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Modelling and Simulation in Engineering
Volume 2011 (2011), Article ID 696947, 12 pages
http://dx.doi.org/10.1155/2011/696947
Research Article

Type-2 Fuzzy Modeling for Acoustic Emission Signal in Precision Manufacturing

Department of Mechanical Engineering, Ecole Polytechnique de Montreal, CP 6079, Succursale Centre-Ville, Montreal, QC, Canada H3C 3A7

Received 17 March 2011; Revised 6 June 2011; Accepted 20 June 2011

Academic Editor: Andrzej Dzielinski

Copyright © 2011 Qun Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents an application of type-2 fuzzy logic on acoustic emission (AE) signal modeling in precision manufacturing. Type-2 fuzzy modeling is used to identify the AE signal in precision machining. It provides a simple way to arrive at a definite conclusion without understanding the exact physics of the machining process. Moreover, the interval set of the output from the type-2 fuzzy approach assesses the information about the uncertainty in the AE signal, which can be of great value for investigation of tool wear conditions. Experiments show that the development of the AE signal uncertainty trend corresponds to that of the tool wear. Information from the AE uncertainty scheme can be used to make decisions or investigate the tool condition so as to enhance the reliability of tool wear.

1. Introduction

Related to advances in machine tools, manufacturing systems and material technology, machining practice is changing from conventional machining to precision machining, even high-precision machining. The scale of precision machining becomes finer and closer to the dimensional scale of material properties. As a result, the acoustic emission (AE) from microscopic sources becomes significant [1].

AE is the class of phenomena whereby transient elastic waves are generated by the rapid release of energy by a localized source or sources within a material, or transient elastic wave(s) so generated (ANSI/ASTM E 610-89). Emissions from process changes, like tool wear, chip formation, can be directly related to the mechanics of the process. AE-based sensing methodologies for tool condition and cutting process monitoring have been studied since 1977 [2]. Signal processing schemes were used to treat AE signal to extract the most useful information, for example, time series analysis [3, 4], Fourier transform [5, 6], Gabor transform [79], and wavelet transform [1013], and so forth. Because the information obtained during the machining process is vague, incomplete or imprecise, these conventional methods need a large number of cutting experiments and additional assumptions in many circumstances for effective uncertainty handling. These requirements reduce the reliability of the models and increase money and time consumption. Moreover, the general mathematical relation cannot be used to map the nonlinear relationship between the AE signal and tool wear condition [14]. Artificial intelligence methods have played an important role in modern tool condition monitoring (TCM) to observe the relation between tool wear and AE signal such as neural networks [15, 16], fuzzy logic [17], and fuzzy neural network [1822]. The increased use of artificial intelligence within TCM has enabled the development of more robust and comprehensive strategies.

It is believed that a relatively uncontaminated AE signal can be obtained because AE frequency range is much higher than that of machine vibrations and environmental noises and does not interfere with the cutting operation. AE can be effectively used for TCM applications at the precision scale. In fact, it is impossible to get an accurate AE signal. It is because that machining process varies considerably depending on the part material, temperature, cutting fluids, chip formation, the tool material, temperature, chatter and vibration, and so forth. Additionally, AE sensors are very sensitive to environmental changes such as changes in temperature, humidity, circuit noise, and even the locating error of the sensors. Moreover, changes of cutting conditions also affect the behaviour of acoustic emission signals. None of previous studies considered the uncertainty in AE signal.

The aim of this paper is to present an innovative type-2 Takagi-Sugeno-Kang (TSK) fuzzy modeling to capture the uncertainties in the AE signal in machining process in order to overcome the challenges in TCM. Type-2 TSK fuzzy modeling method is not only a powerful tool to model high complex nonlinear physical processes, but also a great estimator for the ambiguities and uncertainties associated with the system. It is capable to arrive at a definite conclusion without understanding the exact physics of the machining process. In this paper, type-2 TSK fuzzy modeling is implemented to filter the raw AE signal directly from the AE sensor during turning process. Furthermore, its output interval set assesses the uncertainty information in AE, which is of great value to a decision maker and can be used to investigate the complicated tool wear condition during machining process.

