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Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems
Physics Letters A 361 (2007) 231–237 www.elsevier.com/locate/pla
Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems ✩ Manfeng Hu a,b,∗ , Zhenyuan Xu a , Rong Zhang a,b , Aihua Hu a a School of Science, Southern Yangtze University, Wuxi 214122, China b School of Information Technology, Southern Yangtze University, Wuxi 214122, China
Received 29 December 2005; received in revised form 28 June 2006; accepted 1 August 2006 Available online 5 October 2006 Communicated by A.R. Bishop
Abstract Based on the active control idea and the invariance principle of differential equations, a general scheme of adaptive full state hybrid projective synchronization (FSHPS) and parameters identification of a class of chaotic (hyper-chaotic) systems with linearly dependent uncertain parameters is proposed in this Letter. With this effective scheme parameters identification and FSHPS of chaotic and hyper-chaotic systems can be realized simultaneously. Numerical simulations on the chaotic Chen system and the hyper-chaotic Chen system are presented to verify the effectiveness of the proposed scheme. © 2006 Elsevier B.V. All rights reserved. Keywords: FSHPS; Parameters identification; Invariance principle; Adaptive control idea; Chen system
1. Introduction Chaos control and synchronization has become a hot subject in the field of nonlinear science due to its wide-scope potential applications in various disciplines. Some recent advances about chaos control and different kinds of chaotic synchronization, and a review of relevant experimental applications of these techniques and schemes have been reported in Refs. [1,2]. Chaos control refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior [1]. Chaos synchronization refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy) [2].
Up to now, many types of synchronization phenomena have been presented such as complete synchronization, phase synchronization [3], lag synchronization [4] anti-synchronization [5], partially synchronization [6], generalized synchronization [7], projective synchronization [8–11], Q–S synchronization [12], etc. Complete synchronization is characterized by the equality of the state variables while evolving in time. Antisynchronization is characterized by the vanishing of the sum of relevant variables. Projective synchronization is characterized that the drive and response system could be synchronized up to a scaling factor α (a proportional relation). Recently, We presented a new synchronization in [13], full state hybrid projective synchronization (FSHPS), between two dynamical systems, i.e., for two dynamical systems x(t) ˙ = F (x) ←− drive system,
(a)
y(t) ˙ = G(x, y) ←− response system. (b) ✩
This work was supported by the National Natural Science Foundation of China (No. 10372054) and the Science Foundation of Southern Yangtze University (No. 000408). * Corresponding author. E-mail addresses: [email protected], [email protected] (M. Hu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.092
Definition 1. For the drive system (a) and response system (b), the two dynamical systems are said to be full state hybrid projective synchronization (FSHPS) if there exists constant diagonal matrix H = diag(h1 , h2 , . . . , hn ) ∈ R n×n such that
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limt→∞ y − H x = 0, i.e., limt→∞ |yi − hi xi | = 0, i = 1, 2, . . . , n. Where H is called scaling matrix, h1 , h2 , . . . , hn are called scaling factors. We give some remarks related to Definition 1 as below: Remark 1. In [10,11], authors used the term “full state projective synchronization” to distinguish the topic of [10,11] from the topic of [8,9]. Here we use the term “full state hybrid projective synchronization” to further distinguish our studying topic from the topic of [8–11]. Remark 2. It is easy to see that the definition of full state hybrid projective synchronization encompasses the complete synchronization, anti-synchronization and projective synchronization when scaling matrix H equals I , −I and αI (α is a constant) respectively. Remark 3. It is worth noting that this definition can also encompass the chaos control problem as scaling matrix H = 0 and 0 is the equilibrium of system (b). Furthermore, when some entries (not all) of scaling matrix H are 1 and the others (not all) are 0, the chaos synchronization reduces to partially synchronization which has been discussed in [6]. These two phenomena indicate that the scaling matrix H in our definition cannot be invertible which is different from the analogous study in [14]. Remark 4. When H is invertible, FSHPS is linear generalized synchronization as investigated in [15]. So, from the viewpoint of mathematics, our definition bridge a gap from chaos control to chaos synchronization, to generalized synchronization. The results on FSHPS [13] are derived under the following hypothesis: all the parameters of drive and response systems are precisely known, and the controller is constructed by those known parameters. However, in many practical situation, the values of some systems’ parameters cannot be exactly known a priori, and the synchronization will be destroyed and broken with the effects of these uncertainties. Therefore, studying the FSHPS of chaotic (hyper-chaotic) systems in the presence of unknown parameters is essential. Luckily, adaptive control scheme is an effective method for the synchronization of chaotic systems with unknown parameters. To maintain the synchronization in such uncertain systems, we can use adaptive controllers to compensate the effects of parameters’ uncertainty. There has been some study on this concerned theme [5, 16–26]. An interesting application of chaos synchronization is to estimate the unknown parameter of a chaotic system from time series when partial information about the system is available [5, 16,18,22–26]. As all known, noise is ubiquitous in both manmade and nature systems, chaos control and synchronization of concrete models is unavoidably subject to external and internal noise. Therefore, it is necessary to design an adaptive controller for the FSHPS of chaotic (hyper-chaotic) systems in the presence of noise. Motivated by the above discussions, in this Letter, an adaptive controller with parameters identifica-
tion, which has been used for the complete synchronization of a class of chaotic systems in [16], is extended for the FSHPS of identical chaotic (hyper-chaotic) systems based on the invariance principle of differential equations and active control idea. Also, this scheme is quite robust against the effect of noise. The rest of this Letter is organized as follows. In Section 2, an adaptive controller with parameter identification is proposed for the FSHPS of coupled identical chaotic (hyper-chaotic) systems based on the invariance principle of differential equations and active control idea. In Section 3, the application of results of Section 2 to Chen chaotic systems and hyper-chaotic Chen systems are considered respectively. Concluding remarks are presented in Section 4. 2. Adaptive FSHPS scheme with parameter identification Consider a class of uncertain chaotic system described by x˙ = f (x) + F (x)θ,
(1)
where x = (x1 , x2 , . . . , xn )T ∈ R n is the state vector of the system, f : R n → R n is a continuous vector function, F : R n → R n×p is a function matrix, θ ∈ R p is an unknown parameter vector. Remark 5. Nonlinear dynamical system (1) depends linearly on the parameters and many well-known chaotic (hyperchaotic) system belong to (1), such as Lorenz, Chen, Lü chaotic system, hyper-chaotic Chen, Rössler system, and so on. Let system (1) be the drive system, then the controlled response system is given by y˙ = f (y) + F (y)θˆ + u,
(2)
where y ∈ R n is the state vector, θˆ ∈ R p represents the estimate vector of unknown parameter vector θ , u ∈ R n is a controller to be determined. The goal of control is to find out a appropriate u such that the response system is full state hybrid projective synchronous with the drive system and the unknown parameters are identified simultaneously. Let the vector error state be e(t) = y(t) − H x(t), where H is a n-order diagonal matrix, i.e., H = diag(h1 , h2 , . . . , hn ). Thus the error dynamical system between the drive system (1) and the response system (2) is e(t) ˙ = y˙ − H x˙ = f (y) − Hf (x) + F (y)θˆ − H F (x)θ + u.
