International Journal of Bifurcation and Chaos, Vol. 14, No. 6 (2004) 2133–2141 c World Scientific Publishing Company
ESTIMATION OF MODEL PARAMETERS FROM TIME SERIES BY OPCL AUTO-SYNCHRONIZATION IOAN GROSU Faculty of Medical Bioengineering, University of Medicine and Pharmacy “Gr.T.Popa” Iasi, Romania
[email protected] Received February 6, 2002; Revised June 11, 2002 OPCL autosynchronization ([1997] Phys. Rev. E56, 3709–3711) is used for determination of parameters in 1-D models. Numerical results are given for noisy Duffing and logistic systems. The proposed algorithm works also for models that contain parameters nonlinearly. Numerical results show that the error function has a global minimum. Keywords: Parameter estimation; system identification; global minimum; OPCL.
1. Introduction System identification is a research activity of large interest from science and engineering to biology and medicine [Ljung, 1999; Hayashi & Sakamoto, 1986]. Usually, one deals with a noisy scalar time series obtained from multivariable systems. From these data we need to understand as much as possible. If there are some ideas about the equations governing the dynamical system from which the time series was recorded, it is possible to propose a model, and then to determine the unknown parameters. There are known several methods of system identification [Voss & Kurths, 1997; Timmer et al., 2000; Horbelt, 2001]. Synchronization has known much interest in the last 12 years [Pikovsky, 2001; Mosekilde, 2002]. Also, synchronization was used in system identification [Parlitz, 1996; Parlitz et al., 1996; Dedieu & Ogorzolek, 1997; Maybhate & Amritkar, 1999, 2000]. In 1995 an advanced powerful method of control has been introduced: the Open-Plus-Closed-Loop (OPCL) method [Jackson & Grosu, 1995]. Later, OPCL was used for synchronization of two identical dynamic systems in a master-slave manner [Grosu, 1997]. OPCL offers a driving for a system to achieve any desired dynamics but the price paid is that all
variables are to be known, and to add drive terms in all equations. Let us recall the main lines of OPCL [Jackson & Grosu, 1995; Grosu, 1997]. A general system dx = F(x) , x ∈ Rn (1) dt has to be driven in order to achieve goal dynamics g(t). The driving term is: dg dg D x, g, = D1 g, + D2 (x, g) dt dt dg = − F (g) dt dF(g) (x − g) (2) + A− dg and the driven system: dx dg = F(x) + SD x, g, dt dt
(3)
assures that x(t) → g(t) if kx(0) − g(0)k is small enough. In Eq. (2), A is a Hurwitz matrix [Jackson & Grosu, 1995; Grosu, 1997] and S is 0 or 1 as a switch. If g(t) is the dynamics of an identical system let us say a master system X(t) then D 1 (X, dX/dt) is zero and the slave system has to be driven just 2133
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with the driving term D2 (x, X). Now the synchronization between a master system: dX = F(X) dt and a driven slave system dF(X) dx (x − X) = F(x) + A − dt dX
(4)
(5)
is assured [Grosu, 1997] when kx(0)−X(0)k is small enough. Let us consider that the master system contains the parameters p: dX = F(X, p) dt and the slave (the proposed model)
(6)
dx = F(x, q) (7) dt where the parameters q are unknown. The key idea is to “synchronize” the model (7) to the master (6) with the driving (5), which can be rewritten in this case as: dF(X, q) dx (x − X) (8) = F(x, q) + A − dt dX
2. Identification of a Chaotic Duffing System
and to calculate E(q) =
Z
t2
where T(X, x − X, q) is the rest of the Taylor expansion. In order to have a larger basin of entrainment of the slave to the master dynamics, it is necessary to add in (5) or in (8) as many as possible terms of T. For a polynomial F(x, q), T(X, x − X, q) has a finite number of terms. So for a polynomial F(x, q) the entrainment x(t) to X(t) can be realized for any x(0) and X(0). Unfortunately there is still a disadvantage with this strategy. All components of X(t) are required to be known. How to overpass this drawback is a challenge for the future. In the next sections the identification will be addressed when just one component is necessary. From case to case the driving in (5) is simplified by a proper choice of the matrix A [Grosu, 1997]. The paper is organized as follows. In the next section the above strategy will be applied to the identification of a chaotic Duffing system. Section 3 deals with identification of the logistic dynamics. Section 4 presents the main result namely that a model that depends nonlinearly on parameters has also a global minimum for the error function. Section 5 is for conclusions.
