*3.2. Operational Constraints*

The selection of the control gains of the SOSM is a significant issue where the best gains should guarantee good dynamic performances, stability, and robustness of the DFIG. The tuning of the control parameter by the conventional trial-and-error strategy is a timeconsuming approach. Moreover, other classical methods such as the use of Hurwitz stability criterion for the linearized model of the DC-link voltage dynamics can attain the stability of the control law, but without securing that, the captured gains are the best parameters [26]. To improve the chosen of the control coefficients of the SOSM controller, advanced optimization algorithms are employed to process the tuning problem. In this instance, the restrictions that are forced on the control parameters by the Lyapunov theory are considered as inequality constraints for the optimization problem. This results in an improvement in tracking execution and chattering reduction.

The Lyapunov stability theory of the DFIG dynamics generates nonlinear limitations on the decision variables of the given control problem. By using the control law of Equation (7) and taking into consideration the derivative of the sliding surface as stated in Equation (4), the dynamics of *sVdc*will be defined as:

$$\begin{cases} \frac{d\mathbf{s}\_{V\_{dc}}}{dt} = -\lambda\_{dc} \left| \mathbf{s}\_{V\_{dc}} \right|^{0.5} \text{sgn}(\mathbf{s}\_{V\_{dc}}) + \mathbf{y}\_{dc} + \eta\_{dc} \\\ \frac{d\mathbf{y}\_{dc}}{dt} = -\mathbf{a}\_{d\mathcal{L}} \text{sgn}\left(\mathbf{s}\_{V\_{dc}}\right) \end{cases} \tag{8}$$

Let us consider that the perturbation term *ηdc* of the DC-link voltage dynamics (2) is globally bounded and described as follows [27,28]:

$$|\eta\_{dc}| \le \Psi\_{dc} |s\_{V\_{dc}}|^{0.5} \; ; \Psi\_{dc} \ge 0 \tag{9}$$

To draw sufficient conditions on the robust stability of the studied DC-link voltage dynamics, let us define the following quadratic Lyapunov function:

$$V\_{dc} \left( s\_{V\_{dc'}}, y\_{dc} \right) = \xi\_{dc}^{\text{T}} \mathbf{L}\_{dc} \xi\_{dc} \tag{10}$$

where *ξdc* = [*ξ*1*ξ*2]<sup>T</sup> = #*sVdc* 0.5sgn*sVdc ydc*\$<sup>T</sup> and *Ldc* is a symmetric positive definite matrix, which takes the following form:

$$\mathbf{L}\_{d\mathbf{c}} = \frac{1}{2} \begin{bmatrix} 4\alpha\_{d\mathbf{c}} + \lambda\_{d\mathbf{c}}^2 & -\lambda\_{d\mathbf{c}} \\ -\lambda\_{d\mathbf{c}} & 2 \end{bmatrix} \tag{11}$$

Then, the time derivative of the candidate Lyapunov function *VdcsVdc* , *ydc* along the trajectories of the system (6) is calculated as:

$$\frac{dV\_{dc}}{dt}\left(\mathbf{s}\_{V\_{dc}},y\_{dc}\right) = -\frac{1}{|\mathcal{G}\_1|}\mathfrak{J}\_{dc}^{\mathrm{T}}\mathcal{M}^{\mathrm{T}}\mathfrak{J}\_{dc}^{\mathrm{x}} + \frac{\eta\_{dc}}{|\mathcal{G}\_1|}\mathcal{N}^{\mathrm{T}}\mathfrak{J}\_{dc}^{\mathrm{x}}\tag{12}$$

where *M* = *λdc* 2 2*αdc* + *<sup>λ</sup>*2*dc* − *λdc* −*λdc* 1 and *N*<sup>T</sup> = <sup>2</sup>*αdc* + *<sup>λ</sup>*2*dc* 2 − *λdc* 2 . Using the bounds on the perturbations of Equation (9), it can be demonstrated

