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arxiv: 2512.24493 · v2 · submitted 2025-12-30 · 📡 eess.SY · cs.RO· cs.SY

Bayesian Safety Guarantees for Port-Hamiltonian Systems with Learned Energy Functions

Chi Ho Leung , Philip E. Par\'e This is my paper

Pith reviewed 2026-05-16 18:33 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords Bayesian safetyport-Hamiltonian systemscontrol barrier functionslearned Hamiltonianscredible setsGaussian processessafety filtersmodel uncertainty
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The pith

Uncertainty in a learned Hamiltonian can be split into separate credible sets for energy and drift, yielding a joint safety guarantee of at least 1 minus the sum of the two failure budgets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to keep control barrier functions safe for port-Hamiltonian systems when the Hamiltonian itself is learned from data. It uses a two-stage Bayesian procedure: posterior prediction on the energy function produces credible bands that define high-probability inner approximations to the allowable set, while an independent drift ellipsoid bounds the vector-field uncertainty in the barrier condition. Because the two credible sets are disjoint, their failure probabilities add directly, so the end-to-end guarantee is at least 1 minus the sum of the two chosen budgets. This matters for robots and physical systems that must operate safely with limited or noisy observations of the energy storage. Experiments on a mass-spring oscillator and a planar manipulator confirm that the resulting filter preserves safety and permits larger safe sets than a direct unstructured Gaussian-process barrier.

Core claim

Posterior prediction over the Hamiltonian yields credible bands for energy storage that define Bayesian barriers whose safe sets are high-probability inner approximations with credibility 1 minus eta_ptB; an independent drift credible ellipsoid accounts for vector-field uncertainty with credibility 1 minus eta_dr; since the sets are disjoint the joint safety guarantee is at least 1 minus (eta_dr plus eta_ptB).

What carries the argument

Two-stage Bayesian credible sets, one on the learned Hamiltonian for energy storage and one on the drift vector field, whose failure probabilities add directly because the sets are disjoint.

If this is right

  • The safety filter remains valid with limited noisy observations of the Hamiltonian.
  • Credibility budgets for energy and drift can be tuned independently.
  • The method produces larger safe operating regions than an unstructured GP-CBF on the same system.
  • Safety is preserved on both a mass-spring oscillator and a planar manipulator despite data limitations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The disjoint-credible-set construction could apply to other structured mechanical systems that admit port-Hamiltonian or energy-based models.
  • If the energy and drift uncertainties were correlated, the joint bound would require a more conservative estimate than simple addition.
  • The budgets could be adapted online as new data arrives without recomputing the entire filter.

Load-bearing premise

Energy-storage uncertainty and drift uncertainty can be represented by disjoint credible sets whose failure probabilities add without overlap.

What would settle it

An experiment in which the observed rate of safety violations exceeds the sum of the two budgeted failure probabilities under the same data and model conditions.

Figures

Figures reproduced from arXiv: 2512.24493 by Chi Ho Leung, Philip E. Par\'e.

Figure 1
Figure 1. Figure 1: EB-CBF safety filtering in the Hamiltonian phase [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: EB-CBF safety filtering in the Hamiltonian phase plane [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

Control barrier functions for port-Hamiltonian systems inherit model uncertainty when the Hamiltonian is learned from data. We show how to propagate this uncertainty into a safety filter with independently tunable credibility budgets. To propagate this uncertainty, we employ a two-stage Bayesian approach. First, posterior prediction over the Hamiltonian yields credible bands for the energy storage, producing Bayesian barriers whose safe sets are high-probability inner approximations of the true allowable set with credibility $1 - (\eta_{\mathrm{ptB}})$. Independently, a drift credible ellipsoid accounts for vector field uncertainty in the CBF inequality with credibility $1 - (\eta_{\rm dr})$. Since energy and drift uncertainties enter through disjoint credible sets, the end-to-end safety guarantee is at least $1 - (\eta_{\rm dr} + \eta_{\mathrm{ptB}})$. Experiments on a mass-spring oscillator with a GP-learned Hamiltonian show that the proposed filter preserves safety despite limited and noisy observations. Moreover, we show that the proposed framework yields a larger safe set than an unstructured GP-CBF alternative on a planar manipulator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proposes a two-stage Bayesian method to propagate model uncertainty from a learned Hamiltonian into control barrier functions for port-Hamiltonian systems. Posterior credible bands on the energy yield Bayesian barriers with credibility 1-η_ptB; an independent drift credible ellipsoid accounts for vector-field uncertainty with credibility 1-η_dr. The authors claim that because the two uncertainties enter through disjoint credible sets, the end-to-end safety guarantee is at least 1-(η_dr + η_ptB). Experiments on a mass-spring oscillator with GP-learned Hamiltonian and on a planar manipulator are reported to show safety preservation and larger safe sets than an unstructured GP-CBF baseline.

Significance. If the claimed additive bound holds under the stated disjointness, the framework supplies a structured, tunable-probability safety filter for data-driven port-Hamiltonian models that exploits system structure rather than treating the dynamics as a black-box GP. This could be useful for robotic and physical systems where Hamiltonians are identified from limited noisy data, offering a concrete probabilistic alternative to deterministic robust CBFs.

major comments (1)
  1. Abstract (end-to-end guarantee paragraph): the claim that energy and drift uncertainties enter through disjoint credible sets whose failure probabilities add directly to yield 1-(η_dr + η_ptB) is load-bearing for the central contribution, yet the abstract provides no derivation, no explicit statement of posterior independence, and no discussion of possible dependence induced by the shared data set. Without the full proof, it is impossible to verify whether the union bound is valid or whether additional assumptions (e.g., independent priors or separate data splits) are required.
minor comments (2)
  1. Abstract: the acronyms η_ptB and η_dr are introduced without an explicit sentence defining what each budget controls; a single clarifying clause would improve readability.
  2. Abstract: the experimental claims (safety preservation on the oscillator, larger safe set on the manipulator) are stated without quantitative metrics or reference to any table/figure; the reader cannot assess effect size from the abstract alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to clarify the probabilistic foundation of the end-to-end safety guarantee. We address the single major comment below and indicate the corresponding revision.

read point-by-point responses

  1. Referee: Abstract (end-to-end guarantee paragraph): the claim that energy and drift uncertainties enter through disjoint credible sets whose failure probabilities add directly to yield 1-(η_dr + η_ptB) is load-bearing for the central contribution, yet the abstract provides no derivation, no explicit statement of posterior independence, and no discussion of possible dependence induced by the shared data set. Without the full proof, it is impossible to verify whether the union bound is valid or whether additional assumptions (e.g., independent priors or separate data splits) are required.

    Authors: We agree that the abstract is concise and does not contain a derivation. The end-to-end guarantee is obtained by applying the union bound to two events: (i) the event that the Bayesian barrier fails to contain the true safe set (probability ≤ η_ptB) and (ii) the event that the drift credible ellipsoid fails to contain the true vector field (probability ≤ η_dr). The union bound P(A ∪ B) ≤ P(A) + P(B) holds for any events A and B, irrespective of statistical dependence. Consequently, no assumption of posterior independence, independent priors, or data splitting is required; any dependence induced by the shared data set only makes the bound more conservative. The full construction of the two credible sets and the explicit invocation of the union bound appear in Section 3 of the manuscript. We will revise the abstract to include the phrase “via the union bound applied to the two credible sets” so that the source of the additive bound is stated explicitly. revision: partial

Circularity Check

0 steps flagged

No circularity: standard union bound on independent credible sets

full rationale

The abstract presents a two-stage Bayesian procedure that first obtains credible bands on the learned Hamiltonian (yielding Bayesian barriers with credibility 1-η_ptB) and separately obtains a drift credible ellipsoid (credibility 1-η_dr). The end-to-end guarantee is then stated as at least 1-(η_dr + η_ptB) because the two uncertainty sources enter through disjoint credible sets. This is exactly the standard union bound P(A∪B)≤P(A)+P(B) applied to two independent events; no equation reduces to a fitted quantity by construction, no self-citation is invoked as a load-bearing premise, and no ansatz or renaming is smuggled in. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests on the validity of Bayesian credible sets for the learned Hamiltonian and the assumption that energy and drift uncertainties can be bounded separately.

free parameters (1)
  • credibility budgets η_ptB and η_dr
    Tunable failure probabilities chosen by the designer; they directly set the size of the safe set.
axioms (2)
  • domain assumption Posterior credible bands around the Hamiltonian yield high-probability inner approximations of the true safe set
    Invoked to turn GP posterior into Bayesian barriers.
  • domain assumption Drift uncertainty can be captured by an independent credible ellipsoid
    Allows the additive bound on total failure probability.

