the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Aug 2026
Adaptive multi-mode unity-magnitude input shaping with unknown-input state estimation for floating offshore wind turbines
Abstract. Every operational transition of a floating offshore wind turbine (FOWT) – a start-up, a curtailment change, a grid- support set-point, a change of operating region – moves a control set-point and thereby injects energy into the lightly damped structural modes. Input shaping removes that command-induced excitation by construction, but it is a feedforward mechanism whose guarantees are exact only for a known modal model. This paper gives a complete and verifiable treatment of an adaptive multi-mode unity-magnitude (UM) shaping layer supported by an unknown-input observer, on a coupled control-oriented FOWT model in which the structural motion feeds back into the rotor aerodynamics. Four results are proved: convolving single-mode UM shapers cancels every targeted mode exactly; a non-asymptotic bound limits the residual under amplitude and timing perturbation; the same bound maps a modal-frequency estimation error into a residual bound; and rescaling the impulse instants by the estimated damped frequency preserves cancellation exactly at fixed damping. The estimation layer is placed on a firm footing by an observability analysis and a full-order unknown-input observer with proved decoupled convergence. The numerical study — run on the coupled model, with a second-order pitch actuator, rate and saturation limits, timing quantization, and independent per-mode drift — yields a design result that a decoupled model conceals: because collective pitch strongly damps the platform mode (ζ = 0.32), shaping it is counterproductive. Dropping it gives a shaper that is 9× shorter (0.86 s versus 7.86 s, 9 versus 27 impulses), reduces the command-induced tower damage-equivalent load by 51.4 % instead of 29.8 %, lowers rather than raises the peak tower deflection, and cuts the out-of-band spillover from 19.2 to 6.9. The recommendation is therefore to shape the lightly damped modes and let aerodynamics damp the platform. All model matrices, shaper instants, and observer gains are given so that the study can be reproduced.
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Status: open (until 14 Sep 2026)
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RC1: 'Comment on wes-2026-128', Anonymous Referee #1, 19 Aug 2026
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AC1: 'Reply on RC1', amina Mseddi, 23 Aug 2026
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We thank the referee for a review that was detailed, fair and, in several places, correct about things we had got wrong. Two points below changed the substance of the paper rather than its presentation, and we want to flag them at the outset.
While preparing the sensitivity study requested during review we discovered an error of physical attribution in our own text. The submitted version said that collective pitch control damps the platform mode. It does not: the state matrix used throughout is the open-loop plant and contains no pitch feedback. The damping comes from the dependence of rotor thrust on the relative wind seen by the rotor, acting on the platform through the square of the hub arm; it supplies 91 % of the platform-pitch damping term in the model, with hydrodynamic viscous damping supplying the rest. This is now stated correctly in Sect. 2.1, Sect. 5.1, the abstract and the conclusions.
RC1 – Comment 1 (verdict)
We have chosen the first option and adapted the manuscript. The revision changes what the paper is about at the level of emphasis: the question it now asks in the abstract and the introduction is which structural modes of a floating turbine are worth removing from its own commands, which is a wind-energy design question, and the shaping mathematics is presented as the tool that answers it.
The formal apparatus has been cut where it was not load-bearing and moved where it was. The Definition environments (coupled model, unity-magnitude shaper), the Lemma (residual vibration) and all five Remarks are gone as environments; their content is in the running text. Everything that remains numbered is used quantitatively later in the paper.
We have also added the engineering material whose absence made the paper feel abstract: a physical explanation of why the platform mode is damped and the tower is not (Sect. 2.1), a benchmark against the command shapes an engineer would otherwise use (Sect. 5.3), a study of the actuator duty the method demands (Sect. 5.4), a study of how far the design decision carries (Sects. 5.8–5.9), a check under turbulent wind and irregular waves (Sect. 5.10), and a section on what deployment on a real machine would require (Sect. 6).
Change made in the manuscript:
Structural revision throughout. Sixteen numbered environments reduced to seven (three Assumptions, three Theorems, one Proposition). All proofs moved to Appendix A. Five new subsections and two new sections added, all of them engineering rather than theory.
Location: Whole manuscript; in particular Sects. 2.1, 3.1–3.3, 5.3, 5.4, 5.8–5.12, 6, 7 and Appendix A.
RC1 – Comment 2 (abstract)
the new abstract opens on the engineering problem (a floating turbine changes set-point constantly, and each change rings a lightly damped structure), explains in one sentence what input shaping does in physical terms, states plainly that the study uses a reduced-order control-oriented model and is a proof of concept rather than turbine-level validation, and then gives the engineering finding — the platform-mode trade-off, with what it costs as well as what it buys — before the analytical results. The control-theory vocabulary that remains is limited to naming the tools.
Change made in the manuscript:
The abstract has been completely rewritten and is now three paragraphs of engineering narrative. It states the reduced-order nature of the study in its second paragraph (this also answers Referee 2, technical correction 8) and presents the platform-mode result as a conditional trade-off rather than a rule.
Location: Abstract, p. 1.
RC1 – Comment 3 (limitations in the Introduction)
Thank you. We have kept that habit and extended it. The introduction still states plainly that shaping is feedforward, cannot reject disturbances and is exact only for a known modal model; a new paragraph at the end of the introduction now also separates what is established from what we claim as new (Referee 2, comment 10), and the scope paragraph now says explicitly that design and evaluation share a model.
The limitations themselves have grown into their own Sect. 7, which is franker than the submitted version: it lists what was not done, in order of how much it would matter.
Change made in the manuscript:
Introduction extended with a contribution paragraph and a sharper scope paragraph; Sect. 7 rewritten and extended.
Location: Sect. 1, last three paragraphs; Sect. 7.
RC1 – Comment 4 (applicability to real FOWTs)
This was the most useful comment we received and it prompted most of the new work. Rather than answer it with a paragraph of assurances we have tried to answer it with numbers, and where we could not, to say so.
On constantly varying conditions, the answer has two parts. There is a favourable separation of time scales: a shaped maneuver in the preferred design lasts 0.86 s, while sea state and mean wind move the modes over tens of minutes, so freezing the sequence for one maneuver costs almost nothing, and adaptation needs only one number per mode. But the weak link is getting that number, and we now quantify it: Sect. 5.11 reports what two realistic identifiers achieve on simulated operating records with sensor noise. The tower frequency — the quantity the shaper actually needs — is recovered to better than 0.2 % from twenty minutes of ambient data and to 1–2 % from the response to a single commanded set-point change, which by Proposition 3.2 costs a few per cent of residual vibration. Damping is not recovered reliably, and the drivetrain torsion mode is not identified at all from the assumed sensor set. We report all three outcomes.
On varying operating point, Sect. 5.9 rebuilds the model at seven above-rated operating points using aerodynamic derivatives computed from the public IEA 15 MW rotor-performance tables. The platform-pitch damping ratio moves over 0.12–0.39 across that range, so the conditions do vary materially; the ordering of the two shaper designs does not change.
A new Sect. 6 collects all of this into a discussion of deployment: changing conditions, changing operating point, where the modal parameters come from, online versus between-maneuver adaptation, measurement quality, actuator duty and model fidelity. It states, for example, that a machine with duty-limited pitch bearings should use a ZV shaper rather than the unity-magnitude one.
Change made in the manuscript:
New Sect. 6 (“Applicability to a real floating turbine”), about two pages, with seven headed paragraphs. New numerical Sects. 5.9 (operating points), 5.10 (turbulent wind and irregular waves) and 5.11 (modal identification) supply the evidence it draws on. New Tables 6, 7 and Fig. 7.
Location: Sect. 6; Sects. 5.9–5.11; Tables 6–7; Fig. 7.
RC1 – Comment 5 (presentation style)
We have cut this back substantially.
. Every proof is now in Appendix A.
We kept the three Assumptions as numbered items deliberately, because each is a real constraint on an implementation that is referred to repeatedly, and a reader checking whether the method applies to their machine needs to find them.
Change made in the manuscript:
Environments reduced from sixteen to seven; all proofs relocated to Appendix A; Sect. 3 reorganised into three short subsections, each with the result first and its engineering meaning immediately after.
Location: Sects. 2.1, 2.2, 3.1–3.3, 4.1; Appendix A.
RC1 – Comment 6 (too little explanatory text)
Each result is now followed immediately by a paragraph that answers the question “what does this mean for a floating wind turbine?”.
For the multi-mode cancellation result, the point made is arithmetic and practical: convolving N shapers gives 3ᴺ impulses over the sum of the individual durations, so the slowest mode dominates the command length — and on a floater the platform mode has a period two orders of magnitude longer than the drivetrain mode. That paragraph is where the whole design decision of the paper comes from.
For the perturbation bound, we now say what the bound is for (turning an estimator accuracy specification into a residual guarantee), what its weighting factor implies (late impulses acting on fast modes are the expensive ones, which is why command-grid quantization hurts the drivetrain mode and not the platform mode), and how it should be read (a certificate, not a predictor — it is 1.9 to 21 times conservative).
For the rescaling result, we explain that it reduces adaptation to a stored table divided by one number, with no online solve that could fail while the turbine waits for a command, and — connecting to the new identification study — that the quantity it needs accurately is exactly the quantity operating data give accurately.
The model section now also explains the physics rather than only presenting matrices: why the relative-wind term becomes a damping moment on the platform, why the same mechanism barely damps the tower, and what that asymmetry implies.
