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arxiv: 1907.08254 · v1 · pith:YGMWLAKPnew · submitted 2019-07-18 · 📡 eess.SY · cs.SY

Wave Excitation Force Estimation of Wave Energy Floats Using Extended Kalman Filters

Andrew F. Davis , Brian C. Fabien This is my paper

Pith reviewed 2026-05-24 19:30 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords wave energy converterextended kalman filterexcitation force estimationheaving floatrandom walkharmonic oscillatorwave tank datasensor-based estimation
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The pith

Two Extended Kalman Filter formulations estimate wave excitation force on floats from position and velocity data alone.

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

The paper shows that Extended Kalman Filters can recover the unseen wave force acting on a heaving wave-energy float by treating it either as a random-walk state or as the output of a harmonic oscillator. Both versions run on ordinary sensor readings such as displacement and velocity. The methods are checked on wave-tank records that cover several different wave conditions, and the paper also examines how the estimates change when the internal model, noise level, or sampling rate is altered. If the estimates remain usable, control algorithms that need the force value can operate without a dedicated force sensor.

Core claim

Two different estimation methods using a nonlinear Extended Kalman Filter are tested on experimental wave tank data for a heaving semi-submerged float. The first method relies on directly including the excitation force as a state in the first order dynamics, which allows the random walk of the Kalman filter to identify an estimate of the excitation force. The second method models the wave excitation force as a harmonic oscillator comprised of sinusoidal components. Both methods are evaluated for a variety of incident waves and additional sensitivity analyses are performed to investigate the susceptibility of these estimation methods to changes in the model, measurement noise, and sampling.

What carries the argument

Extended Kalman Filter whose state vector either augments the force as a random-walk variable or augments it as the output of a harmonic oscillator driven by sinusoids.

If this is right

  • Advanced control strategies that require the wave excitation force as input can be implemented using only standard motion sensors.
  • The same estimation approach works across a range of incident wave frequencies and amplitudes in controlled tank conditions.
  • Sensitivity tests quantify how much the estimates degrade when the float model, sensor noise, or data rate changes.

Where Pith is reading between the lines

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

  • The approach could be combined with existing WEC controllers to close the loop without adding hardware.
  • Field deployments might need only modest extra computation if the filter runs on the same embedded processor already used for motion sensing.
  • Similar state-augmentation tricks might apply to estimating other unmeasured loads, such as current forces on tidal devices.

Load-bearing premise

The chosen dynamic model of the float together with the random-walk or harmonic representation of the excitation force is accurate enough for the filter to produce reliable estimates from the available measurements.

What would settle it

Direct comparison of the filter outputs against an independent measurement of the actual excitation force in the same wave-tank runs would show large, persistent errors across the tested wave conditions.

Figures

Figures reproduced from arXiv: 1907.08254 by Andrew F. Davis, Brian C. Fabien.

Figure 1
Figure 1. Figure 1: The wave spectrum of run 17 generated by the Bretschneider Spectrum. Red circles [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time domain plot of estimated excitation force for a sixty second window using run [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time domain plot of estimated water velocity for a sixty second window using run [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The comparison the direct estimation method and five variations of the distur [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The comparison the direct estimation method and five variations of the disturbance [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The excitation force estimation goodness of fit of each run as a function of the peak [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The excitation force estimation goodness of fit of each run as a function of the peak [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The excitation force estimation goodness of fit of each run as a function of the [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The excitation force estimation NMSE using the direct estimation method of each [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The excitation force estimation NMSE using the third order disturbance estimation [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The excitation force estimation NMSE using the direct estimation method of each [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The excitation force estimation NMSE using the disturbance estimation method [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The excitation force estimation NMSE using the direct estimation method of each [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The excitation force estimation NMSE using the disturbance estimation method [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
read the original abstract

In many advanced control strategies the wave excitation force is key to determining the control input. However, it is often difficult to measure the excitation force on a Wave Energy Converter (WEC). The use of Kalman filters to estimate the wave excitation force based on readily available measurement data can potentially fill the gap between the development of WEC control strategies and the data that is available. Two different estimation methods using an nonlinear Extended Kalman Filter are tested on experimental wave tank data for a heaving semi-submerged float. The first method relies on directly including the excitation force as a state in the first order dynamics---which allows the "random walk" of the Kalman filter to identify an estimate of the excitation force. The second method of estimation involves modeling the wave excitation force as a harmonic oscillator comprised of sinusoidal components. Both methods are evaluated for a variety of incident waves and additional sensitivity analyses are performed to investigate the susceptibility of these estimation methods to changes in the model, measurement noise, and sampling rate.

