A New Method for the Design of Interval Type-3 Fuzzy Logic Systems With Uncertain Type-2 Non-Singleton Inputs (IT3 NSFLS-2): A Case Study in a Hot Strip Mill

This paper presents a new method for the construction and training of interval type-3 fuzzy logic systems whose inputs are uncertain type-2 non-singleton numbers (IT3 NSFLS-2). The proposed methodology is divided in two processes: 1) The novel construction of the structure of the IT3 NSFLS-2 systems based on: a) The level-alpha-0 of the interval type-2 fuzzy logic system (IT2-alpha-0 FLS), and on b) The secondary membership function using Gaussian modeling to construct each rule of the alpha-k fuzzy rule base (FRB), the firing intervals of the antecedent and the centroids of the consequent, and 2) The training methodology based on gradient descent algorithm to train the antecedent and consequent parameters of the alpha-0 FRB. The primary membership functions (MF) of the antecedents of the IT3 NSFLS-2 system are modeled as Gaussians with uncertain means and with common standard deviation. The proposal was applied and tested with the prediction of a transfer bar’s surface temperature in an industrial hot strip mill facility located in Monterrey México. The modeling results show that the proposal supports the stability required by this critical process and shows the best performance when compared with similar methods.


I. INTRODUCTION
Recent trends on science have generated an evolution of fuzzy systems. This evolution was marked by the development of new models as in the case of the interval type-3 (IT3) fuzzy logic systems (FLS). This type of system was proposed in 2008 by Rickard, et al. with the term type-3 fuzzy sets [1], [2]. I. [3] the IT3 and the type-n systems models were presented in forecasting application.
In [34, p. 154] the IT3 FLS are defined as: ''The type-3 FLS, is the generalization of the type-2 FLS that has more capacity to cope with uncertainties. In T3-FLSs, the secondary membership function (MF) is also a type-2 MF. Then the upper and lower bounds of memberships are not constant in contrast to the type-2 MFs. These features cause that more level of uncertainties can be handled by type-3 MFs.'' The IT3 model presents several similarities to the general type-2 (GT2) model due to their analogous mathematical foundations. The authors in [10] present a list of characteristics for similar intelligent systems as provided in Table 1. It shows that the IT3 FLS provide better accuracy than the IT2 systems.  [10]. Table 2 shows the challenges to be avoided in the implementation of GT2 models as it is mentioned in [43]. Challenges of the generalization of type-2 fls. adapted from [43].
Based on the information above and according to [10] and [43], the main advantages of IT3FS are: a) Superior accuracy versus IT2 models, b) Better system performance, c) Management of non-uniform uncertainties, d) Management of semantically numerical values that the secondary membership function of IT2 cannot make [58]. The disadvantages are: a) Complex learning process due to the change of the type of the secondary membership function, b) Hard computation and their iterative nature, c) Quantity of alpha planes or slices necessary for the implementation and d) Limited number of applications available in the literature.
In [56], [57], and [58] the authors proposed to handle higher levels of uncertainty using IT3 FLS. Nowadays, several researchers have developed its mathematical foundations and applied it to modeling, predicting, and controlling realworld situations.
In [21] the authors presented the baseline to construct and update the IT3 FLS with singleton inputs using a fractional-order learning algorithm, and only the consequent parameters are adjusted to help to decrease the computational cost.
In the state-of-the-art literature analysis, there are only few papers that used the gradient descent (GD) method for learning: For instance, authors in [40] use only the pure model and in [34] the extended Kalman filter (EKF) with hybridization is used. The main characteristics of the IT3 FLS found in the state-of-the-art literature are shown in Table 3. It is important to note that most of the publications present only a common vector of 4 or 8 fixed equations for the α kcuts calculation in the consequent section (CCS). The use of CCS appears in [7], [15], [20], [21], [22], [24], [28], [30], [34], [35], [41], and [42].
The main contributions of this paper are: 1. An alternative and economical model to construct IT3 NSFLS-2 systems with dynamical structure, where each 2N horizontal levels-α k has its own base of M rules. The output y α k of each level-α k is calculated with the contribution of each i th rule, which only requires both its antecedent's firing intervals f i lα k , f i rα k and its consequent's centroids c i lα k , c i rα k . According to the literature, each output y α k of each level-α k is calculated using the estimation of the alpha cuts at level-α k . This proposal does not estimate each α k -cut of each input x ′ q at each level-α k , that is   calculate the values of the α k -cuts for each input variable, of each rule at each level-α k . e.g. If an IT3 NSFLS-2 fuzzy system has p = 2 input variables x ′ 1 , x ′ 2 , M = 25 rules, and K = 20 levels-α k , then the new proposal does not require the calculation of (24) and (25) for a total of 2000 α k -cuts = 2pMK = 2 × 2×25 × 20 times, for each estimation of y α . 4. To the best knowledge of the authors, this is the first time that the primary MFs of the antecedents section of the IT3 NSFLS-2 are modeled as Gaussians with uncertain mean M i q ∈ M i q 1 , M i q 2 and common standard deviation σ i q , Fig. 1, with such being a more difficult case than the one considered in the state-of-the-art literature that uses the primary MFs modeled as Gaussians with uncertain standard deviation σ i q ∈ σ i q 1 , σ i q 2 and a common mean M i q . 5. The proposal handles the type-2 non-singleton inputs modeled as Gaussians fuzzy numbers with uncertain standard deviations σ i x q ∈ σ i x q1 , σ i x q2 , representing the additive measurement non-stationary noise (Fig. 1). An interval type-3 FLS whose inputs are modeled using type-2 fuzzy numbers is named a type-2 non-singleton type-3 fuzzy logic system (IT3 NSFLS-2). 6. The complete set of equations to update all the parameters of both antecedent and consequent sections of the proposed dynamical structure which are obtained by applying the gradient descent methodology, is presented here. 7. To the best of the knowledge of the authors, this is the first time that the IT3 NSFLS-2 fuzzy systems are applied to predict the transfer bar surface temperature at the entry zone of the finishing scale breaker of a hot strip mill. 8. To the best knowledge of the authors, this is the first time that systematic experiments using IT3 NSFLS-2 fuzzy systems with more than 10 levels-α k are proposed: 100 and 1000 levels-α k .

