The Enhanced Wagner–Hagras OLS–BP Hybrid Algorithm for Training IT3 NSFLS-1 for Temperature Prediction in HSM Processes

Abstract

This paper presents (a) a novel hybrid learning method to train interval type-1 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-1), (b) a novel method, named enhanced Wagner–Hagras (EWH) applied to IT3 NSFLS-1 fuzzy systems, which includes the level alpha 0 output to calculate the output $y$ alpha using the average of the outputs $y$ alpha $k$ instead of their weighted average, and (c) the novel application of the proposed methodology to solve the problem of transfer bar surface temperature prediction in a hot strip mill. The development of the proposed methodology uses the orthogonal least square (OLS) method to train the consequent parameters and the backpropagation (BP) method to train the antecedent parameters. This methodology dynamically changes the parameters of only the level alpha 0, minimizing some criterion functions as new information becomes available to each level alpha $k$. The precursor sets are type-2 fuzzy sets, the consequent sets are fuzzy centroids, the inputs are type-1 non-singleton fuzzy numbers with uncertain standard deviations, and the secondary membership functions are modeled as two Gaussians with uncertain standard deviation and the same mean. Based on the firing set of the level alpha 0, the proposed methodology calculates each firing set of each level alpha $k$ to dynamically construct and update the proposed EWH IT3 NSFLS-1 (OLS–BP) system. The proposed enhanced fuzzy system and the proposed hybrid learning algorithm were applied in a hot strip mill facility to predict the transfer bar surface temperature at the finishing mill entry zone using, as inputs, (1) the surface temperature measured by the pyrometer located at the roughing mill exit and (2) the time taken to translate the transfer bar from the exit of the roughing mill to the entry of the descale breaker of the finishing mill. Several fuzzy tools were used to make the benchmarking compositions: type-1 singleton fuzzy logic systems (T1 SFLS), type-1 adaptive network fuzzy inference systems (T1 ANFIS), type-1 radial basis function neural networks (T1 RBFNN), interval singleton type-2 fuzzy logic systems (IT2 SFLS), interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), type-2 ANFIS (IT2 ANFIS), IT2 RBFNN, general singleton type-2 fuzzy logic systems (GT2 SFLS), general type-1 non-singleton type-2 fuzzy logic systems (GT2 NSFLS-1), interval singleton type-3 fuzzy logic systems (IT3 SFLS), and interval type-1 non-singleton type-3 fuzzy systems (IT3 NSFLS-1). The experiments show that the proposed EWH IT3 NSFLS-1 (OLS–BP) system presented superior capability to learn the knowledge and to predict the surface temperature with the lower prediction error.

1. Introduction

Interval type-3 (IT3) fuzzy logic systems (FLS) represent a very useful technology according to the state-of the-art literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. Nowadays, the implementation of IT3 FLS in real life problems is a blank field given the complications presented in this model, which are analogous to the general type-2 (GT2) FLS based on the definition in [32]:

Definition 1. 

The type-3 FLS is the generalization of the type-2 FLS with more capacity to manage uncertainties. In T3 FLS systems, the secondary membership function (MF) is also a type-2 MF. Then, the upper and lower bounds of memberships are not constant compared to the type-2 MFs. These features cause more uncertainty and can be handled by type-3 MFs [32] (p. 154).

According to the previous definition and the analogy between the GT2 and IT3 systems, both adhere to the mathematical and methodological principles and to the challenges, difficulties, strengths, and weaknesses that authors have defined as complications to face this class of systems [5]. A comprehensive list of challenges to be faced is presented in [41] and are shown in Table 1.

Table 1. Difficulties of GT2 model adapted from [41].

Difficulties	References
Implementation	[42]
Use in practice	[42]
Information is non-functional	[43]
Information is not helpful	[43]
Information not necessary	[43]
Complex learning process	[44,45,46,47,48]
Heavy computation	[44,47,48,49,50,51]
Complexity in the defuzzification	[44,51,52]
Exhaustive computational time	[44,47,48,49,50,51]
Not practical to use	[44]
Method iterative and algorithmic	[53]
Determination of the number of levels-${\alpha }_k$	[49]

A brief survey of recent applications demonstrated that these are only from the theoretical point of view of IT3 singleton fuzzy logic systems (IT3 SFLS) [3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,31,32,33,34,35,36,37,38,39,40] and of interval type-1 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-1) [5,28,29]. This technology presents some challenges and complications in the design and implementation processes; i.e., in [28] the development of a new flowmeter fault detection approach based on optimized IT3 NSFLS-1 with type-1 non-singleton inputs is presented. The introduced method is implemented on an experimental gas industry plant. The faults are detected by the comparison of measured and estimated signals. According to the authors, the level of non-singleton fuzzification and membership parameters are tuned by a maximum correntropy (MC) unscented Kalman filter (KF), and the rule parameters are learned by correntropy KF (CKF) with fuzzy kernel size.

In contrast to the recent developments on automata, drones, and automated remote vehicles (ARVs), among others, which require adaptation, learning, and tuning to obtain the necessary knowledge for adaptation to the changing environments, the applications of IT3 are limited and their analogy with GT2 systems exists, as documented in [5], e.g., the GT2 NSFLS-1 is used as a controller to control and balance a two-wheel mobile robot [54]. The GT2 NSFLS-1 model is used in a proportional, integral, and derivative (PID) controller to obtain effectiveness and robustness in a plan controller affected by external disturbances [55].

The GT2 NSFLS-1 model is used to manage an efficient and energy-conserving permanent magnetic drive [56]. In [57], the GT2 NSFLS-1 is proposed to test and to provide a theoretical framework using the enhanced Karnik–Mendel (KM) algorithm and the Nie-Tan algorithm to see their accuracy. In [58], an adaptive GT2 type-1 non-singleton fuzzy neural network control for motion balance is presented, wherein it is used to adjust a power-line inspection robot. In [59], the authors present GT2 NSFLS-1 classifiers for medical diagnosis. A medical application for regulating glucose levels is proposed in [60]. In [61], a model for synchronizing chaotic systems affected by external disturbances is presented.

The difficulties presented in Table 1 apply in the case of the GT2 and IT3 singleton fuzzy systems in both the Mandami and Takagi–Sugeno–Kang (TSK) models, and it is remarkable that this happens in the singleton form, which is the simplest or most primitive form among fuzzy systems, as seen in [41,42,43,44,45,47,48,49,50,51,53,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89] for GT2 FLS and [90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116] for IT3 FLS. In contrast, in the case of the IT3 NSFLS-1 systems, there are only a limited number of applications [19,20,29,40] and, for interval type-2 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-2), there is only one, [5]. In the up-to-date modern literature, the reference to IT3 NSFLS-1 and IT3 NSFLS-2 is practically nonexistent, but, in contrast, searching the synonym (generalized type-2 non-singleton) for this technology of knowledge acquisition shows that, from 2021 onwards, there are 44 papers dealing with IT3 FLS systems. There are 39 publications named along the lines of “shadowed type-2” fuzzy systems. Using “knowledge acquisition” as a search prompt, one is returned 10 papers dealing with learning, 8 papers with tuning, 11 papers with adaptation, and only 4 papers with updating, as shown on Table 2. Table 3 shows the literature on IT3 FLS in their singleton and non-singleton versions, with 52 papers of type-3 fuzzy logic systems. For knowledge acquisition, there are 4 papers with hybrid learning, 38 papers with learning, 33 papers with tuning, 30 papers with adaptation, and only 5 papers with updating, considering that any of these terms may be mentioned in the same paper.

Table 2. Survey of techniques used to train the GT2 FLS models.

R	GT2		Optimization Model	Knowledge Acquisition Designation				System Designation
	S	N		L	T	A	U	GT2	Generalized Type-2	Shadowed Type-2
[15]		X	Robustness analysis		X					X
[44]	X		Ordered weighted averaging (OWA)	X			X	X
[45]	X		Data-driven			X		X
[47]			Kalman filters	X		X		X
[48]	X		Artificial neural networks (ANN)			X		X
[52]	X		Recursive least squares (RLS), Gradient-based method, hybrid ANN to optimize clustering	X				X
[55]		X	Social spider optimization				X	X
[58]			Particle swarm optimization (PSO)			X		X
[60]	X		Biogeography-based optimization (BBO)	X				X
[63]	X		Least square estimator (LSE), Teaching learning-based optimization (TLBO)			X		X
[71]			Searching algorithms	X	X			X
[72]	X	X	Ant lion optimizer				X	X		X
[78]	X		Hybrid differential evolution algorithm			X		X
[81]		X	Harmony search			X			X	X
[82]		X	Support vector machine (SVM), decision trees, ANN, bagging and boosting, bagging, boosting, GD, fuzzy entropy, PSO	X	X	X	X			X
[83]		X	BP and RLS	X		X			X
[84]		X	Lyapunov function	X	X	X		X	X
[85]		X	Kernel ridge regression (KRR)	X					X
[86]		X	Hierarchically stacked though, gradient descent (GD), gaussian kernel, SVM	X	X	X			X
[87]		X	Multi-objective optimization	X				X
[89]		X	Tuning laws	X	X	X		X

R refers to reference number, GT2 to General type 2 system, S to singleton, N to type-1 non-singleton, L to learning, T to tuning, A to adaptation, and U to updating; “X” means that the paper of the reference contains this description or characteristic.

Table 3. IT3 FLS systems.

