SOS Construction of Compatible Control Lyapunov and Barrier Functions

We propose a novel approach to certify closed-loop stability and safety of a constrained polynomial system based on the combination of Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs). For polynomial systems that are affine in the control input, both classes of functions can be constructed via Sum Of Squares (SOS) programming. Using two versions of the Positivstellensatz we derive an SOS formulation seeking a rational controller that — if feasible — results in compatible CLF and multiple CBFs.