Robust structural control of an underactuated floating wind turbine

This paper investigates the dynamic modeling and robust control of an underactuated floating wind turbine for vibration suppression. The offshore wind turbine is equipped with a tuned mass damper on the floating platform. The Lagrange's equation is employed to establish the limited degree-of-freedom dynamic model. A novel disturbance observer-based hierarchical sliding mode control system is developed for mitigating loads of the underactuated floating wind turbine. In the proposed control scheme, two prescribed performance nonlinear disturbance observers are developed to estimate and counteract unknown disturbances, where the load induced by wave is considered as a mismatched disturbance while the load caused by wind is treated as a matched disturbance. The hierarchical sliding mode controller regulates the states of such an underactuated nonlinear system. In particular, the first-order sliding mode differentiator is used to avoid the tedious analytic computation in the sliding mode control design. The stability of the whole closed-loop system is rigorously ana-lyzed, and some sufficient conditions are derived to guarantee the convergence of the states for the considered system. Numerical simulations deployed on both the design model and the National Renewable Energy Laboratory 5-MW wind turbine model are provided, which demonstrate great effectiveness and strong robustness of the proposed control scheme.

wind turbine structures. Therefore, vibration suppression of floating wind turbines becomes a significant topical area of research in order to reduce their maintenance costs and increase their life cycle.
During the past decade, many efforts have been made to suppress the vibrations of wind turbine by reducing tilt motions of the tower top or the floating platform. The representative methodologies mainly include passive, semi-active, and active structural control. Passive structural control systems do not require the power supply with constant parameters, such as tuned mass damper (TMD). 6 An advantage of TMD-based control is that it does not disturb the power generation while its disadvantage is that it needs extra mass. This disadvantage can be minimized if an existing turbine component can serve as the mass component, such as the reservoir in a hydrostatic wind turbine. 7 By comparison with passive structural control systems, semi-active structural control systems have time-varying parameters that can be tuned during operation, such as semiactive TMD 8 and magnetorheological dampers. 9 Unlike passive control systems, active structural control designs use an actuator to produce a control force on the mass and structure; its advantage is that a greater impact on the platform pitch or the tower bend angle can be achieved. 10 This paper focuses on active control. Active structural control designs with TMDs installed in the nacelle were considered for vibration suppression of the floating offshore wind turbines, 11 where the H ∞ multivariable loop-shaping method was used to design an active structural controller.
To simplify the control architecture, a generalized H ∞ approach was proposed for a TMD on the platform. 12 The TMD-based floating wind turbine was regarded as a simplified linear model with small angle approximations in other studies. 11,12 However, the floating wind turbine is an underactuated nonlinear coupled system with both mismatched and matched disturbances. Therefore, it is useful to develop an advanced controller to handle the nonlinearity effectively.
Sliding mode control is a popular robust nonlinear control technique due to its strong robustness against parameter variations and external disturbances. 13 Over the past two decades, sliding mode control has been successfully applied to many practical underactuated physical systems, such as inverted-pendulum systems, 14 planar vehicles, 15 mobile robots, 16 underwater vehicles, 17 and hypersonic vehicles. 18 Different from the existing sliding mode control structures, [14][15][16][17][18] Wang et al. 19 proposed a hierarchical sliding mode control scheme, which can be accommodated to most second-order underactuated systems. With the aid of the proposed hierarchical sliding mode control, 19 an adaptive controller was developed to stabilize four states of the spherical robot. 20 However, the above papers do not consider adverse effects of mismatched disturbances.
Although Xu et al. 21 considered this issue, their sliding mode control approach was only effective for diminishing disturbance rejections. The TMD-based floating offshore wind turbine is a more complex underactuated nonlinear coupled system with both matched and mismatched disturbances. To the authors' knowledge, there is no results reported to design a sliding mode controller for such a system. Motivated by the aforementioned discussions, a more complete nonlinear dynamic model is established for the TMD-based floating offshore wind turbine system without linearization. A disturbance observer-based adaptive hierarchical sliding mode control algorithm is proposed to regulate both actuated and unactuated degrees of freedom for this nonlinear coupled system with nondiminishing disturbances. Two disturbance observers are independently designed to estimate both matched and mismatched disturbances respectively, where the estimated accuracy of each disturbance observer can be adjusted by only one parameter. The established model is divided into two subsystems; several sliding variables are constructed to develop a hierarchical sliding mode controller for such a system. To reduce the computing burden, the first-order sliding mode differentiator is used to estimate the derivative of the designed disturbance observer. Finally, strong robustness and excellent control performance of the proposed control algorithm are shown by the simulation tests.

