# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 6 Conclusions  
This article introduces a state-feedback controller aiming to stabilize the output voltage of a four-phase interleaved boost converter associated with a PEM fuel cell. Initially, a mathematical model of the system is developed. Subsequently, a model of the proposed controller is constructed based on the state-space representation of the system. The synthesis of the controller gains relies on optimization techniques, notably linear quadratic regulation (LQR), taking convergence conditions into account. The performance of this controller is compared to that of other controllers, including the PI controller, the nonlinear controller DL-STSM, and BS + STSM, using models from the literature. The proposed controller ensures the stabilization of the output voltage of the interleaved boost converter with good performance in both dynamic and steady-state regimes. Additionally, it demonstrates favorable results in rejecting disturbances associated with variations in the current demanded by the load. The results show that the proposed controller also has a response time of 20 ms, offering a reduction in response time of more than 50% compared to the controllers proposed in references ([PERSON] et al., 2018; [PERSON] et al., 2018), as well as a reduction in current and voltage ripple, both below 10%. The proposed method effectively addresses the overshoot/undershoot phenomena commonly observed in hydrogen fuel cells, as highlighted in the reference ([PERSON] et al., 2018). By employing a dual-loop control strategy for the interleaved converter with the system, we ensure the stability of its output voltage and current. The outer loop and the inner loop utilize a state-feedback control approach to manage disturbances in the dynamics of the fuel  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Controller**} & \multicolumn{3}{c}{**Multiple Setpoints of Reference Voltage**} & \multicolumn{3}{c}{**Various Current Loads**} \\ \cline{2-7}  & **Settling** & **Undershoot** & **Overshoot** & **Steady-State** & **Steady-State** & **Undershoot** & **Overshoot** \\  & **Time (ms)** & **(\%)** & **(\%)** & **Error (\%)** & **Error (\%)** & **(\%)** & **(\%)** & **(\%)** \\ \hline PI & 22 & 5.45 & 8 & 0 & 0 & 12 & 14.5 \\ DL-STSM & 80 & 1.81 & 2.75 & 0.1 & 0.1 & 7.9 & 12.5 \\ BS + STSM & 60 & 2.54 & 2.77 & 0.5 & 0.5 & 12.5 & 20.83 \\ Feedback & 20 & 2.76 & 2.76 & 0 & 0 & 2.1 & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance of controllers under multiple setpoints of reference voltage and various current loads.  
Figure 8: Regulating the output voltage under various current loads for an ideal capacitor and for a capacitor with different equivalent series resistance (ESR) values.  
cell. This combined approach allows the system to adapt to rapid changes in load, thereby mitigating the risks of overshoot and undershoot by continuously adjusting the control inputs based on real-time state feedback, as shown in Figures 5 and 6. The explanation for this performance lies in the fact that the state-feedback controller considers the internal states of the system, and the existence of simple mathematical conditions based on the linear model facilitates the synthesis of its gains, clearly demonstrating the advantage of this control method over others.  
Methodology, M.G.; Validation, E.h.E.M.; Formal analysis, M.G. and I.S.; Investigation, M.G.; Writing--original draft, M.G.; Writing--review & editing, I.S., E.h.E.M. and S.D.; Supervision, I.S. and S.D. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Data is contained within the article.  
The authors declare no conflicts of interest.