Complete classification of finite-dimensional estimation algebras of maximal rank

The idea of using estimation algebras to construct finite-dimensional non-linear filters was first proposed by Brockett and dark, and Mitter independently. In his famous talk at the International Congress of Mathematics in 1983, Brockett proposed to classify all finite-dimensional estimation algebras. In this paper we explain why the theory of estimation algebras plays an important role in non-linear filtering. We show how to use the Wei-Norman approach to construct finite-dimensional filters from finite-dimensional estimation algebras. We survey some results in estimation algebras after 1984. We give a self-contained proof of complete classification of finite-dimensional estimation algebras of maximal rank in one place. The proof given here is simpler than those proofs scattered in several papers. This provides the readers with a complete coherent view of the important topic of the classification of finite-dimensional estimation algebras.