Simultaneous Estimation Of The State And Noise Statistics In Linear Dynamical Systems

An optimal procedure for estimating the state of a linear dynamical system when the statistics of the measurement and process noise are poorly known is developed. The criterion of maximum likelihood is used to obtain an optimal estimate of the state and noise statistics. These estimates are shown to be asymptotically unbiased, efficient, and unique, with the estimation error normally distributed with a known covariance. The resulting equations for the estimates cannot be solved recursively, but an iterative procedure for their solution is presented. Several approximate solutions are presented which reduce the necessary computations in finding the estimates. Some of the approximate solutions allow a real time estimation of the state and noise statistics. Closely related to the estimation problem is the subject of hypothesis testing. Several criteria are developed for testing hypotheses concerning the values of the noise statistics that are used in the computation of the appropriate filter gains in a linear Kalman type state estimator. If the observed measurements are not consistent with the assumptions about the noise statistics, then estimation of the noise statistics should be undertaken using either optimal or suboptimal procedures. Numerical results of a digital computer simulation of the optimal and suboptimal solutions of the estimation problem are presented for a simple but realistic example.