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

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## Abstract

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, where*f* 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.

## Key words

Finite-dimensional filters Nonlinear drift Estimation algebra of maximal rank## Preview

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