Design of Begian–Melek–Mendel structure-based interval type-2 fuzzy logic systems optimized with backpropagation algorithms

Abstract

As an emerging technology, the efficient and energy conserving process of permanent magnetic drive (PMD) presents high uncertainties. This paper designs a type of Begian–Melek–Mendel (BMM) structure interval type-2 fuzzy logic systems (IT2 FLSs) for PMD process uncertain parameters forecasting. The antecedent, consequent, and input measurements of systems are all selected as the Gaussian type-2 primary membership functions with uncertain standard deviations. Then the backpropagation algorithms are used to tune the parameters of IT2 FLSs. According to the Monte Carlo simulation studies and convergence analysis, the proposed IT2 FLSs are proved to be superior to two corresponding T1 FLSs in generalization ability.

Keywords

Interval type-2 fuzzy logic systems Begian–Melek–Mendel structure interval type-2 fuzzy sets forecasting convergence

Introduction

As an emerging technology, interval type-2 fuzzy logic systems (IT2 FLSs; Hagras, 2007; Hagras and Wagner, 2012) can better cope with the uncertainties in natural languages; therefore, they have been successfully applied to areas like power systems (Chen et al., 2016; Khosravi et al., 2012), financial systems (Bernardo et al., 2013), permanent magnetic drive (PMD; Barkat et al., 2011; Wang and Chen, 2018), hop strip mill (Méndez and Hernandez, 2013), pattern recognition (Mendoza et al., 2009), fault detection (Safarinejadian et al., 2015), intelligent controllers (Biglarbegian et al., 2011a; Hsu and Juang, 2013; Tao et al., 2012), medical systems (Lees et al., 2010), information systems (Niewiadomski, 2010), and edge detection (Castillo and Melin, 2012; Gonzalez et al., 2016). Especially in recent years, IT2 FLSs have been widely used on forecasting activities. Most recent studies on load forecasting show that IT2 FLSs (Khosravi et al., 2012; Khosravi and Nahavandi, 2014) have superiority approximation capability even better than nonparametric neural networks (Barbounis and Theocharis, 2007; Gao et al., 2014; Gusev and Burkovskii, 2013; Mehdi et al., 2016; Pany and Ghoshal, 2015). Furthermore, IT2 FLSs based on optimization algorithms outperform their T1 counterparts on forecasting.

A T2 FLS (see Figure 1) is usually composed of five blocks as fuzzifier, inference, rules, type-reducer, and defuzzifier. Among them, the block of type-reducer acts as the key role. Compared with the T2 FLS, a T1 FLS does not have the block of type-reducer. In a T2 FLS, there must be at least one T2 FS (fuzzy set) in the antecedent or consequent of fuzzy rules. The output of the inference is a T2 FS, and the block of type-reduction (TR) transforms the T2 FS to T1 FS before executing defuzzification.

Figure 1.

Blocks of a T2 FLS.

T2 FLSs based on Karnik–Mendel (KM) structure (Chen et al., 2018; Chen and Wang, 2017, 2019) are most popular for theoretical studies. However, it is much more complicated for T2 FLSs (Chen et al., 2016; Mendel, 2004, 2013) based on that structure and which may be not suitable for real applications. Recently, IT2 FLS based on Begian–Melek–Mendel (BMM) structure (Biglarbegian et al., 2010, 2011b) is shown to have better stability and robustness (Zhang et al., 2014; Zhang and Wang, 2015) than their T1 counterparts. This paper designs a type of BMM structure-based IT2 FLSs for forecasting. Backpropagation (BP) algorithms are used to optimize the parameters of proposed IT2 FLSs. Furthermore, temperature data for different types of permanent magnets based on the PMD process (Barkat et al., 2011; Wang and Chen, 2018) are employed to study the forecasting problems. Simulation experiments show that the proposed IT2 FLSs are superior to their T1 counterparts on forecasting accuracy.

We organize the rest of this paper as following. The “FLSs” section describes the designed T1 and IT2 FLSs. The “BMM structure-based IT2 FLSs and their BP algorithms” section provides the proposed BP algorithms and their applications on optimizing the parameters of FLSs. Simulation experiments for FLS forecasting approaches are given in the “Simulation studies” section. Finally, the “Conclusion” section is the conclusion and expectation.

FLSs

In this section, we introduce two types of FLS: Mamdani non-singleton T1 FLSs and Mamdani non-singleton T2 FLSs.

Mamdani non-singleton T1 FLSs

For the T1 FLSs, we adopt non-singleton fuzzification, height defuzzification, and the following form of “if–then” fuzzy rules. The four basic blocks are as follows

\begin{matrix} R^{l} : If x_{1} is F_{1}^{l} and \dots and x_{p} is F_{p}^{l}, \\ then y is G^{l} (l = 1, 2, \dots, M) \end{matrix}

(1)

where $M$ is the number of fuzzy rules, $p$ denotes the number of antecedents, $F_{i}^{l} (i = 1, 2, \dots, p)$ is the $i th$ antecedent T1 FS in the $l th$ rule, and $G^{l}$ represents the consequent T1 FS for the $l th$ rule.

Type-1 non-singleton fuzzifier (Mendel, 2001): Input measurement $x_{i} = x'_{i}$ is mapped into a fuzzy number (Kaufman and Gupta, 1985).

Inference: For the T1 input set $A_{x}$ , whose membership function (MF): $μ_{A_{x}} (x) = μ_{X_{1}} (x_{1}) \times μ_{X_{2}} (x_{2}) \times \dots \times μ_{X_{p}} (x_{p})$ , in which × is the T1 t-norm (Mendel, 2001).

The fuzzy relation $R^{l}$ : $F_{1}^{l} \times F_{2}^{l} \dots \times F_{p}^{l} \to G^{l} = A^{l} \to G^{l} (l = 1, \dots, M)$ , whose MF

\begin{matrix} μ_{R^{l}} (x, y) = μ_{A^{l} \to G^{l}} (x, y) = μ_{F_{1}^{l} \times \dots \times F_{p}^{l} \to G^{l}} (x, y) \\ = μ_{F_{1}^{l}} (x_{1}) \times \dots \times μ_{F_{p}^{l}} (x_{p}) \times μ_{G^{l}} (y) \\ = [T_{i = 1}^{p} μ_{F_{i}^{l}} (x_{i})] \times μ_{G^{l}} (y) \end{matrix}

(2)

Then the T1 fuzzy output set $B^{l} = A_{x} ◦ R^{l}$ , whose MF

\begin{matrix} μ_{B^{l}} (y) = μ_{A_{x} ◦ R^{l}} (y) = sup_{x \in X} [μ_{A_{x}} (x) \times μ_{A^{l} \to G^{l}} (x, y)], \\ l = 1, 2, \dots, M \end{matrix}

(3)

where ◦ represents the T1 composition operation (Mendel, 2001).

