Structure and stability analysis of general Mamdani fuzzy dynamic models

Mamdani fuzzy models have always been used as black-box models. Their structures in relation to the conventional model structures are unknown. Moreover, there exist no theoretical methods for rigorously judging model stability and validity. I attempt to provide solutions to these issues for a general class of fuzzy models. They use arbitrary continuous input fuzzy sets, arbitrary fuzzy rules, arbitrary inference methods, Zadeh or product fuzzy logic AND operator, singleton output fuzzy sets, and the centroid defuzzifier. I first show that the fuzzy models belong to the NARX (nonlinear autoregressive with the extra input) model structure, which is one of the most important and widely used structures in classical modeling. I then divide the NARX model structure into three nonlinear types and investigate how the settings of the fuzzy model components, especially input fuzzy sets, dictate the relations between the fuzzy models and these types. I have found that the fuzzy models become type-2 models if and only if the input fuzzy sets are linear or piecewise linear (e.g., trapezoidal or triangular), becoming type 3 if and only if at least one input fuzzy set is nonlinear. I have also developed an algorithm to transfer type-2 fuzzy models into type-1 models as far as their input–output relationships are concerned, which have some important properties not shared by the type-2 models. Furthermore, a necessary and sufficient condition has been derived for a part of the general fuzzy models to be linear ARX models. I have established a necessary and sufficient condition for judging local stability of type-1 and type-2 fuzzy models. It can be used for model validation and control system design. Three numeric examples are provided. Our new findings provide a theoretical foundation for Mamdani fuzzy modeling and make it more consistent with the conventional modeling theory. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 103–125, 2005.