Invariant Kalman Filter for Correlated Wide Band Noises

Abstract

Kalman filtering is a powerful estimation method. One of its weaknesses is related to the white or coloured nature of the disturbing noises in the Kalman filtering model. At the same time, real noises are rarely white or coloured. They are mostly wide band. In this regard, white or coloured noise Kalman filtering makes concessions on adequacy. This pushes system scientists to develop mathematical methods of estimation for systems corrupted by wide band noises. In applications, wide band noises are detected by their autocovariance and cross-covariance functions, which do not allow to model them uniquely. Therefore, it becomes important to develop estimation methods which are independent of a class of wide band noises, but dependent on the unique autocovariance and cross-covariance functions. Such results are called invariant results. In this paper, we prove a complete set of invariant equations for Kalman type filter for a linear signal-observation system corrupted by correlated wide band noises. This filter has a ready form to be used in applications, just respective numerical methods must be developed.