Finite-Dimensional Filters. Part I: The Wei--Norman Technique

Abstract

This two-part paper deals with necessary or sufficient conditions for the existence of finite-dimensional filters. In this first part, we set the problem and propose a construction of such filters by the Wei--Norman technique. After having formulated the problem of finite-dimensional filters in terms of finite-dimensional realizations of input-output mappings, we specify the dependence with respect to the initial measure. We show how different notions of dependence imply different properties of the so-called estimation algebra $\cal E$: $\cal E$ is homomorphic to a Lie algebra of vector fields; $\cal E$ contains only operators of order less than or equal to two; $\cal E$ is finite dimensional and contains only operators of order less than or equal to two. These results depend on a precise definition of a finite-dimensional realization, especially on what concerns the domain of the output function. The last (and most stringent) condition on ${\cal E}$ will be shown to be almost sufficient to recover a family of finite-dimensional realizations thanks to the proof of a Baker--Campbell--Hausdorff formula which allows us to apply the Wei--Norman technique in a quite general setting.