*dVdc dtsVdc* , *ydc* = − 1|*ξ*1|*ξ*<sup>T</sup>*dcQdcξdc* (13)

 that:

where *Qdc* is a symmetric matrix expressed as:

$$\mathbf{Q}\_{d\varepsilon} = \frac{\lambda\_{d\varepsilon}}{2} \begin{bmatrix} 2\mathbf{a}\_{d\varepsilon} + \lambda\_{d\varepsilon}^2 - \left( 4\frac{a\_{d\varepsilon}}{\lambda\_{d\varepsilon}} + \lambda\_{d\varepsilon} \right) \mathbf{\varPsi}\_{d\varepsilon} - \left( \lambda\_{d\varepsilon} + \mathbf{\varPsi}\_{d\varepsilon} \right) \\ - \left( \lambda\_{d\varepsilon} + \mathbf{\varPsi}\_{d\varepsilon} \right) \mathbf{1} \end{bmatrix} \tag{14}$$

when this matrix is positive definite *Qdc* = *Q*<sup>T</sup>*dc* > 0, the stability condition of the dynamics (2) is satisfied in the sense of Lyapunov until . *VdcsVdc*, *ydc* < 0 as given in [29].

To have *Qdc* as a positive definite matrix, the range values of *λdc* and *αdc* should be adjusted as follows:

$$\begin{cases} \lambda\_{dc} > 2\Psi\_{dc} = \lambda\_{dc}^{\min} \\ \varkappa\_{dc} > \lambda\_{dc} \frac{5\lambda\_{dc}\Psi\_{dc} + 4\Psi\_{dc}^2}{2(\lambda\_{dc} - 2\Psi\_{dc})} = \alpha\_{dc}^{\min} \end{cases} \tag{15}$$

To enhance the performance of the DFIG system's control, the optimization theory is adopted to select and tune the effective parameters of the SOSM controllers for the DC-link voltage, stator active/reactive powers, and grid current loops. Hence, the tuning problem related with the STA-SOSM controllers is completely formulated as a constrained optimization problem. Several time-domain performance metrics, i.e., maximum overshoot, steady-state error, rise and/or settling times, as well as the established stability and robustness conditions of Equation (15), are included as inequality constraints for the optimization problem defined as follows:

$$\begin{cases} \text{minimize } f\_{\mathfrak{m}}(\mathbf{x},t), m \in \{IAE, ISE, ITE, ITE\} \\ \mathbf{x} = \left[\lambda\_{Q\_{s}, \mathbf{a}} \alpha\_{Q\_{d}, \mathbf{c}} \lambda\_{dc, \mathbf{a}} \alpha\_{dc, \lambda\_{\mathbf{d}\_{d}}, \mathbf{a}}\right]^{T} \in \mathbb{S} \subseteq \mathbb{R}\_{+}^{6} \\ \text{subject to } : \\ g\_{1}(\mathbf{x},t) = \delta\_{s} - \delta\_{P\_{s}}^{\max} \le 0 \\ g\_{2}(\mathbf{x},t) = \delta\_{Q\_{s}} - \delta\_{Q\_{s}}^{\max} \le 0 \\ g\_{3}(\mathbf{x},t) = \delta\_{d\mathbf{c}} - \delta\_{d\mathbf{c}}^{\max} \le 0 \\ g\_{4}(\mathbf{x},t) = \delta\_{i\_{\mathbf{d}}} - \delta\_{i\_{\mathbf{d}}}^{\max} \le 0 \\ g\_{5}(\mathbf{x},t) = \delta\_{i\_{\mathbf{d}}} - \delta\_{i\_{\mathbf{d}}}^{\max} \le 0 \\ \lambda\_{n,\min} \le \lambda\_{n} \le \lambda\_{n,\max} \end{cases} \tag{16}$$

where *Jm* : <sup>R</sup>6+ → R are the cost functions, *gq* : <sup>R</sup>6+ → R are the problem's inequality constraints, and *δdc*, *<sup>δ</sup>idg* , *<sup>δ</sup>iqg* , *δPs* and *<sup>δ</sup>Qs* are the overshoots of the DC-link voltage, grid current components, and stator power components, respectively. The terms *δ*max *dc* , *δ*max *idg* , *δ*max *iqg* , *δ*max *Ps* and *δ*max *Qs* indicate their maximum specified values. The terms *<sup>λ</sup>Qs*, *<sup>α</sup>Qs* represent the gains of the SOSM controller for the stator power loop, and *<sup>λ</sup>idg* and *<sup>α</sup>idg* are the gains of the controller for the grid current loop.

