Data-Driven Safe Control of Uncertain Linear Systems Under Aleatory Uncertainty

Safe control of constrained uncertain linear systems under aleatory uncertainty is considered. Aleatory uncertainty characterizes random noises and is modeled by a probability distribution function (pdf). 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
$\lambda$
-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-based probabilistic safe control, respectively. Data-based implementations of these probabilistic safe controllers are then considered. 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.