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Control barrier functions: Theory and applications,

    A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” inProc. 18th European Control Conference (ECC). IEEE, 2019, pp. 3420–3431

    work page 2019

  2. [2]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,”IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2017

    work page 2017

  3. [3]

    Safety-critical kinematic control of robotic systems,

    A. Singletary, S. Kolathaya, and A. D. Ames, “Safety-critical kinematic control of robotic systems,”IEEE Control Systems Letters, vol. 6, pp. 139–144, 2021

    work page 2021

  4. [4]

    Control barrier functions for unknown nonlinear systems using Gaussian processes,

    P. Jagtap, G. J. Pappas, and M. Zamani, “Control barrier functions for unknown nonlinear systems using Gaussian processes,” inProc. 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 3699–3704

    work page 2020

  5. [5]

    Pointwise feasibility of Gaussian process-based safety-critical control under model uncertainty,

    F. Castaneda, J. J. Choi, B. Zhang, C. J. Tomlin, and K. Sreenath, “Pointwise feasibility of Gaussian process-based safety-critical control under model uncertainty,” inProc. 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021, pp. 6762–6769

    work page 2021

  6. [6]

    Control barrier functions with real-time Gaussian process modeling,

    R. Gutierrez and J. B. Hoagg, “Control barrier functions with real-time Gaussian process modeling,”arXiv preprint arXiv:2505.06765, 2025

    work page arXiv 2025

  7. [7]

    Robustness of control barrier functions for safety critical control,

    X. Xu, P. Tabuada, J. W. Grizzle, and A. D. Ames, “Robustness of control barrier functions for safety critical control,”IFAC-PapersOnLine, vol. 48, no. 27, pp. 54–61, 2015

    work page 2015

  8. [8]

    Robust control barrier functions for constrained stabiliza- tion of nonlinear systems,

    M. Jankovic, “Robust control barrier functions for constrained stabiliza- tion of nonlinear systems,”Automatica, vol. 96, pp. 359–367, 2018

    work page 2018

  9. [9]

    Robust control barrier–value functions for safety-critical control,

    J. J. Choi, D. Lee, K. Sreenath, C. J. Tomlin, and S. L. Herbert, “Robust control barrier–value functions for safety-critical control,” inProc. 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021, pp. 6814–6821

    work page 2021

  10. [10]

    Port-Hamiltonian systems theory: An introductory overview,

    A. Van Der Schaft, D. Jeltsemaet al., “Port-Hamiltonian systems theory: An introductory overview,”Foundations and Trends® in Systems and Control, vol. 1, no. 2-3, pp. 173–378, 2014

    work page 2014

  11. [11]

    A structure-preserving kernel method for learning Hamiltonian systems,

    J. Hu, J.-P. Ortega, and D. Yin, “A structure-preserving kernel method for learning Hamiltonian systems,”Mathematics of Computation, 2025

    work page 2025

  12. [12]

    Gaussian pro- cess port-Hamiltonian systems: Bayesian learning with physics prior,

    T. Beckers, J. Seidman, P. Perdikaris, and G. J. Pappas, “Gaussian pro- cess port-Hamiltonian systems: Bayesian learning with physics prior,” inProc. 61st IEEE Conference on Decision and Control (CDC). IEEE, 2022, pp. 1447–1453

    work page 2022

  13. [13]

    Data-driven bayesian control of port-Hamiltonian systems,

    T. Beckers, “Data-driven bayesian control of port-Hamiltonian systems,” inProc. 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 8708–8713

    work page 2023

  14. [14]

    Exact infer- ence for continuous-time Gaussian process dynamics,

    K. Ensinger, N. Tagliapietra, S. Ziesche, and S. Trimpe, “Exact infer- ence for continuous-time Gaussian process dynamics,” inProc. AAAI Conference on Artificial Intelligence, vol. 38, 2024, pp. 11 883–11 891

    work page 2024

  15. [15]

    Learning passive continuous-time dynamics with multistep port-Hamiltonian Gaussian processes,

    C. H. Leung and P. E. Par ´e, “Learning passive continuous-time dynamics with multistep port-Hamiltonian Gaussian processes,”arXiv preprint arXiv:2510.00384, 2025

    work page arXiv 2025

  16. [16]

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

    F. Califano, R. Zanella, A. Macchelli, and S. Stramigioli, “The effect of control barrier functions on energy transfers in controlled physical systems,”arXiv preprint arXiv:2406.13420, 2024

    work page arXiv 2024

  17. [17]

