Design of Noniterative Algorithms for Takagi Sugeno Kang Type General Type-2 Fuzzy Logic Systems

: The paper performs the center-of-sets (COS) type-reduction (TR) and defuzzification for Takagi Sugeno Kang (TSK) type general type-2 fuzzy logic systems (GT2 FLSs) on the basis of the  -planes expression of general type-2 fuzzy sets. Actually, comparing the popular Karnik-Mendel (KM) algorithms with other noniterative algorithms is an important question in T2 society. Here the modules of fuzzy inference, COS TR, and defuzzification for TSK type GT2 FLSs are discussed by means of noniterative Nagar-Bardini (NB) algorithms, Nie-Tan (NT) algorithms, and Begian-Melek-Mendel (BMM) algorithms. Simulation instances are constructed to illustrate the performances of three types of noniterative algorithms compared with the KM algorithms. It is proved that, the proposed noniterative algorithms can enhance the computational efficiencies significantly, which afford the potential application value for designers of GT2 FLSs.


Introduction
Interval type-2 fuzzy sets [1] can explain the uncertainties in membership functions (MFs).
However, the secondary membership grades of interval type-2 fuzzy sets (IT2 FSs) are just equal to 1, which must measure the uncertainty of MF uniformly. While the secondary membership grades of GT2 FSs lie between 0 and 1. Therefore, GT2 FSs can be regarded as higher-order uncertain fuzzy set models in contrast to IT2 FSs. Naturally, IT2 and GT2 FLSs use IT2 and GT2 FSs, respectively. As the design degrees of freedom increase, GT2 FLSs [2][3][4][5]15] have advantages over IT2 FLSs [6][7] on many fields subject to uncertainty.
Generally, a GT2 FLS is constituted by five modules as: fuzzifier, fuzzy reasoning (inference), rules, TR and defuzzification (see the Fig. 1). Among which, the module of TR is especially important, which act as the role of varying the T2 fuzzy set to the T1 fuzzy set. Finally the defuzzification block maps the T1 fuzzy set to the output. In the past decades, the calculational costs of GT2 FLSs have been significantly reduced as the  -planes (or say z-Slices [8-10]) descriptions of GT2 FSs were put forward by some well-known researchers. Since then, GT2 FLSs are successfully applied to many fields as edge detection [11][12], intelligent fuzzy control [3,5,10], forecasting [4,[13][14], medical diagnosis [30] and so on. The centroid TR for IT2 FLSs is a very popular theoretical study approach. In the early days, the time-consuming Karnik and Mendel (KM) algorithms [16] were developed to complete the centroid TR. Even so, the iterative property of KM algorithms made them difficult to apply in practical applications. Hence some noniterative algorithms were proposed gradually for perform the centroid TR, they are known as the Greenfield and Chiclana Collapsing Defuzzifier (GCCD [17]), Wu and Mendel uncertainty bound (UB [18]), Nagar and Bardini (NB) algorithms [19][20][21], Nie and Tan algorithms [22][23][24] and Begian and Melek and Mendel (BMM) algorithms [25][26]. In contrast to the centroid TR, studying the COS TR is more beneficial for designing IT2 and GT2 FLSs. Moreover, on the basis of alpha-planes representation of GT2 FSs, it is feasible to expand and improve the centroid TR of IT2 FLSs for performing the COS TR [27][28][29] of more complicated GT2 FLSs.
The paper expands the NB algorithms, NT algorithms and BMM algorithms to perform the COS TR for GT2 FLSs. Simulation experiments are constructed to illustrate the performances of three kinds of noniterative algorithms in contrast to the KM algorithms. The remainder of the paper is arranged as follows. Section two gives the TSK inference structure based GT2 FLS.
Section three provides the proposed noniterative algorithms for performing the COS TR of GT2 FLSs. Six simulation experiments are provided in Section four to illustrate the performances of them. Finally Section five is the conclusions.

TSK GT2 FLSs
Similar to the inference structure of IT2 FLSs, GT2 FLSs are also divided into Mamdani type [13] and Takagi-Sugeno-Kang (TSK) type [2,4,14]. Here we only focus the TSK type. Consider a TSK type GT2 FLS with totally p inputs 1 , and single output Y y  , which can be described by totally N fuzzy rules, in which the form lth rule is as:  denotes the antecedent GT2 FS, and   ) , , represents the crisp consequent parameter. Here the input measurement is also chosen as the GT2 FS, furthermore, the GT2 FLSs are as the " 0 2 C A  "type, i.e., the antecedent is the GT2 FS, and consequent is the crisp number. In addition, the structure of rules doesn't not change when the systems vary from IT2 to GT2, and only the types of FSs are transformed.
Firstly, compute the consequents for each fuzzy rule, that is to say,  can be computed by the KM types of TR algorithms as: Let the whole number of effective  -planes be k , that is to say, the  is equally divided into: The final output of TSK type GT2 FLSs can be got as: here the equation (6) was first put forward by Wagner [10], which can be called as the endpoints average defuzzification method. Although this approach can get the defuzzified value by a relatively simple way, which also need to compute the k  , TSK Y values according to the corresponding  .

