Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data

TL;DR

This paper develops a method for identifying linearized models of nonlinear systems using multiple trajectories under initialization constraints, providing finite data guarantees and highlighting the limitations of single-trajectory approaches.

Contribution

It introduces a deterministic multi-trajectory data collection and regularized least squares method for nonlinear systems, with finite sample error bounds and analysis of nonlinearity and noise trade-offs.

Findings

Finite sample error bounds for learned linearized dynamics

Multi-trajectory approach outperforms single-trajectory methods in nonlinear settings

Numerical experiments validate theoretical results and highlight limitations of traditional methods.

Abstract

The identification of a linear system model from data has wide applications in control theory. The existing work that provides finite sample guarantees for linear system identification typically uses data from a single long system trajectory under i.i.d. random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear, given that there is a certain constraint on the region where one can initialize the experiments. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound shows that one can consistently learn the linearized dynamics, and demonstrates a trade-off between the error due to nonlinearity and the error due to noise. We validate our results through numerical experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d. random inputs, when nonlinearity does exist.