| The Lie algebraic method, pioneered by Brockett and Mitter, provides an important research direction for nonlinear filtering theory. The problem of finding all the finite dimensional estimation algebras, as proposed by Brockett at the International Congress of Mathematics in 1983, is a vital topic in this approach.; In this thesis, the finite dimensional estimation algebras of maximal rank are completely classified. It is shown that the class of the finite dimensional estimation algebras of maximal in fact is a subclass of estimation algebras of Yau filtering systems discovered about 10 years ago.; General considerations and approaches toward the classification of general finite dimensional estimation algebras are proposed. Some structural results are obtained. The properties of Euler operator and the solutions to an under-determined partial differential equation, which inevitably arise in an estimation algebra, are studied. These tools and techniques are applied to the study of finite dimensional estimation algebras with state dimension 2 to obtain a complete classification result.; All the classified finite dimensional estimation algebras support the Mitter Conjecture and Levine Conjecture. |
