Chaos synchronization and parameter estimation from a scalar output

PHYSICAL REVIEW E 76, 027203 共2007兲

Chaos synchronization and parameter estimation from a scalar output signal 1

Maoyin Chen1,2 and Jürgen Kurths2

Department of Automation, Tsinghua University, Beijing 100084, China Institut für Physik, Potsdam Universität, Am Neuen Palais 10, D-14469, Germany 共Received 29 May 2007; published 29 August 2007兲

2

We propose an observer-based approach for chaos synchronization and parameter estimation from a scalar output signal. To begin with, we use geometric control to transform the master system into a standard form with zero dynamics. Then we construct a slaver to synchronize with the master using a combination of slide mode control and linear feedback control. Within a finite time, partial synchronization is realized, which further results in complete synchronization as time tends to infinity. Even if there exists model uncertainty in the slaver, we can also estimate the unknown model parameter by a simple adaptive rule. DOI: 10.1103/PhysRevE.76.027203

PACS number共s兲: 82.40.Bj, 43.72.⫹q, 47.52.⫹j

Chaotic synchronization has been the focus of a growing literature during the last decade 关1–10兴. The original synchronization technique was developed by Pecora and Carroll 关1兴. In their seminal paper, they addressed the synchronization of chaotic systems in the drive-response coupling scheme. A chaotic system, called the master, generates a signal which is sent over a channel to a slaver, which uses this signal to synchronize itself with the master. Recently, chaos synchronization has been successfully used for the estimation of unknown model parameters in chaotic systems. The active-passive-decomposition method 关11兴, the minimization strategy 关12–14兴, the iterative method 关15兴, and statistical methods 关16兴 can be applied to estimate unknown model parameters in chaotic systems. Based on the well-known Lasalle invariance principle of differential equations, Huang and Guo also proposed an adaptive approach to estimate all unknown parameters 关17兴. Freitas et al. treated synchronization as a control problem and utilized geometric control to estimate unknown model parameters 关18兴. In this paper we also consider the estimation of unknown parameters in chaotic systems from a scalar output signal. This signal could be either a variable from the master or a scalar nonlinear combination of all variables from the master. We first transform the master into a standard form with zero dynamics by geometric control theory. Using the combination of slide mode control and linear feedback control, we then construct a slaver to synchronize itself with the master. Within a finite time, partial synchronization is first realized, which further results in complete synchronization as time tends to infinity. Even if there exists model uncertainty in the slaver, we can also estimate the unknown parameter by a simple adaptive rule. Presently, our strategy can estimate a single unknown parameter of the master, but this could be easily extended to many parameters. Consider the chaotic systems x˙ = f共x兲 + g共x兲p, s共t兲 = h共x兲,

共1兲

where f共x兲 , g共x兲 苸 Rn are nonlinear functions, x 苸 Rn is the state, s共t兲 苸 R is the scalar output signal, and p 苸 R is an unknown model parameter. Our problem is to identify the 1539-3755/2007/76共2兲/027203共4兲

unknown parameter p from the time series s共t兲. Throughout the paper, we assume that the parameter p is slowly varying: namely, p˙ ⬇ 0. Within the drive-response coupling scheme 关1兴, system 共1兲 is regarded as the master and the signal s共t兲 is the driving signal to make a slaver synchronize with the master. Therefore, we should construct a suitable slaver, which can be used to estimate the unknown parameter p. For completeness of this paper, we only introduce two concepts in the geometric control theory. A detailed background on geometric control is given in Refs. 关18,19兴. The first concept is the Lie derivative operator L—that is, n

L0f h共x兲

= h共x兲,

L jf h共x兲

=兺 i=1

⳵ L j−1 f h f i共x兲 ⳵ xi

共j ⱖ 1兲.

共2兲

The second concept is the relative degree. The scalar signal h共x兲 has a relative degree r corresponding to the function g共x兲 at point x0 苸 M, where M is a smooth manifold of dimension n, if LgLi−1 f h共x兲 = 0,

共3兲

for all x in a neighborhood of x0 and all i ⬍ r − 1, and LgLr−1 f h共x0兲 ⫽ 0.

