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Traditional statements of the celebrated Kalman filter algorithm focus on theestimation of state, but not the output. For any outputs, measured orauxiliary, it is usually assumed that the posterior state estimates and knowninputs are enough to generate the minimum variance output estimate, given byyn|n = Cxn|n + Dun. Same equation is implemented in most popular control designtoolboxes. It will be shown that when measurement and process noises arecorrelated, or when the process noise directly feeds into measurements, thisequation is no longer optimal, and a correcting term is needed in above outputestimation.  This natural extension can allow designer to simplify noise modeling, reduceestimator order, improve robustness to unknown noise models as well as estimateunknown input, when expressed as an auxiliary output. This is directlyapplicable in motion control 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 andimplemented in Matlab 2016 [2].

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