In the formalism of optimal control theory, the objective functions of (16) are usually described by common performance standards, such as Integral Absolute Error (IAE), Integral Square Error (ISE), Integral Time-Weighted Absolute Error (ITAE), and Integral

Time-Weighted Square Error (ITSE) [3,17,24,25]. Therefore, the global objective function using the IAE index for all controlled dynamics is aggregated as follows:

*JIAE*(*<sup>x</sup>*, *t*) = #*wVdc widg wiqg wQs wPs* \$ ⎡⎢⎢⎢⎢⎢⎢⎢⎣ - *T*0 |*edc*(*<sup>x</sup>*, *t*)|*dt* - *T*0 *eidg* (*x*, *t*)*dt* - *T*0 *eiqg* (*x*, *t*)*dt* - *T*0 *eQs* (*x*, *t*)*dt* - *T*0 |*ePs* (*x*, *t*)|*dt* ⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)

where *T* denotes the total simulation time, *wr* ∈ {*Qs* , *Ps*, *Vdc*, *idg*, *iqg*/ is the weighting coefficient satisfying the convex summation ∑ *r wr* = 1, and *ei*(.) is the tracking error defined as *ePs* (*x*, *t*) = *P*∗*s* − *Ps*(*<sup>x</sup>*, *t*), *eQs* (*x*, *t*) = *Q*<sup>∗</sup>*s* − *Qs*(*<sup>x</sup>*, *t*), *edc*(*<sup>x</sup>*, *t*) = *<sup>V</sup>*<sup>∗</sup>*dc* − *Vdc*(*<sup>x</sup>*, *t*), *eidg*(*x*, *t*) = *i*∗*dg* − *idg*(*<sup>x</sup>*, *t*), and *eiqg* (*x*, *t*) = *i*∗*qg* − *iqg*(*<sup>x</sup>*, *t*).

#### **4. Adaptive Fuzzy ESO-Based Sliding Mode Controller**

In this work, a fuzzy gain-scheduling mechanism is proposed to design an adaptive STA-SOSM controller for the DC-link voltage dynamics. An adaptive extended state observer (ESO) is designed to asymptotically estimate the disturbances and allow the STA-SOSM controller to reject them efficiently. Therefore, the adaptive fuzzy ESO is employed in the DC-link voltage loop to improve the control performance and robustness as shown in Figure 2.

**Figure 2.** Adaptive fuzzy ESO-based SOSM controller for the DC-link voltage.

#### *4.1. Concept of Extended State Observers*

The ESO treats the lumped disturbances of the system as a new system state which is conceived to estimate not only the external disturbances but also the plant dynamics [30,31].

However, for any arbitrary second-order system, an ESO can be written in the following state-space form [15,30]:

$$\begin{cases}
\dot{\mathbf{x}}\_1(t) = \mathbf{x}\_2(t) \\
\dot{\mathbf{x}}\_2(t) = f(\mathbf{x}\_1(t), \mathbf{x}\_2(t), \boldsymbol{u}(t)) + b\_0 \boldsymbol{u}(t) + \boldsymbol{w}(t) \\
\boldsymbol{y}(t) = \mathbf{x}\_1(t)
\end{cases} \tag{18}$$

where *<sup>x</sup>*1(*t*) ∈ R and *<sup>x</sup>*2(*t*) ∈ R are the state variables of the cascade integral form, *f*(*<sup>x</sup>*1(*t*), *<sup>x</sup>*2(*t*), *u*(*t*)) ∈ R defines the dynamics of the system which is a so-called internal disturbance, *u*(*t*) ∈ R is the control variable, *y*(*t*) ∈ R is the output variable, *w*(*t*) ∈ R is an external disturbance, and *b*0 is a given constant, and .*x*(*t*) denotes the time derivative of the variable *<sup>x</sup>*(*t*), i.e., .*<sup>x</sup>*1(*t*) = *dx*1(*t*) *dt*, and so on.

For a given second-order system, a three-order ESO can be used with the disturbance as an additional state variable and defined as follows:

$$\begin{cases} \dot{e}(t) = z\_1(t) - x\_1(t) \\ \dot{z}\_1(t) = z\_2(t) - \beta\_1 e(t) \\ \dot{z}\_2(t) = z\_3(t) + b\_0 u(t) - \beta\_2 e(t) \\ \dot{z}\_3(t) = -\beta\_3 e(t) \end{cases} \tag{19}$$

where *β*1, *β*2, and *β*3 are the gains of the extended state observer.

