At the equilibrium point of this system, it is possible to express the dynamic model of the three degrees of translation as a state-space dynamic equation written as (1)M⋅ξt˙=At⋅ξt+[∑F3×103×1],where ξt is the translational state vector, given by: (2)ξt=[x˙,y˙,z˙,x,y,z]T •

The variables x,y,z are the position relative to the equilibrium point. They refer to surge, sway, and heave of the system, respectively.

Similarly to the translational motion, the dynamic model of the rotational degrees at the equilibrium point can be expressed as, (4)I⋅ξr˙=Ar⋅ξr+[Tg03×1]+[∑T3×103×1]where ξr is the rotational state vector. (5)ξr=[θ˙,ϕ˙,ψ˙,θ,ϕ,ψ]T

The variables θ,ϕ,ψ are rotation angles with respect to axes x, y and z, i.e., roll, pitch and yaw, respectively (Fig. 1).

The term I is the inertia tensor of the system. The equation can be normalized by multiplying by the inverse of this matrix. The matrix Ar is given by, (6)Ar=[−Dr3×3−Kr3×3I3×303×3]where:

Dr=diag(DθDϕDψ) is the rotational damping coefficient diagonal matrix due to the geometry of the floating platform; they include viscous forces and radiation effects.

In Eq. (4), the gyroscopic effect introduces feedback in the system. The gyroscopic torque vector Tg due to the angular moment of rotational components is, (7)Tg=−[θ˙,ϕ˙,ψ˙]×(Lrot+Lgen)where:

Lrot=Irot⋅ωrot, is the angular moment of the rotor, included blades.

where Irot is the rotor inertia tensor, ωrot is the rotor angular velocity, Igen is the generator inertia tensor and ωgen is the generator angular velocity. These angular moments may have a different direction and/or sense, due to rotor orientation, gear transmission and mechanical couplings. If the two angular moments have the same direction, they can be considered as a single angular moment vector. If the rotation axes of the generator and the rotor are aligned with the x-axis, no gyroscopic moment will be generated in this axis, and then the gyroscopic torque vector is described by Eq. (8). (8)Tg=−[0ψ˙⋅(Lrot+Lgen)ϕ˙⋅(Lrot+Lgen)]

Therefore, the model of the FOWT dynamics is represented by two linear vector equations, Eqs. (1) and (4), which represent respectively the translational and rotational motions of the wind turbine. The inputs to these equations are the hydrodynamic (Section 4.1) and aerodynamic (Section 4.2) forces and torques, generated by the external disturbances (waves and wind), to which the control action will then be added.

To study and illustrate the hydrodynamic forces and torques generated by waves, the motion of a semi-submerged object is analyzed (Fig. 2). Due to the internal motion of the water particles, a drag force is produced at the bottom of the wave that attracts the object laterally. The object is pushed upwards but is then dragged laterally and finally is carried towards the bottom of the wave. The rectangle in Fig. 2 represents the FOWT barge, where the Center of Gravity (CoG) must be below the Center of Buoyancy (CoB) for the platform to be stable. In contrast, the Center of Aerodynamics (CoA) can be above both, near the nacelle.

For waves of significant height, i.e., large waves relative to the size of the platform, these forces will be perceived as out of phase with respect to the time at which the wave crest reaches the CoG of the barge. This lag can be modeled as a combination of the wave elongation and its derivative, which is advanced ¼ cycle, as expressed in (9). (9)Fh3×1=Kh3×1⋅ζr+Dh3×1⋅ζ˙rwhere ζr is the wave elevation minus the vertical displacement of the platform, and ζ˙r is its derivative. The modeling of wave forces has been simplified as a function of the waves relative to the platform and their relative velocity, as it is presented in Eqs. (10)–(12), which capture the main behavior, gain and phase, for the purpose of control design.

