Safety-Critical Control of Discontinuous Systems with Nonsmooth Safe Sets

TL;DR

This paper develops a novel control design for discontinuous systems with complex nonsmooth safe sets, using transition functions to ensure safety without unnecessary conservatism, demonstrated through multi-agent reconfiguration.

Contribution

Introduces transition functions for safety control of discontinuous systems, enabling safe transitions between nonsmooth safe set components via convex optimization.

Findings

Convex optimization-based controller guarantees safety in discontinuous dynamics.

Transition functions enable safe system transitions across nonsmooth safe sets.

Method effectively applied to multi-agent reconfiguration problems.

Abstract

This paper studies the design of controllers for discontinuous dynamics that ensure the safety of non-smooth sets. The safe set is represented by arbitrarily nested unions and intersections of 0-superlevel sets of differentiable functions. We show that any optimization-based controller that satisfies only the point-wise active safety constraints is generally un-safe, ruling out the standard techniques developed for safety of continuous dynamics. This motivates the introduction of the notion of transition functions, which allow us to incorporate even the inactive safety constraints without falling into unnecessary conservatism. These functions allow system trajectories to leave a component of the nonsmooth safe set to transition to a different one. The resulting controller is then defined as the solution to a convex optimization problem, which we show is feasible and continuous wherever the system dynamics is continuous. We illustrate the effectiveness of the proposed design approach in a multi-agent reconfiguration control problem.