## 6 Problem Statement
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**MSE** & **Original PSO** & **PSO + T1 FS** & **PSO + IT2 FS** \\ \hline Average & 2.4166 & 3.0426 \(\times\) 10\({}^{-1}\) & **3.2236 \(\times\) 10\({}^{-2}\)** \\ Best & 1.3936 \(\times\) 10\({}^{-1}\) & 4.4127 \(\times\) 10\({}^{-3}\) & **1.9081 \(\times\) 10\({}^{-4}\)** \\ Worst & 6.6763 & 1.5119 & **3.3309 \(\times\) 10\({}^{-1}\)** \\ Standard Deviation & 2.9784 & 3.1949 \(\times\) 10\({}^{-1}\) & **7.1146 \(\times\) 10\({}^{-2}\)** \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of results from the plant without noise. The results shows that an interval type-2 fuzzy system used for parameter adaptation can help PSO to obtain better quality results than other methods such as BA and BCO, even when all these methods use the same methodology for dynamic parameter adaptation and also use a type-1 or interval type-2 fuzzy system for the same task. Aside from the results with the autonomous mobile robot, a comparison against a fuzzy PSO is also presented. In this case, a comparison against a modified PSO proposed by [PERSON] et al. in [PERSON] et al. (2018), where they proposed a PSO that is updated with a fuzzy system, using the current best performance evaluation and the current inertia weight to control or adjust the inertia weight from PSO is performed. The main difference from our proposed fuzzy PSO is that our fuzzy system updates the parameters \(C_{1}\), \(C_{2}\) and _constriction factor_, using _iteration_ and _diversity_ as inputs in an interval type-2 fuzzy system. The following results (contained in Tables 16-18) were obtained using the same parameters and restrictions as the ones published in Shi et al. (2018); the only difference is the dynamic parameters and the fuzzy system used to update these parameters. In addition, results highlighted in bold are the best results. Table 16 contains the results with the Rosenbrock function using the results from Shi et al. (2018) and our proposed PSO method with parameter adaptation using an interval type-2 fuzzy system. From the results in Table 16, it is clear that our proposed approach obtains on average better results when compared with the results of the other method. Table 17 illustrates the results of the comparison between the proposed PSO method against [PERSON] et al. from Shi et al. (2018), with the Rastrigin function. The results in Table 17 clearly show that our proposed approach, which uses an interval type-2 fuzzy system to update the parameters of PSO, can obtain on average better results when compared with the results from [PERSON] et al. (2018). Table 18 shows the results of the comparison between the methods of [PERSON] et al. from [PERSON] et al. (2018) and our proposed approach with an interval type-2 fuzzy system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Rosenbrock Function**} \\ \hline
**Population Size** & **Dimensions** & **Iterations** & **[PERSON] et al. Shi et al. (2018)** & **Our Approach** \\ \hline  & 10 & 1000 & 66.01409 & **2.5431** \\
20 & 20 & 1500 & 108.2865 & **15.8462** \\  & 30 & 2000 & 183.8037 & **38.8491** \\ \hline  & 10 & 1000 & 48.76523 & **1.4956** \\
40 & 20 & 1500 & 63.88408 & **9.5468** \\  & 30 & 2000 & 175.0093 & **27.2133** \\ \hline  & 10 & 1000 & 15.81645 & **2.5656** \\
80 & 20 & 1500 & 45.99998 & **8.5468** \\  & 30 & 2000 & 124.4184 & **22.8496** \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Comparison against Shi et al. Shi et al. (2018).