IC/2005/026
Available at: http://www.ictp.it/~pub− off
United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
ADAPTIVE OBSERVER BASED SYNCHRONIZATION OF A CLASS OF UNCERTAIN CHAOTIC SYSTEMS
Samuel Bowong1 Laboratoire de Math´ematiques Appliqu´ees, D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e de Douala, B.P. 24157 Douala, Cameroun and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Ren´e Yamapi2 D´epartement de Physiques, Facult´e des Sciences, Universit´e de Douala, B.P. 24157 Douala, Cameroun.
Abstract This study addresses the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework. For a class of uncertain chaotic systems with unknown parameters and external disturbances, a robust adaptive observer based response system is constructed to synchronize the uncertain chaotic system. Lyapunov stability theory and Barbalat lemma ensure the global synchronization between the drive and response systems even if Lipschitz constants on function matrices and bounds on uncertainties are unknown. Numerical simulation of the Genesio-Tesi system verifies the effectiveness of the proposed method.
MIRAMARE – TRIESTE May 2005
1 2
Corresponding author. Permanent address: B.P. 8329 Yaound´e, Cameroun.
[email protected] [email protected]
1
Introduction Research in the area of the synchronization of dynamical systems dates back over 300 years.
Huygens, most famous for his studies in optics and the construction of telescopes and clocks, was probably the first scientist who observed and described the synchronization phenomenon as early as in the 17th century. The pioneering paper on the concept of chaos synchronization was not presented until 1990. Pecora and Carroll introduced a method [Pecora & Caroll, 1990] to synchronize two identical chaotic systems with different initial conditions. The idea is to use the output of the drive system to control the response system so that they oscillate in a synchronized manner. Because of their works, chaos synchronization has been intensively studied in the last few years. It has been widely explored in a variety of fields including physical [Lakshmanan & Murali, 1996; Caroll & Pecora, 1991; Bocaletti et al., 2002], chemical and ecological systems [Han et al., 1995; Blasius et al., 1999], secure communications [Cuomo & Oppenheim, 1993; Kocarev & Parlitz, 1995; Hasler, 1995; Morg¨ ul et al., 2003] etc. Hence various synchronization schemes, such as adaptive control [Femat et al., 2002; Chen & L¨ u, 2002], backstepping design [Tan et al., 2003; Bowong & Moukam, 2004], active control [Ho & Hung, 2002; Yassen, 2005], and nonlinear control [Huang et al., 2004; Chen, 2005] have been successfully applied to chaos synchronization. More recently, the synchronization has been regarded as a special case of observer design problem [Nijmeijer & Mareels, 1997; Grassi & Mascolo, 1997; Morg¨ ul & Solak, 1997; Bowong et al., 2004 ]. In this approach, the output (or driving signal) is chosen as a linear or nonlinear combination of the full system state variables. Of particular interest is the problem of synchronizing two or more systems when the designer of the receiver does not know not only the initial state but also some or all parameters. This is a more complicated problem referred to as adaptive synchronization [Liao & Tsai, 2000; Feki, 2003; Chen et al., 2005]. This effect should be taken into account when we want to evaluate the performance of a practical chaos synchronization scheme. As a consequence, adaptive based-observer synchronization in chaotic systems in the presence of unknown parameters and uncertainties is an important issue. In this paper, the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework is studied. In the drive system, not only the Lipschitz constants on function matrices but also the bounds on uncertainties are unknown. In this case, a robust adaptive observer based response system is designed to synchronize the given uncertain chaotic system. If certain conditions are satisfied, several adaptation laws are chosen to estimate unknown constants and uncertain parameters vector respectively and to repress external disturbances. Lyapunov stability theory and Barbalat lemma ensure the global synchronization between the drive and response chaotic systems even if the drive system Lipschitz constants on function matrices and bounds on uncertainties are unknown. The proposed method has three main advantages. First, our scheme covers a large class of chaotic systems that comprise uncertainties, which is rather requested by many synchronizing schemes in literature. Second,
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the scheme proposed here shows high robustness to noise and parameter mismatch. Third, the feedback coupling was given in terms of a series of nonlinear functions of the difference between the outputs of the drive and response systems which can be easily implemented in practice and unknown parameters are correctly estimated. The outline of this paper is as follows. In Section 2, we present the robust adaptive observerbased response system design and we prove its synchronization. In Section 3, we present an illustrative example to demonstrate the effectiveness of the proposed approach. Finally in Section 4, we include some concluding remarks.