Ref.	IT3 System			Learning Algorithm	Knowledge Acquisition Designation
	S	N-1	N-2		Hybrid	L	A	U	T
[1]	X			Classification system does not show a learning algorithm or does not need it		X	X		X
[2]	X			Theoretical paper for modelling and comparing the IT3 and IT2 systems does not involve learning	X	X	X		X
[4]	X			Differential evolution		X	X		X
[5]			X	GD	X	X	X	X	X
[7]	X			Empirical knowledge of experts combined with a trial-and-error approach		X	X		X
[8]	X			Fractal dimensions		X	X		X
[9]	X			Statistical measures, fuzzy c-means clustering, and granular computing used to construct the model not for learning				X
[11]	X			Response aggregation		X	X		X
[12]	X			Backpropagation with momentum learning		X	X		X
[13]	X			Specific adaptation law		X	X		X
[14]	X			Fractional-order model based on restricted Boltzmann machine (RBM) and deep learning contrastive divergence (CD)		X			X
[15]	X			Pitch adjustment rate (PArate) parameter in the original harmony search algorithm (HS)		X			X
[18]		X		Upper bound of approximate error (AE)
[19]		X		Fractional order		X			X
[22]	X			Fuzzy c-regression model clustering algorithm			X		X
[25]	X			Deep reinforcement learning (DRL)		X			X
[26]	X			Unscented Kalman filter (CUKF)	X	X	X	X	X
[27]	X			Surge-guided line-of-sight (SGLOS) and auxiliary dynamics		X	X		X
[28]		X		MC and UKF		X	X		X
[33]	X			Specific control law		X
[39]	X			UKF		X
[40]		X		Lyapunov adaptation rules			X		X
[91]	X			Hybrid learning	X	X
[92]	X			Robust and adaptive command-filtered backstepping control scheme, adaptive laws	X	X
[93]	X			Survey, not a theoretical paper nor an application or development		X	X		X
[95]	X			Bacterial foraging optimization algorithm		X			X
[96]	X			Does not have learning as a classification model		X			X
[97]	X			Spherical fuzzy		X			X
[98]	X			Weighted least square (WLS)					X
[99]	X			Actor-critic learning control algorithm associated with Lyapunov stability examination		X	X		X
[100]	X			+ Nonlinear model predictive control (NMPC)		X			X
[101]	X			+ Marine predator		X	X	X	X
[102]	X			+ Maximum power point tracking (MPPT), genetic algorithm					X
[103]	X			+ Differential evolution			X		X
[104]	X			+ Harmony search		X	X
[105]	X			+ Harmony search and the differential evolution		X	X
[106]	X			Not learning algorithm, the parameters are changed manually		X	X
[107]	X			Terminal sliding mode controller		X	X
[108]	X			Adaptive sliding mode disturbance observer, adaptive laws, output with continuous-time linear systems.	X	X	X
[109]	X			Retained region approach (granulation)		X	X
[110]	X			Multi-objective artificial hummingbird algorithm (MOAHA)		X		X
[111]	X			Enhanced Kalman filter (EKF)				X
[112]	X			* Survey of methods is not an application		X
[113]	X			Extended state space model-based constrained predictive functional control		X
[114]	X			+ Event-triggered control law		X
[115]	X			+ Cartograms to visualize both the expansion and spread		X
[116]	X			+ Non-linear time series		X

S for singleton, N-1 for type-1 non-singleton, N-2 for type-2 non-singleton, L for learning, T for tuning, A for adaptation, and U for updating; + indicates that the learning algorithm is only used to obtain the information for the rule base or for optimization; “X” means that the paper of the reference contains this description or characteristic; * means that this paper is a literature review.

The few applications found in the state-of-the-art literature, the difficulties in optimizing the models, the aspect of multiple calculation for obtaining several planes as mentioned in [82], and the requirement of iterative methods to train the model have led researchers to use different models that stand out principally in GT2 SFLS systems for acquiring knowledge, learning, and tuning in their different definitions.

To the best of the authors’ knowledge, studies of GT2 NSFLS-1 and IT3 NSFLS-1 that use the OLS–BP hybrid learning mechanism as a training method have not been found in the state-of-the-art literature. However, there are publications presented elsewhere referring to the IT2 Mamdani FLS [117,118,119,120] and to the IT2 TSK FLS [121,122] using the proposed hybrid OLS–BP mechanism.

As mentioned earlier, the intention of this article is to present and discuss the proposed OLS–BP hybrid learning algorithm for antecedent and consequent parameter tuning of the novel enhanced Wagner–Hagras (EWH) IT3 NSFLS-1 system and to demonstrate the realizability of its implementation in a real industrial hot strip mill (HSM) application. In this paper, the uncertainty of the inputs is modeled as type-1 non-singleton, the uncertainty of the primary MF of the fuzzifier is modeled as type-2 non-singleton using Gaussians with uncertain mean, and the uncertainty of the secondary MFs is transformed from a type-2 non-singleton with uncertain deviation model to a type-1 non-singleton model.

The main contributions of this paper are:

• The detailed and novel mathematical formulation of the novel hybrid OLS–BP training algorithm applied to the novel EWH IT3 NSFLS-1 fuzzy logic systems.
• A more precise, economical, and novel method to estimate the final value of the IT3 fuzzy logic systems.
• Using a novel method to construct the EWH IT3 NSFLS-1 system with a dynamical structure.
• To the authors’ best knowledge, this is the first time that a hybrid EWH IT3 NSFLS-1 (OLS–BP) fuzzy system is applied to predict the transfer bar surface temperature at the entry zone of the finishing scale breaker of an HSM.

This work is organized as follows. Section 2 presents the foundations for the proposed EWH IT3 FLS system, the BP, and the OLS training methods to allow the reader to contextualize the methodology presented in the same section. Section 3 presents the application and validation of the performance of the proposed methodology applied to the temperature prediction of the transfer table of the hot strip mill facility, and the analysis of the results obtained in the application. Finally, Section 4 provides the conclusions.

2. Materials and Methods

2.1. A New Construction and Calculation of the WH IT3 NSFLS-1 System

The primary foundation for IT3 systems is the uncertainty the horizontal level-${\alpha }_k$ presents to its vertical location or its secondary membership value, ${\mu }_{\tilde{A}}\left(x,u\right)=f_x\left(u\right)={\alpha }_k$, as presented in Figure 1. In the IT3 systems, the interval value, $\left[{\underset{\_}{\alpha }}_k{,\overline{\alpha }}_k\right]$, represents this additional uncertainty. Geometrically, as in [123], this uncertainty is modeled to be between the horizontal levels-${\underset{\_}{\alpha }}_k$ and ${\overline{\alpha }}_k$.

Figure 1. Geometrical view of the IT3 NSFLS-1 with type-1 non-singleton inputs. $x$ is the input variable, u is the primary membership function of $x$, and ${\mu }_{\tilde{A}}\left(x,u\right)=f_x\left(u\right)={\alpha }_k$ is the secondary membership function of $x$ and $u$.

According to the modeling of WH GT2 Mamdani fuzzy systems, which utilize the type reduction center sets and the end-point defuzzification average, Ref. [123], Equation (1), the Wagner–Hagras (WH) IT3 NSFLS-1 can be calculated with more economical and precise results using Equation (2), with $q=1,2,\dots ,p$ being the number of input variables, the number of rules being $i=1,2,\dots ,M$, and $k=1,2,\dots ,N$ being the number of the horizontal levels-${\alpha }_k$.

The classic WH GT2 system uses the weighted average of the contribution of each level:

$$f_{WHIT3NSFLS-1}\left(x^{\prime }\right)=y_{WH-3}=y_{WH\alpha }=\frac{{\sum }_{k=1}^{k_{max}}{\alpha }_k{y}_{{\alpha }_k}}{{\sum }_{k=1}^{k_{max}}{\alpha }_k}$$

(1)

This methodology uses the average of the contribution of each level, which includes the horizontal level-${\alpha }_0$ or IT2${\alpha }_0$ FLS output, $y_{{\alpha }_0}$.

Remark 1. 

Equation (2) presents one of the novelties of this paper, which represents an enhancement of the Wagner–Hagras model by adding the level-${\alpha }_0$, which provides the basis for the evaluation of the overall IT3 system and determines its performance, as in the case of the previous IT2 model.

$$f_{EWHIT3NSFLS-1}\left(x^{\prime }\right)=y_{\alpha }=\frac{{\sum }_{k=0}^N{y}_{{\alpha }_k}}{N+1}$$

(2)

$$y_{{\alpha }_k}=\left[\frac{y_{l{\alpha }_k}+y_{r{\alpha }_k}}{2}\right]$$

(3)

$${}^{}{f}_{EWHIT3NSFLS-1}\left(x^{\prime }\right)=y_{\alpha }=\frac{{\sum }_{k=0}^N\left[\left(y_{l,{\alpha }_k}^{cos}\left(x^{\prime }\right)+y_{r,{\alpha }_k}^{cos}\left(x^{\prime }\right)\right)/2\right]}{N+1}$$

(4)

where $y_{l,{\alpha }_k}^{cos}$ and $y_{r,{\alpha }_k}^{cos}$ are the left and right points of the center of sets of each $y_{{\alpha }_k}$, and its union can be stated as an expansion$y_{\alpha }$ produced by $N+1$ elements $y_{{\alpha }_k}$ corresponding to the $N+1$ horizontal levels-${\alpha }_k$:

$$y_{\alpha }=\frac{1}{N+1}{y}_{{\alpha }_0}+\frac{1}{N+1}{y}_{{\alpha }_1}+\frac{1}{N+1}{y}_{{\alpha }_2}+\cdots +\frac{1}{N+1}{y}_{{\alpha }_k}+\cdots +\frac{1}{N+1}{y}_{{\alpha }_N}$$

(5)

Each weighted output $y_{{\alpha }_k}$, corresponding to each level-${\alpha }_k$, is calculated by the EWH IT3 NFLS-1, and it is modeled with the uncertain level-${\alpha }_k\in \left[{\underset{\_}{\alpha }}_k,{\overline{\alpha }}_k\right]$. The proposed $y_{EWH-3}$ expansion is composed of 2$N+2$ elements, (11).

$$y_{\alpha }=\frac{1}{N+1}\left(\frac{{{\underset{\_}{\alpha }}_{0}y}_{{\underset{\_}{\alpha }}_0}+{\overline{\alpha }}_0{y}_{{\overline{\alpha }}_0}}{{\underset{\_}{\alpha }}_0+{\overline{\alpha }}_0}\right)+\frac{1}{N+1}\left(\frac{{{\underset{\_}{\alpha }}_{1}y}_{{\underset{\_}{\alpha }}_1}+{\overline{\alpha }}_1{y}_{{\overline{\alpha }}_1}}{{\underset{\_}{\alpha }}_1+{\overline{\alpha }}_1}\right)+\cdots +\frac{1}{N+1}\left(\frac{{{\underset{\_}{\alpha }}_{k}y}_{{\underset{\_}{\alpha }}_k}+{\overline{\alpha }}_k{y}_{{\overline{\alpha }}_k}}{{\underset{\_}{\alpha }}_k+{\overline{\alpha }}_k}\right)+\cdots +\frac{1}{N+1}\left(\frac{{{\underset{\_}{\alpha }}_{N}y}_{{\underset{\_}{\alpha }}_N}+{{\overline{\alpha }}_{N}y}_{{\overline{\alpha }}_N}}{{\underset{\_}{\alpha }}_N+{\overline{\alpha }}_N}\right)$$

