Finite-dimensional filters with nonlinear drift,VI: Linear structure of Ω

Ever since the concept of estimation algebra was first introduced by Brockett and Mitter independently, it has been playing a crucial role in the investigation of finite-dimensional nonlinear filters. Researchers have classified all finite-dimensional estimation algebras of maximal rank with state space less than or equal to three. In this paper we study the structure of quadratic forms in a finite-dimensional estimation algebra. In particular, we prove that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂f _j /∂x _i −∂f _i /∂x _j )matrix, wheref denotes the drift term, is a linear matrix in the sense that all the entries in Ω are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank. This research was supported by Army Research Office Grants DAAH 04-93-0006 and DAAH 04-1-0530.