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arxiv: 2605.16037 · v1 · pith:6V25NOV5new · submitted 2026-05-15 · 📡 eess.SP · cs.SY· eess.SY

Enhanced input stacking for non-square MIMO modal identification of aeronautical structures via Fast and Relaxed Vector Fitting

Beatrice E. Bauret Mart\'inez , Gabriele Dessena , Marco Civera , Oscar E. Bonilla-Manrique This is my paper

Pith reviewed 2026-05-20 15:46 UTC · model grok-4.3

classification 📡 eess.SP cs.SYeess.SY
keywords modal identificationvector fittingMIMO systemsground vibration testingaircraft structuresrational approximationstructural dynamicssystem identification
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The pith

An enhanced input-stacking strategy extends Fast and Relaxed Vector Fitting to non-square MIMO systems for accurate modal identification from aircraft ground vibration test data.

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

The paper reformulates Fast and Relaxed Vector Fitting for a Multi-Input Multi-Output framework to identify structural modal parameters from frequency response functions in aeronautical Ground Vibration Testing. It adds an enhanced input-stacking step to produce a rational model even when the number of inputs and outputs differs. Numerical checks on a MIMO beam model confirm that poles are recovered accurately despite added measurement noise. Experimental checks on data from the BAE Systems Hawk T1A aircraft yield modal estimates comparable to those from standard methods. The work concludes that the extended formulation is suitable for practical GVT-based modal analysis.

Core claim

The extended MIMO formulation of FRVF, achieved through an enhanced input-stacking strategy for rational approximation of frequency response functions, allows accurate identification of system poles and subsequent estimation of modal parameters. This holds for non-square MIMO systems and maintains high accuracy even with increasing levels of measurement noise, as validated both numerically and on real GVT data from an aircraft.

What carries the argument

The enhanced input-stacking strategy within the Fast and Relaxed Vector Fitting procedure, which produces a rational model from non-square MIMO frequency response data to extract poles that correspond to structural modes.

If this is right

  • The method identifies system poles and modal parameters with high accuracy from non-square MIMO frequency response data.
  • It maintains performance under increasing measurement noise in both numerical and experimental settings.
  • Results on real aircraft GVT data reach levels comparable to those of existing identification techniques.
  • The approach is directly applicable to structural modal analysis in Ground Vibration Testing campaigns.

Where Pith is reading between the lines

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

  • The stacking technique could be tested on other mechanical systems where input and output counts naturally differ.
  • It may allow experimenters to avoid hardware changes needed to force square MIMO configurations.
  • Similar rational-fitting steps might be combined with time-domain data for hybrid identification workflows.

Load-bearing premise

The poles extracted from the rational model produced by enhanced input-stacking accurately represent the true structural modal parameters of the aircraft without introducing significant spurious modes or bias from the non-square MIMO approximation.

What would settle it

If the modal frequencies and damping ratios extracted from the FRVF poles on the Hawk T1A GVT dataset deviate substantially from reference values obtained by established methods, or if accuracy collapses at moderate noise levels in the beam simulations, the claim of high accuracy and robustness would be falsified.

Figures

Figures reproduced from arXiv: 2605.16037 by Beatrice E. Bauret Mart\'inez, Gabriele Dessena, Marco Civera, Oscar E. Bonilla-Manrique.

Figure 1
Figure 1. Figure 1: MIMO beam model with two excitation forces. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Magnitude of the FRF data, fit, and deviation for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sensitivity analysis of the MIMO beam under [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic diagram showing the accelerometer [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: The BAE Systems Hawk T1A trainer jet aircraft [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: PSD representation across all 91 accelerometer [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: First three mode shapes of the Hawk obtained using the full truncated dataset: (a) first mode, (b) second mode, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

Fast and Relaxed Vector Fitting (FRVF) is a frequency-domain system identification approach that has been widely adopted in electrical system modelling, while its application to mechanical systems has remained relatively unexplored. In this work, FRVF is reformulated for the identification of structural modal parameters of an aircraft based on Ground Vibration Test (GVT) data within a Multi-Input Multi-Output (MIMO) framework. The proposed procedure consists of three stages: (i) rational approximation of frequency response functions via an enhanced input-stacking strategy, (ii) identification of system poles from the resulting rational model, and (iii) estimation of modal parameters from the extracted poles and associated residues. The methodology is first numerically validated on a MIMO beam model, with particular emphasis on accuracy and robustness under increasing measurement noise. Subsequently, experimental validation is conducted using GVT data from the BAE Systems Hawk T1A aircraft. The results obtained demonstrate a level of performance comparable to that achieved by existing methods. Overall, the extended MIMO formulation of FRVF exhibits high accuracy and strong robustness to measurement noise, highlighting its suitability for application in GVT-based modal analysis.

