High-Probability Feedback Controllers from Data with Stochastic Disturbances

A central challenge in data-driven control is effectively leveraging different types of data. This work addresses the design of state feedback controllers for unknown systems using data corrupted by stochastic disturbances. First, we link the distribution of system parameters consistent with noisy data to the stochastic disturbances by reconstructing the data. Based on the distribution of the disturbances, we develop a data-driven formulation that characterizes admissible systems under probability constraints. Then, with this probabilistic description of the underlying system dynamics, a tractable problem in the form of linear matrix inequalities is formulated to evaluate the likelihood that a given feedback gain stabilizes the unknown system. Additionally, a data-driven optimization problem is proposed to co-design the stabilizing probability and feedback gain. This approach facilitates the design of a reliable state feedback controller with a high probability of stabilization. Furthermore, we demonstrate that the proposed co-design method guarantees an increase in the stabilizing probability with the collection of new data. Finally, numerical examples are presented to illustrate the advantages of the proposed controller design approach.