*3.1. Subsystem 1*

The variable *<sup>v</sup>*\**f*α represents the command and *if*α its output. The algorithm is given as follows:

$$\frac{d\dot{t}\_{fa}}{dt} = -\frac{R}{L}\dot{t}\_{fa} - \frac{V\_c}{L} + \frac{1}{L}\upsilon\_{fa}^\* - \frac{1}{L}\upsilon\_{ch\alpha} \tag{14}$$

The error variable *z*1 is given by:

$$z\_1 = \mathbf{i}\_{fa}^\* - \mathbf{i}\_{fa} \tag{15}$$

The error is derived as follows:

$$\frac{d}{dt}(z\_1) = \frac{d}{dt}(i\_{fa}^\*) - \frac{d}{dt}(i\_{fa}) = \frac{d}{dt}(i\_{fa}^\*) + \frac{R}{L}i\_{fa} + \frac{V\_c}{L} - \frac{1}{L}v\_{fa}^\* + \frac{1}{L}v\_{c\ln} \tag{16}$$

Lyapunov's intended function is as follows:

$$v = \frac{1}{2}z\_1^2\tag{17}$$

The derivative of this function is given by:

$$\frac{d}{dt}v = z\_1 + \frac{d}{dt}z\_1 = z\_1 + \frac{d}{dt}(i\_{fa}^\*) + \frac{R}{L}i\_{fa} + \frac{V\_c}{L} - \frac{1}{L}v\_{fa}^\* + \frac{1}{L}v\_{cba} \tag{18}$$

In order to achieve greater stability in the system, the following equality must be achieved:

$$\frac{d}{dt}(\mathbf{i}\_{fa}^{\*}) + \frac{R}{L}\mathbf{i}\_{fa} + \frac{V\_{c}}{L} - \frac{1}{L}\mathbf{v}\_{fa}^{\*} + \frac{1}{L}\mathbf{v}\_{c\mathbf{h}\alpha} = -\mathbf{K}\_{1}\mathbf{z}\_{1} \tag{19}$$

Then the command is as follows:

$$
\sigma\_{fa}^{\*} = L \left[ \frac{d}{dt} (\mathbf{i}\_{fa}^{\*}) + \frac{R}{L} \mathbf{i}\_{fa} + \frac{V\_{\mathcal{E}}}{L} + \mathbf{K}\_{\mathcal{E}} z\_{1} \right] + \upsilon\_{ch\alpha} \tag{20}
$$
