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arxiv: 2512.23203 · v2 · pith:XOVZJXTJnew · submitted 2025-12-29 · 🧮 math.OC

Coefficient-level output-feedback stabilization of linear port-Hamiltonian descriptor systems

Shuo Shi , Juan Zhang This is my paper

Pith reviewed 2026-05-16 19:41 UTC · model grok-4.3

classification 🧮 math.OC
keywords port-Hamiltonian descriptor systemsoutput feedback stabilizationcoefficient-level conditionsproportional feedbackstructure preservationimpulse-free systemsasymptotic stabilitydescriptor systems
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The pith

Coefficient-level conditions stabilize linear port-Hamiltonian descriptor systems via output feedback without explicit representation.

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

The paper shows that for descriptor systems known to admit a port-Hamiltonian structure but given only through their coefficient matrices, proportional output feedback can be designed using direct matrix conditions that guarantee the closed-loop system is regular, impulse-free, asymptotically stable, and remains port-Hamiltonian. These conditions match the solvability criteria that would apply if an explicit port-Hamiltonian form were available, yet avoid computing that form. The framework extends to proportional-derivative feedback, allowing assignment of a prescribed dynamical order while keeping the proportional gain any symmetric positive definite matrix and constructing the derivative gain from coefficient decompositions.

Core claim

For linear port-Hamiltonian descriptor systems supplied only as coefficient matrices, proportional output feedback admits coefficient-level conditions equivalent to the known explicit-port-Hamiltonian solvability criteria; these conditions make the closed-loop system regular, impulse-free, asymptotically stable, and port-Hamiltonian. The same coefficient-based approach extends to proportional-derivative feedback, permitting assignment of any desired dynamical order with the proportional gain chosen freely as any symmetric positive definite matrix.

What carries the argument

Coefficient-level conditions on the system matrices that enforce regularity, impulse-freeness, asymptotic stability, and port-Hamiltonian structure under proportional (and proportional-derivative) output feedback.

If this is right

  • The closed-loop descriptor system is regular and impulse-free.
  • Asymptotic stability holds for any symmetric positive definite proportional gain.
  • The closed-loop system inherits the port-Hamiltonian structure from the open-loop coefficients.
  • Proportional-derivative feedback allows exact assignment of the closed-loop dynamical order.
  • All gains are constructed without solving for an explicit port-Hamiltonian factorization.

Where Pith is reading between the lines

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

  • Numerical linear-algebra packages could implement the conditions as black-box stabilization routines for descriptor models whose energy structure is certified but not exhibited.
  • The coefficient conditions may serve as a template for structure-preserving controllers in other energy-based modeling frameworks that supply only matrix data.
  • If the same coefficient patterns appear in nonlinear or time-varying port-Hamiltonian descriptor systems, direct gain formulas without structure extraction could become feasible.

Load-bearing premise

A port-Hamiltonian representation is known to exist for the given coefficient matrices, but the stabilization conditions must be verifiable directly from those matrices without ever computing the representation.

What would settle it

A concrete set of coefficient matrices satisfying the derived conditions whose closed-loop pencil is either singular, impulsive, unstable, or fails to satisfy the port-Hamiltonian matrix identities.

read the original abstract

This paper studies coefficient-level, structure-preserving output-feedback stabilization of linear port-Hamiltonian (pH) descriptor systems. Existing stabilization conditions generally require explicit pH representations, which may be costly to compute. We consider descriptor systems for which only the coefficient matrices are available and for which a pH representation is known to exist but is not explicitly given. For proportional output feedback, we derive coefficient-level conditions that are equivalent to the known solvability criteria in the explicit pH setting. These conditions ensure that the closed-loop system is regular, impulse-free, asymptotically stable, and remains port-Hamiltonian. We further extend the framework to proportional-derivative output feedback and enable the assignment of a prescribed dynamical order. Under the proposed conditions, the proportional gain may be chosen as any symmetric positive definite matrix, and the derivative gain is constructed from coefficient-based decompositions, without computing a pH representation.

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

2 major / 2 minor

Summary. This paper develops coefficient-level conditions for output-feedback stabilization of linear port-Hamiltonian descriptor systems. For proportional output feedback, it derives conditions equivalent to known solvability criteria that guarantee the closed-loop system is regular, impulse-free, asymptotically stable, and port-Hamiltonian. The approach is extended to proportional-derivative feedback for assigning dynamical order, with gains constructed from coefficient-based decompositions without explicit pH representation.

Significance. If the claimed equivalence holds rigorously, the result would be significant for practical control of descriptor systems in applications such as circuit simulation and mechanical modeling, where explicit pH factorizations are expensive to obtain. Allowing arbitrary symmetric positive definite proportional gains while preserving structure is a potentially useful feature for design flexibility.

major comments (2)
  1. [Abstract] Abstract: The asserted equivalence between the proposed coefficient-level conditions and the known solvability criteria (which presuppose an explicit pH representation with matrices E, Q, J, R) is central to the contribution but lacks a visible derivation or proof of necessity and sufficiency; this must be supplied to confirm that the conditions do not implicitly encode the same positivity or kernel conditions that would require recovering the pH factorization.
  2. [Proportional-derivative feedback results] Proportional-derivative extension: The construction of derivative gains via 'coefficient-based decompositions' needs explicit algorithmic definition and verification that these decompositions enforce regularity, impulse-freeness, and asymptotic stability directly from the given matrices without solving auxiliary Riccati or LMI problems equivalent to computing the pH form.
minor comments (2)
  1. [Introduction] Clarify in the introduction how the assumption that 'a pH representation is known to exist' is used without being computed, and whether this assumption can be checked coefficient-wise.
  2. [Notation and preliminaries] Add a table or explicit mapping relating the original coefficient matrices (E, A, B, C, D) to the pH components to improve readability of the coefficient conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of clarity and rigor that we will address in the revision. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The asserted equivalence between the proposed coefficient-level conditions and the known solvability criteria (which presuppose an explicit pH representation with matrices E, Q, J, R) is central to the contribution but lacks a visible derivation or proof of necessity and sufficiency; this must be supplied to confirm that the conditions do not implicitly encode the same positivity or kernel conditions that would require recovering the pH factorization.

