Identification of a Kalman filter: consistency of local solutions

Prediction error and maximum likelihood methods are powerful tools for identifying linear dynamical systems and, in particular, enable the joint estimation of model parameters and the Kalman filter used for state estimation. A key limitation, however, is that these methods require solving a generally non-convex optimization problem to global optimality. This paper analyzes the statistical behavior of local minimizers in the special case where only the Kalman gain is estimated. We prove that these local solutions are statistically consistent estimates of the true Kalman gain. This follows from asymptotic unimodality: as the dataset grows, the objective function converges to a limit with a unique local (and therefore global) minimizer. We further provide guidelines for designing the optimization problem for Kalman filter tuning and discuss extensions to the joint estimation of additional linear parameters and noise covariances. Finally, the theoretical results are illustrated using three examples of increasing complexity. The main practical takeaway of this paper is that difficulties caused by local minimizers in system identification are, at least, not attributable to the tuning of the Kalman gain.