
"
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P þ 1 I
P þ 1 I
0
# 60
ð22Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m:
ð23Þ
With j and n defined as in Proposition 1. Proof. It follows directly from the procedure in Proposition 1. h
4. Numerical results In this section we consider the synchronization of a Chua’s system considered as a chaotic Liouvillian oscillator. Chua’s circuit [29], shown in Fig. 1, is a simple oscillator circuit which exhibits a variety of bifurcations and chaos. The circuit contains three linear energy-storage elements (an inductor, and two capacitors), a linear resistor, and a single nonlinear resistor NR . The state equations for the Chua’s circuit are as follows:
dv C1 ¼ Gðv C2 v C 1 Þ gðv C 1 Þ; dt dv C2 ¼ Gðv C1 v C 2 Þ þ iL ; C2 dt diL L ¼ v C 2 ; dt
C1
ð24Þ
where G ¼ 1R and gðÞ is a nonincreasing function defined by:
1 gðv R Þ ¼ m0 v R þ ðm1 m0 Þ v R þ Bp v R Bp : 2
ð25Þ
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Fig. 1. Chua’s circuit, with parameter values C 1 ¼ 10 nF, C 2 ¼ 100 nF, R ¼ 1:8 kX; L ¼ 18 mH, m0 ¼ 0:409 mS, m1 ¼ 0:756 mS and Bp ¼ 1:08 V.
Fig. 2. Three-segment piecewise-linear v-i characteristic of the linear resistor in Chua’s circuit. The outer regions have slopes m0 , the inner region has slope m1 . There are two breakpoints at Bp .
Fig. 3. Synchronization diagram.
This relation is shown graphically in Fig. 2, the slopes in the inner and outer regions are m0 and m1 respectively, with m1 < m0 < 0; Bp denote the breakpoints. The nonlinear resistor N R is termed voltage-controlled because the current in the element is a function of the voltage across its terminals. In the first reported study of this circuit, Matsumoto [29] showed by computer simulation that the system possesses a strange attractor called the Double Scroll. Experimental confirmation of the presence of this attractor was made shortly afterwards in [35]. Since then, the system has been studied extensively; a variety of bifurcation phenomena and chaotic attractors in the circuit have been discovered experimentally and confirmed mathematically [36]. In what follows we consider the measured output y ¼ v c2 . From equations of (24) we obtain:
C2 1 y_ þ y þ 1 LG G v c2 ¼ y; Z 1 ydt; iL ¼ L
vc
¼
Z
ydt; ð26Þ
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Fig. 4. Synchronization of
vc
1
.
Fig. 5. Synchronization of
vc
2
.
From (26), the Chua’s system (24) is Liouvillian. This implies that unknown variables v C 1 and iL can be reconstructed with the selected output y ¼ v C 2 . Chua’s system (24) is of the form (6) with fðuÞ ¼ 0; uðÞ ¼ 0,
2 6 A¼6 4
CG1
G C1
G C2
CG2
0
1L
3
2
7 1 7 ; C2 5 0
6 wðxÞ ¼ 4
0
1Þ gðx C1
0
3 7 5;
C ¼ ½ 0 1 0 ;
x¼
vC
1
vC
2
iL
T
:
0
Since gðx1 Þ is nonincreasing and C 1 is a positive constant, then w1 ðxÞ ¼ w1 ðx1 Þ ¼ gðx1 Þ=C 1 is nondecreasing as in (7). Indeed, w2 ðxÞ ¼ w3 ðxÞ ¼ 0 also satisfy property (7), and Chua’s system (24) is Liouvillian, so that, we proceed with the observer design. Taking into account that uðÞ ¼ 0, we will use conditions in Corollary 2 to obtain the observer gains. Using LMI software, observer gains are computed to drive the estimation error to zero. Applying (9), we have the observer for Chua’s system (24),
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10941
Fig. 6. Synchronization of iL .
Fig. 7. Quadratic synchronization errors of
2 6 b x_ ¼ 6 4
CG1
G C1
G C2
CG2
0
1L
vc
1
and
vc
2
.
3
2 gð^x Þ 3 2 3 k1;i C 11 X 7 m 6 7 6 7 2i1 1 7^ xþ4 0 5þ 4 k2;i 5ð½ 0 1 0 eÞ : C2 5 i¼1 k3;i 0 0 0
Hence, the state observer is rewritten as,
G G gð^x1 Þ ^x_ 1 ¼ ^x1 þ ^x2 þ k1;1 e1;2 þ k1;2 ðe1;2 Þ3 þ þ k1;m ðe1;2 Þ2m1 ; C1 C1 C1 G G 1 ^x_ 2 ¼ ^x1 ^x2 þ ^x3 þ k2;1 e1;2 þ k2;2 ðe1;2 Þ3 þ þ k2;m ðe1;2 Þ2m1 ; C2 C2 C2 1 ^x_ 3 ¼ ^x3 þ k3;1 e1;2 þ k3;2 ðe1;2 Þ3 þ þ k3;m ðe1;2 Þ2m1 ; L
ð27Þ
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Fig. 8. Quadratic synchronization error of iL .