Change made in the manuscript:
Interpretation paragraphs added after Theorem 3.1, Proposition 3.2 and Theorem 3.3; a new physical-mechanism paragraph added to Sect. 2.1; a paragraph on the practical signature of a unity-magnitude command added to Sect. 2.2; short justifications added after each of the three Assumptions.
Location: Sects. 2.1, 2.2, 2.3, 3.1, 3.2, 3.3.
RC1 – Comment 7 (insights hidden in remarks)
The observation that the bound is conservative, the observation that late impulses on fast modes dominate the quantization error, the fact that the construction preserves the commanded set-point, the statement that the rescaling result is exact only at fixed damping, and the observation that the unknown-input observer is a standard construction rather than a new one, were all in Remarks. All five are now in the main text at the point where the reader needs them.
No Remark environments remain in the paper.
Change made in the manuscript:
All five Remark environments removed and their content integrated into Sects. 3.1–3.3 and 4.1.
Location: Sects. 3.1–3.3, 4.1.
RC1 – Comment 8 (balance of rigour and insight; move derivations)
Done, and the space freed has been used for engineering material rather than left empty.
All four proofs — multi-mode cancellation, the perturbation bound, the rescaling result and the unknown-input observer — are now in Appendix A, which opens by saying why they are there. Sect. 3 is correspondingly shorter and reads as three design statements with their consequences.
The room that this created, and more, has gone into Sects. 5.3, 5.4, 5.8–5.12 and Sect. 6, which are entirely engineering.
Change made in the manuscript:
New Appendix A collecting all proofs. Sect. 3 restructured. Sect. 5 extended from seven to thirteen subsections, all of the new ones numerical or practical.
Location: Appendix A; Sects. 3 and 5.
RC1 – Comment 9 (emphasise the platform-mode conclusion)
We have promoted it and, at the same time, made it more careful, because Referee 2 was right that as stated it rested on a single damping ratio in a single configuration. We think the combination is stronger than either alone.
It now appears in the third sentence of the abstract, in a dedicated paragraph of the introduction that explains the mechanism in physical terms, as its own numerical subsection (5.2), in a new sensitivity study that asks how far it carries (5.8), in a new operating-point study (5.9), in the turbulent-wind study (5.10) and as the central paragraph of the conclusions.
The message is now stated as a trade-off with both sides quantified: excluding the platform mode makes the command nine times shorter, cuts the delivered pitch travel and the out-of-band amplification to about a third, lowers rather than raises the peak tower deflection and improves the tower-base load reduction from 34 % to 48 %; what it gives up is a platform-pitch load reduction that is 24.9 % in a deterministic maneuver, that falls monotonically as the platform mode becomes better damped, and that in twelve seeds of a moderate sea state is not statistically resolvable at all. We think stating the cost makes the recommendation more usable, not less.
Change made in the manuscript:
Platform-mode message moved to the front of the abstract and given a dedicated introduction paragraph; Sect. 5.2 rewritten as an explicit trade-off; new Sects. 5.8 and 5.9 with Fig. 7 and Table 6; new Sect. 5.10; conclusions restructured around it.
Location: Abstract; Sect. 1, paragraph 4; Sect. 5.2; Sects. 5.8–5.10; Fig. 7; Tables 6–7; Sect. 8.
Citation: https://doi.org/10.5194/wes-2026-128-AC1
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AC1: 'Reply on RC1', amina Mseddi, 23 Aug 2026
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RC2: 'Comment on wes-2026-128', Anonymous Referee #2, 21 Aug 2026
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1. General commentsThe manuscript addresses a relevant problem for floating offshore wind turbines: structural excitation induced by commanded set-point changes. The topic fits well within the scope of Wind Energy Science. The paper is clearly structured, mathematically rigorous in several parts, and explicit about its assumptions and limitations. The integration of multi-mode unity-magnitude input shaping, uncertainty bounds, an unknown-input observer, actuator constraints, and a coupled FOWT model is potentially useful.However, the study remains primarily an analytical and reduced-order design demonstration. The main engineering conclusion, particularly the recommendation to exclude the platform mode from the shaper, is derived from a single seven-state control-oriented model whose parameters are representative rather than identified. Since the same model is used for both design and evaluation, the practical generality of the conclusions has not yet been sufficiently demonstrated.I therefore recommend major revision. The manuscript could become a valuable contribution if the authors strengthen or appropriately limit the validation claims, clarify the modal-identification strategy underlying the adaptive implementation, resolve the unknown-input observer formulation, and improve reproducibility.2. Specific comments1) The validation should be strengthened, or the claims should be softened. The reduced-order model is useful for demonstrating the proposed mechanisms, but it does not provide independent turbine-level validation. The authors should either add tests using independent linearizations or a higher-fidelity aero-hydro-servo-elastic model, or clearly frame the results as a proof-of-concept study.2) The adaptive part is insufficiently specified. Algorithm 1 assumes estimates of modal frequency and damping, but no identification method is presented or validated. The unknown-input observer estimates states, but it is not shown to provide the modal parameters required by the shaper. The authors should explain how these quantities would be obtained in practice, whether online or between maneuvers, and how noise, closed-loop operation, estimation uncertainty, and observation-window length affect the estimates.Relevant closed-loop system-identification literature could also be discussed. One useful methodological example is:Pasquali C., Serafini J., Gennaretti M., Leibbrandt R., "Reduced-order helicopter model identification from closed-loop data", Aerospace Science and Technology, 2024. DOI: 10.1016/j.ast.2024.109419Although this study concerns rotorcraft, it addresses the related problem of reconstructing open-loop reduced-order dynamics from closed-loop data with correlated inputs. The authors are not expected to adopt this specific approach, but discussing how the required modal quantities could realistically be identified from operational data would strengthen the adaptive framework.3) The unknown-input formulation should be clarified. The model defines two disturbances, wind speed and wave moment, whereas Appendix C appears to design the observer only for the wave-moment channel. The treatment of wind-speed variation and the corresponding rank conditions should therefore be stated consistently.4) The conclusion that the platform-pitch mode should be excluded depends strongly on the modeled damping ratio of about 0.32. This should be tested across additional operating points, controller settings, and plausible variations of platform damping and aerodynamic derivatives. Otherwise, the recommendation should be presented as specific to the investigated configuration.5) The comparison set is limited. In addition to the unshaped, three-mode, and two-mode UM cases, the authors should include at least one practical smooth or robust command-shaping baseline, such as an S-curve, low-pass-filtered command, ZV/ZVD shaper, or similar method. This would provide a more meaningful benchmark, especially given the high ideal pitch-rate demand of the UM sequence.6) The actuator analysis is useful but should be more transparent. The authors should show representative commanded and actuator-output pitch trajectories, indicate whether rate or travel limits are active, and clarify whether actuator dynamics are considered during shaper design or only afterwards.7) The DEL reductions should be interpreted cautiously. They are obtained from a short deterministic maneuver sequence and should be described as command-induced short-term load indicators rather than general fatigue reductions. Validation under turbulent wind, irregular waves, or multiple operating points would strengthen this part.8) The practical advantage of the unknown-input observer should be better justified. Since the manuscript notes that a standard observer with the same poles performs similarly in the present numerical case, the authors should explain more clearly when the unknown-input formulation becomes necessary or advantageous.9) Reproducibility should be improved. The appendices provide matrices and gains, but the code and random seeds are only available on request. The authors should consider depositing the scripts, parameters, and data required to reproduce the figures and tables in a public repository.10) The novelty claims should distinguish more clearly between standard components and genuinely new contributions. The convolution-based multi-mode cancellation result and the unknown-input observer structure rely on established principles. The stronger novelty appears to lie in the integration, perturbation/rescaling analysis, and FOWT-specific design conclusions.3. Technical corrections1) Define clearly how “residual cancellation with actuator” in Table 3 is computed.2) In Sect. 4.1, associate the reported transfer-function numerator coefficients explicitly with the corresponding input-output paths.3) Check the notation in Corollary 3.5, where the frequency error and timing error use similar symbols.4) Revise the repeated panel labels in Figure 5.5) Discuss practical issues associated with obtaining tower-top velocity by integrating accelerometer measurements, including bias and drift.6) Check the metadata of online-first references and complete them where possible.7) Define plotted quantities such as residual vibration, cancellation, and out-of-band spillover directly in the relevant captions.8) State more explicitly in the abstract that the numerical study is based on a reduced control-oriented model rather than a full aero-hydro-servo-elastic validation.9) Limit the broad conclusion about letting aerodynamics damp the platform to the tested configuration unless additional evidence is provided.Citation: https://doi.org/
10.5194/wes-2026-128-RC2 -
AC2: 'Reply on RC2', amina Mseddi, 25 Aug 2026
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We thank the referee for a review that was detailed, fair and, in several places, correct about things we had got wrong.
RC2 – Specific comment 1 (independent or higher-fidelity validation)
We have taken both routes. The second one turned out to be open to us after all, and it did not vindicate the paper: it reversed the ranking of the two shaper designs on the load channels. We report that outcome and have rewritten the affected claims.
On framing: the abstract now says in its second paragraph that the study uses a seven-state control-oriented model, that it is a design model and not an aero-hydro-servo-elastic one, and that the results are a proof of concept for the mechanism rather than turbine-level validation. Sect. 5 repeats this before any number is reported, and Sect. 7 states exactly what the one aero-hydro-servo-elastic check does and does not establish, and that no result of the design study should be read as if it were turbine-level validation.