| SYSTEM DESCRIPTION AND PROBLEM FORMULATION
In this section, we briefly introduce the floating wind turbine with the TMD configuration and establish an underactuated nonlinear coupled dynamic model.

| A TMD-based National Renewable Energy Laboratory 5-MW floating wind turbine
We start with a description of the structure of a TMD-based floating wind turbine as depicted in Figure 1, which is primarily composed of a rotor-nacelle assembly (RNA), a tower, an ITI Energy platform, a mooring system, and a TMD. The nacelle houses a few mechanical and electrical components such as the drivetrain, generator, and converters. The tower is mounted on the ITI Energy barge platform. The rigid barge platform is moored by eight catenary lines to alleviate drifting. The TMD is placed on the barge platform. As shown in Figure 1, the platform has six motion degree-of-freedoms (DOFs), which includes three translational DOFs (i.e., surge, sway, and heave) and three rotational DOFs (i.e., roll, pitch, and yaw), where X, Y, and Zrepresent the set of orthogonal axes with their origin denoted by O. The X-axis designates the nominal downwind direction, the XY-plane represents the mean sea level, and the Z-axis points upward opposite to gravity along the centerline of the unbending tower when the platform is undisplaced. In this study, the National Renewable Energy Laboratory (NREL) 5-MW baseline wind turbine 22,23 is used for the analysis and control design. The physical parameters of the NREL 5MW baseline wind turbine and the ITI barge platform are listed in

| Dynamic modeling of a TMD-based floating wind turbine
Note that the fore-aft direction has the largest loading from winds and waves. It is obvious that the highest fatigue damage on the tower is from this direction. Three most-relevant DOFs originated from the tower first bend mode, the platform pitch motion, and the TMD motion. For modeling purposes, the tower is regarded as an inverted pendulum with the structural damping and stiffness, which are modeled as a rotary damper and rotary spring at the base of the rigid body. The effect imposed on the barge platform from the mooring lines and the hydrodynamics is considered as a linear spring and a linear damper. Therefore, the kinetic and potential energies of the TMD-based floating offshore wind turbine can be expressed as where T op is the total kinetic energy, V op is the total potential energy, θ t is the bend angle of the pendulum tower from the Z-axis, θ p is the pitch angle of the platform, x a is the longitudinal displacement of the TMD, I tp represents the inertia moment of the tower & RNA, I bp is the platform pitch inertia, m a is the mass of the TMD, k tp is the equivalent pitch restoring coefficient of the tower and RNA, C hs is the hydrostatic pitch restoring coefficient, C ml denotes the linearized pitch restoring coefficient from mooring lines, m t is the total mass of the tower and RNA, g is the gravitational acceleration, L t is the distance from the mass center of the tower and RNA to the reference point O, m p is the mass of the platform, L p is the distance from the mass center of the platform to the reference point O, k a denotes the stiffness of the TMD, and h c is the distance from the mass center of the TMD to the reference point O. The nonconservative forces acting on the tower first bend mode, the platform pitch motion, and the TMD displacement can be described by where A rad is the added pitch inertia associated with hydrodynamic radiation, B rad is the pitch damping coefficient with respect to hydrodynamic radiation, B vis is the linearized pitch damping coefficient with regard to hydrodynamic viscous drag, d tp represents the equivalent pitch damping coefficient of the tower & RNA, M w denotes the total wave-excitation pitch moment from diffraction applied at the reference point O, F a is the aerodynamic rotor thrust acting on the hub, and L hh is the height of the hub, c a is the damping coefficient of the TMD, and F active is the force delivered by the active control system. According to the Lagrange's equation approach, we have d dt d dt where L op =T op −V op . By virtue of (1)-(8), the overall vibration dynamic model of the TMD-based floating offshore wind turbine can be expressed as For control purposes, we define state variables as x a , and rewrite the whole system (9)-(11) as where Note that x 5 and x 6 are the displacement and velocity of the TMD. The TMD is placed on the platform; its displacement and velocity are necessary to be restricted in scope due to the limited space of the platform (as shown in Table 1). Note that d w and d z are the external excitations from the effects of wave and wind. Assume that d w and d z are the bounded disturbances, which are required to satisfy the conditions |d w | ≤ D w and |d z | ≤ D z , where D w and D z are positive constants. u is the control input. The control objective is to design a single controller u to guarantee the convergence of x 1 , x 2 , x 3 , and x 4 for the underactuated floating offshore wind turbine in the presence of both mismatched and matched disturbances.
Lemma 1. 25 : The first-order sliding mode differentiator is given by where ζ 0 , ζ 1 , and η 0 are the states of the system (15), ϵ 0 and ϵ 1 are the design parameters of such a sliding mode differentiator, and h(t) is a known function. Then, η 0 can approach the differential term _ hðtÞ with arbitrary accuracy if the initial deviations ζ 0 −h(t 0 ) and η 0 − _ hðt 0 Þ are bounded.

