Verification and Synthesis of Compatible Control Lyapunov and Control Barrier Functions

Safety and stability are essential properties of control systems. Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) have been proposed to ensure safety and stability respectively. However, previous approaches typically verify and synthesize the CBFs and CLFs separately, satisfying their respective constraints, without proving that the CBFs and CLFs are compatible with each other, namely at every state, there exists control actions that satisfy both the CBF and CLF constraints simultaneously. There exists some recent works that synthesized compatible CLF and CBF, but relying on nominal polynomial or rational controllers, which is just a sufficient but not necessary condition for compatibility. In this work, we investigate verification and synthesis of compatible CBF and CLF independent from any nominal controllers. We derive exact necessary and sufficient conditions for compatibility, and further formulate Sum-Of-Squares program for the compatibility verification. Based on our verification framework, we also design an alternating nominal-controller-free synthesis method. We evaluate our method in a linear toy, a non-linear toy, and a power converter example.