(3)
Let f˜ = f (y) − Hf (x), F˜ = F (y) − H F (x), and θ˜ = θˆ − θ , then Eq. (3) is e(t) ˙ = f˜ + H F (x)θ˜ + F˜ θˆ + u.
(4)
Let u = −f˜ − F˜ θˆ + Ae,
(5)
where the matrix A is chosen such that it has all its eigenvalues on the left-hand side of the complex plane. From (4) and (5), the error dynamics is described by e˙ = Ae + H F (x)θ˜ .
(6)
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Hence the FSHPS problem becomes the stability of error dynamics (6). If it is globally stabilized at the origin, the FSHPS of drive system (1) and response system (2) can be globally realized. Now, we give our main result. Theorem 1. If there exists a positive symmetric matrix P such that AT P + P A = −Q,
(7)
where Q denotes a positive symmetric matrix, and the parameter estimation update law is chosen as θ˙ˆ = −F (x)T H P e
(8)
then the error system (6) is globally stable. Hence, the response system (2) associated the proposed control law (8) and the drive system (1) can globally asymptotically achieve the FSHPS, and the unknown parameters θ may be dynamically estimated from θˆ in the response system simultaneously. Proof. For the system (6), consider the Lyapunov function candidate V (e, θ˜ ) = eT P e + (θˆ − θ )T (θˆ − θ )
(9)
so the derivative with respect to time is V˙ = e˙T P e + eT P e˙ + (θˆ − θ )T θ˙ˆ + θ˙ˆ T (θˆ − θ ) = eT AT P + P A e = −eT Qe 0.
(10)
It is obvious that V˙ = 0 if and only if ei = 0, i = 1, 2, . . . , n, namely the set E = {(e, θˆ )T ∈ R 2n : e(t) = 0, θˆ = θ} is the largest invariant set contained in M = {V˙ = 0} for Eq. (6). So, according to the well-known invariance principle of functional differential equations [18,27], starting with arbitrary initial values of the system (6) associated the proposed control law (8), the orbit converges asymptotically to the set E, i.e., y → H x and θˆ → θ as t → ∞. Then, the FSHPS and parameter identification of coupled identical chaotic systems is achieved simultaneously. The proof is completed. 2 Remark 6. It is noted that, as H = I and system is hyperchaotic, the synchronization reduced to the one studied in [28]. Further assume F : R n → R n×n is a symmetric matrix, the synchronization problem has been studied in [16]. The main difference between the existing works on synchronization and parameter identification and the present Letter is the synchronization phenomena, i.e., FSHPS here instead of complete synchronization. So our results improve and generalize those existing in the literature. 3. Applications of adaptive FSHPS scheme In this section, chaotic Chen system and hyper-chaotic Chen system are adopted to verify the effectiveness of the adaptive FSHPS scheme with parameter identification obtained in the previous section.
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3.1. Adaptive FSHPS of Chen systems Consider the Chen chaotic system described by the following equations [29]: ⎧ ⎨ x˙1 (t) = a(x2 − x1 ), (11) x˙ (t) = (c − a)x1 + cx2 − x1 x3 , ⎩ 2 x˙3 (t) = x1 x2 − bx3 . And it can be presented in the form of ⎞ ⎛ ⎛ ⎞ ⎛ 0 0 x2 − x1 x˙1 ⎝ x˙2 ⎠ = ⎝ −x1 x3 ⎠ + ⎝ −x1 0 x˙3 x1 x2 0 −x3
⎞⎛ ⎞ 0 a x1 + x2 ⎠ ⎝ b ⎠ . 0 c (12)
Similarly, the response system becomes ⎛ ⎞ ⎛ ⎞ y˙1 0 ⎝ y˙2 ⎠ = ⎝ −y1 y3 ⎠ y˙3 y1 y2 ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 0 y 2 − y1 aˆ u1 ⎝ ⎠ ⎝ ⎠ ⎝ ˆ = −y1 0 y1 + y2 b + u2 ⎠ , (13) 0 −y3 0 u3 cˆ ˆ cˆ are the estimates where a, b, c are unknown parameters, a, ˆ b, of a, b, c, respectively. u = (u1 , u2 , u3 )T is the controller vector. We define the parameters errors as a˜ = aˆ − a, b˜ = bˆ − b, c˜ = cˆ − c, and the errors in the variables as e1 = y1 − h1 x1 , e2 = y2 − h2 x2 , e3 = y3 − h3 x3 . If the matrix A is chosen as A = diag(−1, −1, −1), P is chosen as an identity matrix. Then, according to Eq. (5) and Eq. (8) in the theorem, we get the controller ⎧ u1 = (−y2 + y1 + h1 (x2 − x1 ))aˆ − e1 , ⎪ ⎪ ⎨ u = −h x x + y y + (−y + h x )aˆ 2 2 1 3 1 3 1 2 1 (14) ⎪ − (y2 + y1 − h2 (x1 + x2 ))cˆ − e2 , ⎪ ⎩ u3 = h3 x1 x2 − y1 y2 + (y3 − h3 x3 )bˆ − e3 ˆ cˆ obey the following update laws: and the estimates a, ˆ b, ⎧ ˙ ⎪ ⎨ aˆ = −h1 e1 (x2 − x1 ) + h2 x1 e2 , (15) b˙ˆ = h3 e3 x3 , ⎪ ⎩˙ cˆ = −h2 e2 (x1 + x2 ). Therefore, parameters identification and FSHPS of the two chaotic Chen systems can be achieved simultaneously. In numerical simulations, the “unknown” parameters are chosen to be a = 35, b = 3 and c = 28 so that Chen system has a chaotic attractor, the initial states of the drive system and response system are x1 (0) = 1, x2 (0) = 2, x3 (0) = 3, y1 (0) = 11, y2 (0) = 12, y3 (0) = 13, the parameters have initial conditions a(0) = 1, b(0) = 1, c(0) = 1 and h1 = 1, h2 = −1, h3 = 2. The simulation results are shown in Figs. 1–4. From Fig. 1, one can see the error vector e converge to zero as time t goes to infinity. This shows the FSHPS of two Chen chaotic systems with unknown parameters is realized. Fig. 2 shows the results of parameters identification and demonstrates that the parameters a, b, c adapt themselves to the true values. Figs. 3 and 4 show the slight effect of a noise with the strength 6 added to the
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Fig. 1. FSHPS errors of two Chen systems with unknown parameters.
Fig. 2. Parameters a, b, c of Chen system adapt themselves to the true values.
signal x2 . From the numerical simulation, we can see that the proposed FSHPS scheme is effective, and quite robust against the effect of noise. 3.2. Adaptive FSHPS of hyper-chaotic Chen systems Consider the hyper-chaotic Chen system, generated via state feedback control based on Chen chaotic system, described
by [30] ⎧ x˙1 (t) = a(x2 − x1 ) + x4 , ⎪ ⎪ ⎨ x˙ (t) = dx − x x + cx , 2 1 1 3 2 ⎪ x ˙ (t) = −bx + x x , 3 3 1 2 ⎪ ⎩ x˙4 (t) = −rx4 + x2 x3 ,
(16)
where a, b, c, d, r are unknown parameters. And it can be presented in the form of
M. Hu et al. / Physics Letters A 361 (2007) 231–237
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Fig. 3. The errors under the effect of noise with the strength 6 in time series x2 .
⎛
⎞ ⎛ ⎞ x˙1 x4 ⎜ x˙2 ⎟ ⎜ −x1 x3 ⎟ ⎜ ⎟=⎜ ⎟ ⎝ x˙3 ⎠ ⎝ x1 x2 ⎠ x˙4 x2 x3 ⎛
Eq. (8) in the theorem, we get the controller:
x 2 − x1 ⎜ 0 +⎜ ⎝ 0 0
0 0 −x3 0
0 x2 0 0
0 x1 0 0
⎛ ⎞ ⎞ a 0 ⎜b⎟ ⎜ ⎟ 0⎟ ⎟⎜ c ⎟. ⎟ ⎠ 0 ⎜ ⎝d ⎠ x4 r
⎧ u1 = −y4 + h1 x4 − (y2 − y1 − h1 (x2 − x1 ))aˆ − e1 , ⎪ ⎪ ⎨ u = y y − h x x − (y − h x )cˆ − (y − h x )dˆ − e , 2 1 3 2 1 3 2 2 2 1 2 1 2 ˆ ⎪ u = −y y + h x x − (h x − y ) b − e , 1 2 3 1 2 3 3 3 3 ⎪ ⎩ 3 u4 = −y2 y3 + h4 x2 x3 − (y4 − h4 x4 )ˆr − e4 (19) (17)
⎧ ⎪ a˙ˆ = −h1 e1 (x2 − x1 ), ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎨ bˆ = h3 e3 x3 , c˙ˆ = h2 e2 x2 , ⎪ ⎪ ⎪ ⎪ d˙ˆ = h2 e2 x1 , ⎪ ⎪ ⎩˙ rˆ = h4 e4 x4 .
