Constrained type-2 fuzzy sets

Garibaldi, Jonathan M.; Guadarrama, Sergio

doi:10.1109/t2fuzz.2011.5949562

Cited by 20 publications

(27 citation statements)

References 7 publications

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“…In this section we will provide some formal definition of fuzzy concepts that will be used throughout the paper (definitions taken or rephrased from [8], [9], [13], [14]). Definition 1.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

“…To overcome these issues, Constrained Type-2 (CT2) fuzzy sets have been proposed [9], [10]. The idea behind them is to address the two limitations above by: (i) providing an explicit method for generating the boundaries of the footprint of uncertainty that keeps a shape coherency [9] throughout the generation of the type-2 set, based on an underlying concept modelled by a type-1 set; and (ii) restricting the acceptable embedded sets that may be used to only a subset of all the ESs, in order to process only shapes that may be considered meaningful in that specific context. Even though the concept of CT2 FS has already been formulated [9], [10], some key components are currently lacking formal definitions such as the acceptable embedded sets, constrained inference and centroid defuzzification.…”

Section: Introductionmentioning

confidence: 99%

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Constrained Interval Type-2 Fuzzy Sets

D'Alterio

Garibaldi

John

et al. 2021

IEEE Trans. Fuzzy Syst.

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In many contexts, type-2 fuzzy sets are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation there is no restriction on the shape of the footprint of uncertainty and the embedded sets that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the type-2 fuzzy set and the concept it models. As a consequence, the interpretability of some of the embedded sets and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 fuzzy sets have been proposed. However, no formal definitions for some of their key components (e.g. acceptable embedded sets) and constrained operations have been given. The goal of this paper is to provide some theoretical underpinning for the definition of constrained type-2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real world cases and briefly compared to the standard interval type-2 inference and defuzzification method.

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“…In this section we will provide some formal definition of fuzzy concepts that will be used throughout the paper (definitions taken or rephrased from [8], [9], [13], [14]). Definition 1.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Constrained Interval Type-2 Fuzzy Sets

D'Alterio

Garibaldi

John

et al. 2021

IEEE Trans. Fuzzy Syst.

Self Cite

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“…p e 1 + d e ≤ e ≤ p e 2 − d e (13) which are motivated by the observations that the robustness improvement occurs mainly when the system output is near the setpoint [60], [61]. A graphical illustration of the region is shown in Fig.…”

Section: A Equivalent Proportional-integral Gainsmentioning

confidence: 99%

“…According to the derivations given in Appendix C, when an input (ė, e) satisfies (12), (13), and the following constraint: …”

Section: A Equivalent Proportional-integral Gainsmentioning

confidence: 99%

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On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

2012

IEEE Trans. Fuzzy Syst.

290

111

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Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.Index Terms-Continuity, equivalent generalized type-1 fuzzy sets (EGT1FSs), equivalent PI gains, interval type-2 fuzzy logic controller (IT2 FLC), P-map.

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