Analysis of Footprint of Uncertainty of the IT2 Mamdani Controllers Using Different Type-Reducers


 Appropriate Footprint of Uncertainties (FOUs) are beneficial to the performance of Interval Type-2 (IT2) fuzzy controller, revealing the effect of FOUs is a key problem. In our published work, as the FOUs increase, the IT2 Mamdani and TS fuzzy controllers, using KM or EKM type-reducer (TR), approach the constant and piecewise linear controllers, respectively, while they finally become constant and piecewise linear controllers. To verify the validation of the above results, when a different TR is used, in this study, the effects of other popular TRs (i.e., Nie-Tan, Wu-Mendel, Iterative Algorithm with Stop Condition) on output of IT2 Mamdani fuzzy controller, are explored. We proven that, (1) as the FOUs increase, irrespectively of the TRs used, the IT2 Mamdani fuzzy controllers approach constant controllers, (2) when all the FOUs are equal to 1 (i.e., at their maximum ), the fuzzy controllers using Nie-Tan and Iterative Algorithm with Stop Condition TR become constant controllers. The FOUs of the controllers using Wu-Mendel TR can be infinitely approaching 1 and cannot be equal to 1 (otherwise, the denominator of the TR output expression are equal to 0), hence when FOUs are infinitely approaching 1, the controller will approach the constant controller infinitely. These results imply regardless of which popular TR is used, the IT2 Mamdani fuzzy controller, when using larger FOUs, the fluctuation of the input variables have a limited impact on the output, the ability to deal with system uncertainties will deteriorate. Laboratory control experiments are provided to demonstrate these findings.