2
Robust adaptive synchronization algorithm Chaotic systems are generally described by a set of nonlinear differential equations. It is
very common, however, to be able to separate the dynamics into linear and nonlinear parts. If we furthermore consider that the chaotic system is subjected to unknown parameters and uncertainties, the chaotic dynamics can therefore be described by the following equations: x˙ = A(µ)x + Bf (x) + Bg(x)θ + Bd(x, t),
(1)
y = Cx,
where x ∈ Rn is the state vector, y ∈ Rm is the output vector, µ ∈ Rq is the parameter vector, θ ∈ Rp (p ≤ m) is the uncertain parameter vector, d ∈ R r is the external disturbance vector A(µ) ∈ Rn×n is a matrix that may include parametric perturbations, B ∈ R n×r and C ∈ Rm×n are known matrices. Further, f (.) ∈ R r is a nonlinear function vector, and g(.) ∈ R r×p is a function matrix. Notice that the nonlinear function vector, the uncertainty vector and the external disturbance vector satisfy the matching condition. Generally speaking, the number of the external disturbance d is not greater than that of the 0
output y, namely r ≤ m. If r < m, we can augment B, f (x), g(x) and d into B = [B, 0] with 0
0
0 ∈ Rn×(m−r) , f (x) = [f T (x), 0T ]T with 0 ∈ Rm−r , g (x) = [g T (x), 0T ] with 0 ∈ R(m−r)×p and 0
0
0
0
0
d = [dT , 0T ] with 0 ∈ Rm−r , respectively. Clearly, we have B f (x) = Bf (x), B g (x) = Bg(x) 0
0
and B d = Bd, respectively, and such replacement does not change the chaos nature of system (1). Therefore in system (1) we can suppose that r = m. Among many chaotic systems, Lorenz system can be transformed into the form of system (1):
0 0 0 0 x1 −σ σ 0 x˙ 1 r 0 x˙ 2 = 0 −1 0 x2 + 1 0 −x1 x3 + 1 0 x1 , (2) b 0 −x3 x1 x2 0 1 0 1 x3 0 0 0 x˙ 3
where µ = σ. In addition, Genesio-Tesi system can also be rewritten as follows: 0 0 x1 0 1 0 x˙ 1 2 x˙ 2 = 0 x2 + 0 x1 + 0 (−x3 )a, 0 1 1 1 x3 −c −b 0 x˙ 3 3
(3)
where µ = (c, b)T , f (x) = x21 , g(x) = −x3 and θ = a. Our problem undertaken here is to consider the adaptive synchronization problem of system (1) using the drive-response configuration. That is to say, if the uncertain system (1) is regarded as the drive system, a suitable response system should be constructed to synchronize the drive system (1) with the help of certain driving signal. In order to do so, we must make the following assumptions: Assumption 1: The matrix A(µ) and the nonlinear functions f (x) and g(x) satisfy the following Lipschitz conditions: kA(µ) − A(ˆ µ)k ≤ ka kµ − µ ˆ k,
∀µ, µ ˆ ∈ Rq ,
(4)
kf (x) − f (ˆ x)k ≤ kf kx − x ˆk,
∀x, x ˆ ∈ Rn ,
(5)
∀x, x ˆ ∈ Rn ,
(6)
kg(x) − g(ˆ x)k ≤ kg kx − x ˆk, where ka , kf and kg are appropriate positive constants.
