Abstract
Abstract Fuzzy controllers have been successfully applied to many cases to which conventional control algorithms are difficult to be applied. Even though the representations and the processings of data and information in the fuzzy controller are quite different from other control algorithms, the information processing operation that it carries out is basically a function f:A ⊂ Rn→Rm, from a bounded subset A of n-dimensional Euclidean space to a bounded subset f[A] of m-dimensional Euclidean space, where n and m are the number of measured states and the number of control inputs of controlled system, respectively. Under the assumptions of Mamdani's direct reasoning method, and C.O.G. (center of gravity) defuzzification method, the fuzzy controllers are proven to perform the mappings of any given functions with appropriately defined fuzzy sets. The mapping capabilities of the fuzzy controllers are analyzed in detail for two cases; f:R1 → R1 and g:R2 → R1. Also, it will be shown that the results can be extended to multiple dimensional cases.
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