The properties of reduced-order minimum-variance filters for systems with partially perfect measurements

The problem of the finite-time, reduced-order, minimum variance full-state estimation of linear, continuous time-invariant systems is considered in cases where the output measurement is partially free of corrupting white-noise components. The structure of the optimal filter is obtained and a link between this structure and the structure of the system invariant zeros is established. Using expressions that are derived in closed form for the invariant zeros of the system, simple sufficient conditions are obtained for the existence of the optimal filter in the stationary case. The structure and the transmission properties of the stationary filter for general left-invertible systems are investigated. A direct relation between the optimal filter and a particular minimum-order left inverse of the system is obtained. A simple explicit expression for the filter transfer function matrix is also derived. The expression provides an insight into the mechanism of the optimal estimation.<>