Advanced safety filter based on SOS Control Barrier and Lyapunov Functions

This paper presents a novel safety filter framework based on Control Barrier Functions (CBFs) and Control Lyapunov-like Functions (CLFs). The CBF guarantees forward invariance of the safe set, constraining system trajectories within state constraints, while the CLF guides the system away from unsafe states towards a nominal region, preserving the performance of a nominal controller. The first part of this work focuses on determining compatible CBF and CLF in the presence of linear or quadratic input constraints. This is achieved by formulating the CBF and CLF conditions, along with the input constraints, as Sum of Squares (SOS) constraints using Putinar's Positivstellensatz. For solving the resulting SOS optimization problem, we employ an alternating algorithm that simultaneously searches for a feasible controller in the class of rational functions of the state. The second part of this work details the implementation of the safety filter as a Quadratically Constrained Quadratic Program (QCQP), whose constraints encode the CBF and CLF conditions as well as the input constraints. To avoid the chattering effect and guarantee the uniqueness and Lipschitz continuity of solutions, the state-dependent inequality constraints of the QCQP are selected to be sufficiently regular. Finally, we demonstrate the method on a detailed case study involving the control of a three-phase ac/dc power converter connected to an infinite bus.