Optimal Fault Tolerant Control of Large-Scale Wind Turbines in the Case of the Pitch Actuator Partial Faults

In this paper, an adaptive fault tolerant control strategy is proposed to deal with the three pitch actuator faults in the largescale wind turbines. Firstly, a simultaneous state and fault estimation was performed through a suitable LMI (linear matrix inequality) based optimal strategy. Hereafter, the new control law is designed using the previously estimated fault information. -e actuator efficiency estimator uses as design parameters, respectively, the performance index c against the wind and the learning rate Ξ. of the fault estimation algorithm. -e study shows that the choice of the previous two parameters impacts the response time of the fault estimation and the correlation of the tracking error with the wind. -e aim is to choose a small fault estimation response time while keeping a weak correlation between the tracking error and the wind turbulence noise. Finally, a tuning strategy is elaborated to choose the suitable c and Ξ to match the reconfiguration objective.


Introduction
e increasing energy demand leads to the production of large wind turbines.
us, more and more vibrational behaviors could be expected in the different degrees of freedom (DOFs). Figures 1 and 2 show the most important DOF. e excitation of the DOFs leads to increase in loads and forces on the structure. e aim is to reduce the loads in order to protect the structure against fatigue and damage. e wind turbine operates in two different regions, low and high winds. In the first region, so called partial load region, the aim is to maximize the extracted power [1,2].
is was achieved by controlling the generator torque to maintain an optimum ratio between the tip speed of the blades and the operating wind speed [3]. In the second region, so called full load region, the wind turbine was controlled to reduce loads and keep the power at its rated value. Practically, the constant power was maintained through a fixed speed and a fixed generator torque. is has been obtained by pitching the blades from or toward wind to adjust the speed of the rotor, while applying a constant generator torque [4]. For example, the authors in [5] proposed an anticipative control scheme of a wind turbine working in the maximum power region. e idea is to use the wind speed as a generator of the set point of the MPP (maximal power point) control.
From the discussed papers above, it can be noticed that the wind turbines in the megawatt size are expensive, and hence their reliability is expected to be high to generate as much energy as possible. is type of wind turbines is expected to have reduced downtimes while keeping energy production. Usually, the turbines are turned off even during simple faults to wait for service.