Similarly, the response system becomes ⎛
⎞ ⎛ ⎞ y˙1 y4 ⎜ y˙2 ⎟ ⎜ −y1 y3 ⎟ ⎜ ⎟=⎜ ⎟ ⎝ y˙3 ⎠ ⎝ y1 y2 ⎠ y˙4 y2 y3 ⎛
y 2 − y1 ⎜ 0 =⎜ ⎝ 0 0
0 0 −y3 0
0 y2 0 0
0 y1 0 0
ˆ c, ˆ rˆ are updated according to the foland the estimates a, ˆ b, ˆ d, lowing laws:
⎛ aˆ ⎞
⎞ ⎛ ⎞ 0 u1 ⎜ bˆ ⎟ ⎜ ⎟ ⎜ u2 ⎟ 0⎟ ⎟ ⎜ cˆ ⎟ + ⎜ ⎟ , ⎟ ⎝u ⎠ 0 ⎠⎜ 3 ⎝ dˆ ⎠ y4 u4 rˆ (18)
ˆ c, ˆ rˆ are the estimates of a, b, c, d, r, respectively. where a, ˆ b, ˆ d, u1 , u2 , u3 , u4 are the controllers to be determined. We define the parameters errors as a˜ = aˆ − a, b˜ = bˆ − b, c˜ = cˆ − c, d˜ = dˆ − d, r˜ = rˆ − r, and the errors in the variables as e1 = y1 − h1 x1 , e2 = y2 − h2 x2 , e3 = y3 − h3 x3 , e4 = y4 − h4 x4 . If the matrix A is chosen as A = diag(−1, −1, −1, −1), P is chosen as an identity matrix. Then, according to Eq. (5) and
(20)
Therefore, with the choice of adaptive controller u (19) and the update law (20), parameters identification and FSHPS of the two hyper-chaotic Chen systems can be achieved simultaneously. In numerical simulations, the “unknown” parameters are chosen to be a = 35, b = 3, c = 12, d = 7 and r = 0.5 so that hyper-chaotic Chen system has a chaotic attractor [30], the initial states of the drive system and response system are x1 (0) = 3, x2 (0) = −4, x3 (0) = 2, x4 (0) = 2, y1 (0) = 2, y2 (0) = 1, y3 (0) = 3, y4 (0) = 4, the parameters have initial conditions a(0) = 9, b(0) = 2, c(0) = 7, d(0) = 1, r(0) = 0.1 and h1 = 1, h2 = −1, h3 = 2, h4 = −1/2. The FSHPS errors ˆ c, ˆ rˆ are shown in Fig. 5 and the estimated parameters a, ˆ b, ˆ d, are displayed in Fig. 6. The numerical simulations under the effect of noise can also verify the effectiveness of the proposed scheme. Here we omit them.
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Fig. 4. Estimated parameters under the effect of noise with the strength 6 in time series x2 .
Fig. 5. FSHPS errors of two hyper-chaotic Chen systems with unknown parameters.
4. Conclusions In this Letter, we study the FSHPS of a class of chaotic systems with unknown parameters. Based on the active control idea and the invariance principle of differential equations, a systematic adaptive scheme has been developed to investigate the FSHPS between the drive system and response system in the presence of unknown parameters and to identify the unknown parameters, simultaneously. Numerical simulations on chaotic
Chen system and hyper-chaotic Chen system are presented to verify the theoretical results. As is known, the mainstream of research topic on chaos synchronization is making two chaotic systems be complete synchronization. For achieving complete synchronization, a wide variety of approaches have been proposed, such as linear and nonlinear feedback, time-delay feed-back, adaptive control, impulsive control, observer-based control, fuzzy control, active control theory, the scalar signal method, etc. But from the view-
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Fig. 6. Parameters a, b, c, d, r of the hyper-chaotic Chen system adapt themselves to the true values.
point of application and practical, generalized synchronization may be wider or more practical than those of complete synchronization [31]. The FSHPS is generalized synchronization when scaling matrix H is invertible, so the main contribution of the present Letter is (1): in one hand, realizing FSHPS of chaotic and hyper-chaotic systems with unknown parameters by using a unified method; (2) in the other hand, identifying parameter based on generalized synchronization. In the other side, in the course of using some of the aforementioned chaos control methods, such as linear feedback, to investigate the FSHPS of chaotic (hyper-chaotic) systems, we found that the difficulty went beyond that we had expected. Future work along this line will be focused on the following problems: (i) extending the presented result to a coupled system of more general (non)identical chaotic systems in the presence of parameter mismatch, including those governed by delayed differential equations; (ii) designing some other effective controllers, for example, observer-based controller, to realize the FSHPS of two or many (complex networks) chaotic systems. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. References [1] S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, Phys. Rep. 329 (2000) 103.
[2] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, Phys. Rep. 366 (2002) 1. [3] E.H. Park, M.A. Zaks, J. Kurths, Phys. Rev. E 60 (1999) 6627. [4] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78 (1997) 4193. [5] J. Hu, S.H. Chen, L. Chen, Phys. Lett. A 339 (2005) 455. [6] R. Femat, G. Solis-Perales, Phys. Lett. A 262 (1999) 50. [7] L. Kocarev, U. Parlitz, Phys. Rev. Lett. 76 (11) (1996) 1816. [8] R. Mainieri, J. Rehacek, Phys. Rev. Lett. 82 (1999) 3042. [9] D. Xu, Z. Li, S.R. Bishop, Chaos 11 (3) (2001) 439. [10] G.L. Wen, D. Xu, Phys. Lett. A 333 (2004) 420. [11] G.L. Wen, D. Xu, Chaos Solitons Fractals 26 (2005) 71. [12] Z.Y. Yan, Chaos 15 (2005) 023902. [13] M.F. Hu, Z.Y. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. (2006), in press. [14] G.H. Li, Chaos Solitons Fractals (2006), in press. [15] J.G. Lu, Y.G. Xi, Chaos Solitons Fractals 17 (2003) 825. [16] L.L. Huang, M. Wang, R.P. Feng, Phys. Lett. A 342 (2005) 299. [17] J.D. Cao, J.Q. Lu, Chaos 15 (2005) 043901. [18] D.B. Huang, R.W. Guo, Chaos 14 (2004) 152. [19] M.D. Bernardo, Int. J. Bifur. Chaos 6 (3) (1996) 557. [20] M. Feki, Chaos Solitons Fractals 18 (2003) 141. [21] E.M. Elabbasy, H.N. Agiza, M.M. EI-Dessoky, Chaos Solitons Fractals 21 (2004) 657. [22] U. Parlitz, Phys. Rev. Lett. 76 (1996) 1232. [23] U. Parlitz, L. Junge, L. Kocarev, Phys. Rev. E 54 (1996) 6253. [24] S.H. Chen, J. Hu, C. Wang, J.H. Lü, Phys. Lett. A 321 (2004) 50. [25] S.H. Chen, J.H. Lü, Phys. Lett. A 299 (2002) 353. [26] D.B. Huang, Phys. Rev. E 73 (2006) 066204. [27] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [28] J.W. Feng, S.H. Chen, C.P. Wang, Chaos Solitons Fractals 26 (2005) 1163. [29] G. Chen, T. Ueta, Int. J. Bifur. Chaos 9 (1999) 1465. [30] Y. Li, W.K.S. Tang, G. Chen, Int. J. Bifur. Chaos 10 (2005) 3367. [31] T. Yang, L.O. Chua, Int. J. Bifur. Chaos 9 (1999) 215.
Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems ✩ Manfeng Hu a,b,∗ , Zhenyuan Xu a , Rong Zhang a,b , Aihua Hu a a School of Science, Southern Yangtze University, Wuxi 214122, China b School of Information Technology, Southern Yangtze University, Wuxi 214122, China
Received 29 December 2005; received in revised form 28 June 2006; accepted 1 August 2006 Available online 5 October 2006 Communicated by A.R. Bishop
Abstract Based on the active control idea and the invariance principle of differential equations, a general scheme of adaptive full state hybrid projective synchronization (FSHPS) and parameters identification of a class of chaotic (hyper-chaotic) systems with linearly dependent uncertain parameters is proposed in this Letter. With this effective scheme parameters identification and FSHPS of chaotic and hyper-chaotic systems can be realized simultaneously. Numerical simulations on the chaotic Chen system and the hyper-chaotic Chen system are presented to verify the effectiveness of the proposed scheme. © 2006 Elsevier B.V. All rights reserved. Keywords: FSHPS; Parameters identification; Invariance principle; Adaptive control idea; Chen system
1. Introduction Chaos control and synchronization has become a hot subject in the field of nonlinear science due to its wide-scope potential applications in various disciplines. Some recent advances about chaos control and different kinds of chaotic synchronization, and a review of relevant experimental applications of these techniques and schemes have been reported in Refs. [1,2]. Chaos control refers to a process wherein a tiny perturbation is applied to a chaotic system, in order to realize a desirable (chaotic, periodic, or stationary) behavior [1]. Chaos synchronization refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy) [2].
Up to now, many types of synchronization phenomena have been presented such as complete synchronization, phase synchronization [3], lag synchronization [4] anti-synchronization [5], partially synchronization [6], generalized synchronization [7], projective synchronization [8–11], Q–S synchronization [12], etc. Complete synchronization is characterized by the equality of the state variables while evolving in time. Antisynchronization is characterized by the vanishing of the sum of relevant variables. Projective synchronization is characterized that the drive and response system could be synchronized up to a scaling factor α (a proportional relation). Recently, We presented a new synchronization in [13], full state hybrid projective synchronization (FSHPS), between two dynamical systems, i.e., for two dynamical systems x(t) ˙ = F (x) ←− drive system,
(a)
y(t) ˙ = G(x, y) ←− response system. (b) ✩
This work was supported by the National Natural Science Foundation of China (No. 10372054) and the Science Foundation of Southern Yangtze University (No. 000408). * Corresponding author. E-mail addresses: [email protected], [email protected] (M. Hu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.092
Definition 1. For the drive system (a) and response system (b), the two dynamical systems are said to be full state hybrid projective synchronization (FSHPS) if there exists constant diagonal matrix H = diag(h1 , h2 , . . . , hn ) ∈ R n×n such that
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limt→∞ y − H x = 0, i.e., limt→∞ |yi − hi xi | = 0, i = 1, 2, . . . , n. Where H is called scaling matrix, h1 , h2 , . . . , hn are called scaling factors. We give some remarks related to Definition 1 as below: Remark 1. In [10,11], authors used the term “full state projective synchronization” to distinguish the topic of [10,11] from the topic of [8,9]. Here we use the term “full state hybrid projective synchronization” to further distinguish our studying topic from the topic of [8–11]. Remark 2. It is easy to see that the definition of full state hybrid projective synchronization encompasses the complete synchronization, anti-synchronization and projective synchronization when scaling matrix H equals I , −I and αI (α is a constant) respectively. Remark 3. It is worth noting that this definition can also encompass the chaos control problem as scaling matrix H = 0 and 0 is the equilibrium of system (b). Furthermore, when some entries (not all) of scaling matrix H are 1 and the others (not all) are 0, the chaos synchronization reduces to partially synchronization which has been discussed in [6]. These two phenomena indicate that the scaling matrix H in our definition cannot be invertible which is different from the analogous study in [14]. Remark 4. When H is invertible, FSHPS is