Physics-informed Learning for Passivity-based Tracking Control

Leonardo Colombo; Thomas Beckers

REVIEW 4 major objections 5 minor 26 references

Physics-informed Learning for Passivity-based Tracking Control

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that a Gaussian-process model of a partially unknown port-Hamiltonian system, combined with a modified matching equation and a damping condition, renders the tracking-error equilibrium stable with probability at least…

desk verdict A credible but incremental tracking extension of GP-PHS; the stability theorem is conditional on a strong dissipation inequality that the simulation never verifies. read the letter →

arxiv 2505.01569 v1 pith:QQ2R24F4 submitted 2025-05-02 eess.SY cs.SY

Thomas Beckers , Leonardo Colombo This is my paper · ORCID

classification eess.SYcs.SY MSC 93C1093D30

keywords passivity-basedcontrolport-HamiltoniansystemsGaussianprocessregressiontrackingIDA-PBCprobabilisticstabilitydata-drivensemi-passivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to extend passivity-based tracking control to physical systems whose dynamics are known only partially. It models the unknown part of a port-Hamiltonian system with a Gaussian process, forming a GP-PHS model, and then designs a controller from a modified matching equation rather than from the exact model. The central claim is Theorem 1: if the desired interconnections, damping, and Hamiltonian satisfy the dissipation inequality (8), the tracking error equilibrium $\bar{x}=0$ is stable with probability at least $1-p$. Relaxing (8) to hold only outside a ball gives bounded tracking error and semi-passivity. A simulation on an electrostatic microactuator shows the controller keeping the tracking error in a small neighborhood.

What carries the argument

The central object is the Gaussian process port-Hamiltonian system (GP-PHS), a Gaussian process whose kernel is built from the system's interconnection and dissipation matrices so that its sample paths are port-Hamiltonian dynamics with an unknown Hamiltonian. It supplies the mean prediction and per-component variances that define the confidence set for the model error. The argument is carried by the modified matching equation for tracking, which makes the designed control assign the desired port-Hamiltonian error dynamics, and by the dissipation inequality (8), which lets the desired Hamiltonian $H_d$ serve as a Lyapunov function despite the perturbation $\eta$.

What would settle it

For the microactuator simulation, evaluate inequality (8) at finitely many states along the closed-loop trajectory, using the trained GP-PHS posterior variances with $\eta_i=\beta_i\operatorname{var}(\dot{x}_i|x,\mathcal{D})$ and the chosen $R_d$. If any sampled state violates the inequality, the conditions of Theorem 1 are not met at that point; a violation on a set of positive measure would refute the claimed probabilistic stability guarantee for that particular design.

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Extended reading notes

Core claim

The paper's own terms: data-driven IDA-PBC can be taken from set-point regulation to reference tracking while keeping probabilistic stability and passivity guarantees. With a GP-PHS model, the exact matching equation is replaced by $\hat{G}^{\perp}\mu(\dot{x}|x,\mathcal{D})=\hat{G}^{\perp}\left([J_d-R_d]\nabla_{\bar{x}}H_d+\dot{x}_d\right)$, and the control law cancels the learned dynamics and assigns the desired port-Hamiltonian dynamics. The closed-loop error equation becomes $\dot{\bar{x}}=[J_d-R_d]\nabla_{\bar{x}}H_d+\eta$, with the GP model error $\eta$ bounded by $\beta_i\operatorname{var}(\dot{x}_i|x,\mathcal{D})$ with probability $1-p$. If $[\nabla_{\bar{x}}H_d]^\top\eta\le [\nabla_{\bar{x}}H_d]^\top R_d\nabla_{\bar{x}}H_d$, then $H_d$ is a Lyapunov function and the error equilibrium is stable; weaker forms of the same inequality give boundedness and semi-passivity.

Load-bearing premise

The load-bearing premise is condition (8) (and its relaxed form (15)): the injected damping $R_d$ must dominate the worst-case Gaussian-process model error $\eta$ for every state and every $\eta$ in the confidence set defined by $\beta_i\operatorname{var}(\dot{x}_i|x,\mathcal{D})$. This state-dependent inequality is not derived from the GP model or from $H_d$, is acknowledged as hard to satisfy, and is never verified in the simulation.