This paper is divided into four sections. Section 1 gives a brief overview of previous studies on AE-based TCM. Section 2 introduces TSK fuzzy logic and the type-2 TSK fuzzy modeling algorithm. Section 3 presents a case study where type-2 TSK fuzzy approach is used to filter the raw AE signal directly from the AE sensor and identify the uncertainty interval of AE. The experimental results show the effectiveness and advantages of type-2 TSK fuzzy modeling. Conclusion is given in Section 4.

2. Type-2 TSK Fuzzy Uncertainty Modeling

2.1. TSK Fuzzy Logic

Fuzzy logic has been originally proposed by Zadeh in his famous paper “Fuzzy Sets” in 1965 [23]. Fuzzy logic provides a simple way to obtain a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. TSK fuzzy logic system (FLS) [24, 25] was proposed in an effort to develop a systematic approach to generate fuzzy rules from a given input-output data set. This model consists of rules with fuzzy antecedents and a mathematical function in the consequent part. The antecedents divide the input space into a set of fuzzy regions, while consequents describe behaviour of the system in those regions. TSK FLS has a powerful capability of explaining complex relations among variables using rule consequents which are functions of the input variables. This is due to the model’s properties: on one hand being a general non-linear approximator that can approximate every continuous mapping and on the other hand being a piecewise linear model that is relatively easy to interpret [26] and whose linear submodels can be exploited for control and fault detection [27].

A generalized type-1 TSK model can be described by fuzzy IF-THEN rules which represent input-output relations of the system. For a MISO first-order type-1 TSK model, its 𝑘th rule can be expressed asIF𝑥1is𝑄1𝑘,𝑥2is𝑄2𝑘,,𝑥𝑛is𝑄𝑛𝑘,THEN𝑍is𝑤𝑘=𝑝𝑘0+𝑝𝑘1𝑥1+𝑝𝑘2𝑥2++𝑝𝑘𝑛𝑥𝑛,(1) where 𝑥1,𝑥2,𝑥𝑛 and 𝑍 are linguistic variables; 𝑄1𝑘,𝑄2𝑘,, and 𝑄𝑛𝑘 are the fuzzy sets on universe of discourses 𝑈,𝑉,, and 𝑊, and 𝑝𝑘0,𝑝𝑘1,, and 𝑝𝑘𝑛 are regression parameters.

A Gaussian MF can be expressed by the following formula for the 𝑣th variable:𝑄𝑘𝑣1=exp2𝑥𝑣𝑥𝑣𝑘𝜎2,(2) where 𝑥𝑣𝑘 is the mean of the 𝑣th input feature in the 𝑘th rule for 𝑣[0,𝑛]𝜎 is the standard deviation of Gaussian MF.

Based on Zadeh’s concept of type-2 fuzzy sets and extension principle [28], practical algorithms for conjunction, disjunction, and complement operations of type-2 fuzzy sets are obtained by extending previous studies [29]. Prior work [30] also introduced embedded interval-valued type-2 fuzzy sets and developed a general formula for the extended composition of type-2 relations, which is considered as an extension of the type-1 composition. The characterization in the definition of type-2 fuzzy sets uses the notion that type-1 fuzzy sets can be thought of as a first-order approximation of uncertainty and, therefore, type-2 fuzzy sets provide a second-order approximation. They play an important role in modeling uncertainties that exist in fuzzy logic systems [31] and are becoming increasingly important in the goal of “Computing with Words” [32] and “Computational Theory of Perceptions” [33]. A complete type-2 fuzzy logic theory with the handling of uncertainties was also established [34]. Because of its larger number of design parameters for each rule, it was believed that type-2 FLS have the potential to be used in control [35] and other areas where a type-l model may be unable to perform well [36]. Type-2 TSK FLS and its structures were presented in 1999 [37].