Fault Detection and Tolerant Control in Wind Turbines.
e statistics in [10] show that gearbox faults are the most critical ones in wind turbines, essentially due to its relatively high stopping time and severity on the whole structure. Another type of wind turbine faults is of electrical root. For example, the wind turbine can face short circuit faults in the wires or inverter over voltage faults. It can also experience hard over fault in the generator or some sensor faults like drift or bias. e last reason creates the need to introduce fault detection, isolation, and accommodation systems within the existing classical control systems in order to reduce stop time even with limited energy production [11]. Early detection has been the spot of the control industry studies especially using residual generation and fault analysis. e paper [12] presents an overview of the fault diagnosis method categories, with an application to the wind turbine pitch fault control. For more details, see also [13,14]. e model-based methods are one of the most suitable methods for the study of the fault tolerant control of the wind turbines. In fact, the availability of the large-scale wind turbine model is no longer a problem especially with the existence of highly trusted wind environment software and emulators such as FAST (Fatigue, Aerodynamics, Structures, and Turbulence).
In this paper, the FAST code is used to test the proposed strategy. FAST is an aeroelastic simulator capable of predicting the fatigue loads of two-and three-bladed flexible with 24 degrees of freedom [15,16]. e code uses a modal approach in combination with Kane dynamics to develop the equations of motion. FAST helps considerably researchers and engineers in testing wind load reduction algorithms [17].
is software provides a linearized model parameter about a chosen operating point with more than 40 measures of outputs (rotor speed, generator speed, and tower and blade displacement). e pitch servomotor is the most critical actuator in the wind turbine [18,19]. As discussed earlier, it is used to regulate the power and to attenuate the vibrations in the high wind region. If this actuator loses its efficiency, it will no longer provide the sufficient control efforts to attenuate the vibrations and regulate the power. Based on the last reason, a new control law is to be constructed to deal with the fault.
e new control efforts could be produced by online tuning of the controller parameters or by changing the baseline control law using an estimation of the fault [20][21][22].
In the wind turbines' control field, in the pitch actuator fault, the authors inreference [23] proposed a fault tolerant control strategy to deal with the drop of the hydraulic liquid in the servo pitches of the rotor. In the same field, another type of fault is the imbalance in the rotor due to icing or destruction of one or more blades. is fault not only reduces the aerodynamic efficiency of the turbine and its power output but can also lead to large increases of loads on the drivetrain, blades, and tower. For this, the authors in [24] used the FAST software to model aerodynamic imbalance in a sample 5 MW offshore wind turbine. It has been concluded that the combination of blade and nacelle measurements, obtained from instrumentation already placed on the structure, can be formulated into an algorithm used to detect and locate the imbalance (see also [25]).
Another essential component is the anemometer. e anemometers are used in the feedforward control of the wind turbines. In [26], the authors proposed a new method to estimate the wind speed that could be substituted for anemometers in control loops. Moreover, the authors in [27] proposed a robust observer to generate residuals for the fault turbine moment sensors which provides a measurement of the loads applied on the structure. In [28], the authors have proposed a method to estimate the gearbox efficiency in the wind turbines. e idea is to detect the drop in the efficiency and generate a suitable residual to indicate the fault occurrence (see also [29,30]).
Given the very disturbed wind nature, several optimization methods could be used to minimize the effect of the   disturbance on the outputs. In [31], a multiagent optimal method based on reinforcement learning was proposed to control a grid connected energy system. e fault isolation and tolerant control applied to the wind turbines has been illustrated in many papers. For example, in [32,33], a codesign of the observers and decision mechanisms for satisfying certain trade-off between different isolation performance indices has been established. e performance indices were the false isolation rates, the isolation times, and the minimum size of the isolable faults. e considered fault was additive on the pitch actuator. e paper [34] used jump observers to detect and isolate the faults. e false alarm rate is upper bounded by means of Markov's inequality. e trade-offs between the minimum detectable faults, the false alarm rate, and the response time to faults of the fault diagnoser have been explored. In the previous cited works, the assumption of slow dynamics faults was considered which limits the usage of the strategies to slow faults dynamics.
In this paper, the following contributions have been developed: (i) An integrator-based simultaneous states and fault estimator algorithm is designed. (ii) e stability and the robustness of the algorithm is proven for the partial fault of the pitch actuators using Lyapunov theory. (iii) A fault tolerant control including a baseline multiobjective state feedback regulator is designed. e performance of the method depends on two parameters to be tuned, c and Ξ. (iv) To ensure a best performance of the proposed control solution, two mappings are proposed to tune the design parameters c and Ξ. (v) e evaluation of the proposed control strategy is performed on the 1.5 MW wind turbine benchmark model.
is paper is organized as follows. In Section 2, the wind turbine model used in the benchmark is presented. In Section 3, the wind turbine fault and state estimation is performed and the nominal multiobjective regulator is designed. e wind turbine adaptive fault tolerant control in the faulty production is studied and the tuning strategy is proposed in Section 4. Some concluding remarks are presented in Section 5.

The Wind Turbine Model
(1) e operator δ means a variation about the operating point. A is the dynamics matrix, B is the input matrix, D d is the disturbance matrix, and C is the measured output matrix. e numerical values of the matrices A, B, C, and D d are given in the Appendix. e state vector and the pitch actions are, respectively, represented by the vectors δx and δu: where δx def i and δ _ x def i _ x def i are, respectively, the blade deflection and deflection velocity associated to the i th blade. δy represents the measured rotor speed and deflection velocity. e vector ω is the wind disturbance on the considered states and is given by where V(t) is the instantaneous wind speed and V 0 is the operating wind speed which is equal to 18 m/s in the high wind region. e vector is the rotor speed about the operating point and is given by where Ω r (t) is the measured rotor speed and the operating rotor speed is Ω r0 � 40 rpm. e input pitches about operating point are equal to δβ 1 is the measured pitch angle and the operating pitch angle is β 0 � 9°. e numerical values of variables in the international system of measurement units are given.
It is assumed that the measured outputs are the rotor speed and the blade flap deflection velocities of the three blades. In actual application like wind turbine blade analysis, the deflection velocities could be obtained using the derivative of the deflection displacement measure.
is information could be directly obtained using a MEMS gyroscope sensor as in [35], or estimated using ultrawideband signals as proposed in [36]. e considered turbine is a Wind PACT 1.5 MW large scale. More details about the parameters of this wind turbine could be found in [37,38]. e turbine parameters summary is given in Table 1.