If the wavelength is very large compared to the dimensions of the barge, the signals can be assumed to be in phase with the waves or their derivative. •

A drag force proportional to the relative velocity of the water with respect to the floating object appears in the horizontal plane. This force is ¼ cycle ahead of the vertical force. It can therefore be modeled as a linear function of the derivative of the wave which introduces that lag. (10)Fhx=Dh⋅(ζ˙x−x˙) (11)Fhy=Dh⋅(ζ˙y−y˙)where Dh is a drag coefficient that is function of the floating platform geometry, and ζ˙x and ζ˙y are the projection of the horizontal motion of the wave onto the x and y directions.

where Kh is a buoyancy coefficient that depends on the barge geometry.

The hydrodynamic external torque, Th3x1, can be modeled by a cross product of the vector G2B=CoB−CoG and the hydrodynamic force vector. (13)Th3x1=G2B×Fh3x1,

In addition to the hydrodynamic disturbances caused by waves, the aerodynamic effects generated by wind must be considered. The two components of the aerodynamic forces acting on the wind turbine blades can be expressed in terms of the wind speed (v), measured at the nacelle, the blade pitch angle (β), and the tip-speed ratio (λ=ωrot⋅R/v), where ωrot is the angular speed of blades and R is the blade length: •

The blades lift force (Fal) is function of the lift coefficient (Cl) and the square of the wind speed. (15)Fal=Cl(β,λ)⋅v2

where, Cl(β,λ) and Cd(β,λ) are tabulated functions of the blade pitch angle and the tip-speed ratio.

The values of the local lift and drag coefficients, as well as other parameters, may vary depending on the design and specific operational conditions of the wind turbine. For a control-oriented model, a rotor averaged performance of lift and drag forces around the nominal conditions is more appropriate. The tip speed ration is function of ω and v, therefore the lift and drag coefficients may be linearized at wind turbine nominal conditions as functions of (β,v,ω) as it is shown in (17). (17a)Fal(β,v,ω)≈(Faln+∂Cl∂βn(β−βn)+∂Cl∂vn(1v−1vn)+∂Cl∂ωn(ω−ωn))⋅v2, (17b)Fad(β,v,ω)≈(Fadn+∂Cd∂βn(β−βn)+∂Cd∂vn(1v−1vn)+∂Cd∂ωn(ω−ωn))⋅v2where Faln, Fadn, βn and λn, denote the nominal values used to obtain the nominal lift and drag forces.

For wind aligned with the nacelle, lateral and vertical aerodynamic forces can be neglected, resulting in the vector of aerodynamic forces shown in Eq. (18). (18)Fa3×1≈[Fad(β,v,ω)00]

The aerodynamic forces acting on the wind turbine generate the following torque components:

For small pitch angles, under nominal conditions the external torque vector may be approximated by Eq. (19). (19)Ta3x1≈[R⋅Fal(β,v,ω)|G2A|⋅Fad(β,v,ω)0]

The lateral forces that would appear if there were a yaw alignment error have not been considered, since it is assumed that the nacelle is aligned with the wind by the wind turbine’s yaw control, although in any case these effects will be lower than the considered drag force (Eq. (18)).

In a wind turbine, it is common to control the rotor speed through two signals: the electromagnetic torque, τgen, and the blade pitch angle, β, depending on the operating region in which the wind turbine is working, which in turn depends on the wind speed (low-wind and high-wind regions, respectively) [45]. The forces and torques generated by these actuators are also modeled as a linear combination of the control signal and its derivative, in order to capture the phase shift of the actuators.

The rotor and generator are connected by a gearbox with a ratio of N, and therefore their angular velocities are proportional. Thus, the dynamics of the entire system can be modeled as a function of the rotor’s angular velocity, including the equivalent inertia of the generator. For this purpose, the effective inertia of the system is defined as Iω=Irot+N⋅Igen.

Although the system behaves as a first-order system with respect to the rotor’s angular velocity, the electrical component of the generator is subject to inductive effects. These effects introduce extra inertia into the generator current; therefore, it is necessary to add another order to the model by introducing the derivative of the angular velocity in the system state. The wind turbine’s behavior is represented in Eq. (20). (20)Iω⋅ξ˙ω=Aω⋅ξω+Tav,β+Teω,ewhere:

The wind turbine state vector is given by, (21)ξω=[ω˙,ω]T

The matrix Aω is given by, (22)Aω=[−Lω−Dω10]where:

Dω is the damping coefficient, mainly due to mechanical friction in the system.

The dynamics of the wind turbine are perturbed by a sum of different torques acting on the rotor:

Tav,β is the aerodynamic torque, which is proportional to the effective wind captured by the turbine and the blade pitch angle control signal. The effective wind speed is modified by the degrees of freedom of the wind turbine, primarily the rotational DoFs, which generate large fore–aft oscillations of the nacelle. The blade aerodynamics, which depend on the pitch angle as discussed earlier, also strongly influence this torque.

At nominal conditions the wind torque input may be approximated by Eq. (23), where Fbln is a function of β. (23)Tav,β≈[R⋅Fal(β,v,ω)0]

The electromagnetic torque input may be approximated by Eq. (24), where τgen is the torque control signal. (24)Teω,e≈[−τgen0]

This mathematical development captures the main forces and moments acting on the FOWT.

Once the different forces and moments experienced by a wind turbine have been obtained, this section performs an integration of the different components of the dynamic model to analyze the coupling between the DoFs. Fig. 3 shows a block diagram that considers the main couplings between translational, rotational, and wind-turbine DoFs in response to external disturbances. The system is composed of linear blocks with feedback, which facilitates its identification through classical techniques.

The inputs to the model are wind speed and wave elongation. For cross-swells, the effective waves that excite the degrees of freedom (x, y, θ, ϕ) can be modeled as a function of the wave direction, γ, as described in Eq. (25). (25)ζx=ζϕ=ζ⋅cos⁡(γ)ζy=ζθ=ζ⋅sin⁡(γ)

Furthermore, the surge and sway motions affect the effective wave through the feedback from Eq. (2). Regarding the effective wind speed captured by the wind turbine, it is disturbed by the fore–aft motion of the nacelle, due mainly to the rotational motion of the FOWT, and is therefore represented by feedback from Eq. (4).

The force module, represented by Eqs. (10)–(12) and (18), works with the relative wave elongation, (ζ−z) and the relative horizontal speed (ζ˙−x˙), both supplied by feedback from Eq. (2). The force module also depends on the aerodynamic coefficients, which are functions of the wind speed, the blade pitch angle, and the rotor speed, (β,v,ω), the latter supplied by feedback from Eq. (20). The torque module, Eqs. (14) and (19), works similarly. The coupling of the translational and rotational DoFs occurs through the hydrodynamic torque, which depends on the derivative of the surge and sway. However, yaw is only excited indirectly through gyroscopic coupling with the other rotational degrees of freedom. This gyroscopic coupling is avoided in vertical-axis wind turbines, where all elements rotate with respect to the z-axis.

The wind turbine model, Eq. (20), is excited by the aerodynamic and generator torques, Eqs. (23) and (24), which are functions of the effective wind and the control signals. In addition, the wave disturbance affects these torques as a simplified way to represent the effect of nacelle motion.

It is important to point out that the translational model shown in Eq. (2) influences the rotational model presented in Eq. (4) through feedback in the torque module, but not the other way around. Because they are independent, the translational model can be identified in advance. This significantly reduces the number of parameters to be estimated at each stage of identification. The relative signals used in the model can be measured in an experimental setting for identification purposes.

In a model as this, with so numerous feedback loops, achieving convergence is challenging. Therefore, less significant components were excluded when describing the forces. In addition, the model was further simplified by retaining only the dominant effects. As shown later, these components can be identified separately, leading to a stable model.

The parameters of the general model presented in the previous section are identified for a specific turbine. Synthetic data for all degrees of freedom (DoFs) considered in the simplified model were generated using the high-fidelity simulation software OpenFAST. These data were subsequently used to identify the parameters of the proposed linear model via the MATLAB System Identification Toolbox.

