A Bayesian nonlinear filtering algorithm for linear systems with unknown time-varying noise statistics

The problem of estimating the state of a linear dynamic system driven by additive gaussian noise with unknown time varying statistics is considered. Estimates of the state of the system are obtained which are based on all past observations of the system. These observations are linear functions of the state contaminated by additive white gaussian noise with unknown time varying statistics. The case of scalar measurements is given first and the more general solution for vector measurements is stated. A previously developed algorithm designed for use in the case of stationary noise is modified to allow estimation of an unknown Kalman gain and thus the system state in the presence of unknown time varying noise statistics. The algorithm is inherently parallel in nature and if implemented in a computer with parallel processing capability should only be slightly slower than the stationary Kalman filtering algorithm with known noise statistics. Numerical examples for both the scalar and vector measurement cases are given.