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Distributed Risk-Sensitive Safety Filters for Uncertain Discrete-Time Systems

Armin Lederer, Erfaun Noorani, Andreas Krause

TL;DR

This work tackles safety in uncertain discrete-time multi-agent systems where centralized coordination is impractical.It develops risk-sensitive safety filters by defining control barrier functions via value functions and employing an exponential risk operator to handle model uncertainty.The authors propose distributed safety-filter designs based on worst-case anticipation and proximity to a known safe policy, plus a switching mechanism to ensure feasibility.Numerical evaluations on coupled dynamics and collision-avoidance scenarios demonstrate that the distributed filters closely match centralized baselines while offering tunable safety-performance via the risk parameter.

Abstract

Ensuring safety in multi-agent systems is a significant challenge, particularly in settings where centralized coordination is impractical. In this work, we propose a novel risk-sensitive safety filter for discrete-time multi-agent systems with uncertain dynamics that leverages control barrier functions (CBFs) defined through value functions. Our approach relies on centralized risk-sensitive safety conditions based on exponential risk operators to ensure robustness against model uncertainties. We introduce a distributed formulation of the safety filter by deriving two alternative strategies: one based on worst-case anticipation and another on proximity to a known safe policy. By allowing agents to switch between strategies, feasibility can be ensured. Through detailed numerical evaluations, we demonstrate the efficacy of our approach in maintaining safety without being overly conservative.

Distributed Risk-Sensitive Safety Filters for Uncertain Discrete-Time Systems

TL;DR

This work tackles safety in uncertain discrete-time multi-agent systems where centralized coordination is impractical.It develops risk-sensitive safety filters by defining control barrier functions via value functions and employing an exponential risk operator to handle model uncertainty.The authors propose distributed safety-filter designs based on worst-case anticipation and proximity to a known safe policy, plus a switching mechanism to ensure feasibility.Numerical evaluations on coupled dynamics and collision-avoidance scenarios demonstrate that the distributed filters closely match centralized baselines while offering tunable safety-performance via the risk parameter.

Abstract

Ensuring safety in multi-agent systems is a significant challenge, particularly in settings where centralized coordination is impractical. In this work, we propose a novel risk-sensitive safety filter for discrete-time multi-agent systems with uncertain dynamics that leverages control barrier functions (CBFs) defined through value functions. Our approach relies on centralized risk-sensitive safety conditions based on exponential risk operators to ensure robustness against model uncertainties. We introduce a distributed formulation of the safety filter by deriving two alternative strategies: one based on worst-case anticipation and another on proximity to a known safe policy. By allowing agents to switch between strategies, feasibility can be ensured. Through detailed numerical evaluations, we demonstrate the efficacy of our approach in maintaining safety without being overly conservative.

Paper Structure

This paper contains 11 sections, 5 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Distributed Safety Filters based on CBFs
  4. Centralized Learning of Control Barrier Functions
  5. Distributed Safety Filters via Worst-Case Anticipation
  6. Distributed Safety Filters based on Control Proximity
  7. Feasibility-based Safety-Filter Switching
  8. Numerical Evaluation
  9. Dynamics Coupling between Agents
  10. Agent Coupling via Collision Constraints
  11. Conclusion

Key Result

Lemma 1

Assume there exists a constant $\hat{c}\in\mathbb{R}_+$, such that the cost $c:\mathbb{R}^{Md_x}\rightarrow\mathbb{R}_{0,+}$ satisfies Then, there exists a constant $\xi\in\mathbb{R}_+$, such that the intersection between $\mathbb{V}_{\bm{\pi}}^{\xi}$ and $\overline{\mathbb{X}}_{\mathrm{safe}}$ is empty, i.e., $\mathbb{V}_{\bm{\pi}}^{\xi}\cap\overline{\mathbb{X}}_{\mathrm{safe}}=\emptyset$.

Figures (3)

  • Figure 1: Top: Average trajectories for the nominal control law (dashed lines) overshoot and exhibit significant constraint violations, while our safety filter (full lines) slows down the increase to maintain safety. Shaded areas denote one standard deviation intervals. Bottom: The worst-case safety filter becomes mainly infeasible when one agent comes close to the constraint boundary, but has a high feasibility rate in general. This illustrates that switching in \ref{['eq:overall_filter']} is limited to few situations.
  • Figure 2: The cumulative reward (top) and number of constraint violations (bottom) for our distributed safety filter closely resemble the performance of the centralized baseline from lederer_risk-sensitive_2023. While the rewards only deviate from the nominal policy for small safety thresholds $\xi$, a considerable impact on the number of constraint violations can still be observed for relatively large $\xi$. Large standard deviation intervals (shaded areas) result from few roll-outs with negligible coupling $\theta\approx0$ causing a lack of controllability of the unactuated mass.
  • Figure 3: Top: While the tracking error is barely affected by the risk-aversion for $M=2$ and $M=3$, the increased difficulty of ensuring safety for $M=4$ causes a growth of the MSE with growing $\beta$. Bottom: Irrespective of the number of agents, the number of safety violations reduces with growing risk-aversion $\beta$. Shaded areas denote one standard deviation intervals.

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1: Curi2022
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof