Polynomial systems approach to optimal linear filtering and prediction

Abstract

The solution of the optimal linear estimation problem is considered, using a polynomial matrix description for the discrete system. The filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady-state Kalman filter, or the transfer-function matrix form of the Wiener filter. The pole-zero properties of the optimal filter are more obvious in the polynomial representation, and new insights into the disturbance rejection properties of the filter are obtained. The plant or signal model can be stable or unstable, and allowance is made for both control and disturbance input subsystems, and white and coloured measurement noise (or an output disturbance subsystem). The model structure was determined by the needs of several industrial filtering problems. The polynomial form of filter may easily be included in a self-tuning algorithm, and a simple adaptive estimator is described.