A survey of estimation algebras in application of nonlinear filtering problems

Ever since the technique of Kalman–Bucy filter was popularized, due to its limitations that it needs a Gaussian assumption on the initial data and it acts on linear systems, there has been an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies of last century, the idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett, Clark, and Mitter independently. It has been proven to be an invaluable tool in the study of nonlinear filtering problems. Since then, Yau and his coworkers were devoted to the researches of the classification of finite dimensional estimation algebras (FDEAs) with maximal rank and clarified the complete classification of it. Moreover, they shed some light on the structure of the finite dimensional estimation algebras at most dimension six. In addition, they also got some progress on the classification of FDEAs with non-maximal rank. In this survey, we shall briefly go through the development of the researches on the nonlinear filtering problems, and put emphases on the results of complete classification of FDEAs with maximal rank. And it is also presented that how to use Lie algebra method to the nonlinear filtering problems by Wei–Norman approach. Further, the recent results are given out about the structure of FDEAs with non-maximal rank.

Keywords

finite-dimensional filter, estimation algebras, non-maximal rank, nonlinear filtering problems

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This work was supported by the National Natural Science Foundation of China (11471184), by the Tsinghua University Education Foundation fund (042202008), and by the Tsinghua University start-up fund.

Received 19 May 2018

Published 19 September 2019