Lemma 2. 26 : Consider the continuous and differentiable bounded function
where D is a positive constant.

| MAIN RESULTS
From (12)-(14), it is clear to see that the floating offshore wind turbine with the TMD configuration is a complex underactuated nonlinear coupled system with both matched and mismatched disturbances. The conventional sliding mode control cannot be directly applied in such a system due to the existence of underactuated strong coupled features and mismatched disturbances. In this section, we will present both the controller design and the stability analysis.

| Disturbance observer-based hierarchical sliding mode control design
Note that the nondiminishing disturbances cause big difficulty in the design of the sliding mode control system, especially for the presence of mis- Then, it is indicated from (12) that Using (16), we construct the disturbance observerd w to estimate the mismatched diturbance d w in the following form: where β p > 0.
Lemma 3. By applying the disturbance observer (17) where Δ 1 can converge to an arbitrarily small constant.
Proof. Choose the following Lyapunov-like function candidate The time derivative of V w can be calculated as Substituting (16)- (19) into (22), we have Applying Young's inequality, 27 based on (23) we have From (24), we haved According to (25), we derive Therefore, the estimation errord w can converge to any arbitrarily small constant by appropriately selecting one adjustable parameter. In other words, we can choose a large β p to ensure the excellent estimation performance.
Similarly, a disturbance observer with prescribed performance is designed to estimate the matched disturbance d z for disturbance rejections.
Define − a t5 , a t6 , a t7 , a t8 T , Then, we rewrite (13) as In this case, the disturbance observer can be given bŷ where β z > 0.

