A Duality-Based Optimization Formulation of Safe Control Design with State Uncertainties

TL;DR

This paper introduces a duality-based optimization approach for designing safe controllers under state uncertainties, providing less conservative safety guarantees through a novel set analysis.

Contribution

It formulates a robust safety filter using duality and set convexification, offering a more direct and less conservative safety control method.

Findings

The approach is less conservative than existing methods.

Convexifying the image set does not alter the set of possible inputs.

Simulation results validate the effectiveness of the proposed method.

Abstract

State estimation uncertainty is prevalent in real-world applications, hindering the application of safety-critical control. Existing methods address this by strengthening a Control Barrier Function (CBF) condition either to handle actuation errors induced by state uncertainty, or to enforce stricter, more conservative sufficient conditions. In this work, we take a more direct approach and formulate a robust safety filter by analyzing the image of the set of all possible states under the CBF dynamics. We first prove that convexifying this image set does not change the set of possible inputs. Then, by leveraging duality, we propose an equivalent and tractable reformulation for cases where this convex hull can be expressed as a polytope or ellipsoid. Simulation results show the approach in this paper to be less conservative than existing alternatives.