# Equivalence between Fuzzy PID Controllers and Conventional PID Controllers
## 3 Simulation Results
Alternatively, for practical applications, a FLC could be implemented in a discrete form. Therefore, to enhance the applicability of this research, 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 9. It must be noted that the Fuzzy Logic Controller in Figures 6 and 9 are identical. So, in Figure 9, the gain (\(K\)) in the \"Discrete-Time Integrator\" or \"Discrete Derivative\" does not denote \(K_{P}\) or \(K_{I}\), and is set as 1. The system input and the resulting responses (blue line) are shown in Figure 5, with sampling period  
Figure 6: The equivalent FLC-controlled system in Simulink. set as 0.1 s. Simulation results show the equivalent FLC implemented in a discrete form have similar responses compared with the conventional PID controller. If the sampling period \(T_{s}\) is set to a smaller value, the closer of the two responses will be obtained. Figure 8: The control surface view of the equivalent fuzzy PID controller (\(\int e(t)=0.36\)). Figure 7: Membership functions for fuzzy variables (**a**) \(e(t)\); (**b**) \(\int e(t)\); (**c**) \(\dot{e}(t)\); (**d**) \(u(t)\) in Matlab. We slightly adjusted the membership functions (MFs) of the equivalent FLC design in Figure 7. It could be found that MFs are no longer equally-spaced, which implies that the FLC has become nonlinear, as shown in Figure 10. The 13 singleton values for the output fuzzy variable \(u(t)\) are \(-18\), \(-16.91\), \(-15.05\), \(-12.44\), \(-9.05\), \(-4.91\), \(0\), \(4.91\), \(9.05\), \(12.44\), \(15.05\), \(16.91\), and \(18\). Figure 11 shows that the control surface view is no longer a plane. The resulting error signal, error integral, error derivative, control signal, and system output are shown in Figure 12 (blue line), and can be compared with the previous PID controller, and the equivalent FLC (red lines for both). Figure 4:0 Membership functions adjustment for fuzzy variables (**a**) \(e(t)\); (**b**) \(\int e(t)\); (**c**) \(\dot{e}(t)\); (**d**) \(u(t)\) in Matlab. Figure 4:1 The equivalent FLC-controlled system in discrete form. Figure 4:2 The control surface view of the equivalent fuzzy PID controller (\(\int e(t)=0.36\)) under MFs adjustment. Figure 4:3 The (**a**) step input; (**b**) error signal; (**c**) error integral; (**d**) error derivative; (**e**) control signal and (**f**) system output with PID controller, the equivalent FLC, and the equivalent FLC under MFs adjustment. Table 1 summarizes the response performance, including the rise time (_Tr_), the settling time (Ts), the percentage overshoot (_P.O._), and the steady-state error (\(E_{ss}\)), of the three controllers in Figure 4:4 Which controller has better performance cannot be determined since the performance criterion is not defined. But based on this study, a fuzzy PID controller may outperform a conventional PID controller quickly by fine-tuning the MFs of the fuzzy variables. We have found that some learning-based techniques or evolutionary algorithms have been applied in the optimal FLC design [14; 15; 16]. Experienced researchers should agree on the importance of setting initial values or weights in the learning system, which will greatly influence the learning results and convergence speed. With the proposed equivalence relationship, one can easily and quickly obtain a fuzzy PID controller through a conventional PID controller design, then the derived equivalent FLC can be set as one of the initial designs. This process will result in the optimal FLC design in an efficient way.