General Type-2 Fuzzy Systems
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.
- Castillo, O., L. A.-Angulo, J. R. Castro, and M. G.-Valdez. 2016. A comparative study of type-1 fuzzy logic systems, interval tyoe-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems. Information Sciences 354: 257–274.Google Scholar
- Derrac, J., S. Garcia, D. Molina, and F. Herrera. 2011. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and Evolutionary Computation 1: 3–18.Google Scholar
- Greenfield, S., and R. John. 2009. The uncertainty associated with a type-2 fuzzy set. In Views on fuzzy sets and systems from different perspectives: Philosophy and logic, criticisms and applications, ed. R. Seising, 471–483. Heidelberg: Springer.Google Scholar
- Hamrawi, H., and S. Coupland. 2009. Type-2 fuzzy arithmetic using alpha-planes. In Proceedings of the IFSA/EUSFLAT, 606–611. Portugal.Google Scholar
- Wagner, C., and H. Hagras. 2008. zSlices–Towards bridging the gap between interval and general type-2 fuzzy logic. In Proceedings of the IEEE FUZZ conference, Paper # FS0126. Hong Kong.Google Scholar
- Wagner, C., and H. Hagras. 2013. ZSlices based general type-2 fuzzy sets and systems. In Advances in type-2 fuzzy sets and systems: Theory and applications, eds. Sadeghian, A., J. M. Mendel, and H. Tahayori. New York: Springer.Google Scholar
- Wu, D., and W. W. Tan. 2010. Interval type-2 fuzzy PI controllers: Why they are more robust. In Proceedings of IEEE international conference on granular computing, 802–807. Silicon Valley.Google Scholar