    Authors: We agree that the equivalence is central and that its proof should be presented explicitly. In the revised manuscript we will add a dedicated subsection (in Section 3) containing a complete proof of necessity and sufficiency. The proof proceeds by direct algebraic manipulation of the coefficient matrices, showing that the proposed conditions are equivalent to the known criteria without presupposing or recovering an explicit factorization (E, Q, J, R). We will also include a remark clarifying that no auxiliary positivity or kernel conditions are implicitly encoded beyond what is already verifiable from the given coefficients. revision: yes

  2. Referee: [Proportional-derivative feedback results] Proportional-derivative extension: The construction of derivative gains via 'coefficient-based decompositions' needs explicit algorithmic definition and verification that these decompositions enforce regularity, impulse-freeness, and asymptotic stability directly from the given matrices without solving auxiliary Riccati or LMI problems equivalent to computing the pH form.

    Authors: We acknowledge that the current description of the coefficient-based decompositions is not sufficiently algorithmic. In the revision we will replace the informal description with a precise, step-by-step algorithm (Algorithm 1 in the new Section 4) that operates exclusively on the coefficient matrices. We will also add a theorem (Theorem 4.2) that rigorously verifies, via direct matrix rank and eigenvalue arguments, that the resulting closed-loop system satisfies regularity, impulse-freeness, and asymptotic stability without invoking Riccati equations, LMIs, or any computation equivalent to recovering the pH form. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence claim translates external criteria without self-referential reduction

full rationale

The paper states that coefficient-level conditions are derived to be equivalent to known solvability criteria from the explicit pH setting, while remaining checkable directly from coefficient matrices without recovering the pH factorization. No equations or steps in the abstract or described claims reduce a prediction to a fitted input by construction, nor does any load-bearing premise rest solely on self-citation chains that presuppose the target result. The derivation is presented as a structure-preserving translation that preserves regularity, impulse-freeness, stability, and pH structure, with gains constructed via coefficient-based decompositions. This keeps the central claim self-contained against external benchmarks rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a pH representation exists for the coefficient matrices; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption A port-Hamiltonian representation exists for the descriptor system but is not explicitly given.
    Explicitly stated in the abstract as the setting under which coefficient-level conditions are derived.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

  1. [1]

    Beattie, V

    [1]C. Beattie, V. Mehrmann, and H. Xu,Port-Hamiltonian realizations of linear time invariant systems, arXiv preprint, https://arxiv.org/abs/2201.05355, (2022). [2]C. Beattie, V. Mehrmann, H. Xu, and H. Zwart,Linear port-Hamiltonian descriptor sys- tems, Math. Control Signals Systems, 30 (2018), Art. 17 (27 pages), https://doi.org/10. 1007/s00498-018-0223-...

    work page doi:10.1137/s0363012994272630 2022

  2. [2]

    Kautsky, N

    [21]J. Kautsky, N. Nichols, and E.-W. Chu,Robust pole assignment in singular control sys- tems, Linear Algebra Appl., 121 (1989), pp. 9–37, https://doi.org/10.1016/0024-3795(89) 90689-7. [22]P. Kunkel and V. Mehrmann,Differential-algebraic equations: Analysis and numerical so- lution, EMS Textbk. Math., European Mathematical Society (EMS), Z¨ urich,

    work page doi:10.1016/0024-3795(89 1989

  3. [3]

    [23]C. Mehl, V. Mehrmann, and M. Wojtylak,Linear algebra properties of dissipative Hamil- tonian descriptor systems, SIAM J. Matrix Anal. Appl., 39 (2018), pp. 1489–1519, https://doi.org/10.1137/18M1164275. [24]V. Mehrmann and R. Morandin,Structure-preserving discretization for port-Hamiltonian descriptor systems, in 2019 IEEE 58th Conference on Decision ...

    work page doi:10.1137/18m1164275 2018

  4. [4]

    2025.3553096

    [30]M. Shayman and Z. Zhou,Feedback control and classification of generalized linear systems, IEEE Trans. Automat. Control, 32 (1987), pp. 483–494, https://doi.org/10.1109/TAC. 1987.1104642. [31]V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis,Static output feedback—a survey, Automatica, 33 (1997), pp. 125–137, https://doi.org/10.1016/S0005-109...

    work page doi:10.1109/tac 1987

  5. [5]

    [35]T. Xu, Y. Zeng, L. Zhang, and J. Qian,Direct modeling method of generalized Hamiltonian system and simulation simplified, Procedia Eng., 31 (2012), pp. 901–908, https://doi.org/ 10.1016/j.proeng.2012.01.1119. [36]X. Zhan,Matrix theory, American Mathematical Soc, Providence, RI,

    work page doi:10.1016/j.proeng.2012.01.1119 2012

  6. [6]

    [37]K. Zhou, J. C. Doyle, and K. Glover,Robust and optimal control, Prentice-Hall, Upper Saddle River, NJ, 1995

    work page 1995