Optimal initialization of linear recursive filters

Based on a combination of the Bayesian viewpoint and the classical (non-Bayesian) weighted least squares (WLS) method, an optimal estimator for a linear stochastic system particularly suitable for recursive filter initialization is presented. It accounts for the fact that the data set for initialization in practice consists of the measurements of the time-varying state of the dynamic system, which is random in filtering problems and thus the initialization problem cannot be properly handled either by a Bayesian approach or in the classical WLS formulation. The results are given for both continuous- and discrete-time models of a dynamic system with discrete-time measurements. The proposed estimator is compared with the popular two-point difference technique for initialization, which is a special form of the classical WLS method. Simulation results are provided to support the theoretical results.