operator, as well as other general configurations, approach constant and piecewise linear controllers, respectively. When FOUs are at their maximum, Mamdani FCs become constant controllers and TS FCs become piecewise linear controllers. Hence a key problem is that, it is not clear enough, when different TR is used, whether the increase of FOUs has the same effects on the IT2 FC.
At present, there are many methods of TR in the literature (Wu & Mendel, 2009;Nie & Tan, 2008;Melgarejo, 2007;Wu & Mendel, 2002;Karnik et al., 1999). According to findings in the Web of Science, the most widely used methods are the KM, EKM, Nie-Tan (NT), Wu-Mendel (WM) and Iterative Algorithm with Stop Condition (IASC) TR. However, the KM TR is the most popular choice, when designing an IT2 FC. The number of papers studying TRs is still relatively low (Zhou & Ying et al., 2019;Runkler et al., 2018;Nie & Tan, 2012;Wu, 2012;Chen & Wu et al., 2018;Chen & Wang, 2018;Li et al., 2018;Wu, 2013;Dongrui & Nie, 2011;Wu & Mendel, 2019). Nie and Tan studied the analytical structure of the IT2 FC, using the KM TR and the min() operator, the controller used symmetric rule consequents, to calculate the boundary of switch points of KM TR, while, after firing intervals were determined, the analytical structure was derived (Nie & Tan, 2012). Wu summarized the characteristics of TR methods and compared these using experiments. Nie-Tan and Wu-Tan proved that TRs have a relatively fast response speed (Wu, 2012). Comprehensive descriptions and comparisons of the TRs are given in (Mendel et al., 2020;Chen & Wu et al., 2018;Mendel, 2013). The speed of the algorithms depends on the coding language (Chen & Wu et al., 2018), while the Enhanced Iterative Algorithm with Stop Condition (Dongrui & Nie, 2011) appears to be the fastest one when running in Matlab, coded in C and Java, whereas the Enhanced KM algorithms (Wu & Mendel, 2009) and the optimized direct approach are the fastest ones, when coded in R and Python.
In this study, based on the effects of increasing FOUs on the output of IT2 Mamdani FCs, using KM or EKM TR, the focus is mainly on, the variation of increasing FOUs on the output of IT2 Mamdani FCs, when NT, WM, and IASC TR are used. Detailed mathematical analysis provides these results: although different TRs and different fuzzy operators (i.e., product or min() operator) are used, (1) as FOUs of input sets increase, IT2 Mamdani FCs approach constant controllers; (2) when all Analysis of Footprint of Uncertainty of the IT2 Mamdani Controllers Using Different Type-Reducers 2 FOUs of IT2 input fuzzy sets reach their maximum (i.e., FOUs=1), the IT2 Mamdani FCs, using NT and IASC TR, become constant controllers. When FOUs are infinitely approaching 1, the controller using WM TR will infinitely approach the constant controller behavior. In Section II, the general configuration of IT2 Mamdani FCs is described, with any number of input variables, any number and arbitrary type of IT2 fuzzy sets, two fuzzy AND operators (i.e., product and min()), three widely used TRs (i.e., NT, WM and IASC TRs) and centroid defuzzifier. In Section III, the effects of increasing FOUs on IT2 Mamdani FC are analyzed, when min() operator and one of these TRs are used separately. Section IV shows the effects of the different fuzzy operators on the derived theoretical results. In Section V, real-time control experiments, in laboratory environment, are provided to inspect the theoretical properties. Finally, the derived conclusions are presented in Section VI.
In (1) and (2), respectively. For the IT2 FC, it is necessary to reduce the T2 output fuzzy set to a T1 fuzzy set, before defuzzification. The following three popular TRs (except KM and EKM TR) are used to obtain the output of controller, respectively.
Considering the three commonly used TRs (Nie & Tan, 2008;Melgarejo, 2007;Wu & Mendel, 2002), it is obvious that, whether the k  and k  are sorted, has no effect on the NT TR and the WM TR, but the IASC TR requires the k  and k  to be sorted in ascending order. For the convenience of description, all k  and k  are unified, arranged in ascending while k f is also arranged in the same order as k  and k f as The output of NT TR is an accurate closed-form solution, so there is no need to perform defuzzification process. The output of the IT2 FC is as follows: Note that the rule consequents uk of IT2 FC, using the NT TR, do not need to be sorted, since  as Uncertainty Bound TR) that can effectively reduce the computational burden (Wu & Mendel, 2002) (2) Calculation of the suitable value of ( , ) (3) Computation of ( , ) l u  x and ( , ) Unlike the KM TR, the i  and i  do not need to be sorted, but their maximum and minimum values need to be specified. For convenience, it is assumed 1 In the case of IASC TR (Melgarejo, 2007), iterative termination conditions are used to effectively reduce the computational burden of the KM TR. The iteration process produces the final output of IASC TR as: It should be note that the rule consequents k  and k  are required to be sorted in ascending order.
In the case where the WM or IASC TR is used, the output is a T1 fuzzy set [ Noted that for any kind of TR, In this Section, the min() operator is used to obtain the firing intervals, while the effects of increasing the FOUs, on the output of the IT2 Mamdani FCs, are researched, in the respective cases of NT, WM, and IASC TR use. When the FOUs of all input IT2 fuzzy sets are at their maximum, the following are respectively denoted:

A. Use of NT type-reducer
In order to demonstrate the effects of increasing FOU on the controller, using NT TR, three properties are summarized as follows: c) Combine the proof process of KM TR, as described in the work published in .
In the case of the IT2 Mamdani FC, using KM TR, as the FOUs increase, the following (20) and (21) are either decreasing in a monotonic or non-increase manner, becoming equal to 0, when all the FOUs arrive at their maximum.
For FCs, the output is greater than the minimum value of rule consequents (i.e., 1  ) and less than the maximum value (i.e., M  ). Therefore, Thus, when , when all FOUs become 1. QED Considering that this situation often occurs, when the fuzzy set of the IT2 fuzzy PI and PD controller is configured (Long et al., 2019;Nie & Tan, 2012;Du & Ying, 2010), researchers should avoid the use of NT TR, when designing such controllers.

