Control Barrier Functions for Stochastic Systems under Signal Temporal Logic Tasks

Signal Temporal Logic (STL) offers an expressive formalism for describing complex high-level tasks in dynamical systems. This paper introduces a time-varying Control Barrier Function (CBF) for control-affine nonlinear stochastic systems to fulfill STL specifications. The CBFs are used in a robust optimization problem to provide a lower bound on the satisfaction probability of a given STL specification with a predetermined robustness level. We present an online control synthesis approach to minimize a performance function while having the required satisfaction guarantee. We finally provide a tractable solution to the robust optimization for STL with linear and quadratic predicate functions. To illustrate the effectiveness of the method, we apply it to a simple linear case study and to the path-planning problem for a nonlinear wheeled mobile robot.