Hybrid Fuzzy Controller Design for Position Control System

Complex industrial processes are difficult to control automatically due to their nonlinearities and time-varying behavior. Examining the control strategies utilized by the process operator is another method for controlling a process. The process operators' control strategy is founded on conjecture and practical experience and can be considered a body of knowledge. In 1965, Lotfi A. Zadeh introduces the concepts of fuzzy logic and fuzzy set, which are dependent on linguistic rules; these algorithms were developed by directly evaluating the linguistic rules using digital computers. Fuzzy logic has been developed to provide soft information processing algorithms that permit variables to be partial members of a particular set and use a generalization of the conventional Boolean logical operators to manipulate this data. Today, fuzzy logic systems are used in a variety of engineering applications, medicine, meteorology, and other areas of daily life [1].

Classical PID controllers are regarded as one of the primary techniques for satisfying the needs of control systems. The process of designing PID controllers by adjusting KP, KI, and KD is known as tuning the controller parameters. The fuzzy logic controller is a non-linear method of system stability maintenance. Nevertheless, PID controllers produce a minimal steady-state error. Consequently, the hybridization of the controllers, i.e., the PID controller and the fuzzy logic controller, was devised to take advantage of the advantageous features of both controllers. Hybridization is the process of combining the performance of two distinct controllers to achieve superior performance compared to the performance of the individual controllers. A fuzzy logic controller has a quicker response time and a lower overshoot percentage. PID controller provides a low steady-state error. To design the hybrid model with the two controllers that enhance the performance of the control system [2]. Jang et al. [3] suggest adapting Hybrid PI and PD Type Fuzzy Logic controller to a quad-rotor for controlling the quadrotor’s X-axis (roll), Y-axis (pitch), and Z-axis (yaw). Hybrid PI and PD Type Fuzzy Logic Controller combines PI Fuzzy Logic Controller and PD Type Fuzzy Logic Controller and its purposes to make better performance than using single PI, PD Fuzzy Logic Controller.

Ismail et al. [4] regulate the speed of a DC Series Wound Motor with an FLC. FLC provides the best performance in terms of system response time compared to a dc motor without a controller. Hoang et al. [5] describes a method for developing a fuzzy logic controller that defines the PID parameters to improve the quality of an asynchronous three-stage wind power generator compared to conventional PID controllers. The simulation results indicate that the system's performance has been maintained in synchronous modes and when the active and reactive power is adjusted in response to load demands.

Fadaei et al. [6] proposed a hybrid fuzzy-PD controller for an inverted, wheeled pendulum. The controller is intended to stabilize the inverted pendulum, and its parameters are optimized using a genetic algorithm. The simulation results indicate that, with the force applied to the cart, the system is adequately balanced and would be stable over time. Chawpattnaik et al. [7] introduced the hybrid Fuzzy Fractional Order Proportional Derivative-Fractional Order Proportional Integral controller. The controller is proposed to address AGC issues in multiple interconnected areas of the electrical power system. Zerikat and Chekroun [8] proposed a tracking application control system. By adjusting the external force, the proposed control system displays a rapid response. Salim et al. [9] examined FLC for a buck converter. The simulation results indicate that FPD plus Integral is a suitable controller when the steady-state error must be minimized and that a two-level fuzzy hybrid is the optimal control method when the settling time must be kept to a minimum. Chaudhary et al. [10] compare the performance of PI controller-based DVR, fuzzy controller-based DVR, and PI-fuzzy controller-based DVR in a compensated feeder. Brehm and Rattan [11] presented a hybrid fuzzy PID controller that takes advantage of the properties of the fuzzy PI and PD controllers. Samad et al. [12] proposed a fuzzy logic controller tuned for both a Linear Quadratic Regulator (LQR) and a proportional derivative (PD) controller, which is applied in stabilizing a multi-link robotic system. Paramasivam and Arumugam [13] show that a hybrid controller for switched reluctance motor (SRM) drive has fast tracking capability, less steady-state error, and is robust to load disturbance. Shouran et al. [14] develop and implements a design of the Fuzzy Proportional Integral Derivative with filtered derivative mode (Fuzzy PIDF) for Load Frequency Control (LFC) of a two-area interconnected power system. The Bees Algorithm (BA) and other optimization tools are used to accomplish this task, he simulation results indicate that the contribution of the BA tuned the proposed fuzzy control structures in alleviating the overshoot, undershoot, and the settling time of the frequency in both areas and the tie-line power oscillations. Humaidi et al. [15] present a control design based on an Interval Type-2 Fuzzy Logic (IT2FL) for the trajectory tracking of a 3-RRR (3-Revolute-Revolute-Revolute) planar parallel robot. Abed et al. [16] presented two fuzzy logic (FL) schemes for speed-controlled brushless DC motors. The first controller is a Type 1 FL controller (T1FLC), whereas the second controller is an interval Type 2 FL controller (IT2FLC). The two proposed controllers were compared in terms of system dynamics and performance. Ghanim et al. [17] propose a Fuzzy Logic Controller (FLC) for temperature control, the controller combines the advantages of controller structure and fuzzy inference for air heater temperature control applications. Chintu et al. [18] present a study on an adaptive differential evolution tuned hybrid Fuzzy Proportional Derivative-Proportional Integral structure suggested for automatic generation control (AGC) of power systems. Nguyen and Nguyen [19] this paper introduce the Fuzzy-PI integrated control algorithm to control the system’s efficiency. The Fuzzy algorithm adjusts the KP and KI parameters of the controller. This response depends on the oscillation state of the vehicle. As a result, the vehicle’s stability and comfort are greatly enhanced when the integrated controller is used. Besides, the change of KP and KI parameters is also completely suitable for the vehicle’s driving conditions. In this work [20], the model is obtained for a hybrid intelligent controller for improving the stability of the pressure process station. Performance analysis is done for PID, Fuzzy PID, and Hybrid fuzzy PID controller. The hybrid controller gives reduced settling time better than other controllers. Noaman and Mohammad [21] propose an adaptive Hybrid Neural Fuzzy Controller using the augmented Error Method.

