Finite-Dimensional Filters with Nonlinear Drift, VI: Linear Structure of Ω

摘要

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 = (((partial deriv)f)_j/((partial deriv)x)-i-((partial deriv)f)_i/((partial deriv)x)_j) matrix, where / denotes the drift term, is a linear matrix in the sense that all the entries in Q are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank.