A Heuristic Reference Recursive Recipe for the Menacing Problem of Adaptively Tuning the Kalman Filter Statistics. Part-1. Formulation and Simulation Studies

Since the innovation of the ubiquitous Kalman filter more than five decades back it is well known that to obtain the best possible estimates the tuning of its statistics $X_0$, $P_0$, $\Theta$, $R$ and $Q$ namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The earlier tweaking and other systematic approaches are reviewed but none has reached a simple and easily implementable approach for any application. The present reference recursive recipe based on multiple filter passes through the data leads to a converged `statistical equilibrium' solution. It utilizes the pre, post, and smoothed state estimates and their corresponding measurements and the actual measurements as well as their covariances to balance the state and measurement equations and form generalized cost functions. The filter covariance at the end of each pass is heuristically scaled up by the number of data points and further trimmed to provide the $P_0$ for subsequent passes. A simultaneous and proper choice for $Q$ and $R$ based on the filter sample statistics and certain other covariances leads to a stable filter operation providing the results after few iterations. When only $R$ is present in the data by minimizing the `innovation' cost function using the non filter based Newton Raphson optimization results served as an anchor for matching and tuning the filter statistics. When both $R$ and $Q$ are present in the data the consistency between the injected noise sequences and their statistics provided a simple route and confidence in the present approach. A typical simulation study of a spring, mass, damper system with a weak non linear spring constant shows the present approach out performs earlier techniques. The Part-2 of the paper further consolidates the present approach based on an analysis of real airplane flight test data.