Editorial extensions

If this is right

If Theorem 1 holds, a tracking controller can be designed without knowing the Hamiltonian, provided training data make the GP error small and the damping satisfies inequality (8).
Stability is probabilistic with an explicit probability $1-p$, so data quantity and quality enter directly into the safety claim.
Under the relaxed condition (15), the tracking error is ultimately bounded in a neighborhood of the desired trajectory, and the neighborhood size can be estimated by level-set methods.
Corollary 2 means the closed-loop system is semi-passive with respect to an external input, allowing interconnection with other passive systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the modified matching equation (7) can be solved for $H_d$ and $R_d$ in general; the simulation chooses $H_d$ by hand as the posterior mean evaluated at $\bar{x}$, so the general solvability of the PDE is untested.
A direct test of the load-bearing condition would be to compute $\beta_i\operatorname{var}(\dot{x}_i|x,\mathcal{D})$ along the simulated trajectory and check (8) pointwise; the paper does not report this check.
The confidence set for $\eta$ is symmetric and componentwise state-independent; if the true model error is biased or heteroscedastic, inequalities (8) and (15) could fail even when the GP marginal variances look small.
The approach suggests a design loop the paper does not pursue: jointly optimize $H_d$ and $R_d$ to satisfy both (7) and (8), which would make the damping condition easier to meet and the guarantees constructive.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real, incremental extension of the authors' earlier GP-PHS work from set-point regulation to trajectory tracking, with a coherent Lyapunov argument that is conditional on a strong dissipation inequality the simulation never verifies. I would send it to review, but the paper needs a corrected uncertainty bound and an honest evaluation before it can be published.

What is actually new: the paper takes the modified IDA-PBC matching equation from Wang and Goldsmith and applies it to a learned GP-PHS model, producing a tracking controller with probabilistic stability, boundedness and semi-passivity statements. The split between Theorem 1 (stable equilibrium under a strict dissipation condition) and Lemma 1/Corollary 2 (bounded error and semi-passivity under a relaxed condition) is a useful and honest way to handle model error. The proof structure is standard and, once you read μ as the posterior mean of the Hamiltonian drift rather than the full derivative, the algebra from (7) to (12) is sound.

The soft spots are real but mostly addressable. The notation around μ is never fixed; equation (10) and the control law make the intended meaning clear, but a reader should not have to guess. More importantly, condition (8) is strong, state-dependent, and there is no design rule for H_d and R_d that guarantees it. The paper itself concedes this. The simulation then never verifies (8) or the relaxed condition (15), and the evaluation text explicitly relies on Lemma 1 rather than Theorem 1. So the numerical example supports boundedness, not the stable-equilibrium claim. That gap should be closed or the claims scaled down. A secondary technical issue: the stated GP confidence bound uses β_i var(˙x_i|x,D) while the cited bound [23] is of the form β_i^{1/2} σ_i(x). As written, the η-set is mis-sized, so condition (8) is checked against the wrong uncertainty model. The simulation also ships no code, data, or GP hyperparameters, which limits reproducibility.

Net: this is a plausible starting point for data-driven passivity-based tracking control, and the relaxed boundedness/semi-passivity portion is worth keeping. With a corrected bound, a check (or design procedure) for (8)/(15), and a reproducible simulation, I would accept it. As it stands, it deserves a serious referee but is not ready as-is.

Referee Report

4 major / 5 minor

Summary. Summary: The paper proposes a data-driven passivity-based tracking controller for port-Hamiltonian systems with partially unknown dynamics. An unknown Hamiltonian is learned with a Gaussian process port-Hamiltonian system (GP-PHS) model, and a modified matching equation extends IDA-PBC to reference tracking. Theorem 1 states that if the desired Hamiltonian and damping satisfy the dissipation domination inequality (8), the tracking error equilibrium of the closed loop is stable with probability at least 1-p. Lemma 1 relaxes (8) to hold only outside an epsilon-ball, yielding probabilistic boundedness, and Corollary 2 gives semi-passivity under an external input. The approach is demonstrated on an electrostatic microactuator.

Significance. Significance: The paper addresses a genuine gap—extending IDA-PBC to tracking under unstructured uncertainty—and the use of GP-PHS is a principled way to carry uncertainty into the control design. The Lyapunov argument is transparent, and the relaxed result in Lemma 1 is a sensible fallback when the strong inequality (8) cannot be certified. The simulation illustrates the closed-loop behavior. However, the central stable-equilibrium claim is only conditional on (8), which is neither constructively designed nor verified; the demonstrated result is boundedness. With a verification procedure and a correction of the confidence-set scaling, the paper would make a solid contribution.

major comments (4)

