⚡ electrical engineering

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

This paper reformulates Fast and Relaxed Vector Fitting (FRVF) with an enhanced input-stacking strategy for non-square MIMO modal identification of aeronautical structures, demonstrating its high accuracy and noise robustness through numerical simulations and experimental validation on a BAE Systems Hawk T1A aircraft.

Original authors: Beatrice E. Bauret Martínez, Gabriele Dessena, Marco Civera, Oscar E. Bonilla-Manrique

Published 2026-05-18

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CC BY 4.0

Original authors: Beatrice E. Bauret Martínez, Gabriele Dessena, Marco Civera, Oscar E. Bonilla-Manrique

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out how a giant, complex airplane vibrates when it shakes. Engineers need to know exactly how it moves, where it bends, and how fast it wobbles to make sure it's safe to fly. This process is called Ground Vibration Testing (GVT).

In this paper, the authors introduce a new, smarter way to analyze these vibrations using a mathematical tool called Fast and Relaxed Vector Fitting (FRVF). Here is the breakdown of what they did and why it matters, using simple analogies.

The Problem: Too Many Microphones, Too Few Speakers

Usually, to understand how a structure vibrates, you need to shake it (inputs) and listen to how it responds (outputs).

The Old Way: Most computer programs used for this were designed for electrical circuits. In those circuits, you usually have the same number of "speakers" (inputs) as "microphones" (outputs). It's like a perfect dance where every partner has a match.
The Reality: In airplane testing, you have very few places to attach shakers (maybe 5), but you have dozens or even hundreds of sensors (accelerometers) listening to the vibrations. It's like having 5 people shouting instructions to a choir of 100 singers. The old computer programs got confused because the numbers didn't match (a "non-square" problem).

The Solution: The "Super-Stacking" Trick

The authors created a clever workaround they call Enhanced Input Stacking.

Imagine you have 5 different people shouting instructions to that choir. Instead of trying to listen to each person separately, the new method takes all 5 voices, mixes them together into one giant, super-loud "virtual" voice, and then listens to how the choir responds to that combined sound.

Why this works: Even though the voices are mixed, the notes the choir sings (the natural frequencies) stay exactly the same. The only thing that changes is how loud the notes are. By stacking the inputs, the signal becomes stronger and clearer, making it much easier for the computer to hear the true vibration patterns even if there is background noise.

The Tool: FRVF (The "Tuner")

The method they use, FRVF, is like a highly advanced auto-tuner for the airplane.

It listens: It takes the messy data from the sensors.
It guesses: It tries to fit a mathematical model to the data, looking for specific "poles" (which represent the vibration frequencies).
It relaxes: Unlike older methods that get stuck if the starting guess is slightly off, this "Relaxed" version is flexible. It can adjust its guesses quickly and accurately, even if the data is a bit noisy (like trying to tune a guitar in a windy room).

The Tests: From a Beam to a Real Jet

The authors tested this new method in two ways:

The Simulation (The Beam): They created a digital model of a metal beam. They added artificial "noise" (static) to the data to see if the method would break.
- Result: Even with a lot of noise, the method found the correct vibration frequencies almost perfectly. It was very robust.
The Real World (The Hawk Jet): They applied the method to real data from a BAE Systems Hawk T1A trainer jet. This plane was tested with 5 shakers and 91 sensors.
- Result: The new method successfully identified 27 out of 32 vibration modes that other advanced methods had found. It was much better than older, standard methods (like LSCE), which failed to find most of the modes or gave very inaccurate results.

The Bottom Line

The paper claims that by using this "stacking" trick, they can now use a powerful, fast, and noise-resistant mathematical tool (FRVF) on real-world airplane data, even when the number of sensors doesn't match the number of shakers.

In short: They figured out how to make a tool designed for perfect electrical circuits work perfectly on messy, real-world airplane data by mixing the inputs together to make the signal clearer. This helps engineers identify the airplane's vibration patterns more accurately and reliably than before.

Technical Summary: Enhanced Input Stacking for Non-Square MIMO Modal Identification of Aeronautical Structures via Fast and Relaxed Vector Fitting

Problem Statement
Accurate identification of structural modal parameters (natural frequencies, damping ratios, and mode shapes) is critical for aircraft certification, finite element model updating, and aeroelastic stability prediction. While Ground Vibration Testing (GVT) provides the necessary experimental data, modern aerospace structures often require Multi-Input Multi-Output (MIMO) frameworks to capture coupled and closely spaced vibration modes.

