Review of fuzzy system models with an emphasis on fuzzy functions

Fuzzy system modelling (FSM) is one of the most prominent tools that can be used to identify the behaviour of highly non-linear systems with uncertainty. In the past, FSM techniques utilized Type 1 fuzzy sets in order to capture the uncertainty in the system. However, since Type 1 fuzzy sets express the belongingness of a crisp value x^' of an input variable x in a fuzzy set A by a crisp membership value μ_A(x^'), they cannot fully capture the uncertainties associated with higher-order imprecisions in identifying membership functions. In the future, we are likely to observe higher types of fuzzy sets, such as Type 2 fuzzy sets. The use of Type 2 fuzzy sets and linguistic logical connectives has drawn a considerable amount of attention in the realm of FSM in the last two decades. In this paper, we first review Type 1 fuzzy system models known as Zadeh, Takagi— Sugeno and Turkşen models; then we review potentially future realizations of Type 2 fuzzy systems again under the headings of Zadeh, Takagi—Sugeno and Turkşen fuzzy system models, in contrast to Type 1 fuzzy system models. Zadeh's and Takagi—Sugeno's models are essentially fuzzy rule base (FRB) models, whereas Turkşen's models are essentially fuzzy function (FF) models. Type 2 fuzzy system models have a higher predictive power. One of the essential problems of Type 2 fuzzy system models is computational complexity. In data-driven FSM methods discussed here, a fuzzy C-means (FCM) clustering algorithm is used in order to identify the system structure, ie, either the number of fuzzy rules or alternately the number of FFs.