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Passivity-Preserving Safety-Critical Control using Control Barrier Functions

Federico Califano

TL;DR

This work addresses whether safety-critical control based on control barrier functions (CBFs) preserves passivity when applied to a passivity-based controller (PBC). It derives a concrete criterion that links the passivity dissipation, the CBF, and the CBF-induced safety input, within an energetic framework, and specializes the results to port-Hamiltonian (pH) mechanical systems. By casting EB-PBC in the pH setting, the authors obtain a closed-form damping-injection mechanism and show how energy-based CBFs can implement nontrivial damping that is task-aware rather than purely stabilizing. The approach is validated on a cart-pole system, illustrating how safety filtering can enforce physical or kinematic safety while preserving passive interconnections and enabling energy shaping.

Abstract

In this letter we propose a holistic analysis merging the techniques of passivity-based control (PBC) and control barrier functions (CBF). We constructively find conditions under which passivity of the closed-loop system is preserved under CBF-based safety-critical control. The results provide an energetic interpretation of safety-critical control schemes, and induce novel passive designs which are less conservative than standard methods based on damping injection. The results are specialised to port-Hamiltonian systems and simulations are performed on a cart-pole system.

Passivity-Preserving Safety-Critical Control using Control Barrier Functions

TL;DR

This work addresses whether safety-critical control based on control barrier functions (CBFs) preserves passivity when applied to a passivity-based controller (PBC). It derives a concrete criterion that links the passivity dissipation, the CBF, and the CBF-induced safety input, within an energetic framework, and specializes the results to port-Hamiltonian (pH) mechanical systems. By casting EB-PBC in the pH setting, the authors obtain a closed-form damping-injection mechanism and show how energy-based CBFs can implement nontrivial damping that is task-aware rather than purely stabilizing. The approach is validated on a cart-pole system, illustrating how safety filtering can enforce physical or kinematic safety while preserving passive interconnections and enabling energy shaping.

Abstract

In this letter we propose a holistic analysis merging the techniques of passivity-based control (PBC) and control barrier functions (CBF). We constructively find conditions under which passivity of the closed-loop system is preserved under CBF-based safety-critical control. The results provide an energetic interpretation of safety-critical control schemes, and induce novel passive designs which are less conservative than standard methods based on damping injection. The results are specialised to port-Hamiltonian systems and simulations are performed on a cart-pole system.
Paper Structure (12 sections, 5 theorems, 23 equations, 5 figures)

This paper contains 12 sections, 5 theorems, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Passivity and passivity-based control
  4. Control-barrier functions and safety-critical control
  5. Passivity Preserving Safety-Critical Control
  6. EB-PBC for port-Hamiltonian systems with Safety-Filtering
  7. Port-Hamiltonian systems and EB-PBC
  8. Mechanical systems
  9. Simulations
  10. Limiting kinetic energy
  11. Safety-critical kinematic control
  12. Conclusions

Key Result

Theorem 1

Let $h(x)$ be a CBF on $\mathcal{D}$ for (eq:nonlinearaffinesystem). Any locally Lipschitz controller $u(x)=k(x)$ such that $L_{f}h(x)+L_g h(x)k(x) \geq - \alpha(h(x))$ provides forward invariance of the safe set $\mathcal{C}$. Additionally the set $\mathcal{C}$ is asymptotically stable on $\mathcal

Figures (5)

  • Figure 1: The interconnection view of passivity
  • Figure 2: Graphical support to Theorem \ref{['th:main']}.
  • Figure 3: The cart-pole system and the physical representation of its control effects.
  • Figure 4: Safety critical filtering effect on the lossless system of Fig. \ref{['fig:cart']} with $h(q,p)=-K_e(q,p)+\bar{E}$ for different $\bar{E}$.
  • Figure 5: Safety critical filtering effect on the lossless system of Fig. \ref{['fig:cart']} with $h(q,p)=-K_e(q,p)+\alpha_E(\bar{q}_1-q_1)$ for different $\bar{q}$ and $\alpha_E=10$.

Theorems & Definitions (9)

  • Theorem 1: Ames2017ControlSystems
  • Lemma 1: Xu2015RobustnessControlSingletary2021Safety-CriticalSystems
  • Theorem 2
  • proof
  • Theorem 3
  • Corollary 1
  • proof
  • Remark 1: Safety-Critical Kinematic Control
  • Remark 2: Physical Safety