Fuzzy controllers design via fixed point theorem (Q1820097)

The paper is dedicated to a problem of control of a system described by the input-output equation \(y=f(x)\) which is realized in presence of fuzziness. Fuzziness is attached to the problem formulated by introducing a fuzzy relation \(\mu_{\epsilon}\) defined in the Cartesian product of the output space (equipped with a metric d(\(\cdot,\cdot))\), say \(\mu_{\epsilon}:\) \b{Y}\(\times \underline Y\to [0,1]\). This relation expresses a grade of satisfaction of the inequality \(d(y,z)<\delta\), with \(\epsilon\) standing for a nonnegative constant for any pair of elements y,z\(\in \underline Y\). In such a framework a control formula \(u=r(y)\), u being an element of the space of controls, is derived by means of a fixed point theorem.