On strengthening: we ran the higher-fidelity code. The unshaped, two-mode and three-mode commands were executed on the public OpenFAST model of the IEA-15-240-RWT-UMaineSemi system in OpenFAST v4.2.1, driven through the OpenFAST Simulink interface so that ServoDyn takes the collective-pitch command and the generator torque straight from Simulink (PCMode = 4, VSContrl = 4) — which is what allows an arbitrary shaped set-point to be commanded, the built-in pitch-maneuver facility being limited to ramps. The second-order pitch actuator and the twelfth-order CRONE rotor-speed regulator of the paper were discretized and inserted without re-tuning any coefficient, so the controller is the one whose margins Sect. 5.13 reports. The harness checks out: the regulator holds rated rotor speed to 0.024 % across the analysis window, the commanded torque never reaches the plausibility bound placed on it, the aerodynamic torque OpenFAST develops at the imposed trim pitch lands 3.1 % from the rotor-performance tables the trim was derived from, and the unshaped maneuver delivers 12.060 degrees of pitch travel against 12.061 on the design model.
The result did not confirm the paper. The cancellation mechanism transferred — the two-mode command removes 84.2 % of the tower fore–aft band energy and the three-mode command a further 63.2 % of the platform-pitch band — and so did the cost, at 1.87 and 5.64 times the unshaped pitch travel against 1.86 and 5.57 predicted, together with the predicted out-of-band spillover, the three-mode command amplifying the once-per-revolution band by a factor of 3.7. But the ranking reversed. The three-mode command reduces the tower-base fore–aft equivalent load by 20.0 % where the two-mode command reduces it by 2.3 %, reduces the platform channel by 22.2 % against 0.2 %, and lowers rather than raises the peak tower-top excursion. The cause is the load path: on the aeroelastic model the command-induced tower-base moment is dominated by platform motion, the band below 0.03 Hz carrying about 190 times the energy of the tower fore–aft band, whereas our proxy M_tb = k_t q_fa L_t is proportional to the tower modal deflection alone and the seven-state model has no surge degree of freedom. This is the referee’s objection, confirmed and quantified.
Alongside that, we also bring genuinely external information into the reduced-order model itself. Three elements of the revised study are independent of our own parameter choices. (i) The six aerodynamic derivatives are recomputed at seven above-rated operating points from the published rotor-performance tables of the IEA-15-240-RWT-UMaineSemi configuration, by differentiating the tabulated thrust and torque coefficient surfaces along the published steady pitch schedule; reconstructed thrust and torque match the tabulated values to 1–3 %. (ii) The wave forcing in the new stochastic study is the platform’s own pitch wave-excitation transfer function taken from its public hydrodynamic database, driven by a JONSWAP spectrum. (iii) The nominal hydrostatic pitch stiffness, the platform inertia and the representation of hydrodynamic damping were checked against the published model, and it is those checks that set the ranges of the new sweeps — for instance we found that the reference platform’s hydrodynamic damping is quadratic rather than linear, so a linear coefficient is a sea-state-dependent equivalent, and that our lumped pitch inertia is at the low end of what the reference system implies, which is why the sweep extends it to three times nominal.
What this does and does not establish: the parameter studies show that the conclusions are not artefacts of our particular aerodynamic derivatives or of one platform-damping value, and that the model’s hydrostatic stiffness is right to 1 %; the aero-hydro-servo-elastic cross-check shows that the shaping mechanism itself survives a change of structural model but that the load-channel ranking does not. None of it is turbine-level validation: the cross-check is one operating point, steady wind, still water, one maneuver, our own control layer, and no design-load-case coverage, so it establishes the direction of the discrepancy and not its magnitude. We say exactly this in Sect. 7 and in Appendix F.
Change made in the manuscript:
New Sects. 5.8, 5.9 and 5.10; new Table 6 (derivatives and modal properties at seven operating points); new Fig. 7; new Appendix E documenting how the external data were used. Proof-of-concept framing added to the abstract, to the opening of Sect. 5 and to Sect. 7. New Appendix F documenting the OpenFAST/Simulink coupling, the transplanted control layer, the settings, the band decomposition and the full comparison, with new Table F1 and Fig. F1. The reversal it uncovers is reported in the third paragraph of the abstract, at the end of Sect. 5.2, in the model-fidelity paragraph of Sect. 6, in Sect. 7 and in the conclusions.
Location: Abstract, paragraphs 2 and 3; Sect. 5 opening; Sect. 5.2; Sects. 5.8–5.10; Table 6; Fig. 7; Sect. 6 (model fidelity); Sect. 7; conclusions; Appendices E and F.
RC2 – Specific comment 2 (the adaptive part and modal identification)
This was a real gap and it has been filled in two places, one conceptual and one numerical.
The conceptual part is a new Sect. 4, which begins by separating three things that the submitted version ran together: state estimation, disturbance decoupling, and modal-parameter identification. We now state explicitly that the observer estimates states and decouples disturbances, that it does not estimate modal frequency or damping, and that no such claim is made — an observer built on a fixed A cannot detect that A has changed. Section 4.2 then states what is assumed in this paper (the shaper is synthesized from the exact eigenvalues of the model, so the parameters are known, not identified), describes the two routes by which they would be obtained on a machine, and states that only between-maneuver adaptation is claimed.
The numerical part is a new Sect. 5.11, which actually runs both routes on simulated records of exactly the four sensors of Assumption 3, sampled at 20 Hz, in the turbulent-wind and irregular-wave environment of Sect. 5.10, with white sensor noise at 0, 2 and 10 % of each channel’s standard deviation. The first route is ambient, output-only covariance-driven stochastic subspace identification with a stabilization diagram, run in two frequency bands; the second is a least-squares ARX fit from the known collective-pitch command over a window containing one to six set-point changes. Record length is swept from 10 to 40 minutes for the ambient method and from one to six maneuvers for the ARX method.
The results are mixed and we report them as such (Table 7, Fig. 8). The tower fore–aft frequency is recovered to better than 0.2 % by the ambient method at every record length and noise level tested, and to 0.7–2.1 % by the ARX method from a single maneuver. Damping is much worse: the ambient method reaches -2 % on the tower mode from a 40 min record but is in error by -89 % on the platform mode, and the ARX damping error reaches +86 % at 10 % noise. The platform mode is identified poorly by both, for physical reasons we now explain — a mode with ζ ≈ 0.32 gives a broad, low spectral peak that carries little information about its own damping, and the wave peak appears as a competing pole. The drivetrain torsion mode is not identified at all from this sensor set, because rotor-averaged wind and waves put almost no energy at 2.85 Hz.
What makes this usable rather than merely discouraging is that the quantity the shaper needs accurately is the frequency, by Theorem 3.2, and frequency is precisely what both identifiers deliver. Propagating the measured frequency errors through Proposition 3.1 (Fig. 8b) gives residual vibrations below 0.5 % for the ambient route and 1.7–3.4 % for the maneuver-based route. Damping enters only through the offline map φ(ζ), whose sensitivity the existing Monte-Carlo study already quantifies. We now say all of this explicitly, including the negative result about the drivetrain mode and what would be needed to fix it (a generator-side measurement, which the assumed sensor set does not include).
On the suggested reference: we read it and have cited it, but for a specific reason rather than as a courtesy. Our shaped signal is a supervisory set-point, not a signal generated by the feedback loop from the measurements (Assumption 1), so it is uncorrelated with the wind and wave disturbances and a direct least-squares fit is consistent even in closed loop. That is the easy case. The harder problem — recovering open-loop reduced-order dynamics from ordinary closed-loop records with correlated inputs — is exactly what Pasquali et al. treat for rotorcraft, and it is the problem a deployment would face if it wanted to track the modes between commanded maneuvers with no exogenous excitation. We say this, and we say that we do not adopt their method because our architecture places us in the easier case.
Change made in the manuscript:
New Sect. 4 opening that separates state estimation, disturbance estimation and modal identification. New Sect. 4.2 (“Where the modal parameters come from”). New numerical Sect. 5.11 with Table 7 and Fig. 8. Algorithm 1 now points to Sect. 4.2 for the source of its inputs. Reference Pasquali et al. (2024) added and discussed in Sect. 4.2. Identification method details in Appendix E.
Location: Sect. 4 opening; Sect. 4.2; Sect. 5.11; Table 7; Fig. 8; Algorithm 1; Appendix E, last subsection.
RC2 – Specific comment 3 (unknown-input formulation)
The referee is right and the inconsistency was real: the model carried E = [E_V E_M] while the appendix designed the observer for E_M alone, and Sect. 4.7 of the submitted version reported rank(CE) = rank(E) = 1, which is the rank of the wave column only.
Rather than paper over this by restricting the model, we checked whether a two-channel design is admissible, and it is. For the numerical realization of Appendix C with the full matrix E = [E_V E_M] we obtain rank(E) = rank(CE) = 2; the decoupling identity TE = 0 holds to 1.5 × 10⁻¹⁷; and pole placement on (TA, C) returns a Hurwitz F with the assigned spectrum, which establishes the detectability hypothesis constructively. The observer is therefore now stated, designed and reported for the complete disturbance vector.
We have also added the physical interpretation, since the cost of decoupling is not free: with four measurements and two decoupled channels, T has rank five, and the two lost directions are exactly the state combinations through which wind and wave enter. The wind column acts on the rotor-speed, tower-rate and platform-rate equations simultaneously, so those three states must be reconstructed from the remaining measurement combinations.
