A heuristic reference recursive recipe for 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 \({\mathbf{X}}_{\mathbf{0}}\), \({\mathbf{P}}_{\mathbf{0}}\), \(\Theta \), R and Q namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The manual and other approaches have not matured to a routine approach applicable for any general problem. The present reference recursive recipe (RRR) utilizes the prior, posterior, and smoothed state estimates as well as their covariances to balance the state and measurement equations and thus 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 \({\mathbf{P}}_{\mathbf{0}}\) for subsequent passes. The importance of \({\mathbf{P}}_{\mathbf{0}}\) as the probability matching prior between the frequentist approach via optimization and the Bayesian approach of the Kalman filter is stressed. A simultaneous and proper choice for Q and R based on the filter sample statistics and other covariances leads to a stable filter operation after a few iterations. A typical simulation study of a spring, mass and damper system with a weak nonlinear spring constant by RRR shows it to be better than earlier techniques. Part-2 of the paper further consolidates the present approach based on an analysis of real flight test data.