High Order Control Lyapunov-Barrier Functions for Temporal Logic Specifications

Recent work has shown that stabilizing an affine control system to a desired state while optimizing a quadratic cost subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). In our own recent work, we defined High Order CBFs (HOCBFs) for systems and constraints with arbitrary relative degrees. In this paper, in order to accommodate initial states that do not satisfy the state constraints and constraints with arbitrary relative degree, we generalize HOCBFs to High Order Control Lyapunov-Barrier Functions (HOCLBFs). We also show that the proposed HOCLBFs can be used to guarantee the Boolean satisfaction of Signal Temporal Logic (STL) formulae over the state of the system. We illustrate our approach on a safety-critical optimal control problem (OCP) for a unicycle.