  Data-driven control based on the fundamental lemma by Willems et al. is
frequently considered for deterministic LTI systems subject to measurement
noise. However, besides measurement noise, stochastic disturbances might also
directly affect the dynamics. In this paper, we leverage Polynomial Chaos
Expansions (PCE) to extend the deterministic fundamental lemma towards
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.
