# An Optimal Fuzzy PID Controller Design Based on Conventional PID Control and Nonlinear Factors
## 3 Simulation Results
### Simulation 2: DC Motor Speed Control  
High-performance motor drives are very important in many industrial applications. Based on the design procedure in Simulation 1, a separately excited dc motor with the physical parameters listed in Table 3 is used as the controlled plant [20], and Figure 12 shows the constructed model in Simulink. This is a speed control problem with the desired speed at 1200 rpm [20]. The control structure is shown in Figure 13 for both the PID controller and the proposed optimal fuzzy PID controller. The resulting speed responses of the initial conventional PID controller design with \(K_{P}=50\), \(K_{I}=60\), and \(K_{D}=3\), and the proposed optimal fuzzy PID controller are shown in Figure 14. The search domains of the operating ranges for this case are confined to \(0<a_{e}\leq 20,000\), \(0<a_{i}\leq 50,000\), \(0<a_{d}\leq 200,000\), \(0<a_{u}\leq 5,000,000\), and \(0.01<\gamma_{(e,j,d,u)}\leq 7\). The resulting eight optimized parameters are shown in Table 4, and Table 5 summarizes some response parameters of [PERSON]'s optimized fuzzy-GA controller [20] and the proposed optimal fuzzy  
Figure 14: The motor speed control of the conventional PID controller and the proposed optimal fuzzy PID controller. Figure 12: The block diagram of the DC motor. Figure 13: The control system of the PID controller and the proposed optimal fuzzy PID controller. In this case, the sample-and-hold unit and the zero-order hold unit, which can be used to model A/D (analog-to-digital) and D/A (digital-to-analog) converters, are added to make a discretized FLC, as shown in Figure 18. It must be noted that the gain (K) in the \"discrete-time integrator\" and \"discrete derivative\" is set as 1 [16]. The system input and the resulting responses are shown in Figure 19, with sampling period \(ts\) set as 0.1 ms. The system responses in Figure 19 include the initial conventional PID controller design (\(K_{P}=0.42\), \(K_{I}=25.127\), \(K_{D}=0.012\)), and the proposed optimal fuzzy PID controller design. Furthermore, since we cannot obtain the response waveform of [PERSON]'s design [21], we run a similar response to [PERSON]'s design by the proposed FPID. Table 7 shows the eight optimized parameters of the proposed FPID design. The search domains of the operating ranges are confined to \(0<a_{e}\leq 20,000\), \(0<a_{i}\leq 50,000\), \(0<a_{d}\leq 200,000\), \(0<a_{u}\leq 5,000,000\), and \(0.01<\gamma_{(e,i,d,u)}\leq 7\). Then the performance comparison between [PERSON]'s Fuzzy PID with KF (Kalman Filter) [21] and the proposed optimal FPID is summarized in Table 8. Simulation results demonstrate that the proposed optimal FPID has smaller _Tr_ and _Ts_ than [PERSON]'s design, but has a larger overshoot. Furthermore, the proposed design has good performance in disturbance rejection. Figure 16: Desired motor speed. Figure 17: Load torque as disturbance. Figure 19: The motor speed input and system responses. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Optimized Operating Ranges**} & \multicolumn{4}{c}{**Optimized Nonlinear Factors**} \\ \hline \hline  & \(a_{e}\) & \(a_{i}\) & \(a_{d}\) & \(a_{u}\) & \(\gamma_{e}\) & \(\gamma_{i}\) & \(\gamma_{d}\) & \(\gamma_{u}\) \\ \hline
**The proposed** & 11,412 & 3346.4 & 183,980 & 3,251,100 & 6.5110 & 5.9331 & 0.4719 & 2.5169 \\
**optimal FPID** & & & & & & & \\ \hline
**The proposed FPID** & & & & & & & \\ **with a similar** & & & & & & & \\ **response to** & & & & & & & \\ **Al-Maliki’s results** & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The eight optimized parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & _Tr (s) 10–90\%_ & _Ts(s) \(\pm\)2\%_ & _P.O. (\%)_ \\ \hline Al-Maliki’s FLC-PID with KF [21] & \multicolumn{2}{c}{\(\sim\)0.15} & 0.257 & 0.5 \\ \hline The proposed optimal FPID & 0.0409 & 0.0516 & 1.38 \\ \hline The proposed FPID with a similar & & & & & \\ response to Al-Maliki’s results & & & & & \\ \hline \hline \end{tabular}
\end{table}