Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 801475, 21 pages
Research Article

A Bayes Estimator of Parameters of Nonlinear Dynamic Systems

State Institute of Aviation Systems, Physical-Technical Institute, Russia

Received 28 November 2008; Revised 28 February 2009; Accepted 18 May 2009

Academic Editor: David Chelidze

Copyright © 2009 I. A. Boguslavsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new multipolynomial approximations algorithm (the MPA algorithm) is proposed for estimating the state vector θ of virtually any dynamical (evolutionary) system. The input of the algorithm consists of discrete-time observations Y. An adjustment of the algorithm is required to the generation of arrays of random sequences of state vectors and observations scalars corresponding to a given sequence of time instants. The distributions of the random factors (vectors of the initial states and random perturbations of the system, scalars of random observational errors) can be arbitrary but have to be prescribed beforehand. The output of the algorithm is a vector polynomial series with respect to products of nonnegative integer powers of the results of real observations or some functions of these results. The sum of the powers does not exceed some given integer d. The series is a vector polynomial approximation of the vector E(θY), which is the conditional expectation of the vector under evaluation (or given functions of the components of that vector). The vector coefficients of the polynomial series are constructed in such a way that the approximation errors uniformly tend to zero as the integer d increases. These coefficients are found by the Monte-Carlo method and a process of recurrent calculations that do not require matrix inversion.