# Review of Recent Type-2 Fuzzy Controller Applications
## 2 T2 Fls
### It2 Fls
#### 2.2.1 Karnik-Mendel Method  
The KM method was first developed by [PERSON] and [PERSON] [13; 14]. According to [15], this method is the most widely-used method for TR of an IT2 FS. TR is a major road block in computing the output of a T2 FLS.  
Many enhanced versions have been proposed that use the KM method; however, they work with a different strategy. [PERSON] and [PERSON] presented the enhanced iterative algorithm with stop Condition (EIASC), where five variations are compared experimentally, and the most efficient one is the optimum algorithm also known as EIASC [16].  
[PERSON] and [PERSON] [17] introduced the \(\alpha\)-cut algorithm for solving the fuzzy weighted average (FWA) problem. This algorithm uses the KM algorithm to compute the FWA \(\alpha\)-cut endpoints. Another work proposed a boundary function based on the KM TR method of IT2 PID controllers [18]. This KM method generated a boundary function KM, which was proved to be superior and more feasible. Other algorithms were also developed that improve on existing KM algorithms. This is called the  
Figure 1: Structure of a GT2 FLS, with permission from [9].  
enhanced Kamik-Mendel (EKM) originally presented by [PERSON] and [PERSON] [19]. It was found that on average, the EKM method was able to reduce computational time by 39% compared to KM. In [20], [PERSON] and [PERSON] developed two hardware architectural proposals for the EKM method. The first approach uses a sequential online arithmetic, whereas the second approach includes operators based on the CORDIC (coordinated rotation digital computer) algorithm. The CORDIC algorithm showed better performance in terms of speed and accuracy.  
The fundamental KM algorithms converge to exact solutions monotonically and at a super-exponential rate. They can also run in parallel because they are independent [21]. For IT2 FLS, \(Y^{i}\) in the above rule base is interval Type-2 sets and is represented by two intervals of \(\left\lfloor y^{i},\overline{y}^{i}\right\rfloor\). To calculate the IT2 FLS output, we first arrange \(y^{i}_{r}\) and \(y^{i}_{l}\) in ascending order. The following steps show how to compute left and right intervals for the KM algorithm [11; 13]:  
1. [label=(0),leftmargin=*]
2. Arrange \(y^{i}_{r}\) in ascending order.
3. \(y_{r}\) is calculated as: \[y_{r}=\frac{\sum_{i=1}^{M}f^{i}_{r}y^{i}_{r}}{\sum_{i=1}^{M}f^{i}_{r}}\] (1) \[f^{i}_{r}=\frac{\overline{f^{i}}+f^{i}_{r}}{2};\text{let }y_{r} = y_{r}\text{ (note that }\overline{f^{i}}\text{, }\underline{f^{i}}\text{ are the firing intervals).}\] (3) Find \(S\), such that \(y^{S}_{r}\leqslant y^{\prime}_{r}\leqslant y^{\prime}_{r}\)\({}^{+}\).
4. Find \(y^{\prime}_{r}=\frac{\sum_{i=1}^{M}f^{i}_{r}y^{i}_{r}}{\sum_{i=1}^{M}f^{i}_{r}}\) with \(f^{i}_{r}=\underline{f^{i}}\) for \(i\leqslant S\) and \(f^{i}_{r}=\overline{f^{i}}\) for \(i>S\); now, let \(y^{\prime\prime}_{r}=y_{r}\).
5. If \(y^{\prime\prime}_{r}\
eq y^{\prime}_{r}\), go to Step 6. If \(y^{\prime\prime}_{r}=y^{\prime\prime}_{r}\), set \(y_{r}=y^{\prime\prime}_{r}\), and stop.
6. Let \(y^{\prime}_{r}=y^{\prime\prime}_{r}\), and go to Step 3.  
The algorithm for computing \(y_{1}\) is similar to that of the \(y_{r}\) algorithm, except that all instances of \(y_{r},y^{\prime}_{r},y^{\prime\prime}_{r}\), \(y^{i}_{r}\), \(y^{R}_{r}\), \(y^{R}_{r}\)\({}^{+}\)1, \(f^{i}_{r}\) and \(R\) are replaced with \(y_{l},y^{\prime\prime}_{l},y^{\prime\prime}_{l}\), \(y^{i}_{l}\), \(y^{L}_{l}\), \(y^{L}_{l}\)\({}^{+}\)1, \(f^{i}_{l}\) and \(L\), respectively. Finally, the defuzzified output is the average of \(y_{r}\) and \(y_{l}\) as follows:  
\[Y_{KM}=\left(\frac{y_{l}+y_{r}}{2}\right) \tag{2}\]