This work employs the National Renewable Energy Laboratory (NREL) 5 MW floating offshore wind turbine [46], specifically mounted on the ITI Energy barge platform [47]. The main characteristics of this FOWT are summarized in Table 2. The turbine features an 8-line mooring system (two lines at each barge corner) and is a variable-speed, collective pitch-controlled machine.

Model identification requires exciting the system around its nominal operating point. To this end, the FOWT was subjected to realistic input signals. To ensure sufficient excitation of the relevant dynamics, the OpenFAST simulations are performed under combined wind and wave loading conditions. The wind field is generated using a turbulent wind model, while wave excitation corresponds to a predefined sea state. Fig. 4 shows the excitation signals over two intervals: 0–600 s (used for model identification) and 600–1200 s (used for validation). The inputs comprise wind, wave, and actuator control signals, as shown in Fig. 4.

The FOWT was excited with high-irregular waves corresponding to Sea State Number (SSN) 6 on the Douglas Sea Scale, generated using a JONSWAP spectrum [48] to induce significant barge motion. Waves were introduced at a 30∘ angle of attack (α) relative to the surge axis to simultaneously excite multiple DoFs. This oblique incidence projects wave forces onto the surge, sway, pitch, and roll DoFs.

Wind speed, generated around 14 m/s, incorporates small step changes (Fig. 4, orange line) to perturb the system’s gain and dynamics.

Actuator signals—collective blade-pitch angle and generator torque—were designed as chirp signals comprising segments of varying frequency and amplitude (Fig. 4, yellow and purple lines). This design excites the system over a broad frequency range to facilitate comprehensive dynamic identification.

The OpenFAST response of the 5 MW FOWT to these excitations is shown in Fig. 5 (top) depicts the rotor speed (rpm, blue line) and generator output power (MW, orange line). External disturbances from wind and waves significantly affect both variables. Step-like changes in wind speed produce corresponding transient responses. Furthermore, nacelle motion modulates the effective wind speed, inducing oscillations in rotor speed and power, primarily driven by wave-induced rotational dynamics. These simulations were performed without feedback control; consequently, increased wind speed leads to uncontrolled rotor acceleration and higher power generation.

Wave excitation perturbs all DoFs (Fig. 5, middle and bottom). As anticipated, wind primarily affects surge (Fig. 5, middle, blue line) and induces a mean offset in pitch (Fig. 5, bottom, orange line). Even more, as shown in Fig. 5 (the middle one), the surge can cause the FWOT structure to shift approximately 20 m. This occurs when the wind begins to push and the turbine moves backward, because the mooring lines allow this drift. Therefore, to identify the model, it will be necessary to use the wave displaced to the FOWT position, which will be handled by the effective wave block in Fig. 3.

Rotor speed variations must be included in the model due to their influence on the gyroscopic effect’s angular momentum and the blades’ aerodynamic coefficients. For translational DoFs: surge exhibits a strong, wind-dependent response; sway shows a transient followed by an oscillatory, wave-dominated response; and heave correlates directly with wave elevation. For rotational DoFs: roll and pitch are highly correlated with waves, while yaw responds at distinctly different frequencies. Additional coupling effects between DoFs, though not visually apparent, will be captured via mathematical identification.

Control actuators are incorporated as model inputs. For the 5 MW barge-type turbine, as for most wind turbines, two control signals are available: generator electromagnetic torque (τgen) and collective blade-pitch angle (β). The operational region—determined by wind speed (below or above rated)—dictates which actuator is primary [45]. The forces and torques from these actuators are modeled as a linear combination of the control signal and its derivative to capture actuator phase dynamics.

While rotor speed (ω) is primarily excited by wind, it is also coupled with other DoFs through changes in the effective rotor-swept area and relative wind speed. The normalized rotor speed equation to be identified is: (26)ω¨=(LωDω)⋅(ω˙ω)+(TfbdζTfbkζ)⋅(ζ˙xζx)+Tav+(TdβTkβ)⋅(β˙β)+Teτgenwhere coefficients are normalized by the wind turbine effective inertia Iω. Terms Lω and Dω represent rotor inertia and damping. Tfbdζ and Tfbkζ represent a simplified translational-DoF feedback in effective wave computation, capturing phase shift. Ta quantifies wind speed influence. Tdβ and Tkβ capture the blade-pitch actuator dynamics. Te represents the generator torque influence. Rotor speed couples into other FOWT DoFs, primarily those affected by drag and lift coefficients (e.g., surge in Eq. (18)).

