Solvability Considerations for Non-Elementary Fuzzy Relational Equations
Siegfried Gottwald · 1994
Determining a fuzzy controller is a two step process. First one has to fix the control rules which in general have the form: IF α =A iTHENβ=B i ,i= 1,. . . ,Nfor some (perhaps multidimensional) input variable α and output variableβ. And then one has to convert these control rules into a fuzzy relationRand to apply this relation via the compositional rule of inference to any fuzzy input value of or. Ideally the rule inputsA ishould transform into the rule outputsB i =A i oR. Thus, mathematically this second step can be seen as the problem to solve a system of fuzzy relational equation for an unknown fuzzy relation. Besides these simple, “elementary” types of equations also other, more complicated types of equations have been discussed in fuzzy control theory. The present paper discusses some aspects of how results from the solvability theory of elementary fuzzy relational equations can be extended to discuss also more involved types of relational equations.