An example of a type-2 MF, whose vertices have been assumed to vary over some interval of value, is depicted in Figure 1. The footprint of uncertainty (FOU) associated with this type-2 MF is a bounded shaded region. FOU represents the entire interval type-2 fuzzy set 𝑄. Upper MF and Lower MF are two type-1 MFs that are bounds for the FOU of a type-2 set 𝑄. The intersections of crisp input 𝑥0 show that there are lower MF degree 𝑢 and upper MF degree 𝑢 with respective lower and upper MFs. Detailed type-2 fuzzy sets and interval type-2 FLS background material can be found in [38].

696947.fig.001
Figure 1: Type-2 Gaussian MF.

A generalized 𝑘th rule in the first-order type-2 TSK fuzzy MISO system can be expressed as IF𝑥1𝑄is1𝑘,𝑥2𝑄is2𝑘,,𝑥𝑛𝑄is𝑛𝑘,𝑤THEN𝑍is𝑘=̃𝑝𝑘0+̃𝑝𝑘1𝑥1+̃𝑝𝑘2𝑥2++̃𝑝𝑘𝑛𝑥𝑛,(3) where ̃𝑝𝑘0,̃𝑝𝑘1,,̃𝑝𝑘𝑛 are consequent parameters, 𝑤𝑘 is the output from the 𝑘th IF-THEN rule in an 𝑀-rule FLS, 𝑄1𝑘,𝑄2𝑘𝑄,,𝑛𝑘 are fuzzy sets on the universe of discourse.

One way to obtain a type-2 model directly form a type-1 model is by extending the cluster center, 𝑥𝑗𝑘, from a point to a constant-width interval-valued fuzzy set, ̃𝑥𝑗𝑘 as shown in Figure 2. The size of the interval is 2a:̃𝑥𝑗𝑘=𝑥𝑗𝑘1𝑎𝑘𝑗,𝑥𝑗𝑘1+𝑎𝑘𝑗.(4)

696947.fig.002
Figure 2: Spread of cluster center.

Consequent parameter ̃𝑝𝑘𝑗 is obtained by extending the consequent parameter 𝑝𝑘𝑗 from its type-1 counterpart using the following expression:̃𝑝𝑘𝑗=𝑝𝑘𝑗𝑠𝑘𝑗,𝑝𝑘𝑗+𝑠𝑘𝑗,(5) where 𝑗[0,𝑛], and 𝑠𝑘𝑗 denotes the spread of fuzzy numbers ̃𝑝𝑘𝑗.

Hence, the premise MF is changed from type-1 fuzzy set into type-2 fuzzy set, that is,𝑄𝑗𝑘1=exp2𝑥𝑗𝑥𝑗𝑘1±𝑎𝑘𝑗𝜎𝑘𝑗2,(6) where 𝜎𝑘𝑗 is the standard deviation of Gaussian MF.

Type-2 FLSs are very useful in circumstances in which it is difficult to determine an exact membership function for a fuzzy set. They can be used to handle rule uncertainties and even measurement uncertainties. Type-2 FLSs moves the world of FLSs into a fundamentally new and important direction. To date, type-2 FL moves in progressive ways where type-1 FL is eventually replaced or supplemented by type-2 FL [39].

2.2. Type-2 Fuzzy Modelling Algorithm

The diagram of type-2 TSK fuzzy modelling algorithm is shown in Figure 3. This algorithm is initially presented in [40]. Type-2 TSK fuzzy approach includes two steps: the first step is type-1 fuzzy modelling to eliminate noise components in the AE signal and the second consists of expanding the type-1 fuzzy system to its type-2 counterpart to obtain the information of uncertainty in AE signal. The first step is type-1 TSK fuzzy approach. Subtractive clustering method [41] (see Appendix A) combined with a least-square estimation algorithm is used to cope with the nonlinearity of the AE and the uncertainty of imprecise data from measurement. A detailed description for this type-1 fuzzy modelling can be found in [42, 43]. The second step is type-2 TSK fuzzy approach. The type-1 MFs are considered as the principal MFs of a type-2 FLS, the antecedent MFs are extended as interval type-2 fuzzy memberships by assigning uncertainty to cluster centers using (4), and the consequent parameters are extended as fuzzy numbers by assigning uncertainty to consequent parameter values using (5). The type-2 TSK fuzzy inference engine is presented in Appendix B. Through enumerative search of optimum values of spreading percentage of cluster centers and consequent parameters, the best approach for analysing AE signal is obtained. The Detailed description for this modelling algorithm can be found in [44]. Examples of application of this algorithm can be found in [4547].