linear generalized synchronization as investigated in [15]. So, from the viewpoint of mathematics, our definition bridge a gap from chaos control to chaos synchronization, to generalized synchronization. The results on FSHPS [13] are derived under the following hypothesis: all the parameters of drive and response systems are precisely known, and the controller is constructed by those known parameters. However, in many practical situation, the values of some systems’ parameters cannot be exactly known a priori, and the synchronization will be destroyed and broken with the effects of these uncertainties. Therefore, studying the FSHPS of chaotic (hyper-chaotic) systems in the presence of unknown parameters is essential. Luckily, adaptive control scheme is an effective method for the synchronization of chaotic systems with unknown parameters. To maintain the synchronization in such uncertain systems, we can use adaptive controllers to compensate the effects of parameters’ uncertainty. There has been some study on this concerned theme [5, 16–26]. An interesting application of chaos synchronization is to estimate the unknown parameter of a chaotic system from time series when partial information about the system is available [5, 16,18,22–26]. As all known, noise is ubiquitous in both manmade and nature systems, chaos control and synchronization of concrete models is unavoidably subject to external and internal noise. Therefore, it is necessary to design an adaptive controller for the FSHPS of chaotic (hyper-chaotic) systems in the presence of noise. Motivated by the above discussions, in this Letter, an adaptive controller with parameters identifica-
tion, which has been used for the complete synchronization of a class of chaotic systems in [16], is extended for the FSHPS of identical chaotic (hyper-chaotic) systems based on the invariance principle of differential equations and active control idea. Also, this scheme is quite robust against the effect of noise. The rest of this Letter is organized as follows. In Section 2, an adaptive controller with parameter identification is proposed for the FSHPS of coupled identical chaotic (hyper-chaotic) systems based on the invariance principle of differential equations and active control idea. In Section 3, the application of results of Section 2 to Chen chaotic systems and hyper-chaotic Chen systems are considered respectively. Concluding remarks are presented in Section 4. 2. Adaptive FSHPS scheme with parameter identification Consider a class of uncertain chaotic system described by x˙ = f (x) + F (x)θ,
(1)
where x = (x1 , x2 , . . . , xn )T ∈ R n is the state vector of the system, f : R n → R n is a continuous vector function, F : R n → R n×p is a function matrix, θ ∈ R p is an unknown parameter vector. Remark 5. Nonlinear dynamical system (1) depends linearly on the parameters and many well-known chaotic (hyperchaotic) system belong to (1), such as Lorenz, Chen, Lü chaotic system, hyper-chaotic Chen, Rössler system, and so on. Let system (1) be the drive system, then the controlled response system is given by y˙ = f (y) + F (y)θˆ + u,
(2)
where y ∈ R n is the state vector, θˆ ∈ R p represents the estimate vector of unknown parameter vector θ , u ∈ R n is a controller to be determined. The goal of control is to find out a appropriate u such that the response system is full state hybrid projective synchronous with the drive system and the unknown parameters are identified simultaneously. Let the vector error state be e(t) = y(t) − H x(t), where H is a n-order diagonal matrix, i.e., H = diag(h1 , h2 , . . . , hn ). Thus the error dynamical system between the drive system (1) and the response system (2) is e(t) ˙ = y˙ − H x˙ = f (y) − Hf (x) + F (y)θˆ − H F (x)θ + u.
(3)
Let f˜ = f (y) − Hf (x), F˜ = F (y) − H F (x), and θ˜ = θˆ − θ , then Eq. (3) is e(t) ˙ = f˜ + H F (x)θ˜ + F˜ θˆ + u.
(4)
Let u = −f˜ − F˜ θˆ + Ae,
(5)
where the matrix A is chosen such that it has all its eigenvalues on the left-hand side of the complex plane. From (4) and (5), the error dynamics is described by e˙ = Ae + H F (x)θ˜ .
(6)
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Hence the FSHPS problem becomes the stability of error dynamics (6). If it is globally stabilized at the origin, the FSHPS of drive system (1) and response system (2) can be globally realized. Now, we give our main result. Theorem 1. If there exists a positive symmetric matrix P such that AT P + P A = −Q,
(7)
where Q denotes a positive symmetric matrix, and the parameter estimation update law is chosen as θ˙ˆ = −F (x)T H P e
(8)
then the error system (6) is globally stable. Hence, the response system (2) associated the proposed control law (8) and the drive system (1) can globally asymptotically achieve the FSHPS, and the unknown parameters θ may be dynamically estimated from θˆ in the response system simultaneously. Proof. For the system (6), consider the Lyapunov function candidate V (e, θ˜ ) = eT P e + (θˆ − θ )T (θˆ − θ )
(9)
so the derivative with respect to time is V˙ = e˙T P e + eT P e˙ + (θˆ − θ )T θ˙ˆ + θ˙ˆ T (θˆ − θ ) = eT AT P + P A e = −eT Qe 0.
(10)
It is obvious that V˙ = 0 if and only if ei = 0, i = 1, 2, . . . , n, namely the set E = {(e, θˆ )T ∈ R 2n : e(t) = 0, θˆ = θ} is the largest invariant set contained in M = {V˙ = 0} for Eq. (6). So, according to the well-known invariance principle of functional differential equations [18,27], starting with arbitrary initial values of the system (6) associated the proposed control law (8), the orbit converges asymptotically to the set E, i.e., y → H x and θˆ → θ as t → ∞. Then, the FSHPS and parameter identification of coupled identical chaotic systems is achieved simultaneously. The proof is completed. 2 Remark 6. It is noted that, as H = I and system is hyperchaotic, the synchronization reduced to the one studied in [28]. Further assume F : R n → R n×n is a symmetric matrix, the synchronization problem has been studied in [16]. The main difference between the existing works on synchronization and parameter identification and the present Letter is the synchronization phenomena, i.e., FSHPS here instead of complete synchronization. So our results improve and generalize those existing in the literature. 3. Applications of adaptive FSHPS scheme In this section, chaotic Chen system and hyper-chaotic Chen system are adopted to verify the effectiveness of the adaptive FSHPS scheme with parameter identification obtained in the previous section.