II. PROBLEM DESCRIPTION
The Hot Strip Mill (HSM) process presents many complexities and uncertainties involved in rolling operations. Fig. 2 shows the HSM sub-processes: The reheat furnace, the roughing mill (RM), the transfer tables, the scale breaker (SB), the finishing mill (FM), the round out tables, and the coiler (CLR). The most critical subprocess is the FM. There are several mathematical model-based systems for setting up the FM, like the finishing mill setup (FSU) model which calculates the working references required to obtain the target strip gauge, target strip width and target strip temperature at the exit zone of the FM. The FSU model takes as inputs the FM target strip gage, the target strip width, the target strip temperature, the slab steel grade, the hardness ratio from slab chemistry, the FM load distribution, the FM gauge offset, the FM temperature offset, the FM roll diameters, the FM load distribution, the input transfer bar gauge, the input transfer bar width, and the most critical variable, the input transfer bar temperature.
The FSU model requires knowing accurately what the input transfer bar temperature is at the entry zone of the FM. A minimum entry temperature error will propagate through the entire FM and produce a coil out of the required quality. For the estimation of this FM entry temperature, the math models require knowledge of the transfer bar surface temperature, which is measured by a pyrometer located at the RM exit side, and the time taken to translate the transfer bar from the RM exit zone to the FM SB entry zone.
These pyrometers' measurements are affected by the noise produced by the surface scale growth, environment water steam, the pyrometer's location, calibration, resolution, repeatability, and by the recalescence phenomenon occurring at the RM exit in the body of the transfer bar [67]. The time required by the transfer bar to move its head end from the RM exit to the FM entry zones, is estimated by the FSU model, when calculating the required transfer bar thread speed to reach the strip target temperature at the exit zone of the FM. This time estimation is affected by the free air radiation phenomenon occurring during the transfer bar translation and by the inherent uncertainty of the kinematic and dynamic modeling.
The FSU model parameters are adjusted using both, the uncertain surface temperature measured by pyrometers located at the FM entry zone, and the uncertain surface temperature at the FM entry zone estimated by the FSU model. The proposal was off-line tested using real data from an industrial hot strip mill facility located in Monterrey, México, which is currently using a certain type of fuzzy system for this estimation.

