An approximate gradient-descent method for joint parameter estimation and synchronization of coupled chaotic systems

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1 Physics Letters A 351 (2006) An approximate gradient-descent method for joint parameter estimation and synchronization of coupled chaotic systems Inés P. Mariño a Joaquín Míguez b a Nonlinear Dynamics and Chaos Group Departamento de Matemáticas y Física Aplicadas y Ciencias de la Naturaleza Universidad Rey Juan Carlos Móstoles Madrid Spain b Departamento de Teoría de la Señal y Comunicaciones Universidad Carlos III de Madrid Avda. de la Universidad Leganés Madrid Spain Received 30 August 2005; received in revised form 25 October 2005; accepted 1 November 2005 Available online 9 November 2005 Communicated by A.R. Bishop Abstract We address the problem of estimating the unknown parameters of a primary chaotic system that produces an observed time series. These observations are used to drive a secondary system in a way that ensures synchronization when the two systems have identical parameters. We propose a new method to adaptively adjust the parameters in the secondary system until synchronization is achieved. It is based on the gradientdescent optimization of a suitably defined cost function and can be systematically applied to arbitrary systems. We illustrate its application by estimating the complete parameter vector of a Lorenz system Elsevier B.V. All rights reserved. PACS: Xt; Gg; Pq; j Keywords: Parameter estimation; Chaos synchronization; Chaos control 1. Introduction The problem of estimating the unknown parameters of a chaotic system from an observed time series has recently received considerable attention [1 16]. Alargeclassoftheproposed methods are based on the iterative processing of the complete set of observations in order to obtain a sequence of estimates of the unknown parameters as it is typically the case with the so-called multiple-shooting techniques [12] and various statistical [3 5] and Monte Carlo [67] procedures. However iterative processing is computationally expensive and inadequate for problems where the observations must be handled online. An elegant alternative is to exploit the synchronization properties of chaotic systems to attain parameter estimation as suggested by Parlitz et al. [8].Assumethattheobservedtimeseries * Corresponding author. Tel.: ; fax: addresses: ines.perez@urjc.es (I.P. Mariño) jmiguez@ieee.org (J. Míguez). is produced by a primary system which is known except for (a subset of) its fixed parameters. We can set up a secondary system with the same functional form as the primary one and coupled with the latter in a way that ensure synchronization when the parameter sets in the two systems are identical. If we adaptively adjust the parameters of the secondary system until it becomes synchronized with the primary one then the final values of these parameters are accurate estimates of their counterparts in the primary system. Several variants of this approach have been proposed [8 12] and they usually reduce to the (often ad hoc) design differential equations that govern the evolution of the adjustable parameters until they reach their steady state (in other words the parameters to be adjusted are converted into dynamic state variables). In this Letter we investigate a new method for adaptive parameter estimation based on the synchronization of coupled chaotic systems. Instead of looking for a suitable differential equation we propose a discrete-time (i.e. periodical) update of any adjustable parameters in order to minimize a suitable cost function that involves the derivatives of the state /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.physleta