Fig. 9. Phase Diagram.
Fig. 3 shows the general diagram of the synchronization of Chua’s circuit (24) and the exponential observer (27) in master–slave configuration. 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 [33] and these are: C 1 ¼ 10 nF, C 2 ¼ 100 nF, R ¼ 1:8 kX; L ¼ 18 mH, m0 ¼ 0:409 mS, m1 ¼ 0:756 mS and Bp ¼ 1:08 V. The Matlab-SimulinkÒ program uses the Dormand–Prince integration algorithm, with the integration step set to 1 105 . We fix m ¼ 2 in the observer (27). The LMI (22) is feasible with e ¼ 0:001 and 1 ¼ 0:001, a solution is
2
0:0008
6 P ¼ 4 0:0006 0:1021
0:0006 0:0005 0:0805
0:1021
3
7 0:0805 5; 15:0959
2
k1;1
3
2
1:5
3
6 7 6 7 K 1 ¼ 4 k2;1 5 ¼ 4 0:5 5 k3;1
and K 2 is chosen such that (23) is satisfied, then we obtain
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2
k1;2
3
2
7:1573
10943
3
6 7 6 7 K 2 ¼ 4 k2;2 5 ¼ 4 14:1040 5: 0:0268
k3;2
Figs. 4–6 show the obtained results for the initial conditions x1 ¼ x2 ¼ x3 ¼ 0; ^ x1 ¼ 1; ^ x2 ¼ 0:5 and ^ x3 ¼ 0:002, the synchronization results achieved with the polynomial observer are good. The performance index (quadratic synchronization error) of the corresponding synchronization process is calculated as [37]
JðtÞ ¼
1 t þ 0:001
Z 0
t
jeðtÞj2Q 0 ;
Q 0 ¼ I;
Figs. 7 and 8 illustrate the performance index, which has a tendency to decrease. Finally, Fig. 9 presents the synchronization in a phase diagram, where clearly is observed the chaotic behavior of the Chua’s circuit. 5. Concluding remarks 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 and nonincreasing 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 exponentially synchronized with the chaotic system (Chua’s circuit). A reduced set of measurable state variables were needed to achieve the synchronization with this approach. The effectiveness of the suggested methodology was shown by means of numerical simulations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. H. Nijmeijer, I.M.Y. Mareels, An observer looks at synchronization, IEEE Trans. Circuits Syst. I (44) (1997) 882–890. M. Feki, Observer-based exact synchronization of ideal and mismatched chaotic systems, Phys. Lett. A 309 (2003) 53–60. O. Morgül, M. Feki, A chaotic masking scheme by using synchronized chaotic systems, Phys. Lett. A 251 (1999) 169–176. A. Fradkov, Cybernetical Physics: From Control of Chaos to Quantum Control, Springer, Berlin, 2007. Y. Ushio, Synthesis of synchronized chaotic systems based on observers, Int. J. Bifurcation Chaos 9 (1999) 541–546. T.L. Carroll, L.M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst. I (38) (1991) 453–456. O. Morgül, E. Solak, Observer based synchronization of chaotic systems, Phys. Rev. E 54 (1996) 4803–4811. C. Aguilar-Ibañez, R. Martínez-Guerra, R. Aguilar-López, J.L. Mata-Machuca, Synchronization and parameter estimations of an uncertain Rikitake system, Phys. Lett. A 374 (2010) 3625–3628. Xiaoshan Zhao, Zhenbo Li, Shuang Li, Synchronization of a chaotic finance system, Appl. Math. Comput. 217 (2011) 6031–6039. M. Boutayeb, M. Darouach, H. Rafaralahy, Generalized state observers for chaotic synchronization and secure communication, IEEE Trans. Circuits Syst. I (49) (2002) 345–349. R. Martínez-Guerra, J.L. Mata-Machuca, R. Aguilar-López, A. Rodríguez-Bollain, Chaotic synchronization and its applications in secure communications, in: S. Banerjee, M. Mitra, L. Rondoni (Eds.), Applications Chaos and Nonlinear Dynamics in Engineering, Understanding Complex Systems, vol. 1, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 231–272 (Chapter 8). R. Martínez-Guerra, W. Yu, Chaotic communication and secure communication via sliding-mode observer, Int. J. Bifurcation Chaos 18 (2008) 235–243. J. Mata-Machuca, R. Martínez-Guerra, R. Aguilar-López, An exponential polynomial observer for synchronization of chaotic systems, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 4114–4130. R. Martínez-Martínez, J.L. Mata-Machuca, R. Martínez-Guerra, J.A. León, G. Fernández-Anaya, Synchronization of nonlinear fractional order systems, Appl. Math. Comput. 218 (2011) 3338–3347. J.L. Mata-Machuca, R. Martínez-Guerra, Asymptotic synchronization of the Colpitts oscillator, Comput. Math. Appl. 63 (2012) 1072–1078. Zhengqiang Zhang, Hanyong Shao, Zhen Wang, Hao Shen, Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems, Appl. Math. Comput. 