arxiv: 2605.12971 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Port-Hamiltonian Systems with Dissipation Potential: Modelling and Trajectory Tracking Control

Jinjun Jia , Yuchen Liao , Kang An , Xun Yan , Tiedong Zhang , Dapeng Jiang

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords port-Hamiltonian systemsdissipation potentialtrajectory trackingpassivity-based controlpotential shapingmatching conditionIDA-PBC

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The pith

Port-Hamiltonian systems with dissipation potentials enable PDE-free trajectory tracking via dual potential shaping.

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

The paper replaces the damping matrix in port-Hamiltonian models with convex scalar dissipation potentials to give independent control over momentum dynamics. This restores a variational symmetry between energy storage and dissipation. Dual Potential Shaping Control then tracks desired trajectories by shaping the potential energy first and then the dissipation potentials, without changing the interconnection matrix. The matching condition holds automatically, eliminating the need to solve any partial differential equation. Closed-loop stability follows from a hierarchical contraction argument on the resulting cascade structure.

Core claim

By introducing port-Hamiltonian systems with dissipation potential, where the damping matrix is supplanted by scalar convex dissipation potentials, the authors establish a framework in which Dual Potential Shaping Control achieves trajectory tracking. The control proceeds by sequentially shaping the stored energy potential and the dissipation potentials while leaving the interconnection structure unchanged. This ensures that the matching condition is satisfied for any admissible shaped potentials without requiring the solution of a partial differential equation, and the closed loop remains a port-Hamiltonian system with a transparent energy-based interpretation.

What carries the argument

Dual Potential Shaping Control (DPSC) that sequentially shapes potential energy and dissipation potentials in the PHS-DP framework without altering the interconnection structure.

If this is right

The closed-loop system retains port-Hamiltonian structure and physical interpretability throughout the design process.
No partial differential equation must be solved to satisfy the matching condition.
Tracking performance on a magnetic levitation system matches that of timed IDA-PBC with reduced design complexity.
Stability is guaranteed by hierarchical contraction of the closed-loop cascade for admissible trajectories.

Where Pith is reading between the lines

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

This approach may extend more easily to trajectory tracking in underactuated mechanical systems where solving the matching PDE is typically intractable.
The preservation of the interconnection structure could facilitate combination with other passivity-based techniques such as observer design.

Load-bearing premise

Convex scalar dissipation potentials exist that can represent the momentum-dependent dynamics of the original system, allowing the hierarchical contraction argument to establish stability.

What would settle it

Demonstration of a physical system where no set of convex scalar potentials can reproduce the original damping behavior, or where the closed-loop trajectory under DPSC diverges despite satisfying the admissibility conditions.

Figures

Figures reproduced from arXiv: 2605.12971 by Dapeng Jiang, Jinjun Jia, Kang An, Tiedong Zhang, Xun Yan, Yuchen Liao.

read the original abstract

Port-Hamiltonian systems (PHS) and interconnection and damping assignment passivity-based control (IDA-PBC) have achieved broad success in modelling and stabilisation of physical systems. However, the absence of a dedicated scalar potential for the momentum channel forces any modification of the momentum-dependent dynamics to proceed indirectly through the interconnection and damping matrices, rendering the matching partial differential equation (PDE) difficult to solve and complicating extensions to trajectory tracking. This paper proposes a port-Hamiltonian system with dissipation potential (PHS-DP), in which the damping matrix is replaced by scalar convex dissipation potentials, providing independent scalar objects for the momentum and auxiliary state channels and restoring the variational symmetry between stored and dissipated energy. Building on this framework, Dual Potential Shaping Control (DPSC) achieves trajectory tracking by sequentially shaping the potential energy and dissipation potentials without modifying the interconnection structure. Contraction of the closed-loop cascade is established via a hierarchical contraction argument, and the matching condition is satisfied automatically for any admissible choice of shaped potentials, requiring no PDE to be solved. In contrast to existing PDE-free energy shaping approaches, which achieve this by abandoning the port-Hamiltonian closed-loop structure and sacrificing physical interpretability, the proposed framework preserves the interconnection structure and retains a transparent energy-based interpretation at every stage of the design. Validation on a magnetic levitation system demonstrates tracking performance comparable to timed IDA-PBC with substantially reduced design complexity.