Defuzzifier: Height defuzzifier is selected as following.

The output $y (x) = f_{ns} (x) = \sum_{l = 1}^{M} {\bar{y}}^{l} ϕ_{l} (x)$ , in which $μ_{B^{l}} ({\bar{y}}^{l})$ is the maximum membership grade of the $l th$ output set and $μ_{B^{l}} ({\bar{y}}^{l}) = μ_{G^{l}} ({\bar{y}}^{l}) \times [μ_{F_{1}^{l}} (x'_{1}) \times \dots \times μ_{F_{p}^{l}} (x'_{p})]$ . In addition, $ϕ_{l} (x)$ is the Fuzzy-Based Function (FBF), and $ϕ_{l} (x) = μ_{B^{l}} ({\bar{y}}^{l}) / \sum_{l = 1}^{M} μ_{B^{l}} ({\bar{y}}^{l})$ .

BMM structure-based IT2 non-singleton T2 FLSs

For the BMM structure-based IT2 non-singleton T2 FLSs, we design them by means of non-singleton fuzzification, center-of-set TR, and the following form of “if–then” fuzzy rules

\begin{matrix} {\tilde{R}}^{l} : If x_{1} is {\tilde{F}}_{1}^{l} and \dots and x_{p} is {\tilde{F}}_{p}^{l}, \\ then y is {\tilde{G}}^{l} (l = 1, 2, \dots, M) \end{matrix}

(4)

where $M$ is the number of fuzzy rules, $p$ represents the number of antecedents, ${\tilde{F}}_{i}^{l} (i = 1, 2, \dots, p)$ is the $i th$ antecedent IT2 FS in the $l th$ rule, and ${\tilde{G}}^{l}$ denotes the consequent IT2 FS for the $l th$ rule.

Type-2 non-singleton fuzzifier (Mendel, 2001): Input measurement $x_{i} = x'_{i}$ is mapped into a T2 fuzzy number, that is, a T2 MF is associated with it.

Inference: For the T2 input set ${\tilde{A}}_{x}$ , whose MF: $μ_{{\tilde{A}}_{x}} (x) = ⋂_{i = 1}^{p} μ_{{\tilde{X}}_{i}} (x_{i})$ , in which ∩ denotes the minimum t-norm or product t-norm (Mendel, 2001). Furthermore, the fuzzy relation ${\tilde{R}}^{l}$ : ${\tilde{F}}_{1}^{l} \times \dots \times {\tilde{F}}_{p}^{l} \to {\tilde{G}}^{l} = {\tilde{A}}^{l} \to {\tilde{G}}^{l} (l = 1, 2, \dots, M)$ , whose MF is as

\begin{matrix} μ_{{\tilde{R}}^{l}} (x, y) = μ_{{\tilde{A}}^{l} \to {\tilde{G}}^{l}} (x, y) = μ_{{\tilde{F}}_{1}^{l} \cap \dots \cap {\tilde{F}}_{p}^{l} \to {\tilde{G}}^{l}} (x, y) \\ = μ_{{\tilde{F}}_{1}^{l}} (x_{1}) \cap \dots \cap μ_{{\tilde{F}}_{p}^{l}} (x_{p}) \cap μ_{{\tilde{G}}^{l}} (y) = [\cap_{i = 1}^{p} μ_{{\tilde{F}}_{i}^{l}} (x_{i})] \cap μ_{{\tilde{G}}^{l}} (y) \end{matrix}

(5)

For the $l th$ rule, the IT2 fuzzy output set ${\tilde{B}}^{l} = {\tilde{A}}_{x} ◦ {\tilde{R}}^{l}$ , whose MF

\begin{matrix} μ_{{\tilde{B}}^{l}} (y) = μ_{{\tilde{A}}_{x} ◦ {\tilde{R}}^{l}} (y) = \cup_{x \in X} [μ_{{\tilde{A}}_{x}} (x) \cap μ_{{\tilde{A}}^{l} \to {\tilde{G}}^{l}} (x, y)] \\ = μ_{{\tilde{G}}^{l}} (y) \cap {[μ_{{\tilde{X}}_{1}} ({x'}_{1}) \cap μ_{{\tilde{F}}_{1}^{l}} (x_{1})] \cap \dots \cap \\ [μ_{{\tilde{X}}_{p}} ({x'}_{p}) \cap μ_{{\tilde{F}}_{p}^{l}} (x_{p})]} = μ_{{\tilde{G}}^{l}} (y) \cap F^{l} (x') \end{matrix}

(6)

where $y \in Y (l = 1, \dots, M)$ , ◦ denotes the composition operation, and ∪ represents the maximum t-conorm operation.

Suppose that $μ_{{\tilde{Q}}_{i}^{l}} (x_{i}) \equiv μ_{{\tilde{X}}_{i}} (x_{i}) \cap μ_{{\tilde{F}}_{i}^{l}} (x_{i})$ , then

\begin{array}{l} μ_{{\tilde{B}}^{l}} (y) = μ_{{\tilde{G}}^{l}} (y) \cap {[\cup_{x_{1} \in X_{1}} μ_{{\tilde{Q}}_{1}^{l}} (x_{1})] \cap [\cup_{x_{2} \in X_{2}} μ_{{\tilde{Q}}_{2}^{l}} (x_{2})] \cap \dots \cap \\ [\cup_{x_{p} \in X_{p}} μ_{{\tilde{Q}}_{p}^{l}} (x_{p}) . \\ = μ_{{\tilde{G}}^{l}} (y) \cap {\cup_{x \in X} [\cap_{i = 1}^{p} μ_{{\tilde{Q}}_{i}^{l}} (x_{i})]} \end{array}

(7)

Here, we define $μ_{{\tilde{Q}}^{l}} (x) \equiv \cap_{i = 1}^{p} μ_{{\tilde{Q}}_{i}^{l}} (x_{i})$ and $F^{l} (x') \equiv \cup_{x \in X} [\cap_{i = 1}^{p} μ_{{\tilde{Q}}_{i}^{l}} (x_{i})] = \cup_{x \in X} μ_{{\tilde{Q}}^{l}} (x)$ , so that, $μ_{{\tilde{B}}^{l}} (y) = μ_{{\tilde{G}}^{l}} (y) \cap F^{l} (x'), y \in Y$ .