Assumption 2: The uncertain parameter µ, the unknown parameter θ and the external disturbance d(x, t) are norm bounded by three unknown positive constants µ m , θm and dm , respectively. Assumption 3: Suppose that the pair (C, A(µ)) is observable. Further, there exists a constant vector L ∈ Rn×1 to make the transfer function H(s) = C(sI n − (A(µ) − LC))−1 B be strictly positive real. Some comments regarding the above assumptions are in order. First, the Lipschitz properties are satisfied locally if A(µ) is differentiable with respect to µ. Let U ⊂ R n be a region which contains the chaotic attractor of (1) and let M ⊂ R q be a region containing the relevant parameter values for which (1) exhibits chaotic behavior. The following analysis will remain valid if (1) is satisfied locally for x ∈ U and µ ∈ M. On the other hand, the Lipschitz constants ka , kf and kg are often required to be known for the control design purpose. However, it is often difficult to obtain precise values of k a , kf and kg in some practical systems, hence the Lipschitz constants are often selected to be larger, which will induce the control gain to be higher, and the obtained results would be conservative. Second, from Assumption 3 and Kalman-YakubovichPopov lemma [Marino & Tomei, 1995], there exist two positive definite matrices P = P T and Q = QT such that the following algebraic equations hold: (A(µ) − LC)T P + P (A(µ) − LC) = −Q,
(7)
B T P = C.
(8)
and
Note that the equality (8) implies that the span of rows of B T P belongs to the span of rows of C.
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The robust adaptive observer for system (1) is constructed as follows: w
1 X ˆ α ˆ k (y − C x ˆ)2k−1 + βB(y − Cx ˆ), x ˆ˙ = A(ˆ µ)ˆ x + Bf (ˆ x) + Bg(ˆ x)θˆ + B 2
(9)
k=1
where µ ˆ ∈ Rp is the parameter vector, α ˆ k and θˆ are solutions of adaptation laws to be determined, and βˆ satisfied some condition to be determined in the sequel. Let e = x − x ˆ be the synchronization error. From Eqs. (1) and (9), by adding and subtracting likewise terms, the error dynamics is described by e˙ = [A(µ) − LC]e + LCe + [A(µ) − A(ˆ µ)]ˆ x + B[f (x) − f (ˆ x] ˆ + Bd(t) + B[g(x) − g(ˆ x)]θ + Bg(ˆ x)(θ − θ)
(10)
w
−
1 X T ˆ α ˆ k (B T P e)2k−1 − βBB P e. B 2 k=1
Hence, the synchronization problem becomes the stability of the error dynamics (10). If it is globally stabilized at the origin, the response system (9) can globally synchronize the drive system (1). Consider the Lyapunov function candidate w