(6)

$$y_{\alpha }=\frac{1}{N+1}\left(\frac{{\underset{\_}{\alpha }}_0}{{\underset{\_}{\alpha }}_0+{\overline{\alpha }}_0}\right)y_{{\underset{\_}{\alpha }}_0}+\frac{1}{N+1}\left(\frac{{\overline{\alpha }}_0}{{\underset{\_}{\alpha }}_0+{\overline{\alpha }}_0}\right)y_{{\overline{\alpha }}_0}+\frac{1}{N+1}\left(\frac{{\underset{\_}{\alpha }}_1}{{\underset{\_}{\alpha }}_1+{\overline{\alpha }}_1}\right)y_{{\underset{\_}{\alpha }}_1}+\frac{1}{N+1}\left(\frac{{\overline{\alpha }}_1}{{\underset{\_}{\alpha }}_1+{\overline{\alpha }}_1}\right)y_{{\overline{\alpha }}_1}+\cdots +\frac{1}{N+1}\left(\frac{{\underset{\_}{\alpha }}_k}{{\underset{\_}{\alpha }}_k+{\overline{\alpha }}_k}\right)y_{{\underset{\_}{\alpha }}_k}+\frac{1}{N+1}\left(\frac{{\overline{\alpha }}_k}{{\underset{\_}{\alpha }}_k+{\overline{\alpha }}_k}\right)y_{{\overline{\alpha }}_k}+\cdots +\frac{1}{N+1}\left(\frac{{\underset{\_}{\alpha }}_N}{{\underset{\_}{\alpha }}_N+{\overline{\alpha }}_N}\right)y_{{\underset{\_}{\alpha }}_N}+\frac{1}{N+1}\left(\frac{{\overline{\alpha }}_N}{{\underset{\_}{\alpha }}_N+{\overline{\alpha }}_N}\right)y_{{\overline{\alpha }}_N}$$

(7)

$$y_{\alpha }={K_{{\underset{\_}{\alpha }}_0}y}_{{\underset{\_}{\alpha }}_0}+{K_{{\overline{\alpha }}_0}y}_{{\overline{\alpha }}_0}+{K_{{\underset{\_}{\alpha }}_1}y}_{{\underset{\_}{\alpha }}_1}+{K_{{\overline{\alpha }}_1}y}_{{\overline{\alpha }}_1}+\cdots +{K_{{\underset{\_}{\alpha }}_k}y}_{{\underset{\_}{\alpha }}_k}+{K_{{\overline{\alpha }}_k}y}_{{\overline{\alpha }}_k}+\cdots +{K_{{\underset{\_}{\alpha }}_N}y}_{{\underset{\_}{\alpha }}_N}+{K_{{\overline{\alpha }}_N}y}_{{\overline{\alpha }}_N}$$

(8)

Now, $y_{\alpha }$, the output of the EWH IT3 NSFLS-1, can be modeled as an EWH GT2 NSFLS-1 system composed of 2$N+2$ elements, where

$$\begin{gathered} K_{{\alpha }_0} =\frac{1}{N+1}\left[\frac{{\underset{\_}{\alpha }}_0}{{\underset{\_}{\alpha }}_0+{\overline{\alpha }}_0}\right] \\ {K}_{{\alpha }_1} =\frac{1}{N+1}\left[\frac{{\overline{\alpha }}_0}{{\underset{\_}{\alpha }}_0+{\overline{\alpha }}_0}\right] \\ {K}_{{\alpha }_2} =\frac{1}{N+1}\left[\frac{{\underset{\_}{\alpha }}_1}{{\underset{\_}{\alpha }}_1+{\overline{\alpha }}_1}\right] \\ {K}_{{\alpha }_3} =\frac{1}{N+1}\left[\frac{{\overline{\alpha }}_1}{{\underset{\_}{\alpha }}_1+{\overline{\alpha }}_1}\right] \\ ⋮ \\ {K}_{{\alpha }_k} =\frac{1}{N+1}\left[\frac{{\underset{\_}{\alpha }}_k}{{\underset{\_}{\alpha }}_k+{\overline{\alpha }}_k}\right] \\ {K}_{{\alpha }_{k+1}}=\frac{1}{N+1}\left[\frac{{\overline{\alpha }}_k}{{\underset{\_}{\alpha }}_k+{\overline{\alpha }}_k}\right] \\ ⋮ \\ {K}_{{\alpha }_{2N+1}}=\frac{1}{N+1}\left[\frac{{\underset{\_}{\alpha }}_{N+1}}{{\underset{\_}{\alpha }}_{N+1}+{\overline{\alpha }}_{N+1}}\right] \\ {K}_{{\alpha }_{2N+2}}=\frac{1}{N+1}\left[\frac{{\overline{\alpha }}_{N+1}}{{\underset{\_}{\alpha }}_{N+1}+{\overline{\alpha }}_{N+1}}\right] \end{gathered}$$

(9)

and

$$\begin{gathered} y_{{\alpha }_0}=y_{{\underset{\_}{\alpha }}_0} \\ {y}_{{\alpha }_1}=y_{{\overline{\alpha }}_0} \\ ⋮ \\ {y}_{{\alpha }_k}=y_{{\underset{\_}{\alpha }}_k} \\ {y}_{{\alpha }_{k+1}}=y_{{\overline{\alpha }}_{k+1}} \\ ⋮ \\ {y}_{{\alpha }_{2N+1}}=y_{{\underset{\_}{\alpha }}_{N+1}} \\ {y}_{{\alpha }_{2N+2}}=y_{{\overline{\alpha }}_{N+1}} \end{gathered}$$

(10)

then

$$y_{\alpha }=K_{{\alpha }_0}\left[\frac{y_{l{\underset{\_}{\alpha }}_0}+y_{r{\underset{\_}{\alpha }}_0}}{2}\right]+K_{{\alpha }_1}\left[\frac{y_{l{\overline{\alpha }}_0}+y_{r{\overline{\alpha }}_0}}{2}\right]+\cdots +K_{{\alpha }_k}\left[\frac{y_{l{\underset{\_}{\alpha }}_k}+y_{r{\underset{\_}{\alpha }}_k}}{2}\right]+K_{{\alpha }_{k+1}}\left[\frac{y_{l{\overline{\alpha }}_{k+1}}+y_{r{\overline{\alpha }}_{k+1}}}{2}\right]+\cdots +K_{{\alpha }_{2N+1}}\left[\frac{y_{l{\underset{\_}{\alpha }}_N}+y_{r{\underset{\_}{\alpha }}_N}}{2}\right]+K_{{\alpha }_{2N+2}}\left[\frac{y_{l{\overline{\alpha }}_N}+y_{r{\overline{\alpha }}_N}}{2}\right]$$

(11)

$$y_{\alpha }={\sum }_{k=0}^{2N+2}{K}_{{\alpha }_k}{y}_{{\alpha }_k}$$

(12)

Now, using Equations (13) and (14), the centroids are calculated based on the KM algorithm for any left endpoint, $y_{l{\alpha }_k}$:

$$y_{l{\alpha }_k}=\frac{{\sum }_{n=1}^L{\overline{f}}_{{\alpha }_k}^n\ast c_{l{\alpha }_k}^n+{\sum }_{n=L+1}^M{\underset{\_}{f}}_{{\alpha }_k}^n\ast c_{l{\alpha }_k}^n}{{\sum }_{n=1}^L{\overline{f}}_{{\alpha }_k}^n+{\sum }_{n=L+1}^M{\underset{\_}{f}}_{{\alpha }_k}^n}$$

(13)

for any right endpoint, $y_{r{\alpha }_k}$,

$$y_{r{\alpha }_k}=\frac{{\sum }_{n=1}^R{\underset{\_}{f}}_{{\alpha }_k}^n\ast c_{r{\alpha }_k}^n+{\sum }_{n=R+1}^M{\overline{f}}_{{\alpha }_k}^n\ast c_{r{\alpha }_k}^n}{{\sum }_{n=1}^R{\underset{\_}{f}}_{{\alpha }_k}^n+{\sum }_{n=R+1}^M{\overline{f}}_{{\alpha }_k}^n}$$

(14)

where $\left[{\underset{\_}{f}}_{{\alpha }_k}^n,{\overline{f}}_{{\alpha }_k}^n\right]$ is the estimated firing interval and $\left[c_{l{\alpha }_k}^n,c_{r{\alpha }_k}^n\right]$ is the estimated consequent centroid of the rule $n$ of the level-${\alpha }_k$.

2.1.1. Input Variables, Rules, and Levels-${\mathsf{\alpha }}_{\mathrm{k}}$

To start the building of the EWH IT3 NSFLS-1 system with the design and construction of the IT2${\alpha }_0$ FLS, the designer must select $q=1,2,\dots ,p$ as the input variables, $i=1,2,\dots ,M$ as the number of rules, and $k=1,2,\dots ,N$ as the initial number of horizontal levels-${\alpha }_k$.

The $p$ inputs are type-1 non-singleton numbers modeled as a Gaussian with the mean $x_q^{\prime }$ and standard deviation ${\sigma }_{X_q}$. The well-known type-1 non-singleton Gaussian model is used as primary MF:

$${\mu }_{{\tilde{X}}_q}^{}(x_q)=exp\left[-\frac{1}{2}{\left[\frac{x_q-x_q^{\prime }}{{\sigma }_{X_q}}\right]}^2\right]$$

(15)

Each input’s universe of discourse (UOD) must be covered with the required number of MFs.

