On the geometric structure of Bryson-Johansen filter and stochastic observer

Dividing the state space into the noisy, noise-free state spaces in terms of a projection operator, we will show that the Bryson-Johansen filter [1] changes into the stochastic observer [2]-[5] by the rotation of the coordinate system, and both estimators are completely characterized by the filter gains which are invariant under any coordinate systems. The projection operator also serves to allow for any time-variant singularity in the output error covariance matrix [7], [8].