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arxiv: 1907.01697 · v1 · pith:RBD6CKWRnew · submitted 2019-07-03 · 💻 cs.AI · cs.SY· eess.SY

Recommendations on Designing Practical Interval Type-2 Fuzzy Systems

Dongrui Wu , Jerry Mendel This is my paper

Pith reviewed 2026-05-25 10:48 UTC · model grok-4.3

classification 💻 cs.AI cs.SYeess.SY
keywords interval type-2 fuzzy systemsdesign recommendationssingleton fuzzifiermembership functionstype reductiont-normoptimizationpractical design
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The pith

Recommended defaults for fuzzifiers, membership functions, and inference methods make interval type-2 fuzzy systems easier to design from the start.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to establish a set of representative starting choices that address the main design questions for interval type-2 fuzzy systems. A sympathetic reader would care because the added complexity of these systems, compared with type-1 versions, can discourage new users even when the systems have shown strong results in applications. If the choices prove useful, designers could move more quickly from concept to working prototype without first resolving every parameter through exhaustive search. The authors focus on practical accessibility rather than claiming optimality for every case.

Core claim

The authors recommend singleton fuzzifier, a moderate number of membership functions per input, Gaussian or piecewise linear membership functions, either Mamdani or TSK inference, product t-norm, use of type-reduction, and standard optimization methods as concrete starting points that lower the barrier for new interval type-2 fuzzy system designers.

What carries the argument

A list of default answers to the core design questions (singleton vs non-singleton fuzzifier, membership function count and shape, inference type, t-norm choice, type-reduction, and optimization) that together form an initial practical configuration.

If this is right

  • Beginners gain a concrete path to build a working interval type-2 system without first mastering every design option.
  • Initial prototypes become more reproducible across different research groups.
  • Type-reduction and optimization steps can be applied directly on top of the defaults to refine performance.
  • The overall entry cost for experimenting with interval type-2 systems drops, expanding the set of people who can test them.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These defaults could be encoded into design software so that users receive a ready-to-tune initial system rather than a blank slate.
  • In applications with very high noise, a follow-up test comparing the defaults against non-singleton fuzzifiers would show when the starting choice needs adjustment.
  • The same default list might serve as a baseline for comparing interval type-2 systems against other uncertainty-handling methods such as probabilistic or evidential approaches.

Load-bearing premise

That the suggested starting choices serve as broadly useful defaults across applications without requiring case-by-case validation or extensive prior domain knowledge.

What would settle it

A study in which the recommended defaults produce markedly inferior results compared with other initial choices across several unrelated applications, forcing designers to select different parameters before any optimization begins.

Figures

Figures reproduced from arXiv: 1907.01697 by Dongrui Wu, Jerry Mendel.

Figure 1
Figure 1. Figure 1: Cumulative number of Google Scholar publications on [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Structures of and design choices for (a) T1 fuzzy syst [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fuzzifiers and their choices: two choices for a T1 fuzz [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Rulebase: Gaussian or piecewise linear MFs or FOUs? H [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example input-output mappings of 2-input IT2 fuzzy s [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rulebase: Zadeh or TSK rules? THEN y(x) is Y˜ n , n = 1, . . . , N (8) where Y˜ n is an IT2 fuzzy set. A TSK rule is of the form R˜n : IF x1 is X˜ n 1 and . . . and xp is X˜ n p , THEN y n (x) = [y n , y n ] = [c n 0 + c n 1 x1 + · · · + c n p xp, c n 0 + c n 1 x1 + · · · + c n p xp] where y n, y n , {c n k }k=0,...,p and {c n k }k=0,...,p are crisp num￾bers. More complicated nonlinear functions can also b… view at source ↗
Figure 7
Figure 7. Figure 7: Inference: Minimum or product used to compute firing l [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Several efficient methods have been proposed for computing yl(x ′ ) and yr(x ′ ) [14], [21], [22], [25], [29], [53], [58], [8], including the well-known Karnik-Mendel (KM) algorithms [25], [35]. Comprehensive descriptions and comparisons of the methods are given in [33], [52], [9]. The speeds of the algorithms are programming language dependent [9]. The Enhanced Iterative Algorithm with Stop Condition (EIA… view at source ↗
Figure 8
Figure 8. Figure 8: Output Processing for Mamdani and TSK architectures [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimization of fuzzy systems. H. Summary Our recommendations for practical T1 fuzzy system design are summarized in [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Summary of recommendations for designing (a) T1 and [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Firing levels of the T1 fuzzy controller, and firing i [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
read the original abstract

Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They have demonstrated superior performance in many applications. However, the operation of an IT2 fuzzy system is more complex than that of its type-1 counterpart. There are many questions to be answered in designing an IT2 fuzzy system: Should singleton or non-singleton fuzzifier be used? How many membership functions (MFs) should be used for each input? Should Gaussian or piecewise linear MFs be used? Should Mamdani or Takagi-Sugeno-Kang (TSK) inference be used? Should minimum or product $t$-norm be used? Should type-reduction be used or not? How to optimize the IT2 fuzzy system? These questions may look overwhelming and confusing to IT2 beginners. In this paper we recommend some representative starting choices for an IT2 fuzzy system design, which hopefully will make IT2 fuzzy systems more accessible to IT2 fuzzy system designers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript offers experience-based recommendations for several key design decisions in interval type-2 (IT2) fuzzy systems—singleton vs. non-singleton fuzzifier, number of membership functions per input, Gaussian vs. piecewise-linear MFs, Mamdani vs. TSK inference, min vs. product t-norm, whether to use type-reduction, and optimization approach—with the goal of providing accessible starting points for beginners.

Significance. If the suggested defaults prove broadly serviceable, the paper could meaningfully reduce the initial design burden for new IT2 practitioners by replacing an open-ended list of questions with concrete, representative choices; the contribution is modest because the recommendations are framed as experience-derived heuristics rather than validated optima or parameter-free results.

minor comments (1)
  1. [Abstract] Abstract: the list of design questions is clear, but the abstract does not preview any of the actual recommended choices (e.g., singleton fuzzifier, moderate MF count); adding one sentence summarizing the main suggestions would give readers an immediate sense of the paper’s concrete output.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the recommendation of minor revision. The report provides a helpful summary of the paper's goals but does not list any specific major comments requiring detailed responses or changes.

Circularity Check

0 steps flagged

No significant circularity; advisory recommendations without derivations or self-referential reductions

full rationale

The paper presents practical starting choices for IT2 fuzzy system design (e.g., singleton fuzzifier, moderate MF counts) framed explicitly as guidance to lower barriers for beginners. No derivation chain, equations, parameter fitting, or theorems are claimed. Self-citations to prior IT2 work exist but are not load-bearing for any 'prediction' or uniqueness result; the central content is experiential advice, not a reduction to inputs by construction. This matches the default expectation of no circularity for non-derivational papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that interval type-2 systems have demonstrated superior performance in applications and that design choices can be generalized from experience; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)
  • domain assumption Interval type-2 fuzzy systems have demonstrated superior performance in many applications compared to type-1 counterparts.
    Stated directly in the abstract as background for the need for design guidance.

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Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    M. B. Begian, W. W. Melek, J. M. Mendel, Stability analysi s of type- 2 fuzzy systems, in: Proc. IEEE Int’l Conf. on Fuzzy Systems, Hong Kong, 2008

    work page 2008

  2. [2]

    Bustince, E

    H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez, Z. Xu, B. Bedre- gal, J. Montero, H. Hagras, F. Herrera, B. D. Baets, A histori cal account of types of fuzzy sets and their relationships, IEEE Trans. o n Fuzzy Systems 24 (1) (2016) 179–194

    work page 2016

  3. [3]

    Castillo, P

    O. Castillo, P . Melin, Type-2 Fuzzy Logic Theory and Appl ications, Springer-V erlag, Berlin, 2008

    work page 2008

  4. [4]

    Cervantes, O

    L. Cervantes, O. Castillo, D. Hidalgo, R. Martinez-Soto , Fuzzy dynamic adaptation of gap generation and mutation in genetic optimi zation of type 2 fuzzy controllers, Advances in Operations Research 2 018

    work page

  5. [5]

    S. S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, I EEE Trans. on Systems, Man, and Cybernetics SMC-2 (1) (1972) 30– 34

    work page 1972

  6. [6]

    C. Chen, R. John, J. Twycross, J. M. Garibaldi, An extende d ANFIS architecture and its learning properties for type-1 and int erval type-2 models, in: IEEE Int’l Conf. on Fuzzy Systems, V ancouver, Ca nada, 2016

    work page 2016

  7. [7]

    C. Chen, R. John, J. Twycross, J. M. Garibaldi, Type-1 and interval type-2 ANFIS: A comparison, in: IEEE Int’l Conf. on Fuzzy Sys tems, Naples, Italy, 2017

    work page 2017

  8. [8]

    C. Chen, R. John, J. Twycross, J. M. Garibaldi, A direct ap proach for determining the switch points in the KarnikMendel algorith m, IEEE Trans. on Fuzzy Systems 26 (2) (2018) 1079–1085

    work page 2018

  9. [9]

