On lie algebras and finite dimensional filtering

Abstract

A Lie algebra L(Σ) can be associated with each nonlinear filtering problem, and the realizability or, better, the representability of L(σ) or quotients of L(σ) by means of vector fields on a finite dimensional manifold is related to the existence of finite dimensional recursive filters. In this paper, the structure and representability properties of L(σ) are analyzed for several interesting and/or well known classes of problems. It is shown that, for certain nonlinear filtering problems, L(σ) is given by the Weyl algebra It is proved that neither W _n nor any quotient of W _n can be realized with C ^∞ or analytic vector fields on a finite dimensional manifold, thus suggesting 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 algebras of this class are of a type which suggest that “sufficiently many” statistics are exactly computable. Other examples are presented, and the structure of their Lie algebras is discussed.