Synchronization and Control Based Parameter Identification

Introduction

For many experimental or field data mathematical models exist whose structure is derived from first principles and thus known, but relevant model parameter values have to be estimated from the data. Furthermore, in general not all state variables of the dynamical system of interest are available, but a scalar time series has been measured, only. And, last not least, the data are discretely sampled in time and often perturbed by noise. How can unknown parameter values be efficiently estimated under these conditions? During the past decades different solutions for this problem have been devised. For example, (multiple) shooting methods [1, 2, 3, 4] aim at reconstructing trajectory segments of the available model that fit to the observed data. This approach results in some (high dimensional) optimization algorithms that provide not only proper initial values but also the desired parameter values. Another class of methods are so-called nonlinear filters [2, 5] that are based on a probabilistic framework for estimating states and parameters, in particular for noisy data. For linear systems this approach is also known as Kalman filters and for nonlinear or even chaotic processes many generalizations have been suggested (extended and unscented Kalman filters, exponential families, particle filters, etc.). In the following we shall focus on parameter estimation methods that are connected to synchronization phenomena of dynamical systems and control theory. This approach was developed during the past decade and proved to be particularly elegant and efficient for systems exhibiting chaotic dynamics. The starting point is the fact that many interacting dynamical system exhibit synchronous periodic of chaotic oscillations, even in cases of one-way coupling [6, 7].