X 1 1 ˆα ˆ T (θ − θ) ˆ + V (e, θ, ˆ k ) = eT P e + (θ − θ) (αk − α ˆ k )2 , 2γ 2δk
(11)
k=1
where δk are free positive constants and αk are positive constants to be determined. Let kˆ xk ≤ x ˆm be satisfied for some x ˆm > 0. Moreover, let us define ∆µ = kµ − µ ˆk. So, the time-derivative ˆα of V (e, θ, ˆ k ) with respect to time is ˆα V˙ (e, θ, ˆ k ) = eT [(A(µ) − LC)T P + P (A(µ) − LC)]e + eT P LCe + 2eT P [A(µ) − A(ˆ µ)]ˆ x + 2eT P B[f (x) − f (ˆ x)] ˆ + 2eT P B[g(x) − g(ˆ x)]θ + 2eT P Bg(ˆ x)(θ − θ) + 2eT P Bd(x, t) −
w P
ˆ T P ek2 α ˆ k kB T P ek2k − 2βkB
k=1 w
−
X 1 2 ˆ T θˆ˙ − (θ − θ) (αk − α ˆ k )α ˆ˙ k , γ δk k=1
≤ −eT Qe + 2kB T P ek kLT P ek + 2ka ∆µ x ˆm kBP ek + 2kB T P ek kf kek + 2θm kB T P ek kg kek + 2dm kB T P ek w ˆ T P ek2 ˆ −Pα ˆ k kB T P ek2k − 2βkB + 2eT P Bg(ˆ x)(θ − θ) k=1
w
−
X 1 2 ˆ T θˆ˙ − (θ − θ) (αk − α ˆ k )α ˆ˙ k . γ δk k=1
5
(12)
By Assumption 1, we get the following inequalities: 2kB T P ek kf kek ≤
≤
kf2 ε1
kB T P ek2 + ε1 kek2 ,
w P
k=1
kf2k ε1
(13) kB T P ek2k + ε1 kek2 ,
(kg θm )2 T kB P ek2 + ε2 kek2 , ε2
2θm kB T P ek kg kek ≤
w (k θ )2k P g m ≤ kB T P ek2k + ε2 kek2 , ε2 k=1
(14)
1 kB T P ek2 + ε3 kLT P ek2 ε3
2kB T P ek kLT P ek ≤
w 1 P kB T P ek2k + ε3 λmax (P LLT P )kek2 , ≤ ε k=1 3
2ka ∆µ x ˆm kB T P ek ≤
(ka ∆µ x ˆ m )2 T kB P ek2 + β1 , β1
w (k ∆µ x P ˆm )2k T a kB P ek2k + β1 , ≤ β 1 k=1
and
2dm kB T P ek ≤
(15)
(16)
d2m T kB P ek2 + β2 , β2 (17)
w d2k P m ≤ kB T P ek2k + β2 , β 2 k=1
where ε1 , ε2 , ε3 , β1 and β2 are five suitable positive constants and λ max (P LLT P ) is the maximum eigenvalue of P LLT P . Then, we get ˆα V˙ (e, θ, ˆ k ) ≤ −eT Q − (ε1 + ε2 + ε3 λmax (P LLT P ))In e +
w P
k=1
kf2k ε1
(kg θm )2k 1 (ka ∆µ x ˆm )2k d2k + + + + m ε2 ε3 β1 β2
ˆ − + β1 + β2 + 2eT P Bg(ˆ x)(θ − θ)]
w P
!
kB T P ek2k
ˆ T P ek2 α ˆ k kB T P ek2k − 2βkB
(18)
k=1 w
−
X 1 2 ˆ T θˆ˙ − (θ − θ) (αk − α ˆ k )α ˆ˙ k , γ δk k=1
Let αk =
kf2k ε1
+
(kg θm )2k 1 (ka ∆µ x ˆm )2k d2k + + + m ε2 ε3 β1 β2
6
and
β = β 1 + β2 .
(19)
Then, Eq. (18) may be rewritten as follows: ˆα V˙ (e, θ, ˆ k ) ≤ −eT Q − (ε1 + ε2 + ε3 λmax (P LLT P ))In e ˆT + 2(θ − θ) w P
−
g T (ˆ x)B T P e
(αk − α ˆk)
k=1
1 ˆ˙ − θ γ
kB T P ek2k
1 ˙ − α ˆk δk
(20)
ˆ T C T Ce. + β − 2βe
Should, we choose the following adaptation laws ˙ θˆ = γg T (ˆ x)(y − C x ˆ),
(21)
and α ˆ˙ k = δk ky − C x ˆk2k ,
k = 1, 2, . . .