In the state-space form (19), the ESO takes the system's output *y*(*t*) = *<sup>x</sup>*1(*t*) and control variable *u*(*t*) as input and gives the state variables (*<sup>z</sup>*1(*t*), *<sup>z</sup>*2(*t*), *<sup>z</sup>*3(*t*)) which represent the estimations of system state variables (*<sup>x</sup>*1(*t*), *<sup>x</sup>*2(*t*))*<sup>T</sup>* ∈ R<sup>2</sup> and total disturbances *<sup>w</sup>*(*t*), respectively, i.e., *<sup>z</sup>*1(*t*) = *<sup>x</sup>*ˆ1(*t*), *<sup>z</sup>*2(*t*) = *<sup>x</sup>*ˆ2(*t*), and *<sup>z</sup>*3(*t*) = *<sup>w</sup>*<sup>ˆ</sup>(*t*). By properly selecting the feedback coefficients *β*1, *β*2, and *β*3, the estimation error *e*(*t*) converges asymptotically to a small value. Besides, from Equation (19), it is clear that the ESO does not depend on the system parameters and thus provides strong robustness for observer dynamics [30,31].

#### *4.2. Extended State Observer for the DC-Link Voltage Loop*

To design an extended state observer (19) for the DC-link voltage loop, the first state variable of such an observer is chosen as *<sup>z</sup>*1(*t*) = *Vdc*(*t*). Therefore, the dynamics of the DC-link voltage (2) can be reformulated as:

$$\dot{z}\_1(t) = \frac{dV\_{dc}(t)}{dt} = G\_{dc}u\_{dc0}(t) - d(t) \tag{20}$$

where *udc*0(*t*) = *idg*(*t*), ∀*t* ≥ 0 and *d*(*t*) = <sup>−</sup>*ηdc*(*t*).

The ESO treats the external disturbance as an extended state. Therefore, a second-order ESO is used for the outer DC-link voltage loop, which is adopted as follows:

$$\begin{cases} \varepsilon\_1(t) = z\_1(t) - \dot{z}\_1(t) \\ \dot{\hat{z}}\_1(t) = G\_{dc}u\_{dc0}(t) - \hat{d}(t) + \beta\_1 e\_1(t) \\ \dot{\hat{d}}(t) = -\beta\_2 e\_1(t) \end{cases} \tag{21}$$

where *z*ˆ1(*t*) is an estimate of the output, *<sup>z</sup>*1(*t*) ˆ*d*(*t*) is an estimate of the total disturbance *d*(*t*) = <sup>−</sup>*ηdc*(*t*), and *β*1 and *β*2 are the ESO gains chosen as follows [13,30]:

$$\left[\beta\_1, \beta\_2\right] = \left[2\omega\nu\_0\left\omega\_0^2\right] \in \mathbb{R}\_+^2 \tag{22}$$

In Equation (22), the real *ω*0 > 0 denotes the observer's bandwidth that becomes the only tuning parameter of the extended state observer (21). Based on the above studies and the designed sliding-control law given in Equation (7), the proposed ESO-based control of the DC-link voltage loop is achieved by:

$$\begin{cases} \begin{array}{l} i\_{d\xi}^{\*} = \frac{1}{G\_{dc}} \left( \left( -\lambda\_{dc} \middle| s\_{V\_{dc}} \right|^{0.5} \text{sgn} \left( s\_{V\_{dc}} \right) + y\_{dc} + \frac{dV\_{dc}^{\*}}{dt} \right) + \hat{d}(t) \right) \\\ \frac{dy\_{dc}}{dx} = -a\_{d\varepsilon} \text{sgn} \left( s\_{V\_{dc}} \right) \end{array} \tag{23}$$

The choice of the adequate bandwidth of a given ESO is a difficult task action [32,33]. The appropriate selection can improve the closed-loop system's performance, while the poor selection could degrade the time-domain performances and robustness of the controlled system. In general, the larger the ESO bandwidth, the more accurate the estimation of states will be achieved. On the other hand, the increase of such a bandwidth may lead to vulnerability against noise and loss of robustness. The design of an observer is always a trade-off between the estimation dynamics performance and the noise vulnerability [13,15]. Typically, the bandwidth of the ESO is selected to be 5 to 15 times the DC-link voltage controller's bandwidth. However, this last is limited from 1/1000 to 1/100 of the

switching frequency as discussed in [34]. In this research work, the idea to design a fuzzy gains-scheduling-based observer is proposed to overcome such a complex tuning problem. Such an adaptive fuzzy supervisor is proposed to avoid the aforementioned drawbacks and achieve high convergence performance and robustness of the designed ESO under operational disturbances and high-frequency noises. Figure 2 shows such a proposed adaptive-fuzzy extended state observer for the studied wind energy converter.

