A Stochastic Fundamental Lemma with Reduced Disturbance Data Requirements

TL;DR

This paper introduces a new stochastic fundamental lemma variant that predicts system trajectories without needing past disturbance data, leveraging causality and polynomial chaos expansions.

Contribution

It proposes a fundamental lemma variant that reduces data requirements for disturbances, enhancing data-driven control of stochastic systems.

Findings

The new lemma predicts trajectories without past disturbance data.

Polynomial chaos expansions are used to model disturbance distributions.

Numerical example demonstrates the lemma's effectiveness in control applications.

Abstract

Recently, the fundamental lemma by Willems et al. has been extended towards stochastic LTI systems subject to process disturbances. Using this lemma requires previously recorded data of inputs, outputs, and disturbances. In this paper, we exploit causality concepts of stochastic control to propose a variant of the stochastic fundamental lemma that does not require past disturbance data in the Hankel matrices. Our developments rely on polynomial chaos expansions and on the knowledge of the disturbance distribution. Similar to our previous results, the proposed variant of the fundamental lemma allows to predict future input-output trajectories of stochastic LTI systems. We draw upon a numerical example to illustrate the proposed variant in data-driven control context.