Bridging a Gap in Applied Kalman Filtering: Estimating Outputs When Measurements Are Correlated with the Process Noise [Focus on Education]

Traditional statements of the Kalman filter focus on the estimation of states rather than outputs. An output of interest may contain feedthrough from both known inputs u and unknown inputs w, yn=Cxn+Dun+Hwn. It is usually assumed that the posterior state estimate and known inputs are enough to generate the minimum-variance output estimate, given by which is the same equation implemented in most popular control-design toolboxes.