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General Type-2 Fuzzy Systems

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Uncertain Rule-Based Fuzzy Systems

Abstract

This chapter explores many aspects of the general type-2 fuzzy system that was introduced in Chap. 1. As was done for interval type-2 fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of general type-2 fuzzy systems that may be used in real-world applications—GT2 Mamdani and GT2 TSK fuzzy systems. Importantly, it also distinguishes between GT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification.

The coverage of this chapter focuses on singleton fuzzification and the use of the horizontal-slice representation of a GT2 FS, and includes: GT2 rules, horizontal-slice formulas for firing sets and fired-rules output sets, horizontal-slice first- and second-order rule partitions, combining or not combining fired-rule output sets on the way to defuzzification, horizontal-slice type-reduction (centroid and center-of-sets) for horizontal-slice GT2 Mamdani and TSK fuzzy systems, defuzzification (this is where horizontal slices are aggregated), a summary of the computational steps for two horizontal-slice Mamdani and two horizontal-slice TSK GT2 fuzzy systems, horizontal-slice versions of the NT and BMM direct defuzzification methods, GT2 fuzzy basis functions which provide a mathematical description of a GT2 fuzzy system from its input to its output, remarks and insights about a GT2 fuzzy system, what exactly “design of a GT2 fuzzy system” means as well as a tabular way for making the choices that are needed to fully specify a GT2 fuzzy system, two approaches to design—the partially dependent approach and the totally independent approach, but only for singleton GT2 fuzzy systems—requirements that need to be met in the study of real-world applications of GT2 fuzzy systems, and a case study of GT2 fuzzy logic control. Ten examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sects. 11.9 and 11.11.

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Notes

  1. 1.

    In the Wagner and Hagras references, the term “zSlice” is used instead of horizontal-slice. See Sect. 6.7.3 for a discussion about why this book uses “horizontal-slice” instead of zSlice.

  2. 2.

    In a GT2 fuzzy system, in (11.5) and other places, one needs to keep track of which antecedent is referred to (the first subscript i), which rule is referred to (the superscript l), and which \( \alpha{\text{-cut}} \) is referred to (the second subscript \( \alpha ) \); this unavoidably leads to very heavy subscript and superscript notations.

  3. 3.

    Almaraashi et al. (2016) perform type-reduction followed by defuzzification by first computing the COG of vertical slices (some interpolation of the COGs is also used) and then defuzzifying the resulting T1 FS. Although this kind of type-reduction is ad hoc, it does adhere to the basic design requirement that when all MF uncertainties disappear the output of a T2 fuzzy system must become the same as the output of a T1 fuzzy system. If one is only interested in the defuzzified output of a GT2 fuzzy system, and not a measure of the uncertainties that have flowed through that system, then their vertical slice centroid type-reducer (VSCTR) is arguably a very viable alternative to the WH approach that is taken in this chapter.

  4. 4.

    Height type-reduction for a WH GT2 fuzzy system is left as an exercise (Exercise 11.4).

  5. 5.

    Strictly speaking, Wagner and Hagras (2008, 2010, 2013) are only for Mamdani GT2 fuzzy systems; however, their approach is also applicable to GT2 TSK fuzzy systems, which is why the title of this section includes “WH.”

  6. 6.

    “At least” is used here because it is conceivable that even some sort of GT2 FS functions could be used for the GT2 TSK rule consequents. How to do this is an open research issue.

  7. 7.

    Wagner and Hagras (2010) was the first to make this observation.

  8. 8.

    Of course, when \( \alpha \) is sampled very finely then \( y_{1} ,y_{2} ,\ldots, y_{2k} \) will approach a uniformly sampled set of samples; however, in a WH GT2 fuzzy system one does not want to use too many horizontal slices, or else computational complexity greatly increases as does computation time.

  9. 9.

    This step depends on the design choices made for the secondary MFs, and is returned to in Sect. 11.14. In practice, \( X_{i} \) and Y are sampled, and (11.46) and (11.47) are computed for \( x_{i} \in X_{id} \) and \( y \in Y_{d} \). This is also true for later computations.

  10. 10.

    For notational simplification, \( y_{{l,\alpha_{k} }}^{COS} \) and \( y_{{l,\alpha_{k} }}^{COS} \) are shortened to \( y_{{l,\alpha_{k} }} \) and \( y_{{r,\alpha_{k} }} \), respectively.

  11. 11.

    The extension of approximate type-reduction + defuzzification (Wu-Mendel uncertainty bounds), that is given in Sect. 9.8, is left to the reader as an exercise (Exercise 11.12) because its formulas are much more complicated than the ones for the extensions of the NT and BMM formulas.

  12. 12.

    This step depends on the design choices made for the secondary MFs, and is returned to in Sect. 11.14.

  13. 13.

    Equation (11.62) is excluded here because (see, also footnote 14 in Chap. 3) this author does not feel that the number of FBFs should depend on the discretization, N, of the primary variable.

  14. 14.

    Adaptiveness and novelty are defined in Sect. 9.13.2.

  15. 15.

    A squishing technique leads to a nested set of horizontal slices. Two such techniques are described in Sect. 6.7.4. See also Exercises 6.18–6.20.

  16. 16.

    Recall that the FOU of a GT2 FS is its \( \alpha = 0 \) horizontal slice.

  17. 17.

    For a different kind of design that also uses triangle secondary MFs, see Starczewski (2009). Since the advent of the horizontal-slice decomposition, it is arguably no longer necessary to use the approximate approach to type-reduction that is advocated in this paper. Instead, one only needs to design a WH GT2 fuzzy system that uses a few horizontal slices.

  18. 18.

    Because formulas have only been given in this chapter for singleton fuzzification, this is the only kind of fuzzification that is used in the discussions below. Exercises 11.24 and 11.25 ask the reader to extend these discussions to T1 non-singleton and IT2 non-singleton WH GT2 fuzzy systems.

  19. 19.

    Because formulas have only been given in this chapter for singleton fuzzification, the non-singleton steps that are listed in Sect. 10.2.5 are not included here. Of course, once the formulas for T1 or IT2 non-singleton fuzzification have been obtained (Exercise 11.1) then more steps can be added to the present list, as has been done in Sect. 10.2.5.

  20. 20.

    This section, as well as Sects. 11.16.3 and 11.16.4 were prepared by Prof. Tufan Kumbasar.

  21. 21.

    The design methodology presented in the first three lines of (4.81) cannot be employed here because of the nonlinear nature of the system in (11.93).

  22. 22.

    All GA designs were achieved by using the MATLABⓇ GA toolbox, and used 20 iterations and 20 populations-default MATLAB GA parameter settings. MATLAB is a registered trademark of The MathWorks, Inc.

  23. 23.

    Note that the squishing parameters for all of these horizontal slices can be deduced from the entries into Table 11.9; each is the coefficient of the left-end of the stated interval.

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  1. University of Southern California, Los Angeles, CA, USA

    Jerry M. Mendel

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Mendel, J.M. (2017). General Type-2 Fuzzy Systems. In: Uncertain Rule-Based Fuzzy Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-51370-6_11

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  • DOI: https://doi.org/10.1007/978-3-319-51370-6_11

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