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The effect of control barrier functions on energy transfers in controlled physical systems

Federico Califano, Riccardo Zanella, Alessandro Macchelli, Stefano Stramigioli

TL;DR

This work analyzes how safety-critical control implemented via control barrier functions (CBFs) influences energy transfers in port-Hamiltonian (pH) systems. It derives a closed-form expression for the power injected or extracted by the safety controller, enabling energy-aware designs that can enact damping or intentional energy injection to achieve desired closed-loop energetics. The authors propose and validate energy-aware CBF schemes, including damping-injection and energy-injection paradigms, through mechanical simulations (mass-spring and double pendulum) that demonstrate energy bounding, limit cycles, and locomotion-like behavior. The results provide practical tools for shaping energy flow in safety-critical control, with potential impact on collaborative robotics and other energy-sensitive physical systems.

Abstract

Using a port-Hamiltonian formalism, we show the qualitative and quantitative effect of safety-critical control implemented with control barrier functions (CBFs) on the power balance of controlled physical systems. The presented results will provide novel tools to design CBFs inducing desired energetic behaviors of the closed-loop system, including nontrivial damping injection effects and non-passive control actions, effectively injecting energy in the system in a controlled manner. Simulations validate the stated results.

The effect of control barrier functions on energy transfers in controlled physical systems

TL;DR

This work analyzes how safety-critical control implemented via control barrier functions (CBFs) influences energy transfers in port-Hamiltonian (pH) systems. It derives a closed-form expression for the power injected or extracted by the safety controller, enabling energy-aware designs that can enact damping or intentional energy injection to achieve desired closed-loop energetics. The authors propose and validate energy-aware CBF schemes, including damping-injection and energy-injection paradigms, through mechanical simulations (mass-spring and double pendulum) that demonstrate energy bounding, limit cycles, and locomotion-like behavior. The results provide practical tools for shaping energy flow in safety-critical control, with potential impact on collaborative robotics and other energy-sensitive physical systems.

Abstract

Using a port-Hamiltonian formalism, we show the qualitative and quantitative effect of safety-critical control implemented with control barrier functions (CBFs) on the power balance of controlled physical systems. The presented results will provide novel tools to design CBFs inducing desired energetic behaviors of the closed-loop system, including nontrivial damping injection effects and non-passive control actions, effectively injecting energy in the system in a controlled manner. Simulations validate the stated results.
Paper Structure (17 sections, 5 theorems, 22 equations, 3 figures)

This paper contains 17 sections, 5 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Port-Hamiltonian systems
  4. Power-preserving structure
  5. Dissipation structure
  6. Passivity
  7. Control
  8. Control barrier functions
  9. Putting energy back in control through CBFs
  10. CBF-based damping injection
  11. CBF-based energy-aware designs
  12. Mechanical systems and energy-based CBFs
  13. Simulations and Discussion
  14. Bounding total energy from below and above
  15. Bounding kinetic energy
  16. ...and 2 more sections

Key Result

Theorem 1

Let $h(x)$ be a CBF on $\mathcal{D}$ for (eq:pH). Any locally Lipschitz controller $u=k(x)$ such that $\{h,H \}_J -[h,H ]_R + \partial_x ^{\top}h (x)g(x)k(x) \geq -\alpha (h(x))$ provides forward invariance of $\mathcal{C}$. Additionally $\mathcal{C}$ is asymptotically stable on $\mathcal{D}$.

Figures (3)

  • Figure 1: Mass-Spring system. Bounding energy from above (left) and below (right). From above: i) phase space trajectory, ii) CBF, iii) control input, and iv) total energy.
  • Figure 2: Limiting kinetic energy from inside (left) and outside (right) the safe set. From above: i) phase space trajectory, ii) CBF, iii) control input, and iv) total energy.
  • Figure 3: Double pendulum. CBF imposing total energy lower limit is activated after 10 s. From above, left column: i) CBF, ii) safety-critical control input, iii) total energy, and iv) power terms in (\ref{['powerMech']}), detailed on its right. Top right: cartesian snapshots of the system before (gray) and after (black) the CBF activation, and joint trajectory below.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Example 1: Lower/Upper bound on total energy
  • Remark 1
  • Proposition 2
  • proof