[Section IV, Theorem 1 and Lemma 1] The stability guarantee in Theorem 1 depends entirely on the dissipation domination condition (8), but the paper provides no design rule for choosing H_d and R_d that makes (8) hold, and Section IV never verifies it. The simulation is justified with Lemma 1, yet the relaxed condition (15) is also not evaluated: the paper does not report the epsilon for which (15) holds on the set X=[-2,2]^3, nor the resulting bound on norm(x-x_d). As a consequence, the numerical example supports boundedness but not the stable-equilibrium claim of Theorem 1. Please add a verification procedure for (8) or, for the relaxed claim, compute the region where (15) holds and the corresponding epsilon.
[Section III-A, Theorem 1 and Lemma 1] The confidence set is stated as |eta_i(x)| <= beta_i var(xdot_i | x,D), but the standard GP error bound cited from [23] is of the form beta_i sigma_i(x), where sigma_i^2 = var(xdot_i | x,D). With the variance in place of the standard deviation, the admissible eta-set is mis-sized and the 'with probability at least 1-p' statement in Theorem 1 and Lemma 1 is not justified by the cited bound. Please replace var(xdot_i | x,D) by the posterior standard deviation, or provide a separate derivation for the variance-scaled bound.
[Section III-B, proof of Theorem 1] In Eq. (10), the model is decomposed as xdot = mu(xdot | x,D) + Ghat(x)u + eta(x), but mu(xdot | x,D) is the posterior mean of the GP-PHS model (6), which already contains the input-dependent term Ghat(x)u. If mu includes Ghat u, the control input is double-counted in (11), and the matching equation (7) must be re-derived. Please define mu precisely (e.g., as the posterior mean of the Hamiltonian drift with Ghat u separated out) and rewrite (7) and (10)-(12) consistently.
[Section II-B, Property 1 and Theorem 1] Property 1 requires only that H_d has its minimum at x=x_d, but the proof of Theorem 1 uses H_d as a Lyapunov function to conclude stability. Lyapunov stability at the equilibrium bar-x=0 also requires H_d to be positive definite with respect to bar-x (at least locally) and, for attractivity, to have an isolated minimum; these conditions are not stated or verified. Corollary 1 adds an isolated minimum only for asymptotic stability. Please state explicit positive-definiteness and regularity assumptions on H_d in Property 1.

minor comments (5)

[Section III-B, Corollary 2] The phrase 'than the conditions in Lemma 1 are satisfied' should read 'then the conditions in Lemma 1 are satisfied.'
[Section IV] The desired trajectory components x_d,2 and x_d,3 are defined only implicitly as solutions of (23); the paper should give the explicit expressions or the numerical solution procedure so that the simulation is reproducible.
[Section IV, Eq. (22)] The relationship between the plant damping b and the desired-dynamics parameter bhat used in (22) should be stated; the text says the only change is 1/r_d, but the matrix also contains bhat.
[Section III-A] The meaning of beta_i is left ambiguous; in [23] the confidence parameter depends on the information gain and the chosen failure probability. Please define beta_i explicitly and state how it is computed for the PHS kernel.
[Figure 3 caption] The caption 'Top 1 and 2: Closed-loop system with the proposed tracking control law' is grammatically unclear; consider separating the two panels or rewording the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the control-law derivation is conditional on explicitly stated matching and dissipation inequalities, and the main guarantees are not obtained by fitting a parameter to the claimed output.

full rationale

The paper's central claims are conditional theorems, not the result of fitting a parameter and then renaming the fit a prediction. In Theorem 1, the control law (9) is constructed from the GP-PHS mean model and the desired dynamics (Jd, Rd, Hd) that satisfy the modified matching equation (7). The stability argument uses Hd as a Lyapunov-like function; the derivative in (13) is non-positive precisely when the stated dissipation inequality (8) holds. That inequality is an explicit sufficient condition on the design variables (Hd, Rd) and the uncertainty set, not an artifact of the learning procedure. The relaxed Lemma 1 and Corollary 2 similarly rest on the explicit condition (15). The simulation section does not verify (8) or (15), and the paper itself acknowledges that (8) 'might be challenging to satisfy'; this is an evidentiary or rigor gap in the numerical demonstration, not circular reasoning. The paper adopts GP-PHS from the authors' prior work [19], [21], and cites [23] for Gaussian-process confidence bounds, but these citations are not used to define the target result or to forbid alternative designs; they supply a modeling framework and an uncertainty bound, which are independent inputs. The chosen desired Hamiltonian is a design choice, and no fitted constant is silently converted into the claimed stability conclusion. Therefore no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central theorem depends on conditions (7) and (8) that are assumed rather than constructed; the GP hyperparameters and design parameters are fitted or chosen by hand, and the RKHS norm assumption is inherited from the GP framework.

free parameters (3)