    C. Chen, D. Wu, J. M. Garibaldi, R. John, J. Twycross, J. M. Mendel, A comment on A direct approach for determining the switch poin ts in the KarnikMendel algorithm, IEEE Trans. on Fuzzy Systems 26 (6) (2018) 3905–3907

    work page 2018

  10. [10]

    T. W. Chua, W. W. Tan, GA optimisation of non-singleton f uzzy logic system for ECG classification, in: Proc. IEEE Congress on Evo lutionary Computation, Singapore, 2007

    work page 2007

  11. [11]

    Coupland, R

    S. Coupland, R. I. John, Geometric type-1 and type-2 fuz zy logic systems, IEEE Trans. on Fuzzy Systems 15 (1) (2007) 3–15

    work page 2007

  12. [12]

    Dereli, A

    T. Dereli, A. Baykasoglu, K. Altun, A. Durmusoglu, I. B. Turksen, Industrial applications of type-2 fuzzy sets and systems: A concise review, Computers in Industry 62 (2011) 125–137

    work page 2011

  13. [13]

    X. Du, H. Ying, Derivation and analysis of the analytica l structures of the interval type-2 fuzzy-PI and PD controllers, IEEE Trans . on Fuzzy Systems 18 (4) (2010) 802–814

    work page 2010

  14. [14]

    Duran, H

    K. Duran, H. Bernal, M. Melgarejo, Improved iterative a lgorithm for computing the generalized centroid of an interval type-2 fu zzy set, in: Proc. NAFIPS, New Y ork, 2008

    work page 2008

  15. [15]

    C. I. Gonzalez, P . Melin, J. R. Castro, O. Castillo, O. Me ndoza, Optimization of interval type-2 fuzzy systems for image edg e detection, Applied Soft Computing 47 (2016) 631–643

    work page 2016

  16. [16]

    Gorzalczany, Decision making in signal transmissio n problems with interval-valued fuzzy sets, Fuzzy Sets and Systems 23 (1987 ) 191–203

    M. Gorzalczany, Decision making in signal transmissio n problems with interval-valued fuzzy sets, Fuzzy Sets and Systems 23 (1987 ) 191–203

    work page 1987

  17. [17]

    Greenfield, F

    S. Greenfield, F. Chiclana, S. Coupland, R. John, The col lapsing method of defuzzification for discretised interval type-2 fuzzy se ts, Information Sciences 179 (13) (2008) 2055–2069

    work page 2008

  18. [18]

    Gupta, S

    N. Gupta, S. K. Jain, Comparative analysis of fuzzy powe r system stabilizer using different membership functions, Interna tional Journal of Computer and Electrical Engineering 2 (2) (2010) 262–267

    work page 2010

  19. [19]

    Hagras, Type-2 FLCs: A new generation of fuzzy contro llers, IEEE Computational Intelligence Magazine 2 (1) (2007) 30–43

    H. Hagras, Type-2 FLCs: A new generation of fuzzy contro llers, IEEE Computational Intelligence Magazine 2 (1) (2007) 30–43. 14

    work page 2007

  20. [20]

    Hagras, C

    H. Hagras, C. Wagner, Towards the wide spread use of type -2 fuzzy logic systems in real world applications, IEEE Computational Int elligence Magazine 7 (3) (2012) 14–24

    work page 2012

  21. [21]

    H. Hu, Y . Wang, Y . Cai, Advantages of the enhanced opposi te direction searching algorithm for computing the centroid of an interv al type-2 fuzzy set, Asian Journal of Control 14 (6) (2012) 1–9

    work page 2012

  22. [22]

    H. Z. Hu, G. Zhao, H. N. Y ang, Fast algorithm to calculate generalized centroid of interval type-2 fuzzy set, Control Decis. 25 (4) (2010) 637– 640

    work page 2010

  23. [23]

    Huang, M

    J. Huang, M. Ri, D. Wu, S. Ri, Interval type-2 fuzzy logic modeling and control of a mobile two-wheeled inverted pendulum, IEEE Trans. on Fuzzy Systems 26 (4) (2018) 2030–2038

    work page 2018

  24. [24]

    J. R. Jang, ANFIS: adaptive-network-based fuzzy infer ence system, IEEE Trans. on Systems, Man, and Cybernetics 23 (3) (1993) 66 5–685

    work page 1993

  25. [25]

    N. N. Karnik, J. M. Mendel, Centroid of a type-2 fuzzy set , Information Sciences 132 (2001) 195–220

    work page 2001

  26. [26]

