FINITE-DIMENSIONAL FILTERS .1. THE WEI-NORMAN TECHNIQUE

This two-part paper deals with necessary or sufficient conditions for the existence of finite-dimensional filters. In this first part. we se t the problem and propose a construction of such filters by the Wei-No rman technique. After having formulated the problem of finite-dimensio nal filters in terms of finite-dimensional realizations of input-outpu t mappings, we specify the dependence with respect to the initial meas ure. We show how different notions of dependence imply different prope rties of the so-called estimation algebra epsilon: epsilon is homomorp hic to a Lie algebra of vector fields; epsilon contains only operators of order less than or equal to two: epsilon is finite dimensional and contains only operators of order less than or equal to two. These res ults depend on a precise definition of a finite-dimensional realizatio n, especially on what concerns the domain of the output function. The last (and most stringent) condition on epsilon will be shown to be alm ost 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.