III. CONSTRUCTION OF THE IT3 NSFLS-2
The main foundation of IT3 systems is the uncertainty presented by the horizontal level-α k with respect to its vertical location or its secondary membership value µÃ (x, u) = f x (u) = α k , as is shown in Fig. 1. In the IT3 systems, this additional uncertainty is represented by the interval value α k , α k . Geometrically as in [21], it is interpreted as shown in Fig. 3. This uncertainty is modeled to be between the horizontal level α k and the horizontal level α k , as in Fig. 4.
Based on the economical modeling of WH GT2 Mamdani fuzzy systems that use the type reduction center sets and the end-point defuzzification average [60], [61], [62], [63], the IT3 NSFLS-2 can be calculated as in [60], with q = 1, 2, . . . , p the number of input variables, i = 1, 2, . . . , 2M the number of rules, k = 1, 2, . . . , N , and the number of  initial horizontal levels-α k : Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where y cos l,α k and y cos r,α k are the left and right points of the center of sets of each y α k , and its union can be expressed as an expansion y WH −3 composed by N elements y α k corresponding to the N horizontal levels-α k : Each weighted output y α k corresponding to each level α k can be calculated using the IT3 NFLS-2 modeling with the uncertain level α k ∈ α k , α k . Now the proposed y WH −3 expansion is composed by 2N elements, as in Fig. 4.
Now y α of the IT3 NSFLS-2 can be modeled as any GT2 NSFLS-2 system as it is shown in Fig. 4. where then: The centroids can be calculated with the centroid equations using the Karnik-Mendel (KM) algorithm for any left endpoint y lα k : and for any right endpoint y rα k : , f n α k is the estimated firing interval and c n lα k , c n rα k is the estimated consequent centroid of the rule n of the level-α k .

A. INPUT VARIABLES, RULES, AND LEVELS-α k
The designer must select q = 1, 2, . . . , p the input variables, i = 1, 2, . . . , M the number of rules, k = 1, 2, . . . , N , the initial number of horizontal levels-α k to start the construction of the IT3 NSFLS-2 system. The p inputs are type-2 non-singleton numbers modeled as a Gaussian with common mean x ′ q and an interval of standard deviations σX q ∈ σ X q1 , σ X q2 . The well-known type-2 nonsingleton Gaussian model [61 and 68] is used as primary MF: Each input must cover its universe of discourse (UOD) with the required number of MF.

B. UNIVERSE OF DISCOURSE AND MF
The number M of rules is determined by the array of required MFs of each input. If there are two inputs, and the UOD of X 1 andX 2 are covered by five MFs each, then the rule base has M = 5 × 5 = 25 rules.
As it is shown in Fig. 5, each consequent MF is modeled as Gaussian with uncertain means M i q ∈ M i q 1 , M i q 2 and common standard deviation σ i q : The GT2 Mamdani fuzzy rule base model has p inputs x 1 ∈ X 1 , . . . , x p ∈ X p , one output ∈ Y , and a rule base of size M of the form:

C. RULE BASE
The rule base of the horizontal level-α 0 of M rules, is constructed assigning the initial values of each of the M consequent centroids c i lα 0 , c i rα 0 . These values can be fixed by an expert or with initialization fixed at zero.

R i
:

D. ALPHA CUTS
The M firing intervals f i lα 0 , f i rα 0 of the horizontal level-α 0 or IT2α 0 FLS are calculated based on (23) using the α 0 -cuts or the intersection of x ′ q and the MF of each input and each rule. Only the α 0 -cuts of level-α 0 are calculated, not the α kcuts of any other level-α k , as shown in Table 3. with x i q,max and x i q,max are determined according to the locations of x ′ q with respect to M i q1 and M i q2 as it is shown in Table 4.