218 (2012) 7614–7621. E. Elabbasy, H. Agiza, M. El-Dessoky, Global chaos synchronization for four scroll attractor by nonlinear control, Sci. Res. Essay 1 (2006) 65–71. C. Wang, S. Ge, Adaptive backstepping control of uncertain Lorenz system, Int. J. Bifurcation Chaos 11 (2001) 1115–1119. D. Ghosh, A. Chowdhury, P. Saha, On the various kinds of synchronization in delayed Duffing–Van der Pol system, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 790–803. G. Grassi, D.A. Miller, Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal, Appl. Math. Comput. 218 (2012) 6118–6124. H.B. Fotsin, J. Daafouz, Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification, Phys. Lett. A 339 (2005) 304–315. J.L. Mata-Machuca, R. Martínez-Guerra, Ricardo Aguilar-López, C. Aguilar-Ibañez, A chaotic system in synchronization and secure communications, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1706–1713. M. Ayati, H. Khaloozadeh, Stable chaos synchronisation scheme for nonlinear uncertain systems, IET Control Theory Appl. 4 (2010) 437–447. Yaoyao Lan, Qingguo Li, Chaos synchronization of a new hyperchaotic system, Appl. Math. Comput. 217 (2010) 2125–2132. R. Aguilar-López, J.L. Mata-Machuca, R. Martínez-Guerra, Observability and Observers for Nonlinear Dynamical Systems: Nonlinear Systems Analysis, LAMBERT Academic Publishing, Germany, 2011, ISBN 978-3-8454-3171-0. R. Martínez-Guerra, J.J. Rincón-Pasaye, Synchronization and anti-synchronization of chaotic systems: a differential and algebraic approach, Chaos Solitons Fract. 28 (2009) 511–517. R. Martínez-Guerra, J. Mendoza-Camargo, Observers for a class of Liouvillian and, non-differentially flat systems, IMA J. Math. Control Inf. 21 (2004) 493–509. T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Trans. Circuits Syst. 31 (12) (1984) 1055–1058. L.0. Chua, Chua’s circuit: 10 years later, IEICE Trans. Fundam. E77-A (1994) 1811–1822. G. Kolumban, G. Kis, Performance evaluation of FM-DCSK modulation scheme, in: Proc. of the 1998 Int. Svmposium on Nonlinear Theory and its Applications (NOLTA’98), vol. 1, Crans-Montana, Switzerland, 1998, pp. 81–84. S. Jankowski, A. Londei, C. Mazur, A. Lozowski, Synchronization and association in a large network of coupled Chua circuits, Int. J. Electron. 79 (1995) 823–828.
10944 [33] [34] [35] [36] [37]
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M.P. Kennedy, Robust op amp realization of Chua’s circuit, Frequenz 46 (3–4) (1992) 66–68. H. Khalil, Nonlinear Systems, third ed., Prentice-Hall, Englewood Cliff, NJ, 2002. G.Q. Zhong, F. Ayron, Experimental confirmation of chaos Chua’s circuit, Int. J. Circuit Theory Appl. 13 (11) (1985) 93–98. L.O. Chua, M. Komuro, T. Matsumoto, The double scroll family, parts I and II, IEEE Trans. Circuits Syst. 33 (11) (1986) 1073–1118. R. Martínez-Guerra, A. Poznyak, V. Díaz, Robustness of high-gain observers for closed-loop nonlinear systems: theoretical study and robotics control application, Int. J. Syst. Sci. 31 (2000) 1519–1529.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Synchronization of chaotic Liouvillian systems: An application to Chua’s oscillator Rafael Martínez-Guerra a, Dulce M.G. Corona-Fortunio a, Juan L. Mata-Machuca a,b,⇑ a
Departamento de Control Automático, Cinvestav-IPN, A.P. 14-740, Av. IPN 2508, 07360 D.F., Mexico Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Academia de Mecatrónica, Av. IPN 2580, 07340 D.F., Mexico b
a r t i c l e
i n f o
Keywords: Chaotic Liouvillian systems Chua’s circuit Exponential polynomial observer Numerical simulations
a b s t r a c t In this paper we deal with the synchronization of chaotic oscillators with Liouvillian properties (chaotic Liouvillian system) based on nonlinear observer design. The strategy consists of proposing a polynomial observer (slave system) which tends to follow exponentially the chaotic oscillator (master system). The proposed technique is applied in the synchronization of Chua’s circuit using Matlab-SimulinkÒ and MatlabÒ-LMI programs. Simulation results are used to visualize and illustrate the effectiveness of Chua’s oscillator in synchronization. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Since Pecora and Carroll’s observation on the possibility of synchronizing two chaotic systems [1] (so-called drive-response configuration), several synchronization schemes have been developed [2–5]. Synchronization can be classified into mutual synchronization (or bidirectional coupling) [6] and master–slave synchronization (or unidirectional coupling) [1,7]. In the past years, synchronization of chaotic systems problem has received a great deal of attention among scientists in many fields [8–11]. As it is well known, the study of the synchronization problem for nonlinear systems has been very important from the nonlinear science point of view, in particular, the applications to biology, medicine, cryptography, secure data transmission and so on [12,13]. In general, synchronization research has been focused on the following areas: nonlinear observers [2,14–17], nonlinear control [18], feedback controllers [5], nonlinear backstepping control [19], time delayed systems [12,20], directional and bidirectional linear coupling [21], adaptive control [9], adaptive observers [22,23], sliding mode observers [13,24], active control [25], among others. 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 first 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 finite 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. ⇑ Corresponding author at. Departamento de Control Automático, Cinvestav-IPN, A.P. 14-740, Av. IPN 2508, 07360 D.F., Mexico E-mail addresses: [email protected] (R. Martínez-Guerra), [email protected] (D.M.G. Corona-Fortunio), [email protected] (J.L. Mata-Machuca). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.056
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Chua’s circuit is a nonlinear electronic chaotic oscillator. This circuit is easily constructed [29] and has been employed in a variety of applications [30], e.g. communication systems [31]. The chaotic associative memory architecture proposed in [32] the use of a network of Chua’s oscillators coupled via piecewise linear conductances. In this paper the Chua’s oscillator is viewed as a chaotic system with some Liouvillian properties [26,28], refered 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 exploited as an observability criterion, that is to say, by this property we can know whether a variable can be reconstructed with the measurable output. This paper is organized as follows: In Section 2 we give some definitions about differential-algebraic approach and Liouvillian systems. In Section 3 we treated the synchronization problem and its solution by means of an exponential polynomial observer. In Section 4 we presented the synchronization of the Chua’s circuit [33] and we show some numerical simulations. Finally, in Section 5 we close the paper with some concluding remarks.
2. Definitions We start with some basic definitions for the understanding of Liouvillian systems, the following definitions are presented. Definition 1 (Algebraic Observability Condition – AOC). Let us consider a nonlinear dynamical system with input u, output y, and state vector x ¼ ðx1 ; x2 ; . . . ; xn ÞT . A state variable xi 2 R is said to be algebraically observable if it is algebraic over Rhu; yi,1 that is to say, xi satisfies a differential algebraic polynomial in terms of fu; yg and some of their time derivatives, i.e.,
_ . . . ; y; y; _ . . .Þ ¼ 0; Pi ðxi ; u; u;
i 2 f1; 2; . . . ; ng;
ð1Þ
with coefficients in Rhu; yi. Example 1. Consider the following nonlinear system
x_ 1 ¼ x2 þ x23 ; x_ 2 ¼ x3 ; x_ 3 ¼ u;
ð2Þ
If we define y ¼ x2 , then
x2 ¼ y; _ x3 ¼ y;
ð3Þ
x_ 1 ¼ y þ y_ 2 ; The above system is not algebraically observable since x1 cannot be expressed as a differential algebraic polynomial in terms of fu; yg. Motivated by this fact, we present the next definition. Definition 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.
Example 2. We consider the nonlinear system as in Example 1. From (3) we can observe that, although x1 does not satisfy the AOC we can obtain it by means of the integral
x1 ¼
Z
ðy þ y_ 2 Þ;
Therefore the nonlinear system (2) is Liouvillian. Example 3. Consider the following nonlinear second order system that models a prey–predator situation,
x_ 1 ¼ x1 x2 x1 þ k; x_ 2 ¼ x1 x2 x2 ; y ¼ x1 ; so, we have: 1
Rhu; yi denotes the differential field generated by the field R, the input u, the measurable output y, and the time derivatives of u and y.