2 I.P. Mariño J. Míguez / Physics Letters A 351 (2006) variables of both the primary and the secondary systems. We show that working with the derivatives (instead of the state variables themselves as in [8 12]) allows to derive simple gradient-descent cost optimization algorithms in a systematic manner. The application of the method is illustrated including the derivation of the gradient-descent algorithm and its assessment by computer simulations by means of an example using the Lorenz system. The remaining of the Letter is organized as follows. The parameter estimation problem is explicitly stated in Section 2. Then the general proposed method is introduced in Section 3 and its application to the particular case of the Lorenz system presented in Section 4. Computersimulationresultsareshown in Section 5 and finally conclusions are drawn in Section Problem statement Let ẋ = f(x p) represent the primary system with state variables x R n and (partially or totally) unknown parameters p R m.noticethat ẋ represents the temporal derivative of x. Forconcisenesswe usually leave the time dependence of dynamic variables implicit. If the functional form of (1) is known we can build the secondary system as ẏ = f(y x q) where y R n are the state variables q R m are the parameters and h(x) : R n R k is the time series we observed from the primary system. Function f is equal to fexceptforthecoupling through some of the variables in h(x) (see Section 4 for an example). We assume that this coupling is unidirectional and enables asymptotic synchronization of the secondary system (2)i.e.y xaslongasq = p.noticethatseveraltechniques are now available to design unidirectional schemes which guarantee synchronization [17]. Thesystemin(2) is fully observed and we assume the ability to periodically change the value of the parameter vector q every T time units (t.u.). Our aim is to devise an algorithm to adaptively adjust the parameters in the secondary system quntilthesystemvariables yandtheparametersthemselvesconvergetotheircounterparts in the primary system i.e. both y x and q p. Inthisway synchronization between both systems is achieved and the parameters of the primary system are estimated. 3. Parameter estimation method We propose a gradient-descent optimization of an adequate cost function J thatinvolvestheobservedtimesseriesandthe secondary system dynamics. In general we consider functions of the form J(q I) = 1 T I e(τ) 2 dτ (1) (2) (3) where I is a time interval of length T e is a (possibly multidimensional) error-signal consisting of a difference between temporal derivatives of h(x) and h(y) (e.g. e = ḣ(x) ḣ(y)) and denotes vector norm. Obviously e depends on q through the argument of h(y) and therefore the secondary system parameters can be adjusted by minimizing J with respect to q i.e. we select ˆq = arg min J(q I). q If the period T is long enough and e 0impliessynchronization then ˆq is an accurate estimate of p. However solving the optimization problem (4) directly can be hard and does not provide an adaptive method for estimating p. Inordertoovercomebothdifficultiesletusconsider a sequence of non-overlapping consecutive intervals I n n = IftheresultingcostfunctionsJ(q I n )areconvex and have their minima at the same value of q thenwecan approximate the solution of (4) using a simple gradient-descent algorithm of the form q n = q n 1 µ ˆ q J(q I n ) q=qn 1 where µ 1isastep-sizeparameterand [ J ˆ q J q J =... J ] T q 1 q m is an estimate of the gradient of J with respect to the parameter vector q R m. The use of an error signal that consists of the differences between derivatives of the state variables of the primary and secondary systems (instead of the straightforward differences between the state variables themselves) is advantageous because the analytical expression of ẏ is known from (2). Asan example assume the simple time series h(x) = x i wherex i is the ith component of x. Thentheerrorsignale =ẋ i ẏ i is an explicit function of the components of q included in the ith dynamic equation. 1 This is important because the computation of ˆ q J becomes considerably simplified. Specifically we show by way of an example in the next section that ˆ q J can be found analytically in a straightforward manner by considering a few adequate approximations. 4. Application example: the Lorenz system Let us consider a primary system that obeys the Lorenz equations ẋ 1 = σ 1 (x 1 x 2 ) ẋ 2 = r 1 x 1 x 2 x 1 x 3 1 In practice we may need to obtain the derivative ẋ i from the observation of x i.thisisusuallyasimpleoperationalthoughtheresultcanbedistorted by additive noise contaminating x i.anobviousremedyistofiltertheobserved signal beforehand. Moreover we must note that mismatches in the instantaneous derivative ẋ i only affect the instantaneous gradient of the error signal e(τ) 2 whichisintegratedintheintervaliaccordingto(3) thusattaininga convenient smoothing of any errors in ẋ i.thelongertheintervali thebetter this smoothing. (4) (5)