For the BMM structure-based IT2 non-singleton T2 FLSs, whose firing set $F^{l} (x')$ is a T1 FS, that is, $F^{l} (x') = [{\underline{f}}^{l} (x'), {\bar{f}}^{l} (x')]$ , in which

{\bar{f}}^{l} (x') = \cap_{i = 1}^{p} {\bar{f}}_{i}^{l} ({x'}_{i}) = \cap_{i = 1}^{p} {\bar{μ}}_{{\tilde{Q}}_{i}^{l}} ({\bar{x}}_{i, max}^{l})

(8)

and

{\underline{f}}^{l} (x') = \cap_{i = 1}^{p} {\underline{f}}_{i}^{l} ({x'}_{i}) = \cap_{i = 1}^{p} {{}^{-}μ}_{{\tilde{Q}}_{i}^{l}} ({{}^{-}x}_{i, max}^{l})

(9)

where

{\bar{f}}_{i}^{l} ({x'}_{i}) = sup_{x_{i}} {\bar{μ}}_{{\tilde{Q}}_{i}^{l}} (x_{i})

(10)

{\underline{f}}_{i}^{l} ({x'}_{i}) = sup_{x_{i}} {{}^{-}μ}_{{\tilde{Q}}_{i}^{l}} (x_{i})

(11)

Here, we let ${\bar{x}}_{i, max}^{l}$ and ${{}^{-}x}_{i, max}^{l}$ be the values of $x_{i}$ that are corresponding to $sup_{x_{i}} {\bar{μ}}_{{\tilde{Q}}_{i}^{l}} (x_{i})$ and $sup_{x_{i}} {{}^{-}μ}_{{\tilde{Q}}_{i}^{l}} (x_{i})$ , respectively.

For the rule ${\tilde{R}}^{l}$ , the fired output consequent set $μ_{{\tilde{B}}^{l}} (y)$ is a T1 FS, that is

μ_{{\tilde{B}}^{l}} (y) = \int_{b^{l} \in [{\underline{f}}^{l} \cap {\underline{μ}}_{{\tilde{G}}^{l}} (y), {\bar{f}}^{l} \cap {\bar{μ}}_{{\tilde{G}}^{l}} (y)]} 1 / b^{l}

(12)

where ${\bar{μ}}_{{\tilde{G}}^{l}} (y)$ and ${{}^{-}μ}_{{\tilde{G}}^{l}} (y)$ represent the upper and lower membership grades of $μ_{{\tilde{G}}^{l}} (y)$ .

Let $N$ of $M$ fuzzy rules in the systems be fired, then we combine all the fired output consequent FSs to obtain the output T1 FS, that is, $μ_{\tilde{B}} (y) = \cup_{l = 1}^{M} μ_{{\tilde{B}}^{l}} (y), y \in Y$ , so that

\begin{matrix} μ_{\tilde{B}} (y) = \\ \int_{b \in [{\underline{f}}^{1} \cap {{}^{-}μ}_{{\tilde{G}}^{1}} (y)] \lor \dots \lor [{\underline{f}}^{N} \cap {\underline{μ}}_{{\tilde{G}}^{N}} (y)], [{\bar{f}}^{1} \cap {\bar{μ}}_{{\tilde{G}}^{1}} (y)] \lor \dots \lor [{\bar{f}}^{N} \cap {\bar{μ}}_{{\tilde{G}}^{N}} (y)]} 1 / b^{l}, y \in Y \end{matrix}

(13)

Here, the sign ∫ does not denote the integral operation, but as the union overall admissible.

BMM structure-based IT2 FLSs and their BP algorithms

In this section, we provide how to use the BP algorithms to design and optimize the BMM structure-based IT2 FLSs. Most of the references proposed their optimizations just by showing the process with some datasets, while this paper gives mathematical forecasting formulas for two types of FLSs (Mamdani non-singleton T1 FLSs and BMM structure-based IT2 non-singleton T2 FLSs) to enhance their theory and application process.

Let a collection of $D - p$ input–output data pairs $(x^{(1)} : y^{(1)}), (x^{(2)} : y^{(2)}), \dots, (x^{(D - p)} : y^{(D - p)})$ from the data points be given, in which $D$ is the number of data points for training and $p$ is the number of antecedents. We first define the FLSs completely by the training data.

Here, we define the error function as $e^{(t)} = \frac{1}{2} [f_{ns 1} (x^{(t)}) - y^{(t)}]^{2} (t = 1, 2, \dots, n)$ , in which $f_{ns 1} (x^{(t)})$ represents the T1 FLSs output and $y^{(t)}$ is the desired output.

For the Mamdani T1 non-singleton FLSs, the antecedents and input measurements are all selected as Gaussian T1 MFs. The antecedent MF of T1 FS $F_{i}^{l}$ is the form: $μ_{F_{i}^{l}} (x_{i}) = \exp [- \frac{1}{2} {(\frac{x_{i} - m_{F_{i}^{l}}}{σ_{F_{i}^{l}}})}^{2}] (i = 1, \dots, p; l = 1, \dots, M)$ , and the input measurement MF T1 FS $X_{i}$ is the form: $μ_{X_{i}} (x_{i}) = \exp [- \frac{1}{2} {(\frac{x_{i} - {x'}_{i}}{σ_{X_{i}}})}^{2}] (i = 1, \dots, p; l = 1, \dots, M)$ . That is to say, $m_{F_{i}^{l}} (i = 1, \dots, p; l = 1, \dots, M)$ is the mean of antecedent T1 MF, $σ_{F_{i}^{l}} (i = 1, \dots, p; l = 1, \dots, M)$ is the standard deviation of antecedent T1 MF, and $σ_{X_{i}} (i = 1, \dots, p)$ is the standard deviation of input measurement T1 MF. According to the product t-norm, all these parameters can be tuned by the BP algorithms as the following formulas

\begin{matrix} m_{F_{i}^{l}} (t + 1) = m_{F_{i}^{l}} (t) - α_{m} \frac{\partial e^{(t)}}{\partial m_{F_{i}^{l}}} |_{t} = m_{F_{i}^{l}} (t) - α_{m} [f_{ns 1} (x^{(t)}) - y^{(t)}] \\ \times [{\bar{y}}^{l} (t) - f_{ns 1} (x^{(t)})] \times [\frac{x_{i}^{(t)} - m_{F_{i}^{l}} (t)}{σ_{X_{i}}^{2} (t) + σ_{F_{i}^{l}}^{2} (t)}] ϕ_{l} (x^{(t)}) \end{matrix}

