Maximum likelihood estimates of linear dynamic systems

TLDR

The paper addresses state estimation for linear dynamic systems corrupted by additive Gaussian noise. Using maximum‑likelihood, the authors derive difference equations for filtering and smoothing estimates and their error covariances, extending the results from discrete to continuous systems. The resulting equations are computationally tractable, and a numerical example demonstrates that smoothing markedly reduces estimation errors.

Abstract

This paper considers the problem of estimating the states of linear dynamic systems in the presence of additive Gaussian noise. Difference equations relating the estimates for the problems of filtering and smoothing are derived as well as a similar set of equations relating the covariance of the errors. The derivation is based on the method of maximum likelihood and depends primarily on the simple manipulation of the probability density functions. The solutions are in a form easily mechanized on a digital computer. A numerical example is included to show the advantage of smoothing in reducing the errors in estimation. In the Appendix the results for discrete systems are formally extended to continuous systems.