General type-2 fuzzy logic systems (GT2 FLSs) have drawn great attentions since the alpha-planes representation of general type-2 fuzzy sets (GT2 FSs) was proposed. The iterative of type-reduction (TR) algorithms are difficult to apply in practical applications. In the enhanced types of algorithms, the Nagar-Bardini (NB) algorithms decrease the computation complexity greatly. In terms of the Newton-Cotes quadrature formulas of numerical integration techniques, the paper extends the NB algorithms to three different forms of weighted NB (WNB) algorithms according to the comparisons between the sum operation in NB algorithms and the integral operation in continuous version of NB (CNB) algorithms. The NB algorithms just become a special case of the WNB algorithms. Four simulation examples are used to illustrate and analyze the performances of the WNB algorithms while performing the centroid TR of GT2 FLSs. It also shows that, in general, the WNB algorithms have smaller absolute error and faster convergence speed compared with the NB algorithms, which provides the potential value for T2 FLSs designers and users.
MendelJ.M., Type-2 fuzzy sets and systems: An overview, IEEE Computational Intelligence Magazine2 (2007), 20–29.
2.
HagrasH. and WagnerC., Towards the wide spread use of type-2 fuzzy logic systems in real world applications, IEEE Computational Intelligent Magazine7 (2012), 14–24.
3.
ChenY., WangD.Z. and TongS.C., Forecasting studies by designing Mamdani interval type-2 fuzzy logic systems: with combination of BP algorithms and KM algorithms, Neurocomputing174 (2016), 1133–1146.
4.
MaldonadoY., CastilloO. and MelinP., Particle swarm optimization of interval type-2 fuzzy systems for FPGA applications, Appllied Soft Computing13 (2013), 508–496.
5.
GreenfieldS. and ChiclanaF., Accuracy and complexity evaluation of defuzzification strategies for the discretised interval type-2 fuzzy set, International Journal of Approximate Reasoning54 (2013), 1013–1033.
6.
NieM. and TanW.W., Ensuring the centroid of an interval type-2 fuzzy set, IEEE Transactions on Fuzzy Systems23 (2015), 950–963.
7.
HagrasH. and WagnerC., Towards general type-2 fuzzy logic systems based on zSlices, IEEE Transactions on Fuzzy Systems18 (2010), 637–660.
8.
ChenY. and WangD.Z., Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted Karnik-Mendel algorithms, Soft Computing22 (2018), 1361–1380.
9.
MendelJ.M., General type-2 fuzzy logic systems made simple: A tutorial, IEEE Transactions on Fuzzy Systems22 (2014), 1162–1182.
10.
SanchezM.A., CastilloO. and CastroJ.R., Generalized type-2 fuzzy systems for controlling a mobile robot and a performance comparison with interval type-2 and type-1 fuzzy systems, Expert Systems with Applications42 (2015), 5904–5914.
11.
ChenY. and WangD.Z., Forecasting by general type-2 fuzzy logic systems optimized with QPSO algorithms, International Journal of Control, Automation and Systems15 (2017), 2950–2958.
12.
CastilloO., Amador-AnguloL., CastroJ.R. and Garcia-ValdezM., A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems, Information Sciences354 (2016), 257–274.
13.
MelinP., GonzalezC.I., CastroJ.R., MendozaO. and CastilloO., Edge-detection method for image processing based on generalized type-2 fuzzy logic, IEEE Transactions on Fuzzy Systems22 (2014), 1515–1525.
14.
ChenY., WangD.Z. and NingW., Forecasting by TSK general type-2 fuzzy logic systems optimized with genetic algorithms, Optimal Control Applications & Methods39 (2018), 393–409.
15.
MendelJ.M., Alpha-plane representation for type-2 fuzzy sets: Theory and applications, IEEE Transactions on Fuzzy Systems17 (2009), 1189–1207.
16.
MendelJ.M. and WuH.W., Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 1, forward problems, IEEE Transactions on Fuzzy Systems14 (2006), 781–792.
17.
MendelJ.M., Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Englewood Cliffs, NJ, USA: Prentice-Hall, 2001.
18.