Wind Turbine Control Strategy in the
Nominal Production e nominal controller is designed for two objectives. On the one hand, the rotor speed should be regulated despite the wind changes. On the other hand, the blade's deflections should be attenuated to prevent the structure from being damaged. e control scheme is summarized in Figure 3. e blades' deflection velocity and the rotor speed are both measured and fed to the robust state observer. Finally, the Complexity 3 state feedback controller uses the estimated states to generate the necessary pitch angle variations responsible for regulating rotor speed and attenuating blade deflections. e adjective nominal means the baseline or the fault free situation. e system is linearized about the nominal operating point, and then the nominal controller is derived based on the pole placement method to achieve the nominal production. e opposite to nominal is faulty, which means the situation in which the fault occurs.

e Global Wind Turbine State Space Model.
e global state space representation in the equation (5) takes into account the pitch actuator faults. e actuator effectiveness fault is modeled as a multiplication of the input matrix B by an unknown fault matrix F, and the faulty system is then given by where x(t) ∈ R n is the state vector; u(t) ∈ R m is the input vector; y(t) ∈ R p is the measured output with p ≥ m; and C and B have full rank (rank(C) � p and rank(B) � m).
is the vector of faults on the actuators.

e Observer and the Baseline Controller
Design. e three-blade flap deflection amplitude reduction and the rotor speed regulation are performed through a pole placement state feedback [39] as in equation (6). In fact, the nominal control law in equation (6) is based on the fault free (nominal) situation, in which the fault matrix F(t) equals the identity matrix which is constant.
e variable x in equation (6) is the state vector to estimate. e gains of controller K are design parameters to achieve the multiobjective wind turbine control. e variable x should be estimated robustly in the next paragraph.
In Figure 3, T g0 is the generator torque input at the operating point and is kept constant in the region of high winds, β 0 is the pitch angle input at the operating point, Δβ i c is the pitch angle input variations from the operating point (i � 1, 2, 3), K is the control gain, Δ _ x def i are the deflection velocity variations from the operating point (i � 1, 2, 3), ΔΩ r is the rotor speed variations from the operating point, Δβ i is the measured pitch angles and are input to the state observer, and F(t) is the fault matrix.

e Robust State Observer.
e structure of the state observer is given by system (7) and illustrated by Figure 4: where y(t) corresponds to the vector containing the measured deflection velocities Δ _ x def i and rotor speed ΔΩ r . e vector u(t) corresponds to the measured three blade pitch angles. F(t) is the matrix of the faults. x(t) is the output of the observer.
x(t) ∈ R n are the observer states, y(t) ∈ R n are the observed outputs, F(t) ∈ R m×m is the estimate of F(t), and L(t) ∈ R n×p is the estimator gain to be designed.

e Fault Matrix Estimator.
e fault matrix F(t) has to be estimated. is estimate is F(t) ∈ R m×m and has the following diagonal form: where f i is the estimation of the pitch fault on the actuator of index i. e following integrator-based algorithm is proposed: where i � 1, 2, 3; Ξ � diag(ξ 1 , ξ 2 , . . . , ξ m ); Ξ ∈ R m×m is called the learning rate; and r i ∈ R 1×p . e parameter p is the number of measurements. u i (t) is the control input in the direction to the pitch actuator number i. e adaptability of the estimation consists in the update of the fault estimate dynamics with time by acting on the learning rate Ξ or the matrix R � r 1 r 2 r 3 T . e matrix R is an additional degree of freedom that can be used to refine the approximation B T P � RC in equation (15) in Section 3.2.3. An application of this matrix could be to refine equation (15) only in the direction of a selected measurement from matrix C for a fixed η. For example, to refine equation (15)   Observer + − equation (6) equation (7) ∆x_ defi .
In the resulting matrix RC, the second row vector became null. Indeed, this row represents the contribution of the rotor speed in the product matrix RC. e remaining vector rows 3, 5, and 6 represent the blade deflection velocities contribution in the product matrix RC. Another alternative could be to compute an initial guess of R using YALMIP and refine it according to the need. In the present paper, R was extracted using YALMIP. e learning rate Ξ is selected and handled by the user to ensure the fault estimation performance requirements. It allows to achieve a compromise between a quick fault accommodation and the disturbance amplification. Indeed, for high values of Ξ, high frequency information, namely, disturbances, is integrated in the learning process and vice versa. e variable e y (t) is the output tracking error and is equal to the difference y(t) − y(t).

