Data-driven Safe Control of Linear Systems Under Epistemic and Aleatory Uncertainties

Safe control of constrained linear systems under both epistemic and aleatory uncertainties is considered. The aleatory uncertainty characterizes random noises and is modeled by a probability distribution function (PDF) and the epistemic uncertainty characterizes the lack of knowledge on the system dynamics. Data-based probabilistic safe controllers are designed for the cases where the noise PDF is 1) zero-mean Gaussian with a known covariance, 2) zero-mean Gaussian with an uncertain covariance, and 3) zero-mean non-Gaussian with an unknown distribution. Easy-to-check model-based conditions for guaranteeing probabilistic safety are provided for the first case by introducing probabilistic contractive sets. These results are then extended to the second and third cases by leveraging distributionally-robust probabilistic safe control and conditional value-at-risk (CVaR) based probabilistic safe control, respectively. Data-based implementations of these probabilistic safe controllers are then considered. It is shown that data-richness requirements for directly learning a safe controller is considerably weaker than data-richness requirements for model-based safe control approaches that undertake a model identification. Moreover, an upper bound on the minimal risk level, under which the existence of a safe controller is guaranteed, is learned using collected data. A simulation example is provided to show the effectiveness of the proposed approach.