3 264 I.P. Mariño J. Míguez / Physics Letters A 351 (2006) ẋ 3 = b 1 x 3 + x 1 x 2 where x =[x 1 x 2 x 3 ] T R 3 is the dynamic system state and p =[σ 1 r 1 b 1 ] T R 3 contains the parameters. The secondary system is ẏ 1 = σ 2 (y 1 x 2 ) ẏ 2 = r 2 y 1 y 2 y 1 y 3 ẏ 3 = b 2 y 3 + y 1 y 2 where y =[y 1 y 2 y 3 ] T are the state variables and q =[σ 2 r 2 b 2 ] T are the parameters that can be adjusted in order to estimate p and attain synchronization. Notice that the coupling between both systems is carried out by substituting the variable y 2 of the secondary system by the corresponding one of the primary system i.e. x 2 inthedynamicalevolutionequationofthevariable y 1.Thistypeofcouplingguaranteessynchronizationbetween both systems when their parameters are identical [8]. We assume for this example that the observed time series is the complete state of system (6) i.e.h(x) = x andtherefore the error signal can be defined as e = ẋ ẏ. Inordertoapply algorithm (5) weneedtocomputethegradientestimate ˆ q J(q I n ) = 1 T I n ˆ q e(τ) 2 dτ which in turn reduces to finding an approximate expression for q e(τ) 2 in the interval I n.initiallyitiseasytoseethat e(τ) 2 = 2(ẋ 1 ẏ 1 ) ẏ 1 σ 2 σ 2 e(τ) 2 = 2(ẋ 2 ẏ 2 ) ẏ 2 e(τ) 2 = 2(ẋ 3 ẏ 3 ) ẏ 3 (9) b 2 b 2 but unfortunately the derivatives with respect to σ 2 r 2 and b 2 in the right-hand side of (9) cannot be expressed in closed form because of the complex implicit dependence of the dynamic variables y i i = on the parameters. However if we consider only the explicit derivatives (8) becomes very simple. Specifically if we use notation to denote explicit derivation (meaning that implicit dependencies of the variables on the parameters are neglected) we obtain (6) (7) (8) e(τ) 2 = 2(ẋ 1 ẏ 1 )(x 2 y 1 ) σ 2 e(τ) 2 = 2(ẋ 2 ẏ 2 )y 1 e(τ) 2 = 2(ẋ 3 ẏ 3 )y 3. (10) b 2 Taking (7) (8) and (10) togetheryieldsthe desired approximate gradient ˆ q J. In order to verify the appropriateness of the described approximations we have carried out numerical simulations to approximate both J and its derivatives as functions of the model parameters σ 2 b 2 and r 2.Forthesakeofaclearvisualization we have not tried to depict the four-dimensional plot of J with respect to σ 2 b 2 and r 2 jointly (and the corresponding partial derivatives). Instead we have restricted our attention to the one-dimensional representation of J with respect to a single parameter (e.g. σ 2 )withalltheothersbeingfixed(e.g.r 2 b 2 and the interval I). In particular we have considered the average univariate cost functions J(σ 2 ) J(r 2 ) J(b 2 ) = 1 N i N j = 1 N i N j = 1 N i N j N i N j i=1 j=1 N i N j i=1 j=1 N i N j i=1 j=1 J(σ 2 b 2 r 2 I) b2 =b 1 +η (i) b r 2=r 1 +η (i) r I=I j (11) J(σ 2 b 2 r 2 I) σ2 =σ 1 +η (i) σ r 2 =r 1 +η (i) r I=I j (12) J(σ 2 b 2 r 2 I) σ2 =σ 1 +η (i) σ b 2 =b 1 +η (i) b I=I j (13) and their corresponding explicit derivatives J σ 2 J and J b 2 respectively. The intervals in the sequence {I j } N j j=1 are consecutive and non-overlapping and η σ (i) η r (i) and η (i) are independent and identically distributed random Gaussian perturbations with zero mean and standard deviations s σ = σ s r = r and s b = b respectively.wehaveconsideredthese(small)random perturbations of the fixed parameters in order to verify the robustness of our approach i.e. to check whether the minima of the cost functions are close to the desired values even when the fixed parameters of the secondary system are mismatched with the primary one. We should note that strictly speaking this is only sufficient to