Stability and Robustness of the Observer.
Consider the states tracking error e x (t) � x(t) − x(t), the output tracking error e y (t) � y(t) − y(t), and the fault tracking error e states' tracking error dynamics are given by the state space representation in the following equation: e first objective is to have e x (t) ⟶ 0 and e f (t) ⟶ 0 when t ⟶ +∞. e tracking errors' convergence and stability could be ensured using the Lyapunov function V e (t): e condition of validity of V e (t) is that Ξ is chosen invertible.
P is a symmetric positive definite matrix. e idea of the Lyapunov function choice is as the learning rate increases, the quadratic fault tracking error e f (t) decreases and the fault estimate tends to the actual fault. Finally, as the P matrix decreases, the quadratic state tracking error e y (t) decreases and tends to the null vector. e second objective is to minimize the effect of the wind turbulence on the output tracking error e y (t), using a performance index c > 0 as e minimization of the wind turbulence ω(t) effect on the output error could be performed using the following objective function: Writing equation (8) as a matrix form gives the following: To eliminate the term containing both e T x (t) and u(t), equation (15) could be adopted. Indeed, by adopting (15) the second and fourth term in the development of the derivative of the Lyapunov cancel each other.
In order to solve the condition (15), its LMI version is used as in the following lemma.
Lemma 1 (see [40]). e condition B T P � RC has an LMI equivalent writing: where η is a dimensionless constant used to approximate B T P � RC as an LMI. e smaller η is, the accurate B T P � RC is in LMI writing.
In the sequel, the symbol ' * ' means the transposition of the term in the same diagonal as " * ." For example, in equation (16), " * " is the transposition of the term B T P − RC.
e derivative of the Lyapunov function becomes By replacing the Lyapunov function derivatives in the cost function expression, the following inequality is obtained: ere is no direct action of the wind on the outputs; then, D 2 � 0. e objective function inequality: where Finally, to ensure the stability of the states' tracking error dynamics and the minimization of the wind turbulence on the output tracking error, the following inequality should hold: Using the Schur complement, the same previous inequality becomes where Y � PL, and then the error dynamic (9) satisfies the H ∞ performance. (8) is proposed with the assumption that the input vector u i (t) satisfies the persistent excitation condition, which means that there exist positive scalars μ 1 , μ 2 , and t 0 such that for all t, the following inequality holds: e poles of the designed observer could be placed through the linear matrix inequality in the following lemma.
Lemma 2 (see [41]). e eigenvalues of closed loop observed system dynamic matrix (A − LC) ∈ R n×n A ∈ R n ×n belong to the circular region D(α, τ)with center α + j0 and radius τ if and only if there exists a symmetric positive definite matrix P ∈ R n×n P ∈ R n×n such that the following condition holds: e final integrated problem to be solved for the symmetric positive definite matrix P is as the following lemma. Lemma 3. Assume D 2 is null matrix and let a prescribed H∞ performance level c be given. If there exists a symmetric positive definite matrix P ∈ R n×n and matrices Y ∈ R n×p and R ∈ R m×p such that the following condition holds: where Y � PL, then the error dynamic (9) satisfies the H∞ performance and the poles of the robust state observer are within the region D(α, τ).

