Efficient Learning of a Linear Dynamical System with Stability Guarantees

TL;DR

This paper introduces an efficient method to project any square matrix onto the set of asymptotically stable matrices, enabling stable system identification with explicit error bounds from limited data.

Contribution

It presents a novel, information-theoretic projection technique based on linear quadratic regulation to learn stable linear dynamical systems from a single trajectory.

Findings

Projection is optimal in an information-theoretic sense.

Method guarantees stability of the estimated system matrix.

Provides explicit statistical bounds on estimation error.

Abstract

We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers explicit statistical bounds on the estimation error.