(14)

\begin{matrix} {\bar{y}}^{l} (t + 1) = {\bar{y}}^{l} (t) - α_{{\bar{y}}^{l}} \frac{\partial e^{(t)}}{\partial {\bar{y}}^{l}} |_{t} \\ = {\bar{y}}^{l} (t) - α_{{\bar{y}}^{l}} [f_{ns 1} (x^{(t)}) - y^{(t)}] ϕ_{l} (x^{(t)}) \end{matrix}

(15)

\begin{matrix} σ_{F_{i}^{l}} (t + 1) = σ_{F_{i}^{l}} (t) - α_{σ} \frac{\partial e^{(t)}}{\partial σ_{F_{i}^{l}}} |_{t} = σ_{F_{i}^{l}} (t) - α_{σ} [f_{ns 1} (x^{(t)}) - y^{(t)}] \\ \times [{\bar{y}}^{l} (t) - f_{ns 1} (x^{(t)})] \times σ_{F_{i}^{l}} (t) \\ \times {[\frac{x_{i}^{(t)} - m_{F_{i}^{l}} (t)}{σ_{X}^{2} (t) + σ_{F_{i}^{l}}^{2} (t)}]}^{2} ϕ_{l} (x^{(t)}) \end{matrix}

(16)

\begin{matrix} σ_{X_{i}} (t + 1) = σ_{X_{i}} (t) - α_{X} \frac{\partial e^{(t)}}{\partial σ_{X}} |_{t} = σ_{X_{i}} (t) - α_{X} [f_{ns 1} (x^{(t)}) - y^{(t)}] \\ \times [{\bar{y}}^{l} (t) - f_{ns 1} (x^{(t)})] \times σ_{X} (t) \times {[\frac{x_{i}^{(t)} - m_{F_{i}^{l}} (t)}{σ_{X}^{2} (t) + σ_{F_{i}^{l}}^{2} (t)}]}^{2} ϕ_{l} (x^{(t)}) \end{matrix}

(17)

Here, $i$ represents the number of antecedents, and t is the number of iterations. In addition, ${\bar{y}}^{l}$ is the centroid of consequent T1 FS, and $α_{m}, α_{\bar{y}}, α_{σ}, α_{X}$ are all learning parameters. We turn all these parameters to obtain a minimum or relative minimum value of the error.

For the Mamdani non-singleton T2 FLSs with BMM structure, the output of IT2 FLSs is as

\begin{matrix} y_{BMM, IT 2} = a \frac{\sum_{l = 1}^{M} y^{l} {{}^{-}μ}_{{\tilde{B}}^{l}} (y^{l})}{\sum_{l = 1}^{M} {{}^{-}μ}_{{\tilde{B}}^{l}} (y^{l})} + b \frac{\sum_{l = 1}^{M} y^{l} {\bar{μ}}_{{\tilde{B}}^{l}} (y^{l})}{\sum_{l = 1}^{M} {\bar{μ}}_{{\tilde{B}}^{l}} (y^{l})} \\ = a \frac{\sum_{l = 1}^{M} y^{l} {\underline{f}}^{l}}{\sum_{l = 1}^{M} {\underline{f}}^{l}} + b \frac{\sum_{l = 1}^{M} y^{l} {\bar{f}}^{l}}{\sum_{l = 1}^{M} {\bar{f}}^{l}} \\ = a \sum_{l = 1}^{M} y^{l} ϕ_{l_{1}} (x) + b \sum_{l = 1}^{M} y^{l} ϕ_{l_{2}} (x) \end{matrix}

(18)

where $a$ and $b$ are two adjustable coefficients; in addition, $ϕ_{l_{1}} (x)$ and $ϕ_{l_{2}} (x)$ are two FBFs.

For further study, we define the specific footprint of uncertainties (FOUs) for MFs. Here, the antecedent primary MF is chosen as the form of Gaussian type with uncertain standard deviation, that is

\begin{matrix} μ_{i}^{l} (x_{i}) = \exp {- \frac{1}{2} {(\frac{x_{i} - m_{i}^{l}}{σ_{i}^{l}})}^{2}} (i = 1, \dots, p; \\ l = 1, \dots, M), σ_{i}^{l} \in [σ_{i 1}^{l}, σ_{i 2}^{l}] \end{matrix}

(19)

And the input measurement primary MF is also selected as the form of Gaussian type with uncertain standard deviation, that is

\begin{matrix} μ_{i} (x_{i}) = \\ \exp {- \frac{1}{2} {(\frac{x_{i} - {x'}_{i}}{σ_{i}})}^{2}}, (i = 1, \dots, p; l = 1, \dots, M), σ_{i} \in [σ_{i 1}, σ_{i 2}] \end{matrix}

(20)

The upper and lower firing strengths of the $l th$ fuzzy rule are as

\begin{matrix} {\bar{f}}^{l} = \cap_{i = 1}^{p} {\exp [- \frac{1}{2} \frac{{(x_{i} - m_{i}^{l})}^{2}}{σ_{i 2}^{2} + σ_{i 2}^{l^{2}}}]}, \\ {\underline{f}}^{l} = \cap_{i = 1}^{p} {\exp [- \frac{1}{2} \frac{{(x_{i} - m_{i}^{l})}^{2}}{σ_{i 1}^{2} + σ_{i 1}^{l^{2}}}]} \end{matrix}

(21)

where $m_{i}^{l} (i = 1, \dots, p; l = 1, 2, \dots, M)$ denotes the mean of primary MF antecedent T2 FS, $σ_{i 1}^{l} (i = 1, \dots, p; l = 1, \dots, M)$ represents the lower standard deviation of MF of antecedent T2 FS, $σ_{i 2}^{l} (i = 1, \dots, p; l = 1, \dots, M)$ is the upper standard deviation of MF of antecedent T2 FS, $σ_{i 1} (i = 1, \dots, p)$ denotes the lower standard deviation of MF of input measurement T2 FS, $σ_{i 2} (i = 1, \dots, p)$ represents the upper standard deviation of MF of input measurement T2 FS, and $y^{l} (l = 1, \dots, M)$ is the centroid of consequent IT2 FS.