2.1.2. The Membership Functions and UOD

The array of the required MFs for each input determines the number of rules, $M$. For example, if there are two inputs, and the UOD of ${\tilde{X}}_1$and the UOD of ${\tilde{X}}_2$are covered by five MFs each, then the rule base has $M$ = $5\times 5$ = 25 rules. Each MF is modeled as a Gaussian function with uncertain means, $M_q^i\in \left[M_{q_1}^i,M_{q_2}^i\right]$, and a common standard deviation, ${\sigma }_q^i$:

$${\mu }_{{\tilde{A}}_q^i}^{}(x_q)=exp\left[-\frac{1}{2}{\left[\frac{x_q-M_q^i}{{\sigma }_q^i}\right]}^2\right]$$

(16)

where $q=1,2,\dots ,p$ is the number of inputs and $i=1,2,\dots ,M$ is the number of rules.

The IT3 Mamdani fuzzy rule base model has the following form:

$${\tilde{R}}^i:IF{x}_{1}is {\tilde{A}}_1^{i}and\dots and{x}_{p}is {\tilde{A}}_p^{i}THENyis {\tilde{G}}^i$$

(17)

where it has one output $\in Y$, $p$ inputs $x_1\in X_1$, …, $x_p\in X_p$, and a rule base $i$ of size $M$.

2.1.3. The Rule Base

The horizontal level-${\alpha }_0$ has a rule base that is built by assigning the initial values of each of the $pM$ membership functions, ${\tilde{A}}_1^i,{\tilde{A}}_2^i\dots {\tilde{A}}_p^i$, and the $M$ consequent centroids, $\left[{\underset{\_}{c}}_{l{\alpha }_0}^i,{\overline{c}}_{r{\alpha }_0}^i\right]$.

$${\tilde{R}}^i:IF{x}_{1}is {\tilde{A}}_1^{i}and\dots x_{p}is {\tilde{A}}_p^{i}THENyis \left[{\underset{\_}{c}}_{l{\alpha }_0}^i,{\overline{c}}_{r{\alpha }_0}^i\right]$$

(18)

2.1.4. Alpha ${\alpha }_k$-Cuts

The $M$ firing intervals $\left[{\underset{\_}{f}}_{l{\alpha }_0}^i,{\overline{f}}_{r{\alpha }_0}^i\right]$ of the horizontal level-${\alpha }_0$, or IT2${\alpha }_0$ FLS, are calculated using Equation (19) based on the ${\alpha }_0$-cuts or the intersection of $x_q^{\prime }$ and the MF of each input and each rule. Only the ${\alpha }_0$-cuts of level-${\alpha }_0$ are calculated, but not the ${\alpha }_k$-cuts of any other level-${\alpha }_k$.

$$\left[{\underset{\_}{f}}_{l{\alpha }_0}^i,{\overline{f}}_{r{\alpha }_0}^i\right]=\left[{\prod }_{q=1}^p{a}_{q{\alpha }_0}^i\left({\underset{\_}{x}}_{q,max}^i\right),{\prod }_{q=1}^p{b}_{q{\alpha }_0}^i\left({\overline{x}}_{q,max}^i\right)\right]$$

(19)

with

$$a_{q{\alpha }_0}^i\left({\underset{\_}{x}}_{q,max}^i\right)={\underset{\_}{\mu }}_{{\tilde{X}}_q}\left({\underset{\_}{x}}_{q,max}^i\right){\underset{\_}{\mu }}_{{\tilde{A}}_q^i}\left({\underset{\_}{x}}_{q,max}^i\right)$$

(20)

and

$$b_{q{\alpha }_0}^i\left({\overline{x}}_{q,max}^i\right)={\overline{\mu }}_{{\tilde{X}}_q}\left({\overline{x}}_{q,max}^i\right){\overline{\mu }}_{{\tilde{A}}_q^i}\left({\overline{x}}_{q,max}^i\right)$$

(21)

where ${\underset{\_}{x}}_{q,max}^i$ and ${\overline{x}}_{q,max}^i$ are determined according to the locations of $x_q^{\prime }$ with respect to $M_{q1}^i$ and $M_{q2}^i$.

2.1.5. Firing Intervals

Each firing interval $\left[{\underset{\_}{f}}_{l{\alpha }_0}^i,{\overline{f}}_{r{\alpha }_0}^i\right]$ of the horizontal level-${\alpha }_0$ or IT2${\alpha }_0$ NSFLS-1 is utilized to estimate the antecedent’s firing interval of each level-${\alpha }_k\in \left[{\underset{\_}{\alpha }}_k,{\overline{\alpha }}_k\right]$. As illustrated in [5], the Gaussian model of the vertical slice at $x_{q,max}^{\prime }$ is used to calculate the firing interval $\left[{\underset{\_}{f}}_{l{\alpha }_k}^i,{\overline{f}}_{r{\alpha }_k}^i\right]$ of each level-${\alpha }_k$, as:

$${\mu }_{f_{{vs\alpha }_k}^i}={\alpha }_k=exp\left[-\frac{1}{2}{\left[\frac{x_q^{\prime }-m_{{fvs\alpha }_0}^i}{{\sigma }_{{fvs\alpha }_0}^i}\right]}^2\right]$$

(22)

where

$$m_{f_{vs{\alpha }_0}}^i=\frac{{\underset{\_}{f}}_{l{\alpha }_0}^i+{\overline{f}}_{r{\alpha }_0}^i}{2}$$

(23)

$${\sigma }_{f_{{vs\alpha }_0}}^i=\frac{{{\overline{f}}_{r{\alpha }_0}^i-\underset{\_}{f}}_{l{\alpha }_0}^i}{Z}$$

(24)

$$\left[{\underset{\_}{f}}_{l{\alpha }_k}^i,{\overline{f}}_{r{\alpha }_k}^i\right]=\frac{{\underset{\_}{f}}_{l{\alpha }_0}^i+{\overline{f}}_{r{\alpha }_0}^i}{2}\mp \frac{{\overline{f}}_{r{\alpha }_0}^i-{\underset{\_}{f}}_{l{\alpha }_0}^i}{Z}\sqrt[2]{-2\ln \left({\alpha }_k\right)}$$

(25)

with $z$ = 1, 2, …, n being an integer number estimated by trial and error. The squishing technique represented by Equation (25) creates the horizontal-slice representation of the IT3 NSFLS-1 fuzzy system generating its optimal design, Ref. [123]. The magnitude of the standard deviation of the model is a fraction of the interval of the means.

2.1.6. Consequent Centroids

Each consequent’s centroids, $\left[{\underset{\_}{c}}_{l{\alpha }_0}^i,{\overline{c}}_{r{\alpha }_0}^i\right]$, of the horizontal level-${\alpha }_0$ are used to estimate the $M$ consequents’ centroid of the level-${\alpha }_k\in \left[{\underset{\_}{\alpha }}_k,{\overline{\alpha }}_k\right]$. As depicted in [5], the vertical slice’s Gaussian model at $x_{q,max}^{\prime }$, used to calculate the centroid $\left[{\underset{\_}{c}}_{l{\alpha }_k}^i,{\overline{c}}_{r{\alpha }_k}^i\right]$ of each level-${\alpha }_k$, is:

$${\mu }_{c_{{vs\alpha }_k}^i}={\alpha }_k=exp\left[-\frac{1}{2}{\left[\frac{x_q^{\prime }-m_{{cvs\alpha }_0}^i}{{\sigma }_{{cvs\alpha }_0}^i}\right]}^2\right]$$

(26)

where

$$m_{{cvs\alpha }_0}^i=\frac{{\underset{\_}{c}}_{l{\alpha }_0}^i+{\overline{c}}_{r{\alpha }_0}^i}{2}$$

(27)

$${\sigma }_{{cvs\alpha }_0}^i=\frac{{{\overline{c}}_{r{\alpha }_0}^i-\underset{\_}{c}}_{l{\alpha }_0}^i}{Z}$$

(28)

$$\left[{\underset{\_}{c}}_{l{\alpha }_k}^i,{\overline{c}}_{r{\alpha }_k}^i\right]=\frac{{\underset{\_}{c}}_{l{\alpha }_0}^i+{\overline{c}}_{r{\alpha }_0}^i}{2}\mp \frac{{{\overline{c}}_{r{\alpha }_0}^i-\underset{\_}{c}}_{l{\alpha }_0}^i}{Z}\sqrt[2]{-2\ln \left({\alpha }_k\right)}$$

(29)

2.1.7. Expansion of the Level-${\mathsf{\alpha }}_{\mathrm{k}}$

The proposed EWH IT3 NSFLS-1 algorithm solves the processing of the uncertainty of the secondary grade of each level-${\alpha }_k$ by replacing this level with its two levels-${\alpha }_k$, which represent the uncertainty in the secondary membership: the lower level-${\underset{\_}{\alpha }}_k$ and the upper level-${\overline{\alpha }}_k$. Now, the expanded number of the horizontal levels-${\alpha }_k$ is $2N+2$, transforming the EWH IT3 NSFLS-1 into a EWH GT2 NSFLS-1 system by applying the EWH GT2 methodology to $2N+2$ levels-${\alpha }_k$ (8).

2.1.8. Calculation of ${\mathrm{y}}_{\mathsf{\alpha }}$

For each input–output training data pair $(x^{\prime },y)$, $y_{\alpha }$ can be estimated using Equation (12). The proposed EWH IT3 NSFLS-1 is dynamically constructed because its structure is calculated for each input variable, $x_q^{\prime }$. The horizontal level-${\alpha }_0$ or IT2${\alpha }_0$ NSFLS-1 is used as the baseline to estimate the structure of each horizontal level-${\alpha }_k$ or IT2${\alpha }_k$. Regardless of whether it is the low level-${\underset{\_}{\alpha }}_k$ or upper horizontal level-${\overline{\alpha }}_k$, it requires the same procedure: in each level-${\alpha }_k$, an IT2${\alpha }_k$ NSFLS-1 is constructed with its corresponding antecedent firing interval $\left[{\underset{\_}{f}}_{l{\alpha }_k}^i,{\overline{f}}_{r{\alpha }_k}^i\right]$ and its corresponding consequent centroid $\left[{\underset{\_}{c}}_{l{\alpha }_k}^i,{\overline{c}}_{r{\alpha }_k}^i\right]$. An important characteristic is the estimated parameters of the antecedent and consequent sections of each rule of all the levels-${\alpha }_k\in \left[{\underset{\_}{\alpha }}_k,{\overline{\alpha }}_k\right]$ are dynamic and temporal, and only the parameters of the level-${\alpha }_0$ or IT2${\alpha }_0$ are permanent. Only the level-${\alpha }_0$ has MF parameters of its Gaussians models, while any other level-${\alpha }_k$ temporarily has the corresponding estimated firing intervals $\left[{\underset{\_}{f}}_{l{\alpha }_k}^i,{\overline{f}}_{r{\alpha }_k}^i\right]$ and the estimated centroids $\left[{\underset{\_}{c}}_{l{\alpha }_k}^i,{\overline{c}}_{r{\alpha }_k}^i\right]$ both required to calculate its contribution to the final value $y_{\alpha }$. We propose the average using each output $y_{{\alpha }_k}$: $y_{\alpha }=\frac{{\sum }_{k=1}^{k_{max}}{y}_{{\alpha }_k}}{N}$ to estimate the value $y_{\alpha }$.

