Safety-Critical Control of Discontinuous Systems with Nonsmooth Safe Sets

TL;DR

The paper tackles safety-critical control for discontinuous dynamics with nonsmooth safe sets defined as nested unions and intersections of $0$-superlevel sets. It first shows that an active-component QP controller can be unsafe at points of non-differentiability, then introduces an all-components QP controller with transition functions to allow safe transitions between components, ensuring feasibility and continuity; an adaptive variant further tunes safety parameters online. The authors provide rigorous sufficiency and feasibility analyses, proving that the all-components and adaptive formulations yield safe, single-valued, and continuous controllers wherever the dynamics are continuous. They validate the approach via a multi-agent reconfiguration application, illustrating the method's ability to handle complex, nested disjunctive/conjunctive safety specifications in real time.

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.

Figures (3)

• Figure 1: Illustration for Example \ref{['ex:unsafeDisc']} of the closed-loop dynamics under the controller $u_{\operatorname{act}}$ at the corner point of the safe set. The controller prevents trajectories from violating safety constraints at points of smoothness of $h$, cf. Theorem \ref{['thm:unsafeNondiff']}, but does not prevent violating safety from points of non-smoothness.
• Figure 2: Illustration of the role of the transition function. The safe set is $\mathcal{C}_1 \cup \mathcal{C}_2$. The continuous function $\beta_1$ is positive at the boundary points of $\mathcal{C}_1$ only if they are in $\operatorname{int}(\mathcal{C}_2)$, and zero otherwise in $\mathcal{C}_1$. The constraints in the all-components QP controller $u_{\operatorname{all}}$ defined in \ref{['eq:qp2']} allow trajectories to leave one set only to the other, while remaining in the safe set.
• Figure 3: Illustration of the adaptive all-components controller $u_{\operatorname{adp}}$ acting in a multi-agent reconfiguration problem. From left to right, different snapshots of the agent evolution as time progresses are portrayed. The agents (black circles) travel from initial points (empty circles) to final destinations (crosses) while avoiding the unsafe region (gray area) and colliding with each other. This illustrates how the proposed control design handles nested disjunctive and conjunctive constraints.

Theorems & Definitions (16)

• Definition 2.1
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Corollary 4.3
• Example 4.4
• Remark 4.5
• Theorem 4.6
• Theorem 4.7
• Lemma 4.8