共4兲

If the signal h共x兲 has a relative degree r for all x 苸 M, we can transform the master 共1兲 into a standard form by a coordinate transformation 关19兴. This idea has been already used in Ref. 关18兴. However, there are several restrictions in this work. On the one hand, the estimation of the unknown parameter 关namely, the controller ˜u given by Eq. 共15兲 in Ref. 关18兴兴 requires all states of the master. On the other hand, the relative degree corresponding to the gain H共x兲 of the control is n, which is denoted by Eq. 共10兲 in Ref. 关18兴. In fact, within the driving-response scheme, it is better that only one scalar signal is required to drive the slaver, and the unknown parameter should be estimated by this scalar signal. Let z1 = ␾1共x兲 = h共x兲 , z2 = ␾2共x兲 = L f h共x兲 , . . . , zr = ␾r共x兲 = Lr−1 h共x兲. It can be shown 关19兴 that there always exist n f − r functions ␾i 共i = r + 1 , . . . , n兲 such that Lg␾i共x兲 = 0 and the Jacobian of the matrix function ⌽共x兲 = 关␾1共x兲 , . . . , ␾n共x兲兴T = 关z1 , . . . , zn兴T is defined in the manifold M. Then the master 共1兲 can be transformed into the following dynamics:

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z˙ j = z j+1,

is unknown but bounded time-varying disturbance. We construct a linear feedback control

j = 1,2, . . . ,r − 1,

z˙r = a共z, ␩兲 + b共z, ␩兲p,

u = k共␰r − zˆr兲 + ␦2共␰r − zˆr兲,

␩˙ = f 0共z, ␩兲,

共5兲

where the output y = z1 , z = 共z1, . . . ,zr兲,

␩ = 共zr+1, . . . ,zn兲, a共z, ␩兲 = Lrf h„⌽−1共z, ␩兲, ␩…, and

where k ⬎ 0. Therefore we should choose suitable parameters wi to make the partial synchronization be realized within a finite time; namely, the errors ei共t兲 = zi共t兲 − zˆi共t兲 are zero after a finite time. Let 兩ei兩max = maxtⱖ0 兩 ei共t兲兩. For i = 1 , 2 , . . . , r − 1, consider the sliding surface ␴i : ei = 0 and the Lyapunov function Vi = 21 e2i . It is clear that V˙i = eiei+1 − wiei sgn共ei兲 ⬍ 0 if wi is chosen such that wi ⬎ 兩ei+1兩max. Thus ␴i is attractive and can be reached in a finite time T ⬎ 0. During the sliding surface ␴i, we also get e˙i = 0. Hence it follows that for i = 1 , 2 , . . . , r − 1 and for all t ⬎ T, 0 = zi+1 − zˆi+1 − wi sgn共ei兲,

−1 b共z, ␩兲 = LgLr−1 f h„⌽ 共z, ␩兲, ␩….

Moreover, the dynamics ␩˙ = f 0共0 , ␩兲 is called the zero dynamics 关19兴, which means that limt→⬁关␩共t兲 − ␩⬘共t兲兴 = 0 with ␩共t兲 and ␩⬘共t兲 being the solutions of ␩˙ = f 共␰ , ␩兲 and ␩˙⬘

zi+1 = zˆi+1 + wi sgn共ei兲, zi+1 = ␰i+1 .

0

= f 0共␰ , ␩⬘兲, respectively. In fact, the meaning of zero dynamics has been already implied in a PC decomposition-based synchronization 关1兴. For an autonomous n-dimensional chaotic system u˙ = f共u兲, we first divide the system into two subsystems v˙ = g共v , w兲 and w˙ = h共v , w兲, where v = 共u1 , . . . , um兲, g = (f 1共u兲 , . . . , f m共u兲), w = 共um+1 , . . . , un兲, and h = (f m+1共u兲 , . . . , f n共u兲). Then we create a new subsystem w˙⬘ = h共v , w⬘兲, and we obtain an aug˙ = h共v , w兲 , w ˙ ⬘ = h共v , w⬘兲. If the mented system v˙ = g共v , w兲, w dynamics w˙ = h共v , w兲 is asymptotically stable for all v, w⬘ can approach w as time tends to infinity. It means that subsystems w and w⬘ are synchronized with the help of the driving signal v. Generally speaking, if the driving signal is chosen suitably, many chaotic systems including the Lorenz system and Rössler system have zero dynamics 共see Table I in Ref. 关1兴兲. For the master 共1兲 and its transformed form 共5兲, we construct the model 共a slaver兲 zˆ˙ j = zˆ j+1 + w j sgn共␰ j − zˆ j兲,

