The novel classes of finite dimensional filters with non-maximal rank estimation algebra on state dimension four and rank of one

Abstract

Ever since the technique of Kalman-Bucy filter was popularised, finding new classes of finite dimensional recursive filters has drawn much concern. The idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently in the late 1970s, which has been proven an invaluable tool in tackling nonlinear filtering (NLF) problems. Once the estimation algebra is finite dimensional, one can construct the finite dimensional filters (FDFs) for NLF problems by Wei–Norman approach. In this paper, we give the construction of finite dimensional estimation algebra (FDEA) with state space dimension 4 and linear rank equal to 1, and further obtain a new class of NLF systems with FDFs. Importantly, we show that there is a class of polynomial FDF system in state space dimension 4 with linear rank one, but the coefficients in Wong's Ω-matrix are polynomials of degree two, or higher. In particular, these are the first examples of polynomial filtering systems not of Yau type (i.e. the drift term is not gradient plus affine functions) but with FDFs. Furthermore, we write down several easily satisfied sufficient conditions for the construction of more special classes of FDFs. Additionally, we derive the FDFs for the proposed NLF systems by using the Wei–Norman approach.