Characterizing Smooth Safety Filters via the Implicit Function Theorem

Abstract-- Optimization-based safety filters, such as control barrier function (CBF) based quadratic programs (QPs), have demonstrated success in controlling autonomous systems to achieve complex goals. These CBF-QPs can be shown to be continuous, but are generally not smooth, let alone continuously differentiable. This characterization leads to families of smooth universal formulas for safety-critical controllers that quantify the conservatism of the resulting safety filter, the utility of which is demonstrated through illustrative examples. Over the past decade, control barrier functions (CBFs) [1] have proven to be a powerful tool for designing controllers enforcing safety on nonlinear systems. Most often, such safety filters are instantiated via optimization problems - typically a quadratic program adapt smooth universal formulas for CLFs [12] to CBFs.