Simple Kalman filter alternative: the multi-fractional order estimator

This study is an extension of a presentation given at the 2012 Radar Conference in Glasgow, UK entitled: Simple Disambiguation of Orthogonal Projection in Kalman's Filter Derivation. It is shown therein that Kalman misunderstood and consequently misapplied orthogonal projection in his derivation. Shocking analysis shows the Kalman filter (KF) to be truly optimum only in the absence of specific additive statistical measurement noise it is explicitly designed to filter out, rendering it a deterministic curve-fitting algorithm. Also included are two examples demonstrating that the KF is not the universally asserted minimum variance linear unbiased estimator. These consequences are based on the newly introduced multi-fractional order estimator (MFOE). Included herein are an additional background derivation and supplementary material as well as several new examples to better illustrate internal MFOE mechanics. Examples also show major problems in fine tuning the KF with a trade-off between process and measurement noises. Finally, an example shows a very simple, automatic, seamless, smooth, optimal, non-recursive second to third-order estimator transition which successfully overcomes inadequate and complicated KF adaptivity, including thresholding, feedback, manoeuvre detection, deweighting matrices, manoeuvre probability, dual filter weighting, variable dimension models, state augmentation, multiple models, interactive multiple models, interactive acceleration compensation and so on.