## 2 Materials and Methods
### Parameters of the LED Current Controller and Batteries  
The energy requirement of the light system is mainly imposed by the current of the LED and the desired service time. Considering that the current of the groups of LEDs is regulated at 1.6 A to ensure a uniform 2000 lm illumination per group, each group of LEDs will consume 1.6 Ah to provide the same luminous flux for one hour. Table 2 reports examples of the service time provided by some commercial batteries for a single group of LEDs. For example, the CA1233 battery will ensure 2000 lm for two hours (one active group of LEDs), 4000 lm for one hour (2 active groups of LEDs), or 6000 lm for 40 min (3 active groups of LEDs). Similarly, the PC36-12 NB battery will supply 2000 lm for 24 h (1 active group of LEDs), 4000 lm for 12 h (2 active group of LEDs) or 6000 lm for 8 h (3 active groups  
\begin{table}
\begin{tabular}{l c} \hline
**Parameters** & **Value** \\ \hline Input Voltage \(E=v_{\text{in}}(t)\) & 12 VDC \\ Source Resistance \(r_{\text{in}}\) & 0.3 \(\Omega\) \\ Inductor \(L\) & 0.784 mH \\ Inductor Resistance \(r_{L}\) & 0.2 \(\Omega\) \\ Diode Forward Voltage \(V_{fd}\) & 0.7 V \\ Diode Resistance \(r_{d}\) & 0.03 \(\Omega\) \\ MOSFET Switch Resistance \(r_{\text{sw}}\) & 0.0175 \(\Omega\) \\ Capacitance \(C\) & 132 \(\mathrm{\SIUnitSymbolMicro F}\) \\ Capacitor Resistance \(r_{c}\) & 0.05 \(\Omega\) \\ Impedance per group \(Z_{L}\) & 3.9375 \(\Omega\) \\ Control parameter with ZAD \(k_{1}\) & \(-0.0032\) \\ Switching Frequency \(f_{\text{sw}}=F_{s}\) & 20 kHz \\ PI Controller Proportional Gain \(k_{p}\) & 3.4905 A/A \\ PI Controller Integral Gain \(k_{i}\) & 5627.1 A/s \\ Thermal Resistance (steady state) \(R_{\text{th}}\) & 2 \({}^{\circ}\)C/W \\ Ambient Temperature \(T_{a}\) & 30 \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table}
Table 1: Parameter used in the simulation of the circuit with the buck-boost converter.  
of LEDs). Therefore, the number of active groups of LEDs can be adjusted depending on the light (luminous flux) and service time required. It should be noted that the PC36-12 NB battery has a nominal voltage equal to 10.5 V, which can be used since the proposed power circuit is based on a step-up/down converter (buck-boost), thus providing flexibility in the selection of the battery voltage. Finally, it is observed that the largest battery (PC36-12 NB) provides 12 times the service time of the smaller battery (CA1233) with 7 times the cost, while the medium battery (BW 1280 F1) provides 2.5 times the service time of CA1233 with 2.2 times the cost. Therefore, increasing the battery capacity produces a proportional increase in service time with a smaller proportion of the cost increase.  
Concerning the parameters of the LED current controller, it is designed to achieve a settling time \(t_{s}\) lower than 3 ms and a damping ratio \(\rho=0.8\), which provides a balance between the response time and the maximum overshoot. The first step is to verify the conditions \(r_{c}\) specified in Equation (29) for the simplification of the transfer function \(G_{LED}\): \(\frac{rc}{Z_{L}}=0.0127<<1\) and \(r_{c}\omega_{n}C=0.0107<<1\), where \(\omega_{n}=1.6250\)\(\mathrm{\,\,\,\mathrm{krad}/s}\) is calculated from (32). Figure 3(a) shows the comparison between the exact output node model presented in Equation (28) and the approximation reported in Equation (29), where the correct behavior of the approximated model is evident, hence the control design equations are precise.  
Finally, \(k_{p}=3.4905\)\(A/A\) is calculated from (33) and \(k_{i}=5.6271\)\(kA/s\) is calculated from (34). The normalized closed-loop response of the output node, under the action of the designed LED current controller (30), is tested using exact and approximate models. Figure 3(b) shows this simulation results, which confirms the correct operation of the controller with both models; in fact, both models achieve the same desired settling time with the same overshoot. Therefore, it can be used with the real power circuit.