Wind-speed variation is therefore no longer treated by omission: it is one of the two decoupled unknown inputs. Appendix D now gives both designs — the wave-only gains, which are retained because the convergence figure of the submitted version uses them, and the new two-channel gains — and states which result each supports.
Change made in the manuscript:
The observer construction and its surrounding text rewritten for the two-channel case; a paragraph on the rank conditions and their physical meaning added after it; Sect. 5.12 updated to report rank(CE) = rank(E) = 2 and the numerical decoupling residual; Appendix D restructured into two subsections with both gain sets; notation E = [E_V E_M] introduced in Sect. 2.1.
Location: Sect. 2.1 (definition of E); Sect. 4.1; Sect. 5.12, first paragraph; Appendix D.
RC2 – Specific comment 4 (dependence on the damping ratio of 0.32)
We have run the tests, and we have also limited the recommendation. Both were needed.
Section 5.8 sweeps the three quantities that set the platform-pitch damping ratio in the model, one at a time, and repeats the entire design comparison at every point: the aerodynamic thrust derivative k_V over 0.15–2.5 times nominal, the hydrodynamic damping B_vis over 0.25–16 times nominal, and the platform pitch inertia I_p over 0.7–3 times nominal. Together these move ζ_p from 0.075 to 0.76 and the platform frequency from 0.030 to 0.061 Hz. Section 5.9 additionally rebuilds the model at seven above-rated operating points with aerodynamic derivatives from the public reference tables, where ζ_p ranges over 0.12–0.39.
The ranges were not chosen arbitrarily. They follow from checking our nominal values against the published reference model: the hydrodynamic damping there is quadratic, so any linear coefficient is a sea-state-dependent equivalent and must be treated as uncertain; and our lumped pitch inertia is at the low end of what the reference system implies, which is why the inertia sweep goes to three times nominal, giving platform periods up to 34 s.
The result is more informative than we expected. The three axes collapse onto a single curve when plotted against ζ_p, which says that what governs the trade-off is the damping ratio of the platform mode and not whether the damping is aerodynamic, hydrodynamic or the consequence of a heavier platform. On the tower channel the two-mode shaper is better at every point tested, by 8 to 15 percentage points over ζ_p ∈ [0.08, 0.61], the two converging only near ζ_p ≈ 0.75. On the platform channel the three-mode shaper is always better, but by a margin that decays monotonically: 52 points at ζ_p = 0.08, 25 at the nominal 0.32, 15 at 0.46 and 2 at 0.75.
We have therefore rewritten the recommendation as a conditional one. It now reads that excluding the platform mode is the better compromise when the platform-pitch load channel is not the design driver and when the platform mode is well enough damped that little is forgone — on the evidence here, roughly ζ_p > 0.3 — and that below ζ_p ≈ 0.15 the balance reverses for a design driven by platform-pitch loads. We state that we have no evidence outside the ranges tested and that the conclusion is specific to a shaper acting on the collective-pitch set-point of this configuration.
One item we did not test is controller settings, and we should be straightforward about why. The shaping layer is a feedforward pre-filter on the supervisory set-point and is not inside the feedback loop (Assumption 1), so the cancellation properties and the shaper instants do not depend on the regulator at all; the regulator affects the simulated response but not the design decision. Varying it would have produced a sweep whose outcome we could predict, so we spent the effort on the parameters that do enter the modal properties. We have made this reasoning explicit rather than leaving the omission silent.
Change made in the manuscript:
New Sect. 5.8 with Fig. 7 (three panels). New Sect. 5.9 with Table 6. Sect. 5.2 rewritten to present the choice as a trade-off and to point forward to the sweep. Recommendation limited in the abstract, in Sect. 5.8 and in the conclusions. A paragraph on how representative the nominal parameters are, with the three checks against the public reference model, added at the start of Sect. 5.8.
Location: Sect. 5.2, last paragraph; Sect. 5.8; Sect. 5.9; Fig. 7; Table 6; abstract, paragraphs 2 and 3; Sect. 8.
RC2 – Specific comment 5 (limited comparison set)
We added five baselines, not one, and the comparison changed how we present our own method.
The new Sect. 5.3 compares the unshaped command, the two- and three-mode UM shapers, ZV, ZVD and extra-insensitive shapers designed for the same two lightly damped modes, a second-order critically damped low-pass pre-filter, and an S-curve (raised-cosine) ramp, on exactly the same maneuver, model, actuator and metrics, produced in a single pass of one script. To avoid choosing baselines after seeing the results we fixed them in advance by two stated rules: impulse-based baselines target the same two modes as the proposed shaper, and one low-pass and one S-curve variant are tuned to add the same command delay as the proposed shaper, with a slower variant of each included to show what the extra delay buys.
Three findings come out and only the first favours us. The unity-magnitude sequence gives the shortest command for a given load reduction: −48 % on the tower-base indicator with 0.86 s of delay, against 1.29 s for ZV, 2.58 s for ZVD and EI, and 2.2-2.8 s for the smooth pre-filters. But it is bought with actuator duty: the delivered pitch travel is 1.86 times the unshaped value against 0.97 for ZV, so a plain ZV shaper achieves the same load reduction with about half the pitch travel at the cost of a 50 % longer command. And shaping clearly beats naive smoothing at equal delay: at 0.86 s the low-pass reaches −21 % and the S-curve −10 %, against −48 % for the shaped commands, so the benefit of placing zeros on the modal poles is worth about a factor of three in command latency here.
We have changed the paper’s claim accordingly. The unity-magnitude design is now presented as one point on a trade-off curve — the right choice when command latency has a value, the wrong one when pitch-bearing duty does — rather than as the preferred command shape. Sect. 6 states outright that a machine with duty-limited pitch bearings should use the ZV or ZVD variant.
On the referee’s specific point about the ideal pitch-rate demand: we now explain that the 250° s⁻¹ figure is the largest commanded step divided by the 10 ms command grid interval, and that it is the same for the unshaped command, because a ±1 impulse train moves the command by one full step at a time. The rate demand is therefore not what distinguishes the unity-magnitude sequence; the travel is.
Change made in the manuscript:
New Sect. 5.3 with new Table 4 (ten commands, eight metrics) and new Fig. 3 (load reduction against actuator duty and against command delay). Baseline definitions and the two selection rules stated in Sect. 5.3 and Appendix E. The EI shapers are solved numerically from their defining conditions rather than from tabulated polynomial fits, and the insensitivity tolerance and the resulting instants are published in Appendix E. Claims about the unity-magnitude choice moderated in Sect. 5.3, Sect. 5.4, Sect. 6 and the conclusions.
Location: Sect. 5.3; Table 4; Fig. 3; Sect. 5.4; Sect. 6 (“Actuator”); Sect. 8, last paragraph.
RC2 – Specific comment 6 (actuator analysis transparency)
All three points are now addressed explicitly.
On the design order, which was the item most needing clarification: the actuator is not part of the shaper synthesis. The impulse instants are computed from the modal parameters alone, the sequence is convolved with the set-point, and only then is the command passed through the actuator with its limits. Sect. 5.4 now opens by saying this, explains why the ordering is deliberate (it keeps the design analytic so that the exact-cancellation result applies), and notes that this makes the a posteriori check a necessary part of the procedure rather than an afterthought — a sequence that cancels perfectly on paper is useless if the actuator cannot follow it.
On the trajectories, a new figure shows the commanded and the delivered collective pitch and the delivered pitch rate around one set-point change, for the unshaped, three-mode and two-mode commands, with the rate limit drawn. The ±1 impulse train is clearly visible in the command and is low-passed by the actuator into a smooth trajectory.
On whether limits are active: for every command in the new comparison of Table 4, neither the 8° s⁻¹ rate limit nor the ±30° travel limit becomes active. In fact the largest actuator rate anywhere in the comparison is the one produced by the unshaped command, because the actuator bandwidth rather than the command sets the rate peak. A tight 2° s⁻¹ limit does bind, as the submitted version reported. We have also added the observation that what the actuator does not remove is the duty: the pitch travel is delivered, not filtered away, and it is the quantity a pitch-bearing duty assessment would care about.
Change made in the manuscript:
Sect. 5.4 rewritten with a new opening paragraph on the design order and a new closing paragraph on duty; new Fig. 4 with commanded and delivered pitch and pitch rate; explicit statement that no limit is active across the whole baseline comparison; travel and rate columns added to Table 4.
Location: Sect. 5.4; Fig. 4; Table 4, last three columns.
RC2 – Specific comment 7 (interpretation of the DEL reductions)
We agree and have changed the terminology throughout, and we have also run the suggested check, which produced a result we did not expect.
On terminology: the quantity is now called a short-term equivalent load or a command-induced load indicator wherever it appears — in Table 1, in the table and figure captions, in the text and in the conclusions. Sect. 7 states that it is a load indicator for comparing commands and not a lifetime or certification quantity, and that a fatigue statement would require the standardized design load cases, a distribution of wind speeds and sea states, many seeds per case and lifetime weighting.
On validation: the new Sect. 5.10 drives the same model with a stochastic environment — a rotor-effective wind speed from the IEC 61400-1 Kaimal spectrum at 14.11 m s⁻¹ for class IB, and a wave-induced pitch moment from a JONSWAP spectrum passed through the reference platform’s own pitch wave-excitation transfer function. Twelve seeds were run, each twice, with and without the maneuver on the identical wind and wave realization, so that the command-induced part of the load can be separated from the environmental part and the comparison between commands is paired on the realization.