696947.fig.003
Figure 3: Diagram of subtractive clustering-based type-2 TSK fuzzy approach.

Compared with traditional methods and its type-1 counterpart, type-2 fuzzy modeling can not only obtain a modeling result directly from the input-output data sets, but it can also capture the uncertainty interval of the result [34, 48]. The information about uncertainties in the type-2 interval output is very helpful for decision making.

3. Experimental Study

3.1. Experimental Setup

The experiment described in this paper was taken on the BOEHRINGER CNC lathe. The workpiece material was Titanium Metal matrix Composite (Ti MMC) 10% wt. TiC/Ti-6Al-4V where the microstructural response of cast Ti-6Al-4V-based composite contains 10 vol.-% TiC reinforcement. This kind of material is widely used in aerospace and military applications for its high hardness, light weight, high bending strength, fracture toughness, higher modulus, and elevated temperature resistance and high wear resistance. Consequently, its machining is very difficult.

The cutting tool insert was carbide from SECO tools (CNMG 120408 MF1 CP200). Turning test was done on a cylinder of Ti MMC 2.5 diameter in dry machining conditions. The machining speed was chosen to exceed the manufacturer’s recommendation in order to see and check the tool cutting speed limits (80 m/min). The cutting depth was kept at 0.15 mm and the cutting feed was 0.1 mm.

The aim of this study was to find out the relation between AE and tool wear. During the test, every time when cutting length reached 10 mm, the machine was stopped to manually measure the tool wear parameter (𝑉𝐵𝐵). Figure 4 shows one example of a raw AE signal AEraw directly from AE sensor. During the first 5~8 s, the cutting tool is approaching the workpiece and gradually reaching the cutting depth. After 30 s, the cutting tool leaves the surface of workpiece. The middle period is the steady cutting period, which contains the most useful information for tool wear condition investigation. In the experiment, five AE signal sets were recorded according to different cutting sections: 0~10 mm, 10~20 mm, 20~30 mm, 30~40 mm, and 40~50 mm. This paper focuses on filtering and capturing the uncertainty on AE signals during the five continuous cutting periods using type-2 TSK fuzzy modeling.

696947.fig.004
Figure 4: AE signal from cutting process.
3.2. Data Processing

First, type-1 TSK fuzzy filtering (top part of Figure 3) was used to eliminate noise components in the AE signal. Demirli’s extended subtractive clustering identification algorithm [49] was used to estimate both the number and initial location of cluster centers and extract the TSK fuzzy rules from input/output data. The clustering parameters are preinitialized. The cluster radius is confined to the range [0.15; 1.0] with a step size of 0.15. The accept ratio and the reject ratio are both considered in the range [0; 1.0] with a step size of 0.1. The squash factor is considered in the range [0.05; 2] with a step size of 0.05. Combined with a least-square estimation algorithm, the fuzzy systems for each cutting length were identified. Table 1 lists the number of rules identified and the standard deviation used for the five AE signal sets. List of cluster centres can be found in [17].

tab1
Table 1: Number of rules and parameters of type-1 TSK modeling.

