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Safe Control Synthesis for Hybrid Systems through Local Control Barrier Functions

Shuo Yang, Mitchell Black, Georgios Fainekos, Bardh Hoxha, Hideki Okamoto, Rahul Mangharam

TL;DR

This work addresses safety guarantees for hybrid dynamical systems by moving beyond a single global CBF to multiple local CBFs, one per dynamical mode. It introduces a switching-safety framework that identifies safe and unsafe transition regions, uses Hamilton-Jacobi reachability to compute backward-unsafe sets, and employs dynamic-programming-based refinement to obtain a refined local CBF h_{q,q'} for each potential mode transition. The approach yields global safety guarantees by intersecting mode-specific safe sets and control laws, and is validated with adaptive cruise control and Dubins car collision-avoidance case studies, demonstrating improved safety with reduced conservatism compared to switch-unaware and global-CBF baselines. The methodology enables scalable, switching-aware safety enforcement for hybrid systems, with potential extensions to broader hybrid models and non-deterministic jumps.

Abstract

Control Barrier Functions (CBF) have provided a very versatile framework for the synthesis of safe control architectures for a wide class of nonlinear dynamical systems. Typically, CBF-based synthesis approaches apply to systems that exhibit nonlinear -- but smooth -- relationship in the state of the system and linear relationship in the control input. In contrast, the problem of safe control synthesis using CBF for hybrid dynamical systems, i.e., systems which have a discontinuous relationship in the system state, remains largely unexplored. In this work, we build upon the progress on CBF-based control to formulate a theory for safe control synthesis for hybrid dynamical systems. Under the assumption that local CBFs can be synthesized for each mode of operation of the hybrid system, we show how to construct CBF that can guarantee safe switching between modes. The end result is a switching CBF-based controller which provides global safety guarantees. The effectiveness of our proposed approach is demonstrated on two simulation studies.

Safe Control Synthesis for Hybrid Systems through Local Control Barrier Functions

TL;DR

This work addresses safety guarantees for hybrid dynamical systems by moving beyond a single global CBF to multiple local CBFs, one per dynamical mode. It introduces a switching-safety framework that identifies safe and unsafe transition regions, uses Hamilton-Jacobi reachability to compute backward-unsafe sets, and employs dynamic-programming-based refinement to obtain a refined local CBF h_{q,q'} for each potential mode transition. The approach yields global safety guarantees by intersecting mode-specific safe sets and control laws, and is validated with adaptive cruise control and Dubins car collision-avoidance case studies, demonstrating improved safety with reduced conservatism compared to switch-unaware and global-CBF baselines. The methodology enables scalable, switching-aware safety enforcement for hybrid systems, with potential extensions to broader hybrid models and non-deterministic jumps.

Abstract

Control Barrier Functions (CBF) have provided a very versatile framework for the synthesis of safe control architectures for a wide class of nonlinear dynamical systems. Typically, CBF-based synthesis approaches apply to systems that exhibit nonlinear -- but smooth -- relationship in the state of the system and linear relationship in the control input. In contrast, the problem of safe control synthesis using CBF for hybrid dynamical systems, i.e., systems which have a discontinuous relationship in the system state, remains largely unexplored. In this work, we build upon the progress on CBF-based control to formulate a theory for safe control synthesis for hybrid dynamical systems. Under the assumption that local CBFs can be synthesized for each mode of operation of the hybrid system, we show how to construct CBF that can guarantee safe switching between modes. The end result is a switching CBF-based controller which provides global safety guarantees. The effectiveness of our proposed approach is demonstrated on two simulation studies.
Paper Structure (14 sections, 6 theorems, 11 equations, 4 figures, 1 algorithm)

This paper contains 14 sections, 6 theorems, 11 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Safety for hybrid system
  4. Stability for hybrid system
  5. Preliminary and Problem Formulation
  6. Control Barrier Functions
  7. Hybrid Automaton
  8. Problem Formulation
  9. Safe control for hybrid systems
  10. Case studies
  11. Adaptive cruise control
  12. Dubins car collision avoidance
  13. Conclusion and Discussion
  14. Acknowledgment

Key Result

Theorem 1

( ames2016control) Assume $h(x)$ is a CBF on $D \supset \mathcal{C}$ and $\frac{\partial h}{\partial x}(x) \neq 0$ for all $x \in \partial \mathcal{C}$. Then any Lipschitz continuous controller $u(x)$ such that $u(x) \in K_{cbf}(x)$ for all $x \in \mathcal{C}$ will render the set $\mathcal{C}$ forwa

Figures (4)

  • Figure 1: Hybrid adaptive cruise control.
  • Figure 2: One hybrid automaton example where safe switching controller exists but Theorem \ref{['thm:global-safety']} is not applicable.
  • Figure 3: Trajectories of our approach, switch-unaware CBF approach, and global CBF approach. The unsafe area (red) is defined by the safety constraint function $c(x)$.
  • Figure 4: Dubins car is reaching a goal and avoiding two obstacles. The white and green regions are dry and wet surfaces, respectively. Gray boxes are obstacles. Nominal controller provides reference trajectory for CBF-based approaches. Global CBF method is not applicable in this case.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 1
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 15 more