Two results follow and only the first is favourable. The maneuver adds about 7 % to the short-term tower-base equivalent load of the same 600 s record; the rest is weather. Shaping removes a definite fraction of that increment: paired over realizations, the two-mode shaper reduces it by 43.4 % with a 95 % confidence interval of 28 to 59 %, and the three-mode shaper by 35.5 % (19 to 51 %). But on the platform channel nothing survives: the increment is 3.3 % smaller with the three-mode shaper and 2.2 % larger with the two-mode one, and both confidence intervals contain zero. The platform-pitch load benefit that the deterministic study credits to shaping the platform mode is simply not resolvable once the platform is being driven by waves at a comparable amplitude.
We report this because it matters for the paper’s own conclusion, and it is one of the reasons the deterministic platform-load figures are now presented as a command-induced indicator rather than as a fatigue benefit. We also state the limits of the new study: the wind is a rotor-averaged scalar rather than a turbulence field, the wave forcing is a linear excitation moment without radiation memory or second-order effects, and it is one sea state at one wind speed. It tests whether the effect is visible against realistic broadband excitation; it is not a load campaign.
Change made in the manuscript:
Terminology changed from “damage-equivalent load” to “short-term equivalent load” / “command-induced load indicator” throughout, including Table 1, the captions of Tables 3, 4 and 6 and of Figs. 2, 3 and 7. New Sect. 5.10 with twelve seeds and paired confidence intervals. Environmental spectra and the paired procedure documented in Appendix E. Caveat paragraph added to Sect. 7.
Location: Table 1; Sect. 5.2 and its captions; Sect. 5.10; Sect. 7, paragraph 2; Appendix E.
RC2 – Specific comment 8 (practical advantage of the unknown-input observer)
Investigating this produced the most useful correction in the revision. The reason a standard observer performed comparably is that we were decoupling the wrong channel.
The wave moment enters the platform equation through 1/I_p ≈ 4.5 × 10⁻¹¹, so even a large wave moment is a small quantity in state units and there is correspondingly little for the decoupling to remove. The wind-speed column is entirely different: it acts on the rotor-speed, tower-rate and platform-rate equations directly and with much larger gains, so an observer that ignores it inherits a bias proportional to the wind fluctuation.
The new Table 8 makes the comparison explicit on a 400 s record for three observers with identical assigned poles. Under the wave moment alone, all three are equivalent — which is what the submitted version observed and reported honestly. Once a turbulent wind fluctuation of 1.4 m s⁻¹ standard deviation is present, the standard observer and the wave-only unknown-input observer both carry a rotor-speed estimation error of about 0.10 rad s⁻¹, which is 13 % of rated rotor speed, while the two-channel unknown-input observer reduces it by three and a half orders of magnitude. With sensor noise on the integrated tower-top velocity channel the gain falls to about two orders of magnitude, which we report in the same table rather than only in the best case.
We also report what the decoupling costs, because it is not free. The feedthrough term Hy passes measurement noise straight into the state estimate, and the row of H associated with the tower-top velocity has a coefficient close to unity. With 1 mm s⁻¹ of noise on that channel the tower-rate error of the two-channel observer rises from 1.8 × 10⁻⁶ to 1.0 × 10⁻³ m s⁻¹ — still better than the standard observer, but no longer by orders of magnitude. This connects directly to the accelerometer-integration issue raised in technical correction 5.
The statement in the paper is therefore now conditional and, we think, more useful than either the original claim or its retraction would have been: the unknown-input formulation is worth its cost when a disturbance channel with large state-space gain is present and unmeasured, which on a floating turbine is the rotor-effective wind speed rather than the wave moment, and it is worth less as sensor noise on the decoupled measurement channels grows.
Change made in the manuscript:
Sect. 5.12 rewritten; new Table 8 comparing a standard observer, the wave-only unknown-input observer and the two-channel one under three excitation conditions; a sentence added to Sect. 4.1 and to the conclusions identifying the wind channel as the case in which the formulation is worth its cost.
Location: Sect. 4.1; Sect. 5.12, paragraphs 2 and 3; Table 8; Sect. 8, paragraph 2.
RC2 – Specific comment 9 (reproducibility)
We have not deposited the code, and the Code and data availability statement says so plainly. We have instead made the paper itself sufficient to reproduce the work, which we recognize is the weaker of the two options.
The new Appendix E documents, in one place: the augmented simulated plant and its discretization and time step, with a convergence statement; how a shaped command is constructed exactly rather than on a grid; the definition of both load channels and of the load metric, including the Wöhler slope, the equivalent cycle count, the half-cycle weighting and the discarded transient; the definitions of the residual-vibration index, the actuator quantities and the spillover measure; the swept ranges of every sensitivity study; the formulae by which the aerodynamic derivatives are obtained from the public coefficient tables; the environmental spectra with all their parameters, the seeding scheme and the paired statistical procedure; and the settings of both identification algorithms, including block rows, model orders, stabilization criteria and ARX orders.
Appendix D now additionally gives the two-channel observer gains, and the availability statement lists the public sources of the reference-turbine data used in the new studies. Together with the existing Appendices B and C this covers the settings behind every number and figure in the paper. We accept that a repository would be better and intend to prepare one, but we did not want to claim it before it exists.
Change made in the manuscript:
New Appendix E (“Evaluation pipeline of the additional studies”), eight subsections. Appendix D extended with the two-channel gains. Code and data availability statement rewritten to state plainly that no repository has been created.
Location: Appendix E; Appendix D; Code and data availability statement.
RC2 – Specific comment 10 (novelty claims)
We agree with this reading and have adopted it.
A new paragraph in the introduction states explicitly that the convolution property of multi-mode shapers, the residual-vibration functional and the full-order unknown-input observer of Darouach et al. are established results, restated here because the design rests on them and because the conditions under which they may be used are often left unstated. It then lists what we do claim: the integration into one adaptive collective-pitch layer for a coupled FOWT model together with the perturbation and rescaling analysis; the two-channel unknown-input formulation and the identification of the disturbance conditions under which it pays; the account of where the modal parameters come from, with the numerical study of two identification routes; and the FOWT-specific design study.
In the body we went further than attribution in a sentence. The two results that are not ours no longer appear as numbered theorems at all: the multi-mode cancellation property is given in running text, cited to Singer and Seering (1990) and Singhose (2009), with a line saying that we claim nothing new in it and state it only because the design uses it quantitatively; the observer construction is given the same treatment and cited to Darouach et al. (1994), with its proof left in that reference. What remains numbered is what we derived.
Change made in the manuscript:
New contribution paragraph in the introduction separating established components from claimed contributions; the two borrowed results demoted from numbered theorems to cited statements in running text, with their proofs removed; the abstract reordered so that the engineering finding comes before the analytical results.
Location: Sect. 1, second-to-last paragraph; Sect. 3.1, last paragraph; Sect. 4.1, last paragraph.
Technical corrections
RC2 – Technical correction 1
Defined in full in the note printed under Table 3 and again in Appendix E. It is the reduction, relative to the unshaped command, of the residual vibration of the tower fore–aft deflection, where the residual vibration is the peak deviation from the quasi-static value in a window that begins after the last impulse of the sequence has been applied and ends one minute after the set-point change, averaged over the six set-point changes, with the second-order pitch actuator in the loop.
Change made in the manuscript:
Definition added to the note under Table 3; the same quantity defined operationally in Appendix E under “Derived quantities”, including the window offsets and the quasi-static reference. Because the metric had to be defined rather than recovered, the row was regenerated with the new definition, together with every other simulated number in the paper; see the third paragraph of the opening remarks.
Location: Note under Table 3; Appendix E.
RC2 – Technical correction 2
Done. The six numbers are now given as a labelled display, each attached to its path, with the units stated: ν_max(ω_r/T_g) = 7.9×10⁻⁶, ν_max(ω_r/β) = 4.7×10², ν_max(q_fa/T_g) = 4.0×10⁻⁷, ν_max(q_fa/β) = 2.1×10³, ν_max(θ_p/T_g) = 2.6×10⁻⁸ and ν_max(θ_p/β) = 1.4×10². We also note in the text that the submitted version listed the same six numbers without saying which path each belonged to.
Change made in the manuscript:
The list replaced by a labelled equation with units; a sentence added acknowledging the ambiguity of the original presentation.
Location: Sect. 5.1 (formerly Sect. 4.1), Eq. (10).
RC2 – Technical correction 3
Corrected. The relative modal-frequency error is now ε and the timing error of impulse i remains δt_i, so the mapping reads δt_i = −ε t_i / (1 + ε) instead of using δ for both. We have added a sentence saying that both were written with the same letter in the submitted version.
Note that the corollary is no longer a separate environment: following Referee 1 it has been folded into Proposition 3.1, of which it is a two-line consequence.
Change made in the manuscript:
Symbol δ for the frequency error replaced by ε throughout the statement, the proof and the figure axis; an explanatory sentence added after the proposition; the former Corollary 3.5 merged into Proposition 3.1.
Location: Sect. 3.2, Proposition 3.1 and the sentence following it; Appendix A; Fig. 8b axis label.
RC2 – Technical correction 4
Corrected. The problem was that the right-hand image carried its own internal panel titles “(a) Open-loop magnitude” and “(b) Nichols locus” while the LaTeX sub-captions also used (a) and (b), so the letters appeared twice in one figure.