(x(t) − X(t))2 dt
(9)
t1
where X(t) is the solution of (6) and x(t) is the solution of (8), t1 has to be large enough to be sure that the transients decay away. The time t 2 can be changed at our desire. The function E(q) has a minimum E(p) = 0. Numerical results will show that E(q) has a global minimum E(p) = 0. In order to find this minimum, the parameters q can be modified in an adaptive manner: qk+1 = qk −
η dE(qk ) dqk
(10)
where η has to be chosen in order to assure that E(qk ) is decreasing. In (5) the synchronization achieved when kX(0) − x(0)k is small enough is a disadvantage. To overpass this drawback the driving in (8) has to be completed. Let us consider the Taylor expansion: F(x, q) = F(X + (x − X), q) (x − X)dF (X, q) dX + T(X, x − X, q)
= F(X) +
(11)
An example of OPCL synchronization [Grosu, 1997] is the synchronization of two identical chaotic Duffing systems: adX d2 X + + X 3 = B cos (t) dt2 dt d2 x adx + + x3 = B cos (t) dt2 dt + (3X 2 − p)(x − X)
(12)
(13)
This synchronization looks practical because all that is required is for the signal X(t) to be known. The above systems (12) and (13) synchronize for any p > 0 and X(0)−x(0) small enough. In order to enlarge the basin of entrainment (13) is completed with a quadratic term: d2 x adx + + x3 = B cos (t) + (3X 2 − p)(x − X) dt2 dt + 3X(x − X)2 . (14) Now (12) and (14) give two synchronized dynamics for any X(0) − x(0). Using (12) and (14) we wish to determine the coefficients a and B. We record X(t)
Estimation of Model Parameters from Time Series by OPCL Auto-Synchronization
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(a)
(b) Fig. 1.
Figure 1
Numerical values of E(a1 , B1 ) (16) with no noise (a) (a1 , B1 ) on large intervals, (b) (a1 , B1 ) around (0.1, 10).
from (12) for a = 0.1 and B = 10 and propose a model (unknown a1 and B1 ) driven as in (14):
We calculate the error: Z E(a1 , B1 ) =
dx d2 x + a1 + x3 = B1 cos (t) + (3X 2 − p)(x − X) 2 dt dt + 3X(x − X)2 (15)
E(a1 , B1 ) is represented in Fig. 1. We can observe that E(a1 , B1 ) has a global minimum for a1 = 0.1 and B1 = 10.
t2
(x(t) − X(t))2 dt
(16)
t1
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(a)
Figure 2 Figure 2 (b)
Fig. 2. Numerical values of E(a1 , B1 ) (16) for noisy time series z(t) = X(t) + 0.01 (random −0.5): (a) (a1 , B1 ) on large intervals, (b) (a1 , B1 ) around (0.1, 10).
Estimation of Model Parameters from Time Series by OPCL Auto-Synchronization
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(a)
(b) Fig. 3.
Numerical values of E(a1 , a2 ) (20) with no noise (a) (a1 , a2 ) on large intervals and (b) (a1 , a2 ) around (3.8, −3.8).
Figure 33 Figure
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(a)
Figure 4 Figure 4 (b)
Fig. 4. Numerical values of E(a1 , a2 ) (20) for zn = Xn + 0.005 (random −0.5) (a) (a1 , a2 ) on large intervals, (b) (a1 , a2 ) around (3.8, −3.8).
Estimation of Model Parameters from Time Series by OPCL Auto-Synchronization
Next, we consider that X(t) has not been measured but z(t) = X(t) + 0.01 (random −0.5), where the last term is a uniform noise between −1 and 1. In (15) and (16) X(t) was replaced by z(t). In Fig. 2 we observe again that E(a1 , B1 ) has a minimum around a1 = 0.1 and B1 = 10. To decide the global minimum we can change several times t 2 in (16). The global minimum will be a minimum every time.
3. Identification of Logistic Dynamics A robust synchronization of two identical logistic maps has been proposed [Grosu, 1997]: Xn+1 = cXn (1 − Xn )
(17)
xn+1 = cxn (1 − xn ) + (0.1 − c(1 − 2Xn ))(xn − Xn ) .
(18)
Again, (17) and (18) assure synchronization for x0 − X0 small enough. We complete (18) with a quadratic term: xn+1 = cxn (1 − xn ) + (0.1 − c(1 − 2Xn ))(xn − Xn ) − c(xn − Xn )2
4. Identification of Systems with Nonlinear Dependence on Parameters In the above two sections we observed that the error function has a global minimum and we conjecture that the present algorithm drives the model in such a way that the error function has always a global minimum. We do not have at the moment a general rigorous proof of this result. The advantage of the OPCL auto-synchronization based method of system identification can be more clearly seen in the case when the time series is obtained from a dynamical system that has a nonlinear dependence on parameters. Simple least-squares methods of system identification [Oprisan, 2002] work well for systems where the proposed model depends linearly on parameters but they cannot work in the nonlinear case. We consider the model: 10xn (21) xn+1 = f (xn , b) = 1 + (xbn ) This model has different types of behavior. A bifurcation diagram is shown in Fig. 5. We consider the driven model:
(180 )
In this way, (17) and (180 ) assure the synchronization for any x0 − X0 . Based on this, we propose the identification of the system (17) knowing the time series Xn . For this, we propose a model (unknown a1 and a2 ) driven like in (180 ):
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yn+1 = f (yn , b1 ) + (0.02 − D1 )(yn − xn ) − D2 (yn − xn )2
(22)
where D1 =
df (xn , b1 ) dxn
(23)
D2 =
0.5d2 f (xn , b1 ) dx2n
(24)
xn+1 = a1 xn + a2 x2n + (0.1 − (a1 + 2a2 Xn ))(xn − Xn ) − a2 (xn − Xn )
2
where Xn is the dynamics from (17). We calculate n2 X E(a1 , a2 ) = (xn − Xn )2
(19)
We record xn from (21) with b = 4.3 and calculate E(b1 ) =
with Xn from (17) with c = 3.8. In Fig. 3, we observe that E(a1 , a2 ) has a minimum for a1 = 3.8 and a2 = −3.8. If we measure zn = Xn + 0.005 (random −0.5), then we replace Xn with zn in (19) and (20). In Fig. 4 the minimum of E(a1 , a2 ) is around (a1 , a2 ) = (3.8, −3.8). To decide more exactly the global minimum we need to modify n2 in (20); the global minimum will be a minimum for every value of n2 .
(xn − yn )2
(25)
n1
(20)
n1
n2 X
and 0
E (b1 ) =
n2 X
(xn − zn )2
(26)
n1
where zn+1 = f (zn , b1 )
(27)
The last term represents a surrogate of the expression used in least square methods of system identification. In Fig. 6 are represented E(b 1 ) and E 0 (b1 ). We can observe that E(b1 ), given by the
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Fig. 5.
4 0.8
4.1
0.7
Bifurcation diagram of model (17).
4.2
b1 4.3
4.4
4.5
Figure 5
0.6
4.6 800 700 600
E'(b1)
E(b1)
0.5 500
0.4 03 0.3
400
0.2
300
0.1 4
200 4.1 Fig. 6.
4.2
b1 4.3
4.4
4.5
4.6
E(b1 ) (25) and E 0 (b1 ) (26) for n1 = 50, n2 = 100.
OPCL auto-synchronization has a global minimum for b1 = 4.3 and approaches the minimum smoothly and continuously. The term E 0 (b1 ) has also a minimum at b1 = 4.3 but it approaches it in an irregular manner, which does not look to be even continuous. By our best knowledge the other reported results dealt with systems that depend linearly on the
parameters. There are very new results [Sakaguki, 2002] that use the random optimization method for the same reason. The present algorithm can be applied for identification of any 1-D population models [Sakai, 2001]. As mentioned before, it is necessary to extend the present algorithm to multivariable time series. Preliminary results show that the extension is not straightforward.
Figure 6
Estimation of Model Parameters from Time Series by OPCL Auto-Synchronization
5. Conclusions In this work, OPCL control strategy was used as a rigorous method of auto-synchronization based on system identification. In the case of 1-D models, both continuous and discrete, the error function had a global minimum, and approaches it smoothly and continuously. The effectiveness of the present method is seen in the case of a Duffing system and a logistic map. The main advantage of this method over usual synchronization or least-square based strategies is illustrated in the case of a system with nonlinear dependence on parameters. Our method shows a smooth global minimum.
Acknowledgments The author is grateful for stimulating discussion and help in the preparation of this work from A. I. Lerescu, and assistance with illustrations from M. Negoita, L. Slabu, and assistance with the language from C. Grosu.
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