*3.2. Subsystem 2*

The magnitude *v*∗*f*βrepresents the command and *if* β its output. The algorithm is given as follows:

$$\frac{d\dot{t}\_{f\beta}}{dt} = -\frac{R}{L}\dot{\imath}\_{f\beta} - \frac{V\_c}{L} + \frac{1}{L}\upsilon\_{f\beta}^\* - \frac{1}{L}\upsilon\_{ch\beta} \tag{21}$$

The error variable *z*1 is given by:

$$z\_2 = i\_{f\emptyset}^\* - i\_{f\emptyset} \tag{22}$$

The error is derived as follows:

$$\frac{d}{dt}(z\_2) = \frac{d}{dt}(\mathring{\mathbf{i}}\_{f\beta}) - \frac{d}{dt}(\mathring{\mathbf{i}}\_{f\beta}) = \frac{d}{dt}(\mathring{\mathbf{i}}\_{f\beta}^\*) + \frac{R}{L}\mathring{\mathbf{i}}\_{f\beta} + \frac{V\_c}{L} - \frac{1}{L}\mathbf{v}\_{f\beta}^\* + \frac{1}{L}\mathbf{v}\_{\text{clpl}\beta} \tag{23}$$

Lyapunov's intended function is as follows:

$$v = \frac{1}{2}z\_2^2\tag{24}$$

The derivative of this function is given by:

$$\frac{d}{dt}\upsilon = z\_2 + \frac{d}{dt}z\_2 = z\_2 + \frac{d}{dt}(i\_{f\beta}^\*) + \frac{R}{L}i\_{f\beta} + \frac{V\_c}{L} - \frac{1}{L}v\_{f\beta}^\* + \frac{1}{L}\upsilon\_{ch\beta} \tag{25}$$

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

$$\frac{d}{dt}(\vec{i}\_{f\beta}^\*) + \frac{R}{L}\dot{i}\_{f\beta} + \frac{V\_\varsigma}{L} - \frac{1}{L}v\_{f\beta}^\* + \frac{1}{L}v\_{cl\eta\beta} = -K\_2z\_2\tag{26}$$

Then, the command is as follows:

$$
\sigma\_{f\beta}^{\*} = L \left[ \frac{d}{dt} (i\_{f\beta}^{\*}) + \frac{R}{L} i\_{f\beta} + \frac{V\_c}{L} + K\_2 z\_2 \right] + v\_{cl\beta} \tag{27}
$$