e State Observer and Fault Simulation
Results. e natural frequency of the blade is f n � (1/2π) ������� � (D b /M b ) � 1.511 Hz. M b is the blade mass and D b is the stiffness of the blade. It gives a natural pulsation of ω n � 9.49 rad/s. All the frequencies above the natural frequency should be filtered by the observer. However, the observer should amplify the gain corresponding to ω n for the estimation of the blades' deflections. e Bode diagrams of the designed observer in the system (7) are illustrated in Figures 5(a) and 5(b). e considered channel links the pitch actuator's input (u 1 (t) � Δβ 1 in Figure 4) with the blade deflection observed state (y 3 (t) � Δx def 1 in Figure 4). Figures 5(a) and 5(b) illustrate the designed observer Bode diagram function of the performance level c corresponding to the system in equation (7). e Bode diagrams are divided into two main regions. e first region is about the natural pulsation of the blades ω n � 9.49 rad/s. In this region, the magnitude presents maximal values, which means that the blades' deflections are perfectly observed. e second region is where the pulsations are less than the natural pulsation of the blades. As c becomes smaller (closer to values near 0), the gain of the observer is reduced and the low frequencies of the wind speed are not amplified. e advantage is to prevent penetrating the amplified low frequencies contained in the rotor speed to the fault estimation algorithm through the residual e y (t).
e effect of the amplified low frequencies is reducing the convergence speed of the estimation algorithm in equation (8). As a conclusion, the convergence speed of the fault estimation could be tuned 6 Complexity by the performance index c in addition to the tuning by the learning rate Ξ as illustrated in Figure 6  In Figure 6(a), the effectiveness of actuator number 1 means 1 − f 1 . f 1 represents the decrease in the output of the actuator.
In Figures 7 and 8, the notations e y1 correspond to the rotor speed component and e y2 corresponds to the blade deflection velocity component in the tracking error vector e y . Figures 6(a) and 8 correspond to the first faulty scenario of Section 4 and Figure 7 corresponds to the third faulty scenario. Figure 6(b) illustrates the fault estimation of the blade 1 actuator for several values of learning rate ξ 1 . As the learning rate increases, the estimate of the fault converges rapidly to the actual fault (magenta).
Remark: ere is a coupling effect between the fault estimation and the state observer due to the fault tolerant control (FTC) correction. Such an effect can be neglected as long as the state observer is slower than the fault estimation.
is can be achieved by choosing properly the learning rate.

Fault Scenario with Dynamics Faster than the System
Dynamics. In this section, a sinusoidal loss of efficiency is considered with frequency higher than 1.5 Hz of blades. Figures 6(c) and 6(d) show the result of fault estimation in that case. In the proposed strategy, by only tuning the learning rate to suitable higher values, the faults with dynamics faster than the system dynamics could be reconfigured without need to redesign the fault estimate algorithm as recommended in [33]. e amplitude of the fault is 0.1 and the frequency is 12 rad/s higher than 9.49 rad/s of the blades.

Complexity
In Figure 7, the most important issue is the change of the mean rotor speed error. is information is the only one needed by the algorithm about the fault on the rotor speed. In Figure 8, the change is obvious on the amplitude of the blades at the natural frequency 1.5 Hz. is last information could be easily deduced by computing the value 1/period. Indeed, in Figure 8, the period of the signals is equal to 0.66 seconds giving the blades a natural frequency of 1.5 Hz.
In the faulty case, this increase in the mean and the standard deviation of the tracking error is integrated by the algorithm in equation (8) to deduce the fault amplitude.

