3.1.1. Input/Output Linearization

Input/output linearization of WT speed dynamics in system Equation (6) can be represented as

$$y\_1^{(2)} = \frac{1}{f\_{\text{tot}}} (\dot{T}\_{\text{m}} - \dot{T}\_{\text{e}}) \tag{8}$$

As the electromagnetic torque has much faster response than the mechanical torque, from the perspective of control of WT, *T* ˙ e 0. Equation (8) can be expressed as

$$\begin{array}{rcl} y\_1^{(2)} &=& \frac{1}{J\_{\text{tot}}} \mathcal{T}\_{\text{m}} \\ &=& F\_1(\mathbf{x}) + B1(\mathbf{x})u\_1 \end{array} \tag{9}$$

where

$$\begin{array}{rcl} F\_1(\mathbf{x}) &=& A[-\frac{\mathbf{C}\_\mathbf{P}}{\omega\_\mathbf{m}} - \frac{RV}{F^2}E] \frac{\mathbf{d}\omega\_\mathbf{m}}{\mathbf{d}t} \\ &- \frac{AE\beta}{\tau\_\beta}[-0.088e^{-12.5\tau} - \frac{0.08V^2}{F} + \frac{0.105\beta^2}{(1+\beta^3)^2}] \frac{\mathbf{d}\beta}{\mathbf{d}t} \end{array} \tag{10}$$

$$B1(x) \quad = \frac{AE\beta}{\tau\_{\beta}}[-0.088e^{-12.5\tau} - \frac{0.08V^2}{F} + \frac{0.105\beta^2}{(1+\beta^3)^2}] \tag{11}$$

where

$$\begin{array}{rcl} A &=& \frac{\rho \pi R^2 V^3}{2\omega\_{\rm m}}\\ E &=& (39.27 - 319\tau + 1.1\beta)e^{-12.5\tau} \\ F &=& \omega\_{\rm m}R + 0.08\beta V \\ \tau &=& \frac{1}{\lambda + 0.08\beta} - \frac{0.035}{\beta^3 + 1} \end{array} \tag{12}$$

Please note that d*Vdt* is not included in the FLC design, which cannot be directly measured. As det[*B*<sup>1</sup>(*x*)] = *AEβ τβ* [−0.088*e*<sup>−</sup>12.5*<sup>τ</sup>* − 0.08*V*<sup>2</sup> *F* + 0.105*β*<sup>2</sup> (1 + *β*<sup>3</sup>)<sup>2</sup> ] = 0 when *V* = 0 and *β* = 0, i.e., *<sup>B</sup>*<sup>1</sup>(*x*) is nonsingular for all nominal operation points. Therefore, the FLC is expressed as

$$\mu\_1 \quad = \quad B1(\mathbf{x})^{-1}(-F\_1(\mathbf{x}) + v\_1) \tag{13}$$

And the nonlinear system is linearized as

$$y\_1^{(2)} = v\_1 \tag{14}$$

$$w\_1 \quad = \quad \ddot{y}\_{1\mathfrak{r}} + k\_{11}\dot{e}\_1 + k\_{12}e\_1 \tag{15}$$

where *v*1 is input of linear systems, *k*11 and *k*12 are gains of linear controller, *y*1r is the output reference, and *e*1 = *y*1r − *y*1 as tracking error. The error dynamic is

$$k\ddot{\varepsilon}\_1 + k\_{11}\dot{\varepsilon}\_1 + k\_{12}\varepsilon\_1 = 0\tag{16}$$

### 3.1.2. Definition of Perturbation and State

For this subsystem, a perturbation term including all subsystem uncertainties, nonlinearities and interactions among subsystems is defined.

Define perturbation term <sup>Ψ</sup>1(*x*) as:

$$S\_1: \quad \begin{cases} \quad \Psi\_1(\mathbf{x}) &= \quad F\_1(\mathbf{x}) + (B1(\mathbf{x}) - B1(\mathbf{0}))u\_1 \\ \quad B1(\mathbf{0}) &= \quad \frac{AE\beta}{\tau\_\beta}[-0.088e^{-12.5\tau} - \frac{0.08V^2}{F} + \frac{0.105\beta^2}{(1+\beta^3)^2}] \end{cases} \tag{17}$$

where *B*1(0) is nominal value of *<sup>B</sup>*<sup>1</sup>(*x*).

Defining the state vectors as *z*11 = *y*1, *z*12 = *y*(1) 1 , *z*13 = Ψ1, and control variable as *u*1 = *β*r. The dynamic equation of the subsystem *S*1 becomes as

$$\begin{array}{rcl} \varphi\_1: & \begin{cases} z\_{11} &=& y\_1 \\ \dot{z}\_{11} &=& z\_{12} \\ \dot{z}\_{12} &=& \Psi\_1(x) + B1(0)u\_1 \\ \dot{z}\_{13} &=& \Psi\_1(x) \end{cases} \end{array} \tag{18}$$

For subsystem *S*1, several types of perturbation observers, e.g., linear Luenberger observer, sliding mode observer and high-gain observer, have been proposed [19,29,36]. High-gain observers proposed in [29] are used to estimate states and perturbations in this paper.

### 3.1.3. Design of States and Perturbation Observer

When the system output *y*1 is available, a third-order SPO is employed for estimations of states and perturbation of the subsystem, which is designed as

$$\begin{array}{rcl} S1: \begin{cases} \dot{z}\_{11} &=& \dot{z}\_{12} + l\_{11}(z\_{11} - \dot{z}\_{11})\\ \dot{\dot{z}}\_{12} &=& \dot{z}\_{13} + l\_{12}(z\_{11} - \dot{z}\_{11}) + B1(0)u\_1\\ \dot{z}\_{13} &=& l\_{13}(z\_{11} - \dot{z}\_{11}) \end{cases} \end{array} \tag{19}$$

where *z*ˆ11, *z*ˆ12 and *z*ˆ13 are the estimations of *z*11, *z*12 and *z*13, respectively, and *l*11, *l*12 and *l*13 are gains of the observers, which are designed as

$$l\_{i\bar{j}} = \frac{\mathfrak{a}\_{i\bar{j}}}{\mathfrak{e}\_{i}^{\bar{j}}} \tag{20}$$

where *i* = 1, 2, 3; *j* = 1, ··· ,*ri* + 1, *i* is a scalar chosen to be within (0,1) for representing times of the time-dynamics between the real system and the observer, and parameters *<sup>α</sup>ij* are chosen so that the roots of

$$s^{r\_i+1} + \mathfrak{a}\_{i1}s^{r\_i} + \dots + \mathfrak{a}\_{ir\_i}s + \mathfrak{a}\_{i(r\_i+1)} = 0 \tag{21}$$

are in the open left-half complex plane.

### 3.1.4. Design of Nonlinear Adaptive Controller

The estimated perturbation is used for compensating the real perturbation, and control laws of subsystem *S*1 can be obtained as follows:

$$\mu\_1 \quad = \quad B1(0)^{-1}(-\sharp\_{13} + \upsilon\_1) \tag{22}$$

where *v*1 is defined as

$$
v\_{1} = -\ddot{z}\_{11\text{r}} + k\_{12}(z\_{11\text{r}} - \dot{z}\_{11}) + k\_{11}(\dot{z}\_{11\text{r}} - \dot{z}\_{12})\tag{23}$$
