Bulanık kümelerde optimizasyon problemi ve çözüm yöntemleri

Abstract

Bu çalışma bulanık kümelerde optimizasyon probleminin çözümü üzerine yapılmıştır. Optimizasyon problemi, bulanık küme teorisinin yapısı gereği kolaylık la çözül ememektedir. Bu çözümü elde etmek amacıyla ilk olarak bula nık küme teorisinin genel tanımlan incelenmiştir. En basit yapılı bulanık kontrolörün tanıtımından sonra, problemin çözümünde temel oluşturan bulanık ilişki denk lemlerinin çözümleri incelenmiştir. Çalışma boyunca üç ayrı bulanık kontrolör yapısı göz önüne alınmıştır. Bunlardan ilki kendini düzenleyen kontrolördür. ikincisi ise optimizasyon problemi için elde edilen ilk çözüm yöntemidir. Son olarak ise ilk ikisine oranla daha başarılı sonuç veren alternatif çözüm yöntemi tanıtılmıştır.

This work is on the solution of the optimiza tion problem on fuzzy systems. To reach a Plausible solution a wide study was done on the fuzzy set theory. Some basic notations are necessary to go with the study. Also the solution lies on the basis of solutions to fuzzy relational equations. A fuzzy set A defined in universe of discourse X is expressed by its membership function. A s *?*[(), 1] where A(x) expresses the extent which x fulfills the category specified by A. Any fuzzy set can be represented by the sum of its elements. Therefore A(x) can be shown as, ?/. A(x) or A = SrAL*>. With A and B, two fuzzy sets defined in X, the following can be defined A(x) - 1 - A(x) (AlİB) (x) = max(A(x),B(x) ) (Afifi) (x) - min(A(x), B(x) ) men t s such that By a t - norm we mean afunction of two argu- t : [0,1] x [0,1]-[0,1] a) For x£y,w*z,xtw £ ytz b) It is commutative. c) It is associative. d) It satisfies x t 0 = 0 and x t 1 = x VIII By an s - norm, we mean a function of two arguments t : [0,1] X [0,1]-[0,1] such that a) for x*y,w£z,xsw*ysz b) it is commutative. c) it is associative. d) it satisfies xs0 = x;xs1 = 1. By a fuzzy relation R, defined in the carte sian product X x Y, we mean a mapping R : *xY- [0,1] (2.10) Thus, to each pair of elements (x,y) a number, which expresses the strength of ties, is assigned. For a given R and X couple Y is gathered by their composition. Most frequently used compositions are i ) sup - t ; r(y) = (x°r) (y) = supr [X(x) tR(x,y)] ii ) inf - s ; Y(y) = (XOR) (y) = infr [X(x) sR(x,y)] If the sup-t composition, Y = X. R and its dual y «x o R is given two main problems can be taken into considera tion; i) determine R for given X, Y ii) determine X for given R, Y IX X can be accepted as the input of a system, while Y is the output and R is the characteristic of it. The following theorems with the following definitions give the solutions to above questions. AtpB » 8Upzc(A t C £ B) and ApB - infıc(A s c * B) Theorem 1. : (1) If XeF(X) and Y?F(Y) fulfil Y = X. R the greatest fuzzy relation satisfying the formula can be given by if = X Q Y (2) If RSF(XxY) and YeF(Y) satisfy Y = X. R the maximum input can be given by the equation £ <* R t, t ? (0.11 2. Interactions of control rules There is interaction between control rules if the following holds 3. Consistency of control rules. The points given above are for the simple fuzzy controller. Moreover, a different approach to fuzzy controller is reached by fuzzy modelling. Let X, U, Y be state, control and output spaces respectively. Therefore, a system of order p can be modelled by Yk+p " Xk*p * & Here, R : U x X x X (p times) x X - [0,1] and S : X x Y - [0,13 XI For the problem given here, the system is said to be strictly known. Therefore R and S is clear for the problem. The performance index is given by the above equation J - 2?.i B Yi