Finite Dimensional Nonlinear Estimation in Continuous and Discrete Time.

Abstract

Abstract : It has been shown that, for certain classes of nonlinear stochastic systems in both continuous and discrete time, the optimal conditional mean estimator of the system state given the past observations can be computed with a recursive filter of fixed finite dimension. The typical nonlinear system in these classes consists of a linear system with linear measurements and white Gaussian noise processes, which feeds forward into a nonlinear system described by a certain type of Volterra series expansion or by a bilinear or state-linear system satisfying certain algebraic conditions. The purpose in this paper is to consider estimation problems similar to those presented before, to present simpler proofs that the estimators are indeed finite dimensional, to provide deeper insight into these problems by relating them to the homogeneous chaos of Wiener and to orthogonal polynomial expansions, to explain the similarities and differences between the continuous and discrete time cases, and to prove some extensions of previous results. The existence of polynomials in the innovations in the discrete time recursive estimator, in contrast to the continuous time estimator, is interpreted in terms of the homogeneous chaos.