LQG control for hydrodynamic compensation on large floating wind turbines

This work proposes a novel Linear Quadratic Gaussian (LQG)-based blade pitch control method for floating offshore wind turbines, in which a state-space model of the turbine and water hydrodynamics is included in the LQG design. The actuation considered is collective blade pitch control with the objective of generator power stabilisation and platform motion reduction. A linear Kalman filter is used to estimate un-measurable states relating to wave excitation and radiation through measurements of generator speed, platform pitch, and wind disturbance. Controller design models were validated with the full order nonlinear model under various testing conditions. The new controller design is tested on a nonlinear high-fidelity simulation model of the 15 Mega-Watt (MW) floating semi-submersible wind turbine. In simulations with realistic stochastic wind and wave disturbances, the new controller achieves 32% lower generator speed Root Mean Square Error (RMSE) and 16% lower platform pitch RMSE compared to a standard LQG controller that does not include hydrodynamic states, for equivalent levels of pitch actuation and with a 2 ◦ /sec rate limit on pitch. The inclusion of hydrodynamics in the controller design not only reduced platform pitching fluctuation, but also had a strong effect of hub-height factors such as the generator speed.


Introduction
Wind turbines are currently growing and increasingly being deployed offshore as there is more potential for power generation [1].This comes at the price of complexity in modelling and controller designs.Control objectives for turbines differ according to wind speed.At below rated wind speed, the control objective is to maximise power output of the turbine.In above rated speeds, the objective is to pitch the blades to maintain the generated power at the rated power of the turbine's generator [2].
Blade pitching can be done collectively, with all blades at the same pitch angle, or individually, though collective pitching is more common in industry [3].Currently, the common control method for variable speed wind turbines is a Proportional Integral (PI) control.A PI is designed based on a single Degree-Of-Freedom (DOF) model for the drive-train [4].The P and I terms are gain scheduled according to dynamic variations caused by the varying wind speed [5].As wind speed increases, more blade pitching is required to avoid collecting more wind energy and keep the generated power at the rated value.
When looking at Floating Wind Turbines (FWT), the most widely investigated floaters are the semi-submersible, spar buoy, and barge [6].The COREWIND (COst REduction and increased performance of floating WIND technology) project developed a floating concept for the IEA (International Energy Agency) 15 MW reference FWT named Activefloat based on a semi-submersible floater [7] as shown in Fig. 1 [8].The lowest natural frequencies of a FWT are related to the six platform DOFs, with platform pitching and surging having the most influence on generated power [9].
Jonkman in [10] proposed a similar PI controller for FWTs as for onshore turbines.The PI was detuned to have a bandwidth below the floaters' mode frequencies.Such PI produced lower power and platform motion fluctuations compared to an onshore PI controller applied on a FWT.Other PI-based methods include adding an additional feedback loop related to the tower-top displacement for platform motion reduction.This method was applied in a reference PI controller discussed in [3].However, tower-top motion is difficult to accurately measure, and typically estimation of tower-top displacement from other sensors is required.
Advanced controllers were investigated on FWTs by Namik and Stol in [11].Authors tested a Linear Quadratic Regulator (LQR) controller on a barge type platform for a 5 MW turbine model.Authors in [12] tested an LQI (LQR with integral action) design on a FWT of 5 MW rating.The results gave better performance for the generator speed, platform pitch, and tower-top deflection compared to the baseline PI controller but with increased pitch actuation.Model  Generator speed state (MPC) was also investigated in literature for its power in handling hard and soft constraints on inputs and states [13].Authors in [14,15], and [16] proposed MPC controllers to reduce platform motion and power fluctuation on a barge-type 5 MW FWT.However, most of these studies did not consider state estimation, which is needed in practice as we typically do not have sensors available to measure all turbine states.Recently, researchers have started incorporating hydrodynamics in simulation models of offshore turbines.The most widely used is implemented in the simulation tool OpenFAST [17] developed by the National Renewable Energy Laboratory (NREL) which will be relied on in this work.Authors in [18] developed a linearised model incorporating structural and wave radiation state approximations in their design for generator power improvement.The controller was tested on a 5 MW FWT simulation model.Platform motion control was not considered.In our work, we include both wave excitation and radiation in State Space (SS) form.We address platform motion and generator power improvement through LQG (Linear Quadratic Gaussian) control of the combined turbine and hydrodynamic system.Carrying out controller testing on a nonlinear simulation of the IEA 15 MW FWT.
Therefore, the contributions of the work are the following: 1. To evaluate LQG controllers on a large scale FWT of 15 MW rating.Optimal controllers have been investigated in literature on smaller turbine models such as the 5 MW, but so far there have been very few studies using 15 MW models and we believe this to be the first investigation of LQG-type controllers on a 15 MW floating turbine using detailed nonlinear aeroelastic simulation tools such as OpenFAST.2. To introduce a controller design method incorporating wave excitation and radiation state space approximations for optimal control of platform pitch and generator speed on a FWT.Hydrodynamics were considered in literature in the simulation model, i.e., the testing model.In our work, we incorporate hydrodynamics in the controller design model, in addition to the simulation model.Showing that hydrodynamic effects can affect both hub-height (generator speed) and floater (platform pitch) responses.Wave radiation loads were considered in [18], but platform motion reduction was not attempted, and the controller additionally required actuation of generator torque.3. To demonstrate that Kalman Filtering on a reduced-order model can estimate non-measurable states for control of hydrodynamics through the measurements of generator speed, platform pitch, and wind speed only, for the 15 MW turbine under realistic stochastic wind and wave disturbances.Wave elevations are assumed to be un-measurable and our completed controller achieves hydrodynamic compensation for this turbine without direct measurement of water waves.
This work is organised as follows: The next section gives a theoretical background on hydrodynamics.Section 3 describes the method taken in the work.Section 4 validates the controller design models in open-loop.Section 5 shows the simulation setup and results for regular and irregular sea states under stochastic wind.Finally, conclusions are drawn in Section 6.

