Optimal Minimal-Order Observers for Discrete-Time Systems -- A Unified Theory

Abstract

Luenberger's minimal-order observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. The general solution to the problem of constructing the optimal minimal-order observer is presented for systems having white noise disturbances. In the special case of no measurement noise the observer estimation errors are shown to be identical with those of the corresponding Kalman filter. Estimation errors comparable with the Kalman filter are obtained when measurement noise is not excessive. The observer solution is extended to systems for which the noise disturbances are time-wise correlated processes of the Markov type. In considering correlated noise inputs, the system state equations are not augmented as is done in the usual Kalman filtering theory. The observer solution, modified appropriately to account for the time-wise correlation of the noise inputs, yields minimum mean-square estimates of the state vector.