Observer design for classes of nonlinear port-Hamiltonian systems

Alessandro Macchelli; ENSMM; FEMTO-ST); Filippo Ugolini; Ning Liu (UMLP; Yann Le Gorrec (UMLP; Yongxin Wu (UMLP

arxiv: 2604.03151 · v1 · submitted 2026-04-03 · 🧮 math.OC

Observer design for classes of nonlinear port-Hamiltonian systems

Filippo Ugolini , Ning Liu (UMLP , ENSMM , FEMTO-ST) , Yongxin Wu (UMLP , Yann Le Gorrec (UMLP , Alessandro Macchelli This is my paper

Pith reviewed 2026-05-13 18:25 UTC · model grok-4.3

classification 🧮 math.OC

keywords port-Hamiltonian systemsobserver designLPV embeddingLMI conditionsexponential convergencenonlinear systemsgain-scheduled observerselectromechanical systems

0 comments

The pith

An LPV polytopic embedding enables LMI-based design of exponential observers for nonlinear port-Hamiltonian systems with state-dependent input matrices.

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

The paper develops a systematic method to design observers for nonlinear port-Hamiltonian systems in which the input matrix depends on the state. It applies an integral mean value representation to rewrite the nonlinear error dynamics as a convex combination of linear vertex systems. This polytopic embedding converts the observer design into a set of linear matrix inequalities whose solution yields gains that guarantee exponential convergence of the estimation error. Both constant-gain and gain-scheduled observers are obtained, with the scheduled version certifying higher decay rates. The approach targets models of electromechanical devices such as magnetic levitation systems, MEMS, and electro-active polymer actuators.

Core claim

The nonlinear error dynamics of the port-Hamiltonian system are represented as a convex combination of linear vertex systems using an integral mean value representation. This representation enables the systematic computation of observer gains via LMI conditions that guarantee exponential convergence of the estimation error. Both constant-gain and gain-scheduled observers are derived within this framework.

What carries the argument

The LPV polytopic embedding of the nonlinear error dynamics based on the integral mean value theorem, which converts the nonlinear observer design into solvable LMIs for the gains.

If this is right

Constant-gain observers are obtained directly by solving LMIs derived from the vertex systems.
Gain-scheduled observers certify significantly larger decay rates than constant-gain designs.
The method covers electromechanical systems modeled under quasi-static electrical assumptions.
Numerical examples confirm that the computed gains produce exponential error convergence in simulation.

Where Pith is reading between the lines

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

The same embedding could support joint observer-based controller synthesis for the same class of systems.
Similar integral-mean-value polytopic representations may apply to nonlinear systems outside the port-Hamiltonian structure.
The framework could be extended to include robustness margins against model uncertainties or disturbances.

Load-bearing premise

The nonlinear port-Hamiltonian system admits an exact LPV polytopic embedding via the integral mean value theorem that remains valid over the operating domain and yields solvable LMIs.

What would settle it

A concrete nonlinear port-Hamiltonian system possessing an exponentially convergent observer for which the proposed LMI conditions admit no feasible solution at any positive decay rate.

Figures

Figures reproduced from arXiv: 2604.03151 by Alessandro Macchelli, ENSMM, FEMTO-ST), Filippo Ugolini, Ning Liu (UMLP, Yann Le Gorrec (UMLP, Yongxin Wu (UMLP.

**Figure 1.** Figure 1: Scenario 1 (Fair comparison, λ = 0.0897): (a) Position estimation q vs. ˆq. (b) Position error ˜q (0–0.4 s zoom). (c) Momentum estimation p vs. ˆp. (d) Momentum error ˜p (0–0.4 s zoom). (e) Observer gains L1(t) and L2(t). (f) Observer scheduled gains L1(t) and L2(t). LMI conditions for significantly larger values of λ, achieving approximately a fivefold improvement in the maximum certifiable decay rate (λ… view at source ↗

**Figure 2.** Figure 2: Scenario 2 (Different λ values): (a) Position estimation. (b) Position error (0–0.8 s zoom). (c) Momentum estimation. (d) Momentum error (0– 0.8 s zoom). TABLE III PERFORMANCE COMPARISON — SCENARIO 2 (DIFFERENT λ VALUES) Metric Const Sched Sched (λ=0.897) (λ=0.897) (λ=4.554) Peak |q˜| [µm] 200 200 200 Peak |p˜| [g·m/s] 2.90 2.88 2.00 Peak ∥x˜∥ [g·m/s] 2.90 2.88 2.01 RMS ∥x˜∥ [g·m/s] 0.298 0.295 0.198 Settl… view at source ↗