E. FIRING INTERVALS
Each firing interval f i lα 0 , f i rα 0 of the horizontal level-α 0 or IT2α 0 NSFLS-2 is used to estimate the antecedent's firing interval at each level-α k ∈ α k , α k . As it is shown in Fig. 6, the Gaussian model of the vertical slice at x ′ q,max used to calculate the firing interval f i lα k , f i rα k of each level-α k is: where m i with z = 1, 2, . . . , n being an integer number estimated by trial and error. The magnitude of the standard deviation of the model is a fraction of the interval of the means.  level-α k ∈ α k , α k . As shown in Fig. 7, the Gaussian model of the vertical slice at x ′ q,max used to calculate the centroid c i lα k , c i rα k of each level-α k is: where

G. EXPANSION OF THE LEVEL-α k
The proposed IT3 NSFLS-2 solves the processing of the uncertainty of the secondary grade of each level-α k , Fig. 3, by replacing this level by its two levels-α k that represent the uncertainty in the secondary membership: The lower levelα k and the upper level-α k . Now the expanded number of the horizontal levels-α k is 2N , transforming the IT3 NSFLS-2 into a GT2 NSFLS-2 system, Fig. 4, by applying the WH GT2 methodology to 2N levels-α k (8).

H. CALCULATION OF y α
For each input-output training data pair (x ′ , y), y α can be estimated using (16). The proposed IT3 NSFLS-2 is dynamically constructed because its structure is calculated for each input vector x ′ q . The horizontal level-α 0 or IT2α 0 NSFLS-2 is used as the base line to estimate the structure of each horizontal level-α k or IT2α k . Regardless of it being either the low horizontal level-α k or the upper horizontal level-α k , it requires the same procedure: In each level-α k an IT2α k NSFLS-2 is constructed with its corresponding antecedent firing interval f i lα k , f i rα k and its corresponding consequent centroid c i lα k , c i rα k . An important characteristic is that the estimated parameters of the antecedent and consequent sections of each rule of all the levels-α k ∈ α k , α k are dynamic and temporal, and only the parameters of the level-α 0 or IT2α 0 are permanent. Only the level-α 0 has MF parameters of its Gaussians models, while any other level-α k temporarily has the corresponding estimated firing intervals f i lα k , f i rα k and the estimated centroids c i lα k , c i rα k both required to calculate its weighted contribution to the y α final value.

IV. TRAINING ALGORITHM
An objective function E (θ) may have a non-linear form with respect to an adjustable parameter θ. In the interactive descent methods, the next point θ (new) is determined by one step down from the current point θ (now) in the negative direction of the gradient of the function E (θ now ). The K learning rates are selected by trial and error while meeting the selected criteria of minimizing the error. (35) K is the training rate, and g is the vector of the first partial derivatives of E (θ) and is equivalent to ∂E ∂θ now : , . . . , ∂E ∂θ nnow T (36) Each rule of the level-α 0 uses (35) to update three θ antecedent parameters, M i q1α 0 (37), M i q2α 0 (38), and σ i qα 0 (39), and two θ consequent parameters, c i lα 0 (40), and c i rα 0 (41). Equation (35) requires finding the partial derivatives used to update all the parameters of the antecedent and consequent sections of each rule of only the IT2α 0 NFLS-2 located at level-α 0 .
where K M q1α 0 , K M q2α 0 , K σ q α 0 , K c l α 0 , and K c r α 0 are the training rates of its corresponding parameter. The quadratic error function to minimize is: where: y is the output value of the L input-output data pairs. The error function is: As an example, the logic sequence of the math steps to obtain the partial derivatives of the objective function E with respect to the antecedent parameter M i q1α 0 are illustrated from (44) to (46).
Each level-α k ∈ α k , α k previously defined during the construction process, contributes only by updating the parameters of the permanent level-α 0 . No parameters of the level-α k have training only have it the level-α 0 parameters.
A similar procedure can be used to calculate the equations for training: M i q2α 0 , σ i qα 0 , c i lα 0 , and c i rα 0 of the IT2α 0 NSFLS-2.
As shown in Table 4, the final equations for training the parameters of the antecedent and the consequent depend on the relative position of x ′ q with respect to M i q1α 0 and M i q2α 0 positions. Table 5 shows the complete set of equations for parameters M i q1α 0 , M i q2α 0 and σ i qα 0 , with training under y l contribution. Table 6 also shows the complete set of equations for training these three antecedent parameters under y r contribution. Tables 7 and 8 show the equations for training c i   lα 0 and c i rα 0 consequent parameters by adding the contribution of each horizontal level-α k .

