Complete Classification of Finite Dimensional Estimation Algebras With State Dimension n, Linear Rank n-1, and Constant Wong Matrix

Abstract

Ever since the Lie algebra method was introduced to construct finite dimensional nonlinear filters by Brockett and Mitter independently, there has been an intense interest in classifying all finite dimensional estimation algebras and finding new classes of finite dimensional recursive filters. The estimation algebra method has been proven to be an invaluable tool in the nonlinear filtering theory. This paper considers the finite dimensional estimation algebras derived from a nonlinear filtering system with state dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> , linear rank <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n-1$</tex-math></inline-formula> and constant Wong matrix. Related theories of the underdetermined partial differential equations and the Euler operator are applied to classify the estimation algebras. It is proved that the Mitter conjecture holds and the dimension of the finite dimensional estimation algebras must be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2n$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2n+1$</tex-math></inline-formula> with above conditions. Therefore, we can construct the explicit solution of filtering systems by Wei-Norman approach. This result is of great significance because it is the first classification of non-maximal rank finite dimensional estimation algebras with arbitrary state dimension.