Traditionally, the AE signal is characterized using AE root-mean-square (RMS) measurement in well-controlled tensile tests. To compare AE signal obtained by fuzzy filtering with the one by traditional filter, AE RMS values (illustrated in Figure 5) and AE mean values (depicted in Figure 6) are calculated for both cases. The dotted curves represent the values obtained by traditional filtering, and the solid ones represent the values obtained by fuzzy filtering. In Figure 5, the dotted curves are above the solid curves. This means that the AE RMS values obtained using the traditional filter are larger than those obtained with fuzzy filtering. The difference could be caused by computation of different algorithm, where fuzzy filtering generates fuzzy rules directly from the input-output data acquired from sensor, without traditional user-defined high-pass and low-pass filters. The mean values obtained by the two methods are almost same as shown in Figure 6.

fig5
Figure 5: AE RMS value for different cutting section.
fig6
Figure 6: AE mean value for different cutting section.

The second step consists of expanding the type-1 fuzzy system to a type-2 system. Because the AE signals used are relatively uncontaminated, uncertainty in the AE signal is much smaller than the raw AE signal value. The spreading percentage for clusters is confined to the range [0.0%; 0.01%] with a step size of 0.0001%. The spreading percentage for the consequent parameters is considered as 2%. Spreading percentage for clusters and consequent parameters can be found in [17]. The information on uncertainty in the five AE signal sets is shown in Figure 7 between the type-2 fuzzy output upper boundary AE (dotted curve) and the lower boundary AE (dashed curve). The overall identified AE signal value AEfuzzy is shown as solid curve.

fig7
Figure 7: Uncertainties in AE signal in different cutting sections.

As indicated on Figure 7, there is the uncertainty interval for each cutting section. The most significant is shown on Figure 7(e). Table 2 summarizes the value of maximum and minimum variations between AE and 𝐸fuzzy, AE and AEfuzzy, AE and AE, also AEfuzzy and the raw AE signal value AEraw, where, representatively, 𝑉1=|||AEAEfuzzy|||,𝑉2=||AEAEfuzzy||,𝑉3=|||AEAE|||,𝑉4=||AErawAEfuzzy||.(7)

tab2
Table 2: The variations in modeling results from the five AE signal sets.

The greatest variation of each cutting instant is 𝑉3 between the upper boundary and lower boundary of type-2 interval output. The biggest value is 24.7279 mv in the final cutting section, as same as that on Figure 7(e). The last colon on Table 2 lists 𝑉𝐵𝐵 measured in the end of each cutting section.

As shown in Figures 8 and 9, the development trends of maximum and minimum variations are the same as the tool wear trend. The maximum changes in tool wear condition and AE signal both occurred during the period when the cutting length was changed from 40 mm~50 mm. It is observed that during the initial cutting period (cutting length from 0~40 mm), the variations of AE signal correspond to the initial stages of wear occurring. The period with the most significant variations (cutting section 40~50 mm) corresponds to the period of relatively rapid wear or failure of the cutting tool. Along with the increasing of uncertainty in AE signal, the development of wear is continuous and monotonically increasing. The sufficient information from AE uncertainty scheme can be used to make decision or investigate tool condition so as to enhance the reliability of tool wear estimation.

fig8
Figure 8: Maximum and minimum variations in different cutting sections.
696947.fig.009
Figure 9: Tool wears in different cutting sections.

4. Conclusion

This paper presented a type-2 fuzzy modelling method to determine AE in precision machining. The interval output of type-2 fuzzy modelling provides the information of uncertainty in AE estimations. The sufficient information from AE uncertainty scheme can be used to make decision or investigate tool condition so as to enhance the reliability of tool wear estimation. By applying type-2 fuzzy logic to AE-based tool conditions monitoring, it is possible to automate online diagnosis of cutting tool condition.

Type-2 FLS can model and analyse the uncertainties in machining from the vague information obtained during machining process. The estimation of uncertainties can be used for proving the conformance with specifications for products or autocontrolling of machine system. The application of type-2 fuzzy logic on uncertainty estimation in high precision machining can enable the unmanned use of flexible manufacturing systems and machine tools. It has great meaning for continuous improvement in product quality, reliability, and manufacturing efficiency in machining industry.

