3.2.1. Input/Output Linearization

Input/output linearization of Equation (6) is represented as

$$
\begin{bmatrix} y\_2^{(1)} \\ y\_3^{(1)} \end{bmatrix} = \begin{bmatrix} F\_2(\mathbf{x}) \\ F\_3(\mathbf{x}) \end{bmatrix} + \mathcal{B}2(\mathbf{x}) \begin{bmatrix} u\_2 \\ u\_3 \end{bmatrix} \tag{24}
$$

where

$$F\_2(\mathbf{x}) \quad = \quad \frac{1}{L\_{\rm md}}(-i\_{\rm md}R\_{\rm s} + \omega\_{\rm e}L\_{\rm mq}i\_{\rm mq}) \tag{25}$$

$$F\_3(\mathbf{x}) \quad = \quad -\frac{R\_\mathbf{s}}{L\_\mathbf{mq}} i\_{\mathbf{mq}} - \frac{1}{L\_\mathbf{mq}} \omega\_\mathbf{e} (L\_\mathbf{m} i\_{\mathbf{m}d} + K\_\mathbf{e}) \tag{26}$$

(27)

$$B2(\mathbf{x}) = \begin{bmatrix} B\_2(\mathbf{x}) \\ B\_3(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} \frac{1}{L\_{\rm md}} & 0 \\ 0 & \frac{1}{L\_{\rm mq}} \end{bmatrix} \tag{28}$$

As det[*<sup>B</sup>*2(*x*)] = 1 *<sup>L</sup>*md*L*mq = 0, i.e., *<sup>B</sup>*(*x*) is nonsingular for all nominal operation points. Therefore, the FLC controller is represented as

$$B\begin{pmatrix} u\_2 \\ u\_3 \end{pmatrix} = B2(\mathbf{x})^{-1} \begin{pmatrix} -F\_2(\mathbf{x}) + v\_2 \\ -F\_3(\mathbf{x}) + v\_3 \end{pmatrix} \tag{29}$$

$$B2(x)^{-1} = \begin{bmatrix} L\_{\rm md} & 0 \\ 0 & L\_{\rm mq} \end{bmatrix} \tag{30}$$

And the nonlinear system is linearized as

$$
\begin{bmatrix} y\_2^{(1)} \\ y\_3^{(1)} \end{bmatrix} = \begin{bmatrix} v\_2 \\ v\_3 \end{bmatrix} \tag{31}
$$

where

$$
v\_{2} = -\dot{y}\_{2\text{r}} + k\_{21}\varepsilon\_{2} \tag{32}$$

$$
\omega\_3 \quad = \ y\_{3\mathbf{r}} + k\_{31}\mathbf{e}\_3 \tag{33}
$$

where *v*2 and *v*3 are inputs of linear systems, *k*21 and *k*31 are gains of linear controller, *y*2r and *y*3r the output references. Define *e*2 = *y*2r − *y*2 and *e*3 = *y*3r − *y*3 as tracking errors, the error dynamics are

$$
\dot{\varepsilon}\_2 + k\_{21}\varepsilon\_2 = 0\tag{34}
$$

$$k\_3 + k\_{31}c\_3 = 0\tag{35}$$

### 3.2.2. Definition of Perturbation and State

Define perturbation terms <sup>Ψ</sup>2,3(*x*) as:

$$\begin{aligned} S\_2: \quad \left\{ \begin{array}{lcl} \Psi\_2(\mathbf{x}) &=& F\_2(\mathbf{x}) + (B\_2(\mathbf{x}) - B\_2(\mathbf{0})) \begin{bmatrix} u\_2 \\ u\_3 \end{bmatrix} \\ B\_2(\mathbf{0}) &=& \begin{bmatrix} \frac{1}{L\_{\text{m}\text{d}}} & \mathbf{0} \end{bmatrix} \end{aligned} \right. \\\\ S\_3: \quad \left\{ \begin{array}{lcl} \Psi\_3(\mathbf{x}) &=& F\_3(\mathbf{x}) + (B\_3(\mathbf{x}) - B\_3(\mathbf{0})) \begin{bmatrix} u\_2 \\ u\_3 \end{bmatrix} \\ B\_3(\mathbf{0}) &=& \begin{bmatrix} 0 & \frac{1}{L\_{\text{m}\text{d}}} \end{bmatrix} \end{aligned} \right. \end{aligned} \tag{36}$$

where *L*md0 and *<sup>L</sup>*mq0, *<sup>B</sup>*2(0) and *<sup>B</sup>*3(0) are nominal values of *L*md, *<sup>L</sup>*mq, *<sup>B</sup>*2(*x*) and *<sup>B</sup>*3(*x*), respectively.

Defining the state vectors as *z*21 = *y*2, *z*22 = Ψ2 and *z*31 = *y*3, *z*32 = Ψ3, and control variables as *u*2 = *V*md and *u*3 = *<sup>V</sup>*mq. The dynamic equations of the two subsystems *S*2 and *S*3 become as

$$\begin{aligned} S\_{2}: \quad \left\{ \begin{array}{rcl} z\_{21} &=& \Psi\_{2} \\ z\_{21} &=& \Psi\_{2}(\mathbf{x}) + \mathcal{B}\_{2}(\mathbf{0}) \begin{bmatrix} u\_{2} \\ u\_{3} \end{bmatrix} \\ z\_{22} &=& \Psi\_{2}(\mathbf{x}) \end{bmatrix} , \\\\ S\_{3}: \quad \left\{ \begin{array}{rcl} z\_{31} &=& \Psi\_{3} \\ z\_{31} &=& \Psi\_{3}(\mathbf{x}) + \mathcal{B}\_{3}(\mathbf{0}) \begin{bmatrix} u\_{2} \\ u\_{3} \end{bmatrix} \\ z\_{32} &=& \Psi\_{3}(\mathbf{x}) \end{aligned} \right\} , \end{aligned} \tag{37}$$
