Linear Filtering for Wide Band Noise Driven Observation Systems

Abstract

Filtering of wide band noise driven systems accounts the following problem. Given an autocovariance function, there are infinitely many wide band noise processes, which have this autocovariance function. Each of them produces its own best estimate. The problem is a selection of the best one of these best estimates. A similar problem arises in control theory as a selection of optimal one of the optimal controls. In this paper we investigate this problem for a wide class of wide band noises. It is proved that in the case of independent wide band and white noises corrupting, respectively, the signal and observations, the best estimates and the optimal controls in the linear filtering and LQG problems are independent of the respective wide band noises. We present a complete set of formulae for the best estimate and, respectively, for the optimal control in terms of the system parameters and autocovariance function of the wide band noise disturbing the signal system.

Invariant Kalman Filter for Correlated Wide Band Noises

“…In invariant maximum principle in the Pontryagin's form and controllability results for linear and nonlinear systems are established. Invariant Kalman filter is also obtained in the special cases when the signal noise is wide band but the observation noise is non‐degenerate white , and when the signal noise is white but the observations are corrupted by the sum of wide band and non‐degenerate white noises . In fact, this paper generalises these results to the case when both the signal and observation systems are corrupted by the sum of white and wide band noises.…”

“…Finally, we make some remarks about the wideness of the invariance of the filter from Theorem . In it is proved that the invariance of the signal wide band noise φ 1 can be extended (at least in a special case) to all square integrable relaxing functions Φ 1 . But its proof method is not extendable to the invariance of the observation wide band noise φ 2 , which still remains within differentiable relaxing functions Φ 2 with square integrable derivatives and .…”

Impulsive semilinear heat equation with delay in control and in state

“…Bashirov in [], discovered that under general conditions, a wide band noise can be modeled as distributed delay of a white noise. Bashirov et al in [] developed different problems on the basis of this discovery; in spite of this it is not part of our work since it moves us to the stochastic systems where noise is a significant component in control systems. This is the reason that we mention the delays in noises to emphasize the importance of them in processes.…”