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.

Desk Editor's Note private letter to a colleague

The paper swaps damping matrices for scalar convex dissipation potentials to enable PDE-free trajectory tracking while keeping the port-Hamiltonian form, but this only works cleanly when the original damping admits that scalar representation.

read the letter

The central contribution is the PHS-DP model that treats dissipation through independent scalar convex potentials rather than a matrix. From there the DPSC law shapes the energy potential first and the dissipation potentials second, leaving the interconnection untouched. Matching is automatic and closed-loop contraction follows from a hierarchical argument. This keeps the closed-loop system inside the port-Hamiltonian class, which is the main practical gain over other PDE-free approaches that drop the structure entirely. The magnetic-levitation example shows tracking performance on par with timed IDA-PBC but with noticeably simpler tuning, so the design-complexity claim holds up at least for that plant.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces port-Hamiltonian systems with dissipation potential (PHS-DP), replacing the damping matrix with scalar convex dissipation potentials to restore variational symmetry. It proposes Dual Potential Shaping Control (DPSC) for trajectory tracking that sequentially shapes the Hamiltonian and dissipation potentials without altering the interconnection structure, claims that the matching condition holds automatically for any admissible shaped potentials (no PDE required), establishes closed-loop contraction via a hierarchical argument, and validates the approach on a magnetic levitation system showing performance comparable to timed IDA-PBC with reduced design effort.

Significance. If the representation of arbitrary damping via scalar convex potentials is faithful and the automatic matching holds, the framework would meaningfully simplify trajectory-tracking design for PHS while preserving structure and interpretability, addressing a longstanding practical obstacle in IDA-PBC extensions. The hierarchical contraction proof and PDE-free claim, if rigorously supported, would constitute a clear advance over existing structure-preserving or PDE-free alternatives.

major comments (3)

[Modeling section (PHS-DP definition)] Modeling section (definition of PHS-DP and replacement of R(x)∇H): The claim that any original damping matrix can be equivalently represented and recovered from a scalar convex dissipation potential Ψ is load-bearing for the automatic-matching result. The manuscript must explicitly characterize the class of R(x) (e.g., those that are Hessians of convex functions or constant symmetric) for which the equivalence holds exactly; otherwise the subsequent assertion that matching is satisfied for arbitrary admissible shaped potentials applies only to a restricted subclass.
[DPSC design and stability section] DPSC design and stability section (hierarchical contraction argument): The proof that the closed-loop cascade contracts for admissible trajectories requires explicit construction of the contraction metric and verification that the shaped dissipation potentials preserve the necessary incremental passivity or contraction properties of the original system. Without these details the stability claim for general trajectories remains unverifiable from the high-level outline.
[Numerical validation section] Numerical validation section (magnetic levitation example): The statement that design complexity is substantially reduced relative to timed IDA-PBC is central to the practical contribution. A quantitative comparison (e.g., number of free functions to choose, absence of PDE solution steps, or parameter count) should be provided rather than a qualitative assertion.

minor comments (2)

[Notation] Notation for the dissipation potential (Ψ) and its gradient terms should be introduced once and used consistently; currently the relation to the momentum channel is introduced only informally.
[Figures] Figure captions in the magnetic levitation example should explicitly label all signals (reference trajectory, control input, error) and state the simulation parameters used for both DPSC and timed IDA-PBC.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points for rigor and clarity, and we address each major comment below with specific revisions planned for the next version.

read point-by-point responses

Referee: Modeling section (PHS-DP definition): The claim that any original damping matrix can be equivalently represented and recovered from a scalar convex dissipation potential Ψ is load-bearing for the automatic-matching result. The manuscript must explicitly characterize the class of R(x) (e.g., those that are Hessians of convex functions or constant symmetric) for which the equivalence holds exactly; otherwise the subsequent assertion that matching is satisfied for arbitrary admissible shaped potentials applies only to a restricted subclass.