Appendices

A. Subtractive Clustering

Subtractive clustering identification algorithm [41] is to estimate both the number and initial location of cluster centers and extract the TSK fuzzy rules from input/output data. Subtractive clustering operates by finding the optimal data point to define a cluster centre based on the density of surrounding data points. This method is a fast clustering method designed for high-dimension problems with a moderate number of data points. This is because its computation grows linearly with the data dimension and as the square of the number of data points. A brief description of Chiu’s subtractive clustering method is as follows.

Consider a collection of 𝑞 data points {𝑥1,𝑥2,𝑥𝑛} specified by 𝑚-dimensional 𝑥𝑗. Without loss of generality, assume the feature space is normalized so that all data are bounded by a unit hypercube. Calculate potential for each point by using equation below:𝑝𝑖=𝑞𝑗=1𝑒𝛼||𝑥𝑖𝑥𝑗2||4,𝛼=𝑟2𝑎,(A.1) where denotes the Euclidean distance. It is noteworthy that only the fuzzy neighbourhood within the radius 𝑟𝑎 contributes to the measure of potential.

After the potential of every data point has been computed, the data point with the highest potential is selected as the first cluster center. Assume 𝑥1 is the location of the first cluster center, and 𝑝1 is its potential value, then revise the potential of each data point 𝑥𝑗 by the formula𝑝𝑖𝑝𝑖𝑝1𝑒𝛽||𝑥𝑖𝑥1||2,(A.2) where 𝛽=4/𝑟2𝑏 and 𝑟𝑏=𝜂𝑟𝑎.

When the potential of all data points have been reduce, by (A.2), the data point with the highest remaining potential is selected as the second cluster center. Then further reduce the potential of each data points. Generally, after 𝑘th cluster center has been obtained, the potential of each data point is revised by formula𝑝𝑖𝑝𝑖𝑝𝑘𝑒𝛽||𝑥𝑖𝑥𝑘||2,(A.3) where 𝑥1 is the location of the 𝑘th cluster center and 𝑝1 is its potential value.

The process of acquiring new cluster center and revising potential repeats by using the Algorithm 1.

alg1
Algorithm 1:

B. Type-2 TSK Fuzzy Inference Engine

For the most general structure of T2 TSK FLS—Model I, antecedents are T2 fuzzy sets and consequents are T1 fuzzy sets. Membership grades are interval sets, that is,𝜇𝑘𝑣=𝜇𝑘𝑣,𝜇𝑘𝑣,(B.1) where 𝜇𝑘𝑣_ and _𝜇𝑘𝑣 are the lower value and upper value of the 𝑣th input variable in the 𝑘th rule.

The explicit dependence of the total firing interval for 𝑘th rule can be computed as𝑓𝑘=𝜇𝑘1𝑥1𝜇𝑘2𝑥2𝜇𝑘𝑛𝑥𝑛,(B.2)_𝑓𝑘=𝜇𝑘1𝑥1𝜇𝑘2𝑥2𝜇𝑘𝑛𝑥𝑛,(B.3) where variable 𝑓𝑘 and _𝑓𝑘 denote lower value and upper value of fire strength. The symbol is a conjunction operator, which is a T-norm. It can be either MIN operator or product operator .

The interval value of the consequent of the 𝑘th rule 𝑤𝑘 is 𝑤𝑘=[𝑤𝑘𝑙,𝑤𝑘𝑟], where𝑤𝑘𝑙=𝑛𝑗=1𝑐𝑘𝑗𝑥𝑗+𝑐𝑘0𝑛𝑗=1𝑠𝑘𝑗||𝑥𝑗||𝑠𝑘0,𝑤𝑘𝑙=𝑛𝑗=1𝑐𝑘𝑗𝑥𝑗+𝑐𝑘0+𝑛𝑗=1𝑠𝑘𝑗||𝑥𝑗||𝑠𝑘0.(B.4)

Here 𝑤𝑙𝑘 and 𝑤𝑘𝑟 denote lower and upper values of consequent output for 𝑘th rule. 𝑐𝑘𝑗 and 𝑠𝑘𝑗 denote the centre (mean) and the spread of fuzzy number ̃𝑝𝑘𝑗.

