Bridging a gap in Kalman filtering output estimation with correlated noises or direct feed-through from process noise into measurements

Traditional statements of the celebrated Kalman filter algorithm focus on the estimation of state, but not the output. For any outputs, measured or auxiliary, it is usually assumed that the posterior state estimates and known inputs are enough to generate the minimum variance output estimate, given by yn|n = Cxn|n + Dun. Same equation is implemented in most popular control design toolboxes. It will be shown that when measurement and process noises are correlated, or when the process noise directly feeds into measurements, this equation is no longer optimal, and a correcting term of Hwn|n . = HE(wn|zn) is needed in above output estimation. This natural extension can allow designer to simplify noise modeling, reduce estimator order, improve robustness to unknown noise models as well as estimate unknown input, when expressed as an auxiliary output. This is directly applicable in motioncontrol applications which exhibits such feed-through, such as estimating disturbance thrust affecting the accelerometer measurements. Based on a proof of suboptimality [1], this correction has been accepted and implemented in Matlab 2016 [2].