    Safe control synthesis with uncertain dynamics and constraints,

    K. Long, V . Dhiman, M. Leok, J. Cort ´es, and N. Atanasov, “Safe control synthesis with uncertain dynamics and constraints,”IEEE Robotics and Automation Letters, vol. 7, no. 3, pp. 7295–7302, 2022

    work page 2022

  18. [18]

    Control barriers in bayesian learning of system dynamics,

    V . Dhiman, M. J. Khojasteh, M. Franceschetti, and N. Atanasov, “Control barriers in bayesian learning of system dynamics,”IEEE Transactions on Automatic Control, vol. 68, no. 1, pp. 214–229, 2021

    work page 2021

  19. [19]

    ProBF: Learning probabilistic safety certificates with barrier functions,

    A. R. Kumar, S. Liu, J. F. Fisac, R. P. Adams, and P. J. Ramadge, “ProBF: Learning probabilistic safety certificates with barrier functions,” arXiv preprint arXiv:2112.12210, 2021

    work page arXiv 2021

  20. [20]

    Inter- connection and damping assignment passivity-based control of port- controlled Hamiltonian systems,

    R. Ortega, A. Van Der Schaft, B. Maschke, and G. Escobar, “Inter- connection and damping assignment passivity-based control of port- controlled Hamiltonian systems,”Automatica, vol. 38, no. 4, pp. 585– 596, 2002

    work page 2002

  21. [21]

    Gaussian processes in machine learning,

    C. E. Rasmussen, “Gaussian processes in machine learning,” inSummer School on Machine Learning. Springer, 2003, pp. 63–71

    work page 2003

  22. [22]

    Gaussian process training with input noise,

    A. McHutchon and C. Rasmussen, “Gaussian process training with input noise,”Advances in Neural Information Processing Systems, vol. 24, 2011

    work page 2011

  23. [23]

    A new kernel-based approach for linear system identification,

    G. Pillonetto and G. De Nicolao, “A new kernel-based approach for linear system identification,”Automatica, vol. 46, no. 1, pp. 81–93, 2010

    work page 2010

  24. [24]

    Hairer, G

    E. Hairer, G. Wanner, and S. P. Nørsett,Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993

    work page 1993

  25. [25]

    ¨Uber die lage der integralkurven gew ¨ohnlicher differ- entialgleichungen,

    M. Nagumo, “ ¨Uber die lage der integralkurven gew ¨ohnlicher differ- entialgleichungen,”Proc. Physico-Mathematical Society of Japan. 3rd Series, vol. 24, pp. 551–559, 1942

    work page 1942

  26. [26]

    Set invariance in control,

    F. Blanchini, “Set invariance in control,”Automatica, vol. 35, no. 11, pp. 1747–1767, 1999

    work page 1999

  27. [27]

    V . I. Arnol’d,Mathematical Methods of Classical Mechanics. Springer Science & Business Media, 2013, vol. 60

    work page 2013

  28. [28]

    Classical mechanics,

    H. Goldstein, C. Poole, J. Safkoet al., “Classical mechanics,” 1980

    work page 1980

  29. [29]

    Lanczos,The Variational Principles of Mechanics

    C. Lanczos,The Variational Principles of Mechanics. Courier Corpo- ration, 2012

    work page 2012

  30. [30]

    R. M. Neal,Bayesian Learning for Neural Networks. Springer Science & Business Media, 2012, vol. 118

    work page 2012

  31. [31]

    Adam: A Method for Stochastic Optimization

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    Gpytorch: Blackbox matrix-matrix Gaussian process inference with gpu acceleration,

    J. Gardner, G. Pleiss, K. Q. Weinberger, D. Bindel, and A. G. Wilson, “Gpytorch: Blackbox matrix-matrix Gaussian process inference with gpu acceleration,”Advances in Neural Information Processing Systems, vol. 31, 2018

    work page 2018

  33. [33]

    OptNet: Differentiable optimization as a layer in neural networks,

    B. Amos and J. Z. Kolter, “OptNet: Differentiable optimization as a layer in neural networks,” inProc. International Conference on Machine Learning. PMLR, 2017, pp. 136–145

    work page 2017

  34. [34]

    Pytorch: An imperative style, high-performance deep learning library,

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antigaet al., “Pytorch: An imperative style, high-performance deep learning library,”Advances in Neural Information Processing Systems, vol. 32, 2019

    work page 2019