## 2 Materials and Methods
### Model of the Buck Converter with a DC Motor  
The DC motor and mechanical load are modeled by Equations (1) and (2), which correspond to the electromechanical part of the system, described by a second order mathematical model, where the state variables are the speed of the DC motor (\(W_{m}\)) in rad/s and the armature current (\(i_{a}\)) in amps. \[\dot{W}_{m}(t)=\frac{-BW_{m}(t)}{I_{eq}}+\frac{k_{t}i_{a}(t)}{I_{eq}}+\frac{-T_ {fric}}{I_{eq}}+\frac{-T_{L}}{I_{eq}} \tag{1}\]  
\[i_{a}(t)=\frac{-k_{e}}{L_{a}}W_{m}(t)+\frac{-R_{a}}{L_{a}}i_{a}(t)+\frac{V_{a}}{ L_{a}} \tag{2}\]  
The term \(k_{e}\) is the voltage constant (V/rad/s) of the motor, \(L_{a}\) is the armature inductance (mH), \(R_{a}\) is the armature resistance (\(\Omega\)), \(V_{a}\) is the voltage applied to the motor (V) and is equal to \(\upsilon_{c}\), \(B\) is the viscous friction coefficient (N\(\cdot\)m/(rad/s)), \(I_{eq}\) is the moment of inertia (kg\(\cdot\)m\({}^{2}\)), \(k_{t}\) is the motor torque constant (N\(\cdot\)m/A), \(T_{fric}\) is the friction torque (N\(\cdot\)m), and \(T_{L}\) is the load torque (N\(\cdot\)m). Figure 2 shows that the complementary part of the electrical subsystem corresponds to the circuit of the buck converter. The mechanical subsystem corresponds to the rotating mechanical part of the motor and the mechanical load connected to it. Here, the way in which the output of the buck converter feeds the DC motor with a regulated voltage is illustrated. The presented configuration allows the speed of the motor to be controlled in a closed loop, manipulating the state of the switch (S = ON and S = OFF); that is, the control gate of MOSFET transistor. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(r_{s}\) & Total resistance at the source & 0.84 \(\Omega\) \\ \(E\) & Power supply voltage & 40.086 V \\ \(V_{fd}\) & Forward voltage & 1.1 V \\ \(L\) & Inductance & 2.473 mH \\ \(r_{L}\) & Total resistance in the inductor & 1.695 \(\Omega\) \\ \(C\) & Capacitance & 46.27 \(\mu\)F \\ \(R_{a}\) & Armature resistance & 2.7289 \(\Omega\) \\ \(L_{a}\) & Armature inductance & 1.17 mH \\ \(B\) & Viscosity friction coefficient & 0.000138 (N\(\cdot\)m/(rad/s)) \\ \(I_{eq}\) & Inertia moment & 0.000115 (kg\(\cdot\)m\({}^{2}\)) \\ \(k_{t}\) & Motor torque constant & 0.0663 (N\(\cdot\)m/A) \\ \(k_{e}\) & Motor voltage constant & 0.0663 (V/rad/s) \\ \(T_{fric}\) & Friction constant & 0.0284 (N\(\cdot\)m) \\ \(T_{L}\) & Load torque & Variable (N\(\cdot\)m) \\ \(W_{mref}\) & Speed reference & Variable (rad/s) \\ \(W_{m}\) & Speed of the DC motor & Variable (rad/s) \\ \(N\) & Control parameter of the FPIC & 1 \\ \(Fc\) & Switching frequency & 6 kHz \\ \(Fs\) & Sampling frequency & 6 kHz \\ \(1T\_p\) & Delay period & 166.6 \(\mu\)s \\ \(K_{51}\) & Control parameter & 2 \\ \(K_{52}\) & Control parameter & 2 \\ \(K_{53}\) & Control parameter & Variable \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the controller and the circuit. The state variables of the buck converter with the motor depends on the state of the switch (S). Thus, if the switch is closed (S = ON) and the diode is not conducting (inactive), the system is represented as described in Equation (3). \[\left[ \begin{array}{c}\dot{W}_{m}\\ i_{a}\\ v_{c}\\ i_{L}\end{array} \right]=\left[ \begin{array}{cccc}\frac{-R}{I_{eq}}&\frac{k_{L }}{I_{eq}}&0&0\\ \frac{-R}{I_{eq}}&\frac{-R_{eq}}{I_{eq}}&\frac{1}{I_{eq}}&0\\ 0&\frac{-1}{C}&0&\frac{1}{C}\\ 0&0&\frac{-1}{L}&\frac{-(r_{L}+r_{L})}{L}\end{array} \right]\left[ \begin{array}[ ]{c}W_{m}\\ i_{a}\\ v_{c}\\ i_{L}\end{array} \right]+\left[ \begin{array}{c}\frac{-(r_{file}+T_{L})}{l_{ eq}}\\ 0\\ 0\\ \frac{E}{L}\end{array} \right] \tag{3}\]  
In a compact form, it can be represented as shown in Equation (4), where \(x_{1}=W_{m}\), \(x_{2}=i_{a}\), \(x_{3}=v_{c}\), and \(x_{4}=i_{L}\). \[\dot{x}=A_{1}x+B_{1} \tag{4}\]  
When the switch is open (S = OFF) and the diode is conducting, the model is described as presented in Equation (5). \[\left[