Analysis of the difference between footprints of uncertainty for interval type-2 fuzzy PI controllers

Performance Analysis of Interval Type-2 Fuzzy X¯ and R Control Charts

“…To simulate ARL, SDRL and RL percentiles of IT2TFN X and R control charts, sample size n = 5, process mean 0.0 and standard deviation 1.0 were selected. L 1 = L 2 = 0.05 and FOU = 0.30 were used in the fuzzification process according to experts, as they provide a fast response while at the same time reducing overshoots according to [18].…”

“…Considering a process whose mean is 0 and standard deviation is 1, 20 samples are randomly generated whose sample subgroups have a size of 5. Using the control limit equations for the traditional 𝑋 and R control charts, as can be seen in [18], we obtain: LCL = −1.477 and UCL = 1.480 for X ; and LCL = 0 and UCL = 5.417 for R. Table 6 shows the situation described, together with the X and R parameters. Thus, IT2TFN X and R control charts offer several overwhelming advantages over traditional X and R control charts.…”

“…An interval type-2 fuzzy set A is represented by a lower membership function (LMF) 𝐴 and an upper membership function (UMF) A , where A = 〈A , A 〉. In particular, when triangular pertinence functions are used, it is illustrated as in Figure 2 Upper membership and lower membership functions are described according to Equations ( 2) and (3), respectively: (3,8,12,15,18) in a three-dimensional space.…”

“…Concerning the most usual FOU values, [17] demonstrate the analytical structure variation in values of FOU (θ) ranging from 0 to 0.5. In order to discuss other values of FOU, [18] reported the differences between FOU values and used PI controllers through interval type-2 fuzzy numbers. It is worth mentioning that the value of the FOU depends on the type of application and what is to be controlled.…”