On the sample complexity of stabilizing linear dynamical systems from data

TL;DR

This paper demonstrates that stabilizing linear systems can be achieved by observing only a number of states equal to the system's McMillan degree, reducing data requirements compared to full system identification.

Contribution

It shows that stabilizing controllers can be constructed from fewer observed states than needed for full model learning, based on the system's intrinsic dimension.

Findings

Stabilization possible from n states for an n-dimensional system

Fewer data needed than for full system identification

Numerical experiments confirm theoretical results

Abstract

Learning controllers from data for stabilizing dynamical systems typically follows a two step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) $n$, then there always exist $n$ states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.