Hydrodynamics: Theory
The following subsections describe the theory behind the hydrodynamic factors that we rely on in our work.

Wave excitation
States related to wave excitation represent the forces applied on the platform due to waves applied on said platform in fixed position.Wave excitation forces can be described by the infinite integral below [19]: where   is the incident wave excitation kernels,  is the wave elevation, and   and ẋ are the approximated wave excitation states and their first time derivatives, respectively.The idea is to approximate the convolution integral shown in Eq. (1) in SS format, where F  is the output force caused by the disturbance, and     and   are the SS approximation matrices.
Authors in [20] proposed an Impulse Response Function (IRF) for the SS model approximation shown above.Obtaining the state matrices can be done via minimising a cost function of the following form [21].
where Q   is the cost, G is the weighting function, and  is the original impulse response.These impulse response models are converted to SS models for analysis and control design.

Wave radiation
States related to radiation represent forces on the platform in still water due to waves generated by the moving platform [19].The convolution integral in the Cummins equation [22] is used to describe radiation forces (  ): where  ∞ is a positive infinite-frequency mass matrix,  is known as the memory matrix [23], and  is the displacement vector of the six platform DOFs.The  term is difficult to obtain as it requires memory.
can be obtained from panel codes such as WAMIT [24] (Wave Analysis MIT).The full theory behind WAMIT and wave radiation approximations can be found in [23,24].It was recommended by the NREL team to use 60 s as the memory time when analysing the nonlinear model.More time will require more computational effort [23].

Wave models
Wave models applied on the floater can have several profiles.It is best to test FWTs under periodic wave motions as they resemble realistic waves.Several wave profiles can be used to test the controller.A more realistic wave model is JONSWAP (JOint North Sea WAve observation Project) [25].The JONSWAP model used in the study can be seen in the simulation section in Fig. 7(a), where it resembles characteristics of real wave data.