e Baseline Controller Design.
Initially, the eigenvalues of the dynamics matrix A of the system are equal to the vector λ 0 as follows: e eigenvalue − 0.1043 corresponds to the rotor speed dynamic. e last three eigenvalues represent the three blades' dynamics. ese eigenvalues are double conjugates because the blades represent an oscillatory system. e objective of the control is to reduce the oscillations of the blades by choosing new eigenvalues bigger in absolute value than the initial eigenvalues and with negative real parts to ensure stability of the closed loop. e design control parameter K is computed so that the closed loop eigenvalues λ (A+BK) match the desired eigenvalues λ d : In the present paper, the desired eigenvalues of the nominal closed loop are e design steps of the nominal control are as follows: (i) e controllability matrix of the pair (A, B) is of full rank 7. So, the system is controllable.
(ii) e control gain is defined as K � e gain K is then substituted in the desired characteristic polynomial equation: where I n is the identity matrix and s is the Laplace operator. e integer n is the number of states; it is equal to 7 in this paper. λ d,j is the jth element of the desired closed loop eigenvalue vector λ d .
(i) Equation (29) is solved by equating the identical powers on both sides. is allows extracting the value of the control matrix K.
Finally, the obtained control gain K is applied to the system according to the scheme of Figure 3. e results of the regulation are depicted in Figures 9-12. Figures 9 and   10 illustrate that the speed regulation objective is achieved. In fact, as the rotor speed increases (decreases), the blade pitch angle increases (decreases) in order to prevent wind from driving the wind turbine (in order to provide more wind to drive the wind turbine). e expected achieved performance is of 90% of attenuation. In the open loop, a difference from the operating point of 10 rpm at the instant 150 s is denoted. After applying the control strategy, the difference becomes 1 rpm, which corresponds to 90% of attenuation. Figures 11 and 12 show that when the deflection increases (the red signal), the pitch action (green) increases to attenuate the deflection as presented in the same figure (blue signal). It can be concluded that the blade pitch action is composed of two signals: a low frequency control action responsible for rotor speed regulation and a high frequency control action responsible for mitigating the blade deflections. Figures 9-12 correspond to the nominal case (healthy case). In this case, no fault is occurring.

Wind Turbine Optimal Adaptive Fault Tolerant Control in the Faulty Operation
e considered fault is the loss of efficiency in the three pitch actuators. e scenario created in the FAST software is a drop of 55% in the effectiveness of the pitch actuator 1 at the instant 90 s, 40% in the effectiveness of the pitch actuator 2 at the instant 150 s, and 60% in the effectiveness of the pitch actuator 3 at the instant 200 s.

e Fault Mechanism.
e considered pitch actuator is hydraulic based. In Figure 13(a), the prefix p denotes the piston side, the prefix r denotes the rod side, and the prefix v denotes the proportional valve side. A p and A r correspond, respectively, to the surfaces of the piston side and the rod side.
e pump pumps oil from the accumulator with a pressure p s which passes across the proportional valve with a mass flow Q v . Finally, the air arrives to the piston side with the pressure p p and thrusts the rod linked to the blades. e trust force is F cyl and is given by where F fric is the friction force of the piston on the cylinder and is linearly dependent to the displacement velocity of the piston.
For the same rod pressure p r and friction F fric , the considered pitch loss of efficiency fault occurs when the pressure p p drops due to the ageing of the pump station. is ageing appears for example because of excessive Joule effect (heat) losses in the wirings of the pump motor. Under these conditions, the thrust force F cyl of the rod on the blade is no longer sufficient to actuate the blade. e thrust force F cyl is represented by the action matrix B in equation (5). e fault is modeled as a loss of efficiency percentage (equal or less than 100%) multiplied by the action matrix B. As a result, the pitch efforts are no longer sufficient for damping the oscillations of the blades. e aim of the algorithm in equation (8) is to track that multiplicative percentage with a sufficient precision and response time.
After that, equation (31) comes to amplify the supplied power (control) to the pump to compensate the lost energy due to Joule effect. e amplification is performed mathematically by dividing the old control signal by the estimated faulty efficiency.

e Fault Tolerant Controller.
e fault tolerant control signal is obtained by dividing the initial baseline control signal by the estimated fault: To avoid the division by zero in the case of zero efficiency, F(t) should be replaced by a small number (for example, 10 − 4 ). e scheme of the global fault tolerant control is given by Figure 13(b).   Before being fed to the pitch actuator, the effectiveness is estimated by the adaptive algorithm. en, the control signal is divided by the estimated efficiency to compensate the drop in the actual efficiency of the pitch actuator.
Remark: e different LMIs of the algorithm are resolved through the YALMIP tool [42].
Δβ i is the pitch angle input variations from the operating point for the i th actuator (i � 1, 2, 3), Δ _ x def i are the deflection velocity variations from the operating point for the i th blade, and ΔΩ r is the rotor speed variations from the operating point.
Δβ i F is the faulty pitch angle input variations from the operating point for each actuator, f i is the estimated efficiency for the i th actuator, and x is the estimated state variations of the wind turbine from the operating point.

e ree Pitch Actuators Loss of Effectiveness Faulty
Scenario. Let us consider the blade without loss of generality. Figure 14 illustrates the blade 1 deflection.
In the faulty situation, the deflections increase from 1.5 meters to 4 meters because the pitch actuator could not answer completely to the corrective action of the baseline regulator against the wind. According to [43], the maximal allowable industrial deflection for the considered wind turbine is of 3 m. In the faulty case, this limit is exceeded which increases the risk of damage of the blade.

