Convex Synthesis of Control Barrier Functions Under Input Constraints

This letter presents a systematic method based on the sum of square (SOS) optimization to synthesize control barrier functions (CBFs) for nonlinear polynomial systems subject to input constraints. The approach consists of two design steps. In the first step, using a linear-like representation of the nonlinear dynamics, an SOS optimization problem is formulated to search for an initial CBF and controller jointly. In the second step, an iterative optimization procedure involving the solution of a series of SOS problems is proposed to alternatively update the CBF and the controller to increase the invariant set defined by the CBF. The efficacy of the proposed approach is validated using numerical examples.