The approach taken here is to exploit the Mamdani type of fuzzy rules and reasoning to generate the appropriate values of the ESO's bandwidth *ω*0 ∈ R+ of Equation (21). Let us consider as inputs of the proposed fuzzy gains scheduler the linguistic variables *e*(*kTs*) = *ek* as the error between the actual and the estimated output, i.e., the sampled signals *Vdc* and *<sup>V</sup>*<sup>ˆ</sup>*dc*, respectively, and <sup>Δ</sup>*e*(*kTs*) = Δ*ek* as the change of error, where *Ts* denotes the sampling period for the fuzzy supervisor. The designed fuzzy inference supervision mechanism produces the tuned bandwidth of the ESO denoted as 0.

For this proposed fuzzy-based tuning mechanism, a Mamdani model is applied as a type of inference mechanism. The decision-making output can be acquired using a Max-Min fuzzy inference where the crisp output is computed by the center of gravity defuzzification approach. As shown in Figure 3, all used membership functions for fuzzy sets are triangular, uniformly distributed, and symmetrical on the universe of discourse.

**Figure 3.** *Cont*.

**Figure 3.** Membership functions of the proposed fuzzy gains-scheduling mechanism.

A set of linguistic rules in the form of Equation (24) is utilized in the fuzzy reasoning inference to define the output 0:

$$\text{If } e\_k \text{ is } A\_i \text{ and } \Delta e\_k \text{ is } B\_i, \text{ then } \mathcal{a} \uplus\_0 \text{ is } \mathcal{C}\_i \tag{24}$$

where *Ai*, *Bi*, and *Ci* are the fuzzy sets of the inputs/output linguistic variables *ek*, Δ*ek*, and 0, respectively. The linguistic levels assigned to the fuzzy inputs and outputs are labeled as follows: negative big (NB), negative (N), zero (ZE), positive (P), and positive big (PB). A set of 25 rules are defined for this fuzzy inference as given in Table 1.

**Table 1.** Fuzzy rules for the 0 parameter's tuning.


As proposed in [32,33], and since it is assumed that the ESO's bandwidth parameter *ω*0 varies in the prescribed range *ω*min 0 , *ω*max 0 , this effective parameter is calculated according to the following linear transformation [34]:

$$
\omega\_0 = \left(\omega\_0^{\text{max}} - \omega\_0^{\text{min}}\right) \omega\_0 + \omega\_0^{\text{min}} \tag{25}
$$

where *ω*max 0and *ω*min 0are the maximum and minimum limits of *ω*0, respectively.

## **5. Results and Discussion**

The proposed STA-SOSM controllers based on an adaptive fuzzy ESO for the DC-link voltage loop are built using MATLAB/Simulink environment. The introduced algorithms GA, PSO, HSA, WCA, GOA, and TEO were switched with the equal values for the mutual factors, i.e., population size *Npop* = 50 and the maximum number of iterations *Niter* = 100, and run on an Intel R CoreTMi5 CPU computer at 2.5 GHz and 8 GB of RAM. The parameters of the DFIG (1.5 MW) used in this work are given in our previous works [3]. The gains of the STA-SOSM controllers for the DC-link voltage and other control loops are tuned thanks to the proposed advanced optimization algorithms for the problems (16) and (17).

Figure 4 shows the convergence curves of the cost functions for the considered timedomain performance indices. Moreover, the TEO algorithm for the ISE and ITSE criteria outperforms the other reported methods in terms of fast and non-premature convergence.

Figure 4a,c show that the TEO for the IAE and ITAE indicators display the better convergence as a second- and third-order, respectively, after the GA one.

**Figure 4.** Convergence curves comparison: (**a**) IAE; (**b**) ISE; (**c**) ITAE; and (**d**) ITSE criterion.

Since the TEO metaheuristic outperformed all the other reported ones, the effective gains *λdc* and *αdc* of the STA-SOSM controller, retained for the rest of the control strategy, are selected as the best results obtained by the TEO algorithm of the optimization problem (16) and (17). Table 2 summarizes such decision variables for each time-domain performance criterion.


