OPTIMAL POLYNOMIAL FILTERING IN SYSTEMS WITH UNCERTAIN OBSERVATIONS

摘要

In this paper we consider the least mean-squared error polynomial estimation problem in systems with uncertain observations, when the uncertainty is modeled by independent random variables. For our purpose we consider an augmented system, suitably defined, such that the optimal linear estimator of the augmented state based on the augmented observations provides the optimal polynomial estimator for the state of the original system. This augmented system satisfies the necessary conditions to apply the Nahi algorithm which provides the optimal linear filter for the state of the system. Finally, as we have already indicated, the optimal polynomial filter for the state of the original system can be obtained from this linear estimator. Stationary systems are also treated and we show that the augmented system is asymptotically stationary provided that the original stationary system is a-symptotically stable. Consequently, we can apply the steady-state form of the Nahi algorithm which presents great advantages from a computational point of view. This allows us to obtain the steady-state linear filter of the augmented state and, from it, the steady-state polynomial filter of the original state.