Learning the Dynamics of Autonomous Linear Systems From Multiple 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 tradeoff 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.

Learning Dynamical Systems by Leveraging Data from Similar Systems

“…The derived sample complexity results typically show how the system identification error goes to zero by increasing the number of samples used in the single trajectory. For the multiple trajectories setup, it is typically assumed that one has access to data generated from multiple independent trajectories [6], [14]- [17]. In practice, the multiple trajectories setup has the advantage of being able to handle unstable systems, and other cases where collecting a single long trajectory is infeasible.…”