# Revised Gravitational Search Algorithms Based on Evolutionary-Fuzzy Systems
## 2 The Gravitational Search Algorithm
The main idea is to tune the parameter \(\alpha\) of the gravitational constant \(G\) through a fuzzy controller optimized via GA, PSO and DE. The first FGSA uses GA to optimize the membership functions' slope of the fuzzy variable \(\alpha\). In the second and third FGSA, PSO and DE define the best range of the fuzzy input/output membership functions. In both cases, the optimization process of the fuzzy controller is accomplished just one time: once the fuzzy controller is optimized, it is inserted in the steps of GSA to design FGSA. In order to compare our outcomes to those of [22; 26; 27; 28; 35], the results are averaged over 30 and 16 runs. The results show that FGSA optimized by GA achieves better results than [22; 26; 27; 28; 35]. The best improvement with respect to [22] is of 19 orders of magnitude. Better results have been achieved by comparing our findings with the findings of [27; 35]: a maximum of 22 and 20 orders of magnitude, respectively, is achieved. The order of magnitude of FGSA-GA is six-times greater than [26] for the test function \(F1\). The results of [28] are improved for all of the functions achieving a maximum order of magnitude of 10. Moreover, FGSA optimized with PSO achieves better results than GSA-PSO. Significant improvements are achieved for test functions \(F1\) and \(F7\), where FGSA-PSO is better than GSA-PSO. For the function \(F13\), FGSA-DE is better than GSA-DE by two orders of magnitude. Comparing the results of FGSA-GA, FGSA-PSO and FGSA-DE, it follows that FGSA-DE is optimal for unimodal functions, whereas FGSA-GA is good for multimodal functions. Future studies will focus on novel adaptive gravitational search algorithms for fuzzy-controlled servo system. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **GSA**[22] & **BGSA[35]** & **GSA[27]** & **FGSA[26]** & **FGSA-GA** & **FGSA-PSO** & **FGSA-DE** \\ \hline \(F_{1}\) & \(7.3\times 10^{-11}\) & \(4.65\times 10^{-5}\) & \(8.85\times 10^{-34}\) & \(2.2\times 10^{-21}\) & \(1.78\times 10^{-27}\) & \(7.93\times 10^{-25}\) & \(2.85\times 10^{-35}\) \\ \hline \(F_{2}\) & \(4.03\times 10^{-5}\) & \(0.0016\) & \(1.16\times 10^{-10}\) & \(2.5\times 10^{-11}\) & \(2.32\times 10^{-13}\) & \(6.16\times 10^{-7}\) & \(4.44\times 10^{-17}\) \\ \hline \(F_{3}\) & \(0.16\times 10^{3}\) & \(26.29\) & \(468.44\) & \(1.1\times 10^{-11}\) & \(3.27\times 10^{-5}\) & \(5.55\times 10^{2}\) & \(9.68\times 10^{-4}\) \\ \hline \(F_{4}\) & \(3.7\times 10^{-6}\) & \(1.28\) & \(0.0912\) & \(4.8\times 10^{-11}\) & \(2.28\times 10^{-11}\) & \(2.85\times 10^{0}\) & \(3.86\times 10^{-11}\) \\ \hline \(F_{10}\) & \(6.9\times 10^{-6}\) & \(0.0400\) & \(6.34\times 10^{-15}\) & – & \(7.88\times 10^{-15}\) & \(7.05\times 10^{-11}\) & \(7.99\times 10^{-15}\) \\ \hline \(F_{11}\) & \(0.29\) & \(0.00409\) & \(4.9343\) & – & \(1.30\times 10^{-9}\) & \(1.35\times 10^{1}\) & \(2.89\times 10^{-2}\) \\ \hline \(F_{12}\) & \(0.01\) & \(0.9001\) & \(0.1103\) & – & \(2.44\times 10^{-21}\) & \(1.12\times 10^{-2}\) & \(1.97\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison between FGSA-GA and revised GSA over 30 runs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **GSA**[28] & **FGSA-GA** & **FGSA-PSO** & **FGSA-DE** \\ \hline \(F_{1}\) & \(8.15\times 10^{-17}\) & \(1.73\times 10^{-27}\) & \(8.91\times 10^{-25}\) & \(2.58\times 10^{-35}\) \\ \hline \(F_{2}\) & \(1.42\times 10^{-12}\) & \(2.01\times 10^{-13}\) & \(7.26\times 10^{-7}\) & \(5.17\times 10^{-17}\) \\ \hline \(F_{3}\) & \(0.946\) & \(3.22\times 10^{-5}\) & \(5.77\times 10^{2}\) & \(2.58\times 10^{-4}\) \\ \hline \(F_{4}\) & \(5.62\times 10^{-5}\) & \(8.29\times 10^{-11}\) & \(2.26\times 10^{0}\) & \(3.11\times 10^{-11}\) \\ \hline \(F_{10}\) & \(8.22\times 10^{-14}\) & \(8

## References

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