The Deterministic Interpretation Of The Kalman Filter*

It is known that the Kalman filter has both stochastic and deterministic interpretations, whereby the deterministic interpretation relates the prediction of the filter to the response of the plant driven by the minimising least squares disturbances acting thereon. Whilst the deterministic interpretation is known, the contribution of this note is to provide an alternative, simple and self-contained proof of these properties in the discrete case. The presentation allows an efficient derivation of the key deterministic properties, i.e. that the residuals computed by the Kalman filter (both forwards and backwards) are identical to the least squares disturbances. Results with variations are given for both zero and non-zero initial conditions. Finally, a numerical example is given to illustrate the deterministic properties.