So, the extended output of the IT2 TSK FLS can be calculated by using following equation:𝑤𝑤=𝑙,𝑤𝑟=𝑤1𝑤1𝑙,𝑤1𝑟𝑤𝑛𝑤𝑛𝑙,𝑤𝑛𝑟×𝑓1𝑓1,𝑓1𝑓𝑀𝑓𝑀,𝑓𝑀1𝑀𝑘=1𝑓𝑘𝑤𝑘/𝑀𝑘=1𝑓𝑘.(B.5)

Hence 𝑤=[𝑤𝑙,𝑤𝑟] is an interval type-1 set, the two endpoints 𝑤𝑙 and 𝑤𝑟 can be obtained by using equations below:𝑤𝑙=𝑛𝑘=1𝑓𝑘𝑙𝑤𝑘𝑙𝑛𝑘=1𝑓𝑘𝑙,𝑤𝑟=𝑛𝑘=1𝑓𝑘𝑟𝑤𝑘𝑟𝑛𝑘=1𝑓𝑘𝑟.(B.6)

This interval set of the output has the information about the uncertainties that are associated with the crisp output, and this information can only be obtained by working with T2 TSK FLS. To compute𝑤, two endpoints 𝑤𝑙 and 𝑤𝑟 must be computed. In order to compute 𝑤𝑙 and 𝑤𝑟, 𝑓𝑘𝑙 and 𝑓𝑘𝑟 have to be determined. 𝑤𝑙 and 𝑤𝑟 can be obtained by using the iterative procedure KM Algorithm [30]. Here, the computation procedure for 𝑤𝑙 and 𝑤𝑟 is briefly provided as follows.

Without loss of generality, assume that the precomputed 𝑤𝑘𝑟 are arranged in ascending order: 𝑤1𝑟𝑤2𝑟𝑤𝑚𝑟, then, we have the following.

Step 1. Compute 𝑤𝑟 in (B.6) by initially setting 𝑓𝑘𝑟=𝑓𝑘+_𝑓𝑘2(B.7) for 𝑘=1,,𝑅, where 𝑓𝑘 and _𝑓𝑘 have been previously computed using (B.2) and (B.3), respectively, and let 𝑤𝑟𝑤𝑟.

Step 2. Find 𝑦 (1𝑦𝑚1) such that 𝑤𝑦𝑟𝑤𝑟𝑤𝑟𝑦+1.

Step 3. Compute 𝑤𝑟 in with 𝑓𝑘𝑟=𝑓𝑘 for 𝑘𝑦 and 𝑓𝑘𝑟=_𝑓𝑘 for 𝑘>𝑦, and let 𝑤𝑟𝑤𝑟.

Step 4. If 𝑤𝑟𝑤𝑟, then go to Step 5. If 𝑤𝑟=𝑤𝑟, then stop. And set 𝑤𝑟𝑤𝑟.

Step 5. Set 𝑤𝑟=𝑤𝑟 and return to Step 2.

The procedure for computing 𝑤𝑙 is very similar to the one just given for 𝑤𝑟. Replace 𝑤𝑘𝑟 by 𝑤𝑘𝑙, and compute 𝑤𝑙. In Step 2 find 𝑧 (1𝑧𝑚1) such that 𝑤𝑧𝑙𝑤𝑙𝑤𝑙𝑧+1. Additionally, in Step 3 compute 𝑤𝑙 with 𝑓𝑘𝑟=𝑓𝑘 for 𝑘𝑧 and 𝑓𝑘𝑟=_𝑓𝑘 for 𝑘>𝑧.

In an interval type-2 TSK FLS, output 𝑤 is an interval type-1 fuzzy set, so the crisp output of any interval type-2 TSK FLS can be obtained by using the average value of 𝑤𝑙 and 𝑤𝑟, that is, 𝑤=𝑤𝑙+𝑤𝑟2.(B.8)

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