Discrete Kalman filtering using a generalized companion form

For an nth order constant system with p outputs, this paper shows that the Kalman filter gain can be described by np-p(p-1)/2 difference equations instead of the usual n(n+1)/2 difference equations. This is the minimum number of difference equations required for the solution of the Kalman filter gain. For the special case of stationary processes, this paper further shows that only np-p(p-1)/2 combinations of the system noise covariance enter into determination of the Kalman filter gain. The results have application in filtering and in the quadratic regulator problems.