⚡ electrical engineering

Physically interpretable modes of a moored wave energy converter via dynamic mode decomposition with exogenous input

This paper presents a physically interpretable, low-order linear model for a moored wave energy converter by applying Dynamic Mode Decomposition with control (DMDc) to experimental data, successfully capturing hydrodynamic memory and identifying distinct physical mechanisms while revealing the quantitative signature of catenary mooring nonlinearity through a divergence between time-domain and spectral performance metrics.

Original authors: Bruno Paduano, Edoardo Pasta, Fabio Carapellese, Guglielmo Papini, Mattia Glorioso, Nicolás Faedo, Giuliana Mattiazzo, Pedro Lomonaco

Published 2026-08-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bruno Paduano, Edoardo Pasta, Fabio Carapellese, Guglielmo Papini, Mattia Glorioso, Nicolás Faedo, Giuliana Mattiazzo, Pedro Lomonaco

Original paper licensed under CC BY 4.0 (https://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 the ocean not as a chaotic, crashing wall of water, but as a giant, invisible orchestra. Every wave is a musician playing a note, and the floating machines we build to harvest their energy are the instruments trying to catch the melody. For decades, scientists have struggled to write the sheet music for these floating machines. They have two main ways to do this: they can try to calculate every single drop of water using super-computers (which takes forever and is too slow for real-time use), or they can use "black box" artificial intelligence. The AI is great at guessing what happens next, but it's like a magician who pulls a rabbit out of a hat but won't tell you how the trick works. If the ocean changes its tune slightly, the AI might get confused, and because it's a black box, engineers can't easily see why.

To fix this, researchers are looking for a middle ground: a method that is as smart as the AI but as transparent as a clear window. This is where a technique called "Dynamic Mode Decomposition" (DMD) comes in. Think of DMD as a way to take a messy, complex video of a dancing object and break it down into its fundamental "dance moves." Instead of seeing a blur of motion, DMD identifies the specific rhythms, the specific wobbles, and the specific spins that make up the dance. It turns a chaotic storm into a simple list of musical notes. The big question is: Can we use this "dance move" finder to understand a floating wave energy converter that is tied to the ocean floor with ropes, even when the ropes go slack and tight again?

This paper by Bruno Paduano and his team at Politecnico di Torino and Oregon State University says "yes, but with a catch." They took a real, physical model of a wave energy device called SWINGO—a floating buoy with a spinning pendulum inside—and tested it in a giant wave tank at a 1:20 scale. They didn't just guess; they measured everything. They recorded how the waves hit the boat, how the boat moved, how the internal pendulum swung, and how the ropes (mooring lines) pulled on the boat.

The team used a special version of the "dance move" finder called DMD with control (DMDc). This version is smart enough to know that the ocean is pushing the boat (the input) and that the boat is reacting (the output). To make sure the math worked perfectly, they used a clever trick called "Hankel embedding." Imagine trying to predict the next step of a dancer, but you only have a blurry photo of their feet. By stacking several photos on top of each other to create a short video clip, you can see the momentum and the history of the movement. That's what they did with the data, allowing them to account for the fact that water has "memory" (it doesn't stop moving instantly) and that their measurements of the waves and the boat might have been slightly out of sync.

The result was a surprisingly simple, low-order model that could predict the boat's behavior with high accuracy. When they broke down the model's "dance moves," they found six distinct clusters, each representing a different physical reality. Some moves were the boat rocking gently with the waves (the main resonance), while others were the boat bobbing up and down, or the internal pendulum swinging. They even found moves that represented the ropes pulling tight.

However, the paper reveals a fascinating limitation. When they tested the model on new, unseen wave conditions (simulating different stormy days), it worked beautifully for the boat's body and the internal engine. But for the ropes, the model was a bit of a split personality. If you looked at the timing of the rope's pull, the model was terrible; it couldn't predict exactly when the rope would go slack or snap tight. But if you looked at the energy or the "vibe" of the rope's movement (its frequency spectrum), the model was actually quite good.

The authors explain this split personality with a vivid physical insight: the ropes have a "unilateral" nature. They can pull, but they can't push. When the boat moves up, the rope goes slack (no force), and when it moves down, the rope goes tight (huge force). This "slack/taut" switching is a non-linear jump that a simple linear model can't perfectly mimic in real-time. The model captures the rhythm of the rope's tension but misses the instant it snaps tight.

In short, the paper proves that you can build a clear, understandable, and mathematically simple model of a complex floating energy device using real-world data. It successfully identifies the specific "dance moves" of the system, separating the smooth, resonant rocking of the hull from the jerky, non-linear behavior of the mooring lines. While it doesn't perfectly predict the exact second a rope goes slack, it captures the overall energy and structure of the system so well that it could be a powerful tool for designing better wave energy converters in the future, offering a transparent alternative to the opaque black boxes of deep learning.

