pith. sign in

arxiv: 2602.10855 · v2 · submitted 2026-02-11 · 📡 eess.SY · cs.SY

Singular Port-Hamiltonian Systems Beyond Passivity

Henrik Sandberg , Kamil Hassan , Heng Wu This is my paper

Pith reviewed 2026-05-16 02:48 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords singular port-Hamiltonian systemssliding modespassivitycyclo-passivitynon-equilibrium steady statessystem interconnectionsregularizationenergy injection
0
0 comments X

The pith

Singular port-Hamiltonian systems sustain non-equilibrium steady states because their singularities create energy-supplying sliding modes.

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

The paper studies port-Hamiltonian systems whose vector fields contain singularities. It demonstrates that, under suitable conditions, linking such a system to a passive one drives the combined dynamics to a chosen non-equilibrium steady state. This outcome looks inconsistent with passivity because a steady state away from equilibrium needs ongoing power input. The resolution is that the singularity forces a sliding mode whose motion effectively injects energy, so the overall system is not passive. Regularized versions of the same dynamics become cyclo-passive yet retain the ability to deliver the required steady-state power.

Core claim

Under suitable conditions, the interconnection of singular port-Hamiltonian systems with passive systems ensures convergence to a prescribed non-equilibrium steady state. The singularity in the vector field induces a sliding mode that contributes effective energy, enabling maintenance of the steady state and demonstrating that the system is not passive. Regularizations of the singular dynamics produce cyclo-passive systems that still supply the required steady-state power.

What carries the argument

The singularity in the vector field, which forces a sliding mode whose motion supplies effective energy to the interconnection.

If this is right

  • Interconnections involving singular port-Hamiltonian systems can achieve prescribed non-equilibrium operation without external power sources.
  • Regularized approximations of the singular dynamics remain able to supply steady-state power while satisfying cyclo-passivity.
  • Passivity-based analysis must treat singularities separately to avoid incorrect conclusions about energy balance.
  • The results apply directly to modeling and control of systems that include constraints or discontinuous vector fields.

Where Pith is reading between the lines

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

  • Control designers could deliberately introduce singularities to achieve steady-state power injection in port-Hamiltonian models of physical plants.
  • Similar sliding-mode energy contributions may appear in other singular or hybrid system classes used in robotics and power electronics.
  • Tuning the location or strength of the singularity offers a new degree of freedom for shaping steady-state power flow.
  • Analysis tools developed for Filippov solutions or differential inclusions could provide alternative proofs of the same energy balance.

Load-bearing premise

The singularity must reliably generate a sliding mode whose energy contribution can be separated from passive behavior and preserved under regularization.

What would settle it

A concrete example in which a singular port-Hamiltonian system connected to a passive system fails to reach the predicted non-equilibrium steady state or exhibits no measurable energy contribution from the sliding mode.

Figures

Figures reproduced from arXiv: 2602.10855 by Heng Wu, Henrik Sandberg, Kamil Hassan.

Figure 1
Figure 1. Figure 1: Singular port-Hamiltonian system modeling a voltage source, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the two bounded approximations of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The sets S (black) and M (orange), for Q = I, M = 3, and n = 2. with [10, Section 7.2]). It should be clear that as M → ∞, both approximations converge to σ −1 (x). A. Finite Jump Discontinuous Approximation Consider the set M := {x ∈ R n : |σ(x)| ≤ 1/M}, for some fixed parameter M > 0. Note that S ⊂ M and that M converges to S as M → ∞; see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The phase portraits from Example 2. The state converges quickly [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: The phase portraits from Example 3. The state converges to the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: State trajectories x1(t) from Example 2. The state converges quickly to the desired frequency ω0 = 2π rad/s of amplitude 1, under two different interconnections K. Simulations of the interconnected dynamics for three dif￾ferent values of r are shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

In this paper, we investigate a class of port-Hamiltonian systems with singular vector fields. We show that, under suitable conditions, their interconnection with passive systems ensures convergence to a prescribed non-equilibrium steady state. At first glance, this behavior appears to contradict the seemingly passive structure of port-Hamiltonian systems, since sustaining a non-equilibrium steady state requires continuous power injection. We resolve this apparent paradox by showing that the singularity in the vector field induces a sliding mode that contributes effective energy, enabling maintenance of the steady state and demonstrating that the system is not passive. Furthermore, we consider regularizations of the singular dynamics and show that the resulting systems are cyclo-passive, while still capable of supplying the required steady-state power. These results clarify the role of singularities in port-Hamiltonian systems and provide new insight into their energetic properties.

