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Figure 14. The motor speed control of the conventional PID controller and the proposed optimal fuzzy PID controller.

Figure 14 The motor speed control of the conventional PID controller and the proposed optimal fuzzy PID controller.

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Figure 1. Graphical definition of MFs for fuzzy variables, e(t), [ e(t), e(t), and u(t). 2.1. The Equivalent FLC from a Conventional PID Controller
Figure 1. Graphical definition of MFs for fuzzy variables, e(t), [ e(t), e(t), and u(t). 2.1. The Equivalent FLC from a Conventional PID Controller
Figure 2. Sliced cube fuzzy associative memory (FAM) representation of the knowledge base.
Figure 2. Sliced cube fuzzy associative memory (FAM) representation of the knowledge base.
Figure 3. Different nonlinear factor for tuning MFs. (a) y = 1; (b) y < 1; and (c) y>1. 2.3. The Efficient Genetic Algorithm
Figure 3. Different nonlinear factor for tuning MFs. (a) y = 1; (b) y < 1; and (c) y>1. 2.3. The Efficient Genetic Algorithm
3. Simulation Results This section evaluates the performance of the proposed optimal fuzzy PID controller by considering a two-order plant and motor control systems. The simulation is verified by use of Matlab /Simulation.
3. Simulation Results This section evaluates the performance of the proposed optimal fuzzy PID controller by considering a two-order plant and motor control systems. The simulation is verified by use of Matlab /Simulation.
Figure 5. The unit feedback control structure: (a) PID control; and (b) optimal fuzzy PID control.
Figure 5. The unit feedback control structure: (a) PID control; and (b) optimal fuzzy PID control.
Figure 6. Membership functions for fuzzy variables: (a) e(t); (b)_[ e(t); (c) e(t); and (d) u(t) in Matlab.
Figure 6. Membership functions for fuzzy variables: (a) e(t); (b)_[ e(t); (c) e(t); and (d) u(t) in Matlab.
Figure 7. The control surface view of the equivalent fuzzy PID controller ([ e(t) = 0.266).
Figure 7. The control surface view of the equivalent fuzzy PID controller ([ e(t) = 0.266).
Figure 8. Performance of GAs across generations: (a) with initial equivalent FLC; and (b) without initial equivalent FLC.
Figure 8. Performance of GAs across generations: (a) with initial equivalent FLC; and (b) without initial equivalent FLC.
Table 1. The eight optimized parameters. Table 2. Response performance of different controllers.
Table 1. The eight optimized parameters. Table 2. Response performance of different controllers.
Figure 9. The step responses: (blue) the conventional PID controller; and (red) the proposed optimal fuzzy PID controller. The step responses with the conventional PID controller (Kp = 50, K; = 250, Kp = 1.5) and the proposed optimal fuzzy PID controller (with initial equivalent FLC) are shown in Figure 9. The resulting eight optimized parameters are shown in Table 1. Furthermore, Table 2 summarizes the response performance, including the rise time (Tr), the settling time (Ts), the percentage overshoot (P.O.), and the steady-state error (E;;), of Pelusi’s GNFC optimized controller [19] and the proposed optimal fuzzy PID controller. Pelusi has provided many kinds of controllers, such as the genetic fuzzy controller (GFC), the neuro fuzzy controller (NFC), and the genetic neuro fuzzy controller (GNFC), but the GNFC has the best performance. Compared with Pelusi’s design, the proposed optimal FPID has smaller Tr and Ts, but a very small overshoot occurred at peak time t = 0.08 s with amplitude 1.0173 as shown in Figure 9. On the other hand, Pelusi’s GNFC controller belongs to a PD-type FLC, so E;, cannot be eliminated.
Figure 9. The step responses: (blue) the conventional PID controller; and (red) the proposed optimal fuzzy PID controller. The step responses with the conventional PID controller (Kp = 50, K; = 250, Kp = 1.5) and the proposed optimal fuzzy PID controller (with initial equivalent FLC) are shown in Figure 9. The resulting eight optimized parameters are shown in Table 1. Furthermore, Table 2 summarizes the response performance, including the rise time (Tr), the settling time (Ts), the percentage overshoot (P.O.), and the steady-state error (E;;), of Pelusi’s GNFC optimized controller [19] and the proposed optimal fuzzy PID controller. Pelusi has provided many kinds of controllers, such as the genetic fuzzy controller (GFC), the neuro fuzzy controller (NFC), and the genetic neuro fuzzy controller (GNFC), but the GNFC has the best performance. Compared with Pelusi’s design, the proposed optimal FPID has smaller Tr and Ts, but a very small overshoot occurred at peak time t = 0.08 s with amplitude 1.0173 as shown in Figure 9. On the other hand, Pelusi’s GNFC controller belongs to a PD-type FLC, so E;, cannot be eliminated.
