## 3 Optimal Design of Fuzzy Systems Approach  
The proposed method is based on the concepts and equations of the original DE and HS algorithms. Both original algorithms have the problem of parameters being fixed throughout the iterations. In previous work, this problem was eliminated using fuzzy logic to perform the dynamic parameter adjustment. This method consisted of a Mamdani fuzzy system with two inputs (iterations and diversity) and two outputs for each method, as shown by [PERSON] et al. in [46].  
Therefore, the main contribution in this paper is that the proposed method uses the original DE and HS algorithms to find the best architecture of the previous fuzzy differential evolution (FDE) and fuzzy harmony search (FHS) fuzzy systems shown by [PERSON] et al. in [46], which are responsible for adjusting parameters along the iterations. The proposed methods are optimized fuzzy differential evolution (OFDE) and optimized fuzzy harmony search algorithm (OFHS). Once the best architectures of the proposed OFDE and OFHS algorithms have been found, two types of problems are considered for optimization: benchmark mathematical functions and a non-linear control problem. Figure 1 shows the general diagram of the proposed method in this paper.  
The proposed fuzzy systems have triangular membership functions at the inputs and outputs in both algorithms. The triangular membership function depends on three parameters which are \(a\), \(b\)  
Figure 1: Proposed method.  
and \(c\) as given by Equation (14), where \(a\) and \(c\) represents the lower parts of the triangle and \(b\) locates the peak. This triangular membership function is represented graphically in Figure 2.  
\[f(x;a,\ b,c)= max\Big{(}\min\Big{(}\frac{x-a}{b-a},\frac{c-x}{c-b}\Big{)},0\Big{)} \tag{14}\]  
Tables 1 and 2 summarize the representation of the knowledge of the inputs and outputs of the fuzzy system with triangular membership functions.  
Figures 3 and 4 illustrate the parameters that form each input and output triangular membership function.  
The fuzzy system was optimized with DE and HS; in the DE, it is necessary to use a chromosome, and in HS, a harmony to optimize the membership functions (MFS) is necessary, and in this case, it is called a vector, as shown in Figure 5. The vector has 44 positions, of which six parameters are fixed and 38 are optimized.  
\begin{table}
\begin{tabular}{c c} \hline Low & \(\mu\text{Low}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.05}{0.17-0.005},\frac{0.3-x}{0.13-0.17}\Big{)},0\Big{)}\) \\ Medium & \(\mu\text{Medium}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.17}{0.05},\frac{1-x}{ 0.15}\Big{)},0\Big{)}\) \\ High & \(\mu\text{High}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.5}{1-0.5},\frac{1-x}{ 1.5-1}\Big{)},0\Big{)}\) \\ \hline \end{tabular}
\end{table}
Table 1: Mathematical representation of the input membership functions.  
\begin{table}
\begin{tabular}{c c} \hline Low. & \(\mu\text{Low}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.05}{0.17-0.005},\frac{0.3-x}{0.13-0.17}\Big{)},0\Big{)}\) \\ Medium Low & \(\mu\text{Medium}\ Low(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.17}{0.05},\frac{1 -x}{0.15}\Big{)},0\Big{)}\) \\ Medium & \(\mu\text{Medium}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.3}{0.5-0.3},\frac{0.7-x}{0.5},0\Big{)},0\Big{)}\) \\ Medium High & \(\mu\text{Medium}\ High(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.5}{0.5-0.7}, \frac{0.85-x}{0.85-(7)},0\Big{)}\) \\ High & \(\mu\text{High}(\text{x})=\max\Big{(}\min\Big{(}\frac{x-0.7}{0.85-0.7},\frac{1 -x}{1-0.85}\Big{)},0\Big{)}\) \\ \hline \end{tabular}
\end{table}
Table 2: Mathematical representation of the output membership functions.  
Figure 2: Points of triangular membership function.  
Figure 4: Parameters of the membership functions of the Output 1 and Output 2.  
Figure 5: Vector for the optimization optimized fuzzy differential evolution (OFDE) and optimal fuzzy harmony search (OFHS).  
Figure 3: Parameters of the membership functions for Input 1 and Input 2.  
The limits for the optimization of each triangular membership function are specified in Table 3.