    Li, TOPSIS-based nonlinear-programming methodolo gy for multi- attribute decision making with interval-valued intuition istic fuzzy sets, IEEE Trans

    D. Li, TOPSIS-based nonlinear-programming methodolo gy for multi- attribute decision making with interval-valued intuition istic fuzzy sets, IEEE Trans. on Fuzzy Systems 18 (2) (2010) 299–311

    work page 2010

  27. [27]

    Liang, N

    Q. Liang, N. N. Karnik, J. M. Mendel, Connection admissi on control in A TM networks using survey-based type-2 fuzzy logic syste ms, IEEE Trans. on Systems, Man, and Cybernetics 30 (3) (2000) 329–33 9

    work page 2000

  28. [28]

    Martinez-Soto, O

    R. Martinez-Soto, O. Castillo, L. T. Aguilar, Type-1 an d type-2 fuzzy logic controller design using a hybrid PSO-GA optimization method, Information Sciences 285 (2014) 35–49

    work page 2014

  29. [29]

    Melgarejo, A fast recursive method to compute the gen eralized centroid of an interval type-2 fuzzy set, in: Proc

    M. Melgarejo, A fast recursive method to compute the gen eralized centroid of an interval type-2 fuzzy set, in: Proc. NAFIPS, S an Diego, CA, 2007

    work page 2007

  30. [30]

    J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems : Introduction and New Directions, Prentice-Hall, Upper Saddle River, NJ, 2001

    work page 2001

  31. [31]

    J. M. Mendel, Computing with words: Zadeh, Turing, Popp er and Occam, IEEE Computational Intelligence Magazine 2 (2007) 1 0–17

    work page 2007

  32. [32]

    J. M. Mendel, Type-2 fuzzy sets and systems: An overview , IEEE Computational Intelligence Magazine 2 (1) (2007) 20–29

    work page 2007

  33. [33]

    J. M. Mendel, On KM algorithms for solving type-2 fuzzy s et problems, IEEE Trans. on Fuzzy Systems 21 (3) (2013) 426–446

    work page 2013

  34. [34]

    J. M. Mendel, General type-2 fuzzy logic systems made si mple: a tutorial, IEEE Trans. on Fuzzy Systems 22 (5) (2014) 1162–11 82

    work page 2014

  35. [35]

    J. M. Mendel, Uncertain rule-based fuzzy systems: intr oduction and new directions, 2nd ed., Springer, 2017

    work page 2017

  36. [36]

    J. M. Mendel, Explaining the performance potential of r ule-based fuzzy systems as a greater sculpting of the state space, IEEE Trans . on Fuzzy Systems 26 (4) (2018) 2362–2373

    work page 2018

  37. [37]

    J. M. Mendel, Type-2 fuzzy sets as well as computing with words, IEEE Computational Intelligence Magazine 14 (1) (2019) 82–95

    work page 2019

  38. [38]

    J. M. Mendel, H. Hagras, H. Bustince, F. Herrera, Commen ts on ‘Interval type-2 fuzzy sets are generalization of interval -valued fuzzy sets: Towards a wide view on their relationship’, IEEE Trans . on Fuzzy Systems 24 (1) (2016) 249–250

    work page 2016

  39. [39]

    J. M. Mendel, D. Wu, Perceptual Computing: Aiding Peopl e in Making Subjective Judgments, Wiley-IEEE Press, Hoboken, NJ, 2010

    work page 2010

  40. [40]

    Miller, The magical number seven plus or minus two: so me limits on the capacity for processing information, Psychological Review 63 (2) (1956) 81–97

    G. Miller, The magical number seven plus or minus two: so me limits on the capacity for processing information, Psychological Review 63 (2) (1956) 81–97

    work page 1956

  41. [41]

    J. G. Monicka, N. Sekhar, K. R. Kumar, Performance evalu ation of membership functions on fuzzy logic controlled AC voltage c ontroller for speed control of induction motor drive, International J ournal of Computer Applications 13 (5)

    work page

  42. [42]

    M. Nie, W. Tan, Analytical structure and characteristi cs of symmetric centroid type-reduced interval type-2 fuzzy PI and PD contr ollers, IEEE Trans. on Fuzzy Systems 20 (3) (2012) 416–430

    work page 2012

  43. [43]

    M. Nie, W. W. Tan, Towards an efficient type-reduction me thod for interval type-2 fuzzy logic systems, in: Proc. IEEE Int’l Co nf. on Fuzzy Systems, Hong Kong, 2008

    work page 2008

  44. [44]

