  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 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 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 and
implemented in Matlab 2016 [2].