The figure has been regenerated with descriptive panel titles instead of letters, so the sub-caption letters are now unique. We verified that the regenerated figure reproduces the reported loop properties exactly: gain crossover 1.20 rad s⁻¹, phase margin 50.2°, modulus margin 0.687, resonant peak 2.29 dB, and a phase margin range of 50.2–52.5° over the stated gain-scaling interval.
Change made in the manuscript:
Figure regenerated without internal panel letters; sub-captions rewritten to name the contents of each panel.
Location: Fig. 9 (formerly Fig. 5).
RC2 – Technical correction 5
Added, and it turned out to connect to two other parts of the revision.
A new paragraph after Assumption 3 states the problem concretely: accelerometer bias integrates into a velocity ramp and low-frequency noise into a random walk, so the raw integral drifts without bound; in practice the integration is combined with a high-pass or complementary filter, which removes the drift at the cost of distorting the signal below the corner. For this application the corner must sit below the tower fore–aft frequency of 0.45 Hz but may sit above the platform-pitch frequency of 0.05 Hz, so a corner near 0.1 Hz is workable for the tower mode — but the platform motion must then come from the platform inertial unit rather than from the tower-top channel, which is one reason the assumed sensor set keeps a separate platform measurement.
The consequences are then quantified in two places: Table 8 shows that noise on this channel is what erodes the advantage of the unknown-input observer, because the observer feeds the measurement through directly with a coefficient close to unity; and Table 7 reports identification accuracy at 2 % and 10 % channel noise.
Change made in the manuscript:
New paragraph after Assumption 3; cross-references added in Sect. 5.12 and Sect. 6; sensor noise made an explicit variable of the new Tables 7 and 8.
Location: Sect. 2.3, after Assumption 3; Sect. 5.12, last paragraph; Sect. 6 (“Measurement quality”); Tables 7 and 8.
RC2 – Technical correction 6
All references were re-checked against Crossref. Three corrections were needed. The Gürleyük entry carried the online-first year 2006 while the article appears in the January 2007 issue; it now carries 2007 with a note giving the online date. A second entry (Knudsen et al.) also carried a wrong year, but it is not cited in the revised text and has been removed from the bibliography rather than corrected. The Bai et al. entry, which is an online-first article without volume or page numbers, now carries an explicit “available online” note.
Two references were added: Pasquali et al. (2024), verified as Aerospace Science and Technology 153, 109419; and the public IEA Wind Task 37 reference-turbine repository, cited with the specific files used in the new studies and the access date.
Change made in the manuscript:
Bibliography corrected and extended; entries checked against Crossref records.
Location: Reference list.
RC2 – Technical correction 7
Done for all three, in the figure captions and in the table notes rather than only in the running text. The main captions were afterwards cut to one line each for readability, so the definitions sit in the panel captions and in the notes printed directly under the tables.
The panel captions of the vibration-sensitivity and spillover figure (Fig. 5) define residual vibration as the amplitude left by the shaper at a mode of the given frequency, normalized to what an unshaped step would produce, and out-of-band spillover as the value of |S(jω)| above unity, that is the factor by which the pre-filter amplifies command content at a frequency it was not designed for. The note under Table 3 defines the spillover measure with its frequency band, and the command-induced load panel of Fig. 2 is now labelled as a short-term command-induced load indicator rather than as a fatigue quantity. Cancellation is the quantity plotted in Fig. 6b, and its panel caption now defines it as one minus the residual vibration expressed as a fraction of the unshaped value, which is the measure the note under Table 3 states in full.
Change made in the manuscript:
Definitions added to the panel captions of Figs. 2 and 5 and to the notes under Tables 3 and 4.
Location: Captions of Figs. 2 and 5; notes under Tables 3 and 4.
RC2 – Technical correction 8
Stated in the abstract’s second paragraph, which now reads that synthesis and evaluation share one seven-state control-oriented model, that it is a design model rather than an aero-hydro-servo-elastic one, and that the design study is a proof of concept for the mechanism rather than turbine-level validation. The same statement is repeated at the opening of Sect. 5 before any number appears. Sect. 7 states what the single aero-hydro-servo-elastic cross-check of Appendix F does and does not establish, and that no result of the design study should be read as turbine-level validation.
Change made in the manuscript:
Sentence added to the abstract; equivalent statements added at the opening of Sect. 5 and in Sect. 7.
Location: Abstract, paragraph 2; Sect. 5 opening paragraph; Sect. 7.
RC2 – Technical correction 9
We have done both: we produced additional evidence and we limited the conclusion to what that evidence supports.
Every statement of the recommendation now carries its condition. The abstract gives the trade-off with both sides quantified and then reports that the aeroelastic cross-check of Appendix F reverses it, so the recommendation is never stated as a rule. Section 5.8 states it as: excluding the platform mode is the better compromise when the platform-pitch load channel is not the design driver and when the platform mode is well enough damped that little is forgone — on the evidence here, roughly ζ_p > 0.3 — and the balance reverses below ζ_p ≈ 0.15 for a design driven by platform-pitch loads. It adds that we have no evidence outside the ranges tested and that the conclusion is specific to a shaper acting on the collective-pitch set-point of this configuration. The conclusions section repeats the condition explicitly.
The closing sentence of the paper, which previously read as a general rule, is now framed as a design procedure rather than a result, and it names the load path rather than the modal damping as what decides the choice: shape the modes that ring, find out which of them your load path responds to, and settle the remainder on actuator duty.
Change made in the manuscript:
Conditional wording introduced in the abstract, Sect. 5.2, Sect. 5.8, Sect. 6 and Sect. 8; the closing sentence rewritten; Table 2 role column reworded to “excluded from the shaper in the preferred design”.
Location: Abstract, paragraph 3; Sect. 5.2, last paragraph; Sect. 5.8, last paragraph; Sect. 8, paragraphs 3–5; Table 2.
Citation: https://doi.org/10.5194/wes-2026-128-AC2
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AC2: 'Reply on RC2', amina Mseddi, 25 Aug 2026
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RC3: 'Comment on wes-2026-128', Anonymous Referee #3, 24 Aug 2026
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General comments
The paper presents an adaptive multi-mode unity-magnitude input-shaping approach for floating offshore wind turbines, combined with an unknown-input observer for state and modal estimation. In general it is difficult for the WES readers to follow the method developed in this paper unless the audiences have very strong theoretical control background and the associated mathematics. I suggest the paper is more suitable for a theoretical control journal or theoretical control application journal to present this new method and its application for floating wind turbines. Or, It should be formulated and presented in such away that more suitable for wind energy science (WES) audiences.Overall comments
- The manuscript is written as a theoretical control paper with all the assumption, definitions, lemmas, theorems, propositions, proofs, and remarks. This makes the paper difficult to be followed by the broader WES audience.
- The possibility and applicability of the presented method are weakly discussed for applying to the real Floating wind turbines. At least, simulations using simulation tools, Bladed or OpenFAST shoud be included to demonstrate its robustness and applicability.
- The shaper synthetic of exclude the platform pitch mode while keeping the tower fore-aft mode is arguable needs to be further checked. See my comments in the Technical comments.
- The numerical verification/study and the shaper design are based on the same reduced-order model. Consider to use full aero-elastic simulation to verify the shaper design which can make the conclusion stronger.
Technical comments
- In table 1, It is not presented at which wind speeds or operating points are the numberical studies performed. This information is very important because the behavior of the aerodynamic damping couples with the collective pitch control changes from different operation points. (e.g., can change from positive damping to negative damping)
- In table 2 and at Line 183 to 186, It is presented that Tower fore-aft mode is lightly damped and the platform pitch mode is strongly aero-damped. Why? From the aerodynamic perspective they all driven by the aerodynamic thrust so they should have similar aerodynamic damping effect at below rated windspeeds, and they all suffer the negative aerodynamic damping issue at rated wind speed and above related wind speeds. This probably shows that your control-oriented 7 states model has some issues. And this will also change your shaper synthesis and the design choice in Section 4.2. Therefore, the rest of the results.
- Line 187, ¨the largest numerator coefficents are ... ¨, some of the values are not large at all, so using ¨dominent¨ should be better.
- Line 186, please explicitly addess the six input - output paths.
- The discussion on Fig.5b is not consistent with the actual figure. Please check. Or Fig. 5b needs to be zoom in around the cross-over frequency to better see the value
- The DEL reductions should be interpreted carefully. This should be computed based on turbulent wind fields, irregular waves with multiple change of set-points would make your results on the DEL reduction stronger.
- The shaper synthetic and the design choice and the numerical studies are based on the same reduced order (control-oriented) model will make your conclusion weaker. Therefore, a full aero-servo-hydro-elastic simulation is needed.
Citation: https://doi.org/10.5194/wes-2026-128-RC3 -
AC3: 'Reply on RC3', amina Mseddi, 25 Aug 2026
reply
We thank the referee his effort and his contructful comments that helped us improve our paper
RC3 – General comment (suitability for the WES readership)
We have taken the second option, and the revision is substantially restructured towards it.
The formal apparatus is reduced from sixteen numbered environments to five: three Assumptions, one Proposition and one Theorem, each used quantitatively later in the paper. Two Definitions, one Lemma and all five Remarks have been dissolved into running text; the two results that are not ours are now stated in running text with their sources cited and their proofs removed; and the two proofs that are ours have been moved to Appendix A, so that Sect. 3 reads as three design statements rather than as a sequence of derivations.