Complexity
After applying the fault tolerant control strategy, the blade deflections are perfectly reconfigured.
Let F(s) be the effect of the pitch angle i on the blade i deflection. e effect of the strategy on the blades deflections is interpreted as an additional zero at the origin in the transfer function F(s). In Figure 15, it can be noticed that when s tends to zero, sF(s) tends to zero. e reconfiguration time for each fault estimation of index i is dictated by the product ξ i × r i . e rapidity of the algorithm depends on the learning rate Ξ and the matrix R. As studied earlier in the paper, in addition to the learning rate Ξ, even the performance level c could influence the response time of the fault estimation through the low frequencies of the wind. In fact, for a given c, a matrix P depending on c could be found through the LMI in equation (22). en, by using the LMI in equation (15), the matrix R is obtained and used to compute the estimation F(t) using equation (8). As a conclusion, the parameter c influences the response time of the algorithm through the matrix R.
e derivative form of the block between Δβ i c and F(s) in Figure 15 has a risk of turbulence amplification. For this reason, a study to choose the learning rate Ξ and the performance level c function of the correlation between the tracking error e y 2 (t) (blade 1 deflection velocity as an example) and the wind turbulence is proposed in the next section.

e Choice of c and Γ Based on the Correlation with Turbulence and the Estimation Response Time Criteria.
In this section, empirical boundaries are proposed for choosing the performance level c and the learning rate Ξ based on two criteria. Namely, the correlation between the wind speed ω and the residual e y is considered. is variable is crucial to investigate because it could be amplified if decided to use the performance level c to tune the estimation response time. e second criterion is the rapidity of convergence which is important to ensure because only short times of convergence are recommended for a fast recovery of the fault.

e Correlation between the Wind Turbulence and the Residual e y .
e expression of the correlation of the two signals is given by where E[x] denotes the expectation of the signal x and SD x denotes the standard deviation of the signal x. e correlation Corr belongs to the interval [− 1, 1]. It obeys to the weak and strong correlation. If the coefficient Corr ∈ − 0.5 0 ∪ 0 0.5 , it is called a weak correlation. If the coefficient Corr ∈ − 1 − 0.5 ∪ 0.5 1 , it is called a strong correlation.

4.4.2.
e Estimation Response Time of the Algorithm. e response time of the fault estimation algorithm in equation (8) for each actuator of index i is given by Many simulations have been made on the FASTsoftware in order to derive empirical data on υ and Corr helping to choose the appropriate values of the two parameters c and Ξ. en, an interpolation between data points has been performed to define mappings linking the tracking error and the wind disturbance. Figures 16 and 17 show the evolution of the correlation and fault estimation response time with the performance level and the learning rate for the considered wind turbine in Table 1. From Figure 18, one can conclude that for c less than 1, this parameter influences the correlation more than the response time of the algorithm. However, when c exceeds 2.5, the correlation becomes more influenced also by Ξ as c increases. For c less than 2.25, the response time of the algorithm is influenced by both Ξ and c. But, as c increases, only Ξ influences the response time of the algorithm. In order to guarantee a fast reconfiguration of the system performance, one should choose a small response time of the algorithm by choosing a suitable pair (c, Ξ). However, one should guaranty that neither the resulting performance level nor the learning rate amplifies the correlation between the tracking error e y and the wind turbulence ω.
For the last reason, the two parameters should obey to the following empirical law. For example, if c is in [0.9, 1.4] interval, Ξ should not be in [2.21, 9.84] (zone 1 in Figure 18) to avoid high correlation, and with the same idea, if c is chosen in [1.4, 2.02], Ξ should not be in the interval [8.317, 9.84] (zone 2 in Figure 18).
An example of a choice of the pair (c, Ξ) leading to a fast and less correlated response time is indicated in Figure 19 by (c, Ξ) � (2.25, 1.25).