assess the problem of estimating one single parameter at a time but it is also a good illustration of how the proposed approximation works. In the next section we present numerical results that show the actual performance of the proposed algorithm when estimating the full vector q =[σ 2 b 2 r 2 ] T. Fig. 1 depicts the average univariate cost functions and their explicit derivatives. To obtain this figure we have assigned the primary Lorenz with the standard parameter values of chaotic behavior σ 1 = 10 b 1 = 8/3 andr 1 = 28 and both the primary and the secondary systems have been numerically integrated using the fourth-order Runge Kutta method with integration step h = 10 2 t.u. The average has been computed over a set of 100 independent simulations (i.e. N i = 100 in Eqs. (11) (13)) and the sequence of intervals has the form I j =[(j 1)T jt ) j = N j witht = 0.5 t.u. and N j = 200. Fig. 1(a) (c) show the shape of the average cost functions J(σ 2 ) J(b 2 ) and J(r 2 )respectively.theminimaareindicated with a vertical line and it is seen that they are located very close to the values of σ 1 (for J(σ 2 )) b 1 (for J(b 2 ))andr 1 (for J(r 2 )). We can also observe that the functions are convex in a relatively large neighborhood of the indicated minima although Fig. 1(a) b

4 I.P. Mariño J. Míguez / Physics Letters A 351 (2006) Fig. 1. (a) The (averaged) cost function J(σ 2 ).(b)the(averaged)costfunctionj(b 2 ).(c)the(averaged)costfunctionj(r 2 ).(d)theexplicitderivative J σ.(e)the 2 explicit derivative J b.(f)theexplicitderivative J 2 r.thelocationoftheminimaofthecorrespondingcostfunctionsareindicatedwithaverticallineinthesix 2 subplots. shows that J(σ 2 ) may exhibit other local minima at locations different from σ 2 σ 1. The validity of the approximation by explicit derivatives is illustrated by Fig. 1(d) (f). The locations of the minima of the corresponding cost functions are indicated with vertical lines and the plots show that the curve obtained by explicit derivation vanish approximately at the true minimum of the cost function (in the three cases). Indeed these computer simulations indicate that the explicit derivatives are proportional to the true derivatives which is enough for the purpose of implementing agradient-descentalgorithm.itshouldbeobservedhowever that the sign of the explicit derivative J is inverted with respect to the true derivative of J(r 2 ).Thisisnotaproblemforthe implementation of the estimation algorithm (we simply need to use a negative step-size parameter for r 2 asshowninthenext section) but it indicates that we have to a priori check the relationship between the true and the approximate gradient for each type of system. 5. Computer simulations Because of the approximate proportionality relationship that we have observed between the explicit derivatives and the true derivatives of the cost function the generic gradient algorithm of Eq. (5) is particularized to σ 2n = σ 2n 1 µ σ J σ 2 b 2n = b 2n 1 µ b J b 2 r 2n = r 2n 1 + µ r J (14) where µ σ µ r and µ b are the specific step-sizes associated to σ 2 r 2 and b 2 respectively.wehaveappliedtheabovealgorithm with µ σ = µ r = 10 3 and µ b = 10 4 chosen to achieve an adequate trade-off between convergence speed and estimation accuracy to the joint estimation of p =[σ 1 b 1 r 1 ] T. The same as in the previous set of simulations the parameter values in the primary system are σ 1 = 10 r 1 = 28 and b 1 = 8/3 and we use a fourth-order Runge Kutta method with step h = 10 2 t.u. to numerically integrate the systems. The initial conditions of the state variables are given by [x 1 x 2 x 3 ]= [ ] and [y 1 y 2 y 3 ]=[ ] andtheyarekept constant for the complete set of simulations. The starting values for the parameter estimates are σ 20 = 5+η σ r 20 = 21+η r and b 20 = η bwhereη σ η r and η b are random Gaussian perturbations with zero mean and standard deviations s σ = σ 1 10 s r = r 1 10 and s b = b 1 10 respectively.hencetheinitialparameter