Technical Summary: Physically Interpretable Modes of a Moored Wave Energy Converter via Dynamic Mode Decomposition with Exogenous Input

Problem Statement
The development of wave energy converters (WECs) is hindered by the complexity of offshore fluid-structure interactions. While high-fidelity computational fluid dynamics (CFD) is accurate, it is computationally prohibitive for control and design iteration. Conversely, mid-fidelity linear potential flow models often fail to capture nonlinear effects (drag, mooring constraints) and experimental responses accurately. Data-driven approaches, particularly deep learning, offer high predictive power but suffer from a lack of physical interpretability, stability guarantees, and generalization. There is a need for data-driven identification methods that balance predictive accuracy with structural transparency, allowing engineers to associate identified modes with specific physical mechanisms (e.g., hydrodynamic resonance, mooring coupling) rather than treating the model as a "black box."

Methodology
The authors propose a framework based on Dynamic Mode Decomposition with control (DMDc), a Koopman-based linear identification method. The study utilizes experimental data from a 1:20 Froude-scaled prototype of the SWINGO wave energy converter (an inertial reaction mass device with a gyropendulum) tested in a wave basin at Oregon State University.

Key methodological components include:

Data Acquisition: The state vector ( $x$ ) is constructed exclusively from sensors deployable in offshore conditions: an Inertial Measurement Unit (IMU) for hull accelerations and pitch, encoders for the Power-Take-Off (PTO) state, and in-line load cells for mooring line tensions. The wave excitation force ( $u$ ) is measured directly via a dedicated "clamped-hull" campaign using a six-axis load cell, avoiding the need for reconstruction via boundary element methods.
Hankel Time-Delay Embedding: To address three specific challenges—non-Markovian state (missing position/velocity from IMU data), hydrodynamic memory (radiation forces), and phase mismatch between the clamped and floating test campaigns—the authors employ a Hankel embedding. Both state and input vectors are stacked over time delays ( $L_x$ and $L_u$ ) to recover the underlying attractor dynamics and allow the identified operator to absorb relative phase shifts.
Modal Analysis and Clustering: The identified DMDc operator is diagonalized to extract eigenvalues and eigenvectors. Modes are filtered based on physical relevance (magnitude and frequency ranges) and then clustered using the Modal Assurance Criterion (MAC) combined with a frequency-similarity index. This groups fragmented numerical modes into distinct physical clusters.
Validation Metrics: The model is validated on unseen JONSWAP sea states using two complementary metrics:
1. Time-domain fit ( $fit_i$ ): Measures instantaneous waveform agreement.
2. Logarithmic spectral Pearson correlation ( $\rho_{log, i}$ ): Measures agreement in spectral structure, independent of phase.

Key Results

Model Stability and Order: The identified operator is stable (all eigenvalues within the unit circle). A truncation retaining 90% of singular value energy yielded 17 components, which were filtered down to 8 physically meaningful modes.
Physical Interpretation of Modes: The clustering procedure successfully associated identified modes with distinct physical mechanisms:
- Cluster 1 & 3: Sub-resonance regimes dominated by hydrostatic stiffness and radiation damping (stiffness-dominated behavior).
- Cluster 2: The principal hydrodynamic resonance of the floater, characterized by low damping and strong coupling between pitch, surge acceleration, and the internal gyropendulum.
- Clusters 4–6: Low-frequency and high-frequency couplings involving mooring forces and surge/heave motions.
Predictive Performance:
- Body and PTO Channels: The model demonstrated strong performance on the hull motion (pitch, heave) and PTO state, with high time-domain fit and spectral correlation. Pitch showed the highest fit, attributed to its role as the most observable degree of freedom due to strong cross-channel coupling.
- Mooring Channels: A distinct asymmetry was observed. While the spectral correlation remained substantial (0.49–0.75), the time-domain fit was poor or negative.
Generalization: A model identified on a single band-limited white-noise training set successfully generalized to three unseen JONSWAP sea states with varying significant wave heights and peak periods, indicating the identified modal content represents the device dynamics rather than the specific training signal.

Significance and Claims
The paper claims to provide the first application of DMDc to a moored floating wave energy converter on experimental data with an explicit focus on the physical interpretation of the identified modes. The primary contributions are:

Structural Transparency: The framework offers a structurally transparent alternative to black-box identification, linking mathematical modes to specific hydrodynamic and mechanical mechanisms (e.g., distinguishing between stiffness-dominated and resonance-dominated regimes).
Diagnostic Validation Protocol: The authors demonstrate that the divergence between time-domain fit and spectral correlation serves as a quantitative signature of specific nonlinearities. Specifically, the asymmetry observed in the mooring channels is interpreted as the signature of the unilateral slack/taut nonlinearity of the catenary mooring. The linear model captures the spectral structure of the constraint but fails to reproduce the instantaneous transitions between slack and taut states.
Feasibility of Offshore-Deployable Sensors: The study validates that a robust, interpretable linear model can be identified using only sensors realistically available in offshore operations (IMU, encoders, load cells), provided the wave excitation force is measured or estimated.

Limitations and Future Work
The authors acknowledge that the linear model cannot represent the slack/taut transitions of the mooring or the friction nonlinearity at the pendulum hinge. They suggest that future work should explore Extended DMD (EDMD) with nonlinear Koopman observables (e.g., encoding the unilateral nature of the mooring constraint) to address these limitations. Additionally, the operational challenge of reconstructing wave excitation forces from wave elevation (rather than direct measurement) remains an unassessed source of error for real-world deployment.

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