Table 3. DC motor parameters [20].
Table 3. DC motor parameters [20].
Figure 10. Membership functions for fuzzy variables: (a) e(t); (b) f e(t e(t); and (d) u(t) in Matlab.
Figure 10. Membership functions for fuzzy variables: (a) e(t); (b) f e(t e(t); and (d) u(t) in Matlab.
Figure 11. The control surface view of the proposed optimal fuzzy PID controller ([ e(#) = 26.3556
Figure 11. The control surface view of the proposed optimal fuzzy PID controller ([ e(#) = 26.3556
Figure 12. The block diagram of the DC motor.
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 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 Kp = 50, K; = 60, and Kp = 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 < ae < 20,000, 0 < a; < 50,000, 0 < ag < 200,000, 0 < ay < 5,000,000, and 0.01 < Yeiau) <7.
Figure 13. The control system of the PID controller and the proposed optimal fuzzy PID controller 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 Kp = 50, K; = 60, and Kp = 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 < ae < 20,000, 0 < a; < 50,000, 0 < ag < 200,000, 0 < ay < 5,000,000, and 0.01 < Yeiau) <7.
Table 4. The eight optimized parameters.
Table 4. The eight optimized parameters.
Table 6. DC motor parameters [21].
Table 6. DC motor parameters [21].
Table 5. Response performance of different controllers. Figure 15. Membership functions for fuzzy variables: (a) e(t); (b) _[ e(t e(t); and (d) u(t) in Matlab.
Table 5. Response performance of different controllers. Figure 15. Membership functions for fuzzy variables: (a) e(t); (b) _[ e(t e(t); and (d) u(t) in Matlab.
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 (Kp = 0.42, Kj = 25.127, Kp = 0.012), and the proposed optimal fuzzy PID controller design. Furthermore, since we cannot obtain the response waveform of Al-Maliki’s design [21], we run a similar response to Al-Maliki’s design by the proposed FPID.
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 (Kp = 0.42, Kj = 25.127, Kp = 0.012), and the proposed optimal fuzzy PID controller design. Furthermore, since we cannot obtain the response waveform of Al-Maliki’s design [21], we run a similar response to Al-Maliki’s design by the proposed FPID.
Table 8. Response performance of different controllers.
Table 8. Response performance of different controllers.
Figure 18. The proposed FLC-controlled system in discrete form.
Figure 18. The proposed FLC-controlled system in discrete form.
Figure 19. The motor speed input and system responses. Table 7. The eight optimized parameters.
Figure 19. The motor speed input and system responses. Table 7. The eight optimized parameters.
Figure 20. Membership functions for fuzzy variables: (a) e(t); (b) f e(t e(t); and (d) u(t) in Matlab.
Figure 20. Membership functions for fuzzy variables: (a) e(t); (b) f e(t e(t); and (d) u(t) in Matlab.
Figure 21. The control surface view of the proposed optimal fuzzy PID controller ( e(f) = 3346.4)
Figure 21. The control surface view of the proposed optimal fuzzy PID controller ( e(f) = 3346.4)
Table 9. Number of tuning parameters for different controllers. 4. Conclusions
Table 9. Number of tuning parameters for different controllers. 4. Conclusions
To summarize of the above simulation examples, the proposed optimal fuzzy PID controller has quicker responses with smaller rising time and settling time. However, we still cannot claim that the proposed optimal FPID has outperformed other methods. In order to compare it with other approaches, we used the same [AE performance index, shown in Equation (6), but it is not practical to only consider the error signal without considering the energy or the power consumption of the controller output. The proposed optimal FPID design features a simple controller structure and a straightforward design approach based on conventional PID control and only eight nonlinear factors. Table 9 shows comparisons with the number of tuning parameters for different controllers.
To summarize of the above simulation examples, the proposed optimal fuzzy PID controller has quicker responses with smaller rising time and settling time. However, we still cannot claim that the proposed optimal FPID has outperformed other methods. In order to compare it with other approaches, we used the same [AE performance index, shown in Equation (6), but it is not practical to only consider the error signal without considering the energy or the power consumption of the controller output. The proposed optimal FPID design features a simple controller structure and a straightforward design approach based on conventional PID control and only eight nonlinear factors. Table 9 shows comparisons with the number of tuning parameters for different controllers.
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