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Generalizations of Backup Control Barrier Functions: Expansion and Adaptation for Input-Bounded Safety-Critical Control

David E. J. van Wijk, Dohyun Lee, Ersin Das, Tamas G. Molnar, Aaron D. Ames, Joel W. Burdick

Abstract

Guaranteeing the safety of nonlinear systems with bounded inputs remains a key challenge in safe autonomy. Backup control barrier functions (bCBFs) provide a powerful mechanism for constructing controlled invariant sets by propagating trajectories under a pre-verified backup controller to a forward invariant backup set. While effective, the standard bCBF method utilizes the same backup controller for both set expansion and safety certification, which can restrict the expanded safe set and lead to conservative dynamic behavior. In this study, we generalize the bCBF framework by separating the set-expanding controller from the verified backup controller, thereby enabling a broader class of expansion strategies while preserving formal safety guarantees. We establish sufficient conditions for forward invariance of the resulting implicit safe set and show how the generalized construction recovers existing bCBF methods as special cases. Moreover, we extend the proposed framework to parameterized controller families, enabling online adaptation of the expansion controller while maintaining safety guarantees in the presence of input bounds.

Generalizations of Backup Control Barrier Functions: Expansion and Adaptation for Input-Bounded Safety-Critical Control

Abstract

Guaranteeing the safety of nonlinear systems with bounded inputs remains a key challenge in safe autonomy. Backup control barrier functions (bCBFs) provide a powerful mechanism for constructing controlled invariant sets by propagating trajectories under a pre-verified backup controller to a forward invariant backup set. While effective, the standard bCBF method utilizes the same backup controller for both set expansion and safety certification, which can restrict the expanded safe set and lead to conservative dynamic behavior. In this study, we generalize the bCBF framework by separating the set-expanding controller from the verified backup controller, thereby enabling a broader class of expansion strategies while preserving formal safety guarantees. We establish sufficient conditions for forward invariance of the resulting implicit safe set and show how the generalized construction recovers existing bCBF methods as special cases. Moreover, we extend the proposed framework to parameterized controller families, enabling online adaptation of the expansion controller while maintaining safety guarantees in the presence of input bounds.
Paper Structure (8 sections, 10 theorems, 41 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. Backup Control Barrier Functions
  5. Generalized Backup Set Method
  6. Adaptive Set Expansion
  7. Numerical Simulations
  8. Conclusion

Key Result

Theorem 1

If $h$ is a CBF for eq:affine-dynamics on $\mathcal{C}_{\rm S}$, then any locally Lipschitz controller $\boldsymbol{k}:\mathcal{X} \to \mathcal{U}$, $\boldsymbol{u}=\boldsymbol{k}(\boldsymbol{x})$ satisfying for all $\boldsymbol{x} \in \mathcal{C}_{\rm S}$ renders the set $\mathcal{C}_{\rm S}$ forward invariant.

Figures (2)

  • Figure 1: Controlled invariant sets for \ref{['eq: db_int']} with three different expansion strategies. $\mathcal{C}_{\rm I}$ is constructed using the closed-loop flow for the backup controller $\boldsymbol{k}_{\rm b}$, whilst $\mathcal{C}^\star_{\rm I,1}$ and $\mathcal{C}^\star_{\rm I,2}$ use the switching controller $\boldsymbol{k}_{\rm s}$ in \ref{['eq:switching_controller']}, which allows for a more flexible set expansion policy. The flows denoted by $\boldsymbol{\phi}_{\rm b}$, $\boldsymbol{\phi}_{\rm s,1}$, and $\boldsymbol{\phi}_{\rm s,2}$ represent the evolution of \ref{['eq: db_int']} under each of the controllers. The viability kernel, denoted by $\mathcal{C}_{\rm V}$, is plotted for comparison.
  • Figure 2: Simulation results of the safe quadrotor landing problem, comparing the standard bCBF (purple), the generalized bCBF (red), and the adaptive bCBF (blue). All three approaches maintain safety (a,$\mkern2mu$b,$\mkern2mu$c) and obey the input constraints (d,$\mkern2mu$e), yet differ in task performance. The bCBF is unable to reach the goal due to the terminal constraint, as the flows (dashed gray) must reach $\mathcal{C}_{\rm B}$ (a). The \ref{['eq:gen-qp']} also does not reach the goal, but achieves a much larger expansion of $\mathcal{C}_{\rm B}$ than the \ref{['eq:bcbf-qp']} (b). Using the adaptation law in \ref{['cor:adaptation_law']}, the adaptive approach reaches the goal via superior expansion of $\mathcal{C}_{\rm B}$ (c) by modifying the backup controller gains (f,$\mkern2mu$g).

Theorems & Definitions (22)

  • Theorem 1: ames_2017
  • Definition 1: Backup Set and Controller
  • Remark 1
  • Lemma 1: gurriet_scalable_2020 tamasACC_ROM_bCBF
  • Theorem 2: gurriet_scalable_2020
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Lemma 3
  • ...and 12 more