Universal Formula for Smooth Safe Stabilization

This paper formulates a safe and stabilizing control state feedback law for a control affine nonlinear system. We assume that there exist a known control Lyapunov function (CLF) and a control barrier function (CBF) that are compatible, i.e., there exists a control choice satisfying the conditions given by both the CLF and CBF at each given state. In contrast to the approach in the literature of finding a minimum-norm control using optimization on the feasible control set, we take a different approach by finding and combining different weighted centroids of the feasible control set. As a result, we can propose a control feedback law with guaranteed smoothness everywhere except at the origin, and with guaranteed continuity at the origin if small control property holds.