2.2. The Backpropagation Method for Antecedent Tuning

In this subsection, we explain the BP method for tuning. An objective function, $E\left(\theta \right)$, can have a non-linear form to an adjustable parameter, $\theta$. In the interactive descent methods, the next point, $\theta \left(new\right)$, is determined by one step down from the current point, $\theta \left(now\right)$, in the negative direction of the gradient of the function $E\left({\theta }_{now}\right)$. The $K$learning rates are selected by trial and error while meeting the selected criteria of minimizing the error.

$$\theta \left(new\right)=\theta \left(now\right)-Kg$$

(30)

$$\theta \left(new\right)=\theta \left(now\right)-K\frac{\partial E}{\partial {\theta }_{now}}$$

(31)

$K$ is the training rate and $g$ is the vector of the first partial derivatives of $E\left(\theta \right)$ and is equivalent to $\frac{\partial E}{\partial {\theta }_{now}}$:

$$g\left(\theta \right)={\left[\frac{\partial E}{\partial {\theta }_{1now}},\frac{\partial E}{\partial {\theta }_{2now}},\dots ,\frac{\partial E}{\partial {\theta }_{nnow}}\right]}^T$$

(32)

Each rule of the level-${\alpha }_0$ applies Equation (32) to update three $\theta$ antecedent parameters: $M_{q1{\alpha }_0}^i$, $M_{q2{\alpha }_0}^i$, and ${\sigma }_{q{\alpha }_0}^i$.

Equation (32) requires finding the partial derivatives used to update all the parameters of the antecedent section of each rule of only the IT2${\alpha }_0$ NFLS-2 located at level-${\alpha }_0$.

$$M_{q1{\alpha }_0}^i\left(new\right)=M_{q1{\alpha }_0}^i\left(now\right)-K_{M_{q1{\alpha }_0}}\frac{\partial E}{\partial M_{q1{\alpha }_0}^i}$$

(33)

$$M_{q2{\alpha }_0}^i\left(new\right)=M_{q2{\alpha }_0}^i\left(now\right)-K_{M_{q2{\alpha }_0}}\frac{\partial E}{\partial M_{q2{\alpha }_0}^i}$$

(34)

$${\mathsf{\sigma }}_{q{\alpha }_0}^i \left(new\right)={\mathsf{\sigma }}_{q{\alpha }_0}^i\left(now\right)-K_{{\sigma }_q{\alpha }_0}\frac{\partial E}{\partial {\sigma }_{q{\alpha }_0}^i}$$

(35)

where $K_{M_{q1{\alpha }_0}}$, $K_{M_{q2{\alpha }_0}}$, and $K_{{\sigma }_q{\alpha }_0}$ are the training rates of its corresponding parameter.

The quadratic error function to minimize is

$$E=\frac{1}{2}{\left(y-y_{\alpha }\right)}^2$$

(36)

where $y$ is the output value of the input–output data pair. The error function is

$$e=y-y_{\alpha }$$

(37)

For example, the methods to obtain the partial derivatives of the objective function, $E$, with respect to the antecedent parameter, $M_{q1{\alpha }_0}^i$, are shown in Equations (38)–(40).

$$M_{q1{\alpha }_0}^i\left(new\right)=M_{q1{\alpha }_0}^i\left(now\right)-K_{M_{q1{\alpha }_0}}\frac{\partial E}{\partial M_{q1{\alpha }_0}^i}$$

(38)

Then

$$\begin{array}{c}\frac{\partial E}{\partial M_{q1{\alpha }_0}^i}=[\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\underset{\_}{\alpha }}_1}}\frac{\partial y_{{\underset{\_}{\alpha }}_1}}{\partial M_{q1{\alpha }_0}^i}+\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\overline{\alpha }}_1}}\frac{\partial y_{{\overline{\alpha }}_1}}{\partial M_{q1{\alpha }_0}^i}+\cdots +\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\underset{\_}{\alpha }}_k}}\frac{\partial y_{{\underset{\_}{\alpha }}_k}}{\partial M_{q1{\alpha }_0}^i}+\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\overline{\alpha }}_k}}\frac{\partial y_{{\overline{\alpha }}_k}}{\partial M_{q1{\alpha }_0}^i}+\cdots \\ +\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\underset{\_}{\alpha }}_N}}\frac{\partial y_{{\underset{\_}{\alpha }}_N}}{\partial M_{q1{\alpha }_0}^i}+\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\overline{\alpha }}_N}}\frac{\partial y_{{\overline{\alpha }}_N}}{\partial M_{q1{\alpha }_0}^i}]\end{array}$$

(39)

which is equivalent to

$$\begin{array}{c}\frac{\partial E}{\partial M_{q1{\alpha }_0}^i}=\\ \left[\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\alpha }_1}}\frac{\partial y_{{\alpha }_1}}{\partial M_{q1{\alpha }_0}^i}+\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\alpha }_2}}\frac{\partial y_{{\alpha }_2}}{\partial M_{q1{\alpha }_0}^i}+\cdots +\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\alpha }_k}}\frac{\partial y_{{\alpha }_k}}{\partial M_{q1{\alpha }_0}^i}+\cdots +\frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\alpha }_N}}\frac{\partial y_{{\alpha }_N}}{\partial M_{q1{\alpha }_0}^i}+\dots \frac{\partial E}{\partial y_{\alpha }}\frac{\partial y_{\alpha }}{\partial y_{{\alpha }_{2N}}}\frac{\partial y_{{\alpha }_{2N}}}{\partial M_{q1{\alpha }_0}^i}\right]\end{array}$$

(40)

Each level-${\alpha }_k\in \left[{\underset{\_}{\alpha }}_k{,\overline{\alpha }}_k\right]$, previously defined during the construction process, contributes only by updating the parameters of the permanent level-${\alpha }_0$. No parameters of the level-${\alpha }_k$ have training; there is only training for the level-${\alpha }_0$ parameters.

A similar procedure can be used to calculate the equations for BP training: $M_{q2{\alpha }_0}^i$ and ${\sigma }_{q{\alpha }_0}^i$of the IT2${\alpha }_0$ NSFLS-1.

The final equations for the BP training of the antecedent’s parameters depend on the relative position of $x_q^{\prime }$ with respect to $M_{q1{\alpha }_0}^i$ and $M_{q2{\alpha }_0}^i$ positions.

2.3. The OLS Method for Consequent Tuning

Suppose that a particular system has one input, $u\left(k\right)$, and one output, $y\left(k\right)$, with an additive noise, $e\left(k\right)$, measured $t$ times every $T$ periods. Then it is possible to describe its dynamic behavior using the next model [124]:

$$y\left(k\right)={\sum }_{j=1}^n{a}_{j}y\left(k-j\right)+{\sum }_{j=0}^n{b}_{j}u\left(k-j\right)+e\left(k\right)$$

(41)

where $k=\mathrm{1,2},\dots t$; $a_j$, $b_j$ $a_j\in R$, $n$ is the order of the system. Equation (41) can be written in compact form:

$$y\left(k\right)=p^{T}z\left(k\right)+e\left(k\right)$$

(42)

with $p^T=\left[b_0,a_1,b_1,\dots ,a_n,b_n\right]$ as the parameters’ estimation matrix of size 2$n+1$ and $z^T\left(k\right)=\left[u\left(k\right),y\left(k-1\right),u\left(k-1\right),\dots ,y\left(k-n\right),u(k-n)\right]$ as the measurements vector. In the case of $t$ input–output data pairs, it can be expressed as

$$Y\left(t\right)=P^{T}Z\left(t\right)+E\left(t\right)$$

(43)

with the output measured transpose vector of size

$$Y^T\left(t\right)=\left[y\left(1\right),y\left(2\right),\dots ,y\left(t\right)\right]$$

(44)

The measurements matrix can be expressed as

$$Z\left(t\right)=\left[\begin{array}{ccc}u\left(1\right),& u\left(2\right),& \begin{array}{cc}\cdots ,& u\left(t\right)\end{array}\\ y\left(0\right),& y\left(1\right),& \begin{array}{cc}\cdots ,& y(t-1)\end{array}\\ \begin{array}{c}u\left(0\right),\\ ⋮\\ \begin{array}{c}y\left(1-n\right),\\ u\left(1-n\right),\end{array}\end{array}& \begin{array}{c}u\left(1\right),\\ ⋮\\ \begin{array}{c}y\left(2-n\right),\\ y\left(2-n\right),\end{array}\end{array}& \begin{array}{c}\begin{array}{cc}\cdots ,& u(t-1)\end{array}\\ \begin{array}{cc}\ddots & ⋮\end{array}\\ \begin{array}{c}\begin{array}{cc}\cdots ,& y(t-n)\end{array}\\ \begin{array}{cc}\cdots ,& y(t-n)\end{array}\end{array}\end{array}\end{array}\right]$$

(45)

while the noise transpose vector is

$$E^T\left(t\right)=\left[e\left(1\right),e\left(2\right),\dots ,e\left(t\right)\right]$$

(46)

One must minimize the next criteria during the estimation of $P$:

$$J={\left(Y\left(t\right)-Z\left(t\right)P\left(t\right)\right)}^{T}I\left(Y\left(t\right)-Z\left(t\right)P\left(t\right)\right)$$