Three kinds of noniterative algorithms
In this section, we obtain the output for TSK type GT2 FLSs according to the NB algorithms, NT algorithms and BMM algorithms.

NB algorithms
Recent studies show that T2 FLSs on the basis of NB algorithms [19] own superior capability in face of environment uncertainties and external disturbances. By means of the fuzzy reasoning [31], let the output COS type-reduced set of TSK type GT2 FLSs at the  level be an interval, Then the two end points of interval can be calculated as: At the related  -level, the COS defuzzified value can be obtained as: Aggregating all the  , NB y to get the COS type-reduced set, i.e., The final ouput of TSK type GT2 FLSs based on NB algorithms can be obtained as the form in equation (6), i.e., As a matter of fact, the simple closed form NB algorithms get the ouput from the linear combination of two different type-1 FLSs: one depends on the upper membership functions of type-2 fuzzy sets, and the other relies on the lower membership functions of T2 FSs.

NT algorithms
For the centroid type-reduction, the latest researches prove that the continuous Nie and Tan (CNT) algorithms [22] are actually an exact approach. Moreover, the sampling-based NT algorithms [32][33] can precisely approach the CNT algorithm. Here the COS type-reduction for GT2 FLSs is studied based on the NT algorithms. At the corresponding  -level, the COS defuzzified value can be computed by the NT algorithms as: Aggregating all the , NT y  to get the COS type-reduced set, i.e., Finally the output of TSK type GT2 FLSs can be calculated as: Actually, just choose the average of upper and lower firing intervals, the closed form of NT algorithms can be got. In addition, this type of closed form of algorithm can perform the type-reduction and defuzzification simultaneously.

BMM algorithms
IT2 FLSs based on BMM algorithms [25][26]   . In like manners, the equation (12) can be transformed to: Then it can be reexpressed as: (17) in which Observing the equations (15) and (17), we can easily get that the BMM algorithms and NT algorithms prove to be the same as the coefficients are chosen as Based on the above mentioned analysis, the conclusion is that the BMM algorithms are more general form of above two kinds of algorithms, i.e., the latter are two special forms of the former.

Simulation experiments
Here we provide six simulation experiments to show the performances of three types of noniterative algorithms to complete the COS TR for TSK type GT2 FLSs. In these experiments, we divide the  equally into  effective values as: Furthermore, suppose that  be varied from one to a hundred with the step size of one. As 100   (the maximum number), the COS type-reduced sets calculated by noniterative algorithms and KM algorithms are studied. In addition, when the number of valid alpha-planes  varies from one to a hundred with the step size of one, the COS defuzzified values are also investigated.
For the proposed TSK type GT2 FLSs, the input measurement set is as: Simulation example two: For each fuzzy rule, the primary MF of GT2 FS is chosen as the Gaussian primary MF with uncertain mean, observe the form in Figure 3, that is to say, the MF expression is as: where 12 [ , ] Right top end point: Then the parameters for antecedent are as: . As for these above six examples, the COS type-reduced sets obtained by the proposed noniterative algorithms and the popular KM algorithms can be seen in Figure 4.    Figure 4).  times of noniterative algorithms are greatly reduced, that is to say, the latter can improve the calculational efficiency a lot. Therefore, it is reasonable to believe that we should adopt noniterative algorithms to investigate the COS type-reduction of GT2 FLSs.

Conclusion
TSK type GT2 FLSs on the basis of three kinds of noniterative are proposed in this paper. We also discuss the modules of inference, COS TR and defuzzification for the GT2 FLSs. As for calculating the COS type-reduced sets and defuzzified values, six experiments are given to show the effective of proposed three types of noniterative algorithms in contrast to the KM algorithms.
As the computational times of former are significantly less than the latter, which can afford the potential values for adopting them in T2 FLSs to deal with uncertain environments.
In the next work, adopting both the iterative and noniterative algorithms for studying the centroid and COS type-reduction [16-26, 28-29, 32-35] of T2 FLSs will be further studied.