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3.1. Adaptive FSHPS of Chen systems Consider the Chen chaotic system described by the following equations [29]: ⎧ ⎨ x˙1 (t) = a(x2 − x1 ), (11) x˙ (t) = (c − a)x1 + cx2 − x1 x3 , ⎩ 2 x˙3 (t) = x1 x2 − bx3 . And it can be presented in the form of ⎞ ⎛ ⎛ ⎞ ⎛ 0 0 x2 − x1 x˙1 ⎝ x˙2 ⎠ = ⎝ −x1 x3 ⎠ + ⎝ −x1 0 x˙3 x1 x2 0 −x3
⎞⎛ ⎞ 0 a x1 + x2 ⎠ ⎝ b ⎠ . 0 c (12)
Similarly, the response system becomes ⎛ ⎞ ⎛ ⎞ y˙1 0 ⎝ y˙2 ⎠ = ⎝ −y1 y3 ⎠ y˙3 y1 y2 ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 0 y 2 − y1 aˆ u1 ⎝ ⎠ ⎝ ⎠ ⎝ ˆ = −y1 0 y1 + y2 b + u2 ⎠ , (13) 0 −y3 0 u3 cˆ ˆ cˆ are the estimates where a, b, c are unknown parameters, a, ˆ b, of a, b, c, respectively. u = (u1 , u2 , u3 )T is the controller vector. We define the parameters errors as a˜ = aˆ − a, b˜ = bˆ − b, c˜ = cˆ − c, and the errors in the variables as e1 = y1 − h1 x1 , e2 = y2 − h2 x2 , e3 = y3 − h3 x3 . If the matrix A is chosen as A = diag(−1, −1, −1), P is chosen as an identity matrix. Then, according to Eq. (5) and Eq. (8) in the theorem, we get the controller ⎧ u1 = (−y2 + y1 + h1 (x2 − x1 ))aˆ − e1 , ⎪ ⎪ ⎨ u = −h x x + y y + (−y + h x )aˆ 2 2 1 3 1 3 1 2 1 (14) ⎪ − (y2 + y1 − h2 (x1 + x2 ))cˆ − e2 , ⎪ ⎩ u3 = h3 x1 x2 − y1 y2 + (y3 − h3 x3 )bˆ − e3 ˆ cˆ obey the following update laws: and the estimates a, ˆ b, ⎧ ˙ ⎪ ⎨ aˆ = −h1 e1 (x2 − x1 ) + h2 x1 e2 , (15) b˙ˆ = h3 e3 x3 , ⎪ ⎩˙ cˆ = −h2 e2 (x1 + x2 ). Therefore, parameters identification and FSHPS of the two chaotic Chen systems can be achieved simultaneously. In numerical simulations, the “unknown” parameters are chosen to be a = 35, b = 3 and c = 28 so that Chen system has a chaotic attractor, the initial states of the drive system and response system are x1 (0) = 1, x2 (0) = 2, x3 (0) = 3, y1 (0) = 11, y2 (0) = 12, y3 (0) = 13, the parameters have initial conditions a(0) = 1, b(0) = 1, c(0) = 1 and h1 = 1, h2 = −1, h3 = 2. The simulation results are shown in Figs. 1–4. From Fig. 1, one can see the error vector e converge to zero as time t goes to infinity. This shows the FSHPS of two Chen chaotic systems with unknown parameters is realized. Fig. 2 shows the results of parameters identification and demonstrates that the parameters a, b, c adapt themselves to the true values. Figs. 3 and 4 show the slight effect of a noise with the strength 6 added to the
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Fig. 1. FSHPS errors of two Chen systems with unknown parameters.
Fig. 2. Parameters a, b, c of Chen system adapt themselves to the true values.
signal x2 . From the numerical simulation, we can see that the proposed FSHPS scheme is effective, and quite robust against the effect of noise. 3.2. Adaptive FSHPS of hyper-chaotic Chen systems Consider the hyper-chaotic Chen system, generated via state feedback control based on Chen chaotic system, described
by [30] ⎧ x˙1 (t) = a(x2 − x1 ) + x4 , ⎪ ⎪ ⎨ x˙ (t) = dx − x x + cx , 2 1 1 3 2 ⎪ x ˙ (t) = −bx + x x , 3 3 1 2 ⎪ ⎩ x˙4 (t) = −rx4 + x2 x3 ,
(16)
where a, b, c, d, r are unknown parameters. And it can be presented in the form of
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Fig. 3. The errors under the effect of noise with the strength 6 in time series x2 .
