Algorithmic aspects of fuzzy control
The study concerns two design issues arising in the realm of fuzzy rule-based computing: its computational complexity and a reasoning carried out in the presence of sparse rulebases. The discussed fuzzy controller assumes either a relational or functional form where the rules read as: \[ \text{if \(X\) is \(A\) then \(y\) is \(B\)}\quad\text{or}\quad\text{if \(X\) is \(A\) then \(y= f(x)\)}, \] where \(A\) and \(B\) are fuzzy sets defined in the corresponding input and output (control) space. The computational complexity of such fuzzy controllers is analyzed in detail. Raised is the problem of reasoning in the case of a sparse rulebase, namely the rules with the union of their antecedents exhibiting some ``gaps'' in the input space (viz. the regions with zero membership values). The proposed interpolation scheme relies on the expression \[ {d(A^*,A_{i1})\over d(A^*,A_{i2})}= {d(B^*,B_{i1})\over d(B^*,B_{i2})}, \] where \(A^*\) is a given input, \(A_{i1}\) and \(A_{i2}\) are the two antecedents that are the closest to \(A^*\), \(d(.,.)\) stands for the distance function between two fuzzy sets and \(B^*\) is looked for.
scientific article; zbMATH DE number 1004300
scientific article