Explicit construction of finite dimensional nonlinear filters with state space dimension 2

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. The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed in Brockett and Clark (1980), Brockett, and Mitter (1979). Tam, Wong and Yau (1990), and Yau have demonstrated that the concept of estimation algebra is an invaluable tool in the study of nonlinear filtering problems. In Chiou and Yau, the concept of an estimation algebra with maximal rank was introduced. Let n be the dimension of the state space. For n=1, it turns out that all nontrivial finite dimensional estimation algebras are with maximal rank. They were classified by the works of Tam-Wong-Yau (1990). For n=2, the authors have classified all finite dimensional estimation algebras with maximal rank. In this paper the authors construct explicitly finite dimensional filters with state space dimension 2 via the Wei-Norman (1964) approach by using the result of Chiou and Yau. From the Lie algebraic point of view, these are the most general finite dimensional filters