Lemma 4. The disturbance observer (28)-(30) is designed to guarantee that the estimation errord
where Δ 2 can converge to an arbitrarily small constant.
Proof. Consider the following Lyapunov-like function candidate With the similar derivation process as the proof of Lemma 3, the time derivative of V z can be computed from (27)-(30), which can be expressed as By virtue of (33), we haved which implies Therefore, we can select a large β z such that the estimation errord z converges to any arbitrarily small constant. According to the property of the considered system (12) and (13), we split the overall system into two subsystems, where one is (x 1 , x 2 , x 3 , x 4 )-subsystem given by (12)- (13) and the other is (x 3 , x 4 )-subsystem described by (13). It is important for us to construct proper sliding variables to regulate the states of two subsystems for vibration suppression. We define the errors The sliding variable of the (x 1 , x 2 , x 3 , x 4 )-subsystem is selected as where c 1 and c 2 are positive constants. The time derivative of s 1 is given by Substituting (12) and (13) into (42) yields Then, we have where ℓ 1 = a p4 a t1 −ðc 2 −a p2 Þa p1 , ℓ 2 = c 1 −a p1 −ðc 2 − a p2 Þa p2 −a p5 x 5 sin x 1 −a p6 cos x 1 + a p4 a t2 , ℓ 3 = ðc 2 −a p2 Þa p3 − a p4 a t3 , ℓ 4 = ðc 2 −a p2 Þa p4 + a p3 − a p4 a t4 , ℓ 5 = ðc 2 −a p2 Þa p5 cos x 1 + a p4 a t8 cos x 3 ℓ 6 = a p5 cos x 1 + a p4 a t6 : Let _ s = 0. The equivalent controller of the (x 1 , x 2 , x 3 , x 4 )-subsystem can be written as u 1eq = 1 a p4 b t1 ½ℓ 1 x 1 + ℓ 2 x 2 + ℓ 3 x 3 + ℓ 4 x 4 + ℓ 5 x 5 + ℓ 6 x 6 −ðc 2 −a p2 Þa p6 sin x 1 + ðc 2 − a p2 Þd w + _ d w −a p4 a t5 cos x 3 sin x 3 + a p4 a t7 sin x 3 + a p4 d z , Note that the controller (45) cannot be used in practice. It is necessary for us to replace the disturbances d w and d z with their estimationsd w andd z , respectively. Thus, we obtain u 1eq = 1 a p4 b t1 ½ℓ 1 x 1 + ℓ 2 x 2 + ℓ 3 x 3 + ℓ 4 x 4 + ℓ 5 x 5 + ℓ 6 x 6 −ðc 2 −a p2 Þa p6 sinx 1 + ðc 2 − a p2 Þd w + _ d w −a p4 a t5 cosx 3 sinx 3 + a p4 a t7 sinx 3 + a p4dz : To avoid the tedious analytic computation of the disturbance observer _ d w , the first-order sliding mode differentiator is introduced to estimate _ d w . Applying Lemma 1, the first-order sliding mode differentiator can be given by where ζ 0 , ζ 1 , and η 0 are the states of the system (47) and ϵ 0 and ϵ 1 are positive constants. By virtue of Lemma 1 and (47), we obtain where Δ 3 is an estimation error of the first-order sliding mode differentiator. It implies from Lemma 1 that jΔ 3 j ≤ Δ 3 with Δ 3 > 0. Replacing _ d w with η 0 , the equivalent controller of the (x 1 , x 2 , x 3 , x 4 )-subsystem can be expressed as The sliding variable of the (x 3 , x 4 )-subsystem can be chosen as where c 3 is positive constant. The time derivative of s 2 can be computed as Substituting (13) into the above expression, one gets _ s 2 = c 3 x 4 + a t1 x 1 + a t2 x 2 − a t3 x 3 − a t4 x 4 − a t5 cos x 3 sin x 3 + a t6 x 6 + a t7 sin x 3 + a t8 x 5 cos Let _ s 2 = 0. The equivalent controller of the (x 3 , x 4 )-subsystem is described as Note that only one control input can be used to control four state variables. This means that we need to use one control input to ensure two sliding surfaces can be reached (i.e., s 1 = 0 and s 2 = 0). The conventional sliding mode control method cannot be used for this underactuded nonlinear coupled system. To achieve this goal, a second-layer sliding variable is developed as where α 1 and α 2 are the sliding mode parameters, which satisfy α 1 > 0 and α 2 > 0. According to the variable structure theory, the switching control part is required to be designed to ensure the states can reach and thereafter stay on the sliding surface. Thus, the complete controller must include some portion of the switching control part, which can be expressed as where u sw is the switching term of the sliding mode controller, which can be given by where ϱ and λ are positive design parameters, which will be given later. Substituting (56) into (55) gives To reduce the chattering, in practice, some smoothing functions such as s jsj + σ and tanhðsÞ are introduced to replace the discontinuous sign function in the sliding mode controller (57), where σ is a small positive constant.
where ρ = ϱ − D > 0. Integrating both sides of (63), we have which indicates From (65), we have s 2 L ∞ . By virtue of (63), we obtain It is concluded from (66) that _ s 2 L ∞ . According to Barbalat's lemma, we have lim where _ x 4 is the angular acceleration of the tower. Hence, this indicates that _ s 2 2 L ∞ . By virtue of (54), we obtain s 1 2 L ∞ and _ s 1 2 L ∞ due to s 2 L ∞ and _ s 2 L ∞ : It is noted that the stability of the system does not depend on the parameters α 1 and α 2 . It is reasonable to define two different sliding variables as follows: where α 1g and α 1h are positive constants with α 1g 6 ¼ α 1h . Without loss of generality, we assume that Then, it implies from (67) that Applying (65), we have It follows from (70) that ρ Ð ∞ 0 js g jdτ < ∞, which indicates s g 2 L 1 . From (69), we obtain Thus, Similarly, we have From the above analysis, we have s 1 Note that s 2 = c 3 e 4 + e 5 = c 3 x 3 + x 4 . As a result of x 4 = _ x 3 in (13), it implies that we have lim t!∞ x 3 = 0. According to (12), (38), and (41), the original system can be expressed as where , E 2 = ½0,s 1 +d w T : Let Q = Q T > 0, then it implies from A is Hurwitz that the Lyapunov equation A T P + PA = −Q has a unique solution P = P T > 0. Choose the Lyapunov function candidate The time derivative of V 1 can be given by where λ min and λ max represent minimum and maximum eigenvalues, respectively. By virtue of (77), we have It follows from where κ = λ min ðQÞ 2λ max ðPÞ , δ = 2λ 2 max ðPÞ λ min ðQÞ jjE 2 jj 2 : Then, we have Note that δ can converge to an arbitrarily small value from (20) and (74). It implies from (80) that e 1 and e 2 can converge to an arbitrarily small value by appropriately choosing the design parameters. In other words, it indicates that both x 1 and x 2 will converge to an arbitrarily small value.
This proof is completed.

| SIMULATION STUDY
In this section, the proposed dynamic model is verified by using NREL 5-MW wind turbine model, and the developed disturbance observer-based hierarchical sliding mode controller is tested on both the design model and the NREL 5-MW wind turbine model within FAST code.

