# Analytical Structure Characterization and Stability Analysis for a General Class of Mamdani Fuzzy Controllers

Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 317)

## Abstract

Stability of a fuzzy control system is closely related to the analytical structure of the fuzzy controller, which is determined by its components such as input and output fuzzy sets and fuzzy rules. We first characterize the mathematical input–output structure of fuzzy controllers and then utilize the structure characteristics to advance stability analysis. We study how the components of a general class of Mamdani fuzzy controllers dictate the controller’s input–output relationship. The controllers can use input fuzzy sets of any types, arbitrary fuzzy rules, arbitrary inference methods, either Zadeh or the product fuzzy logic AND operator, singleton output fuzzy sets, and the centroid defuzzifier. We theoretically prove that regardless of the choices for the other components, if and only if Zadeh fuzzy AND operator and piecewise linear (e.g., trapezoidal or triangular) input fuzzy sets are used, the fuzzy controllers become a peculiar class of nonlinear controllers with the following interesting characteristics: (1) they are linear with respect to input variables; (2) their control gains dynamically change with the input variables; and (3) they become linear controllers with constant gains around the system equilibrium point. These properties make the fuzzy controllers suitable for analysis and design using conventional control theory. This necessary and sufficient condition becomes a sufficient condition if the product AND operator is employed instead. We name the fuzzy controllers of this peculiar class type-A fuzzy controllers. Taking advantage of this new structure knowledge, we have established a necessary and sufficient local stability condition for the type-A fuzzy control systems. It can be used not only for the stability determination, but also for practically designing a type-A fuzzy control system that is at least stable at the equilibrium point even when model of the controlled system is mathematically unknown. Three numerical examples are provided to demonstrate the utility of our new findings.

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