Provably stable learning control of linear dynamics with multiplicative noise

TL;DR

This paper presents a data-driven approach for identifying and controlling linear systems with multiplicative noise, ensuring stability and optimality with probabilistic guarantees using tensor algebra techniques.

Contribution

It introduces a least-squares identification method with confidence bounds and a robust LQR control synthesis for systems with multiplicative noise, leveraging tensor algebra.

Findings

Guaranteed stability with high probability

Sample complexity bounds for system identification

Convergence to the true optimal control

Abstract

Control of linear dynamics with multiplicative noise naturally introduces robustness against dynamical uncertainty. Moreover, many physical systems are subject to multiplicative disturbances. In this work we show how these dynamics can be identified from state trajectories. The least-squares scheme enables exploitation of prior information and comes with practical data-driven confidence bounds and sample complexity guarantees. We complement this scheme with an associated control synthesis procedure for LQR which robustifies against distributional uncertainty, guaranteeing stability with high probability and converging to the true optimum at a rate inversely proportional with the sample count. Throughout we exploit the underlying multi-linear problem structure through tensor algebra and completely positive operators. The scheme is validated through numerical experiments.