## 4 Control Responses
In this study, the center-average defuzzification method is used to calculate the output value, which is defined as follows [27]:  
\[c=\frac{\sum c_{i}\cdot\mu_{c}(z)}{\sum\mu_{c}(z)} \tag{8}\]  
where \(c\), \(c_{i}\), and \(\mu_{c}\) are the output value, the center value of the \(i\)-th output membership function, and the product rule, respectively. Equation (7) can be rewritten as follows:  
\[u(t)=\left(\frac{\sum c_{i}\cdot\mu_{c}(z)}{\sum\mu_{c}(z)}\right)_{p}e(t)+ \left(\frac{\sum c_{i}\cdot\mu_{c}(z)}{\sum\mu_{c}(z)}\right)_{I}\int e(t)+k_{d }\frac{d}{dt}e(t) \tag{9}\]  
\(k_{p}\) and \(k_{i}\) are re-expressed using Equation (8). \(k_{d}\) is determined as 0.0001 according to the experimental results. To evaluate the proposed valve performance, the flow rate is first evaluated using the spool displacement. Figure 10a,b show the experimental results for the flow rate at room temperature for each frequency. Errors are observed in the flow-rate data below 3.78 Lpm (liter per minute) because the flow meter can only detect flow rates ranging from 3.78 to 37.8 Lpm. The flow-rate data properly follow the displacement of the spool, although the sensor exhibits an error at low flow rates, as shown in Figure 10a. However, a large error is observed in the flow-rate data obtained from the flow meter with a time delay, as shown in Figure 10b, because the flow meter and indicator (LA-5-A-V1F1, Flow Technology Inc., Tempe, AZ, USA) have time delays of 4 and 20 ms, respectively. Therefore, in this work, the displacement of the spool is selected as the control target because the flow rate is directly related to the spool displacement; the flow rate can be calculated using the spool displacement and properties of hydraulic oil with temperatures according to Equation (5). It is noted here that the response time of the flow meter is not sufficient to deal with the fast motion between the input and output (about 200 Hz). This is the reason why the gap sensor is used to measure the spool displacement. Figure 10c,d show the control results obtained at room temperature using the normal PID controller. As shown in the figure, the displacement of the spool follows the desired displacement at operating frequencies of 1 and 10 Hz. The same PID gains are observed in the results, which are 0.12, 80, and 0.0001 for the P, I, and D gains, respectively. However, the gains differ at 200 Hz: 0.3, 620, and 0.0001 for P, I, and D, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & \multicolumn{4}{c}{**Temperature**} \\ \cline{2-5}  & **Cold** & **Warm** & **Hot** & **Extreme Hot** \\ \hline Extreme Low & 80 & 80 & 80 & 80 \\ Medium Low & 180 & 180 & 180 & 180 \\ Low & 280 & 280 & 210 & 210 \\ Normal & 310 & 310 & 250 & 250 \\ High & 430 & 360 & 330 & 280 \\ Medium High & 550 & 430 & 360 & 330 \\ Extreme High & 620 & 500 & 430 & 430 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fuzzy rules for I gain (\(k_{i}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & \multicolumn{4}{c}{**Temperature**} \\ \cline{2-5}  & **Cold** & **Warm** & **Hot** & **Extreme Hot** \\ \hline Extreme Low & 0.12 & 0.12 & 0.12 & 0.12 \\ Medium Low & 0.12 & 0.12 & 0.12 & 0.12 \\ Low & 0.12 & 0.12 & 0.12 & 0.12 \\ Normal & 0.12 & 0.12 & 0.18 & 0.18 \\ High & 0.23 & 0.2 & 0.18 & 0.18 \\ Medium High & 0.25 & 0.2 & 0.18 & 0.2 \\ Extreme High & 0.3 & 0.2 & 0.2 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fuzzy rules for P gain (\(k_{p}\)). Figure 11 shows the control results obtained at room and high temperature using the fuzzy PID controller. As shown in Figure 11a,b, the performance results obtained at the frequencies of 1 Hz are similar to those of the normal PID controller operating at a high temperature. The tracking error increases at high temperatures because the heat affects the system, as shown in Figure 11c,d.