NExT-LF: Natural Excitation with Loewner Framework

• NExT-LF is an operational modal analysis technique that integrates NExT’s output-only impulse-response extraction with the Loewner Framework’s tangential interpolation for noise-robust model realization.
• It uses frequency-domain transformation and SVD-based model reduction to accurately extract modal parameters including natural frequencies, damping ratios, and mode shapes from ambient vibration data.
• The method outperforms traditional techniques by offering enhanced stability, superior noise robustness, and reliable mode tracking in both simulated and experimental studies.

The Natural Excitation Technique with the Loewner Framework (NExT-LF) is an operational modal analysis (OMA) methodology that unites the output-only identification approach of NExT with the tangential interpolation and noise-robust model realization capabilities of the Loewner Framework. NExT-LF enables robust estimation of modal parameters—natural frequencies, damping ratios, and mode shapes—from vibration measurements acquired under unknown, ambient (broadband or operational) excitation. The method provides reliable performance in high-noise settings and addresses stability and mode-tracking limitations inherent to standard OMA algorithms such as NExT-ERA and Stochastic Subspace Identification (SSI) (Dessena et al., 2024, Dessena et al., 13 Jan 2026).

1. Theoretical Principles

NExT-LF is based on the premise that under wide-sense stationary, broadband excitation, the cross-correlation of output channels retrieves the impulse-response functions (IRFs) of the underlying dynamical system. Specifically, for response channels $y_i(t)$ and $y_j(t)$, the cross-correlation: $$R_{ij}(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_0^T y_i(t)\, y_j(t+\tau)\, dt$$ converges (up to scaling) to the free-decay IRF from $j$ to $i$.

After extracting these IRFs, a one-sided Fourier transform produces frequency-response function (FRF) samples at discrete frequencies. The Loewner Framework then reconstructs a minimal linear realization (state-space model) directly from these FRF samples using tangential interpolation over selected "left" and "right" frequency points. The eigenstructure of the reduced-order system yields physical modal parameters: poles $\lambda_i$, natural frequencies $\omega_i = |\operatorname{Im} \lambda_i|$, damping ratios $$\zeta_i = -\operatorname{Re} \lambda_i / |\lambda_i|$$, and mode shapes from realization residues (Dessena et al., 2024, Dessena et al., 13 Jan 2026).

2. Stepwise Methodology

The NExT-LF workflow encompasses the following sequence:

1. Acquisition and preprocessing:
   • Collect multi-channel output-only vibration data $y(t)\in\mathbb R^{p}$ under ambient or operational excitation.
   • Detrend, segment, and window the data (e.g., Hann window, overlapping windows).
   • Apply band-pass filtering to suppress drift and high-frequency noise (Dessena et al., 13 Jan 2026).
2. Impulse-response estimation (NExT stage):
   • For each pair (or with respect to a selected reference channel $r$), compute cross-correlations:

   $$\widehat R_{ij}(\tau) = \frac{1}{T} \int_{0}^{T} y_i(t)\, y_j(t+\tau)\, dt$$

   over a lag range $\tau \in [0, \tau_{\max}]$. - Assemble IRFs as lagged vector (or matrix) time series.

3. Frequency-domain transformation:

   • For each synthetic IRF $H_r(\tau)$, compute the one-sided Fourier transform:

   $$F_r(j\omega_k) = \int_{-\tau_{\max}}^{+\tau_{\max}} H_r(\tau) e^{-j \omega_k \tau} d\tau$$

• In practice, fast Fourier transform (FFT) is applied, retaining positive frequencies.

1. Loewner matrix assembly:

   • Select "right" ($\{s_i\}$) and "left" ($\{\widehat s_j\}$) interpolation frequency sets within the mode-containing band.
   • Construct Loewner matrix $\mathcal L$ and shifted Loewner matrix $\mathcal L_\sigma$:

   $$(\mathcal L)_{j,i} = \frac{\widehat f_j - f_i}{\widehat s_j - s_i}, \quad (\mathcal L_\sigma)_{j,i} = \frac{\widehat s_j \widehat f_j - s_i f_i}{\widehat s_j - s_i}$$

2. Model order selection and reduction:

   • Form the Loewner pencil $M(\zeta) = \zeta \mathcal L - \mathcal L_{\sigma}$.
   • Compute the singular value decomposition (SVD), and truncate to the dominant $r$ components based on singular value decay.
   • Extract reduced realization matrices:

   $$E_r = -Y^* \mathcal L X,\; A_r = -Y^* \mathcal L_\sigma X,\; B_r = Y^* V,\; C_r = W X$$

   (using SVD factors $Y, X$; $V, W$ contain frequency response data).

3. Modal parameter extraction:

   • Solve the generalized eigenvalue problem $A_r v = \lambda E_r v$.
   • Compute modal frequencies $\omega_i$, damping ratios $\zeta_i$, and mode shapes $\phi_i$ from $C_r v_i$.
   • Use stabilization diagrams, screening criteria (e.g., frequency and damping intervals), and mode-shape MAC to select physically meaningful modes (Dessena et al., 2024, Dessena et al., 13 Jan 2026).