B. Use of WM type-reducer
In the case where WM TR is used, as FOUs increase, the output of IT2 FC changes as follows: Property 4: When all FOUs are infinitely approaching their maximum, the IT2 Mamdani FC, using WM TR, approximates a constant controller infinitely.  (4), (6), (10) and (11) is 0. Taking this into account, the method of taking the limit to prove is as follows: It is derived that, the output equations of controller, using ISAC TR, are the same as those of one using KM TR; so it inherits the conclusion that the IT2 controller, using KM TR, becomes a constant controller, when FOUs are at maximum. The process for proving this conclusion is similar to the case of using KM TR, so it will not be repeated here.  (35) and (36), after the output of IASC TR is converted into a form, there is no difference from the output of KM TR. Since the output of the IT2 controller, using KM TR, is increasingly similar to the constant controller, as the FOUs increase, the IT2 controller, using IASC TR, follows also the same trend. Due to limited space, the steps of proof will not be repeated here.
QED According to the above analysis, as FOUs increase, the output of the IT2 controller, either using NT TR, WM TR, or IASC TR, is consistent with the controller using KM and EKM TR. In order to better illustrate the effects of increasing FOUs on the output of IT2 Mamdani FC, using different TRs, their respective output surface is shown in Fig. 3. It is evident that, although the control surfaces are slightly different, when the controller uses the four TRs, the output surfaces gradually tend to a constant plane, as the FOUs increase. Furthermore, when θ=0.9, the output is almost a constant plane, using any TR.     8 that, when identical FOUs are used, although the position curve of the motor is different, the position response of the motor gradually slows down, while the overshoot grows smaller, the setting time becomes longer. Moreover, the rule consequents are symmetrically configured about 0, that is, the output of the controller gradually tends to the 0 plane. This occurrence is consistent with the theoretical analysis that shows the output of the controller gradually tending to the constant plane, as the FOUs increase. In previous study presented in (Zhou & Ying et al., 2019), it was demonstrated that, when the FOUs of the input fuzzy sets increased, the output of the IT2 Mamdani FC, using the KM TR, gradually tended to the constant controller, and eventually became one. When the FOUs are larger, the ability of system to resist disturbance obviously deteriorates and recovery time becomes longer. Therefore, one can speculate that when the FOUs are larger, the robustness of the IT2 fuzzy control system, using the NT TR, or WM TR, or IASC TR, will also be poor.

VI. CONCLUSION
By studying the effects of increasing the FOUs on the output of IT2 Mamdani FC, using different TRs, it is proven that, whether KM TR, EKM TR, NT TR, WM TR or IASC TR is used, the controller is gradually converging to the behavior of a constant controller. When all the FOUs are at their maximum, the IT2 Mamdani FC using KM TR, EKM TR, NT or IASC TR becomes a constant controller, the controller using WM TR will approach the constant controller infinitely when FOUs are infinitely approaching their maximum.
The properties (1)- (7), proposed in this study, not only help to understand the role of the five widely used TRs in the IT2 FC, but also facilitate the analysis of the effects of increasing FOUs, while they can also provide guidance for the design of FOUs. Regardless of whether KM TR, EKM TR, NT TR, WM TR or IASC TR is used, the IT2 Mamdani FC, when using larger FOUs, will produce an output similar to the constant; that is, the fluctuation of the input variables have a limited impact on the output, while the ability of IT2 FC to deal with system uncertainties will deteriorate. When the FOUs are maximum (or infinitely approaching their maximum), the output of IT2 Mamdani controller is a constant (or approximates a constant infinitely), which is no longer a function of the input variables and the superiority of the FC disappears. In addition, according to the published study (Zhou & Ying et al., 2019), properties (1)- (7) can be extended to the IT2 TS FC, but as the FOUs increase, this controller gradually tends to be a piecewise linear controller and actually becomes a piecewise linear controller when the FOUs are at their maximum.