Learning Linearized Models from Nonlinear Systems with Finite Data

TL;DR

This paper develops a method to identify linearized models from nonlinear systems using multiple trajectories, providing finite sample error bounds and demonstrating the limitations of single-trajectory approaches.

Contribution

It introduces a deterministic data collection and regularized least squares approach for nonlinear systems, with theoretical error bounds and validation through experiments.

Findings

Finite sample error bounds depend on nonlinearity and noise levels.

Multiple trajectories improve linearized model accuracy over single trajectories.

Linear system identification may be insufficient for nonlinear systems with single trajectories.

Abstract

Identifying a linear system model from data has wide applications in control theory. The existing work on finite sample analysis for linear system identification typically uses data from a single 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. 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 demonstrates a trade-off between the error due to nonlinearity and the error due to noise, and shows that one can learn the linearized dynamics with arbitrarily small error given sufficiently many samples. We validate our results through 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.