Polynomial systems approach to continuous-time weighted optimal linear filtering and prediction

The solution of the optimal weighted minimum-variance estimation problem is considered using a polynomial matrix description for the continuous-time linear system description, which allows for the possible presence of transport delays on the measurements. The filter or predictor is given by the solution of two diophantine equations and is equivalent (in the delay-free case) 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 useful insights into the disturbance rejection properties of the filter are obtained. Allowance is made for both control and disturbance input subsystems and white and colored measurement noise (or an output disturbance subsystem). The model structure was determined by the needs of filtering and prediction problems in the metal processing industries, where, for example, there are delays between the X-ray gauge and the roll gap of the mill.