**Figure 3.** Figure 3: (a) Constant Observer gains, only for λ = 0.897 since for λ = 4.55 the constant L approach fails. (b) Scheduled Observer gains. (c) Scheduled Observer gains zoom. ditions are feasible due to its simpler implementation. However, for systems with strong nonlinearities—particularly in the input matrix—the gain-scheduled approach provides a less conservative alternative that enables observer synthesis when th… view at source ↗

read the original abstract

This paper presents a systematic observer design methodology for a class of port-Hamiltonian (pH) systems with state-dependent input matrices. Such systems can model a wide range of electromechanical systems, including magnetic levitation systems, MEMS devices, and electro-active polymer actuators such as DEA actuators, HASEL actuators, etc. In these applications, state-dependent input matrices naturally arise when the system is modeled under quasi-static electrical assumptions. An LPV polytopic embedding framework, together with LMI-based synthesis conditions, is proposed. The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation, which enables systematic computation of the observer gains that ensures exponential convergence. Both constant-gain and gain-scheduled observers are derived. Numerical results demonstrate the effectiveness of the proposed observer, with the gain-scheduled design achieving a significant increase in the maximum certifiable decay rate compared with constant-gain approaches, thereby reducing conservatism.

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 gives a usable LMI procedure for observers in port-Hamiltonian systems with state-dependent input matrices, but the polytopic embedding needs to handle the extra u-dependence from the g-term to support the general-input claim.

read the letter

This paper gives a usable LMI procedure for observers in port-Hamiltonian systems with state-dependent input matrices, which appear in electromechanical models under quasi-static assumptions. They apply the integral mean-value theorem to the full error dynamics, including the [g(x) - g(hat x)] u term, to obtain a polytopic LPV representation and then solve LMIs for both fixed-gain and scheduled observers that certify exponential convergence. The numerical example shows the scheduled version certifies noticeably higher decay rates, which is the practical payoff. What is actually new is the explicit treatment of state-dependent g(x) inside the pH structure with this embedding; prior work handled constant input matrices. The approach stays inside standard convex optimization, so it is straightforward to implement once the vertices are set up. The soft spot is the u-dependence. Because the mean-value form multiplies the integrated Jacobian of g by the instantaneous u, each vertex matrix is scaled by u. For the claimed polytopic system to be independent of u, either u must be added as a scheduling variable or the polytope must be enlarged over the admissible u-set. The abstract does not mention this step, so the exponential guarantee may hold only for constant or known u rather than arbitrary inputs. If the full derivation augments the vertices accordingly, the claim stands; otherwise the result is narrower than stated. This is aimed at control engineers who already work with nonlinear pH models in hardware and need an off-the-shelf LMI observer. A reader in that niche will get a concrete design method with some conservatism reduction. It is grounded enough to deserve a serious referee, mainly to check the exact vertex construction and confirm the LMI conditions cover general inputs.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an observer design method for nonlinear port-Hamiltonian systems whose input matrix depends on the state. It employs an LPV polytopic embedding of the nonlinear error dynamics obtained via the integral mean-value theorem, expresses the dynamics as a convex combination of linear vertex systems, and derives LMI conditions that certify exponential convergence of the estimation error. Both constant-gain and gain-scheduled observers are obtained; numerical examples illustrate that the scheduled design yields a higher certifiable decay rate.

Significance. If the embedding remains exact and the resulting LMIs are free of hidden input-dependent conservatism, the work supplies a convex-optimization route to observer synthesis for a practically relevant class of electromechanical pH models. The explicit comparison between constant-gain and scheduled designs quantifies the conservatism reduction achievable by scheduling.

major comments (2)

