Safe Control in the Presence of Stochastic Uncertainties

Accurate quantification of safety is essential for the design of autonomous systems. In this paper, we present a methodology to characterize the exact probabilities associated with invariance and recovery in safe control. We consider a stochastic control system where control barrier functions, gradient-based methods, and barrier certificates are used to constrain control actions and validate safety. We derive the probability distributions of the minimum and maximum barrier function values during any time interval and the first entry and exit times to and from any super level sets of the barrier function. These distributions are characterized by deterministic convection-diffusion equations, and the approach used is generalizable to other safe control methods based on barrier functions. These distributions can be used to characterize various quantities associated with invariance and recovery, such as the safety margin, the probability of entering and recovering from safe and unsafe regions, and the mean and tail distributions of failure and recovery times.