The general structure of BP algorithms to tune all MF parameters is as

φ (t + 1) = φ (t) - α_{φ} \frac{\partial e^{(t)}}{\partial φ} |_{t}

(22)

in which $φ$ denotes any one design parameters of IT2 FLSs, and $α_{φ}$ is the learning parameter ( $α_{φ} = 0.2$ for initialization). The error function is defined as

e^{(t)} = \frac{1}{2} {[f_{ns 2 - 2} (x^{(t)}) - y^{(t)}]}^{2} (t = 1, 2, \dots, n)

(23)

Then all the parameters of Mamdani non-singleton T2 FLSs with BMM structure can be turned by the BP algorithms as the following formulas

\begin{matrix} m_{i}^{l} (t + 1) = m_{i}^{l} (t) - α_{m_{i}^{l}} \frac{\partial e^{(t)}}{\partial m_{i}^{l}} |_{t} \\ = m_{i}^{l} (t) - α_{m_{i}^{l}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ \times [y^{l} (t) - f_{ns 2 - 2} (x^{(t)})] \\ \times {a [\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 1}^{2} (t) + σ_{i 1}^{l^{2}} (t)}] ϕ_{l_{1}} (x^{(t)}) + b [\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 2}^{2} (t) + σ_{i 2}^{l^{2}} (t)}] ϕ_{l_{2}} (x^{(t)})} \end{matrix}

(24)

\begin{matrix} y^{l} (t + 1) = y^{l} (t) - α_{y^{l}} \frac{\partial e^{(t)}}{\partial y^{l}} |_{t} \\ = y^{l} (t) - α_{y^{l}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ [a ϕ_{l_{1}} (x^{(t)}) + b ϕ_{l_{2}} (x^{(t)})] \end{matrix}

(25)

\begin{matrix} σ_{i 1}^{l} (t + 1) = σ_{i 1}^{l} (t) - α_{σ_{i 1}^{l}} \frac{\partial e^{(t)}}{\partial σ_{i 1}^{l}} |_{t} \\ = σ_{i 1}^{l} (t) - α_{σ_{i 1}^{l}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ \times [y^{l} (t) - f_{ns 2 - 2} (x^{(t)})] \times σ_{i 1}^{l} (t) \times a \\ \times {[\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 1}^{2} (t) + σ_{i 1}^{l^{2}} (t)}]}^{2} ϕ_{l_{1}} (x^{(t)}) \end{matrix}

(26)

\begin{matrix} σ_{i 2}^{l} (t + 1) = σ_{i 2}^{l} (t) - α_{σ_{i 2}^{l}} \frac{\partial e^{(t)}}{\partial σ_{i 2}^{l}} |_{t} \\ = σ_{i 2}^{l} (t) - α_{σ_{i 2}^{l}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ \times [y^{l} (t) - f_{ns 2 - 2} (x^{(t)})] \times σ_{i 2}^{l} (t) \times b \\ \times {[\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 2}^{2} (t) + σ_{i 2}^{l^{2}} (t)}]}^{2} ϕ_{l_{2}} (x^{(t)}) \end{matrix}

(27)

\begin{matrix} σ_{i 1} (t + 1) = σ_{i 1} (t) - α_{σ_{i 1}} \frac{\partial e^{(t)}}{\partial σ_{i 1}} |_{t} \\ = σ_{i 1} (t) - α_{σ_{i 1}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ \times [y^{l} (t) - f_{ns 2 - 2} (x^{(t)})] \times σ_{i 1} (t) \times a \\ \times {[\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 1}^{2} (t) + σ_{i 1}^{l^{2}} (t)}]}^{2} ϕ_{l_{1}} (x^{(t)}) \end{matrix}

(28)

\begin{matrix} σ_{i 2} (t + 1) = σ_{i 2} (t) - α_{σ_{i 2}} \frac{\partial e^{(t)}}{\partial σ_{i 2}} |_{t} \\ = σ_{i 2} (t) - α_{σ_{i 2}} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \\ \times [y^{l} (t) - f_{ns 2 - 2} (x^{(t)})] \times σ_{i 2} (t) \times b \\ \times {[\frac{x_{i}^{(t)} - m_{i}^{l} (t)}{σ_{i 2}^{2} (t) + σ_{i 2}^{l^{2}} (t)}]}^{2} ϕ_{l_{2}} (x^{(t)}) \end{matrix}

(29)

\begin{matrix} a (t + 1) = a (t) - α_{a} \frac{\partial e^{(t)}}{\partial a} |_{t} \\ = a (t) - α_{a} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \sum_{l = 1}^{M} y^{l} ϕ_{l_{1}} (x^{(t)}) \end{matrix}

(30)

\begin{matrix} b (t + 1) = b (t) - α_{b} \frac{\partial e^{(t)}}{\partial b} |_{t} \\ = b (t) - α_{a} [f_{ns 2 - 2} (x^{(t)}) - y^{(t)}] \sum_{l = 1}^{M} y^{l} ϕ_{l_{2}} (x^{(t)}) \end{matrix}

(31)

where $α_{m_{i}^{l}}, α_{y^{l}}, α_{σ_{i 1}^{l}}, α_{σ_{i 2}^{l}}, α_{σ_{i 1}}, α_{σ_{i 2}}, a$ and $b$ are all learning parameters. Here, we choose $α_{σ_{i 1}^{l}} = α_{σ_{i 2}^{l}} = α_{σ_{i 1}} = α_{σ_{i 2}} = α_{σ}$ .

In the paper, we define two performance indices for evaluating the designed FLSs optimized with BP algorithms. And they are as root mean square error (RMSE; Chen and Wang, 2018) and the mean absolute percentage error (MAPE; Khosravi and Nahavandi, 2014). Then the two formulas are provided as

RMSE = \sqrt{\frac{1}{N - D - p} \sum_{i = D - p + 1}^{N - 2 p} {[y^{(i)} - f (x^{(i)})]}^{2}}

(32)

in which $f (x^{(i)})$ is the output of FLSs approaches, $y^{(i)}$ denotes the actual output, $N$ is the whole number of data points, $p$ represents the number of antecedents, and $D$ is the number of data points for training

MAPE = \frac{1}{N - D - p} \sum_{i = D - p + 1}^{N - 2 p} | \frac{f (x^{(i)}) - y^{(i)}}{y^{(i)}} | \times 100 %

(33)

Simulation studies

Data

Simulation studies employ both PMD samarium cobalt and alnico permanent magnetic temperature data to show the effectiveness of the proposed BP optimized FLSs forecasting approaches. Moreover, noises are added to the data to simulate the affect of uncertainty. Figure 2 provides the schematic view of PMD.

Figure 2.

Schematic view of PMD (Chen et al., 2018; Chen and Wang, 2017, 2019).

Simulation setup

First of all, we observe the temperature data of samarium cobalt permanent magnet. As shown in Figure 3, the unit of horizontal axis is millisecond (ms), while the unit of vertical axis is °C. The design and test of FLS forecasters are based on 1000 noisy data points: $x (1), x (2), \dots, x (1000)$ . We adopt the former 504 data points $x (1), x (2), \dots, x (504)$ to train, that is to say, to design the FLS forecasters; while the latter 496 data points $x (505), x (506), \dots, x (1000)$ are used to test the designs. BP algorithms are used for optimizing the parameters of FLS forecasters.

