On a Stochastic Fundamental Lemma and Its Use for Data-Driven Optimal Control

Data-driven control based on the fundamental lemma by Willems et al. is frequently considered for deterministic linear time invariant (LTI) systems subject to measurement noise. However, besides measurement noise, stochastic disturbances might also directly affect the dynamics. In this article, we leverage polynomial chaos expansions to extend the deterministic fundamental lemma toward stochastic systems. This extension allows to predict future statistical distributions of the inputs and outputs for stochastic LTI systems in data-driven fashion, i.e., based on the knowledge of previously recorded input-output-disturbance data and of the disturbance distribution we perform data-driven uncertainty propagation. Finally, we analyze data-driven stochastic optimal control problems and we propose a conceptual framework for data-driven stochastic predictive control. Numerical examples illustrate the efficacy of the proposed concepts.