# Characterizing Smooth Safety Filters via the Implicit Function Theorem

arXiv: https://arxiv.org/abs/2309.12614 · HTML: https://mlsanity.com/paper/2309.12614/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2023.

## Built on by (most cited first, 1 of 1)

- [Unifying Controller Design for Stabilizing Nonlinear Systems with Norm-Bounded Control Inputs](https://mlsanity.com/paper/2403.03030/) (2024)

## Abstract (arXiv)

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. In this paper, we present a general characterization of smooth safety filters -- smooth controllers that guarantee safety in a minimally invasive fashion -- based on the Implicit Function Theorem. 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.