Mathematical analysis of a simplified general type-2 fuzzy PID controller

Abstract: For the type reduction of general type-2 fuzzy PID controller is time consuming and the mathematical expression of general type-2 fuzzy PID controller is difficult to derived. So, a simplified general type-2 fuzzy PID (SGT2-FPID) controller is studied in this article. The SGT2-FPID controller adopts triangular function as the primary and secondary membership function. Then the primary membership degree of apex for the secondary membership degree will be applied to get the output of SGT2-FPID controller, which can reduce the computation complexity of general type-2 fuzzy controller type reduction. Furthermore, the mathematical expressions of SGT2-FPID controller, type-1 fuzzy PID controller and interval type-2 fuzzy PID controller are discussed. Finally, 4 plants are applied to demonstrate the effectiveness and robustness of SGT2-FPID controller. The simulation results show that when the plants have uncertainty in model structure, measurement and external disturbance, the SGT2-FPID controller can achieve better control performances in contrast to compared controllers.


Introduction
In real control problems, there exited many uncertainties, like model structure, measurement, external disturbance and so on, tradition PID and type-1 fuzzy controller can't deal with these uncertainties [1][2][3][4][5]. Type-2 fuzzy controller can handle uncertainties more robust than PID and type-1 fuzzy controller for it was described by type-2 fuzzy sets proposed by Zadeh in 1975 [6]. Type-2 fuzzy sets mainly included interval type-2 fuzzy sets whose secondary membership degree   A A X (1) where : [0,1]  → A X , and μA (x) represents the membership degree of the element x∈X to the set

A.
A general type-2 fuzzy sets A defined in universal sets X can be described as Eq (2) [77].  (2) u is the primary membership degree and ( , ) xu  A is the secondary membership degree related to input variable x and primary membership degree u.
If the secondary membership degrees ( , ) xu  A are set to 1, then interval type-2 fuzzy sets can be shown as Eq (3). Figure 1 shows the definition of type-1 fuzzy sets, interval type-2 fuzzy sets and general type-2 fuzzy sets whose secondary membership function is triangular.

α-planes representation for general type-2 fuzzy sets
Liu introduced an α-plane representation for general type-2 fuzz sets [34], and pointed that α-plane denoted as  A can be defined as Eq (4).
If assemble all α-planes  A , then general type-2 fuzzy sets A can be described as Eq (5). [0,1] () FOU   = AA (5) The centroid of general type-2 fuzzy sets can be calculated by the centroids of its all α-planes are the left and right end points of interval type-2 fuzzy sets  A whose secondary membership degree is α.

General type-2 fuzzy logic systems
The general structure of fuzzy PID controller can be depicted as Figure 2 [76]. The antecedent parts can be type-1, interval type-2 or general type-2 fuzzy sets and the consequent parameters are crisp values.
Fuzzy logic systems Process Figure 2. Structure of fuzzy PID controller.
In this paper, triangular primary function is applied. The inputs of general type-2 fuzzy PID controller are normalize error (E) defined in [-de-d1,de+d1] and error derivative ( E ) defined in [ 2,2] , which is shown as Figure 3. d1 and d2 decide the footprint of uncertain for primary membership degree, for simplify, de de = and d1 = d2. The consequent parameters are symmetric and from Figure 3, 4 rules will be generated as follows, here H1 > H2 > -H2 > -H1.
By fuzzy inference of interval type-2 fuzzy logic systems and product operation, the fired membership degrees of fuzzy rules can be described as Eq (12

Type-1 fuzzy PID controller
The triangular membership function of type-1 fuzzy PID controller is depicted as Figure 4, also for simplify, de de = .  From Figure 5 and defuzzification process of type-1 fuzzy sets, the output of type-1 fuzzy inference U(t) in Figure 2 can be calculated as Eq (18).
According to Eq (19) and Figure 2, the final output of T1F-PID controller can be expressed as Eq (20).

Interval type-2 fuzzy PID controller
For KM algorithm didn't have analytic solution, so NT type reduction [78,79] algorithm will be applied to get the mathematical expression of IT2F-PID controller. Figure 7 shows an example of upper and lower bounds for fuzzy rules corresponding to consequent parameters using IT2F-PID controller.

Simplified general type-2 fuzzy PID controller
For type reduction of general type-2 fuzzy sets was converted to type reduction of several interval type-2 fuzzy sets, so the number of α-planes will affect the real time of GT2F-PID controller. Figure 9 shows an example of membership degrees for fuzzy rules corresponding to consequent parameters using GT2F-PID controller. Figure 9. General type-2 fuzzy PID membership degrees of fuzzy rules and consequent parameters.
The differences of GT2F-PID and SGT2F-PID controller can be seen from Figure 10. (j=1, 2, …, D) whose secondary membership degree is αj. KM or other interval type-2 type reduction algorithm will be applied for each interval type-2 fuzzy sets. And the last, assembles all the centroids of α-plane and gets the output of GT2F-PID controller.
In this paper, the SGT2F-PID controller adapts the primary membership degree of α-plane (α = 1) as the membership degree of fuzzy rules, which is calculated as Eq (26).
According to Eq (28) and Figure 2, the final output of SGT2F-PID controller can be expressed as Eq (29).
where: Figure 11 shows the shape of control surface of simplified general type-2 fuzzy controller, here H1 = 1, H2 = 0 and w = 0. From the control surface curve of T1-FPID, IT2-FPID and SGT2-FPID controller, when the system error is near the endpoint, the output of SGT2-FPID controller is larger than T1-FPID and IT2-FPID, so the SGT2-FPID controller has the faster rising time. When the error is near zero, the output of SGT2-FPID controller is smoother than T1-FPID and IT2-FPID, so the SGT2-FPID controller has faster steady time and smaller overshoot.

