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arxiv: 2604.25339 · v1 · submitted 2026-04-28 · 🧮 math.OC

Observer-Based State Feedback Controller for a Mindlin Plate Model in port-Hamiltonian framework

Ignacio Diaz Alastuey (UMLP , ENSMM , FEMTO-ST) , Yann Le Gorrec (UMLP , Yongxin Wu (UMLP This is my paper

Pith reviewed 2026-05-07 15:45 UTC · model grok-4.3

classification 🧮 math.OC
keywords port-Hamiltonian systemsMindlin plateobserver-based state feedbackstrictly positive realfinite-difference discretizationboundary controlstabilitytwo-dimensional systems
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The pith

A finite-dimensional observer-based state feedback controller made strictly positive real stabilizes the boundary-controlled Mindlin plate upon interconnection.

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

The paper extends an earlier lumped observer-based state feedback method from one-dimensional to two-dimensional boundary-controlled port-Hamiltonian systems. The 2-D Mindlin plate is discretized with a structure-preserving finite-difference scheme on staggered grids, after which a controllability decomposition isolates the controllable modes. Gains are chosen so the resulting finite-dimensional controller is strictly positive real. This property is shown to guarantee stability of the closed-loop interconnection with the original infinite-dimensional plate model, with numerical simulations used to check performance.

Core claim

After structure-preserving finite-difference discretization of the 2-D port-Hamiltonian Mindlin plate on staggered grids and subsequent controllability decomposition, the state-feedback and observer gains can be selected so that the finite-dimensional observer-based state feedback controller is strictly positive real. Interconnection of this controller with the infinite-dimensional boundary-controlled plate then yields a stable closed-loop system.

What carries the argument

The strictly positive real property of the finite-dimensional observer-based state feedback controller obtained after controllability decomposition of the discretized port-Hamiltonian Mindlin plate model.

If this is right

  • The closed-loop system formed by the finite controller and infinite-dimensional plate remains stable.
  • The design extends the original 1-D lumped OBSF method to 2-D plates while preserving the stability argument.
  • Numerical simulations confirm that the resulting controller regulates plate behavior effectively.
  • The discretization and decomposition steps make gain tuning feasible for the strictly positive real requirement.

Where Pith is reading between the lines

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

  • The same discretization-plus-decomposition route may apply to other two-dimensional distributed systems expressed in port-Hamiltonian form.
  • Practical digital implementation of the finite controller becomes feasible once the strictly positive real property is verified.
  • The stability result could guide boundary-actuated vibration control designs for thin plates in engineering settings.

Load-bearing premise

The structure-preserving finite-difference discretization on staggered grids together with the controllability decomposition permits gain selection that renders the finite controller strictly positive real while keeping the interconnection stable.

What would settle it

A numerical simulation or frequency-domain calculation in which the closed-loop plate displacement grows unbounded, or in which the finite controller transfer function exhibits a negative real part at some frequency, would falsify the stability guarantee.

Figures

Figures reproduced from arXiv: 2604.25339 by ENSMM, FEMTO-ST), Ignacio Diaz Alastuey (UMLP, Yann Le Gorrec (UMLP, Yongxin Wu (UMLP.

Figure 1
Figure 1. Figure 1: Mindlin Plate deformations The deformation assumptions are illustrated in view at source ↗
Figure 2
Figure 2. Figure 2: Example of problem of interest view at source ↗
Figure 3
Figure 3. Figure 3: Example of problem of interest. at the boundaries correspond to the clamped part; the points ψ 0{0,1} q and ψ 3{0,1} q at the boundaries correspond to the free borders; and finally the points ψ {0,1,2,3}2 q cor￾respond to the boundaries where our controller will act. Then, we consider the set of discrete general coordinates Xq : n x mn q = q(t, ψmn q )| ∀ψ mn q ∈ Ψq \ ∂Ω o , Xp : n x mn p = p(t, ψmn p )| ∀… view at source ↗
Figure 4
Figure 4. Figure 4: Rc matrix calculation. 498, which can be reduced to a controllable subspace of dimension 420. Then, to design the controller, we apply the LQR-based control design described in Section 3.3 to compute the state-feedback gain K. Using this gain, a stable closed-loop system is obtained under full-state feedback. The corresponding eigenvalues are shown in view at source ↗
Figure 6
Figure 6. Figure 6: Energy overtime for controlled systems the different systems, the Hamiltonian evolution is shown in view at source ↗
Figure 8
Figure 8. Figure 8: , where several eigenvalues of the closed-loop system with the Luenberger OBSF (blue dots) are located in the right half-plane, leading to instability. In contrast, all eigenvalues of the closed-loop system with the passive OBSF (blue markers) remain in the left half-plane, thereby guaranteeing stability. 5. CONCLUSION In this paper, an observer-based state-feedback (OBSF) control strategy for a boundary-c… view at source ↗
Figure 7
Figure 7. Figure 7: Energy overtime for controlled systems It can be observed that the energy of the system controlled by the Luenberger OBSF begins to diverge toward the end of the simulation. This behavior is attributed to the spillover effect, as also reported in view at source ↗
read the original abstract

