A graphical interpretation and universal formula for safe stabilization

The safe stabilization problem is studied via a graphical approach in this paper. Firstly, the compatibility condition for the control Lyapunov function (CLF) and control barrier function (CBF) is provided by visualizing and analyzing the geometry of safe stabilization. Related graphical interpretations are provided to show the proposed condition’s connections with the current results. Next, the analytical solution of the CLF and CBF-based quadratic program (CLF-CBF-QP) is obtained with a graphical interpretation. Because Sontag’s universal formula for nonlinear stabilization is a special solution of the point-wise minimal norm (PMN) controller, generalized universal formulas for both compatible and incompatible safe stabilization are derived. Afterward, some essential properties of the two proposed universal formulas are discussed, such as Lipschitz continuity, continuity at origin, locally asymptotic stability, safety, etc. Finally, we use the proposed generalized universal formulas to address the safe stabilization problem in adaptive cruise control (ACC) systems. The efficacy of the generalized universal formulas is exhibited with numerical results.