[Error dynamics representation] Error-dynamics section: the term [g(x) − g(ˆx)]u is rewritten by the integral mean-value theorem as an integral of Dg along the line segment multiplied by the instantaneous u. Consequently each vertex matrix is scaled by the current value of u. The LMI conditions must therefore either augment the vertex set by the admissible range of u or schedule the observer gain on both state and input; the manuscript does not state which route is taken.
[LMI conditions] LMI synthesis paragraph: the abstract asserts that the LMIs guarantee exponential convergence, yet the explicit matrix inequalities, the precise definition of the polytopic vertices, and the operating-domain assumptions under which the mean-value embedding is valid are not displayed. Without these expressions it is impossible to verify that no additional conservatism or domain restriction is introduced.

minor comments (2)

[Numerical results] The numerical examples would benefit from tabulated parameter values and explicit statements of the admissible input sets used for each benchmark.
[Notation] Notation for the state-dependent input matrix g(x) should be introduced once and used uniformly; several passages employ inconsistent symbols for its Jacobian.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify key technical aspects of the observer design. We address each major comment below, indicating the revisions that will be incorporated.

read point-by-point responses

Referee: [Error dynamics representation] Error-dynamics section: the term [g(x) − g(ˆx)]u is rewritten by the integral mean-value theorem as an integral of Dg along the line segment multiplied by the instantaneous u. Consequently each vertex matrix is scaled by the current value of u. The LMI conditions must therefore either augment the vertex set by the admissible range of u or schedule the observer gain on both state and input; the manuscript does not state which route is taken.

Authors: We appreciate this observation. In the gain-scheduled observer design, the observer gain is scheduled jointly on the estimated state and the instantaneous input u. This directly incorporates the scaling by u into the polytopic representation without requiring augmentation of the vertex set by the admissible range of u (which would introduce additional conservatism). We will add an explicit statement of this joint scheduling strategy in the revised error-dynamics section, together with a clarification of how the integral mean-value embedding is formed. revision: yes
Referee: [LMI conditions] LMI synthesis paragraph: the abstract asserts that the LMIs guarantee exponential convergence, yet the explicit matrix inequalities, the precise definition of the polytopic vertices, and the operating-domain assumptions under which the mean-value embedding is valid are not displayed. Without these expressions it is impossible to verify that no additional conservatism or domain restriction is introduced.

Authors: We agree that explicit display of the LMIs, vertex definitions, and domain assumptions is necessary for full verification. These elements appear in Theorem 1 and the surrounding text, where the polytopic vertices are obtained from the integral mean-value representation over a compact operating domain on which the embedding is exact. In the revision we will reproduce the complete matrix inequalities, give the precise vertex matrices, and state the domain assumptions explicitly so that readers can confirm the absence of hidden conservatism. revision: yes

Circularity Check

0 steps flagged

No circularity: standard mean-value LPV embedding plus LMI synthesis

full rationale

The derivation applies the integral mean-value theorem to obtain an exact convex combination representation of the error dynamics (including the state-dependent g(x) term) and then solves LMIs for observer gains. This chain relies on external convex-optimization theory and the mean-value theorem rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equation reduces to its own input by construction; the exponential-convergence claim is obtained from the LMI feasibility conditions, which are independent of the target rate. The skeptic concern about u-dependence is a potential correctness or domain-of-validity issue, not a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the nonlinear error system admits an exact convex polytopic representation via the integral mean value theorem and that the resulting LMIs are feasible over the operating region.

axioms (1)

domain assumption The nonlinear error dynamics admit an exact representation as a convex combination of linear vertex systems via the integral mean value theorem.
Invoked to convert the nonlinear observer error equation into an LPV polytopic form amenable to LMI synthesis.

pith-pipeline@v0.9.0 · 5485 in / 1366 out tokens · 53870 ms · 2026-05-13T18:25:59.430875+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 unclear

?

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

The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation... Aγ(t) = ∫ ∂γ/∂x (¯x(s),u(t)) ds
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

polytopic LPV representation: ˙˜x = Σ hi(ˆx,u) Ai(L) ˜x with Nv=2^nk vertices

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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work page doi:10.1016/j.mechatronics.2026.103515 2026

