Recurrent filtering and identification of dynamical system parameters

An investigation is made of questions of convergence of a multimodel recursion algorithm that is nonlinear over the measurement results and used in a filtering process to estimate the vectors of the phase coordinates and the unknown parameters of a dynamical system from the object being controlled and the information sensor. The algorithm is described also for the case of dependent measurement errors. Theorem 1 is formulated for the situation where the vector of the unknown parameters coincides with the vector of the parameters of one of the models. This theorem provides only necessary and only sufficient conditions for convergence in probability of the results of estimation with a multimodel algorithm. Theorem 2 is formulated for the case where the vector of the unknown parameters does not coincide with the vector of the parameters of any of the models. This theorem gives a variant of the sufficient conditions for convergence in probability of the results of estimation by means of the multimodel algorithm to the vector of the parameters of the model that is closest in some sense to the true dynamical system. Theorem 3, which provides sufficient conditions for convergence in probability of the estimation results, is formulated for the proposed basic statistical dichotomy algorithm, which is used if the vector of the unknown parameters belongs to some connected region. A description is given of the variant of the statistical dichotomy algorithm being realized. This variant uses some quadrature formulas or other, and it usually requires realization of a small number (five or more) models of the dynamic system and Kalman filters in theory to obtain an arbitrarily small identification error. Included are an example of application of Theorems 1 and 2, an example of identification in the course of filtering the time constant of an oscillatory link realized using the statistical dichotomy algorithm, and an example of identification, in the course of the filtering, of the time constant and the decrement of damping of that link, done by combining the statistical dichotomy algorithm and the principle of coordinate descent.