⎛
⎞ ⎛ ⎞ x˙1 x4 ⎜ x˙2 ⎟ ⎜ −x1 x3 ⎟ ⎜ ⎟=⎜ ⎟ ⎝ x˙3 ⎠ ⎝ x1 x2 ⎠ x˙4 x2 x3 ⎛
Eq. (8) in the theorem, we get the controller:
x 2 − x1 ⎜ 0 +⎜ ⎝ 0 0
0 0 −x3 0
0 x2 0 0
0 x1 0 0
⎛ ⎞ ⎞ a 0 ⎜b⎟ ⎜ ⎟ 0⎟ ⎟⎜ c ⎟. ⎟ ⎠ 0 ⎜ ⎝d ⎠ x4 r
⎧ u1 = −y4 + h1 x4 − (y2 − y1 − h1 (x2 − x1 ))aˆ − e1 , ⎪ ⎪ ⎨ u = y y − h x x − (y − h x )cˆ − (y − h x )dˆ − e , 2 1 3 2 1 3 2 2 2 1 2 1 2 ˆ ⎪ u = −y y + h x x − (h x − y ) b − e , 1 2 3 1 2 3 3 3 3 ⎪ ⎩ 3 u4 = −y2 y3 + h4 x2 x3 − (y4 − h4 x4 )ˆr − e4 (19) (17)
⎧ ⎪ a˙ˆ = −h1 e1 (x2 − x1 ), ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎨ bˆ = h3 e3 x3 , c˙ˆ = h2 e2 x2 , ⎪ ⎪ ⎪ ⎪ d˙ˆ = h2 e2 x1 , ⎪ ⎪ ⎩˙ rˆ = h4 e4 x4 .
Similarly, the response system becomes ⎛
⎞ ⎛ ⎞ y˙1 y4 ⎜ y˙2 ⎟ ⎜ −y1 y3 ⎟ ⎜ ⎟=⎜ ⎟ ⎝ y˙3 ⎠ ⎝ y1 y2 ⎠ y˙4 y2 y3 ⎛
y 2 − y1 ⎜ 0 =⎜ ⎝ 0 0
0 0 −y3 0
0 y2 0 0
0 y1 0 0
ˆ c, ˆ rˆ are updated according to the foland the estimates a, ˆ b, ˆ d, lowing laws:
⎛ aˆ ⎞
⎞ ⎛ ⎞ 0 u1 ⎜ bˆ ⎟ ⎜ ⎟ ⎜ u2 ⎟ 0⎟ ⎟ ⎜ cˆ ⎟ + ⎜ ⎟ , ⎟ ⎝u ⎠ 0 ⎠⎜ 3 ⎝ dˆ ⎠ y4 u4 rˆ (18)
ˆ c, ˆ rˆ are the estimates of a, b, c, d, r, respectively. where a, ˆ b, ˆ d, u1 , u2 , u3 , u4 are the controllers to be determined. We define the parameters errors as a˜ = aˆ − a, b˜ = bˆ − b, c˜ = cˆ − c, d˜ = dˆ − d, r˜ = rˆ − r, and the errors in the variables as e1 = y1 − h1 x1 , e2 = y2 − h2 x2 , e3 = y3 − h3 x3 , e4 = y4 − h4 x4 . If the matrix A is chosen as A = diag(−1, −1, −1, −1), P is chosen as an identity matrix. Then, according to Eq. (5) and
(20)
Therefore, with the choice of adaptive controller u (19) and the update law (20), parameters identification and FSHPS of the two hyper-chaotic Chen systems can be achieved simultaneously. In numerical simulations, the “unknown” parameters are chosen to be a = 35, b = 3, c = 12, d = 7 and r = 0.5 so that hyper-chaotic Chen system has a chaotic attractor [30], the initial states of the drive system and response system are x1 (0) = 3, x2 (0) = −4, x3 (0) = 2, x4 (0) = 2, y1 (0) = 2, y2 (0) = 1, y3 (0) = 3, y4 (0) = 4, the parameters have initial conditions a(0) = 9, b(0) = 2, c(0) = 7, d(0) = 1, r(0) = 0.1 and h1 = 1, h2 = −1, h3 = 2, h4 = −1/2. The FSHPS errors ˆ c, ˆ rˆ are shown in Fig. 5 and the estimated parameters a, ˆ b, ˆ d, are displayed in Fig. 6. The numerical simulations under the effect of noise can also verify the effectiveness of the proposed scheme. Here we omit them.
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Fig. 4. Estimated parameters under the effect of noise with the strength 6 in time series x2 .
Fig. 5. FSHPS errors of two hyper-chaotic Chen systems with unknown parameters.
4. Conclusions In this Letter, we study the FSHPS of a class of chaotic systems with unknown parameters. Based on the active control idea and the invariance principle of differential equations, a systematic adaptive scheme has been developed to investigate the FSHPS between the drive system and response system in the presence of unknown parameters and to identify the unknown parameters, simultaneously. Numerical simulations on chaotic
Chen system and hyper-chaotic Chen system are presented to verify the theoretical results. As is known, the mainstream of research topic on chaos synchronization is making two chaotic systems be complete synchronization. For achieving complete synchronization, a wide variety of approaches have been proposed, such as linear and nonlinear feedback, time-delay feed-back, adaptive control, impulsive control, observer-based control, fuzzy control, active control theory, the scalar signal method, etc. But from the view-
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Fig. 6. Parameters a, b, c, d, r of the hyper-chaotic Chen system adapt themselves to the true values.
point of application and practical, generalized synchronization may be wider or more practical than those of complete synchronization [31]. The FSHPS is generalized synchronization when scaling matrix H is invertible, so the main contribution of the present Letter is (1): in one hand, realizing FSHPS of chaotic and hyper-chaotic systems with unknown parameters by using a unified method; (2) in the other hand, identifying parameter based on generalized synchronization. In the other side, in the course of using some of the aforementioned chaos control methods, such as linear feedback, to investigate the FSHPS of chaotic (hyper-chaotic) systems, we found that the difficulty went beyond that we had expected. Future work along this line will be focused on the following problems: (i) extending the presented result to a coupled system of more general (non)identical chaotic systems in the presence of parameter mismatch, including those governed by delayed differential equations; (ii) designing some other effective controllers, for example, observer-based controller, to realize the FSHPS of two or many (complex networks) chaotic systems. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. References [1] S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, Phys. Rep. 329 (2000) 103.
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