Kalman Filter - Kalman’s Original Filter Derivation

Kalman’s Original Filter Derivation

Kalman’s own derivation is based on orthogonal projection. In his section “Orthogonal Projections” he gives a short tutorial on statistical orthogonality and assumes an estimate in the form of a linear combination of measurement data, maintaining that this is the “optimal estimate” when the mean square error (MSE) (He uses the equivalent term “loss function”.) is minimized. He then purports to minimize the MSE by projecting the solution onto the measurement data using orthogonal projection.

Later in his paper under the rubric “Solution of the Wiener Problem” Kalman contradicts himself by assigning to his estimate a prescribed solution in the form of a state equation. He projects the measurement data onto his prescribed state equation (the opposite of his assertion above), stating that the “… principal problem of the paper …” is to “… find an estimate … which minimizes the expected loss …”, and further pointing out that this constitutes “… reconstructing all the state variables …”.