A significant limitation exists in applying the Fast and Relaxed Vector Fitting (FRVF) algorithm—a computationally efficient frequency-domain system identification method originally developed for electrical networks—to mechanical systems. Standard FRVF implementations assume square Frequency Response Function (FRF) matrices, where the number of input channels equals the number of output channels. However, in experimental modal analysis of aircraft, the number of measured response channels (sensors) typically far exceeds the number of excitation locations (actuators), resulting in non-square FRF matrices. This mismatch prevents the direct application of existing FRVF formulations to realistic GVT scenarios.

Methodology
The authors propose a reformulation of FRVF to accommodate non-square MIMO configurations through an enhanced input-stacking strategy. The methodology proceeds in three stages:

Enhanced Input Stacking: The core innovation involves reshaping the three-dimensional FRF tensor ( $N_{out} \times N_{in} \times N_{freq}$ $N_{o u t} \times N_{in} \times N_{f r e q}$ ) into a two-dimensional array compatible with FRVF. Instead of the conventional upper-triangular stacking used for square systems, the proposed algorithm performs frequency-wise superposition of input channels. For each output channel, FRFs from all input channels are summed.
- Theoretical Basis: Based on the principle of modal superposition for linear time-invariant systems, this summation creates an effective FRF driven by a virtual, coherent excitation. While this reweights the residues, it preserves the system poles (natural frequencies and damping ratios) and maintains the spatial proportions of the mode shapes, thereby enhancing the signal-to-noise ratio (SNR) for weakly excited modes without altering the physical correlation between inputs and outputs.
System Identification: The stacked FRFs are processed using the FRVF algorithm. This involves iterative rational approximation using QR-based residue estimation, weak inverse-magnitude weighting to handle noise, and efficient pole stabilization.
Parameter Extraction: Modal parameters are extracted from the resulting rational model using stabilization diagrams. Candidate poles are selected based on frequency stability (<1 Hz variation), damping consistency (<0.05 variation), and mode shape correlation (Modal Assurance Criterion > 0.95).

Key Contributions

Generalization of FRVF: The work extends the FRVF algorithm from square systems to arbitrary non-square MIMO systems, addressing a critical gap in applying electrical network modeling techniques to structural dynamics.
Input-Stacking Algorithm: The introduction of a specific stacking procedure (Algorithm 1) that allows for the aggregation of multiple excitation contributions, improving observability of modes without requiring reciprocity assumptions or collocated sensors/actuators.
Validation Framework: The methodology is validated through both numerical simulations on a cantilever beam and experimental testing on a full-scale aircraft.

Results

Numerical Validation (Cantilever Beam):
- The method successfully identified all 10 modes of a simulated beam with negligible frequency errors (<0.1%) and perfect mode shape correlation (MAC = 1) in noise-free conditions.
- Noise Robustness: Under additive white Gaussian noise (AWGN) up to 1%, the method maintained high accuracy. Even at 3–5% noise levels, natural frequencies remained accurate (errors <0.9%) and mode shapes highly correlated (MAC ≥ 0.99), although damping ratio estimates for higher modes showed increased error (up to 40%).
Experimental Validation (BAE Systems Hawk T1A):
- The method was applied to GVT data from a Hawk T1A trainer jet (5 inputs, 91 outputs) within a 5–165 Hz bandwidth.
- Comparison with iLF and LSCE: FRVF identified 27 of the 32 modes reported by the improved Loewner Framework (iLF), with frequency deviations generally within 1%. In contrast, the Least Squares Complex Exponential (LSCE) method identified only four modes with significant deviations in both frequency and damping.
- Mode Shapes: Visual inspection and MAC values confirmed that FRVF recovered physically meaningful mode shapes, particularly for the lower, well-separated modes. Performance degraded slightly for modes above 40 Hz due to increased modal density and reduced observability, consistent with the limitations of equidistant pole initialization in congested bands.

Significance and Claims
The paper claims that the extended MIMO formulation of FRVF, coupled with enhanced input stacking, offers a reliable, accurate, and noise-tolerant approach for modal analysis of aeronautical structures. The authors assert that this approach:

Overcomes the square-system restriction of standard FRVF, making it suitable for realistic aircraft GVT campaigns where sensors outnumber actuators.
Demonstrates performance comparable to established broadband methods like the iLF, while outperforming classical time-domain techniques like LSCE in complex MIMO environments.
Provides a computationally efficient alternative for identifying structural dynamics, suitable for applications requiring high robustness against measurement noise.

The authors conclude that while the method is highly effective, future work could focus on automated mode selection criteria and dynamic initial pole placement to further improve convergence in highly congested frequency bands.

The Problem: Too Many Microphones, Too Few Speakers

The Solution: The "Super-Stacking" Trick

The Tool: FRVF (The "Tuner")

The Tests: From a Beam to a Real Jet

The Bottom Line

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