j = 1,2, . . . ,r − 1,

zˆ˙r = a共␰, ␩⬘兲 + b共␰, ␩⬘兲q + ⌬共␰, ␩⬘兲 + u,

␩˙ ⬘ = f 0共␰, ␩⬘兲,

共6兲

with the estimated partial states

␰1 = s共t兲 = z1 , ␰i+1 = zˆi+1 + wi sgn共␰i − zˆi兲,

i = 1,2, . . . ,r − 1,

共7兲

where wi 共i = 1 , 2 , . . . , r − 1兲 are parameters, ⌬共␰ , ␩⬘兲 is the model uncertainty, u is the external input, and the parameter q is adjusted by q˙ = 共␰r − zˆr兲b共␰, ␩⬘兲.

共9兲

共8兲

We assume that the model uncertainty 兩⌬共␰ , ␩⬘兲 兩 ⱕ ␦共␰ , ␩⬘兲d共t兲, where ␦共␰ , ␩⬘兲 is the known function and d共t兲

共10兲

A detailed proof of the attractiveness of the sliding surfaces ␴i and computation of the instant T are given in 关6兴. It is known that the time evolution of chaotic systems is confined to the chaotic attractor which can be estimated within a hyperellipsoid using the invariance theorem 关20兴. Let Di+1 be the orthogonal projection of the hyperellipsoid onto the 共i + 1兲th axis of Rn space; then, Di+1 is a line segment of length Li+1. It is clear that the maximum divergence of the error ei+1共t兲 is less than Li+1. Therefore, wi = Li+1 is a suitable choice for the sliding observer gain. The idea of the choice of wi is given in Refs. 关6,20兴. Equations 共7兲 and 共10兲 imply that partial synchronization in the systems 共5兲 and 共6兲 can be first realized within a finite time T. After the instant T, we obtain e˙r = a共z, ␩兲 − a共␰, ␩⬘兲 + b共z, ␩兲p − b共␰, ␩⬘兲q − ⌬ − ker − ␦2er , e˙␩ = f 0共z, ␩兲 − f 0共␰, ␩⬘兲,

共11兲

where e␩ = ␩ − ␩⬘. From the above analysis, if we choose the suitable parameters wi, we get ␰i = zi after the instant T. Since the asymptotical stability of the zero dynamics ␩˙ = f 0共␰ , ␩兲, the error e␩ tends to zero as time approaches infinity. This means that e˙r ⬇ b共␰ , ␩⬘兲p − b共␰ , ␩⬘兲q − ⌬ − ker − ␦2er when time t is sufficiently large. For the error er, let a Lyapunov function Vr = 1 / 2er2 + 1 / 2共p − q兲2. When time t is sufficiently large, we get V˙r ⱕ − ker2 − 关␦2er2 − ␦d兩er兩兴 ⱕ − ker2 + d2/4.

共12兲

Therefore, the error er exponentially decays and is ultimately bounded. Since the parameters k can be freely adjusted, a transient performance and final estimation accuracy are guaranteed. Hence q ⬇ p even if there exists a model uncertainty ⌬共␰ , ␩⬘兲 in the model. From the above analysis, Eqs. 共6兲–共10兲 depend on the “sign” function. Fortunately, this function can be approximated by the continuous function

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FIG. 1. Chaos synchronization and parameter estimation without uncertainty in the model 共16兲. 共a兲–共c兲 show the curves of states of the master 共solid lines兲 and the slaver 共dashed lines兲. 共d兲 shows the true parameter ␤ 共solid line兲 and the estimated parameter ␤ˆ 共dashed line兲.