The translational DoF equations are: (27)x¨=(DxKx)⋅(x˙x)+(FhdxFhkx)⋅(ζ˙xζx)+Fadω⋅ω+Favx⋅v+(FadxFakx)⋅(β˙β) (28)y¨=(DyKy)⋅(y˙y)+(FhdyFhky)⋅(ζ˙yζy) (29)z¨=(DzKz)⋅(z˙z)+(FhdzFhkz)⋅(ζ˙ζ)where coefficients are normalized by the FOWT mass and added mass, M, D and K represent the inherent damping and stiffness of each DoF. Fhd(⋅) and Fhk(⋅) model hydrodynamic wave forces, required to capture the phase shift under real conditions. Fadω relates aerodynamic force to rotor speed via the drag coefficient. Favx captures the primary aerodynamic force from wind. Fadx and Fakx represent blade-pitch influence on aerodynamic forces, capturing actuator phase dynamics.

As a result, added mass effects are implicitly included in the normalized model coefficients, even though they are not modeled explicitly as separate terms.

The rotational DoF equations are: (30)θ¨=(DθKθ)⋅(θ˙θ)+(ThdθThkθ)⋅(ζ˙θζθ)+Thθy˙y˙+Taθvv+(TadθTakθ)⋅(β˙β)+Teθτgen (31)ϕ¨=(DϕKϕ)(ϕ˙ϕ)+(FdϕFkϕ)(ζ˙ϕζϕ)+Fhϕx˙⋅x˙+Gϕψ˙ψ˙+Taϕωωb+Taϕvv+(TadϕTakϕ)(β˙β) (32)ψ¨=(DψKψ)⋅(ψ˙ψ)+(FdψFkψ)⋅(ζ˙ζ)+Gψϕ˙ϕ˙+Cψωωwhere coefficients are normalized by the FOWT inertia I. Thd(⋅) and Thk(⋅) represent wave-induced torques. Coupling terms are included: Thθy˙ (roll-surge), Fhϕx˙ (pitch-surge), and gyroscopic couplings Gϕψ˙ and Gψϕ˙ between pitch and yaw. Aerodynamic, and control input coefficients follow a structure similar to the translational case.

The parameters for the selected FOWT were estimated using OpenFAST simulation data and the MATLAB System Identification Toolbox. Discrete transfer functions for the seven Eqs. (26)–(32) were identified using ARX (Autoregressive with Exogenous Input) models [49]. This choice is motivated by the need for a computationally efficient and transparent identification framework that is consistent with the control-oriented nature of the proposed model. Both wind and wave excitations are considered, depending on the degree of freedom being identified. The ARX structure employed two poles to capture the common second-order dynamics and a zero for inputs with derivative influence.

Fig. 6 validates the identified model (orange dashed line) against the OpenFAST nonlinear simulation (blue line). The root mean square error (RMSE) between the signals quantifies the model accuracy.

The identified model replicates the OpenFAST response effectively, capturing the correct frequencies and amplitudes for both rotor speed and power. Output power is proportional to rotor speed and generator torque; the identified proportionality constant encapsulates the gearbox ratio and generator efficiency.

Fig. 7 shows the validation for the translational DoFs, with fit quality assessed via the RMSE. The identified model accurately captures surge behavior despite its distinct frequency response. Sway is well-represented at low frequencies, though some high-frequency components are less precise. Heave is modeled with high fidelity. Cumulative errors for surge and sway are minimized.

Similarly, Fig. 8 presents the validation for rotational DoFs. Roll and pitch, which exhibit predominantly linear behavior, are identified accurately. Yaw dynamics are more nonlinear, resulting in a less precise match; the model captures the frequency but slightly underestimates the amplitude, suggesting a potential need for a more complex nonlinear model for this DoF.