This paper generalises an early lumped observer-based state-feedback (OBSF) control design methodology, originally developed for one-dimensional (1-D) boundary-controlled port-Hamiltonian systems, to a two-dimensional (2-D) boundary-controlled Mindlin plate. To this end, the 2-D port-Hamiltonian Mindlin plate model is first introduced and then discretized using a structure-preserving finite-difference method on staggered grids. A controllability decomposition is subsequently applied to identify the controllable modes of the discretized model. Furthermore, the state-feedback and observer gains are designed so that the OBSF controller is strictly positive real. This guarantees the stability of the closed-loop system when the finite-dimensional OBSF controller is interconnected with the 2-D boundary-controlled Mindlin plate. Numerical simulations are finally presented to illustrate the effectiveness of the proposed method.

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

0 major / 3 minor

Summary. The manuscript generalizes an observer-based state-feedback (OBSF) controller from 1-D boundary-controlled port-Hamiltonian systems to the 2-D boundary-controlled Mindlin plate. The 2-D pH Mindlin plate is discretized via a structure-preserving finite-difference scheme on staggered grids; a controllability decomposition isolates the controllable subsystem; state-feedback and observer gains are chosen so that the resulting finite-dimensional OBSF controller is strictly positive real; and this SPR property is invoked to conclude asymptotic stability of the closed-loop system under the natural power-conserving interconnection with the infinite-dimensional plant. Numerical simulations illustrate the design.

Significance. If the central claims hold, the work supplies a systematic, structure-preserving route to stabilizing controllers for 2-D distributed-parameter systems that extends the 1-D pH OBSF methodology without introducing new free parameters or ad-hoc tuning. The explicit use of staggered-grid discretization to retain the pH structure, followed by controllability decomposition and gain selection that enforces the SPR property, is a clear methodological contribution. The paper includes reproducible numerical examples that demonstrate closed-loop behavior, which strengthens the practical utility of the approach.

minor comments (3)
  1. [Abstract] The abstract would benefit from a single sentence referencing the main stability theorem (e.g., the interconnection result that converts SPR of the controller into asymptotic stability of the closed loop).
  2. [§2] Notation for the port-Hamiltonian variables (energy variables, effort-flow pairs, and boundary ports) is introduced without a compact summary table; a short table in §2 would improve readability for readers coming from the 1-D literature.
  3. [Numerical simulations] In the numerical section, the time-step and spatial grid sizes used for the staggered discretization should be stated explicitly alongside the reported simulation results to facilitate reproduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the recognition of its methodological contributions, and the recommendation for minor revision. The referee's description accurately reflects the paper's scope and approach. No specific major comments were listed in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. We remain available to address any unlisted minor issues or suggestions in a revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external pH theory

full rationale

The paper's chain proceeds from the 2D pH Mindlin plate model, through a structure-preserving staggered-grid discretization that preserves the pH structure by construction, to a controllability decomposition that isolates the controllable subsystem, followed by explicit gain selection to enforce the SPR property on the finite-dimensional OBSF controller. Closed-loop stability is then obtained from the standard power-conserving interconnection with the infinite-dimensional plant and the strict passivity supplied by the SPR controller. All steps invoke established results from the pH literature (passivity, positive-real lemma, structure preservation under discretization) rather than self-citations or fitted parameters that presuppose the target stability claim. No equation reduces to a prior equation by definition, and no uniqueness theorem is imported from the authors' own prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard mathematical properties of port-Hamiltonian systems and strictly positive real transfer functions; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Interconnection of a strictly positive real system with a passive plant yields a stable closed loop
    Invoked to conclude stability from the SPR property of the finite-dimensional controller.

pith-pipeline@v0.9.0 · 5461 in / 1265 out tokens · 49829 ms · 2026-05-07T15:45:21.380238+00:00 · methodology

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Reference graph

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