Type-2 fuzzy elliptic membership functions for modeling uncertainty

Erdal Kayacan*, Andriy Sarabakha, Simon Coupland, Robert John, Mojtaba Ahmadieh Khanesar

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Whereas type-1 and type-2 membership functions (MFs) are the core of any fuzzy logic system, there are no performance criteria available to evaluate the goodness or correctness of the fuzzy MFs. In this paper, we make extensive analysis in terms of the capability of type-2 elliptic fuzzy MFs in modeling uncertainty. Having decoupled parameters for its support and width, elliptic MFs are unique amongst existing type-2 fuzzy MFs. In this investigation, the uncertainty distribution along the elliptic MF support is studied, and a detailed analysis is given to compare and contrast its performance with existing type-2 fuzzy MFs. Furthermore, fuzzy arithmetic operations are also investigated, and our finding is that the elliptic MF has similar features to the Gaussian and triangular MFs in addition and multiplication operations. Moreover, we have tested the prediction capability of elliptic MFs using interval type-2 fuzzy logic systems on oil price prediction problem for a data set from 2nd Jan 1985 till 25th April 2016. Throughout the simulation studies, an extreme learning machine is used to train the interval type-2 fuzzy logic system. The prediction results show that, in addition to their various advantages mentioned above, elliptic MFs have comparable prediction results when compared to Gaussian and triangular MFs. Finally, in order to test the performance of fuzzy logic controller with elliptic interval type-2 MFs, extensive real-time experiments are conducted for the 3D trajectory tracking problem of a quadrotor. We believe that the results of this study will open the doors to elliptic MFs’ wider use of real-world identification and control applications as the proposed MF is easy to interpret in addition to its unique features.

Original languageEnglish
JournalEngineering Applications of Artificial Intelligence
Pages (from-to)170-183
Publication statusPublished - 1 Apr 2018


  • Addition
  • Elliptic membership function
  • Fuzzy arithmetics
  • Fuzzy sets
  • Gaussian
  • Multiplication
  • Triangular
  • Type-2 fuzzy logic theory
  • Uncertainty


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