V. CONVERGENCE ANALISYS
The fuzzy-logic identification approach works for the trajectory tracking for a conventional dynamic system. The HSM is a complex system with a complex mathematical description. The objective is to design an IT3 NSFLS-2 identifier to achieve that the output of the fuzzy model converges to the output of the real system as t → ∞, without any knowledge of the plant except the assumptions that its inputs and outputs are measured by sensors and its values are bounded by the limits of the process operation. In [69] and [70] it is established that, by choosing a σ i qα 0 as small as σ * q the fuzzy system can match all the L input-output data pairs (x ′ , y) to an arbitrary accuracy.
Lemma 1: For arbitrary ϵ > 0, the fuzzy system − y (t) < ϵ where f IT 3NSFLS−2 x ′ is the output y α in the training phase, x ′ (t) is the input training vector, y (t) is the output training value, with t = 1, 2, . . . , L.
Because the proposed IT3 NSFLS-2 is a universal fuzzy identifier, the training algorithm based on gradient descent guarantees that the total error from (43) converges to a value ϵ at every step of training. This can be proved using the general results of the gradient descent algorithm as defined in [69] and applied to the proposed IT3 NSFLS-2 training.
Let f and f be a sequence of real-valued vectors generated respectively by one of the gradient-descent algorithms: where η is the training rate and g : R n → R is a cost function, g ∈ C 2 . Assume that all f and f ∈ D ⊂ R n for some compact D. Then, to train the parameters M i q ∈ M i q 1 , M i q 2 and σ i q , to minimize the squared errors given by (42), and to test for convergence, it is necessary to apply Lemma 1 [69] to each of the lower and upper firing interval functions (23). Both are piecewise differentiable, i.e., each branch is differentiable [60] over its segment domain, and using the Taylor series expansion, it is possible to prove the next points: As for the convergence tuning of the centroid parameters c i lα k , c i rα k the same Lemma 1 applies.

VI. SIMULATIONS
This section presents the experimental testing of the proposal, the prediction of the transfer bar surface temperature in an industrial hot strip mill facility located in Monterrey, México.

A. 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, Table 9, were obtained and used as offline training data, (x ′ 1 , x ′ 2 , y). The inputs were x ′ 1 , the transfer bar surface temperature measured by the pyrometer is located at the RM exit zone, and x ′ 2 , the real time to move 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.

B. ANTECEDENT MEMBERSHIP FUNCTIONS
The primary membership functions for each antecedent of the base IT2α 0 NSFLS-2 were Gaussian functions with uncertain means M i q1α 0 , M i q2α 0 , and with the standard deviation σ i qα 0 , as shown in Tables 10 and 11. An array of two inputs, with five MF each, produces M = 25 rules.

C. FUZZY RULE BASE
The IT3 NSFLS-2 fuzzy rule base consists of a set of IF-THEN rules that represent the model of the complete system. The IT2α 0 NSFLS-2, that is the base of the 3D construction of the proposed fuzzy system, has two inputs x ′ 1 and x ′ 2 and one output y α . The rule base has M = 25 rules of the type shown in Table 12.

