Physica D 220 (2006) 175–182 www.elsevier.com/locate/physd On a recursive method for the estimation of unknown parameters of partially observed chaotic systems In´ es P. Mari˜ no a,∗ , Joaqu´ ın M´ ıguez b a Nonlinear Dynamics and Chaos Group, Departamento de Matem´ aticas y F´ ısica Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, C/ Tulip´ an s/n, 28933 M ´ ostoles, Madrid, Spain b Departamento de Teor´ ıa de la Se ˜ nal y Comunicaciones, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan´ es, Madrid, Spain Received 9 November 2005; received in revised form 17 April 2006; accepted 7 July 2006 Available online 4 August 2006 Communicated by R. Roy Abstract We investigate a recently proposed method for on-line parameter estimation and synchronization in chaotic systems. This novel technique has been shown effective to estimate a single unknown parameter of a primary chaotic system with known functional form that is only partially observed through a scalar time series. It works by periodically updating the parameter of interest in a secondary system, with the same functional form as the primary one but no explicit coupling between their dynamic variables, in order to minimize a suitably defined cost function. In this paper, we review the basics of the method, and investigate its robustness and new extensions. In particular, we study the performance of the novel technique in the presence of noise (either observational, i.e., an additive contamination of the observed time series, or dynamical, i.e., a random perturbation of the system dynamics) and when there is a mismatch between the primary and secondary systems. Numerical results, including comparisons with other techniques, are presented. Finally, we investigate the extension of the original method to perform the estimation of two unknown parameters and illustrate its effectiveness by means of computer simulations. c  2006 Elsevier B.V. All rights reserved. Keywords: Parameter estimation; Chaos synchronization; Chaos control 1. Introduction Many problems in science and engineering reduce to adjusting the parameters of a dynamical model in order to match an observed time series. If we assume that (a) the model is a replica of the system originating the observations, except for the parameters to be adjusted, and (b) after a proper selection of its adjustable parameter values the model dynamics follows the observed time series closely, then the model parameters are estimates of the real system parameters [1]. The methods proposed in the literature to address this problem can be classified as either off-line or on-line techniques. Off-line methods first collect a set of observations (e.g., samples from the received time series) and then process the complete set iteratively to produce a sequence ∗ Corresponding author. Tel.: +34 916647448; fax: +34 914887338. E-mail address: ines.perez@urjc.es (I.P. Mari˜ no). of approximations to the values of the unknown parameters. On-line techniques process the observations sequentially and parameter estimates are computed recursively, i.e., existing estimates are updated using only newly collected observations instead of the complete set of data. Off-line algorithms are computationally heavier, although often more accurate, and unsuitable for applications in which the model system should operate continuously. Different procedures of both classes have been suggested. The so-called multiple shooting methods [2,3] are off-line techniques that address the estimation of the unknown fixed parameters, as well as the values of the dynamic variables of the chaotic system at a grid of sampling times, as a boundary-value problem and have shown to be very effective for several applications. However, they involve the optimization of large dimensional cost functions (not only the parameters are estimated) and can be complex to implement compared to other methods. Some standard 0167-2789/$ - see front matter c  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2006.07.008