GP hyperparameters = not reported
Lengthscales Λ, signal variance σ_f, noise variances and structural parameters φ_J, φ_R, φ_G are fitted to the training data via negative log marginal likelihood; their values determine the confidence intervals used in (8).
Desired damping r_d = 10 (1/r_d = 10)
Hand-chosen design parameter in the target PHS (22); it sets the dissipation that must dominate the model error in (8).
GP bound constants β_i = not reported
The high-probability error bounds depend on β_i from [23]; values are needed to check (8) but are not given.

assumptions (4)

domain assumption The physical system is a port-Hamiltonian system (1) with known structural matrices J, R, G up to unknown parameters.
Section II-B assumes the PHS structure and known parametric form of J, R, G; the GP-PHS model uses this structure as a prior.
domain assumption The unknown Hamiltonian dynamics have bounded RKHS norm with respect to the PHS kernel, ||(J-R)∇H||_{k_phs} < ∞.
Invoked in Section III-A to apply the GP error bound from [23]; limits the class of learnable functions to the RKHS of the chosen kernel.
ad hoc to paper The matching equation (7) has a solution Hd, Jd, Rd satisfying Property 1.
Theorem 1 requires (7); in the simulation it is solved numerically but no general constructive procedure is given.
ad hoc to paper The dissipation domination inequality (8) (or its relaxed version (15)) holds for the designed Hd and Rd and the GP confidence set.
This is the key condition for stability/boundedness; the paper acknowledges it may be hard to satisfy and does not verify it in the example.

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Cite this review

Pith. "Pith review of Physics-informed Learning for Passivity-based Tracking Control." pith.science (2026). https://pith.science/paper/QQ2R24F4

@misc{pith2026250501569,
  author       = {Pith},
  title        = {Pith review of: Physics-informed Learning for Passivity-based Tracking Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQ2R24F4}},
  note         = {Machine review of arXiv:2505.01569}
}

read the original abstract

Passivity-based control ensures system stability by leveraging dissipative properties and is widely applied in electrical and mechanical systems. Port-Hamiltonian systems (PHS), in particular, are well-suited for interconnection and damping assignment passivity-based control (IDA-PBC) due to their structured, energy-centric modeling approach. However, current IDA-PBC faces two key challenges: (i) it requires precise system knowledge, which is often unavailable due to model uncertainties, and (ii) it is typically limited to set-point control. To address these limitations, we propose a data-driven tracking control approach based on a physics-informed model, namely Gaussian process Port-Hamiltonian systems, along with the modified matching equation. By leveraging the Bayesian nature of the model, we establish probabilistic stability and passivity guarantees. A simulation demonstrates the effectiveness of our approach.

Figures

Figures reproduced from arXiv: 2505.01569 by the authors.

**Figure 2.** Electrostatic microactuator with unknown capacity [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Top 1 and 2: Closed-loop system with the proposed tracking control [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The Hamiltonian function of the tracking error dynamics is [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [23]

Information-theoretic regret bounds for Gaussian process optimiza- tion in the bandit setting,

N. Srinivas, A. Krause, S. M. Kakade, and M. W. Seeger, “Information-theoretic regret bounds for Gaussian process optimiza- tion in the bandit setting,” IEEE Transactions on Information Theory , vol. 58, no. 5, pp. 3250–3265, 2012

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R. Ortega, Z. P. Jiang, and D. J. Hill, “Passivity-based control of nonlinear systems: A tutorial,” in Proceedings of the American Control Conference, vol. 5. IEEE, 1997, pp. 2633–2637

work page 1997

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T. Hatanaka, N. Chopra, M. Fujita, and M. W. Spong, Passivity-based control and estimation in networked robotics . Springer, 2015

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[4] [3]

Passivity-based controllers for the stabilization of dc-to-dc power converters,

H. Sira-Ramirez, R. A. Perez-Moreno, R. Ortega, and M. Garcia- Esteban, “Passivity-based controllers for the stabilization of dc-to-dc power converters,” automatica, vol. 33, no. 4, pp. 499–513, 1997

work page 1997

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J. A. Acosta, R. Ortega, A. Astolfi, and A. D. Mahindrakar, “Intercon- nection and damping assignment passivity-based control of mechan- ical systems with underactuation degree one,” IEEE Transactions on Automatic Control, vol. 50, no. 12, pp. 1936–1955, 2005

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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

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F. G ´omez-Estern and A. J. Van der Schaft, “Physical damping in ida- pbc controlled underactuated mechanical systems,” European Journal of control, vol. 10, no. 5, pp. 451–468, 2004

work page 2004