New Classes of Finite Dimensional Filters With Non-Maximal Rank

Ever since the technique of Kalman filter was popularized, there has been an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies, the idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It has been proven to be an invaluable tool in the study of nonlinear filtering problem. For all known finite dimensional estimation algebras, the Wong’s $\Omega $ -matrix has been proven to be a constant matrix. However, the Wong’s $\Omega $ -matrix is shown not necessary to be a constant matrix in this letter when we consider finite dimensional estimation algebras with state dimension 3 and rank equal to 1. Several easily satisfied conditions are established for an estimation algebra of a special class of filtering systems to be finite-dimensional. Finally, we give the construction of finite dimensional filters of non-maximal rank.