Decoupling in fuzzy systems: A cascade compensation approach
The paper addresses a problem of decoupling in systems described by fuzzy relational equations, \(y_ 1=x_ 1\circ x_ 2\circ R_ 1\), \(y_ 2=x_ 1\circ x_ 2\circ R_ 2\) where \(x_ 1,x_ 2\) stand for input fuzzy sets, \(x_ 1:X\to [0,1]\), \(x_ 2:x_ 2\to [0,1]\), \(y_ 1,y_ 2\) are output fuzzy sets described in the appropriate spaces whereas \(R_ 1\) and \(R_ 2\) are fuzzy relations of the system. A compensator working in cascade structure is given by formulas, \(x_ 1=u_ 1\circ u_ 2\circ \tilde R_ 1\), \(x_ 2=u_ 1\circ u_ 2\circ \tilde R_ 2\) where \(u_ 1\) and \(u_ 2\) denote inputs of the compensator and \(\tilde R_ 1\) and \(\tilde R_ 2\) are two fuzzy relations which are looked for in order to compensate the previous system. A necessary and sufficient condition for the existence of a solution of the decoupling problem is studied.
scientific article; zbMATH DE number 4101043
scientific article