The design of optimal reduced order stochastic observers for discrete-time linear systems

The minimum variance state estimation of linear discrete-time systems with random white noise input and partially noisy measurements is investigated. An observer of minimal-order that attains the minimum-variance estimation error is found. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise. An explicit expression is found for the transfer function matrix of the observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained