Lie algebraic and approximation methods for some nonlinear filtering problems

Abstract

State estimation problems for systems involving small parameters are treated by both Lie algebraic and analytical approximation techniques. An asymptotic expansion for the unnormalized conditional density corresponding to the case of observations of a Gauss-Markov process through a (weak) polynomial nonlinearity is computed and a convergence result is derived. The convergence result is based on arguments used recently to prove existence and uniqueness and to estimate the tail behavior of solutions to nonlinear filtering problems with unbounded coefficients. The expansion is related to certain approximations of the associated estimation Lie algebra. Lie algebraic methods are used to compute finite dimensional filters for the terms in the asymptotic expansion.