On the estimation of initial conditions in kernel-based system identification

TL;DR

This paper addresses the challenge of estimating initial conditions in kernel-based system identification, proposing three methods that improve impulse response reconstruction accuracy by jointly estimating initial states and kernel hyperparameters.

Contribution

It introduces three mixed maximum likelihood-a posteriori estimators for initial condition estimation in kernel-based system identification, utilizing EM algorithm for efficient solutions.

Findings

Proposed methods outperform existing schemes that ignore initial conditions.

Numerical experiments demonstrate improved impulse response reconstruction accuracy.

Methods effectively estimate initial conditions even with limited data.

Abstract

Recent developments in system identification have brought attention to regularized kernel-based methods, where, adopting the recently introduced stable spline kernel, prior information on the unknown process is enforced. This reduces the variance of the estimates and thus makes kernel-based methods particularly attractive when few input-output data samples are available. In such cases however, the influence of the system initial conditions may have a significant impact on the output dynamics. In this paper, we specifically address this point. We propose three methods that deal with the estimation of initial conditions using different types of information. The methods consist in various mixed maximum likelihood--a posteriori estimators which estimate the initial conditions and tune the hyperparameters characterizing the stable spline kernel. To solve the related optimization problems, we resort to the expectation-maximization method, showing that the solutions can be attained by iterating among simple update steps. Numerical experiments show the advantages, in terms of accuracy in reconstructing the system impulse response, of the proposed strategies, compared to other kernel-based schemes not accounting for the effect initial conditions.