| NREL computer-aided engineering tools
The NREL FAST code is used to simulate the loads and the dynamic responses of the NREL wind turbine models. The NREL FAST code introduces the aerodynamics, structural (elastic) dynamics, hydrodynamics, control, and servo dynamics, which is primarily composed of InflowWind, HydroDyn, AeroDyn, ElastoDyn, ServoDyn, MoorDyn, and SubDyn modules. InflowWind is used to compute wind velocities with the help of the time series of wind speed vectors. ElastoDyn represents a structural-dynamic model that outputs displacements, velocities, accelerations, and reaction loads to AeroDyn and ServoDyn. ServoDyn involves control and actuator models. For structural control purposes, two independent single degree-of-freedom TMD systems are incorporated into the FAST code (i.e., FASTv8). 23 In this paper, the performance of the designed controller will be evaluated by using FAST (version 8).

| Model validation
To verify the proposed dynamic model for the floating offshore wind turbine, we let the design model (9) Figure 2, the simulation results of the platform pitch angles, the tower bending angles, and the TMD displacements in both models demonstrate a good match between them.

| Simulation results with the design model
In this section, the disturbance observer-based hierarchical sliding mode control architecture is implemented on the design model (12) is achieved with the proposed disturbance observer structures. The pitch angle displacement and velocity of the platform are provided in Figure 5, and the bend angle displacement and velocity of the tower are plotted in Figure 6. Figures 5 and 6 demonstrates that good control performances are achieved with the designed controller.
To evaluate the performance of the first-order sliding mode differentiator (47), the simulations are conducted with and without measurement noises, where the function randn is introduced to produce the measurement noises. The estimation results are plotted in Figure 7, and the estimated errors are shown in Figure 8. It indicates from Figure 8 that the estimated accuracy is degraded due to the adverse effect of the measurement noises.
In addition, multifrequency disturbances are also introduced in the simulations to further illustrate the advantage of the designed disturbance observers, where the mismatched disturbance and the matched disturbance are respectively described by

| Simulation results with the NREL 5-MW wind turbine model
To verify the active control performance, the proposed disturbance observer-based hierarchical sliding mode control algorithm is tested by the NREL 5-MW wind turbine model based on the FAST code, where the optimal parameters of the TMD are given by m a = 400 000 kg, k a = 103 019 N/m, and c a = 60 393 N/(m/s). 12 The wind conditions in all the cases are generated based on the IEC Kaimal spectral model with normal turbulence model in TurbSim. The wave conditions in all the cases are generated by the HydroDyn module based on the JONSWAP spectrum. The peak-spectral period of the incident waves in all the cases is set to 10 s with the significant wave height being 5.5 m. In the first case, the mean hub-height longitudinal wind speed is 18 m/s (above-rated), and the turbulence intensity is Category A. The  Figures 11 and 12 (for comparison purposes, both no TMD and passive control cases are considered), which shows that the proposed active controller has achieved significant improvement on the vibration suppression of the wind turbine compared with the passive control even if there exist model uncertainties and various disturbances such as winds and waves. To evaluate the energy consumption, it can be computed that the consumed average active TMD power accounts for 10% of the rated wind turbine power. A sensible design is that passive control works most of the time while the active control (basically adding an active force on top of the passive control) is activated only if the vibration is over certain limit. This can largely help reduce power consumption. To further validate the advantage of the proposed controller, the mean hub-height longitudinal wind speed of 24 m/s (above-rated) is chosen to conduct the simulation, and the turbulence intensity is Category B. As shown in Figures 13 and 14, the time responses of both the platform pitch angle and the tower bend angle are provided to demonstrate the vibration suppression performance. In this case, the average active energy consumption accounts for 12% of the rated wind turbine power. The simulation results without TMD and with passive control are plotted in Figures 13 and 14, which indicate that the great effectiveness and strong robustness of the proposed controller are achieved.

| CONCLUSION
In this paper, a nonlinear dynamic model for a TMD-based NREL 5-MW floating offshore wind turbine was derived. A disturbance observer-based hierarchical sliding mode control algorithm was proposed to stabilize such an underactuated nonlinear coupled system with both matched and mismatched disturbances. Two prescribed performance disturbance observers were independently constructed to estimate the matched and mismatched disturbances, and each estimation error can be adjusted by only one design parameter. With the disturbance observers, a hierarchical sliding mode controller was designed to suppress the vibration of the floating offshore wind turbine. Some sufficient conditions were derived to ensure the stability of the closed-loop system. The simulation results verified the accuracy of the developed design model and demonstrated the strong robustness and great effectiveness of the proposed control algorithm.