Deterministic, recursive filtering in non-linear differential systems

New algorithms for state estimation in non-linear differential systems are studied on the basis of a previously developed theory, which was applicable to polynomial-type non-linearities. The filtering is formulated as a deterministic optimal control problem. Via Pontryagin´s principle and the invariant imbedding technique the problem is converted into an initial value problem of a partial differential equation. A theoretical solution of the filtering is then constructed with the aid of successive approximations in a modified partial differential equation. A recursive third order algorithm is obtained by applying a non-linear Riccati-type transformation and the method of characteristic curves to the partial differential equation. The basic algorithm is derived for systems with additive process noise. An extension to systems with non-additive process noise is described, too. Applications to the rectilinear orbit problem and to a system where the process noise is not additive are presented.