The Role of Defuzzification Methods in the Application of Fuzzy Control

Implementation of CMOS fuzzy controllers as mixed-signal integrated circuits

“…The inference methods that they implemented were the "min-max" Mamdani's method, which uses a min operator to perform the fuzzy implication and a max operator to aggregate rules [23], and the "product-sum" method, which uses algebraic product and sum instead of min and max [10]. To defuzzify the global fuzzy output, the traditional strategies were the Middle Of Maxima (MOM) and the Center Of Gravity (COG) methods [3,24,25]. This is shown in Figure 4a.…”

“…The studies in [24][25] reveal that the MOM method can make the controller input-output relation non-monotone, discontinuous, and even independent of the input membership functions. Regarding the COG method, some authors [25,27] have detected the problem that the final output is more influenced by ambiguous rather than by precise consequents, because ambiguous consequents are represented by wider membership functions.…”

“…As cited for antecedents, this information concerns the situation of the membership function in the universe of discourse (given by the element with maximum degree of pertenence or by the center of area), its grade of fuzziness (represented by the support or width), and the way of covering that support (represented by the shape) (Figure 2). This can be achieved by using inference methods which first defuzzify each rule's output (generally given by a fuzzy set, B i ') to obtain a representative value f(B i '), and then aggregate all of them via a weighted average [24,25,28,29] (Figure 4b). The following general expression shows how these methods derive the controller output: (1) where h i is the activation degree of the i-th rule.…”

“…In particular, the Weighted Fuzzy Mean (WFM) [24] and Quality Method (QM) [25] take f(B i ') as constants, c i , namely the center of area of the consequent fuzzy sets, B i , while the weights are chosen as the activation degrees scaled by a constant parameter, w(h i , B i ) = h i •γ i . In the first method, the parameter γ i is proportional to the support of the consequent fuzzy set, so that the global output is similar to that obtained by the productsum-COG method, thus suffering from the same drawback of ambiguous consequent dominance previously commented.…”

“…Finally, when w(h i , B i ) is equal to h i and f(B i ') is a constant, c i , the resulting method is known as Fuzzy Mean, Height Method, Singleton, or zero-order Sugeno's Method [22,24,25,30,31], which can be considered as a simplification of all the above methods. In this case, Equation (1) is reduced to the following expression:…”