    M. A. Sanchez, O. Castillo, J. R. Castro, Information gr anule formation via the concept of uncertainty-based information with inte rval type-2 fuzzy sets representation and Takagi-Sugeno-Kang consequ ents opti- mized with Cuckoo search, Applied Soft Computing 27 (2015) 6 02– 609

    work page 2015

  45. [45]

    H. B. Sola, J. Fernandez, H. Hagras, F. Herrera, M. Pagol a, E. Bar- renechea, Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Toward a wider view on their relationship, IEEE T rans. on Fuzzy Systems 23 (5) (2015) 1876–1882

    work page 2015

  46. [46]

    Takagi, M

    T. Takagi, M. Sugeno, Fuzzy identification of systems an d its application to modeling and control, IEEE Trans. on Systems, Man, and Cyb ernetics 15 (1985) 116–132

    work page 1985

  47. [47]

    C. W. Tao, J. S. Taur, C.-W. Chang, Y .-H. Chang, Simplifie d type-2 fuzzy sliding controller for wing rock system, Fuzzy Sets an d Systems 207 (16) (2012) 111–129

    work page 2012

  48. [48]

    Taskin, T

    A. Taskin, T. Kumbasar, An open source Matlab/Simulink toolbox for interval type-2 fuzzy logic systems, in: Proc. IEEE Symposi um Series on Computational Intelligence, Cape Town, South Africa, 20 15

    work page

  49. [49]

    L.-X. Wang, J. M. Mendel, Generating fuzzy rules by lear ning from examples, IEEE Trans. on Systems, Man, and Cybernetics 22 (2 ) (1992) 1414–1427

    work page 1992

  50. [50]

    Wu, On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers, IEEE Trans

    D. Wu, On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers, IEEE Trans. on Fuzzy System s 20 (5) (2012) 832–848

    work page 2012

  51. [51]

    Wu, Twelve considerations in choosing between Gauss ian and trape- zoidal membership functions in interval type-2 fuzzy logic controllers, in: Proc

    D. Wu, Twelve considerations in choosing between Gauss ian and trape- zoidal membership functions in interval type-2 fuzzy logic controllers, in: Proc. IEEE World Congress on Computational Intelligenc e, Brisbane, Australia, 2012

    work page 2012

  52. [52]

    Wu, Approaches for reducing the computational cost o f interval type-2 fuzzy logic systems: Overview and comparisons, IEEE Trans

    D. Wu, Approaches for reducing the computational cost o f interval type-2 fuzzy logic systems: Overview and comparisons, IEEE Trans. on Fuzzy Systems 21 (1) (2013) 80–99

    work page 2013

  53. [53]

    D. Wu, J. M. Mendel, Enhanced Karnik-Mendel Algorithms , IEEE Trans. on Fuzzy Systems 17 (4) (2009) 923–934

    work page 2009

  54. [54]

    D. Wu, J. M. Mendel, On the continuity of type-1 and inter val type- 2 fuzzy logic systems, IEEE Trans. on Fuzzy Systems 19 (1) (20 11) 179–192

    work page

  55. [55]

    D. Wu, J. M. Mendel, Designing practical interval type- 2 fuzzy logic systems made simple, in: Proc. IEEE World Congress on Comput ational Intelligence, Beijing, China, 2014

    work page 2014

  56. [56]

    D. Wu, W. W. Tan, Genetic learning and performance evalu ation of type-2 fuzzy logic controllers, Engineering Applications of Artificial Intelligence 19 (8) (2006) 829–841

    work page 2006

  57. [57]

    D. Wu, W. W. Tan, A simplified type-2 fuzzy controller for real-time control, ISA Trans. 15 (4) (2006) 503–516

    work page 2006

  58. [58]

    Y eh, W.-H

    C.-Y . Y eh, W.-H. Jeng, S.-J. Lee, An enhanced type-redu ction algorithm for type-2 fuzzy sets, IEEE Trans. on Fuzzy Systems 19 (2) (20 11) 227– 240

    work page

  59. [59]

    L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-1, Information Sciences 8 (1975) 19 9–249

    work page 1975

  60. [60]

    H. Zhou, H. Ying, A method for deriving the analytical st ructure of a broad class of typical interval type-2 Mamdani fuzzy contro llers, IEEE Trans. on Fuzzy Systems 21 (3) (2013) 447–458

    work page 2013

  61. [61]

    H. Zhou, H. Ying, Deriving and analyzing analytical str uctures of a class of typical interval type-2 TS fuzzy controllers, IEEE Trans. on Cybernetics 47 (9) (2017) 2492–2503

    work page 2017