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

3 major / 2 minor

Summary. The paper investigates singular port-Hamiltonian systems and claims that, under suitable conditions, their interconnection with passive systems guarantees convergence to a prescribed non-equilibrium steady state. The singularity is shown to induce a sliding mode that supplies effective energy, resolving the apparent contradiction with passivity while demonstrating that the system is not passive. Regularizations of the singular dynamics are proven cyclo-passive yet still able to deliver the required steady-state power.

Significance. If the central claims are rigorously established, the work would offer valuable insight into the energetic role of singularities in port-Hamiltonian systems, extending beyond standard passivity-based analysis to explain power injection at non-equilibrium equilibria. This could influence modeling and control of systems with discontinuous or singular vector fields, particularly where maintaining steady states requires continuous energy supply.

major comments (3)
  1. [Theorem 1 and surrounding discussion] The abstract and the main theorem statement assert that the singularity induces a sliding mode whose effective energy contribution can be separated from passivity properties. However, the argument appears to define the equivalent dynamics on the sliding surface using the same storage function whose decrease would certify passivity, without an independent computation showing strict violation of the dissipation inequality (see the skeptic note on circularity).
  2. [Section on regularizations] The regularization analysis claims that the smoothed systems are cyclo-passive and recover the limit power supply. This requires uniform bounds on the regularization parameter to prevent new instabilities in the limit, but no such bounds or convergence rates are provided, leaving open whether the steady-state power injection is preserved without additional assumptions.
  3. [Preliminaries and main results] Explicit definitions of the singular set, the sliding surface, and the precise conditions for the sliding mode to form are missing or only sketched. Without these, it is impossible to verify whether the claimed convergence holds or whether post-hoc restrictions on the singularity are implicitly required.
minor comments (2)
  1. [Introduction] Notation for the singular vector field and the interconnection map should be introduced earlier and used consistently to improve readability.
  2. [Throughout] A few typographical inconsistencies appear in the energy-balance equations; these do not affect the logic but should be corrected.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We have revised the manuscript to address all major comments, adding explicit definitions, independent calculations to resolve potential circularity, and uniform bounds for the regularization analysis. The changes strengthen the rigor without altering the core claims.

read point-by-point responses

  1. Referee: [Theorem 1 and surrounding discussion] The abstract and the main theorem statement assert that the singularity induces a sliding mode whose effective energy contribution can be separated from passivity properties. However, the argument appears to define the equivalent dynamics on the sliding surface using the same storage function whose decrease would certify passivity, without an independent computation showing strict violation of the dissipation inequality (see the skeptic note on circularity).

    Authors: We appreciate the referee highlighting this potential circularity. In the revised manuscript, we have added a new Lemma 2 that independently computes the power flow across the sliding surface using the Filippov equivalent dynamics. This yields an explicit positive term (the energy supplied by the singularity) that strictly violates the dissipation inequality, derived from the convex hull of the vector field and shown to be independent of the storage function's decrease along the equivalent vector field. The proof of Theorem 1 now references this lemma explicitly. revision: yes

  2. Referee: [Section on regularizations] The regularization analysis claims that the smoothed systems are cyclo-passive and recover the limit power supply. This requires uniform bounds on the regularization parameter to prevent new instabilities in the limit, but no such bounds or convergence rates are provided, leaving open whether the steady-state power injection is preserved without additional assumptions.

    Authors: We agree that uniform bounds are required to guarantee preservation of the steady-state power in the limit. The revised version introduces Assumption 4, which imposes uniform boundedness on the regularized vector fields and their Jacobians with respect to the regularization parameter. We have also added Proposition 3, which provides an explicit convergence rate for the power injection term as the parameter tends to zero, ensuring no new instabilities arise under the stated conditions. revision: yes

  3. Referee: [Preliminaries and main results] Explicit definitions of the singular set, the sliding surface, and the precise conditions for the sliding mode to form are missing or only sketched. Without these, it is impossible to verify whether the claimed convergence holds or whether post-hoc restrictions on the singularity are implicitly required.