Methodology
The following explains the linearisation process adopted in the work and defines the controller design models used for controller design.Description of the control method adopted, and estimation procedure will be shown as well.

Overview
The work tests how beneficial in terms of floater and hub-height factors is including the hydrodynamic states into the controller design.To observe this benefit, we need another controller with the same objective except it does not account for hydrodynamic states.Since the relation between hub-height factors and wind variations is predominant compared to wave elevations, we wanted to test if hydrodynamic states can in fact have a positive impact on the generator speed (hub-height factor).In addition, in improving the floater's response.Based on that, we need two controller design models.The first referred to as the  design model includes only structural Turbine states.The second is named the TW design model, and includes structural Turbine states and hydrodynamic Wave states.State description of the  and TW models is found in Section 3.3.The design models were validated with the full nonlinear model by measuring the R 2 value.The R 2 value is a measure of data fit and can be calculated as follows [26]: where  is the sample number,   and ŷ are the actual and fit data, respectively, and  is the mean of the data.The closer the value is to 1, the more accurate the fit.It is best to test the effectiveness of both controllers using a complex model of the turbine.This was achieved by using a nonlinear model based on all the DOFs of the turbine.Such DOFs include blade flap and edge motions, tower-top side-toside and fore-aft motions, and floating platform DOFs.The complexity also lies in the representation of hydrodynamics in the nonlinear plant.WAMIT [24] is used by OpenFAST to solve the convolution integrals shown in Sections 2.1 and 2.2 to represent the wave radiation and excitation as discussed earlier.
Controller design was based on an LQG controller to find an optimal gain (   ∕  ) for the blade pitching commands.Since we are dealing with a non-ideal system, estimators are essential as there are unmeasurable states related to hydrodynamics (excitation and radiation).Estimation was aided by a linear Kalman filter.
Depending on whether we are using the  or TW controller design model, then the Kalman filter's mathematical model and LQG optimal controller calculation can use either  or TW SS model, hence the subscript T/TW in Fig. 2. The application was to compare both controllers under the same full order nonlinear plant model.The criteria behind which controller outperforms the other was based on a Pareto frontier through iterating the LQG output weighing matrix, and checking which controller had lower actuation yet achieving lower RMSE for the generator speed and platform pitch.
The controller block diagram of the work is shown in Fig. 2. The Kalman filter takes in the measurable nonlinear outputs of the generator speed  and platform pitch  subject to wind speed disturbance .Note that in a practical implementation, the controller would also need to be gain-scheduled based on wind speed  by repeating the controller design using the linearisations at different speeds  and interpolating the results, but this was not implemented for the simulations performed later in Section 5. Based on that information, the filter outputs estimated states x ∕  to generate a collective blade pitch command   ∕  through the optimal controller gain    ∕  .
It is worth noting that the  and TW controller design models are linear models of the studied turbine.In Section 4, we validate those linear models with the full order nonlinear model to investigate how well the linear models fit the nonlinear model.The  and TW linear models are referred to as controller design models since the controllers were designed using those two linear models.

Linearisation
The floating platform has 6 DOFs, and so each DOF has its own effects from wind and wave disturbances.A designer can choose from 1-6 platform DOFs to include, and accordingly define the number of states per DOF to represent the wave excitation and radiation.Linearisation should be performed in still water conditions.This is done to achieve an equilibrium point for the DOFs used.Equilibrium cannot be reached when waves are present since the platform and tower will be displaced from their Operating Point (OP).
In our study, we linearised the 15 MW FWT model at the OP of the applied wind speed.If the applied stochastic wind disturbance had a mean of   , then a total of 36 linearisations were performed at   and then averaged to convert the 36 linear models into 1 at   .The intention was to linearise the model every 10 • of rotor azimuth, then averaging over 360 • to get a more accurate linear model instead of linearising once at a certain rotor azimuth.The state definitions of the controller design models are described in the subsection below.
A post processing tool is used that performs a transformation from rotating to a fixed coordinate system using Multi-Bladed Coordinate transformation (MBC) [27].This is needed to couple the dynamics between the tower-nacelle assembly and the three rotating individual blades as the tower sees the effects from all three blades, hence transformation from rotating to fixed frame is needed.

