Synthesis of Recursive Control Barrier Functions

This paper deals with the enforcement of safety constraints using control barrier functions (CBFs). We focus on dynamic systems that have high relative degree and where the output needs to be differentiated multiple times to obtain an explicit dependence on the control input. As the system approaches operating points that are close to the safety boundary, the CBF shrinks the allowed control action space. To quantify this shrinkage, we propose practical proxy metrics to capture the volume of the CBFs action space. We exploit these proxy metrics to derive sum-of-squares optimization formulations that synthesize CBFs with high-order degree, in particular their extended class $K$ functions. We apply this tuning technique to synthesize CBFs that avoid vehicle collisions in an adaptive cruise control problem. Simulation results demonstrate the ability of our tuning approach to reduce loss of control performance when the system operates close to the safety boundary.