Controller analysis
In summary, the unified T1-FPID, IT2F-PID and SGT2F-PID controller mathematical expressions can be indicated as Eq (31). where: If calculate the derivative of KSGT2 to w, then the partial derivative is Eq (33).
From Eq (33), KSGT2 is a decreasing function of w and in general, w is in range [0,1]. So the ranges of KSGT2 is denoted as Eq (34). Figure 12 shows the curve of KSGT2 as w is rising from 0 to 1.

Mathematical Biosciences and Engineering
Volume 17, Issue 6, 7994-8036. when w = w1, KSGT2 = KT1 and w = w2, KSGT2 = KIT2, then w1 = (de-d1)/(2de) and w2 = 0.5. According to the characteristics of PID controller, the advantage of proportional action is timely. If increase proportional gain, then the system response speed will be enhanced (that is reducing the rising time and steady time) but the system overshoot will be increased. The integral action can eliminate static error, if increase integral gain, the system overshoot will be decreased. The differential action also has the advantage of timely, which is belonging to 'future control'. If increase differential gain, the steady time and system overshoot will be reduced.
From above analysis, if a control system maintains both faster response speed and smaller overshoot, the PID controller should chose larger proportional gain, integral gain and differential gain. Figure 12 shows that if w < w1, then the proportional gain, integral gain and differential gain of SGTF-PID are larger than T1F-PID and IT2F-PID.Thus the controlling efforts of SGTF-PID will be better than T1F-PID and IT2F-PID, which is proved by section 5 of four simulation examples.

Simulations
In simulations, 3 plants and a practical inverted pendulum system are tested to demonstrate the robustness and efficiency of SGT2F-PID. The controlling efforts of SGTF-PID are also compared with PID, T1F-PID, and IT2F-PID controller using NT type reduction algorithm. Step response curve of P1 in case 4. Table 1 summarizes some controlling performance comparisons of SGT2F-PID controller with other 3 controllers. In Table 1, ts is steady state time, tr is rising time, OS is system overshoot and three error integral criterions ISE, ITSE, ITAE.    Step response curve of P2 in case 4.   Step response curve of P3 in case 4. Table 3 shows the P3 controlling performance comparisons of SGT2F-PID controller with other 3 controllers.

Nonlinear inverted pendulum system (P4)
The inverted pendulum system was often applied to demonstrate the reliability of a new controller, as shown in Figure 25.

Mathematical Biosciences and Engineering
Volume 17, Issue 6, 7994-8036.  From case 3 to case 6, we will indicate the controlling effects of SGT2-FPID controller when the system adding uncertainties. Case 3: Pendulum mass uncertainty.
Here, we will add pendulum mass uncertainty (Δmp = 2.7kg) at 2s.  Table 4 shows the P4 controlling performance comparisons of SGT2F-PID controller with other 5 controllers for case 1 and case 2. As compares with controlling performances of [80] and [81], another two error integral criterions are added as follows.   Table 5 shows the P4 controlling performance comparisons of SGT2F-PID controller with other 4 controllers for case 3 to case 6.

Conclusions
We discuss 3 kinds of fuzzy PID controllers and derive the mathematical expressions of TIF-PID, IT2F-PID and SGT2F-PID described by Eq (21), Eq (25) and Eq (30). The SGT2F-PID controller contains more adjustable parameters and only 4 fuzzy rules are generated. For the primary membership degree of α-plane (α = 1) is used to get the defuzzification result of SGT2F-PID controller, thus the SGT2F-PID controller maintains the ability of handing uncertainties as general type-2 fuzzy controller and higher real-time. By the mathematical expressions of TIF-PID, IT2F-PID and SGT2F-PID controller, the controlling performance is discussed and explains why SGT2F-PID controller has better controlling effects than TIF-PID and IT2F-PID controller.
And 4 simulations including a second order linear plant, an unstable first order linear plant and two second order nonlinear plants are tested. In addition, the controller parameters of each plant are the same when the plant parameters are changed, which demonstrate the robustness of SGT2F-PID controller. From the 4 simulation results, when the controlled object changes, the SGT2F-PID controller can still maintain small overshoot, faster response time and stable time. Also the controller performance evaluation indexes (ISE, ITSE, ITAE) of SGT2F-PID controller are better than other 3 compared controllers. The results of simulation 4 indicates that, when the controlled object exists uncertainties of measurement, structure and external disturbance, the SGT2F-PID controller can handle these uncertainties more robust than PID, TIF-PID and IT2F-PID controller.
The next researches will focus on the following 4 aspects: Ⅰ). Although SGT2F-PID controller can achieve better control performances, but the determined parameters are more than other controllers. How to determine the appropriate parameters will be a major work. II). Triangular function is applied as primary and secondary membership function, other membership function like Gaussian, trapezoid will be discussed in the future. III). In this paper, we fix the parameters de and d1 and discuss the influence of w on the controller parameters gains. In the future, we will study the influence of de and d1 on the controller parameters gains. IV). The fractional order simplified general type-2 fuzzy PID controller will be investigated and compared with existing PID and fuzzy PID controllers.