Review of fuzzy system models with an emphasis on fuzzy functions
Ismail Burhan Turksen · Transactions of the Institute of Measurement and Control · 2009
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.