Estimation of Steady-State Optimal Filter Gain with Coloured Observation Noise

In this paper, a direct approach is proposed for obtaining the steady-state Kalman filter gain matrix for linear discrete time systems with the coloured observation noise. The proposed approach does not require priori knowledge of the noise covariance matrices, and does not employ the suboptimal Kalman filter. The two new algorithms for estimating gain matrix K are obtained, and the estimate of gain K is consistent. Based on Fadeeva's scheme for computing the matrix inverse, the problem is transformed into the one of identification of the innovation model described by the vector autoregressive moving average (ARMA) model.