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. The manuscript reformulates Fast and Relaxed Vector Fitting (FRVF) for structural modal parameter identification of aircraft from Ground Vibration Test (GVT) data in a non-square MIMO setting. It introduces an enhanced input-stacking strategy to enable rational approximation of frequency response functions, followed by pole extraction from the resulting common-denominator model and subsequent modal parameter estimation from poles and residues. Numerical validation is performed on a MIMO beam model with added measurement noise, and experimental validation uses GVT data from the BAE Systems Hawk T1A aircraft; the authors report accuracy and noise robustness comparable to existing methods.

Significance. If the input-stacking step is shown to preserve the original system poles exactly, the work would usefully extend FRVF (primarily developed for electrical systems) to mechanical modal analysis. This could benefit GVT workflows for aeronautical structures where non-square MIMO data are common, by providing a frequency-domain rational fitting route with reported robustness to noise. The numerical and experimental results, if supported by the missing invariance argument, would constitute a practical contribution.

major comments (2)
  1. [Abstract and methodology description] The three-stage procedure (enhanced input-stacking followed by FRVF fit and pole extraction) is central to the claim that extracted poles correspond to structural modal parameters. No explicit invariance argument or side-by-side pole comparison (stacked vs. unstacked or vs. known analytical poles) is provided to confirm that the stacking matrix and common-pole enforcement leave the denominator polynomial unchanged. This is load-bearing for the weakest assumption identified in the review.
  2. [Numerical validation] Numerical validation section: while robustness to increasing measurement noise is emphasized, the manuscript does not report quantitative error metrics (e.g., pole relative error, MAC values, or FRF reconstruction error) that would allow direct assessment of whether the fitted poles remain accurate when the input-stacking dimension changes.
minor comments (2)
  1. [Abstract] The abstract states that results are 'comparable to that achieved by existing methods' but does not name the reference methods or provide a table of comparative metrics.
  2. [Methodology] Notation for the stacked transfer function and the weighting in the FRVF cost function should be clarified to make the reformulation reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for identifying areas where additional rigor would strengthen the manuscript. We address each major comment below and propose targeted revisions to clarify the methodology and enhance the quantitative assessment of results.

read point-by-point responses

  1. Referee: [Abstract and methodology description] The three-stage procedure (enhanced input-stacking followed by FRVF fit and pole extraction) is central to the claim that extracted poles correspond to structural modal parameters. No explicit invariance argument or side-by-side pole comparison (stacked vs. unstacked or vs. known analytical poles) is provided to confirm that the stacking matrix and common-pole enforcement leave the denominator polynomial unchanged. This is load-bearing for the weakest assumption identified in the review.

    Authors: We agree that an explicit invariance argument is necessary to rigorously support the claim that the extracted poles correspond to the underlying structural modes. The enhanced input-stacking is constructed to enforce a common denominator while preserving the original system poles, but a formal derivation of this invariance was omitted from the original submission. In the revised manuscript we will add a short subsection (or appendix) that derives the invariance of the denominator polynomial under the stacking operation and includes side-by-side numerical comparisons of poles obtained from stacked versus unstacked formulations against the known analytical poles of the MIMO beam. revision: yes

  2. Referee: [Numerical validation] Numerical validation section: while robustness to increasing measurement noise is emphasized, the manuscript does not report quantitative error metrics (e.g., pole relative error, MAC values, or FRF reconstruction error) that would allow direct assessment of whether the fitted poles remain accurate when the input-stacking dimension changes.

    Authors: We accept that the presentation of numerical results would benefit from explicit quantitative metrics. Although the manuscript already demonstrates robustness through visual comparisons, we will augment the numerical validation section with tables reporting pole relative errors, Modal Assurance Criterion (MAC) values, and FRF reconstruction errors for multiple noise levels and different input-stacking dimensions. These additions will enable direct quantitative evaluation of pole accuracy under varying stacking configurations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with external validation

full rationale

The paper reformulates FRVF for MIMO modal identification via a three-stage procedure (enhanced input-stacking for rational approximation, pole extraction from the fitted model, and modal parameter estimation from poles/residues). This is presented as a methodological extension followed by separate numerical validation on a MIMO beam model (with noise robustness checks) and experimental validation on BAE Systems Hawk T1A GVT data. No load-bearing step reduces by construction to a fitted quantity defined in terms of the output, no self-citation chain justifies a uniqueness claim, and no ansatz or renaming is smuggled in as a derivation. The approach relies on the standard properties of vector fitting and input stacking to preserve system poles, with accuracy assessed against independent benchmarks rather than tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so specific free parameters such as rational model order or weighting factors, domain assumptions about frequency response function quality, or any invented entities cannot be extracted. The method implicitly relies on standard rational approximation assumptions but details are absent.

pith-pipeline@v0.9.0 · 5757 in / 1124 out tokens · 54111 ms · 2026-05-20T15:46:47.009252+00:00 · methodology

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