(47)

with its least-squares solution as

$${\widehat{P}}^T\left(t\right)={\left[Z^T\left(t\right)Z\left(t\right)\right]}^{-1}{Z}^{T}Y(t)$$

(48)

On the other hand, Equation (49) represents a system:

$$Ax=b$$

(49)

where $A$ is a matrix of size $m\times n$, $x$ is a vector of size $n$, $b$ is a vector of size $m$, with $m>n$. This system has a solution if $b$ lies in the range space of $A$or equivalently $\rho \left(A\right)=\rho \left(A,b\right).$ Tacking a decomposition of $b$ as $b=$ $b_1+e,$ then (49) can be expressed as

$$Ax-b_1=e$$

(50)

Let us call $e$ of size $m$ the error. If

$$A^{T}A=F^{T}F$$

(51)

With $F$ being any upper or lower triangular matrix of size $n$, then (49) can be written as

$$F^{T}Fx=A^T{b}_1$$

(52)

A least-square solution can be found using previous Equation (52). The method does not require the explicit factorization of the $A^{T}A$ matrix nor the inverse matrix of $F^T$. If the transformation matrix $F$ is defined as

$$F=\left[\begin{array}{cccccccc} & & & j& k& & & \\ 1& & & & & & & \\ & 1& & & & & & \\ & & \ddots & & & & & \\ & & & c& s& & & \\ & & & -s& c& & & \\ & & & & & \ddots & & \\ & & & & & & 1& \\ & & & & & & & 1\end{array}\right]$$

(53)

then it is easy to check if the values of $c$ and $s$ are selected in such a way that the following condition is fulfilled:

$$c^2+s^2=1$$

(54)

Then the orthogonal transformation or rotational matrix can be defined as

$$F^T=F^{-1}$$

(55)

which is known as a rotational matrix because its application produces a rotation of an $\alpha$ angle in the system coordinates, with $\sin \left(\alpha \right)=c$ and $\cos \left(\alpha \right)=s$.

When an arbitrary $D$ matrix is pre-multiplied by the $F$ matrix, the rows $j$ and $k$ of the product will have the next values:

$$d_j^{\prime }=c{d}_j+s{d}_k$$

(56)

$$d_k^{\prime }=c{d}_k-s{d}_j$$

(57)

An adequate selection of $c$ and $s$ allows one element of the rows $j$ or $k$. The successive application of $m$ transformations of this type allows the cancellation of $m$ row elements, finally obtaining the triangular matrix as a result of successive transformations:

$$∆=F^m\dots F^{‴}{F}^{″}{F}^{\prime }D=FD$$

(58)

The rotational orthogonal transformations method is used to find the least-square solution of sub-determined systems of linear equations.

Rewriting (50) as

$$\left[\begin{array}{cc}A& b_1\end{array}\right]\left[\begin{array}{c}x\\ -1\end{array}\right]=e$$

(59)

If it is defined that $D=\left[\begin{array}{cc}A& b_1\end{array}\right]$ as a matrix of size $mx(n+1)$ and $x^{\prime }=\left[\begin{array}{c}x\\ -1\end{array}\right]$ as a vector of size $(n+1)$, and if the orthogonal transformation matrix $F$ to (49) is applied, the next system is obtained:

$$FD{x}^{\prime }=Fe$$

(60)

Then, it is possible to apply the orthogonal transformation solution to Equation (1) for its parameter’s identification. The last-square solution of (48) can be expressed as

$$\left[Z^T\left(t\right)Z\left(t\right)\right]P^T=Z^{T}Y\left(t\right)$$

(61)

The new estimation of parameter $P^T$ can be calculated by solving the triangular equivalent system:

$$F\left(t\right)P^T\left(t\right)=q\left(t\right)$$

(62)

where the upper triangular matrix, $F\left(t\right)$, of size $2n+1$ is the square root of $Z^T\left(t\right)Z\left(t\right)$ and $q\left(t\right)$ is a vector of size $2n+1$. The composition of $F\left(t\right)$ and $q\left(t\right)$ produces a triangular matrix, 2n + 2, as represented in Figure 2.

Figure 2. Schematic representation of F(t) and q(t) [124].

From the new measurements obtained at time $(t+1)$, it is possible to create a new equation that has the form

$$y(t+1)=z^T\left(t+1\right)P\left(t\right)$$

(63)

with

$$z^T\left(t+1\right)=\left[u(t+1),y\left(t\right),u\left(t\right),\dots ,y\left(t-n+1\right),u(t-n+1)\right]$$

(64)

The new system constituted by $F\left(t\right)$, $q\left(t\right)$, and $z^T\left(t+1\right),$ as represented in Figure 3, can be reduced to a new triangular matrix to obtain it by $F(t+1)$ and $q(t+1)$. For each period, the previous algorithm reduces to zero the compound vector $\left[z^T\left(t+1\right),y(t+1)\right]$, of size $2n+2,$to calculate $F(t+1)$ and $q(t+1)$, as represented in Figure 4. Then, the parameters of ${\widehat{P}}^T(t+1)$ can be calculated by solving the triangular equivalent system of (1):

$$F(t+1){\widehat{P}}^T(t+1)=q(t+1)$$

(65)

$${\widehat{P}}^T(t+1)=F^{-1}(t+1)q(t+1)$$

(66)

Figure 3. Schematic representation of F(t), q(t), and z^T (t + 1) [124].

Figure 4. Schematic representation of F(t + 1) and q(t + 1) [124].

Considering

$$y_{l{\alpha }_k}=\frac{{\sum }_{n=1}^L{\overline{f}}_{{\alpha }_k}^n\ast c_{l{\alpha }_k}^n+{\sum }_{n=L+1}^M{\underset{\_}{f}}_{{\alpha }_k}^n\ast c_{l{\alpha }_k}^n}{{\sum }_{n=1}^L{\overline{f}}_{{\alpha }_k}^n+{\sum }_{n=L+1}^M{\underset{\_}{f}}_{{\alpha }_k}^n}$$

(67)

$$y_{r{\alpha }_k}=\frac{{\sum }_{n=1}^R{\underset{\_}{f}}_{{\alpha }_k}^n\ast c_{r{\alpha }_k}^n+{\sum }_{n=R+1}^M{\overline{f}}_{{\alpha }_k}^n\ast c_{r{\alpha }_k}^n}{{\sum }_{n=1}^R{\underset{\_}{f}}_{{\alpha }_k}^n+{\sum }_{n=R+1}^M{\overline{f}}_{{\alpha }_k}^n}$$

(68)

$${\lambda }_{l{\alpha }_k}={\left[{\overline{f}}_{{\alpha }_k}^1,{\overline{f}}_{{\alpha }_k}^2\dots {\overline{f}}_{{\alpha }_k}^L,{\underset{\_}{f}}_{{\alpha }_k}^{L+1},{\underset{\_}{f}}_{{\alpha }_k}^{L+2},\dots {\underset{\_}{f}}_{{\alpha }_k}^M\right]}^T$$

(69)

$${\lambda }_{r{\alpha }_k}={\left[{\underset{\_}{f}}_{{\alpha }_k}^1,{\underset{\_}{f}}_{{\alpha }_k}^2\dots {\underset{\_}{f}}_{{\alpha }_k}^R,{\overline{f}}_{{\alpha }_k}^{R+1},{\overline{f}}_{{\alpha }_k}^{R+2},\dots {\overline{f}}_{{\alpha }_k}^M\right]}^T$$

(70)

and

$${\theta }_{l{\alpha }_k}={\left[{\underset{\_}{c}}_{l{\alpha }_k}^1,{\underset{\_}{c}}_{l{\alpha }_k}^2,\dots {\underset{\_}{c}}_{l{\alpha }_k}^M\right]}^T$$

(71)

$${\theta }_{r{\alpha }_k}={\left[{\overline{c}}_{r{\alpha }_k}^1,{\overline{c}}_{r{\alpha }_k}^2,\dots {\overline{c}}_{r{\alpha }_k}^M\right]}^T$$

(72)

then

$$y_{l{\alpha }_k}={\lambda }_{l{\alpha }_k}^T{\theta }_{l{\alpha }_k}$$

(73)

$$y_{r{\alpha }_k}={\lambda }_{r{\alpha }_k}^T{\theta }_{r{\alpha }_k}$$

(74)

The OLS method [124] can be used recursively online, starting with the next initial conditions: $F_l\left(0\right)={\lambda }_{l{\alpha }_0}$, $P_l\left(0\right)={\theta }_{l{\alpha }_0}$, $q_l\left(0\right)=y$, $F_r\left(0\right)={\lambda }_{r{\alpha }_0}$, $P_r\left(0\right)={\theta }_{r{\alpha }_0}$, and $q_r\left(0\right)=y$, where $y$ is the output value of the training input–output data pair. The pseudocode of the OLS is shown in Algorithm 1.

Algorithm 1: Parameter estimation using rotational orthogonal transformation
1:	Initialize $n$$,F_l\left(0\right)={\lambda }_{l{\alpha }_0}$, $P_l\left(0\right)={\theta }_{l{\alpha }_0}$, $q_l\left(0\right)=y$, $F_r\left(0\right)={\lambda }_{r{\alpha }_0}$, $P_r\left(0\right)={\theta }_{r{\alpha }_0}$ and $q_r\left(0\right)=y$
2:	Triangulate$\mathrm{the}{F}_l$$\mathrm{and}{F}_r$ matrices
3:	Solve$\mathrm{the}\mathrm{system}\mathrm{equations}.{\widehat{P}}_l^T(t+1)={F_l}^{-1}(t+1)q_l(t+1)$$\mathrm{and}{\widehat{P}}_r^T(t+1)={F_r}^{-1}(t+1)q_r(t+1)$
4:	Assign estimated values. ${{\widehat{\theta }}_{l{\alpha }_0}=\widehat{P}}_l\left(t+1\right)$, ${{\widehat{\theta }}_{r{\alpha }_0}=\widehat{P}}_r\left(t+1\right)$

2.4. The Convergence Analysis

In this section, we prove that the training method developed in this proposal guarantees that the output of the fuzzy model converges as $t\to \infty$ to the real system, considering that we do not have any knowledge of the plant, only the inputs and outputs provided by the sensors, assuming that these values are bounded by the limits of the process operation. In [125], it is established that, by choosing a ${\sigma }_{q{\alpha }_0}^i$ as small as ${\sigma }_q^{\ast },$ the fuzzy system can match all the $L$ input–output data pairs $(x^{\prime },y)$ to an arbitrary accuracy.

