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

• Published in: IEEE transactions on automatic control Vol. 68; no. 10; pp. 1 - 16
• Main Authors: Pan, Guanru, Ou, Ruchuan, Faulwasser, Timm
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Control data (computers); Data-driven control; fundamental lemma; learning systems; Linear systems; model predictive control; Noise measurement; Numerical prediction; Optimal control; polynomial chaos; Polynomials; Predictive control; Random variables; Statistical distributions; Stochastic processes; Stochastic systems; Trajectory; Uncertainty; uncertainty quantification
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3232442

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.