**Table 2.** TEO-based results for the STA-SOSM controllers' gains tuning.

Besides, the bandwidth of the ESO is selected to be between 1/200 and 15/100 of the switching frequency for the GSC circuit. The fuzzy gains-scheduling mechanism is employed to adaptively tune the bandwidth of the ESO within the predefined limits. The obtained fuzzy surface for the bandwidth gain is presented in Figure 5. On the other hand, Figure 6 shows the histories of the scheduled gain of the proposed adaptive fuzzy ESO. Figure 7 presents the performance of the DC-link voltage when different input steps are set.

It can be noted that the chattering phenomenon is reduced using the proposed TEO and fuzzy-based method in comparison with the STA-SOSM and PI controllers-based ones.

**Figure 5.** Fuzzy surface of the observer's bandwidth parameter 0.

**Figure 6.** Time histories of the observer's bandwidth control 0.

**Figure 7.** Time-domain performances' comparison of the DC-link voltage.

Moreover, the STA-SOSM controller based on different types of observers leads to lower DC-link voltage overshoot and faster response compared with the STA-SOSM and classical PI controllers-based cases as summarized in Table 3. Indeed, the proposed design and tuning method is compared to the STA-SOSM controller associated with a linear observer as well as without observers to show its superiority and effectiveness. Also, Figure 8 presents the time-domain performance of the DC-link voltage controller under severe voltage dips conditions. Since a 30% voltage drop is considered, the proposed adaptive fuzzy observer and TEO-based tuning method are successfully able to mitigate the voltage dip.


**Table 3.** Time-domain performances' comparison for the controlled DC-link voltage dynamics.

> Based on these demonstrative results, one can notice that the lower amplitude of the fluctuations exists near the starting of the voltage dips in comparison with the STA-SOSM controller only. However, the proposed robust and intelligent STA-SOSM controllers based on linear and ESO observers present approximately the same performance in the case of DC-link voltage dips. Further comparison in terms of Total Harmonic Distortion (THD) variation is made in Table 4.

**Figure 8.** Time-domain responses of the controlled DC-link voltage under drop conditions.



The currents THD of stator and rotor for the adaptive fuzzy ESO-based SOSM controllers are about 0.26% and 0.28%, respectively. These values are better than the other reported design methods, such as TEO-tuned PI controller, TEO-tuned STA-SOSM controller, and STA-SOSM controller-based linear observer. In addition, these values agree with the limits of the IEEE519-1992 standard which stipulate that the value of the THD does not exceed 5%. The DC-link voltage states and their according estimations and observation errors are shown in Figures 9 and 10 for linear and adaptive fuzzy ESO observers, respectively. The observation errors are obtained as the difference between the actual value of the DC-link voltage and the value estimated by the observers, i.e., *e* = *Vdc*− *V* ˆ *dc*.

The estimation dynamics using the different proposed observers are perfectly ensured. However, the adaptive fuzzy ESO presents a faster convergence and reconstruction of the system's state than in the linear observer-based case, i.e., an observation error reached null value after 0.05 s with ESO compared to 0.3 sec with the linear observer. Moreover, Figure 11 gives the total disturbance estimations for the linear and adaptive fuzzy ESO observers, respectively. Such a convergence rate comparison and total disturbance estimation of observers indicate the effectiveness and superiority of the proposed adaptive fuzzy extended state observer for the DC-link voltage regulation.

**Figure 9.** Convergence's dynamics of the linear observer for the DC-link voltage.

**Figure 10.** Dynamic performance of the adaptive fuzzy ESO for the DC-link voltage.

In a wind conversion system, the external disturbances' upper bound is unknown and is hard to be estimated in an actual doubly-fed induction generator. Therefore, the fuzzy gain-scheduled ESO is proposed in this work to compel observer parameters to vary according to the disturbances' upper bound in real-time. The uncertainty upper limit is required when developing the DC-link voltage second-order sliding mode controller, ye<sup>t</sup> the upper limits are unknown in practical wind conversion systems and are difficult to be estimated. Overestimation of these upper bounds may lead to a conservative choice of controller parameters which will produce more control effectiveness, aggravate control chattering, and shorten the service cycle of wind turbines.

**Figure 11.** Total disturbance estimations' comparison of the reported observers.