ð4Þ
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y_ ¼ yx2 y þ k; x_ 2 ¼ yx2 x2 ;
ð5Þ
we can solve directly (5), and obtain:
y_ ¼ yx2 y þ k; Z x2 ¼ exp ðy þ 1Þdt ; we obtain the parameter k to be:
Z k ¼ y_ exp ðy þ 1Þdt þ y and we say that system (4) is Liouvillian. For further information we recommend to see [26,28]. 3. Problem formulation and main result Let us consider the following chaotic Liouvillian system,
x_ ¼ Ax þ wðxÞ þ uðxÞ þ fðuÞ;
ð6Þ
y ¼ Cx;
where x 2 Rn , is the state vector2; u 2 Rl , is the input vector, l 6 n; y 2 R is the measured output; fðÞ : Rl ! Rn is an input dependent vector function; A 2 Rnn and C 2 R1n are constants; and wðÞ : Rn ! Rn , uðÞ : Rn ! Rn are state dependent nonlinear vector functions. We restrict each wi ðÞ to be nondecreasing, that is, for all a; b 2 R; a > b, it satisfies the following monotone sector condition
06
wi ðaÞ wi ðbÞ ; ab
i ¼ 1; . . . ; n:
ð7Þ
In the same manner, we restrict each ui ðÞ to be nonincreasing, that is, for all a; b 2 R; a > b, it satisfies
ui ðaÞ ui ðbÞ ab
6 0;
i ¼ 1; . . . ; n:
ð8Þ
To show the relation between the observers for nonlinear systems and chaos synchronization we give the observer’s definition. Definition 3 (Exponential Observer). An exponential observer for (6) is a system with state ^ x such that
kx ^xk 6 j expðntÞ; where j and n are positive constants. In the context of master–slave synchronization, x can be considered as the state variable of the master system, and ^ x can be viewed as the state variable of the slave system. Hence, the master–slave synchronization problem can be solved by designing an observer for (6). In what follows, we will solve the synchronization problem by using a exponential polynomial observer based upon the Lyapunov method [34]. To this end, we first 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 (6) is assumed to be a chaotic Liouvillian system, then by Definition 2 all states of (6) can be reconstructed. In this sense, we will propose an observer scheme. The observer structure. The observer for system (6) has the next form
b x þ wðb x Þ þ uðb x Þ þ fðuÞ þ x_ ¼ Ab
m X 2i1 Ki y Cb ; x i¼1
where ^ x 2 Rn , and K i 2 Rn , for 1 6 i 6 m. 2
Mathematically, chaotic systems are characterized by local instability and global boundedness of the trajectories, i.e. kxðtÞk is bounded for all t P 0.
ð9Þ
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Remark 1. The meaning of m can be understood as follows. As it is well known, an Extended Luenberger observer can be seen as a first 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. Observer convergence analysis. In order to prove the observer convergence, we analyze the observer error which is defined as e ¼ x ^ x. From Eqs. (6) and (9), the dynamics of the state estimation error is given by
e_ ¼ ðA K 1 CÞe þ /ðeÞ þ qðeÞ
m X K i ðCeÞ2i1 ;
ð10Þ
i¼2
where /ðeÞ :¼ wðxÞ wð^ xÞ and qðeÞ :¼ uðxÞ uð^ xÞ. It follows from (7) that each component of /ðeÞ satisfies
06
/i ðei Þ ; ei
8ei – 0;
ð11Þ
which implies a relationship between /ðeÞ and e as follows,
eT /ðeÞ ¼
n X
ei /i ðei Þ ¼
i¼1
n X / ðei Þ e2i i ; ei i¼1
by using (11) we have the following condition
0 6 eT /ðeÞ:
ð12Þ
By a similar analysis, from (8) we have
eT qðeÞ 6 0:
ð13Þ
Properties (12) and (13) will allow us prove that the state estimation error eðtÞ decays exponentially. We have the main result. Proposition 1. Consider the chaotic Liouvillian system (6) and the observer (9). If there exists a matrix P ¼ P T > 0, and scalars e > 0; 1 > 0, 2 > 0 satisfying the linear matrix inequality (LMI)
2 6 4
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P þ 1 I
P 2 I
P þ 1 I
0
0
P 2 I
0
0
3 7 560
ð14Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m;
ð15Þ
with M i :¼ PK i C. Then, there exist positive constants j and n such that, for all t P 0,
keðtÞk 6 j expðntÞ; where
j¼
qffiffi b e akeð0Þk, n ¼ 2ba ¼ kmin ðPÞ, and b ¼ kmax ðPÞ.
Proof. We use the following Lyapunov function candidate V ¼ eT Pe. From (10), the time derivative of V is m h i X V_ ¼ eT ðA K 1 CÞT P þ PðA K 1 CÞ e þ 2eT P/ðeÞ þ 2eT PqðeÞ 2 ðCeÞ2i2 eT M i e i¼2
h
i
¼ eT ðA K 1 CÞT P þ PðA K 1 CÞ e þ 2eT P/ðeÞ þ 2eT PqðeÞ
m X
ðCeÞ2i2 eT ðMi þ MTi Þe
i¼2
and, in view of (14) and (15),
V_ 6 eeT e 21 eT /ðeÞ þ 22 eT qðeÞ: By properties (12) and (13) we have
V_ 6 ekek2 :
ð16Þ
We write the Lyapunov function as V ¼ 2
akek 6
kek2P
kek2P ,
then by Rayleigh–Ritz inequality we have that
2
ð17Þ
6 bkek ; þ
where a :¼ kmin ðPÞ, and b :¼ kmax ðPÞ 2 R (because P is positive definite).
