On the Properties of Reduced-Order Kalman Filters

Several known results are unified by considering properties of reduced-order Kalman filters. For the case in which the number of noise sources equals the number of observations, it is shown that the reduced-order Kalman filter achieves zero steady-state variance of the estimation error if and only if the plant has no transmission zeros in the right-half plane, since these would be among the poles of the Kalman filter. The reduced-order Kalman filter cannot achieve zero variance of the estimation error if the number of independent noise sources exceeds the number of observations. It is also shown that the reduced-order Kalman filter achieves the generalized Doyle-Stein condition for robustness without dimining robustneess.