Each of the three results in Sect. 3 is now stated twice: first in plain words, then precisely. The multi-mode result, for instance, opens "Put plainly: shapers designed separately, one per mode, and then convolved into a single sequence, cancel every one of those modes together. There is no cross term to worry about and no simultaneous solve to perform — the modes do not interfere." Each result is followed by a paragraph saying what it means for a floating turbine.
The engineering material that was missing has been added: a physical account of the damping mechanism (Sect. 2.1), a benchmark against the command shapes an engineer would otherwise use (Sect. 5.3), the actuator duty the method demands (Sect. 5.4), how far the design decision carries (Sects. 5.8–5.9), a check in turbulent wind and irregular waves (Sect. 5.10), a section on what deployment on a real machine would require (Sect. 6), and the aero-hydro-servo-elastic cross-check of Appendix F. Of the paper's current length, the large majority is engineering rather than theory.
Change made in the manuscript:
Environments reduced from sixteen to five; the proofs that are ours relocated to Appendix A; plain-language statements added before each result in Sect. 3; new Sects. 5.3, 5.4, 5.8–5.12, Sects. 6 and 7, and Appendices A, E and F.
Location: Sects. 2.1, 3.1–3.3, 5.3, 5.4, 5.8–5.12, 6, 7; Appendices A, E and F.
Overall comments
RC3 – Overall comment 1 (theoretical presentation)
Reviewer comment:
The manuscript is written as a theoretical control paper with all the assumption, definitions, lemmas, theorems, propositions, proofs, and remarks. This makes the paper difficult to be followed by the broader WES audience.
Response:
Addressed as described above: sixteen numbered environments reduced to five, no Definition, Lemma or Remark anywhere in the paper.
The reduction went further than counting. A numbered theorem is now used only where the result is our own. The multi-mode cancellation property (Singer and Seering, 1990; Singhose, 2009) and the full-order unknown-input observer (Darouach et al., 1994) are stated in running text, cited to their sources, and their proofs are not reproduced — for the observer we say plainly that the proof is in that reference. What remains numbered is the residual bound of Proposition 3.1, which turns an identification accuracy into a guarantee on the residual, and the rescaling rule of Theorem 3.2, which is the adaptation law; both are ours and both are used quantitatively in Sect. 5. Their two proofs are in Appendix A.
We have kept the three Assumptions as numbered items, because each is a real constraint on an implementation and is referred to repeatedly, so a reader checking whether the method applies to their machine needs to be able to find them. If the editor prefers, the two remaining results can be demoted to unnumbered statements as well, without any loss of content.
One consequence should be flagged rather than left for the editor to notice. Our reply to Referee 1 was posted before this last change and describes seven environments, three of them Theorems, with all four proofs in Appendix A. That count is superseded by the version submitted here: five environments and two proofs. No result was withdrawn and no statement changed — only their status as ours or borrowed.
Change made in the manuscript:
Sixteen environments reduced to five; two Definitions, one Lemma and five Remarks dissolved into the text; the two borrowed results demoted to cited statements in running text; the two proofs that are ours moved to Appendix A.
Location: Sects. 2.2, 3.1–3.3, 4.1; Appendix A.
RC3 – Overall comment 2 (applicability; simulations in Bladed or OpenFAST)
We have done both parts of this.
On applicability, a new Sect. 6 discusses deployment in seven headed paragraphs: continuously varying conditions, changing operating point, where the modal parameters come from, the update rate of the adaptation, measurement quality, actuator duty and model fidelity. It is deliberately concrete rather than reassuring — it states, for example, that the drivetrain torsion mode cannot be identified at all from the sensor set the paper assumes, and that a machine with duty-limited pitch bearings should use a ZV shaper instead of the unity-magnitude one.
On simulation, the three commands were executed on the public OpenFAST model of the same turbine and platform (IEA-15-240-RWT-UMaineSemi) in OpenFAST v4.2.1, with ElastoDyn, AeroDyn 15, InflowWind, ServoDyn, SeaState, HydroDyn in its potential-flow form and MoorDyn. The commands are delivered through the OpenFAST Simulink interface, with ServoDyn taking the collective-pitch command and the generator torque directly from Simulink (PCMode = 4, VSContrl = 4); this is what allows an arbitrary shaped set-point to be commanded, the built-in pitch-maneuver facility being limited to ramps. The second-order pitch actuator and the twelfth-order regulator of the paper were discretized at the OpenFAST time step and inserted with no coefficient re-tuned.
Change made in the manuscript:
New Sect. 6 (about two pages, seven headed paragraphs). New Appendix F with Table F1 and Fig. F1 documenting the coupling, the settings, the harness checks and the full comparison.
Location: Sect. 6; Appendix F, Table F1, Fig. F1.
RC3 – Overall comment 3 (excluding the platform-pitch mode is arguable)
The referee was right to doubt it, and the check we ran does not support the original recommendation.
On the seven-state model the two-mode design is better on the tower channel because the tower-base load proxy of that model, M_tb = k_t q_fa L_t, is proportional to the tower modal deflection alone. On the aeroelastic model the ordering reverses on both load channels: the three-mode command reduces the tower-base equivalent load by 20.0 % where the two-mode command reduces it by 2.3 %, and the platform channel by 22.2 % against 0.2 %. The cancellation itself transfers unchanged — the two-mode command removes 84.2 % of the tower fore–aft band energy — and so does the cost, at 1.87 and 5.64 times the unshaped pitch travel against 1.86 and 5.57 predicted.
The reason is that on a floater the tower-base moment is governed by platform motion rather than by tower fore–aft deflection: the band below 0.03 Hz carries about 190 times the energy of the tower fore–aft band, and the seven-state model has no surge degree of freedom at all. The recommendation to leave the platform mode unshaped is therefore a property of the load proxy and not of the machine, and the paper now says so wherever the recommendation appears.
Change made in the manuscript:
The reversal is reported in the third paragraph of the abstract, in the introduction, at the end of Sect. 5.2, in Sect. 5.8, in the model-fidelity paragraph of Sect. 6, in Sect. 7 and in the conclusions, and documented in Appendix F.
Location: Abstract, paragraph 3; Sect. 1; Sect. 5.2; Sect. 5.8; Sect. 6; Sect. 7; Sect. 8; Appendix F.
RC3 – Overall comment 4 (design and verification share one model)
We agree, and this is the same point Referee 2 raised. The aero-hydro-servo-elastic execution described under overall comment 2 is exactly this verification, and it is kept deliberately outside the design study so that the two are not confused: the opening of Sect. 5 states that synthesis and evaluation share the seven-state model with the single exception of Appendix F.
It did make the conclusion stronger, but not in the direction we expected: it reversed the load-channel ranking. We report that rather than defend the original claim, and Sect. 7 states plainly that the check is one operating point in steady wind and still water with this paper's own control layer, so it settles the direction of the discrepancy and not its size.
Change made in the manuscript:
New Appendix F; scope sentences added to the introduction and to the opening of Sect. 5; Sect. 7 rewritten around what the check does and does not establish.
Location: Sect. 1; Sect. 5 opening; Sect. 7; Appendix F.
Technical comments
RC3 – Technical comment 1 (operating point not stated in Table 1)
The referee is right that this was missing, and Table 1 now carries it. The deterministic study is performed at a single above-rated point: V_hub = 14.11 m/s, collective pitch 10.2°, constant rated generator torque. Sect. 5.9 repeats the whole design study at seven above-rated points from 11.2 to 25 m/s using aerodynamic derivatives computed from the public rotor-performance tables, and the platform-pitch damping ratio moves over 0.12 to 0.39 across that range.
On the negative-damping mechanism the referee refers to: that phenomenon is a closed-loop effect of a collective-pitch speed regulator on a floater, and it cannot appear in this model, because the state matrix used throughout is the open-loop plant and contains no pitch feedback. Sect. 2.1 now says this explicitly, and the paper cites Larsen and Hanson (2007) for the mechanism. The same is true of the aeroelastic cross-check, where the speed loop is closed on generator torque and the pitch channel carries only the supervisory command. A shaping layer sitting on top of a pitch regulator that does exhibit negative damping would have to be re-examined, and we say so.
Change made in the manuscript:
Operating-point row added to Table 1, naming the wind speed, the collective pitch and the torque setting for the deterministic study and pointing to Sect. 5.9 for the operating-point sweep.
Location: Table 1; Sects. 2.1, 5.9.
RC3 – Technical comment 2 (why the two modes have different damping)
This comment found a real error, and we are grateful for it. The referee is right that both modes are driven by the same aerodynamic thrust and should therefore be similarly aerodynamically damped. They are. Our explanation of the asymmetry was wrong, and it is corrected.
Rebuilding the model with the thrust-damping derivative k_V set to zero separates the aerodynamic from the non-aerodynamic damping of each mode. The platform-pitch mode has a total damping rate of 0.206 s⁻¹, of which 0.185 s⁻¹ (90 %) is aerodynamic. The tower fore–aft mode has a total rate of 0.206 s⁻¹, of which 0.149 s⁻¹ (73 %) is aerodynamic. The two aerodynamic rates differ by a factor of only 1.24. The submitted text attributed the asymmetry to the lever arm — "the same mechanism damps the tower much less, because there the arm is unity" — and that is not correct: the arm accounts for a factor of about 1.24, because L_t² = 22500 m² and I_p/m_t = 18333 m² very nearly cancel.