Figure 3.

(a) The temperature data of samarium cobalt permanent magnet, (b) training is completed by 500 input–output data pairs generated from $x (1), x (2), \dots, x (504)$ , and (c) testing is completed by 496 input–output data pairs generated from $x (505), x (506), \dots, x (1000)$ .

Figure 4 shows the simulation procedure for developing FLS forecasters and verifying their performances. In the paper, every four antecedents $x (n - 3), x (n - 2), x (n - 1), x (n)$ are used to forecast the next $x (n + 1)$ . Therefore, we design a one-step forecaster, whose mathematical model can be viewed as $x (n + 1) = f [x (n), x (n - 1), x (n - 2), x (n - 3)]$ . Two FSs are adopted for characterizing each antecedent, so that the total number of fuzzy rules is 16.

Figure 4.

Simulation procedure for developing FLS approaches.

We adopt the first 504 noisy data $x (1), x (2), \dots, x (504)$ for training, that is, for designing FLS forecasters. Then the rest 496 noisy data $x (505), x (506), \dots, x (1000)$ are used for testing. The temperature data of samarium cobalt permanent magnet (a), training data (b), and testing data (c) are shown in Figure 3.

Here, Gaussian MFs are selected for the Mamdani singleton T1 FLSs and Mamdani non-singleton T1 FLSs, and Gaussian primary MFs with uncertain standard deviation are selected for the Mamdani non-singleton T2 FLSs with BMM structure (Mamdani denotes the inference type for FLSs). For the Mamdani singleton T1 FLSs, each fuzzy rule is characterized by eight antecedent MF parameters and one consequent parameter. For the Mamdani non-singleton T1 FLSs, each fuzzy rule is characterized by eight antecedent MF parameters, four input measurement MF parameters (the standard deviation for each of the four Gaussian MFs), and one consequent parameter. Therefore, the total numbers of parameters for two types of T1 FLSs are 144 and 208, respectively. For the Mamdani non-singleton T2 FLSs with BMM structure, each fuzzy rule is characterized by 12 antecedent MF parameters (the upper and lower bounds on the standard deviation, and the mean for each of the four Gaussian MFs), 8 input measurement MF parameters (the upper and lower bounds on the standard deviation for each of the four Gaussian MFs), and 1 consequent parameter. So that the total number of design parameters for Mamdani non-singleton T2 FLSs with BMM structure is equal to $21 \times 16 + 2$ , that is, 338, where this two means the adjustable coefficients.

Here, we select the product t-norm. After 1000 epochs of BP iterations (for each iteration, 20 times of Monte Carlo simulations (Rezaie et al., 2007) are performed), the simulation graphs of forecasting results are provided as follows (Figures 5 –7).

Figure 5.

Simulation graph of Mamdani singleton T1 FLSs forecasting for samarium cobalt permanent magnet temperature data.

Figure 6.

Simulation graph of Mamdani non-singleton T1 FLS forecasting for samarium cobalt permanent magnet temperature data.

Figure 7.

Simulation graph of Mamdani non-singleton T2 FLSs with BMM structure forecasting for samarium cobalt permanent magnet temperature data.

Next, let us consider the temperature data of alnico permanent magnet as the second case. As shown in Figure 8, the design and test of FLS forecasters are based on another 1000 noisy data points: $x' (1), x' (2), \dots, x' (1000)$ (the notation of prime means that the data are different from the former). Beginning from $x' (1), x' (2), \dots, x' (4)$ , each four antecedents are fed into the FLS forecasters to forecast the next one. We use the former 504 data points $x' (1), x' (2), \dots, x' (504)$ to design the FLS forecasters, and the latter 496 data points to test the design.

Figure 8.

(a) The temperature data of alnico permanent magnet, (b) training is completed by 500 input–output data pairs generated from $x' (1), \dots, x' (504)$ , and (c) testing is completed by 496 input–output data pairs generated from $x' (505), \dots, x' (1000)$ .

After 1000 epochs of BP iterations (for each iteration, 20 times of Monte Carlo simulations are performed), the simulation graphs of forecasting results are given in Figures 9 –11.

Figure 9.

Simulation graph of Mamdani singleton T1 FLS forecasting for alnico permanent magnet temperature data.

Figure 10.

Simulation graph of Mamdani non-singleton T1 FLS forecasting for alnico permanent magnet temperature data.

Figure 11.

Simulation graph of Mamdani non-singleton T2 FLSs with BMM structure forecasting for alnico permanent magnet temperature data.

For these three types of FLS forecasters, all the parameters are tuned by the BP algorithms. Training and testing are performed simultaneously according to 1000 epochs. In order to measure the performances of three types of FLS forecasters comprehensively, we take the average of the values of these RMSEs and MAPEs in terms of 20 Monte Carlo simulations; then the mean of $RMS E_{s 1} (BP)$ , $RMS E_{ns 1} (BP)$ , $RMS E_{ns 2 - 2} (BP)$ , $MAP E_{s 1} (BP)$ , $MAP E_{ns 1} (BP)$ , and $MAP E_{ns 2 - 2} (BP)$ for the above two examples is provided in Figures 12 and 13.

Figure 12.

Mean of RMSE and MAPE for case 1: (a) RMSE and (b) MAPE.

Figure 13.

Mean of RMSE and MAPE for case 2: (a) RMSE and (b) MAPE.

Simulation result analysis

Observing from the case of samarium cobalt permanent magnet temperature data (see Figures 5 –7 and 12), we can find the following:

As for the three types of Mamdani-type FLSs (singleton T1 FLSs, non-singleton T1 FLSs, and non-singleton IT2 FLSs with BMM structure), the graphs of average of RMSEs are all first monotone decreasing and then approach relatively convergence points (in the process of 1000 epochs of iteration, 20 times of Monte Carlo simulations).

The graphs of average of MAPEs are all first monotone decreasing and then approach relatively convergence points (in the process of 1000 epochs of iteration, 20 times of Monte Carlo simulations).

For the above-defined two error indices, the proposed non-singleton IT2 FLSs with BMM structure outperform singleton T1 FLSs and non-singleton T1 FLSs. Furthermore, the IT2 FLSs obtain the fastest convergence speed.

