On the relationship between lie algebras and nonlinear estimation

A Lie algebra L(Σ) can be associated with each nonlinear filtering problem, and the realizability of L(Σ) or quotients of L(Σ) with vector fields on a finite dimensional manifold is related to the existence of finite dimensional recursive filters. In this paper the structure and realizability properties of L(Σ) are analyzed for several interesting classes of problems. It is shown that, for certain nonlinear filtering problems, L(Σ) is given by the Weyl algebra Wn = R < x1,...,xn, ∂/∂x1,..., ∂/∂xn > . It is proved that neither Wn nor any quotient of Wn can be realized with CΣ or analytic vector fields on a finite dimensional manifold, thus showing that for these problems, no statistic of the conditional density can be computed with a finite dimensional recursive filter. For another class of problems (including bilinear systems with linear observations), it is shown that L(Σ) is a certain type of filtered Lie algebra; the implications of this property are discussed.