Fuzzy relation equations and approximate solutions. Some recent results
نویسنده
چکیده
For theoretical fuzzy control it is a well known strategy to transform a system of control rules into a system of relation equations. Because these systems of relation equations are not always solvable, solvability criteria and approximate solutions have been discussed. We reconsider some of the results in this field and extend them using more recent results on t-norm based fuzzy logics and on approximation processes. 1 Fuzzy control and relation equations A fuzzy controller usually is determined by a finite list if α is Ai, then β is Bi, i = 1, . . . , n , (1) of linguistic control rules, which are supposed to be designed to describe some control procedure with input variable α and output variable β. The standard understanding, originating from [15], is that such a fuzzy controller should be realized by a fuzzy relation R which connects fuzzy input information A with fuzzy output information B via the compositional rule of inference CRI B = A ◦R = {y ‖ ∃x(A(x)&R(x, y)} . (2) Each of the control rules from (1) determines the single equation Bi = Ai◦R. Therefore the system (1) of linguistic control rules is to be transformed into a system of fuzzy relation equations Ai ◦R = Bi, for i = 1, . . . , n . (3) Related to the system (3) of relation equations, one considers the fuzzy relation
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