*2.1. Bilinear Switching Model of DC-DC Buck Converter*

The working principle is based on two alternative phenomena: charging and discharging, based on the control signal state, Sc. When considering the continuous conduction mode and the control signal state levels, two modes are to be considered [45–53].

Mode 1:

For the ON mode in which *Sc* = 1, the transistor T is closed and the diode D is open. Based on Kirchhoff's current and voltage laws, we can write:

$$\begin{cases} \frac{di\_l}{dt} = \frac{V\_{dc}}{L} - \frac{V\_{load}}{L} \\\\ \frac{dV\_{load}}{dt} = \frac{i\_l}{C} - \frac{i\_{load}}{C} \end{cases} \tag{1}$$

Mode2:

*Sc* = 0, the transistor T becomes open and the diode D begins closed. The state equations describing the inductor current and the output voltage dynamics are given in (2).

$$\begin{cases} \begin{array}{c} \frac{d\dot{i}\_l}{dt} = -\frac{V\_{load}}{L} \\\\ \frac{dV\_{load}}{dt} = \frac{\dot{i}\_l}{C} - \frac{\dot{i}\_{load}}{C} \end{array} \tag{2}$$

The combination of the two sub models leads to the general buck DC-DC converter model, as illustrated in (3). ⎧

$$\begin{cases} \begin{array}{c} \frac{d\dot{l}\_l}{dt} = S\_c \frac{V\_{dc}}{L} - \frac{V\_{load}}{L} \\\\ \frac{dV\_{load}}{dt} = \frac{\dot{l}\_l}{C} - \frac{\dot{i}\_{load}}{C} \end{array} \tag{3} $$

For a resistive load, (3) begins

$$\begin{cases} \begin{array}{c} \frac{d\dot{i}\_l}{dt} = S\_c \frac{V\_{dc}}{L} - \frac{V\_{load}}{L} \\\\ \frac{dV\_{load}}{dt} = \frac{\dot{i}\_l}{C} - \frac{V\_{load}}{RC} \end{array} \tag{4}$$

#### *2.2. Averaged Dynamic Model of DC-DC Buck Converter*

By using the state-space averaging method [46], Equation (4) can be written as illustrated with Equation (5).

$$\begin{cases} \begin{array}{c} \frac{d \langle i \rangle\_{T\_s}}{dt} = \langle S\_{\mathcal{C}} \rangle\_{T\_s} \frac{\langle V\_{dc} \rangle\_{T\_s}}{L} - \frac{\langle V\_{load} \rangle\_{T\_b}}{L} \\\\ \frac{d \langle V\_{load} \rangle\_{T\_b}}{dt} = \frac{\langle i \rangle\_{T\_s}}{\mathcal{C}} - \frac{\langle V\_{load} \rangle\_{T\_b}}{RC} \end{array} \tag{5}$$

where *ilTs* , *VloadTs* , *ScTs* and *VdcTs* are the averaged values of inductor current, output voltage, control signal, and input voltage, respectively, in a switching period *Ts*.

This can be put into the more compact form of an uncertain nonlinear system, as indicated by (6). .

$$X = f(X) + \mathcal{g}(X)\mathfrak{a} \tag{6}$$

The nonlinear equations *f*(*X*) and *g*(*X*), and averaged state vector *X*, are defined as follows:

$$\mathcal{S}(X) = \begin{pmatrix} \frac{V\_{dc}}{L} \\ 0 \end{pmatrix} \tag{7}$$

$$f(X) = \begin{pmatrix} 0 & -\frac{V\_{load}}{L} \\ \frac{i\_l}{\mathcal{C}} & -\frac{V\_{load}}{RC} \end{pmatrix} \tag{8}$$

$$X = \left[ \left< i\_I \right>\_{T\_s} \left< V\_{load} \right>\_{T\_s} \right]^T \tag{9}$$

Here, *α* is the duty cycle.