(22)
where γ and δk are positive constants, we have ˆ T C T Ce. ˆα V˙ (e, θ, ˆ k ) ≤ −eT Q − (ε1 + ε2 + ε3 λmax (P LLT P ))In e + β − 2βe
(23)
Thus, if the following condition
βˆ ≥
β , 2(y − C x ˆ)T (y − C x ˆ)
(24)
is satisfied, then ˆα V˙ (e, θ, ˆ k ) ≤ −λmin (S)kek2 ,
(25)
where λmin (s) is the minimum eigenvalue of S = Q − (ε 1 + ε2 + ε3 λmax (P LLT P ))In . From the above inequality, free parameters ε 1 , ε2 and ε3 can be selected to be small enough to let ˆα λmin (S) > 0 so that V˙ (e, θ, ˆ k ) < 0. And so the system is Lyapunov stable, whence e ∈ L ∞ and ˆ ∈ L∞ . Therefore, from (10) and (11), we have V (e, θ, ˆα (θ − θ) ˆ k ) ∈ L∞ and e˙ ∈ L∞ . Integrating (25), we obtain Z
t 0
kek2 ≤
ˆ ˆα V (e(0), θ(0), α ˆ k (0)) − V (e, θ, ˆk ) . λmin (S)
ˆ Since V (e(0), θ(0), α ˆ k (0)) is finite, it follows that e ∈ L2 . Hence, using Barbalat’s Lemma [Khalil, 1995] and the fact that e ∈ L ∞ , e˙ ∈ L∞ and e ∈ L2 , it results that lim e(t) = 0. t→∞ R t Moreover, since f and g are Lipschitz, then e˙ is uniformly continuous and the integral 0 edt ˙ = −e(0) is finite. Thus, by Barbalat’s lemma, lim e(t) ˙ = 0. Therefore using (10), we obtain t→∞ ˆ = 0. lim [Bg(x)θ − Bg(ˆ x)θ] t→∞
Since the coupling term is a series of nonlinear functions of the difference between the outputs of the drive and response systems, it can lead to many types of feedback coupling schemes for chaos synchronization. The variation of the type of the coupling term depends on the form of the nonlinear structure. An experimental realization of this coupling term has no difficulties for many practical systems. When the synchronization is achieved, i.e., x(t) → x ˆ(t) as t → ∞, the 7
values of the adaptive parameters α ˆ k will remain at their optimal values which can be used for the practical implementation of the coupling term. We can now summarize our result in the following theorem Theorem 1 : Consider the drive chaotic system (1) satisfying Assumptions 1, 2 and 3. If condition (24) holds, then the response system (9) associated with the adaptation laws (21) and (22) globally synchronizes the drive system (1). It is then important to mention that if dg(x(t))/dt is bounded and g(x(t)) satisfies Z
t+T
g T (x(τ ))B T Bg(x(τ ))dτ ≥ ηI,
t
ˆ = 0. for some T , η > 0 and any t ≥ 0, then lim kθ − θk t→∞
ˆ Remark 1 : Since the error y − C x ˆ tends to zero, the magnitude of βB(y − Cx ˆ) in the response system (9) would increase unboundedly and become infeasible in computation. In practice, we can replace βˆ in the robust adaptive observer (9) with βˆ ≥
β , if 2(y − C x ˆ)T (y − C x ˆ)
ˆ β = 0,
if
˜ kek ≥ β, (26) ˜ kek ≤ β,
where β˜ is a sufficiently small positive constant. Therefore, the state error would be contained within a neighborhood of the origin.
