Geometry of Kalman filters. (English) Zbl 1139.93035

In this paper is presented a geometric explanation of Kalman filters in terms of a symplectic linear space and a special quadratic form on it. It is an extension of the work of Bougerol with application of a different metric introduced earlier. The author’s purpose in this paper is to show that both contraction properties can be understood purely in terms of the symplectic structure and that they are rigidly related. This is accomplished by the introduction of another metric into the space of positive definite matrices, based on the symplectic structure. It is called the sector metric. Several basic properties of this metric were established. In particular it is a Finslerian metric, i.e., at every tangent space there is a norm, rather than a scalar product. The new results are contained in several theorems proved in the paper.


93E11 Filtering in stochastic control theory
93B27 Geometric methods