Design model T
The ideal SS representation is described in Eq. ( 5), where  are the model states,  and  as the model inputs and outputs, respectively.The relation between states, inputs, and outputs is described through the model matrices , , , and .
The  controller design model is only based on the turbine states discarding any hydrodynamics affecting the turbine model.The model consists of 13 states with 1 input and 2 outputs as follows.
... ẋ ẋ ẋ ẋ1 ẋ2 ẋ3 ] where for the states:   ,   ,   are the platform surge, platform pitch, and tower-top fore-aft displacement, respectively. 1 ,  2 ,  3 are the 1st three blade flap-wise modes, and   is the generator speed.The input of the model is the collective blade pitch command   .The outputs for our study are the generator speed  and platform pitch .Including the tower fore-aft, platform pitch and surge states was needed in the controller design model as they are among the lowest structural mode frequencies [8].An additional three states relating to the three blade flap-wise modes were included [ 1 ,  2 ,  3 ] to investigate how stochastic wind can affect the outputs when accounting for the stochastic flap-wise motion of the three blades.

Design model TW
Controller design model TW has 53 states in total, the 13 structural turbine states   described above, and 16 wave excitation and 24 wave radiation states.Those hydrodynamic states are polynomial coefficients used to estimate the nonlinear hydrodynamic effects.The 40 hydrodynamic states are denoted as   .A toolbox developed by OpenFAST [28], which is based in MATLAB translates the theory in Sections 2.1 and 2.2 in code to generate state approximations.Depending on how accurate the designer wants the states to approximate the nonlinear hydrodynamic responses, the generated number of states can vary.The hydrodynamic approximation was based on an R 2 of 95% for both wave excitation and radiation.These hydrodynamic states (  ) are combined to a state vector    = [    ]  .The 53 states were reduced to 13 via a balanced reduction algorithm.Such algorithm relies on the Hankel singular value decomposition [29], which gives the energy contribution per state in the model.In this way, we compare two controller design models of equal complexity, although the TW model still accounts for the hydrodynamics while the  model does not.This also improves numerical conditioning of the design models and the state space model used internally within the Kalman Filter, as the  matrix of the full 53 state TW model has a mixture of very large and very small eigenvalues.
From open-loop validation of the  and TW models, it was seen that a high correlation can be achieved with 13 states when compared with the full nonlinear model in open-loop.Validation will be discussed in Section 4. Note that the outputs are the same as the  model.The input is the collective blade pitch command    .

State feedback formulation
Optimisation is performed through minimising a cost function  related to the states and inputs of our system as shown is Eq.(7b).Tuning of such function can be done by the state weighting matrix Q, whereas penalising the controller actions can be done by tuning the  matrix or scalar () depending on how many control commands we have.Matrix Q is used to weight the states, meaning that depending on our control objective, we can put more emphasis on a certain state(s) to improve their performance.In our case, we are using two control models that have 13 states, but the definitions of the states per model are different.When comparing those two controllers together, it would be more applicable to place weights on the outputs rather than the states.Since our system has two outputs, then the   output weighting matrix will be 2 × 2 as shown in Eq. ( 8), where   and   are the generator speed and platform pitch weights, respectively.The input weighting matrix is a scalar (  ) since we are using collective pitching, i.e. we generate a single control command. where: Eq. (7a) shows the cost in output form.We need to transform our output weighting matrix into a square matrix having the same size as our model states.That is we need to convert the 2 × 2   into a 13 × 13   .The transformation is derived in the following: +     =  T  T    + 2 T  T    + T    +     .
Comparing the quadratic function in Eq. (7b) +    + 2   with Eq. ( 9), we have: The Q matrix is used to place weights on the system states or in our case outputs, hence is multiplied by the states/outputs  of the system.The R matrix places penalties on the control input, hence multiplied by the system input(s) .Finally,  is known as the cross term since it is multiplied by both the states and input(s).