3
Numerical simulation In this section, three-dimensional Genesio-Tesi system is illustrated to show the effectiveness
of the proposed robust adaptive observer scheme. The dynamics of Genesio-Tesi system is given by system (3) . When parameter a, b and c are chosen to be 0.44, 1.1 and 1, Genesio-Tesi system behaves chaotically as shown in Fig. 1 for the initial conditions (x 1 (0), x2 (0), x3 (0)) = (0.1, 0, 0). Here, we assume that the parameters a, b and c are unknown and a is disturbed by ∆d = 0.01 sin(2t). We will use the solution x 1 of (3) as the signal to be transmitted to the response system, i.e., y = x1 . Thus, in system (3), matrices A(µ), B and C are chosen as 0 0 1 0 and C = [1, 0, 0]. B= 0 A(µ) = 0 0 1 , 1 −c −b 0
Further, f (x) = x21 , g(x) = −x3 and d(x, t) = ∆dx3 . Clearly, the pair (A, C) is observable, thereby permitting the choice of the gain matrix L = (3, 1.9, −3.3) T to make the transfer function H(s) = C[sI3 − (A(µ) − LC)]−1 B = 8
s3
+
1 + 3s + 1
3s2
be strictly positive real. As derived earlier, a robust adaptive observer based response system is designed as follows: ˙ 0 1 0 x ˆ1 x ˆ1 0 0 ˙ˆ2 = 0 x 0 1 x ˆ2 + 0 x ˆ21 + 0 (−ˆ x3 )ˆ a ˆ ˙x x ˆ3 1 1 −ˆ c −b 0 ˆ3 (27) 0 1 ˆ 1−x α1 (x1 − x ˆ1 ) + α ˆ 3 (x1 − x ˆ1 )3 + β(x ˆ1 )], + 0 [ˆ 2 1
where the adaptation laws a ˆ, α ˆ 1 and α ˆ 3 are chosen as
a ˆ˙ = −γ x ˆ3 (x1 − x ˆ1 ),
(28)
α ˆ˙ 1 = δ1 (x1 − x ˆ 1 )2 ,
(29)
α ˆ˙ 3 = δ3 (x1 − x ˆ 1 )6 ,
(30)
and
with γ, δ1 and δ3 three positive constants and β , if βˆ ≥ 2(x1 − x ˆ 1 )2 ˆ β = 0, if
˜ |e1 | ≥ β, (31) ˜ |e1 | ≤ β,
with β˜ a sufficiently small positive constant.
The Initial conditions and parameters of the drive system are those of Fig. 1. The The robust adaptive observer based response system (27) was with the following initial conditions: ˆ = 0.02. We also simulated (ˆ x1 (0), xˆ2 (0), xˆ3 (0)) = (0.2, 0, 0), α ˆ 1 (0) = 0.01, α ˆ 3 (0) = 0.02 and θ(0) (27) with ˆb = 1.1 and cˆ = 1.2. Note that in this case, ∆b = ∆c = |b − ˆb| = |c − cˆ| = 0.1 and p ∆µ = (∆b)2 + (∆c)2 = 0.1414. The parameters γ = δ1 = δ3 = β = 1 and β˜ = 0.1 were chosen.
Figure 2 presents the time evolution of the synchronization error. It clearly appears that after a transient period, the synchronization error converges exactly to the origin. This implies that the response system (27) globally synchronizes the driving Genesio-Tesi system (3) though there exist unknown Lipschitz constants on function matrices and unknown bounds on uncertainties. In other words, the proposed adaptive observer based response system (9) can effectively synchronize the uncertain chaotic system (1). In order to add evidence of the effectiveness and efficiency of the proposed synchronization scheme, we have plotted x1 versus x ˆ1 , x2 versus x ˆ2 and x3 versus x ˆ3 without and with feedback couplings. The projections of the attractors from the six-dimensional phase space onto the planes (x1 , xˆ1 ), (x2 , x ˆ2 ) and (x3 , x ˆ3 ), respectively for α ˆ1 = α ˆ 3 = 0 are shown on the left hand side of Fig. 3 (uncontrolled evolution). These projections clearly indicate that these oscillations 9
are not synchronized neither phase nor frequency (see [Femat & Solis-Perales, 1999] for details concerning definition of synchronization kinds). The relation between the states of the drive and response systems under feedback couplings is depicted on the right hand side of Fig. 3. Note that the phases of the master and slave systems are locked, which is a common measure of the degree of synchronization. Thus, it clearly appears that the manifolds x 1 = x ˆ 1 , x2 = x ˆ2 and x3 = x ˆ3 are stable, and one can conclude that the chaotic oscillations of the drive and response systems are synchronized in complete sense and the synchronization objective is attained. Figures 4(a), 4(b) and 4(c) present the time evolution of the adaptive parameters α ˆ 1, α ˆ 3 and a ˆ, respectively. Note that the estimated gains α ˆ 1 and α ˆ 3 increase with time and then saturate at their optimal values which are optimal parameters suitable for the implementation process. Note also that the unknown parameter a is quickly estimated to the right value (see Fig. 4(c)).