3. Algorithmic Summary and Implementation

A representative high-level pseudocode is as follows (Dessena et al., 2024):

Input: y(t) ∈ ℝ^{p×T} // p output channels, T samples r // chosen reference channel τ_max // maximum lag for correlation {s_i}, {ŝ_j} // interpolation frequencies 1. For each i=1…p: R_{r,i}(τ) <- (1/T) ∫_0^T y_r(t)·y_i(t+τ) dt, τ∈[0,τ_max] 2. H_r(τ) <- [R_{r,1}(τ),...,R_{r,p}(τ)] F_r(ω) <- FFT{ H_r(τ) } 3. For ℓ=1…k: f_i <- F_r(s_i) For m=1…q: f̂_j <- F_r(ŝ_j) For j=1…q, i=1…k: L_{j,i} ← (f̂_j − f_i)/(ŝ_j − s_i) Lσ_{j,i} ← (ŝ_j f̂_j − s_i f_i)/(ŝ_j − s_i) 4. [Y,Σ,X] <- svd( Lσ − ζ·L ) // pick ζ, truncate to r largest Σ 5. E_r ← −Y^*·L·X A_r ← −Y^*·Lσ·X B_r ← Y^*·[f̂_1;…;f̂_q] C_r ← [f_1 … f_k]·X 6. Solve A_r v = λ E_r v ω_i ← |Im(λ_i)|, ζ_i ← −Re(λ_i)/|λ_i|, ϕ_i ← C_r v_i Output: {ω_i, ζ_i, ϕ_i}_{i=1}^r

Preprocessing steps (detrending, windowing, filtering) and detailed workflow, including candidate model order ranges and screening criteria, are provided in specific experimental studies (Dessena et al., 13 Jan 2026).

4. Validation and Performance in Numerical and Experimental Studies

Validation of NExT-LF encompasses both simulated (Euler-Bernoulli beam) and experimental campaigns:

• In the numerical beam study (Dessena et al., 2024), NExT-LF consistently estimates all eight modes up to 1% additive Gaussian noise, achieving frequency errors below 0.1% and mode-shape MAC scores above 0.9.
• Benchmarking against NExT-ERA demonstrated that the latter fails to identify higher modes and exhibits MAC as low as 0.65 at merely 0.1% noise, whereas NExT-LF exhibits markedly superior robustness.
• In the Sheraton Universal Hotel experiment (Dessena et al., 2024), NExT-LF extracted stable low-frequency modes and produced fewer spurious estimates than NExT-ERA. This represents the first OMA application to this structure.

In propeller-driven vibration testing (PVT) of an aluminum spar with seven accelerometers under various propeller excitation regimes (Dessena et al., 13 Jan 2026):

• Dominant resonance frequencies (2.5 Hz, 13.4 Hz, 24.1 Hz) remained detectable under both baseline and propeller excitation.
• The first two modes showed excellent repeatability (MAC > 0.99, frequency deviation < 1%), while the third mode had increased scatter (frequency down-shift of −5.23%, MAC = 0.827), consistent with torsion coupling and sweep non-stationarity.
• NExT-LF remained effective despite non-ideal excitation and highlighted capacity to resolve coupling effects in higher order dynamics.

5. Comparative Assessment and Limitations

The principal strengths of NExT-LF, drawn directly from experimental findings and methodological analysis (Dessena et al., 2024, Dessena et al., 13 Jan 2026), are summarized below:

Feature	NExT-LF Performance	Note
Noise robustness	Consistent identification up to high noise levels	Outperforms NExT-ERA, SSI in noisy data
Output-only capability	No external input or force measurement required	Enables PVT, OMA in inaccessible systems
Mode stability	Delivers fewer spurious modes, robust low-frequency poles	Mode tracking via stabilization diagrams
Setup efficiency	Rapid deployment (e.g., single motor-propeller in PVT)	No need for shakers, force transducers

Challenges include sensitivity of higher modes to non-ideal, narrowband, or non-stationary excitation and the masking of structural modes at specific harmonics (low-throttle PVT); mitigation strategies involve excitation sweeping, automated scheduling, and pre-test coupling modeling (Dessena et al., 13 Jan 2026).

6. Future Directions

Several research directions are motivated by these studies:

• Automated Excitation Control: Feedback-controlled RPM scheduling and systematic sweep protocols to ensure broadband excitation and reduce mode masking.
• Coupling-Aware Test Planning: Pre-test modeling to quantify and mitigate propeller-induced torque and gyroscopic coupling, thus improving high-order mode identification.
• Benchmarking: Extension to full-scale aircraft structures and comparison with Ground Vibration Testing (GVT) and in-flight OMA, employing standardized MAC-based validation (Dessena et al., 13 Jan 2026).
• Algorithmic Enhancements: Further tuning of Loewner-based screening criteria and integration with other OMA modalities.

A plausible implication is that adoption of NExT-LF may further lower the barrier for operational modal analysis of complex systems, particularly in aerospace and civil engineering, where ambient or operational excitation is intrinsic and forced-input testing is impractical.

7. Impact and Context within Modal Analysis

NExT-LF represents a significant methodological advancement in output-only modal parameter extraction. It preserves the physical interpretability and practical convenience of correlation-based OMA while introducing noise-robust, rank-revealing model realization via tangential interpolation. By consistently delivering stable and accurate modal estimates across a wide range of scenarios and excitation environments, the method offers a viable replacement or complement to classical approaches such as NExT-ERA and covariance-driven SSI, especially under challenging noise or operational constraints (Dessena et al., 2024, Dessena et al., 13 Jan 2026).