Generalizing the Basic Discrete Kalman Filter

To derive the discrete Kalman filter in Chap. 9 , we make several assumptions that restrict its usefulness. In particular, we assume that the measurement and process noise processes were white and uncorrelated. In this chapter, we show how both colored noise and correlated measurement and process noise can be handled using shaping filters. For colored measurement noise, this results in a measurement equation that does not include noise. This “perfect” measurement required the use of reduced-order Kalman filters. To implement Kalman filters in practice, it is advantageous to reduce the computational load. This can be accomplished using sequential computation to eliminate matrix inversion in the corrector, where one measurement is processed at a time. To reduce computational errors, we can propagate the square root of the covariance matrices in square root filtering. This comes at the expense of complicating the Kalman filter algorithm.