Identification of a Kalman filter: consistency of local solutions

TL;DR

This paper proves that local solutions in Kalman filter identification are statistically consistent and asymptotically unique, easing concerns about local minima in system tuning.

Contribution

It demonstrates that local minimizers for estimating the Kalman gain are consistent and unique asymptotically, providing new insights into Kalman filter tuning.

Findings

Local solutions are statistically consistent estimates of the true Kalman gain.

As dataset size increases, the objective function becomes unimodal with a unique minimizer.

Guidelines are provided for designing optimization problems for Kalman filter tuning.

Abstract

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.