## 2 Materials and Methods
### Boost Converter  
The basic diagram of the boost converter is presented in Figure 1. The term \(V_{in}\) is the input voltage, \(i\) is the current in the inductor \(L\), S is the switch, \(D\) the diode, \(C\) the capacitance of the capacitor, and \(v\) the voltage at the load or output voltage. When the switch S is closed (ON), the coil \(L\) stores energy from the voltage source \(V_{in}\), while the load is supplied through the capacitor \(C\). When the switch is open (OFF), the current passes through the diode \(D\) and feeds the capacitor \(C\) and the load resistance \(R\). According to the diagram in Figure 1, the boost converter is governed by the differential expressions defined in Equations (1) and (2), where \(u\) is the control signal. The state space in which the system described by the boost converter evolves is a subset \(\mathbb{S}^{1}\times\mathbb{R}^{2}\) (where \(\mathbb{S}^{1}\) is the unit circle). \[\frac{dv}{dt}=-\frac{1}{RC}v+\frac{1}{C}i(1-u), \tag{1}\]  
\[\frac{di}{dt}=-\frac{1}{L}v(1-u)+\frac{V_{in}}{L}. \tag{2}\]  
Equations (3)-(5) consider that \(x_{1}\), \(x_{2}\), and \(s\) are functions of the boost converter parameters, \(V_{in}\), \(v\), \(i\), and the time \(t\). In addition, those equations depend on the different parameters of the boost converter presented in Figure 1 and represented mathematically in Equations (1) and (2). \[x_{1}=\frac{v}{V_{in}}, \tag{3}\]  
\[x_{2}=\sqrt{\frac{L}{C}}\frac{i}{V_{in}}, \tag{4}\]  
\[s=\sqrt{LC}t. \tag{5}\]  
Next, a new expression is obtained by performing the first derivative of the variables defined in Equation (3) and represented in Equation (4) as \(\dot{x}_{1}\) and \(\dot{x}_{2}\). This equation can consider a parameter \(\gamma=\sqrt{\frac{L}{R^{2}C}}\) to simplify the mathematical expression. \[\left( \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right)=\left( \begin{array}{cc}-\gamma&1-u\\ u-1&0\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)+\left( \begin{array}{c}0\\ 1\end{array} \right). \tag{6}\]  
From a theoretical perspective, as the boost converter is a voltage booster, \(v>V_{in}\), so \(x_{1}>1\). Therefore, when simulating the system, it is stated that the initial voltage condition must be strictly greater than 1. When \(u\) changes in the set \(\{0,1\}\), a piecewise linear system is obtained, as represented in Equations (7) and (8). \[\dot{X}(t)=A_{1}X(t)+B,\text{corresponding to }u=1, \tag{7}\]  
Figure 1: Diagram of the boost converter. \[\dot{X}(t)=A_{2}X(t)+B,\text{corresponding to }u=0. \tag{8}\]  
This last expression is simplified by considering that the matrices \(A_{1},\ A_{2},\) and \(B\) have the terms presented in Equations (9)-(11). The terms consider the previously established parameter \(\gamma\). \[A_{1}=\left( \begin{array}{cc}-\gamma&0\\ 0&0\end{array} \right), \tag{9}\]  
\[A_{2}=\left( \begin{array}{cc}-\gamma&1\\ -1&0\end{array} \right), \tag{10}\]  
\[B=\left( \begin{array}{c}0\\ 1\end{array} \right). \tag{11}\]  
Equations (7) and (8) represent the differential expressions given by the general Equation (12). This new expression is known as the control system, where \(u\in\mathbb{R}\) is the variable control or system input [18]. \[\dot{X}=A_{1}X+B+(A_{2}-A_{1})Xu. \tag{12}\]  
The result of this expression is defined according to the topology of the system. Thus, the solution for each topology (\(u\) in the set \(\{0,1\}\)) can establish a solution in time by integrating the terms between \(t_{0}\) and \(t\) (Equation (13)), where \(\phi_{i}(t-t_{0})=e^{A_{i}(t-t_{0})}\) and \(\psi_{i}(t-t_{0})=\int_{t_{0}}^{t}e^{A_{i}(t-\tau)}Bd\tau\)[19]. \[X_{i}(t)=\phi_{i}(t-t_{0})X(t_{0})+\psi_{i}(t-t_{0}). \tag{13}\]  
With each topology and the calculation of the exponential matrix, a new expression is established, as presented in Equation (14). Herein, the term \(I_{2}\) is the identity matrix with a size of \(2\times 2\).