Lemma 1. 

Let $p(k+1)$ be a sequence of real-valued vectors generated by the GD algorithm:

$$p\left(k+1\right)=p\left(k\right)-\alpha \nabla f\left(p\left(k\right)\right)$$

(75)

where $f:R^n\to R$ is a cost function and $f\in C^2$ (i.e., $f$ has continuous second derivative). Assume that all $p\left(k\right)\in D\subset R^n$ for some compact $D$, then there exist $ϵ>0$ and $L>0$ such that

$$0<ϵ<\alpha \le \frac{2-ϵ}{L}$$

(76)

Equations (33)–(35) have the same format as Equation (75) and Lemma 1 can be applied to prove the convergence of the parameters when training, as shown in [5].

3. Results and Discussion

3.1. The Problem: Industrial Process Description

The HSM process presents the many complexities and uncertainties involved in rolling operations. Figure 5 shows the HSM sub-processes: the reheat furnace, the transfer tables, the roughing mill (RM), the scale breaker (SB), the finishing mill (FM), the round out tables, the cooling banks, and the coiler (CLR).

Figure 5. Schematic representation of HSM.

There are several mathematical models to configure the FM, which is the most critical subprocess since the necessary work references are calculated to obtain the caliber of the target strip, the width, and the temperature of the target strip in the exit zone of the FM. The mathematical model takes as inputs the FM target strip gauge, target strip width, target strip temperature, slab steel grade, slab chemistry hardness ratio, the distribution of the FM load capacity, FM gauge offset, FM offset temperature, FM roller diameters, FM load distribution, inlet transfer bar gauge, transfer bar width inlet, and the most critical variable, the inlet transfer bar temperature.

The model requires determining precisely what the temperature of the transfer bar is in the FM inlet zone. A minimum error in this input temperature will result in a coil without the required quality. To estimate this inlet temperature, it is necessary to know the surface temperature of the transfer bar, which is measured by a pyrometer located on the outlet side of RM. In addition, it is necessary to know the time needed to move the transfer bar from the RM in the exit area to the entry area of the FM SB.

The measurements of these pyrometers are necessarily affected not only by the calibration, resolution, and repeatability of the sensor but also by the noise produced during the growth of scale on the metal surface, water vapor in the environment, and the physical location of the pyrometer. Also contributing to the noise is the recalescence phenomenon, which occurs at the MR output in the body of the transfer bar [126]. The mathematical model estimates the time required by the transfer bar to move its head from the exit zone RM to the entry zone FM. This estimated time is affected by the free-air radiation phenomenon during transfer bar translation and the inherent uncertainty of kinematic and dynamic modeling.

The mathematical model’s parameters are fitted using the uncertain surface temperature measured by the pyrometers located in the FM inlet zone and the uncertain surface temperature in the FM inlet zone estimated by the model. The methodology estimates the inlet transfer bar temperature in the FM inlet zone, which was tested offline using real data at an HSM industrial facility in Monterrey, Mexico.

3.2. Simulation

This section presents the proposed methodology’s experimental testing for predicting the transfer bar surface temperature.

3.2.1. Input–Output Data Pairs

From an industrial HSM process, one hundred and seventy-five noisy input–output data pairs of three different types of coils, with different target gages and target widths and the same steel grade, were obtained and used as offline training data, $(x_1^{\prime },x_2^{\prime },y)$. The inputs were $x_1^{\prime }$, the transfer bar surface temperature measured by the pyrometer located at the RM exit zone, and $x_2^{\prime }$, the real-time to move the transfer bar end from the RM exit zone to the SB entry zone. The output y was the transfer bar surface temperature measured by the pyrometers located at the SB entry zone and was used to calculate the temperature prediction error.

3.2.2. Antecedent Membership Functions

The primary membership functions for each antecedent of the base IT2${\alpha }_0$ NSFLS-2 system were Gaussian functions with uncertain means $M_{q1{\alpha }_0}^i$ and $M_{q2{\alpha }_0}^i$, with the standard deviation ${\sigma }_{q{\alpha }_0}^i$.

3.2.3. Fuzzy Rule Base

The EWH IT3 NSFLS-1 fuzzy rule base is constituted by a set of IF-THEN rules that represent the model of the system. The IT2${\alpha }_0$ NSFLS-1 is the base for the 3D construction of the proposed fuzzy system and has two inputs, $x_1^{\prime }$ and $x_2^{\prime }$, and one output, $y_{\alpha }$. The rule base has $M=25$ rules generated as indicated in Equation (17).

A flowchart for the implementation of the proposed algorithm applied to the solution of the HSM surface temperature prediction is show in Appendix A, Figure A1.

3.3. Results and Discussion

This paper used data sets from a mill coil with three sets divided into two sets, the first for an initial adjustment and tuning process and the second for a setup validation process. For type A, we used eighty-three; for type B, we used sixty-five; for type C, twenty-seven input–output data pairs were used for the initial offline training process; and seven input–output data pairs were used for testing. A Dell PC i7, 16 GB RAM and 2.8 GHz using Win 11 OS, was used to execute the fuzzy systems.

Seven input–output data pairs were used to test the offline SB entry temperature estimation. The root mean square error (RMSE) for the prediction obtained with T1 FLS, T1 ANN (ANFIS and RBFNN), IT2 FLS, IT2 ANN (ANFIS and RBFNN), GT2, and the proposed EWH IT3 system using only one ${\alpha }_k$-cuts, all of them trained with the BP–BP algorithm, were used as benchmarks, as shown in Table 4 and Figure 6. We used Equation (77) to calculate the RMSE:

$$RMSE=\sqrt{\frac{{\sum }_{K=1}^n{\left(y-y_{\alpha }\right)}^2}{n}}$$

(77)

where $y$ is the output value of the input–output data pair (the measured output value), $y_{\alpha }$ is the estimation obtained by the fuzzy system, and $n$ is the number of data pairs.

Table 4. Comparison between the benchmark models T1 SFLS, IT2 SFLS, IT2 NSFLS-1, T1 and IT2 ANFIS, T1 RBFNN and IT2 RBFNN, and GT2 models with BP–BP learning using the classic WH algorithm and the EWH algorithm.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1
T1 SFLS	3.39596
T1 ANFIS	3.36958
T1 RBFNN	4.28737
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
IT2 ANFIS	3.37824
IT2 RBFNN	3.47980
WH GT2 SFLS (BP–BP)	1.4515
EWH GT2 SFLS (BP–BP)	1.4497
WH GT2 NSFLS-1 (BP–BP)	1.0383
EWH GT2 NSFLS-1 (BP–BP)	1.0383

Figure 6. RMSE of prediction of T1, IT2, RBFNN, ANFIS, and GT2 systems with BP–BP learning.

The ANN models show a bigger rate of error due to their characteristics and architectures without worrying if their configuration in pure (RBFNN) or hybrid (ANFIS). On the other hand, the network type also does not improve the prediction, regardless of whether or not it is type-1 or type-2, as documented in the literature. The author of [127] claims that the ANN’s present accuracy levels are close to 80%.

The EWH algorithm shows an enhancement of 0.2% in relation to the classic WH model that used a BP–BP learning model for GT2 SFLS systems. On other hand, the WH algorithm using IT3 SFLS models shows an enhancement of 0.36% for classic WH singleton and 0.41% for the EWH proposed algorithm in singleton as compared with the IT2 singleton model shown in Table 5 and Figure 7.

Table 5. Comparison between the benchmark models with BP–BP learning using the classic WH algorithm and the EWH algorithm.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	2
T1 SFLS	3.39596
T1 ANFIS	3.36958
T1 RBFNN	4.28737
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
IT2 ANFIS	3.37824
IT2 RBFNN	3.47980
IT2 NSFLS-1	1.2542
WH IT3 SFLS (BP–BP)		1.4212
EWH IT3 SFLS (BP–BP)		1.4192
WH IT3 NSFLS-1 (BP–BP)		0.9729
EWH IT3 NSFLS-1 (BP–BP)		0.8761

Figure 7. RMSE of prediction of benchmarking fuzzy systems with BP–BP learning.

For non-singleton cases, both models, WH and EWH, present the same prediction in the case of the GT2 models. In contrast, the IT3 models show a bigger enhancement in comparison with the IT2 NSFLS-1 models. In the first case, the WH IT3 NSFLS-1 (BP–BP) showed an enhancement of 22.5% for the WH algorithm and 30.2% for the EWH proposed algorithm, as shown in Table 5 and Figure 7.

The RMSE prediction of the GT2 and the proposed EWH algorithm using different levels-${\alpha }_k$, as shown in Table 6 and Figure 8, show that the GT2 SFLS-1 (BP–BP) with the proposed model (EWH) outperforms the WH GT2 SFLS with only 10 ${\alpha }_k$-cuts in an order of 19.1% for the WH GT2 singleton with BP–BP learning algorithm and of 19.5% for the IT3 with the EWH algorithm with BP–BP learning, as shown in Table 7. On other hand, with the non-singleton models, the enhancement is 2.3% for the WH algorithm and 17% for the EWH algorithm, considering that both implement BP–BP learning. The best results are obtained with 100 ${\alpha }_k$-cuts, which show an enhancement of 12.3% for the WH algorithm and 17.5% for the EWH algorithm, both with BP–BP learning, as shown in Table 6 and Figure 8.

Table 6. Comparison between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and GT2 models with BP–BP learning using the classic WH algorithm and the EWH algorithm with different numbers of ${\alpha }_k$-cuts.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	10	100	1000
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH GT2 SFLS (BP–BP)	1.4515	1.1501	1.4912	1.5727
EWH GT2 SFLS (BP–BP)	1.4497	1.1433	1.4852	1.5166
WH GT2 NSFLS-1 (BP–BP)	1.0397	1.2338	1.097	1.3325
EWH GT2 NSFLS-1 (BP–BP)	1.0383	1.1534	1.0321	1.326

Figure 8. RMSE of prediction of GT2 systems with BP–BP learning.