5 266 I.P. Mariño J. Míguez / Physics Letters A 351 (2006) Fig. 2. (a) Time evolution of the first state variables of the primary and secondary systems x 1 and y 1 respectively.(b)timeevolutionoftheparameters in the secondary system. values are different (and independently drawn) for each simulation run. For a single simulation trial Fig. 2 shows how the parameters are estimated and the secondary system is brought into identical synchronization with the primary one. In particular Fig. 2(a) shows the time evolution of the first state variable in each system (x 1 and y 1 ). It is seen how y 1 tightly matches x 1 after 6 t.u. in spite of a very different initial value. This is a consequence of the relatively fast convergence of the parameter estimates as shown in Fig. 2(b). In order to better study the accuracy of the parameter estimation algorithm we can use as a figure of merit the normalized absolute error between the estimated parameter and its desired value i.e. we compute the discrete-time error signals ε σ (n) = σ 2n σ 1 σ 1 ε r (n) = r 2n r 1 r 1 ε b (n) = b 2n b 1 b 1. (15) Since the slope and final values of ε σ (n) ε r (n) and ε b (n) can possibly depend of the starting parameter estimates we have run 20 independent simulations (thus with randomly chosen initial estimates) and averaged the obtained errors. This is shown in Fig. 3. Fig. 3(a) shows the (average) evolution of ε σ (n) with time (the error values at t = nt are actually plotted). It is observed how the normalized error decreases quickly and consistently Fig. 3. (a) Time evolution of the average value of the normalized absolute error ε σ (n). (b)timeevolutionoftheaveragevalueofthenormalizedabsolute error ε r (n). (c)timeevolutionoftheaveragevalueofthenormalizedabsolute error ε b (n). for the complete simulation period of 2000 t.u. Similar results are shown for ε r (n) and ε b (n) in Fig. 3(b) and (c) respectively. 6. Conclusions We have introduced a simple adaptive algorithm for the estimation of the parameters of a (primary) chaotic system that drives the dynamics of a coupled (secondary) system. Unlike similar existing techniques the proposed algorithm operates in discrete-time hence it is more suitable for implementation with digital circuitry.

6 I.P. Mariño J. Míguez / Physics Letters A 351 (2006) The method is aimed at the minimization of a cost function that involves the derivatives of the dynamic variables of both systems and it can be systematically applied to arbitrary systems (although its convergence properties may need to be studied for specific cases). The definition of the cost function in terms of derivatives (instead of the system state variables themselves) is useful because they can be explicitly written in terms of the desired parameters. We have exemplified the use of the algorithm by addressing the estimation of the full parameter vector of a Lorenz system. For this particular case we have analytically approximated the gradient of the cost function and derived simple parameter updating rules. The validity of the approach has been demonstrated through computer simulations. Acknowledgements This work has been supported by the Spanish Ministry of Science and Technology (project BFM ) and Universidad Rey Juan Carlos (project PPR ). References [1] E. Baake H.G. Bock K.M. Briggs Phys. Rev. A 42 (1992) [2] A. Ghosh V. Ravi Kumar B.F. Kulkarni Phys. Rev. E 64 (2001) [3] J. Timmer Chaos Solitons Fractals 11 (2000) [4] A. Sitz U. Schwarz J. Kurths H.U. Voss Phys. Rev. E 66 (2002) [5] V.F. Pisarenko D. Sornette Phys. Rev. E 69 (2004) [6] H. Singer J. Comput. Graph. Stat. 11 (2002) 972. [7] H. Sakaguchi Phys. Rev. E 65 (2002) [8] U. Parlitz L. Junge L. Kocarev Phys. Rev. E 54 (1996) [9] U. Parlitz Phys. Rev. Lett. 76 (1996) [10] A. Maybhate R.E. Amritkar Phys. Rev. E 59 (1999) 284. [11] C. Zhou C.-H. Lai Phys. Rev. E 59 (1999) [12] A. d Anjou C. Sarasola F.J. Torrealdea R. Orduna M. Grana Phys. Rev. E63(2001) [13] R. Konnur Phys. Rev. E 67 (2003) [14] D. Huang Phys. Rev. E 69 (2004) [15] C. Tao Y. Zhang G. Du J.J. Jiang Phys. Rev. E 69 (2004) [16] U.S. Freitas E.E.N. Macau C. Grebogi Phys. Rev. E 71 (2005) [17] R. Brown N.F. Rulkov Phys. Rev. Lett. 78 (1997) 4189.