MendelJ.M., On KM algorithms for solving type-2 fuzzy set problems, IEEE Transactions on Fuzzy Systems21 (2013), 426–446.
19.
WuD.R. and MendelJ.M., Enhanced Karnik-Mendel algorithms, IEEE Transactions on Fuzzy Systems17 (2009), 923–934.
20.
MendelJ.M. and WuH.W., Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: part 1, forward problems, IEEE Transactions on Fuzzy Systems14 (2006), 781–792.
21.
MendelJ.M. and LiuF.L., Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set, IEEE Transactions on Fuzzy Systems15 (2007), 309–320.
22.
LiuX.W., MendelJ.M. and WuD.R., Study on enhanced Karnik-Mendel algorithms: Initialization explanations and computation improvements, Information Sciences184 (2012), 75–91.
23.
EI-NagarA.M. and EI-BardiniM., Simplified interval type-2 fuzzy logic system based on new type-reduction, Journal of Intelligent & Fuzzy Systems27 (2014), 1999–2010.
24.
LiJ.W., JohnR., CouplandS. and KendallG., On Nie-Tan operator and type-reduction of interval type-2 fuzzy sets, IEEE Transactions on Fuzzy Systems26 (2018), 1036–1039.
25.
BiglarbegianM., MelekW.W. and MendelJ.M., On the robustness of type-1 and interval type-2 fuzzy logic systems in modeling, Information Sciences181 (2011), 1325–1347.
26.
BiglarbegianM., MelekW.W. and Mendel.J.M., On the stability of interval type-2 TSK fuzzy logic systems, IEEE Transactions on Cybernetics40 (2010), 798–818.
27.
GreenfieldS., ChiclanaF., CouplandS. and JohnR., The collapsing method of defuzzification for discretised interval type-2 fuzzy sets, Information Sciences179 (2009), 2055–2069.
28.
ChenY., Study on weighted Nagar-Bardini algorithms for centroid type-reduction of interval type-2 fuzzy logic systems, Journal of Intelligent & Fuzzy Systems34 (2018), 2417–2428.
29.
WangT., ChenY. and TongS.C., Fuzzy reasoning models and algorithms on type-2 fuzzy sets, International Journal of Innovative Computing, Information and Control4 (2008), 2451–2460.
30.
ChenY., WangD.Z. and NingW., Studies on centroid type-reduction algorithms for general type-2 fuzzy logic systems, International Journal of Innovative Computing, Information and Control11 (2015), 1987–2000.
31.
MendelJ.M. and JohnR.B., Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems10 (2002), 117–127.
32.
KlirG.J. and YuanB., Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, Upper Saddle River, NJ.
33.
LiuF.L., An efficient centroid type-reduction strategy for general type-2 fuzzy logic system, Information Sciences178 ( (2008), 2224–2236.
34.
ChenY. and WangD.Z., Type-reduction of interval type-2 fuzzy logic systems with weighted Karnik-Mendel algorithms, Control Theory and Applications33 (2016), 1327–1336.
35.
MathewsJ.H. and FinkK.D., Numerical Methods Using Matlab, Prentice-Hall, Upper Saddle River, NJ.
36.
KhanesarM.A., JalalianA., KaynakO. and GaoH., Improving the speed of center of sets type reduction in interval type-2 fuzzy systems by eliminating the need for sorting, IEEE Transactions on Fuzzy Systems25 (2017), 1193–1206.
37.
GonzalezC., CastroJ.R., MelinP. and CastilloO., An edge detection method based on generalized type-2 fuzzy logic, Soft Computing20 (2016), 773–784.
38.
Ontiveros-RoblesE., MelinP. and CastilloO., Comparative analysis of noise robustness of type-2 fuzzy logic controllers, Kybernetika54 (2018), 175–201.
39.
HsuC.H. and JuangC.F., Evolutionary robot wall-following control using type- 2 fuzzy controller with species-de-activated continuous ACO, IEEE Transactions on Fuzzy Systems21 (2013), 100–112.
40.
Ontiveros-RoblesE., MelinP. and CastilloO., New methodology to approximate type-reduction based on a continuous root-finding Karnik Mendel algorithm, Algorithms10 (2017), 1–21.