In this paper, the response of a control system was tested through different types of controllers such as PD controller, a fuzzy controller with a different rule base, and a hybrid fuzzy controller.

3.1 PD controller

This section describes the design and simulation of the PD controller. PD controller will be implemented in light of the fact that PI or PID controllers will not be implemented because there is no steady-state error. The block diagram of the PD controller is depicted in Figure 4. The following formula describes the PD controller:

$u(t)=K_p \times e(t)+K_d \times \Delta e(t)$                    (1)

It is preferred to ascertain the controller gains K_p and K_d that provide the desired response for our PD controller; therefore, through trial and error, it was determined that K_p = 2.5 and K_d = 0.15. Figure 5 depicts the unit step response of the PD controller. Applying the PD controller improved the system performance through less rise time from 0.283 sec to 0.26 sec, less settling time from 0.911 sec to 0.58 sec, and the overshoot reduced to 5.3% from 9.9%. As shown in Figure 3 the system has an acceptable response and overshoot. To investigate the response of the system due to different controller types. First, we apply a PD controller to improve its response and as a result, the overshoot was reduced according to the K_p and K_d gains selected.

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Figure 4. Block diagram for the position PD controller

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Figure 5. Step response for PD-controlled position system

3.2 PD fuzzy controller

This section introduces the formulation and construction of four FCs for gradually increasing the number of rules. These are the rule bases:

• 9-rule base FC (each input has 3 membership functions (MF)).
• 25-rule base FC (each input has 5 MF)
• 49-rule base FC (each input has 7 MF)
• 81-rule base FC (each input has 9 MF)

The same inputs were chosen for all of them which are:

(Position Error) ER = c – r

(Error Change) ERC = (ER₂ – ER₁) / $\Delta \mathrm{t}$

where c is the angular displacement of the output shaft, r is the angular displacement of the reference input shaft, ER1 and ER2 are the position control system's errors at t1 and t2, and t = (t2 - t1).

As shown in Figure 6, all the rules were derived from the control action of the proportional and derivative signals. In addition, the scaling factors (G1, G2, and G3) were adjusted through iterative trial and error, and the optimal response was obtained by varying the parameters and increasing the number of rules.

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Figure 6. Proportional and derivative control actions

3.2.1 Controller with a 9-rule foundation fuzzy

As depicted in Figure 7, three triangular MFs of each input and output (I/P and O/P) are utilized and uniformly distributed across their universes of discourse. The range [-1 1] constitutes the effective universe of discourse. Negative (N), zero (ZE), and positive (P) are the linguistic values for these MFs for I/P and O/P, respectively. Table 1 displays the PD fuzzy controller's nine-rule base table. Figure 8 depicts the block diagram for the 9-Rule FC, with the tuned gains (scaling factors) G1 = 3, G2 = 0.6, and G3 = 5. Figure 9 depicts this controller's step response.

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Figure 7. Input /output membership functions mapping of the 9-rules FC

Table 1. Fuzzy controller rule matrix (9-rules)

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Figure 8. The position controller with 9–rules FC

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Figure 9. The step response for a system with a 9-rule FC

3.2.2 Controller with a 25-rule foundation fuzzy

As depicted in Figure 10, five triangular MFs are used for each I/P and O/P and are uniformly distributed across their discourse universes.

