*3.3. Subsystem 3*

The third subsystem is used for the setting of *Vdc*. It contains a single error variable that is between the *DC* bus voltage and its reference value *z*3. The error variable is defined by:

$$
\omega\_3 = \upsilon\_{dc}^\* - \upsilon\_{dc} \tag{28}
$$

The derivative of the error is as follows:

$$\frac{d}{dt}z\_3 = \frac{d}{dt}v\_{dc}^\* - \frac{d}{dt}v\_{dc} = \frac{1}{\mathcal{C}\_{dc}}i\_{dc}^\* = \frac{d}{dt}v\_{dc}^\* - \frac{P\_{dc}^\*}{\mathcal{C}\_{dc}v\_{dc}}\tag{29}$$

The Lyapunov function is given by:

$$v = \frac{1}{2}z\_3^2\tag{30}$$

The derivative of this function is given by:

$$\frac{d}{dt}\upsilon = z\_3 \frac{d}{dt} z\_3 = z\_3 \left[ \frac{d}{dt} \upsilon\_{dc}^\* - \frac{P\_{dc}^\*}{\mathbb{C}\_{dc} \upsilon\_{dc}} \right] \tag{31}$$

If we achieve the equality of the equations below, we obtain a better stability of the system:

$$\frac{d}{dt}\upsilon\_{dc}^\* - \frac{P\_{dc}^\*}{\mathbb{C}\_{dc}\upsilon\_{dc}} = -\mathsf{K}\_3\mathsf{z}\_3\tag{32}$$

Then, the command is as follows:

$$\begin{array}{l}P\_{dc}^\* = \mathbb{C}\_{dc} \boldsymbol{\upsilon}\_{dc} K\_3 \mathbf{z}\_3\\ \dot{\mathbf{r}}\_{dc}^\* = \mathbb{C}\_{dc} K\_3 \mathbf{z}\_3\end{array} \tag{33}$$