Where the referee's conclusion does not follow is that the model is at fault. The quantity that differs by an order of magnitude is the damping ratio, not the damping: ζ is the damping rate divided by 2ω_n, and the tower frequency is 8.75 times the platform frequency, which is almost exactly the ratio of the two damping ratios (8.74). The asymmetry in Table 2 is a normalization, and both entries in that table are correct eigenvalues of the same state matrix. No number in the paper changes.
Nor does the shaper synthesis or the design choice change, and the reason is worth stating because it is a better argument than the one it replaces. What decides whether a mode is worth shaping is how many cycles it rings after a commanded step, and that is measured by ζ, not by the damping rate. The platform mode decays to 1/e in 0.50 of a cycle, the tower in 4.36 cycles and the drivetrain in 5.30. The two modes that ring are the two that are shaped in the preferred design. We have rewritten the physical explanation in Sect. 2.1 and Sect. 5.1 around this, and marked it as a correction so that a reader comparing versions can see what changed.
We note that this is the second attribution error of this kind that the review process has caught in this paper — the submitted version also credited the platform damping to the pitch controller, which is likewise impossible for an open-loop state matrix. Both are now corrected, and the numerical content was unaffected in both cases because every quantity is computed from the state matrix rather than from the narrative.
One consequence should be stated openly. This correction supersedes two descriptions in our response to Referee 1, which was finalized and submitted before the present review reached us: comment 1 of that response speaks of "why the platform mode is damped and the tower is not", and comment 6 of "why the same mechanism barely damps the tower". Neither is right, and the manuscript now carries the corrected account wherever the mechanism is discussed. We would rather flag the discrepancy here than leave a reader to find it between the two letters.
Change made in the manuscript:
The damping explanation in Sect. 2.1 rewritten with the aerodynamic and non-aerodynamic parts of both modes given separately, the near-cancellation of the lever arm and the inertia ratio stated, and the asymmetry attributed to the frequency ratio. Sect. 5.1 rewritten to match and to give decay per cycle for all three modes.
Location: Sect. 2.1, second consequence; Sect. 5.1; Table 2.
RC3 – Technical comment 3 (“largest” numerator coefficients)
Agreed and changed. The sentence now reads "Writing ν_max for the dominant numerator coefficient of each path, so that the six can be compared with one another rather than read as absolute magnitudes, the values are …". The added clause is there because the point of the list is the comparison between paths, not the size of any one of them.
Change made in the manuscript:
Wording changed from "largest" to "dominant", with a clause stating that the values are for comparison between paths.
Location: Sect. 5.1, sentence preceding Eq. (10).
RC3 – Technical comment 4 (address the six input–output paths explicitly)
Done. The sentence now names them: the two inputs are generator torque and collective pitch, the three responses are rotor speed, tower fore–aft deflection and platform pitch, and the text states what the coupling means physically — generator torque reaches the tower and the platform, and collective pitch reaches the rotor speed. Equation (10) then gives the six dominant numerator coefficients in the same order, and Fig. 1b shows the tower response to a generator-torque impulse, which is non-zero precisely because the model is coupled.
Change made in the manuscript:
Sentence rewritten to name the two inputs, the three outputs and the physical meaning of each non-zero path.
Location: Sect. 5.1, sentence preceding Eq. (10); Eq. (10); Fig. 1b.
RC3 – Technical comment 5 (Fig. 5b and its discussion)
We checked the numbers and they are right: recomputing the spectra gives a three-mode peak of 19.35 at 9.43 Hz and a two-mode peak of 6.92 at 9.40 Hz, which are the values quoted. What the referee is pointing at is that they cannot be read off the figure: the crossover region, which is the part a designer actually uses, is compressed against the axis by the 6–10 Hz spikes and the y-scale needed to show them.
We have taken the referee's second suggestion. Figure 5b now carries an inset that zooms the 0.02–1.2 Hz band on a linear scale around unity, with the two crossover frequencies marked: 0.07 Hz for the three-mode sequence and 0.69 Hz for the two-mode one. This also fixes a related imprecision elsewhere: the text in Sect. 6 quoted the 0.7 Hz crossover without saying that it belongs to the two-mode command only, the three-mode sequence being nine times longer and crossing an order of magnitude lower. That is now stated.
Change made in the manuscript:
Fig. 5b regenerated with a crossover inset marking both crossing frequencies; the crossover sentence in Sect. 6 attributed to the two-mode command.
Location: Fig. 5b; Sect. 5.5; Sect. 6 (“Model fidelity”).
RC3 – Technical comment 6 (interpretation of the load reductions)
We agree on both counts and have acted on both.
On interpretation, the quantity is no longer called a damage-equivalent load anywhere in the paper. It is a short-term command-induced load indicator, defined in the note under Table 3 and in Appendix E, and Sect. 7 states that it is an indicator for comparing commands rather than a lifetime or certification quantity and that a fatigue statement would need the standardized design load cases.
On turbulence and waves, Sect. 5.10 repeats the maneuver in Kaimal turbulence (V_hub = 14.11 m/s, class IB) and JONSWAP waves (H_s = 3 m, T_p = 9 s, γ = 3.3) over twelve seeds, with all six set-point changes in every seed. The result refines the claim rather than confirming it: the tower-channel benefit survives the stochastic environment, and the platform-channel benefit of the three-mode shaper is not statistically resolvable across the twelve seeds. We report the second outcome as plainly as the first.
What remains undone, and Sect. 7 says so, is a load campaign over a distribution of wind speeds and sea states with many seeds per design load case. The aeroelastic cross-check of Appendix F is also in steady wind and still water, which is deliberate — it isolates the command-induced response — but it means that check establishes the direction of the discrepancy it uncovers and not its magnitude.
Change made in the manuscript:
Terminology changed throughout; Sect. 5.10 with twelve seeds of turbulent wind and irregular waves; interpretation caveats in Sect. 7.
Location: Table 1; note under Table 3; Sect. 5.10; Sect. 7; Appendix E.
RC3 – Technical comment 7 (full aero-servo-hydro-elastic simulation needed)
This is the same requirement as overall comment 4 and it has been met, with the outcome described there: the aero-hydro-servo-elastic execution reversed the load-channel ranking of the two designs, and the paper is written around that result rather than around the original one.
For completeness, the harness checks reported in Appendix F are: the transplanted regulator holds the mean rotor speed to 0.024 % of rated across the analysis window; the commanded generator torque never reaches the plausibility bound placed on it; the aerodynamic torque OpenFAST develops at the imposed trim pitch is within 3.1 % of the value the published rotor-performance tables give at the same pitch and speed; and the unshaped maneuver delivers 12.060° of pitch travel against 12.061° on the design model, so the actuator and its limits transplant exactly.
Change made in the manuscript:
New Appendix F with the coupling, the settings, the harness checks, the band decomposition, Table F1 and Fig. F1.
Location: Appendix F.
Citation: https://doi.org/10.5194/wes-2026-128-AC3
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Adaptive multi-mode unity-magnitude input shaping with unknown-input state estimation for floating offshore wind turbines
The paper presents an adaptive multi-mode unity-magnitude input-shaping approach for floating offshore wind turbines, combined with an unknown-input observer for state and modal estimation. The method is developed theoretically and demonstrated on a reduced-order coupled FOWT model. The main engineering conclusion by the authors is that shaping only the lightly damped modes, while excluding the strongly aerodynamically damped platform mode, yields superior performance.
VERDICT: The topic is potentially valuable to the floating wind community, and I appreciate that the limitations of the proposed approach are clearly stated in the introduction. However, the manuscript reads more like a theoretical control-engineering paper than a wind-energy (control) paper. The extensive use of definitions, lemmas, theorems, propositions, proofs, and remarks makes the paper difficult to follow for the broader WES audience. In addition, when establishing the theoretical framework, these are often presented with limited engineering interpretation. This makes it unclear why certain results matter from a practical FOWT perspective, and how the approach scales to real FOWTs operating across a wide range of conditions. The paper would benefit from a stronger focus on physical interpretation and engineering relevance. The most interesting contribution is the practical design insight that the platform mode should not be shaped by aerodynamic damping, but this message is somewhat obscured by the strong theoretical emphasis. In its current form, the manuscript is better suited to a theoretical control-engineering journal. I therefore recommend either substantially adapting the manuscript for the WES audience or considering submission to a more control-engineering-oriented journal.
Overall comments
- The abstract is difficult to follow after the opening sentence and becomes highly technical quickly. Consider rewriting it with greater emphasis on the engineering problem, the practical contribution, and the key findings for a wind-energy audience.
- I appreciate that the limitations of the proposed method are clearly stated in the Introduction.
- Please provide a more concrete discussion of the applicability of the proposed method to real FOWTs. How robust and applicable is the approach for a FOWT of which the (environmental) operating conditions vary constantly?
- The presentation style based on definitions, lemmas, propositions, theorems, proofs, and remarks is more suitable for a theoretical control-engineering journal than for WES.
- The manuscript contains relatively little explanatory text around the theoretical developments. Additional context and engineering interpretation would significantly improve readability and help readers understand the relevance of the presented results.
- Important insights are often presented in remarks rather than integrated into the main narrative.
- The balance between mathematical rigor and engineering insight could be improved. Several derivations could potentially be shortened or moved to appendices, allowing more room for discussion of the practical implications.
- The practical design conclusion regarding the exclusion of the platform mode from the shaping strategy is interesting and should be emphasized more strongly.