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Figure 10. Input/output membership functions mapping of the 25-rules FC

The range [-1 1] constitutes the effective universe of discourse. Negative-Big (NB), Negative-Small (NS), Zero (ZE), Positive-Small (PS), and Positive-Big (PB) are the linguistic values (for I/P and O/P) for these MFs. Table 2 displays the rule table for the 25-rules controller.

Figure 11 depicts the block diagram for the 25-Rule FC. The tuned gains (scaling factors), where G1 = 3, G2 = 0.5, and G3 = 2, whereas G1 = 3, G2 = 0.5, and G3 = 2. Figure 12 depicts the controller's unit step response.

Table 2. Fuzzy controller rule matrix (25-rules)

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Figure 11. The block diagram for the position control system with 25-rules

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Figure 12. The step response for a system with a 25-rule FC

3.2.3 The 49-rule fuzzy controller

As depicted in Figure 13, seven triangular MFs of each I/P and O/P are utilized and uniformly distributed across their universes of discourse.

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Figure 13. Input/output membership functions mapping of the 49–rules FC

Table 3. Fuzzy controller rule matrix (49-rules)

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Figure 14. The block diagram for the position control system with 49–rules FC

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Figure 15. The step response for the system with a 49-rule FC

The range [-1 1] constitutes the effective universe of discourse. For I/P and O/P, these MFs have the following linguistic values: Negative Pig (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PM), and Positive Big (PB). Table 3 displays the 49-rules table controller.

Figure 14 depicts the block diagram for the 49-Rule FC. While G1=2.5, G2=0.4, and G3=4, the tuned gains (scaling factors) are: G2=0.4, G3=4. Figure 15 depicts this controller's step response.

3.2.4 The 81-rule fuzzy controller

As given in Figure 16, nine triangular MFs of each I/P and O/P are utilized and uniformly distributed across their universes of discourse.

The range [-1, 1] constitutes the effective universe of discourse. The following linguistic values are applicable to these membership functions: (for I/P and O/P) Negative Pig (NB), Negative Medium (NM), Negative Small (NS), Negative (N), Zero (ZE), Positive (P), Positive Small (PS), Positive Medium (PM), and Positive Big (PB). Table 4 displays the rule table for the 81-rules controller.

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Figure 16. Input/output membership functions mapping of the 81–rules FC

Table 4. Fuzzy controller rule matrix (81-rules)

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Figure 17. The block diagram for the position control system with 81–rules

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Figure 18. The step response for the system with an 81-rule FC

Figure 17 depicts the block diagram for the 81-Rule FC. While G1=3.8, G2=0.22, and G3=7.4, the tuned gains (scaling factors) are: G2=0.22, G3=7.4. Figure 18 represents this controller's step response.

3.3 Hybrid fuzzy controller

To make use of the advantage of the PD controller and FC and eliminate their disadvantages, this can be done by using a hybrid fuzzy controller which combines both two controllers.

In this work, the PD controller and FC are combined so that the control variable (U_HY) in this controller is:

$U_{H Y}=\left\{\begin{array}{lr}U_{P D} & \text { if }|E R|>e_{l i m} \text { or }|E R C|>e_{l i m} \\ U_{P D}+U_F & \text { otherwise }\end{array}\right.$                    (2)

where, U_PD represents the output of the PD controller, UF represents the output of the FC, and e_lim represents the input domain of the FC. Eq. (2) indicates that if the angle error or the change in angle error is outside the input domain of FC, then no fuzzy controller rules will be activated, and the only control variable will come from the PD controller. Figure 19 depicts the block diagram of the hybrid fuzzy controller. Kp = 3.1 and Kd = 3.8 are the parameters of the PD controller, and G1 = 4.5, G2 = 0.25, and G3 = 7.4 are the scaling factors for FC with nine MFs (81 rules). Figure 20 depicts the resulting optimized unit step response.

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Figure 19. Block diagram for the position control system with hybrid fuzzy controller

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Figure 20. Step response for the system with hybrid fuzzy controller

The following conclusions can be drawn based on the findings:

After constructing a mathematical model of the system and determining its transfer function, the conventional controller (PI, PD, and PID) can be used to control any linear control system.

Fuzzy logic controllers provide the capacity to represent, manipulate, and implement human knowledge regarding system control. The rules for a controller must be established by the designer, and the information could come from the human decision-maker performing the control task or from an understanding of the plant dynamics. In addition to being able to control the nonlinearity that occurs in the plant, the expansion of the rules causes the fuzzy controller to make effective decisions and be very effective at controlling the system.

The implementation of a hybrid fuzzy controller provides the advantages of the fuzzy controller and other conventional controllers while eliminating their disadvantages. This indicates that the hybrid fuzzy controller possesses both the speed characteristic of the conventional controller and the good transient response characteristic of the fuzzy controller.

This study can be extended for future work by conducting a comparison study between the proposed FL controller with other control techniques in the literature [23-30].