Introduction to the Kalman Filter and Tuning its Statistics for Near Optimal Estimates and Cramer Rao Bound

Abstract: This report provides a brief historical evolution of the concepts in the Kalman filtering theory since ancient times to the present. A brief description of the filter equations its aesthetics, beauty, truth, fascinating perspectives and competence are described. For a Kalman filter design to provide optimal estimates tuning of its statistics namely initial state and covariance, unknown parameters, and state and measurement noise covariances is important. The earlier tuning approaches are reviewed. The present approach is a reference recursive recipe based on multiple filter passes through the data without any optimization to reach a `statistical equilibrium' solution. It utilizes the a priori, a posteriori, and smoothed states, their corresponding predicted measurements and the actual measurements help to balance the measurement equation and similarly the state equation to help form a generalized likelihood cost function. The filter covariance at the end of each pass is heuristically scaled up by the number of data points is further trimmed to statistically match the exact estimates and Cramer Rao Bounds (CRBs) available with no process noise provided the initial covariance for subsequent passes. During simulation studies with process noise the matching of the input and estimated noise sequence over time and in real data the generalized cost functions helped to obtain confidence in the results. Simulation studies of a constant signal, a ramp, a spring, mass, damper system with a weak non linear spring constant, longitudinal and lateral motion of an airplane was followed by similar but more involved real airplane data was carried out in MATLAB. In all cases the present approach was shown to provide internally consistent and best possible estimates and their CRBs.