D. FEEDBACK AND SIMULATION PROCESS
Three different sets of data for three different coil types were taken from a real mill. Each of these data sets was split into two sets: One for the initial adjustment and tuning process, and the other for the setup validation process. Eighty-three of type A, sixty-five of type B and twenty-seven of type C input-output data pairs were used for the initial offline training process, and seven input-output data pairs were used for testing. The production gage and width coil targets of the training data with the steel grade are shown in Table 9. In this initial offline process, computational time was not an issue. Table 13 shows the predicted temperature by the proposed 44074 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.     IT3 NSFLS-2 and as compared to type-1 (T1), IT2 NSFLS-2 and GT2 NSFLS-2 fuzzy systems. A Dell PC i7, 16 GB    Seven input-output data pairs were used to test the offline SB entry temperature estimation. The prediction results obtained with T1 systems, and IT2 benchmark models are shown in Fig. 8, while the prediction of the GT2 and the proposed IT3 systems using different levels-α are shown in Fig. 9. Table 14, Fig. 10, and Fig. 11 show the root mean square error (RMSE) behavior of the GT2 systems vs. the performance of the proposed IT3 systems using 1, 10, 100 and 1000 levels-α. Fig. 10 shows the performance using only 1 level-α, while Fig. 11 shows the performance using 1,  10, 100 and 1000 levels-α. The time used to calculate the temperature and to update its parameters is shown in Tabl. 15 and Fig. 12.
Twenty epochs of training were chosen for offline tuning. The exclusive usage of validated and bounded input-output data pairs guarantees the convergence of the proposed IT3 NSFLS-2, as proved experimentally in this research. The proposed training method gave the IT3 NSFLS-2 presented the better performance, but computational time was higher than that of the IT2 NSFLS-2 and the GT2 NSFLS-2, as shown in Fig. 12.
The results show that the best estimation is obtained by the IT3 NSFLS-2 model using 100 levels-α with a RMSE = 0.8194 • . Both the IT3 NSFLS-2 using 10 and 1000 levels-α presented the values of RMSE=0.9258 • C and RMSE=0.8280 • C, respectively. An unexpected result is shown with the RMSE produced using 100 levels-α: It is better than that produced by the IT2 NSFLS-2, GT2 NSFLS-2, and the result obtained by the T1 radial basis function neural network (T1 RBFNN) and by the type-1 singleton fuzzy logic system (T1 SFLS).

VII. CONCLUSION
Based on mathematical foundations for classical fuzzy systems, this paper presents a new method to construct and train IT3 NSFLS-2 systems whose inputs are modeled as type-2 non-singleton numbers. A real-world problem of predicting the surface temperature of a transfer bar in a hot strip mill as a case study was introduced and compared with similar methods.
Benchmark results showed that our method presented superior performance capabilities when compared to conventional IT2, GT2 and IT3 NSFLS-2 systems. It was shown that our method used few training epochs, showed stable convergence and lower RMSE values. Best results for the proposed system occurred when using 100 levels-α. The lower error was obtained when using the first level-α 0 to estimate the firing intervals and the centroids instead of the traditional calculation process used in conventional models.
Experimental results also showed that computational times for the IT2 NSFLS-2, GT2 NSFLS-2, and the proposed IT3 NSFLS-2 were less than 1 second, which is considered appropriate for controlling industrial processes.
The knowledge gap of using type-2 non-singleton numbers to model the system's inputs, and the use of Gaussians with variable mean and fixed standard deviation to model the MFs of antecedents and consequent has been covered with our method. As future work, we have envisaged the application of our method to other industrial cases as well as further mathematical analysis in control theory regarding its robustness and stability proof using Lyapunov theory. in diverse industrial fields, such as an advisor of metal-mechanic and automotive and work as a Professor with UANE Campus Saltillo on Industrial Engineering and Systems Program, in 2016. His current research interests include the applications of computational intelligence techniques to modeling quality inspection systems, uncertain process modeling, classification problems, and learning methods. He was a member of the Mexican Society of Artificial Intelligence. He is an active member of the Mexican Logistics Association. He is a reviewer of the IEEE TRANSACTIONS ON FUZZY SYSTEMS journal and IEEE LATIN AMERICA TRANSACTIONS journal.