Identification of a Kalman filter: consistency of local solutions

TL;DR

The study addresses Kalman gain identification via Prediction Error Methods in linear dynamical systems, where non-convex optimization can yield spurious local minima. It proves that, for Kalman gain estimation, the PEM objective is asymptotically unimodal, yielding a unique local (and global) minimizer $L^\bstar$ as data grow, and that any interior local minimizer of the finite-sample objective converges to $L^\bstar$ almost surely. The results are supported by three numerical examples showing spurious minima vanish with more data and by an extension discussion for joint estimation of noise covariances and linear parameters. Practically, this provides robust statistical guarantees for Kalman-gain tuning and informs the design of PEM problems, including stability constraints, with potential algorithmic refinements such as alternating updates.

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.

Figures (3)

• Figure 1: Objective $V_N(L)$ and its limit $\bar{V}(L)$ for a one-dimensional system with $A = 0.9$, $C = 1, L^\star = 0.8$ and $e_k \sim {\mathcal{N}}(0, \, 1)$.
• Figure 2: Optimization iterates and solutions for the two-state example with $\alpha=0.02$, and the model parameters $\mu = 0.1$, $\Delta t = 0.1$, $\sigma_f = 10$, and $\sigma_v = 1$.
• Figure 3: Estimation error for the three-state example with two measurements, for different realizations and different values of $N$. The model parameters are as in Figure \ref{['fig-opti']}, with in addition: ${\mathrm{Cov}}\left[v^{{\mathrm{acc.}}}_k\right] = 1$, ${\mathrm{Cov}}\left[v^{{\mathrm{pos.}}}_k\right] = 2$, $p = 0.1$, $\sigma_w^2 = 10$, $a_f = 0.9$.