Finite-dimensional recurrent algorithms for optimal nonlinear logical–dynamical filtering

Abstract

The problem of the most accurate estimation of the current state of a multimode nonlinear dynamic observation system with discrete time based on indirect measurements of this state is considered. The general case when a mode indicator is available and the measurement errors depend on the plant disturbances is investigated. A comparative analysis of two known approaches is performed—the conventional absolutely optimal one based on the use of the posterior probability distribution, which requires the use of an unimplementable infinite-dimensional estimation algorithm, and a finitedimensional optimal approach, which produces the best structure of the difference equation of a low-order filter. More practical equations for the Gaussian approximations of these two optimal filters are obtained and compared. In the case of the absolutely optimal case, such an approximation is finitedimensional, but it differs from the approximation of the finite-dimensional optimal version in terms of its considerably larger dimension and the absence of parameters. The presence of parameters, which can be preliminarily calculated using the Monte-Carlo method, allows the Gaussian finite-dimensional optimal filter to produce more accurate estimates.