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

3 major / 2 minor

Summary. The manuscript proposes and experimentally tests two Extended Kalman Filter formulations for estimating wave excitation force on a heaving semi-submerged WEC float from position and velocity measurements alone. The first augments the first-order hydrodynamic model with the excitation force as a random-walk state; the second represents the force as a sum of harmonic oscillators. Both are evaluated on wave-tank data across multiple regular and irregular wave conditions, with additional sensitivity tests on model parameters, measurement noise, and sampling rate.

Significance. If the estimates are shown to be reliable, the work would support advanced WEC control strategies that require real-time excitation-force information without dedicated force sensors. The use of physical tank experiments across varied wave conditions and the inclusion of sensitivity analyses are positive features that strengthen the practical relevance of the claim.

major comments (3)
  1. [Results section] Results section (and associated figures): quantitative performance metrics such as RMSE, bias, or correlation between estimated and reference excitation forces are not reported. Without these, the claim that the filters 'can estimate' the force rests on qualitative visual agreement whose strength cannot be assessed.
  2. [Model description] Model description (likely §2–3): the full state-space equations, including numerical values or identification method for added mass, radiation damping, and hydrostatic restoring coefficients, are not supplied. This prevents independent assessment of whether the assumed first-order dynamics are sufficiently accurate for the EKF to recover unbiased force estimates.
  3. [Validation approach] Validation approach: because excitation force cannot be measured directly, the experimental results test consistency between the filter output and the assumed hydrodynamic model rather than absolute accuracy. The manuscript should explicitly discuss this limitation and any steps taken to mitigate model mismatch (e.g., viscous drag or higher-order wave effects).
minor comments (2)
  1. Notation for the two EKF variants should be made consistent across text and figures to avoid reader confusion between the random-walk and harmonic formulations.
  2. Figure captions should state the exact wave conditions (height, period, regular/irregular) and sampling rate used in each panel.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Results section] Results section (and associated figures): quantitative performance metrics such as RMSE, bias, or correlation between estimated and reference excitation forces are not reported. Without these, the claim that the filters 'can estimate' the force rests on qualitative visual agreement whose strength cannot be assessed.

    Authors: We agree that quantitative metrics strengthen the evaluation. The revised manuscript will include RMSE, bias, and correlation coefficients between estimated and reference excitation forces for all regular and irregular wave cases, added to the Results section and figures/tables. revision: yes

  2. Referee: [Model description] Model description (likely §2–3): the full state-space equations, including numerical values or identification method for added mass, radiation damping, and hydrostatic restoring coefficients, are not supplied. This prevents independent assessment of whether the assumed first-order dynamics are sufficiently accurate for the EKF to recover unbiased force estimates.

    Authors: We acknowledge the need for full reproducibility. The revised manuscript will supply the complete state-space equations along with the numerical values and identification methods (from potential flow analysis or tank tests) for added mass, radiation damping, and hydrostatic restoring coefficients. revision: yes

  3. Referee: [Validation approach] Validation approach: because excitation force cannot be measured directly, the experimental results test consistency between the filter output and the assumed hydrodynamic model rather than absolute accuracy. The manuscript should explicitly discuss this limitation and any steps taken to mitigate model mismatch (e.g., viscous drag or higher-order wave effects).

    Authors: This limitation is inherent and valid. The revised manuscript will add an explicit discussion of the model-consistency nature of the validation, potential mismatches (viscous drag, higher-order effects), and mitigation approaches such as testing across varied wave conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: standard EKF estimation validated on independent experimental data

full rationale

The paper applies two standard EKF formulations (random-walk state augmentation and harmonic oscillator model) to estimate excitation force from heave position/velocity measurements using a first-order hydrodynamic model. Performance is assessed directly against experimental tank data for multiple wave conditions, with sensitivity checks on model parameters, noise, and sampling. No step reduces a reported estimate or prediction to a fitted input by construction, no self-citation is load-bearing for the central claim, and no uniqueness theorem or ansatz is imported from prior author work. The derivation chain is self-contained against external measurements.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper rests on standard EKF assumptions plus two domain-specific models for the unmeasured force; no new entities are introduced.

free parameters (1)
  • Process-noise covariance for excitation-force state
    Tuned to allow the random-walk model to track the unknown force; value not reported in abstract.
axioms (2)
  • domain assumption The float dynamics can be adequately captured by a first-order model driven by the excitation force.
    Invoked when the force is appended as a state in the first method.
  • domain assumption Wave excitation force can be represented as a finite sum of sinusoids at known frequencies.
    Basis of the second estimation method.

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Reference graph

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