Kalman filters
The SS model depicted in Eq. ( 5) is under the assumption of ideal states (measurable).In real life, such SS representation carries with it two kinds of noise.The first is the process noise, which relates to the uncertainty in our mathematical model.The other noise is the observation or measurement noise, which relates to the uncertainty in measuring tools.The Kalman filter's objective is to estimate unmeasurable states from information about measurable data given a mathematical model.The non-ideal SS representation is shown below.
where  is the Kalman gain, x are the estimated states,  and  are the process and observation uncorrelated white noises, respectively with zero mean and   and   variances. is the process matrix,  and ŷ are the actual and estimated outputs, respectively, where the actual outputs are the generator speed and platform pitch as shown in the block diagram in Fig. 2. A measure of the estimated states' uncertainty is the covariance matrix  , solved using the algebraic Riccati equation: Tuning the Kalman filter is done through the matrices   and   , where   is a square matrix of the number of states in our model, and   is a square matrix of the number of outputs in our model.For ease of design, the   matrix was set as an identity matrix, while the   matrix was tuned as shown below.Further information on Kalman filters can be found in [30].
In real life, the platform pitch can be measured through a 9-axis Inertial Measurement Unit (IMU).Such IMU includes an accelerometer, a gyroscope, and a magnetometer which is able to give measurements of the floater's position and orientation such as the platform pitch.Wind speed can be measured through an anemometer at hub height, while the generator speed can be measured through a rotational speed sensor.The work does not rely on measurements of wave states applied on the platform.Such factor was assumed to be un-measurable and therefore the Kalman filter estimates un-measurable hydrodynamic states through the measurements of the wind speed, generator speed, and platform pitch using appropriate sensors.

Hydrodynamics: Implementation
We presented some theoretical background of wave excitation and radiation.Implementation of such effects can be achieved using Open-FAST and WAMIT [23].Eqs. ( 1)-( 2) are solved using fitting tools to minimise the cost functions.Those fitting tools were developed by NREL and can be found in [28].They are coded in MATLAB for a better user interface.
The approximation is based on calculating an R 2 value each approximation iteration.This R 2 value is the fit between the state space approximation and the nonlinear hydrodynamics obtained from WAMIT.A designer can enter a certain R 2 value.Once the required R 2 is reached, the fitting tool exports the state matrices as a compatible file that can be included with the other linear state space models related to the structure of the turbine.The higher the R 2 requested, the higher the number of generated states.Both wave excitation and radiation state space matrices were generated for an R 2 of 95%.
OpenFAST encompasses all the dynamics affecting the turbine.A high-fidelity nonlinear model can therefore be tested using OpenFAST.Definitions of each module or input file can be found in [17].In terms of the hydrodynamics, an input file named HydroDyn is used to define the wave states, and floating platform settings.
• Wave conditions: One can define the wave characteristics applied on the platform, be it still water, periodic (regular or irregular), stochastic, or user defined.One can also have the option to include 2nd order wave excitation [31], and currents.In this work, second order excitation and currents are not included.• Floating platform: One can choose to include wave excitation, radiations, or both using either the potential flow theorem [32], or a frequency domain solver like WAMIT.
Offshore wind turbines of large scale are placed on a floating platform and attached to mooring cables to the bottom of the sea [33].Mooring lines are designed from polyester or nylon, with nylon being the preferred design for its low stiffness and tension reduction [34].The mooring system as any input file in OpenFAST can be incorporated in the full nonlinear model of the turbine.MoorDyn [35], which is the mooring tool used in offshore turbine analysis uses a lumped mass technique for the mooring cables.From the definition of the IEA 15 MW FWT [8], the mooring system consists of 3 lines of 0.33 m diameter, length of 850 m, and a stiffness of 3270MN.