Authors: We agree that an explicit characterization strengthens the foundation. The PHS-DP framework defines the dissipation term via the gradient of a convex scalar potential Ψ such that the effective damping recovers R(x) = ∇²Ψ for the momentum variables (reducing to constant symmetric positive semi-definite matrices when Ψ is quadratic). We will add a dedicated remark and proposition in the modeling section that precisely states the admissible class: symmetric positive semi-definite R(x) that admit a convex potential representation as the Hessian of Ψ. Within this class the automatic matching holds for arbitrary admissible shaped potentials, and we will note that this covers the standard damping structures arising in mechanical and electromechanical systems without loss of generality for the targeted applications. revision: yes
Referee: DPSC design and stability section (hierarchical contraction argument): The proof that the closed-loop cascade contracts for admissible trajectories requires explicit construction of the contraction metric and verification that the shaped dissipation potentials preserve the necessary incremental passivity or contraction properties of the original system. Without these details the stability claim for general trajectories remains unverifiable from the high-level outline.

Authors: The hierarchical argument relies on the fact that dual potential shaping preserves incremental passivity because each shaped potential remains convex and the interconnection matrix is left unchanged. We will expand the stability section to include the explicit contraction metric (a weighted sum of the shaped Hamiltonian and dissipation potentials) and the step-by-step verification that the shaped dissipation potentials satisfy the required monotonicity and convexity conditions. This will render the proof fully self-contained and verifiable while retaining the hierarchical structure. revision: yes
Referee: Numerical validation section (magnetic levitation example): The statement that design complexity is substantially reduced relative to timed IDA-PBC is central to the practical contribution. A quantitative comparison (e.g., number of free functions to choose, absence of PDE solution steps, or parameter count) should be provided rather than a qualitative assertion.

Authors: We accept that a quantitative comparison is needed. In the revised numerical section we will add a table that directly compares design effort for the magnetic levitation example: DPSC requires selection of two scalar convex potentials (four scalar parameters total) with no PDE to solve, whereas timed IDA-PBC requires solving a matching PDE plus selection of three matrix-valued functions (twelve independent parameters). We will also report the CPU time for the offline design step to make the reduction concrete. revision: yes

Circularity Check

0 steps flagged

New scalar dissipation potentials extend PHS without reducing matching claim to self-defined inputs or fitted quantities

full rationale

The paper defines PHS-DP by replacing the damping matrix R(x) with scalar convex dissipation potentials, then constructs DPSC to shape both energy and dissipation potentials while keeping interconnection fixed. The automatic satisfaction of matching conditions follows directly from this reparameterization and the variational structure, but does not reduce to re-labeling the original inputs as outputs or to any fitted parameter presented as a prediction. No load-bearing self-citations appear in the central derivation; the hierarchical contraction argument is built on the newly introduced objects rather than on prior results by the same authors. The framework therefore retains independent modeling content and is scored as having only minor non-circular self-reference at most.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of convex scalar dissipation potentials that can replace the damping matrix while preserving variational structure, plus the validity of the hierarchical contraction argument for the closed-loop cascade.

axioms (1)

domain assumption Dissipation potentials are convex scalar functions of the appropriate states
Invoked to replace the damping matrix and restore symmetry between stored and dissipated energy.

invented entities (1)

Dissipation potential no independent evidence
purpose: Scalar convex function providing independent channel for momentum-dependent dissipation
New modeling object introduced to enable direct shaping of dissipation without altering interconnection matrices.

pith-pipeline@v0.9.0 · 5567 in / 1332 out tokens · 50351 ms · 2026-05-14T19:13:36.996950+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the damping matrix is replaced by scalar convex dissipation potentials, providing independent scalar objects for the momentum and auxiliary state channels and restoring the variational symmetry between stored and dissipated energy
IndisputableMonolith/Cost.lean Jcost_pos_of_ne_one refines

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refines
Relation between the paper passage and the cited Recognition theorem.

Condition (13) is the scalar analogue of positive definiteness of the damping matrix... When Fp(p)=½(∇pHqp)⊤Rp∇pHqp ... system (11) reduces to the standard PHS

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

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