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Energy-Aware Bayesian Control Barrier Functions for Physics-Informed Gaussian Process Dynamics

Chi Ho Leung, Philip E. Paré

TL;DR

This paper addresses safe control for systems with dynamics learned by Gaussian processes, specifically leveraging energy structure in port-Hamiltonian systems. It introduces a Bayesian control barrier function (B-CBF) framework and instantiates it as energy-aware EB-CBFs that construct conservative energy barriers from a Hamiltonian GP posterior and a vector-field GP, then embeds a closed-form drift bound into a CBF-QP safety filter. The approach yields high-probability safety guarantees by operating over posterior credible model sets and provides practical, minimally invasive safety enforcement demonstrated on a mass-spring oscillator. The results highlight how energy-aware barriers capture velocity-dependent safety margins and improve safety calibration under data uncertainty, with potential impact for robotics and physical systems where energy constraints are central.

Abstract

We study safe control for dynamical systems whose continuous-time dynamics are learned with Gaussian processes (GPs), focusing on mechanical and port-Hamiltonian systems where safety is naturally expressed via energy constraints. The availability of a GP Hamiltonian posterior naturally raises the question of how to systematically exploit this structure to design an energy-aware control barrier function with high-probability safety guarantees. We address this problem by developing a Bayesian-CBF framework and instantiating it with energy-aware Bayesian-CBFs (EB-CBFs) that construct conservative energy-based barriers directly from the Hamiltonian and vector-field posteriors, yielding safety filters that minimally modify a nominal controller while providing probabilistic energy safety guarantees. Numerical simulations on a mass-spring system demonstrate that the proposed EB-CBFs achieve high-probability safety under noisy sampled GP-learned dynamics.

Energy-Aware Bayesian Control Barrier Functions for Physics-Informed Gaussian Process Dynamics

TL;DR

This paper addresses safe control for systems with dynamics learned by Gaussian processes, specifically leveraging energy structure in port-Hamiltonian systems. It introduces a Bayesian control barrier function (B-CBF) framework and instantiates it as energy-aware EB-CBFs that construct conservative energy barriers from a Hamiltonian GP posterior and a vector-field GP, then embeds a closed-form drift bound into a CBF-QP safety filter. The approach yields high-probability safety guarantees by operating over posterior credible model sets and provides practical, minimally invasive safety enforcement demonstrated on a mass-spring oscillator. The results highlight how energy-aware barriers capture velocity-dependent safety margins and improve safety calibration under data uncertainty, with potential impact for robotics and physical systems where energy constraints are central.

Abstract

We study safe control for dynamical systems whose continuous-time dynamics are learned with Gaussian processes (GPs), focusing on mechanical and port-Hamiltonian systems where safety is naturally expressed via energy constraints. The availability of a GP Hamiltonian posterior naturally raises the question of how to systematically exploit this structure to design an energy-aware control barrier function with high-probability safety guarantees. We address this problem by developing a Bayesian-CBF framework and instantiating it with energy-aware Bayesian-CBFs (EB-CBFs) that construct conservative energy-based barriers directly from the Hamiltonian and vector-field posteriors, yielding safety filters that minimally modify a nominal controller while providing probabilistic energy safety guarantees. Numerical simulations on a mass-spring system demonstrate that the proposed EB-CBFs achieve high-probability safety under noisy sampled GP-learned dynamics.
Paper Structure (24 sections, 7 theorems, 105 equations, 2 figures)

This paper contains 24 sections, 7 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Port-Hamiltonian Systems
  5. MS-PHS GP Posteriors Vector-field and Hamiltonian
  6. Posterior Vector-Field
  7. Posterior Hamiltonian
  8. Control Barrier Functions
  9. Bayesian-CBFs and Problem Formulation
  10. Bayesian-CBF Theory
  11. Problem Formulation
  12. Energy-Aware Bayesian-CBFs
  13. Posterior Kinetic and Potential Energy
  14. Energy-Aware Admissible Barriers
  15. Energy-Aware Bayesian Barriers from the Hamiltonian Posterior
  16. ...and 9 more sections

Key Result

Theorem 1

Suppose $h\in C^1(\mathcal{D})$ is a CBF and $\nabla h\neq 0$ on $\partial\mathcal{S}$. If $\mathcal{K}_{\mathrm{cbf}}(x)\neq\emptyset$ for all $x\in \mathcal{D}$, then any Lipschitz continuous feedback $u(x)\in\mathcal{K}_{\mathrm{cbf}}(x)$ renders $\mathcal{S}$ forward invariant. Moreover, $\mathc

Figures (2)

  • Figure 1: EB-CBF safety filtering in the Hamiltonian phase plane $(q,p)$ with a single kinematic constraint $q \geq -1$. Gray flowlines show the mass-spring vector field $f(q, p)$. The nominal trajectory (blue, dashed) would enter Unsafe Region 1 (hashed pink), ending at the blue dot. With the CBF filter (orange), the trajectory respects Energy Barrier 1 (solid pink shading) and remains safe, ending at the orange dot.
  • Figure 2: EB-CBF safety filtering in the Hamiltonian phase plane $(q,p)$ with mixed barriers: (i) $q \geq -1$, (ii) $0.15 \leq H(q, p)$, and (iii) $0.75 \geq H(q, p)$. The energy-aware Bayesian barriers (shaded) areas covered the unsafe region (hashed), indicating that energy-aware Bayesian barriers provide a conservative estimates of the actual unsafe regions.

Theorems & Definitions (26)

  • Definition 1: Forward invariance
  • Definition 2: Control barrier functions
  • Theorem 1: Safety via CBFs
  • Remark 3.3.1: Allowable set does not directly admit a CBF
  • Definition 3: Credible model set
  • Definition 4: Bayesian forward invariance
  • Remark 4.1.1: Connection to robust-CBFs
  • Theorem 2: Bayesian safety via B-CBFs
  • proof
  • Lemma 5.1.1: Kinetic energy and relative degree
  • ...and 16 more