Model validation
Before testing the resulting controller designs in OpenFAST, validation of the controller design models was performed in open-loop to evaluate how well they approximate the full nonlinear model.Figs.3-4 show the TW model validated with the full order nonlinear model under a wind step disturbance case from 20-21 m/s.A second case was validated when applying a periodic wave elevation of 5 m with a period of 10 s to show the difference when applying waves into the models.Fig. 3 shows a higher correlation than in Fig. 4. Since the controller design model is based on SS approximation of hydrodynamic forces, then difference will arise when applying wave disturbances into the models.Nevertheless, a high correlation was still achieved when including waves in the validation.Note that there are some oscillations at a frequency of about 0.37 Hz seen in Fig. 3(a) for the nonlinear model.This is related to the revolving three blades of the turbine.As the rotor disk revolves, the shaft sees the effect of the three blades combined.This is referred to as the 3p frequency.The rotor rotates at rated speed, which is 7.55 rpm, and hence the 3p frequency will be 7.55(3/60) = 0.378 Hz.On the other hand, the linear model (controller design model) uses an average of the oscillations, therefore this frequency is not present.
Since our work was tested on turbulent wind speed, then further validation was performed under such case.In addition, several cases for wave periods were tested to check how the controller design models adapted when changing the frequency of the applied waves.The amplitude of the applied wave was set at 5 m peak-to-peak (pk-pk) for all testing cases, while the periods were set to 8 s, 10 s, and 12 s as shown in Table 1.All cases were subject to a stochastic wind speed of 20 m/s mean.Note that further validation is performed in the simulation section when testing the controllers using the JONSWAP wave model.
Compared to the  model, the TW model had a higher R 2 value for both, the generator speed and platform pitch indicating a better fit to the full order nonlinear model.The platform pitch had a higher R 2 difference between both models than for the generator speed as platform pitching is more influenced by hydrodynamic effects.Including such effects into the SS model reflected a higher correlation (TW) to the nonlinear model than excluding hydrodynamics (T) for floater and hub-height factors.

Simulation setup
All results discussed below are based on the full order nonlinear model.The nonlinear plant models had all DOFs activated to properly test the effectiveness of the controllers.A periodic wave disturbance was applied with fixed amplitude for the first testing case, while a JONSWAP model was adopted for the second testing case.A normal Turbulence Model (NTM) wind disturbance with class B intensity as defined by the International Electrotechnical Commission (IEC) [36] was used with means of 20 m/s and 16 m/s.
The proposed work tested the reliability of the two controllers (T and TW) through Pareto analysis.In our case we wanted to compare the generator speed and platform pitch.The criteria was to fix the   input weighing scalar in the LQG design, and vary the   output weighing matrix at each iteration in the Pareto frontier.If both   and   were varied at each iteration, then the Pareto will not be very accurate as the   which represents the penalty on the control action will increase as we increase our   , meaning that little to no change will occur per iteration.Both performance factors were plotted against the blade pitch RMSE coming from the control commands.The LQG controllers used (for both  and TW design models) had: Both output weights were logarithmically varied from 10 0 to 10 2 in 10 steps (iterations), where  is from 1 to 10.The controller was designed to find a multiobjective solution to both control objectives, which is maintain the generator speed at the rated value under the least platform pitching possible.This equal emphasis is done by setting both   () and   () equal at each iteration.
When dealing with highly variable wind input, the actuation needs to be limited as not to add excessive loading on the structure, hence the use of a rate limiter in the simulation.A rate limit of ±2 • ∕s was used, which is the same rate recommended in the definition of the IEA 15 MW turbine model [8].
Results will show both controllers under stochastic wind and regular periodic wave elevation along with Kalman filter estimations.Irregular sea state results will be shown as well.Note that in the figures, GS is the generator speed, PPIT is the platform pitch, and Bld1 is blade 1.

