On a necessary and sufficient condition for finite dimensionality of estimation

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has proven to be an invaluable tool in the study of nonlinear filtering problems. In this paper, a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite-dimensional. Also presented is a rigorous proof of the Wei–Norman program which allows one to construct finite-dimensional recursive filters from finite dimensional estimation algebras.