Observing from the case of alnico permanent magnet temperature data (see Figures 9 –11 and 13), we can find the following:

As for the three types of Mamdani-type FLSs, the graphs of RMSE are all first monotone decreasing and then approach relatively convergence points (in the process of 1000 epochs of iteration, 20 times of Monte Carlo simulations).

The graphs of average of MAPEs are all first monotone decreasing and then approach relatively convergence points (in the process of 1000 epochs of iteration, 20 times of Monte Carlo simulations).

As there may exist more uncertainties for this case, the data fluctuation ranges are comparatively larger than the former case. For the above-defined two error indices, the proposed non-singleton IT2 FLSs with BMM structure outperform singleton T1 FLSs and non-singleton T1 FLSs. In addition, the IT2 FLSs obtain the fastest convergence speed.

Based on the above analysis, we can obtain strong evidences that the proposed BP optimized IT2 FLSs with BMM structure are better choice for forecasting than both singleton T1 FLSs and non-singleton T1 FLSs. Especially for forecasting problem with uncertainties, it is reasonable to conclude that Mamdani-type IT2 FLSs with BMM structure may have stronger flexibility and approximation ability than their T1 counterparts. However, it is much more difficult to design and apply IT2 FLSs as the number of degrees of freedom is greater than T1 FLSs. Forecasting studies by designing and applying IT2 FLSs is more applicable than T1 FLSs.

Conclusion

The paper designs a type of non-singleton IT2 FLSs according to the BP algorithms. We choose the antecedents, consequents, and input measurement primary MFs of fuzzy rules as Gaussian type with uncertain standard deviations. Simulation studies and convergence analysis based on the datasets of samarium cobalt and alnico permanent magnet temperature are used to show the reliability of the proposed BP optimized Mamdani-type IT2 FLSs with BMM structure. Compared with two types of Mamdani-type T1 FLSs, the forecasting performances of Mamdani-type IT2 FLSs are better. So it is reasonable to design and optimize high-level FLSs by evolutionary algorithms for real applications.

There are so many interesting works lie ahead, including studying the center-of-set TR (Khanesar et al., 2017) of T2 FLSs, initializing the search space partition for centroid TR algorithms (Chen, 2019a, 2019b; Chen et al., 2020; Chen and Wang, 2018; Greenfield and Chiclana, 2013; Li et al., 2018; Liu and Mendel, 2011; Mendel and Liu, 2013; Ontiveros-Robles et al., 2017; Wu, 2013), investigating global optimization algorithms such as simulated annealing and ant colony optimization for optimizing the parameters of T2 FLSs, and further developing and deploying T2 FLSs on forecasting, control, and fuzzy identification problems (Castillo et al., 2016, 2019; Cervantes and Castillo, 2015; Gaxiola et al., 2016; Hernandez et al., 2015; Ontiveros-Robles et al., 2018; Tong and Li, 2010, 2014) affected by uncertainties. Future studies will be primarily focused on T2 FLS design and applications.

Footnotes

Acknowledgements

The author is very grateful to Professor Jerry Mendel, who has provided the author some valuable suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The paper is sponsored by the National Natural Science Foundation of China (No. 61973146, No. 61773188, No. 61903167, and No. 61803189), the Youth Fund of Education Department of Liaoning Province (LJKQZ2021143), the Doctoral Start-up Foundation of Liaoning Province (No. 2021-BS-258), and the Talent Fund Project of Liaoning University of Technology (No. xr2020002).

ORCID iD

Yang Chen

References

Barbounis

Theocharis

(2007) Locally recurrent neural networks for wind speed prediction using spatial correlation. Information Sciences 177(24): 5775–5797.

Barkat

Tlemcani

Nouri

(2011) Noninteracting adaptive control of PMSM using interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 19(5): 925–936.

Bernardo

Hagras

Tsang

(2013) A genetic type-2 fuzzy logic based system for the generation of summarized linguistic predictive models for financial applications. Soft Computing 17(12): 2185–2201.

Biglarbegian

Melek

Mendel

(2010) On the stability of interval type-2 TSK fuzzy logic systems. IEEE Transactions on Systems, Man and Cybernetics. Part B, Cybernetics 40(3): 798–818.

Biglarbegian

Melek

Mendel

(2011a) Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: Theory and experiments. IEEE Transactions on Industrial Electronics 58(4): 1371–1384.

Biglarbegian

Melek

Mendel

(2011b) On the robustness of type-1 and interval type-2 fuzzy logic systems in modeling. Information Sciences 181(7): 1325–1347.

Castillo

Amador-Angulo

Castro

, et al. (2016) A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems. Information Sciences 354(c): 257–274.

Castillo

Melin

(2012) A review on the design and optimization of interval type-2 fuzzy controllers. Applied Soft Computing 12(4): 1267–1278.

Castillo

Melin

Ontiveros

, et al. (2019) A high-speed interval type 2 fuzzy system approach for dynamic parameter adaptation in metaheuristics. Engineering Applications of Artificial Intelligence 85: 666–680.

10.

Cervantes

Castillo

(2015) Type-2 fuzzy logic aggregation of multiple fuzzy controllers for airplane flight control. Information Sciences 324: 247–256.

11.

Chen

(2019a) Study on centroid type-reduction of interval type-2 fuzzy logic systems based on noniterative algorithms. Complexity 2019: 7325053.

12.

Chen

(2019b) Study on sampling based discrete Nie-Tan algorithms for computing the centroids of general type-2 fuzzy sets. IEEE Access 7(1): 156984–156992.

13.

Chen

Wang

(2017) Forecasting by general type-2 fuzzy logic systems optimized with QPSO algorithms. International Journal of Control, Automation and Systems 15(6): 2950–2958.

14.

Chen

Wang

(2018) Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted enhanced Karnik-Mendel algorithms. Soft Computing 22(4): 1361–1380.

15.

Chen

Wang

(2019) Forecasting by designing Mamdani general type-2 fuzzy logic systems optimized with quantum particle swarm optimization algorithms. Transactions of the Institute of Measurement and Control 41(10): 2886–2896.

16.

Chen

Wang

Ning

(2018) Forecasting by TSK general type-2 fuzzy logic systems optimized with genetic algorithms. Optimal Control Applications & Methods 39(1): 393–409.

17.

Chen

Wang

Tong

(2016) Forecasting studies by designing Mamdani interval type-2 fuzzy logic systems: With the combination of BP algorithms and KM algorithms. Neurocomputing 174(b): 1133–1146.

18.

Chen

Lan

(2020) Study on reasonable initialization enhanced Karnik-Mendel algorithms for centroid type-reduction of interval type-2 fuzzy logic systems. AIMS Mathematics 5(6): 6149–6168.

19.