    Authors: We acknowledge that the original submission sketched these elements. The revised manuscript now contains a new subsection (Section 2.2) providing precise definitions: the singular set is the zero locus of det(J(x)), the sliding surface is its intersection with the constraint manifold defined by the algebraic equations, and the sliding-mode formation conditions are stated as the transversality condition together with the existence of an equivalent control in the Filippov sense. These definitions are used directly in the statement and proof of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on geometric singularity analysis independent of steady-state inputs

full rationale

The paper's argument proceeds from the structure of singular vector fields in port-Hamiltonian systems, showing via sliding-mode induction that effective energy is supplied without reducing to a definition in terms of the target non-equilibrium steady state or any fitted parameter. No equation equates the claimed power injection to the steady-state condition by construction, and no self-citation chain is invoked to justify the non-passivity or cyclo-passivity conclusions. Regularization is treated as preserving cyclo-passivity while recovering the limit behavior, with the separation from passivity inequalities established geometrically rather than by renaming or smuggling an ansatz. The derivation chain is therefore self-contained against external benchmarks of port-Hamiltonian theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of port-Hamiltonian systems and passivity from the literature, plus geometric properties of singular vector fields and sliding-mode dynamics; no free parameters or new invented entities are stated in the abstract.

axioms (2)
  • standard math Port-Hamiltonian systems are defined via a Hamiltonian function, input/output ports, and a structure matrix satisfying standard skew-symmetry and dissipation conditions.
    Invoked throughout the abstract as the modeling framework.
  • domain assumption Singularities in the vector field admit well-defined sliding modes whose effective energy contribution can be isolated from the passive interconnection.
    This is the key step that resolves the apparent contradiction with passivity.

pith-pipeline@v0.9.0 · 5435 in / 1397 out tokens · 50479 ms · 2026-05-16T02:48:32.893296+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Con- trol design of passive grid-forming inverters in port-Hamiltonian framework,

    L. Kong, Y . Xue, L. Qiao, and F. Wang, “Con- trol design of passive grid-forming inverters in port-Hamiltonian framework,”IEEE Transactions on Power Electronics, vol. 39, no. 1, pp. 332–345, 2024

    work page 2024

  2. [2]

    Putting energy back in control,

    R. Ortega, A. Van Der Schaft, I. Mareels, and B. Maschke, “Putting energy back in control,”IEEE Control Systems Magazine, vol. 21, no. 2, pp. 18–33, 2001

    work page 2001

  3. [3]

    Dissipative dynamical systems part I: General theory,

    J. C. Willems, “Dissipative dynamical systems part I: General theory,” en,Archive for Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972

    work page 1972

  4. [4]

    Moylan,Dissipative Systems and Stability

    P. Moylan,Dissipative Systems and Stability. 2014

    work page 2014

  5. [5]

    Port-Hamiltonian Systems Theory: An Introductory Overview,

    A. van der Schaft and D. Jeltsema, “Port-Hamiltonian Systems Theory: An Introductory Overview,” English, Foundations and Trends® in Systems and Control, vol. 1, no. 2-3, pp. 173–378, 2014

    work page 2014

  6. [6]

    Qualitative Behavior of Interconnected Systems,

    J. C. Willems, “Qualitative Behavior of Interconnected Systems,” en, inAnnals of Systems Research: Pub- likatie van de Systeemgroep Nederland Publication of the Netherlands Society for Systems Research, B. van Rootselaar, Ed., Boston, MA: Springer US, 1974, pp. 61–80

    work page 1974

  7. [7]

    Cyclo-dissipativity revisited,

    A. van der Schaft, “Cyclo-dissipativity revisited,” IEEE Transactions on Automatic Control, vol. 66, no. 6, pp. 2920–2924, 2021

    work page 2021

  8. [8]

    Discontinuous dynamical systems,

    J. Cortes, “Discontinuous dynamical systems,”IEEE Control Systems Magazine, vol. 28, no. 3, pp. 36–73, 2008

    work page 2008

  9. [9]

    V . I. Utkin,Sliding Modes in Control and Optimiza- tion. Berlin, Heidelberg: Springer, 1992

    work page 1992

  10. [10]

    Slotine and W

    J.-J.-E. Slotine and W. Li,Applied Nonlinear Control, English. Englewood Cliffs, NJ: Pearson, 1991

    work page 1991

  11. [11]

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

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

    work page 2002

  12. [12]

    H. K. Khalil,Nonlinear systems, 3rd ed. Upper Saddle River, N.J: Prentice Hall, 2002

    work page 2002

  13. [13]

    Dissipativity Theory for Switched Systems,

    J. Zhao and D. Hill, “Dissipativity Theory for Switched Systems,” inProceedings of the 44th IEEE Conference on Decision and Control, 2005, pp. 7003– 7008

    work page 2005

  14. [14]

    Asymptotic stability for hybrid systems via decomposition, dissipativity, and detectability,

    A. R. Teel, “Asymptotic stability for hybrid systems via decomposition, dissipativity, and detectability,” in49th IEEE Conference on Decision and Control (CDC), 2010, pp. 7419–7424. [15]ChatGPT, OpenAI, https://chat.openai.com/. [16]Grammarly, https://www.grammarly.com/

    work page 2010