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.