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Conclusion In this paper, the problem of adaptive synchronization of a class of uncertain chaotic systems
is considered in the drive-response framework. For a class of uncertain chaotic systems with unknown Lipschitz constants on function matrices and unknown bounds on uncertainties, a robust adaptive observer-based response system is constructed to globally synchronize the uncertain chaotic drive system. Numerical example of the Genesio-Tesi system is considered to show the efficiency and effectiveness of this scheme. A fairly good agreement is obtained between the analytical and numerical results.
Acknowledgments This work was done during the visit of Samuel Bowong at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. He would like to thank the Condensed Matter group for the invitation, hospitality and financial support.
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0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x3
1
x2
1
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8 −0.6
−0.4
−0.2
0
0.2
x1
0.4
0.6
0.8
1
−0.8 −0.6
1.2
−0.4
−0.2
0
(a)
0.2
x1
0.4
0.6
0.8
1
1.2
1 0.8 0.6 0.4
x3
0.2 0 −0.2 −0.4 −0.6 −0.8 −0.8
−0.6
−0.4
−0.2
0
x2
0.2
0.4
0.6
0.8
1
(c)
Figure 1: Chaotic attractor of the Genesio-Tesi system. The system parameters are provided in text.
13
(b)
0.6
0.4 0.3
0.4
0.2 0.2
0.1 0
e2
e1
0
−0.2
−0.1 −0.2 −0.3
−0.4
−0.4 −0.6
−0.8
−0.5 0
20
40
60
80
100
Time(s)
120
140
160
180
−0.6
200
0
20
40
60
(a)
80
100
Time(s)
120
140
160
180
200
(b)
0.5 0.4 0.3 0.2
e3
0.1 0 −0.1 −0.2 −0.3 −0.4
0
20
40
60
80
100
Time(s)
120
140
160
180
200
(c)
Figure 2: Time evolution of the synchronization error. (a) e1 = x1 − xˆ1 , (b) e2 = x2 − xˆ2 and e3 = x3 − xˆ3 .
14
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
1
1
0.4
x
x
0.4
0.2
0.2 0
0
−0.2
−0.2
−0.4
−0.4 −0.6 −0.6
−0.4
−0.2
0
0.2
x
0.4
0.6
0.8
1
−0.6 −0.6
1.2
−0.4
−0.2
0
0.2
(a)
1
x
0.4
0.6
0.8
1
1.2
(d)
1
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
x2
x2
0.2 0.2
0 0 −0.2
−0.2
−0.4
−0.4
−0.6
−0.6 −0.8 −0.8
−0.6
−0.4
−0.2
0
x2
0.2
0.4
0.6
0.8
−0.8 −0.8
1
−0.6
−0.4
−0.2
0
(b)
1.5
x2
0.2
0.4
0.6
0.8
1
(e)
1 0.8
1
0.6 0.4
0.5
x3
x3
0.2 0 0 −0.2 −0.4
−0.5
−0.6 −1 −0.8
−0.6
−0.4
−0.2
0
x3
0.2
0.4
0.6
0.8
−0.8 −0.8
1
(c)
−0.6
−0.4
−0.2
0
x3
0.2
0.4
0.6
0.8
1
(f)
Figure 3: Relation between the drive and response systems: without coupling terms (Figures on the left-hand side) and with coupling terms (Figures on the right-hand side).
15
2
0.04
1.5
0.035
α3
0.045
α1
2.5
1
0.03
0.5
0.025
0
20
40
60
80
100
Time(s)
120
140
160
180
0.02
200
0
20
40
60
(a)
80
100
Time(s)
120
140
160
0.9 0.8 0.7 0.6 0.5
a
0
0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
Time(s)
120
140
160
180
200
(c)
Figure 4: Time evolution of the adaptive parameters. (a) αˆ 1 , (b) αˆ 3 and (c) aˆ.
16
180
200
(b)
