## 5 Comparison
### Comparison between Analytical and Numerical Solutions before and after the Controller  
Emphasizing the robust agreement and consistency observed between numerical simulations (solid line) and approximate solutions (dashed line) for the uncontrolled system and the system with the PPF controller.  
Figure 16 showcases a significant correlation between the numerical and the approximate results for the uncontrolled system. This agreement underscores the compatibility and alignment of both methods in representing the behavior of the system without control. Furthermore, in Figure 17, a particularly strong correlation between numerical and approximate solutions is evident, specifically highlighting the impact of integrating the PPF controller into the system. This reinforces the reliability and consistency of both numerical simulations and approximate methods in depicting the system's behavior when the PPF controller is implemented. While on one hand Figure 17a presents the previous comparison of the main system amplitude, on the other hand Figure 17b presents the same comparison but for the PPF controller amplitude. These observations underscore the robustness and accuracy of both numerical and approximate approaches in portraying the system's behavior across various control scenarios, further validating their reliability in analyzing system dynamics.  
\begin{table}
\begin{tabular}{|c|c c c|} \hline \(\mathbf{\sigma_{1}}\) & **RK-4 Solution** & **FRC** & **1 RK4 Solution\(-\)FRC\(\mid\)** \\ \hline \(-3\) & 0.463124975460432 & 0.100187814076129 & 0.362937161384303 \\ \(-2.7\) & 0.463124975460483 & 0.111368846389836 & 0.351756129070647 \\ \(-2.4\) & 0.463124975460878 & 0.125367175821846 & 0.337757799639032 \\ \(-2.1\) & 0.463124975460678 & 0.143405678035335 & 0.319719297425343 \\ \(-1.8\) & 0.463124975460543 & 0.16753881881606 & 0.295586156578937 \\ \(-1.5\) & 0.46331265829522 & 0.201510240778341 & 0.26815010611811 \\ \(-1.2\) & 0.463124975460479 & 0.252961129489559 & 0.21016384597092 \\ \(-0.9\) & 0.463124975460708 & 0.340411579702025 & 0.1227213395758683 \\ \(-0.6\) & 0.463124975460622 & 0.524477232033863 & 0.061352256573241 \\ \(-0.3\) & 0.463124975461229 & 1.22474096509607 & 0.761615989634841 \\ \(0\) & 0.463124975461430 & 0.063492063492063 & 0.399632911969367 \\ \(0.3\) & 0.463124975460857 & 1.226201951876355 & 0.763076976415498 \\ \(0.6\) & 0.463569183270159 & 0.52449787640375 & 0.060928696372916 \\ \(0.9\) & 0.463124975460831 & 0.34041157970205 & 0.1227213395758806 \\ \(1.2\) & 0.463124975460638 & 0.252961129489559 & 0.210163845971079 \\ \(1.5\) & 0.463124975460543 & 0.201510240778341 & 0.261614734682202 \\ \(1.8\) & 0.463124975460607 & 0.167538818881606 & 0.295586156579001 \\ \(2.1\) & 0.463124975461053 & 0.100187814076129 & 0.362937161384924 \\ \(2.4\) & 0.463124975460740 & 0.111368846389836 & 0.351756129070904 \\ \(2.7\) & 0.463703390512664 & 0.125367175821846 & 0.338336214690818 \\ \(3\) & 0.463124975461298 & 0.143405678035335 & 0.319719297425963 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the results obtained for the FRC and RK-4 solution.  
Figure 16: Comparison between approximate solution (dashed line) and numerical solution (solid line) for the vibration amplitude of uncontrolled main system.