Verification and Synthesis of Compatible Control Lyapunov and Multiple High-Order Control Barrier Functions

Abstract

Stability and safety are important properties of control systems. control Lyapunov functions (CLF) and control barrier functions (CBF) have been proposed to enforce stability and safety properties, respectively. However CLFs and CBFs could fail to guarantee joint safety and stability when there does not exist a common control action that simultaneously satisfies both the CBF and CLF constraints. In this article, for control-affine dynamical systems both with and without input limits, we propose the necessary and sufficient condition for compatible CLFs and multiple high-order control barrier functions. We derive the compatibility condition using Farkas’ Lemma and Positivstellensatz, and formulate an sum-of-squares (SOS) program to numerically verify the compatibility condition. Based on the proposed verification framework, we propose a bilinear-alternating method to synthesize compatible CLFs and CBFs with a large safe region and region-of-attraction. Moreover, we derive the necessary and sufficient condition for the continuity of the CLF-CBF-quadratic program controller, which we can verify through solving an SOS program. We showcase our approach in several numerical examples, including adaptive cruise control, an inverted pendulum, and a 13-state quadrotor.