Discussion with respect to Some Existing Methods.
e proposed strategy is not complicated and uses an integral structure with one Lyapunov function to prove stability and convergence of both state estimation error and fault estimation error. e proposed approach uses a reduced number of measurements. Indeed, only the rotor speed and the individual blades' deflection velocity are needed. is reduces the information needs with respect to other techniques such as the one in [44] (requiring also the generator speed measurement). It does not either require the information about the presumed fault size as in [45]. e residual-based techniques in [45][46][47] deal only with fault detection tasks. Contrariwise, the proposed approach deals with both fault estimation and fault isolation suitable for active fault tolerant control strategies. Indeed, the individual deflection velocity measurement of each blade allows a natural isolation of the fault blade without the need of additional isolation methods. is method avoids the intrinsic delays of online least-squares methods, which could have a negative impact on the fault estimation response time.
e proposed strategy is equipped with a tuning method to avoid amplification of wind turbulence in the direction of the high learning rates. is issue is not sufficiently discussed in other works related to wind turbines.
Contrariwise to other strategies, the proposed method could deal with faults having dynamics faster than the system dynamics only by tuning the learning rate suitably.

Comparison of the Proposed Strategy with the Virtual
Actuator with State Residual Integrator. A new scenario of pitch fault on blade 1 is considered for the comparison between the proposed strategy and a virtual actuator-based strategy (details of this method are not the objective of the paper, [48]). In this section, the considered fault happend at the instant 15 seconds. is fault represents pitch actuator loss of efficiency of 55%. Figure 20 illustrates the structure of the strategy used for comparison. e principle of this strategy is to use a model reference and to compute the difference between the estimated faulty states and the model reference states. is difference is integrated and used as new control law. Equation (8) was used to derive the faulty matrix B f . e difference with respect to the proposed strategy is in the integration of the state residual x Δ instead of the division in equation (30) and the use of reference model. Figure 21 illustrates a comparison between the proposed strategy and the virtual actuator with integrating the state residual in terms of blade fault reconfiguration. In some regions where the wind speed makes high amplitudes of high frequencies, which means the effort needed for integration is higher, the performance of the virtual actuator becomes poor in the interval [16 s; 26 s]. is could be explained by the existence of two integrators, the first presented in equation (8) used to estimate the fault needed by B f and the second to integrate the state residual. In conclusion, the

Digital Implementation of the Strategy.
e equations to be implemented are the baseline control law in equation (6), the observer in equation (7), and the fault estimation in equation (8). In order to digitalize the continuous equations, the first-order approximation in equation (34) is used. If _ z is the derivative, it could be approximated by where T is the sampling step. Hence, the strategy could be digitalized for each iteration of index k as follows. e digitalized baseline controller (from equation (6)) is given by e digitalized robust observer (from equation (7)) is given by where A d , B d , and L d are, respectively, the state matrix, the input matrix, and the observer matrix. e expressions of the previous matrix are given by A d � I n + TA; B d � TB; and L d � TL. e digitalized fault estimation (equation (8)) for each actuator of index i is given by f k+1,i � f k,i + Tξ k,i r k,i e y,k,i u k,i .

Conclusion
In this paper, the partial failure of pitch actuators in wind turbine was considered. e LMI-based fault tolerant control was applied to the considered fault. e method presents the advantages that it estimates at the same time the states and the fault. e method convergence is proved through Lyapunov theory. Two key design parameters were identified in the integrated design, namely, the performance level and the learning rate. It has been shown that the performance level c of the observer and the learning rate Ξ impact indirectly the loads on the turbine. e parameters have an effect on the availability time of the fault estimation. In some zones of the pair (c, Ξ), it has been found that as the learning rate increases, the fault becomes quickly available and the fault tolerant strategy accommodates rapidly the prefault load reduction. In other zones, the robustness c enhances the loads reduction. Moreover, the performance level c provides another degree of freedom to the adaptive algorithm in addition to the learning rate Ξ. An empirical design method was proposed to impose both a given correlation of the measuring error to the wind turbulence and a given response time of the effectiveness estimation algorithm. e proposed method helps choosing the desired short