Gao

Huang

Wang

, et al. (2014) Exploiting a modified gray model in back propagation neural networks for enhanced forecasting. Cognitive Computation 6(3): 331–337.

20.

Gaxiola

Melin

Valdez

, et al. (2016) Optimization of type-2 fuzzy weights in backpropagation learning for neural networks using GAs and PSO. Applied Soft Computing 38: 860–871.

21.

Gonzalez

Castro

Melin

, et al. (2016) An edge detection method based on generalized type-2 fuzzy logic. Soft Computing 20(2): 773–784.

22.

Greenfield

Chiclana

(2013) Accuracy and complexity evaluation of defuzzification strategies for the discretised interval type-2 fuzzy set. International Journal of Approximate Reasoning 54(8): 1013–1033.

23.

Gusev

Burkovskii

(2013) A neural network forecasting model for integrated economic indicators. Automation and Remote Control 74(9): 1567–1572.

24.

Hagras

(2007) Type-2 FLSs: A new generation of fuzzy controllers. IEEE Computational Intelligence Magazine 2(1): 30–43.

25.

Hagras

Wagner

(2012) Towards the wide spread use of type-2 fuzzy logic systems in real world applications. IEEE Computational Intelligence Magazine 7(3): 14–24.

26.

Hernandez

MdlA

Melin

Méndez

, et al. (2015) A hybrid learning method composed by the orthogonal least-squares and the back-propagation learning algorithms for interval A2-C1 type-1 non-singleton type-2 TSK fuzzy logic systems. Soft Computing 19(3): 661–678.

27.

Hsu

Juang

(2013) Evolutionary robot wall-following control using type-2 fuzzy controller with species-de-activated continuous ACO. IEEE Transactions on Fuzzy Systems 21(1): 100–112.

28.

Kaufman

Gupta

(1985) Introduction to Fuzzy Arithmetic Theory and Applications. New York: Van Nostrand Reinhold.

29.

Khanesar

Jalalian

Kaynak

(2017) Improving the speed of center of set type-reduction in interval type-2 fuzzy systems by eliminating the need for sorting. IEEE Transactions on Fuzzy Systems 25(5): 1193–1206.

30.

Khosravi

Nahavandi

(2014) Load forecasting using interval type-2 fuzzy logic systems: Optimal type reduction. IEEE Transactions on Industrial Informatics 10(2): 1055–1063.

31.

Khosravi

Nahavandi

Creighton

, et al. (2012) Interval type-2 fuzzy logic systems for load forecasting: A comparative study. IEEE Transactions on Power Systems 27(3): 1274–1282.

32.

Lees

Wang

Hagras

(2010) Type-2 fuzzy ontology and its application to personal diabetic-diet recommendation. IEEE Transactions on Fuzzy Systems 18(2): 316–328.

33.

John

Coupland

, et al. (2018) On Nie-Tan operator and type-reduction of interval type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 26(2): 1036–1039.

34.

Liu

Mendel

(2011) Connect Karnik-Mendel algorithms to root-finding for computing the centroid of an interval type-2 fuzzy set. IEEE Transactions on Fuzzy Systems 19(4): 652–665.

35.

Mehdi

Alfred

Jonathan

(2016) Drought forecasting using Markov chain model and artificial neural networks. Water Resources Management 30(7): 2245–2259.

36.

Mendel

(2001) Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Englewood Cliffs, NJ: Prentice Hall.

37.

Mendel

(2004) Computing derivatives in interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 12(1): 84–98.

38.

Mendel

(2013) On KM algorithms for solving type-2 fuzzy set problems. IEEE Transactions on Fuzzy Systems 21(3): 426–446.

39.

Mendel

Liu

(2013) Simplified interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 21(6): 1056–1069.

40.

Méndez

Hernandez

MdlA

(2013) Hybrid learning mechanism for interval A2-C1 type-2 non-singleton Takagi-Sugeno-Kang fuzzy logic systems. Information Sciences 220(1): 149–169.

41.

Mendoza

Melin

Castillo

(2009) Interval type-2 fuzzy logic and modular networks for face recognition applications. Applied Soft Computing 9(4): 1377–1387.

42.

Niewiadomski

(2010) On finity, countability, cardinalities, and cylindric extensions of type-2 fuzzy sets in linguistic summarization of databases. IEEE Transactions on Fuzzy Systems 18(3): 532–545.

43.

Ontiveros-Robles

Melin

Castillo

(2017) New methodology to approximate type-reduction based on a continuous root-finding Karnik Mendel algorithm. Algorithms 10(3): 77–96.

44.

Ontiveros-Robles

Melin

Castillo

(2018) Comparative analysis of noise robustness of type 2 fuzzy logic controllers. Kybernetika 54(1): 175–201.

45.

Pany

Ghoshal

(2015) Dynamic electricity price forecasting using local linear wavelet neural network. Neural Computing and Applications 26(8): 2039–2047.

46.

Rezaie

Amalnik

Gereie

, et al. (2007) Using extended Monte Carlo simulation method for the improvement of risk management: Consideration of relationships between uncertainties. Applied Mathematics and Computation 190(2): 1492–1501.

47.

Safarinejadian

Ghane

Monirvaghefi

(2015) Fault detection in non-linear systems based on type-2 fuzzy logic. International Journal of Systems Sciences 45(3): 394–404.

48.

Tao

Taur

Chang

, et al. (2012) Simplified type-2 fuzzy sliding controller for wing rocket system. Fuzzy Sets and Systems 207(16): 111–129.

49.

Tong

(2010) Robust adaptive fuzzy backstepping output feedback tracking for nonlinear system with dynamic uncertainties. Science China Information Sciences 53(2): 307–324.

50.

Tong

(2014) Observer-based adaptive fuzzy backstepping control of uncertain pure-feedback systems. Science China Information Sciences 57(1): 1–14.

51.

Wang

Chen

(2018) Study on permanent magnetic drive forecasting by designing Takagi Sugeno Kang type interval type-2 fuzzy logic systems. Transactions of the Institute of Measurement and Control 40(6): 2011–2023.

52.

(2013) Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: Overview and comparisons. IEEE Transactions on Fuzzy Systems 21(1): 80–99.

53.

Zhang

Wang

(2015) State estimation of discrete-time Takagi-Sugeno fuzzy systems in a network environment. IEEE Transactions on Cybernetics 45(8): 1525–1536.

54.

Zhang

Shi

Wang

(2014) On energy-to-peak filtering for nonuniformly sampled nonlinear systems: A Markovian jump system approach. IEEE Transactions on Fuzzy Systems 22(1): 212–222.