A note on estimation algebras on nonlinear filtering theory

Abstract

The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters. In his talk at the International Congress of Mathematicians in 1983, Brockett proposed to classify all finite dimensional estimation algebras. In this paper we consider some filtering systems. In a special filtering system: (1) We have some structure results. (2) For any arbitrary finite dimensional state space, under the condition that the drift term is a linear vector field plus a gradient vector field, we classify all finite dimensional estimation algebras with maximal rank. (3) We classify all finite dimensional estimation algebras with maximal rank if the dimension of the state space is less than or equal to three. A more general filtering system is considered. The above three results can be ‘used’ locally. Therefore from the algebraic point of view, we have now understood generically some finite dimensional filters.