Simulation results for regular sea state
This case tests the controllers under a NTM with 20 m/s mean, and a periodic wave elevation of 5 m pk-pk and a 10 s period.It can be seen in Fig. 5 that for a certain blade pitching RMSE, lower RMSE for the generator speed and platform pitch can be achieved for the TW model.As the iterations increase, more actuation is requested due to an increased   , resulting in a better closed-loop performance for both design models.For the last 5 iterations, the  controller seems to approach instability since increasing the actuation lead to an increase in the output RMSEs.
Fig. 6 shows the estimation of the generator speed and platform pitch aided by a linear Kalman filter.There were some differences between actual and estimated outputs since as mentioned earlier that the SS models include approximations of wave excitation and radiation, while the nonlinear model relies on WAMIT.

Simulation results for irregular sea state
This case tests the controllers under NTMs of 20 m/s and 16 m/s means, and a JONSWAP wave elevation model shown in Fig. 7(a).Validation with the full nonlinear model under the 20 m/s mean case was performed.The generator speed's R 2 values for the TW and  models were 0.873 and 0.851, respectively.The platform pitch's R 2 values for the TW and  models were 0.744 and 0.634, respectively.Starting with the 20 m/s NTM in Fig. 7, a Pareto frontier was constructed for 10 iterations.Similar to the regular wave case, the TW model responded more robustly to stochastic wind and wave disturbances compared to the  controller.
To give a better visual on the results, the time domain responses are shown in Fig. 8 for iteration 5 of the Pareto frontier.For the TW model,    both the generator speed and platform pitch showed improvements over the  model.What was also noticed is that the generator speed was trying to minimise the oscillatory behaviour caused by the applied waves.It can be seen that hub-height factors such as the generator speed are directly influenced by wave elevations, where appropriate compensation is needed to reduce fluctuations caused by the waves.
Controllers were tested on a different NTM with a mean of 16 m/s to generalise the results.Note that the  and TW models were linearised at 16 m/s (mean speed of NTM) for this case as discussed in Section 3.2.Validation with the full nonlinear model was performed.The generator speed's R 2 values for the TW and  models were 0.830 and 0.818, respectively.The platform pitch's R 2 values for the TW and  models were 0.727 and 0.638, respectively.
A Pareto frontier was formed under the same JONSWAP model.Note that Fig. 7(a) and 9(a) are the same.Similar behaviour was seen for the 16 m/s mean NTM, where for a lower actuation RMSE, a lower generator speed and platform pitch fluctuation was achieved for the TW design model.Note that 5 iterations were performed as shown.It is seen that the  model is approaching instability, hence for higher iterations, the system will go unstable, while the TW will remain functional.Such higher iterations were not included in the figure for a proper stable controllers' testing.The closed-loop results for iteration 5 are shown in Fig. 10.For a lower actuation RMSE, the generator speed and platform pitch can achieve less RMSE compared to the  controller.Table 2 shows the percentage reduction in RMSE for the generator speed and platform pitch when adopting the TW controller over the T. The reduction shown is at a blade pitch actuation RMSE of 2 • for the two NTM wind cases.

Conclusion
This work addressed a novel LQG control method for the IEA 15 MW FWT.The application was to investigate the improvements of the floaters' and hub-height responses when accounting for hydrodynamics in the controller design.The work showed two control design methods.It is concluded that including hydrodynamic states in the controller design model not only improved the platform pitching, but also hubheight factors such as the generator speed.The use of Kalman filters helped extract estimates of un-measurable hydrodynamic states.
Further improvement on the closed-loop performance can be achieved by including additional states in the controller design such as blade edgewise, and DOFs related to the floater other than surging and pitching.This work used two DOFs related to the floater as they have the strongest effect on hub-height related factors.Future work should investigate individual pitch control perhaps under the use of an LQ with an estimator to further enhance the performance and reduce asymmetric loading.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Table 1 R
2percentages for the validation of the controller design models with the full order nonlinear model of the 15 MW FWT.

Table 2
Percentage reduction in RMSE for the TW controller compared to the T controller at a blade pitch actuation RMSE of 2 • subject to JONSWAP wave model and NTMs.