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By using (17) we obtain the following upper bound of (16)
e V_ 6 kek2P : b
ð18Þ
Taking the time derivative of V ¼ kek2P and replacing in inequality (18), we obtain
d e kekP 6 kekP : dt 2b Finally, the result follows with
keðtÞk 6 j expðntÞ;
qffiffi e. where j ¼ abkeð0Þk, and n ¼ 2b
ð19Þ h
Corollary 1. Let us consider wðÞ 0. Then system (9) is an exponential observer of system (6) if there exists a matrix P ¼ P T > 0, and scalars e > 0; 2 > 0 satisfying
"
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P 2 I
P 2 I
0
# 60
ð20Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m;
ð21Þ
With j and n defined as in Proposition 1. Proof. The result is proven as in Proposition 1. h Corollary 2. Let us consider uðÞ 0. Then system (9) is an exponential observer of system (6) if there exists a matrix P ¼ P T > 0, and scalars e > 0; 1 > 0 satisfying the LMI
"
ðA K 1 CÞT P þ PðA K 1 CÞ þ eI
P þ 1 I
P þ 1 I
0
# 60
ð22Þ
and
kmin M i þ M Ti P 0;
i ¼ 2; . . . ; m:
ð23Þ
With j and n defined as in Proposition 1. Proof. It follows directly from the procedure in Proposition 1. h
4. Numerical results In this section we consider the synchronization of a Chua’s system considered as a chaotic Liouvillian oscillator. Chua’s circuit [29], shown in Fig. 1, is a simple oscillator circuit which exhibits a variety of bifurcations and chaos. The circuit contains three linear energy-storage elements (an inductor, and two capacitors), a linear resistor, and a single nonlinear resistor NR . The state equations for the Chua’s circuit are as follows:
dv C1 ¼ Gðv C2 v C 1 Þ gðv C 1 Þ; dt dv C2 ¼ Gðv C1 v C 2 Þ þ iL ; C2 dt diL L ¼ v C 2 ; dt
C1
ð24Þ
where G ¼ 1R and gðÞ is a nonincreasing function defined by:
1 gðv R Þ ¼ m0 v R þ ðm1 m0 Þ v R þ Bp v R Bp : 2
ð25Þ
R. Martínez-Guerra et al. / Applied Mathematics and Computation 219 (2013) 10934–10944
10939
Fig. 1. Chua’s circuit, with parameter values C 1 ¼ 10 nF, C 2 ¼ 100 nF, R ¼ 1:8 kX; L ¼ 18 mH, m0 ¼ 0:409 mS, m1 ¼ 0:756 mS and Bp ¼ 1:08 V.
Fig. 2. Three-segment piecewise-linear v-i characteristic of the linear resistor in Chua’s circuit. The outer regions have slopes m0 , the inner region has slope m1 . There are two breakpoints at Bp .
Fig. 3. Synchronization diagram.
This relation is shown graphically in Fig. 2, the slopes in the inner and outer regions are m0 and m1 respectively, with m1 < m0 < 0; Bp denote the breakpoints. The nonlinear resistor N R is termed voltage-controlled because the current in the element is a function of the voltage across its terminals. In the first reported study of this circuit, Matsumoto [29] showed by computer simulation that the system possesses a strange attractor called the Double Scroll. Experimental confirmation of the presence of this attractor was made shortly afterwards in [35]. Since then, the system has been studied extensively; a variety of bifurcation phenomena and chaotic attractors in the circuit have been discovered experimentally and confirmed mathematically [36]. In what follows we consider the measured output y ¼ v c2 . From equations of (24) we obtain:
C2 1 y_ þ y þ 1 LG G v c2 ¼ y; Z 1 ydt; iL ¼ L
vc
¼
Z
ydt; ð26Þ
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Fig. 4. Synchronization of
vc
1
.
Fig. 5. Synchronization of
vc
2
.
From (26), the Chua’s system (24) is Liouvillian. This implies that unknown variables v C 1 and iL can be reconstructed with the selected output y ¼ v C 2 . Chua’s system (24) is of the form (6) with fðuÞ ¼ 0; uðÞ ¼ 0,
2 6 A¼6 4
CG1
G C1
G C2
CG2
0
1L
3
2
7 1 7 ; C2 5 0
6 wðxÞ ¼ 4
0
1Þ gðx C1
0
3 7 5;
C ¼ ½ 0 1 0 ;
x¼
vC
1
vC
2
iL
T
:
0
Since gðx1 Þ is nonincreasing and C 1 is a positive constant, then w1 ðxÞ ¼ w1 ðx1 Þ ¼ gðx1 Þ=C 1 is nondecreasing as in (7). Indeed, w2 ðxÞ ¼ w3 ðxÞ ¼ 0 also satisfy property (7), and Chua’s system (24) is Liouvillian, so that, we proceed with the observer design. Taking into account that uðÞ ¼ 0, we will use conditions in Corollary 2 to obtain the observer gains. Using LMI software, observer gains are computed to drive the estimation error to zero. Applying (9), we have the observer for Chua’s system (24),
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10941
Fig. 6. Synchronization of iL .
Fig. 7. Quadratic synchronization errors of
2 6 b x_ ¼ 6 4
CG1
G C1
G C2
CG2
0
1L
vc
1
and
vc
2
.