The fitted discrete models were converted to continuous state-space to extract the parameters in Eqs. (26)–(32). For the rotational DoFs (Eqs. (30)–(32)), the resulting identified transfer functions yield the following continuous representations (coefficients rounded for brevity): (33)θ¨=(−0.0182−0.2819)(θ˙θ)+(−0.52800.01895)(ζ˙θζθ)+0.00089v+(0.021840.00335)⋅(β˙β)+(0.0076860.002594)⋅(τ˙genτgen) (34)ϕ¨=(−0.1048−0.2678)(ϕ˙ϕ)+(0.4314−0.0662)(ζ˙ϕζϕ)+0.0214v+(0.1906−0.0244)(β˙β) (35)ψ¨=(−0.01222−0.02913)(ψ˙ψ)+(−0.13160.04354)(ζ˙ζ)+0.03978ϕ˙+(0.15420.00808)(ω˙ω)

For pitch and roll, the identification achieved optimal results considering only disturbances (waves, wind) and relevant control signals (blade-pitch and, for roll, generator torque). Other inputs provided negligible improvement. Yaw dynamics are distinct, being directly perturbed only by waves. While control signals have no direct effect, gyroscopic coupling with pitch and rotor speed are significant perturbations.

As a summary, it is possible to conclude that

Roll and pitch are linear and well-identified.

This section analyzes the system response to wave perturbations in the frequency domain [50]. The primary objective is to identify which degrees of freedom (DoFs) most critically affect energy production and which pose the greatest risk to structural stability and safety. Ensuring structural integrity and preventing emergency shutdowns is paramount, with particular concern for the pitch and roll DoFs. Oscillations in these modes induce fore-aft and side-to-side nacelle movements, leading to accelerated fatigue and potential structural damage.

Waves represent the dominant external disturbance, especially under aggressive sea states common in many offshore locations. The analysis focuses on the frequency response of the rotational DoFs (roll, pitch, yaw) to wave excitation characterized by a JONSWAP spectrum for Sea State Number 6 (SSN6), which provides significant energy to excite these modes.

Fig. 9 compares the power spectral density (PSD) of SSN6 waves with the magnitude (gain) of the transfer functions from wave excitation to the rotational DoFs, as derived from the identified models (Eqs. (33)–(35)).

The resonance frequencies for roll (red) and pitch (yellow) align closely with the peak of the wave energy spectrum. In contrast, the yaw mode (purple) resonates at a significantly lower frequency. This confirms that roll and pitch are more susceptible to direct wave excitation, as observed in the time-domain simulations. Furthermore, when waves approach at an oblique heading angle, the effective wavelength increases (and the effective frequency decreases), potentially shifting the spectral peak closer to the pitch and roll natural frequencies and thereby amplifying oscillations.

The gyroscopic coupling from pitch to yaw (green), calculated by applying the pitch gain to the identified pitch-to-yaw transfer function, exhibits two minor peaks at the natural frequencies of yaw and pitch. Its magnitude decays rapidly across most of the wave frequency range, indicating a secondary influence compared to direct wave excitation. This coupling between yaw and pitch through the gyroscopic effect means that if we reduce the pitch oscillation, we will also automatically reduce the yaw oscillations. Nevertheless, in light of the results, as explained in the roll control problem, the gyroscopic moments of the rotor and the generator could have cancelled each other out by design. To give greater weight to the gyroscopic effect, it would be necessary to redesign the gearbox to avoid a change in the direction of rotation.

Altering the resonant frequencies of critical DoFs would require structural redesign—a core aspect of control co-design that relies on a deep understanding of the system dynamics. The presented frequency-domain analysis provides a tool for iteratively evaluating new designs and formulating mathematical criteria for performance optimization.

An alternative solution is to implement active control loops that modify the dynamic response of the resonant DoFs, attenuating peak gains and reducing damaging oscillations. To demonstrate this approach, a simple control loop is designed using the identified linear model and subsequently validated on the high-fidelity nonlinear simulation.

