Algebraic and Geometric Methods in Nonlinear Filtering

Abstract

The purpose of this paper is to show how the structure of the recursive nonlinear filtering problem leads naturally to the use of methods from nonlinear system theory and the theory of Lie algebras, and to illustrate the application of these methods to a number of specific nonlinear filtering problems. The paper is expository in nature and provides sufficient review of the requisite background in nonlinear systems and Lie algebras and enough examples of the application of tools from these fields to nonlinear filtering problems, so that the value of these methods in nonlinear filtering can be understood by researchers in all of these fields. The application of these methods to nonlinear filtering problems has led to a number of new results concerning finite dimensional filters and to a deeper understanding of the structure of nonlinear filtering problems in general. In particular, new finite dimensional and lower dimensional filters have been obtained; it has been shown that certain problems are inherently infinite dimensional; the understanding of some known filters has been enhanced; and these methods, along with asymptotic methods, have led to new interesting suboptimal filters. We conclude by outlining a procedure for using these tools and by posing some open problems.