Abstract
This paper is concerned with a least-squares state estimation problem for noisy observations with feedback. The statistics of the observation noise is assumed to be dependent on the state variables so that we can accept a wider class of applications than the case of usual white Gaussian noise.
Firstly, a form of equivalent data transformation is shown which transforms noisy data, via feedback, into one with a conevnient format for signal processing. It is also shown that any equivalent transformation can be represented in this form.
Secondly, nonlinear filtering formulas are derived for the observation obtained through an equivalent feedback transformation. The result previously obtained by the author for feedbackless observations is applied to get a generalized Bayes' formula for the optimal estimate, and approximate formulas in the Bayes' fashion and in the stochastic differential equation. The approximation is made by applying additional noise, and it is shown that the approximate estimate converges to the optimal one as the variance of the additional noise approaces to zero.