Table 7. Comparison between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and IT3 models with BP–BP learning using the classic WH algorithm and the EWH algorithm with different numbers of ${\alpha }_k$-cuts.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	2	22	202	2002
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH IT3 SFLS (BP–BP)		1.4212	1.0573	1.4063	1.4568
EWH IT3 SFLS (BP–BP)		1.4192	1.0528	1.4016	1.4239
WH IT3 NSFLS-1 (BP–BP)		0.9729	1.1107	1.0547	1.2197
EWH IT3 NSFLS-1 (BP–BP)		0.8761	1.0125	1.0275	1.168

In contrast, when the IT3 fuzzy systems are used, the results show a reduction in the error rates in every number of ${\alpha }_k$-cuts tested. e.g., with 202 ${\alpha }_k$-cuts, the enhancement from the results of IT2 SFLS using the WH learning is in the order of 1.4%, and, for the EWH IT3 NSFLS-1 (BP–BP), it is in the order of 27.9%, as shown in Table 7 and Figure 9.

Figure 9. RMSE of prediction of IT3 systems with BP–BP learning.

The values of the RMSE of the GT2 using the proposed hybrid learning (OLS–BP) with the WH and the proposed EWH are presented in Table 8 for only 1 ${\alpha }_k$-cut. Their results show an enhancement of 34.2% comparing the IT2 SFLS with the WH GT2 SFLS using OLS–BP learning and show an enhancement of 33.9% when comparing the IT2 SFLS against the EWH GT2 SFLS (OLS–BP) system (see Table 8 and Figure 10). The results show that the tested systems WH GT2 SFLS (OLS–BP) and EWH GT2 SLFS (OLS–BP) outperform the IT2 SFLS, with only 1 ${\alpha }_k$-cut. In a complementary form, the WH GT2 NSFLS-1 (OLS–BP) presents an enhancement of 28.7% for the WH GT2 NSFLS-1 (OLS–BP) learning and 30.5% for the EWH GT2 NSFLS-1 (OLS–BP) system.

Table 8. Comparison between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and GT2 models with OLS–BP learning using the classic WH algorithm and the EWH algorithm.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH GT2 SFLS (OLS–BP)	0.9389
EWH GT2 SFLS (OLS–BP)	0.9424
WH GT2 NSFLS-1 (OLS–BP)	0.8952
EWH GT2 NSFLS-1 (OLS–BP)	0.8724

Figure 10. RMSE of prediction of GT2 systems with OLS–BP learning.

On other hand, the RMSE for prediction using both the WH and the EWH IT3 models with the OLS–BP learning algorithms shows that the error rates are reduced significantly to 34.2% for the WH IT3 SFLS (OLS–BP) and 33.9% for the EWH algorithm in the IT3 SFLS (OLS–BP), as shown in Table 9 and Figure 11. For the IT3 with only two ${\alpha }_k$-cuts, the IT3 NSFLS-1 model presents continuous enhancements when compared with IT2 and with GT2 systems. The WH IT3 NSFLS-1 (OLS–BP) system presents better performance, with a reduction of 28.7% and 30.5% for the EWH IT3 NSFLS-1 (OLS–BP) model, as shown in Table 9 and Figure 11.

Table 9. Comparison between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and IT3 models with OLS–BP learning using the classic WH algorithm and the EWH algorithm.

Fuzzy Systems∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	2
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH IT3 SFLS (OLS–BP)		0.9389
EWH IT3 SFLS (OLS–BP)		0.9424
WH IT3 NSFLS-1 (OLS–BP)		0.8952
EWH IT3 NSFLS-1 (OLS–BP)		0.8724

Figure 11. RMSE of prediction of IT3 systems with OLS–BP learning.

Table 10 and Figure 12 show the RMSE of the GT2 systems using the OLS–BP learning algorithm with a varied number of ${\alpha }_k$-cuts. The results show a significant performance reduction in comparison with the IT2 SFLS systems, presenting a 34.2% reduction in comparison with that of WH GT2 SFLS (OLS–BP) learning, 33.9% in comparison with that of IT2 SFLS, and a 33.2% reduction in comparison with that for the EWH GT2 SFLS (OLS–BP) system. In contrast, when comparing the non-singleton models, there is a 38.5% and 29.5% reduction for the WH GT2 NSFLS-1 (OLS–BP) and EWH GT2 NSFLS1 (OLS–BP), respectively. In a complementary form, the WH GT2 NSFLS-1 (OLS–BP) presents an enhancement of 28.7% for the WH GT2 NSFLS-1 (OLS–BP) learning and 30.5% for the EWH GT2 NSFLS-1 (OLS–BP) system. Compared with the IT3 models, the RMSE showed a reduction in the error of prediction of 34.2% for the WH IT3 SFLS (OLS–BP) and a reduction of 33.9% for the EWH IT3 SLFS (OLS–BP) models. For non-singleton models, a reduction of 38.7% for the IT3 NSFLS-1 (OLS–BP) and a reduction of 30.15% in comparison with the IT2 NSFLS-1 were obtained, respectively.

Table 10. Comparison from1 to 1000 ${\alpha }_k$-cuts between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and GT2 models with OLS-BP learning using the classic WH and the EWH algorithms.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	10	100	1000
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH GT2 SFLS (OLS–BP)	0.9424	0.9332	0.9356	0.9631
EWH GT2 SFLS (OLS–BP)	0.9521	0.9438	0.9458	0.9658
WH GT2 NSFLS-1 (OLS–BP)	0.8979	0.9316	0.8888	1.002
EWH GT2 NSFLS1 (OLS–BP)	0.8851	0.9183	0.8659	0.9967

Figure 12. RMSE of predictions of the GT2 systems using from 1 to 1000 ${\alpha }_k$-cuts with OLS-BP learning.

The values of RMSE prediction for the IT3 using the proposed learning OLS–BP with the WH algorithm and the proposed EWH algorithm are presented in Table 11 using different quantities of ${\alpha }_k$-cuts. The results show an enhancement of 34.2% in a comparison of the IT2 SFLS with the WH IT3 SFLS algorithm using OLS–BP learning. It also showed an enhancement of 33.9% when comparing the IT2 SFLS to the EWH IT3 SFLS (OLS–BP) system, as shown in Table 11 and Figure 13, demonstrating that the tested systems WH IT3 SFLS (OLS–BP) and EWH IT3 SLFS (OLS–BP) outperform the IT2 SFLS with only two ${\alpha }_k$-cuts.

Table 11. Comparison from 1 to 1000 ${\alpha }_k$-cuts between the benchmark models (IT2 SFLS and IT2 NSFLS-1) and IT3 models with OLS-BP learning using the classic WH and the EWH algorithms.

Fuzzy System∖${\mathit{\alpha }}_{\mathit{k}}$-Cuts	1	2	10	100	1000
IT2 SFLS	1.4249
IT2 NSFLS-1	1.2542
WH IT3 SFLS (OLS–BP)		0.9389	0.9247	0.9177	0.9461
EWH IT3 SFLS (OLS–BP)		0.9424	0.9184	0.9231	0.9556
WH IT3 NSFLS-1 (OLS–BP)		0.8952	0.9127	0.8795	0.9666
EWH IT3 NSFLS-1 (OLS–BP)		0.8724	0.8905	0.8634	0.9408

Figure 13. RMSE of prediction of IT3 systems using from 1 to 2002 ${\alpha }_k$-cuts with hybrid OLS-BP learning.

For offline tuning, twenty training epochs were used with validated and bounded input–output data pairs, which guarantees the convergence of the proposed EWH IT3 NSFLS-1, as experimentally demonstrated in this research.

With the proposed OLS–BP hybrid training method, the IT3 NSFLS-1 was the one that presented the best performance. The results obtained by the GT2 systems are better than those of the IT2 models, but not better than those of the IT3 systems, as shown in Figure 14.

Figure 14. RMSE of prediction of GT2 systems with hybrid OLS–BP learning.

The results show that the best estimation is obtained by the proposed EWH IT3 NSFLS-1 (OLS–BP) model using 202 levels-$\alpha$ with a RMSE = 0.8634 $°\mathrm{C}$. The IT3 NSFLS-1, using any number of levels-$\alpha$, presented values of RMSE below 1 °C, as shown in Figure 15.

Figure 15. RMSE of prediction of IT3 systems using from 1 to 1000 ${\alpha }_k$-cuts with hybrid OLS-BP learning.

4. Conclusions

This work presents a novel hybrid learning method for parameter tuning of the novel EWH method for IT3 NSFLS-1 output estimation. The consequent parameters are tuned using the OLS training algorithm, while the antecedent parameters are tuned using the classic BP algorithm. The proposed EWH fuzzy systems use the average, instead of the weighted average, to estimate the final output value of the fuzzy system, $y_{\alpha }$, where the contribution of the horizontal level-${\alpha }_0$ or IT2${\alpha }_0$ FLS output $y_{{\alpha }_0}$ improves the accuracy of this estimation. Each horizontal level-${\alpha }_k$ contributes 100% with its estimation of its output, $y_{{\alpha }_k}$.

The simulation results show that the proposed EWH IT3 NSFLS-1 (OLS–BP) hybrid algorithm implies better performance in temperature estimation when compared with that of BP–BP training. The better performance is obtained by the proposed EWH fuzzy systems as compared with the classic WH fuzzy systems. In addition, the comparisons between several types of fuzzy systems showed that those of the IT3 NSFLS-1 type are the best among the IT3 SFLS, GT2 NSFLS-1, GT2 SFLS, T1 SFLS, T1 RBFNN, IT2, RBFNN, and T1 and IT2 ANFIS systems.

For future work, we plan to apply the hybrid algorithm and the EWH to the GT2 fuzzy systems and to apply this system to the FM exit gage, the FM exit width, and to FM exit temperature estimation of the head strip. Furthermore, we plan to make a comparative benchmarking of the performance of the EWH IT3 NSFLS-1 (OLS–BP) using Gaussian, triangular, and trapezoidal functions for modeling the input MFs, the primary MFs, and the secondary MFs.