Approximation theory of fuzzy systems based upon genuine many-valued implications -- SISO cases (Q1855478)

The authors prove that the single input and single output (SISO) fuzzy systems based upon genuine many-valued implications are universal approximators [for the MIMO cases, see ibid. 130, 159-174 (2002; Zbl 1017.93057), reviewed below]. The study on approximation theory of fuzzy systems is very important and necessary. In most applications of fuzzy systems, the main design objective can be transformed to finding a designed mapping from the input spaces to the output spaces, which may also be denoted as a function. Thus, the problem of designing fuzzy systems can be considered as approximation problems of functions. It is shown theoretically that fuzzy control systems based upon genuine many-valued implications are equivalent to those based upon t-norm implications, and the general approach to construct fuzzy systems is given. It is also shown that a defuzzifier based upon the center of areas is not appropriate for fuzzy systems based upon genuine many-valued implications.