A control strategy is proposed to modify the turbine’s dynamic response by feeding back rotational states to the available actuators. The design is based on the identified transfer functions relating the control signals (collective blade-pitch angle, β, and generator torque, τgen) and disturbances to the rotational DoFs, specifically Eqs. (34) and (35) for roll and pitch.

Initial analysis considers proportional (P) feedback. The root locus for roll control (Fig. 10) indicates that, in principle, stable closed-loop poles can be achieved using either the blade-pitch or generator torque actuator.

However, the FOWT presents a strongly under-actuated control problem: the authority of the actuators is severely limited relative to the disturbance forces acting on the platform. Realistic maximum feedback gains must be estimated based on actuator saturation limits and expected oscillation amplitudes. Assuming actuation margins of ±8° for blade-pitch and ±15 kNm for generator torque, and desired control ranges of ±10° for roll and ±5° for pitch, the maximum feasible gains are calculated (Table 3).

Applying these maximum gains to the roll loop results in closed-loop poles (blue squares in Fig. 10) virtually overlapping the open-loop poles, indicating negligible performance improvement. The blade-pitch effect on roll is inherently weak. The similarly weak effect of generator torque was unexpected and is likely due to the gearbox configuration. Typically, the generator rotor turns opposite to the blades to mitigate gyroscopic effects, causing the generator’s electromagnetic torque to be compensated by the aerodynamic torque on the blades, nullifying its net effect on the roll moment. This insight highlights how such analysis can inform co-design, suggesting that a gearbox redesign could enhance this control pathway.

In contrast, proportional control for the pitch DoF shows greater promise. The root locus (Fig. 11, left) allows for meaningful pole relocation. The achievable closed-loop poles (blue squares), constrained by the maximum gain from Table 3, provide a damping ratio of approximately 0.4.

Performance is improved by adding a derivative (D) term to provide phase lead, increasing the damping ratio. The resulting PD controller places the closed-loop poles closer to the real axis (blue triangles, Fig. 11, right). With the actuator saturation constraint, the achievable closed-loop system has a damping ratio of 0.27 and a natural frequency of 0.45 rad/s, moving it farther from the peak wave excitation frequency. The designed PD controller is: (36)β=βnom+85(0.7ϕ˙+0.7ϕ)

The effectiveness of the PD controller is first evaluated in the frequency domain (Fig. 12). The closed-loop transfer function (yellow, top) shows a significant reduction and shift of the resonance peak compared to the open-loop system (orange). The bottom graph plots the product of the wave PSD and the pitch gain, representing the propagated energy density. The reduction in the area under this curve (the integral) provides a direct index for control tuning or co-design optimization.

Time-domain verification using the linear model (Fig. 13) confirms the controller’s performance. Under combined wind and irregular wave disturbances, the PD controller (black line, top) reduces the pitch oscillation amplitude from approximately 6° to 3°, compared to the uncontrolled response (blue line). The phase shift observed in both responses is induced by the wind disturbance. To discount this shift, the standard deviation is used as a performance index, yielding a reduction factor of 0.43 with PD control. The corresponding control signal (bottom) remains within the specified ±8° limit, saturating only rarely.

Finally, the controller is validated on the high-fidelity OpenFAST nonlinear model (Fig. 14). The PD control law (Eq. (36), with βnom=8∘) is implemented in Simulink, and the system is subjected to the same disturbance sequence.

The results closely match the linear model verification. The PD controller reduces pitch oscillations from ~6° to ~3°, with brief intervals near 4° due to strong wind events that induce a mean pitch offset—an effect the controller cannot fully reject due to performance limitations. Consequently, the reduction in the standard deviation index is slightly less pronounced than in the linear case. The control signal operates within its bounds, oscillating around the nominal value and only deviating significantly to counteract sustained wind-induced moments. This confirms the appropriateness of the gain limitation and demonstrates the successful translation of the linear design to the complex nonlinear system.