3
2 gð^x Þ 3 2 3 k1;i C 11 X 7 m 6 7 6 7 2i1 1 7^ xþ4 0 5þ 4 k2;i 5ð½ 0 1 0 eÞ : C2 5 i¼1 k3;i 0 0 0
Hence, the state observer is rewritten as,
G G gð^x1 Þ ^x_ 1 ¼ ^x1 þ ^x2 þ k1;1 e1;2 þ k1;2 ðe1;2 Þ3 þ þ k1;m ðe1;2 Þ2m1 ; C1 C1 C1 G G 1 ^x_ 2 ¼ ^x1 ^x2 þ ^x3 þ k2;1 e1;2 þ k2;2 ðe1;2 Þ3 þ þ k2;m ðe1;2 Þ2m1 ; C2 C2 C2 1 ^x_ 3 ¼ ^x3 þ k3;1 e1;2 þ k3;2 ðe1;2 Þ3 þ þ k3;m ðe1;2 Þ2m1 ; L
ð27Þ
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Fig. 8. Quadratic synchronization error of iL .
Fig. 9. Phase Diagram.
Fig. 3 shows the general diagram of the synchronization of Chua’s circuit (24) and the exponential observer (27) in master–slave configuration. 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 [33] and these are: C 1 ¼ 10 nF, C 2 ¼ 100 nF, R ¼ 1:8 kX; L ¼ 18 mH, m0 ¼ 0:409 mS, m1 ¼ 0:756 mS and Bp ¼ 1:08 V. The Matlab-SimulinkÒ program uses the Dormand–Prince integration algorithm, with the integration step set to 1 105 . We fix m ¼ 2 in the observer (27). The LMI (22) is feasible with e ¼ 0:001 and 1 ¼ 0:001, a solution is
2
0:0008
6 P ¼ 4 0:0006 0:1021
0:0006 0:0005 0:0805
0:1021
3
7 0:0805 5; 15:0959
2
k1;1
3
2
1:5
3
6 7 6 7 K 1 ¼ 4 k2;1 5 ¼ 4 0:5 5 k3;1
and K 2 is chosen such that (23) is satisfied, then we obtain
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2
k1;2
3
2
7:1573
10943
3
6 7 6 7 K 2 ¼ 4 k2;2 5 ¼ 4 14:1040 5: 0:0268
k3;2
Figs. 4–6 show the obtained results for the initial conditions x1 ¼ x2 ¼ x3 ¼ 0; ^ x1 ¼ 1; ^ x2 ¼ 0:5 and ^ x3 ¼ 0:002, the synchronization results achieved with the polynomial observer are good. The performance index (quadratic synchronization error) of the corresponding synchronization process is calculated as [37]
JðtÞ ¼
1 t þ 0:001
Z 0
t
jeðtÞj2Q 0 ;
Q 0 ¼ I;
Figs. 7 and 8 illustrate the performance index, which has a tendency to decrease. Finally, Fig. 9 presents the synchronization in a phase diagram, where clearly is observed the chaotic behavior of the Chua’s circuit. 5. Concluding remarks 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 and nonincreasing 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 exponentially synchronized with the chaotic system (Chua’s circuit). A reduced set of measurable state variables were needed to achieve the synchronization with this approach. The effectiveness of the suggested methodology was shown by means of numerical simulations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. H. Nijmeijer, I.M.Y. Mareels, An observer looks at synchronization, IEEE Trans. Circuits Syst. I (44) (1997) 882–890. M. Feki, Observer-based exact synchronization of ideal and mismatched chaotic systems, Phys. Lett. A 309 (2003) 53–60. O. Morgül, M. Feki, A chaotic masking scheme by using synchronized chaotic systems, Phys. Lett. A 251 (1999) 169–176. A. Fradkov, Cybernetical Physics: From Control of Chaos to Quantum Control, Springer, Berlin, 2007. Y. Ushio, Synthesis of synchronized chaotic systems based on observers, Int. J. Bifurcation Chaos 9 (1999) 541–546. T.L. Carroll, L.M. Pecora, Synchronizing chaotic circuits, IEEE Trans. Circuits Syst. I (38) (1991) 453–456. O. Morgül, E. Solak, Observer based synchronization of chaotic systems, Phys. Rev. E 54 (1996) 4803–4811. C. Aguilar-Ibañez, R. Martínez-Guerra, R. Aguilar-López, J.L. Mata-Machuca, Synchronization and parameter estimations of an uncertain Rikitake system, Phys. Lett. A 374 (2010) 3625–3628. Xiaoshan Zhao, Zhenbo Li, Shuang Li, Synchronization of a chaotic finance system, Appl. Math. Comput. 