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A. Hammoud, N. Liu, Y . Le Gorrec, Y . Civet, and Y . Perriard, “Energy- based modeling and robust position control of a dielectric elastomer cardiac assist device,” IFAC-PapersOnLine, vol. 58, no. 6, pp. 25–30, 2024, doi: 10.1016/j.ifacol.2024.08.251

work page doi:10.1016/j.ifacol.2024.08.251 2024

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Y . Yeh, N. Cisneros, Y . Wu, K. Rabenorosoa and Y . Le Gorrec, “Modeling and Position Control of the HASEL Actuator via Port- Hamiltonian Approach,” IEEE Robotics and Automation Letters , vol. 7, no. 3, pp. 7100-7107, July 2022, doi: 10.1109/LRA.2022.3181365

work page doi:10.1109/lra.2022.3181365 2022

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N. E. Cisneros Pinto, “Contribution to design energy-based modeling and control of HASEL actuators,” Ph.D. dissertation, ´Ecole Nationale Sup´erieure de M ´ecanique et des Microtechniques, Besanc ¸on, France, 2025

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van der Schaft,L2-Gain and Passivity Techniques in Nonlinear Control, ser

A. van der Schaft, “L2-gain and Passivity Techniques in Non- linear Control”, Springer International Publishing AG , 2017, https://doi.org/10.1007/978-3-319-49992-5

work page doi:10.1007/978-3-319-49992-5 2017

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A. van der Schaft and B. Maschke, “The Hamiltonian formulation of energy conserving physical systems with external ports,” Archiv f ¨ur Elektronik und ¨Ubertragungstechnik, vol. 49, pp. 362–371, 1995

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[8] [8]

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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Control by Interconnection and Standard Passivity-Based Con- trol of Port-Hamiltonian Systems,

R. Ortega, A. van der Schaft, F. Castanos and A. Astolfi, “Control by Interconnection and Standard Passivity-Based Con- trol of Port-Hamiltonian Systems,” IEEE Transactions on Auto- matic Control , vol. 53, no. 11, pp. 2527-2542, Dec. 2008, doi: 10.1109/TAC.2008.2006930

work page doi:10.1109/tac.2008.2006930 2008

[10] [10]

Full-order observer design for a class of port-Hamiltonian systems,

A. Venkatraman and A. J. van der Schaft, “Full-order observer design for a class of port-Hamiltonian systems,” Automatica, vol. 46, no. 3, pp. 555–561, 2010

work page 2010

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Observer design for a class of nonlinear port-Hamiltonian systems,

T. Pfeifer and T. Meurer, “Observer design for a class of nonlinear port-Hamiltonian systems,” IEEE Transactions on Automatic Control , vol. 66, no. 8, pp. 3842–3849, 2021

work page 2021

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Observer design for a class of nonlinear port-controlled Hamiltonian systems,

M. Rojas, C. Granados-Salazar, and G. Espinosa-P ´erez, “Observer design for a class of nonlinear port-controlled Hamiltonian systems,” International Journal of Control, vol. 94, no. 10, pp. 2707–2718, 2021

work page 2021

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Observer design for a class of nonlinear Hamiltonian systems based on energy function structure,

C. Granados-Salazar, M. Rojas, and G. Espinosa-P ´erez, “Observer design for a class of nonlinear Hamiltonian systems based on energy function structure,” IFAC PapersOnLine, vol. 58, no. 6, pp. 178–183, 2024

work page 2024

[14] [14]

Structure-preserving ob- servers for port-Hamiltonian systems via contraction analysis,

D. Spirito, Y . Le Gorrec, and B. Maschke, “Structure-preserving ob- servers for port-Hamiltonian systems via contraction analysis,” IEEE Transactions on Automatic Control , 2024

work page 2024

[15] [15]

Observer for Lipschitz nonlinear systems: Mean value theorem and sector non- linearity transformation,

D. Ichalal, B. Marx, J. Ragot, D. Maquin, and S. Mammar, “Observer for Lipschitz nonlinear systems: Mean value theorem and sector non- linearity transformation,” IEEE Transactions on Automatic Control , vol. 57, no. 5, pp. 1269–1273, 2012

work page 2012

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Shanghai Lectures on Multivariable Analysis,

W. G. Faris, “Shanghai Lectures on Multivariable Analysis,” Lecture notes, Oct. 18, 2016

work page 2016

[17] [17]

109, 2025

Cisneros, Nelson and Wu, Yongxin and Rabenorosoa, Kanty and Le Gorrec, Yann, ”Dynamic Modeling of a Curling HASEL Actuator Us- ing the Port-Hamiltonian Framework with Experimental Validation,” Mechatronics, vol. 109, 2025

work page 2025