Completeness of fuzzy controller carrying a mapping f:R1→R1

Even though the representation and processing of data and information in a fuzzy controller are quite different than in other control algorithms, the information processing operation that it carries out is basically a function f:A⊂Rⁿ →R^m, 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 a controlled systems, respectively. Under the assumptions of Mamdani´s direct reasoning method and the COA (center of area) defuzzification method, fuzzy controllers are proved to perform the mapping of any given function with physically meaningful fuzzy sets. The mapping capabilities of the fuzzy controllers are analyzed in detail for the case f:R¹→R¹