217 (2011) 6031–6039. M. Boutayeb, M. Darouach, H. Rafaralahy, Generalized state observers for chaotic synchronization and secure communication, IEEE Trans. Circuits Syst. I (49) (2002) 345–349. R. Martínez-Guerra, J.L. Mata-Machuca, R. Aguilar-López, A. Rodríguez-Bollain, Chaotic synchronization and its applications in secure communications, in: S. Banerjee, M. Mitra, L. Rondoni (Eds.), Applications Chaos and Nonlinear Dynamics in Engineering, Understanding Complex Systems, vol. 1, Springer-Verlag, Berlin, Heidelberg, 2011, pp. 231–272 (Chapter 8). R. Martínez-Guerra, W. Yu, Chaotic communication and secure communication via sliding-mode observer, Int. J. Bifurcation Chaos 18 (2008) 235–243. J. Mata-Machuca, R. Martínez-Guerra, R. Aguilar-López, An exponential polynomial observer for synchronization of chaotic systems, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 4114–4130. R. Martínez-Martínez, J.L. Mata-Machuca, R. Martínez-Guerra, J.A. León, G. Fernández-Anaya, Synchronization of nonlinear fractional order systems, Appl. Math. Comput. 218 (2011) 3338–3347. J.L. Mata-Machuca, R. Martínez-Guerra, Asymptotic synchronization of the Colpitts oscillator, Comput. Math. Appl. 63 (2012) 1072–1078. Zhengqiang Zhang, Hanyong Shao, Zhen Wang, Hao Shen, Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems, Appl. Math. Comput. 218 (2012) 7614–7621. E. Elabbasy, H. Agiza, M. El-Dessoky, Global chaos synchronization for four scroll attractor by nonlinear control, Sci. Res. Essay 1 (2006) 65–71. C. Wang, S. Ge, Adaptive backstepping control of uncertain Lorenz system, Int. J. Bifurcation Chaos 11 (2001) 1115–1119. D. Ghosh, A. Chowdhury, P. Saha, On the various kinds of synchronization in delayed Duffing–Van der Pol system, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 790–803. G. Grassi, D.A. Miller, Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal, Appl. Math. Comput. 218 (2012) 6118–6124. H.B. Fotsin, J. Daafouz, Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification, Phys. Lett. A 339 (2005) 304–315. J.L. Mata-Machuca, R. Martínez-Guerra, Ricardo Aguilar-López, C. Aguilar-Ibañez, A chaotic system in synchronization and secure communications, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1706–1713. M. Ayati, H. Khaloozadeh, Stable chaos synchronisation scheme for nonlinear uncertain systems, IET Control Theory Appl. 4 (2010) 437–447. Yaoyao Lan, Qingguo Li, Chaos synchronization of a new hyperchaotic system, Appl. Math. Comput. 217 (2010) 2125–2132. R. Aguilar-López, J.L. Mata-Machuca, R. Martínez-Guerra, Observability and Observers for Nonlinear Dynamical Systems: Nonlinear Systems Analysis, LAMBERT Academic Publishing, Germany, 2011, ISBN 978-3-8454-3171-0. R. Martínez-Guerra, J.J. Rincón-Pasaye, Synchronization and anti-synchronization of chaotic systems: a differential and algebraic approach, Chaos Solitons Fract. 28 (2009) 511–517. R. Martínez-Guerra, J. Mendoza-Camargo, Observers for a class of Liouvillian and, non-differentially flat systems, IMA J. Math. Control Inf. 21 (2004) 493–509. T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Trans. Circuits Syst. 31 (12) (1984) 1055–1058. L.0. Chua, Chua’s circuit: 10 years later, IEICE Trans. Fundam. E77-A (1994) 1811–1822. G. Kolumban, G. Kis, Performance evaluation of FM-DCSK modulation scheme, in: Proc. of the 1998 Int. Svmposium on Nonlinear Theory and its Applications (NOLTA’98), vol. 1, Crans-Montana, Switzerland, 1998, pp. 81–84. S. Jankowski, A. Londei, C. Mazur, A. Lozowski, Synchronization and association in a large network of coupled Chua circuits, Int. J. Electron. 79 (1995) 823–828.
10944 [33] [34] [35] [36] [37]
R. Martínez-Guerra et al. / Applied Mathematics and Computation 219 (2013) 10934–10944
M.P. Kennedy, Robust op amp realization of Chua’s circuit, Frequenz 46 (3–4) (1992) 66–68. H. Khalil, Nonlinear Systems, third ed., Prentice-Hall, Englewood Cliff, NJ, 2002. G.Q. Zhong, F. Ayron, Experimental confirmation of chaos Chua’s circuit, Int. J. Circuit Theory Appl. 13 (11) (1985) 93–98. L.O. Chua, M. Komuro, T. Matsumoto, The double scroll family, parts I and II, IEEE Trans. Circuits Syst. 33 (11) (1986) 1073–1118. R. Martínez-Guerra, A. Poznyak, V. Díaz, Robustness of high-gain observers for closed-loop nonlinear systems: theoretical study and robotics control application, Int. J. Syst. Sci. 31 (2000) 1519–1529.