sgn␧共ei兲 ⬇ ei/共兩ei兩 + ␧兲,

x˙2 = ␤x1 − x2 − x1x3,

x˙3 = − ␪x3 + x1x2 , 共14兲

where ␴, ␤, and ␪ are parameters. In order to show the efficiency of our approach, assume that the parameters ␴ and ␪ are known and the parameter ␤ is unknown. Hence the Lorenz system becomes x˙ = f共x兲 + g共x兲␤ where x = 共x1 , x2 , x3兲, f共x兲 = 共␴共x2 − x1兲 , −x2 − x1x3 , −␪x3 + x1x2兲, and g共x兲 = 共0 , x1 , 0兲. For the case of the driving signal s共t兲 = h共x兲 = x1, we get Lgh共x兲 = 0, L f h共x兲 = ␴共x2 − x1兲, and LgL f h共x兲 = ␴x1. Hence the scalar signal h共x兲 = x1 has a relative degree r = 2 corresponding to g共x兲. In addition, we choose the transformation z1 = ␾1 = x1, z2 = ␾2 = ␴共x2 − x1兲, z3 = ␾3 = x3. Then the transformed system is given by z˙1 = z2,

z˙2 = a共z, ␩兲 + b共z, ␩兲␤,

␩˙ = f 0共z, ␩兲

共15兲

where z = 共z1 , z2兲, ␩ = z3, a共z , ␩兲 = −␴z1z3 − ␴z1 − 共␴ + 1兲z2, b共z , ␩兲 = ␴z1, and f 0共z , ␩兲 = −␪␩ + z21 + z1z2 / ␴. For the system 共15兲, we construct a slaver z˙ˆ 1 = zˆ2 + w sgn共␰1 − zˆ1兲, zˆ˙2 = a共␰, ␩⬘兲 + b共␰, ␩⬘兲␤ˆ + ⌬共t兲 + u,

FIG. 2. Chaos synchronization and parameter estimation with uncertainty ⌬共t兲 = 兩␩ 兩 sin共t兲 in the model 共16兲. 共a兲–共c兲 show the curves of states of the master 共solid lines兲 and the slaver 共dashed lines兲. 共d兲 shows the true parameter ␤ 共solid line兲 and the estimated parameter ␤ˆ 共dashed line兲.

共13兲

where ␧ ⬎ 0 is a sufficiently small number. For a suitable choice of ␧, the continuous action 共13兲 can approach the discontinuous sign function very well. In order to verify our main results, we choose Lorenz system as an example. The dynamics of the Lorenz system is described by x˙1 = ␴共x2 − x1兲,

10 0 (d)

80

共16兲

␩˙ ⬘ = f 0共␰, ␩⬘兲,

with the uncertainty ⌬共t兲. Further, the estimated partial states ␰1 = z1, ␰2 = zˆ2 + w sgn共␰1 − zˆ1兲 and the estimated parameter ␤ˆ

˙ is adjusted by ␤ˆ = ␴␰1共␰2 − zˆ2兲. In this simulation the parameters ␴ = 10, ␪ = 8 / 3, and ␤共t兲 = 28+ 2 sin共0.5t兲. We first consider the case where there exists no uncertainty in the model 共16兲—i.e., ⌬共t兲 = 0. In this case we choose a linear feedback control u = k共␰2 − zˆ2兲 and the parameters w = 0.1, k = 1.5, and ␧ = 2. Simulation results are shown in Fig. 1. The second case is that there is the uncertainty ⌬共t兲 = 兩␩ 兩 sin共t兲 in the model 共16兲. Hence 兩⌬共t兲 兩 = ␦d共t兲 with ␦ = 兩␩兩 and d共t兲 = 1. In this case we choose a linear feedback control u = k共␰2 − zˆ2兲 + ␦2共␰2 − zˆ2兲 and the parameters w = 0.1, k = 1.5, and ␧ = 2. Simulation results are plotted in Fig. 2. From these figures, we show that our approach is very effective to estimate the unknown model parameter in chaotic systems. Compared with Refs. 关11–18兴, our approach has the following merits. 共i兲 We can choose a complex nonlinear function h共x兲 as a driving signal, provided that the transformation ⌽共x兲 exists. The more complex this function is, the more secure the synchronization is. 共ii兲 Even if the model 共6兲 includes the model uncertainty, we can also choose a suitable parameter k to approximately estimate the unknown parameter. In the case of the uncertainty, the parameter estimation is difficult in Refs. 关11,14,17,18兴. 共iii兲 The parameter estimation 共8兲 is adjusted adaptively, and it only requires a scalar output signal. This may also estimate the time-varying parameter. 共iv兲 The speed of partial synchronization can be adjusted by the parameters wi 共i = 1 , 2 , . . . , r兲. In real applications, we can choose suitable values to make the fast synchronization. The authors would like to acknowledge the support of the Alexander von Humboldt Foundation.

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