# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 3 Results and Discussion
### A Comparison between PSO and GA Algorithms  
In this section, a comparison between the proposed PSO method and GA algorithm is performed in order to demonstrate the capability of the proposed PSO method to improve the performance of shunt-damping circuits. Each algorithm is run 10 times with the same number of particles/population (100) and the same number of maximum iterations/generations (250). The remaining parameters for the GA algorithm are the default parameters of the GA solver in MATLAB, while, for PSO, the same parameters as mentioned above are used.  
Tables 9 and 10 show the best fitness value found, the total time (in s) the iterations/generations executed for the algorithm take in order to start to converge, the total number of iterations and the total function evaluations required to find the optimal solution for both algorithms in each run time. Based on the results tabulated in Tables 8 and 9, most best fitness values found by PSO are lower than the ones found by GA in most of the 10 test runs. In fact, 70% of the fitness values reached by PSO are lower than \(-86\), a value that is never reached by GA. For PSO, the lowest fitness value of \(-86.34\) is reached with 10,600 objective function evaluations. Nevertheless, GA reaches its best value \(-84.77\) using a computational effort relative to 23,855 evaluations, as shown in Tables 8 and 9. Additionally, it is observed that the average time required by PSO to find the optimal circuit design (in 10 test runs) is shorter compared to GA, which shows that PSO can perform faster than GA.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Model 1** & **Model 2** & **Model 3** \\ \hline Mode 2 & 39.60 & 25.47 & 24.07 \\ Mode 3 & 55.92 & 47.08 & 44.22 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Amplitude reduction (dB).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of** & **Best Fitness** & **Time (s)** & \
\begin{tabular}{c} **Total** \\ **Iterations** \\ \end{tabular} &
\begin{tabular}{c} **Function** \\ **Evaluations** \\ \end{tabular} \\ \hline
1 & \(-84.77\) & 8326.053 & 250 & 23,855 \\
2 & \(-70.5\) & 6513.69 & 250 & 23,855 \\
3 & \(-80.96\) & 7823.02 & 250 & 23,855 \\
4 & \(-69.33\) & 3644.80 & 111 & 10,650 \\
5 & \(-83.34\) & 7231.36 & 250 & 23,855 \\
6 & \(-70.72\) & 7508.04 & 250 & 23,855 \\
7 & \(-70.24\) & 4252.65 & 189 & 18,060 \\
8 & \(-69.6\) & 2168.00 & 72 & 6945 \\
9 & \(-80.7\) & 7050.17 & 250 & 23,855 \\
10 & \(-83.58\) & 7220.28 & 250 & 23,855 \\ \hline
**Avg Total** & \(\mathbf{-76.374}\) & \(\mathbf{6173.81}\) & \(\mathbf{212.2}\) & \(\mathbf{20,264}\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Fitness, computation time and iterations of GA.  
Figure 13 shows a comparison of the PSO and GA methodologies' convergence during the search process of the optimal parameters for the test run (execution) with the best fitness value. It is clear from this figure that PSO converges at a faster rate (around 45 iterations) than GA (around 232 generations). Another interesting outcome of this figure is that, in GA, although the best solution is improved relatively fast in the first 70 iterations, in the remaining iterations, it is trapped in a local optimum, which is worster from the solution found using the PSO. Thus, this comparison demonstrates the capability of PSO to reach better values for this kind of problem and exert less computational effort to reach the \"optimum\" when compared to GA. Nevertheless, a more comprehensive study is needed to establish this statement, which is out of the scope of this study.