Finite-dimensional estimation algebra on arbitrary state dimension with nonmaximal rank: linear structure of Wong matrix

Ever since Brockett, Clark and Mitter introduced the estimation algebra method, it becomes a powerful tool to classify the finite-dimensional filtering system. In this paper, we investigate finite-dimensional estimation algebra with non-maximal rank. The structure of Wong matrix omega will be focused on since it plays a critical role in the classification of finite-dimensional estimation algebras. In this paper, we first consider general estimation algebra with non-maximal rank and determine the linear structure of the submatrix of omega by using rank condition and property of Euler operator. In the second part, we proceed to consider the case of linear rank n-1 and prove the linear structure of omega. Finally, we give the structure of finite-dimensional filters which implies the drift term must be a quadratic function plus a gradient of a smooth function.