Bulanık kümelerde optimizasyon problemi ve çözüm yöntemleri
Optimization problem and solution methods in fuzzy sets
- Tez No: 39317
- Danışmanlar: DOÇ.DR. İBRAHİM EKSİN
- Tez Türü: Yüksek Lisans
- Konular: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol, Computer Engineering and Computer Science and Control
- Anahtar Kelimeler: Bulanık kümeler, Optimizasyon, Fuzzy sets, Optimization
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET 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. VII
Özet (Çeviri)
SUMMARY OPTIMIZATION PROBLEM AND SOLUTION METHODS IN FUZZY SETS 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 VIIIBy 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 IXX 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 <p Y Theorem 2. : (1) If the set of solutions to y=J!ro/?with respect to R for X ? F(X) and Y ? F(Y) is nonempty then least element of R is given by (2) If the solutions of y«*oJ?with respect to X is nonempty the least X is equal to J?= R P Y Adjoint fuzzy relation is given by Y(y) = infix [ X(x) if R(x,y) ) and Theorem 3. : If X and Y fulfil adjoint fuzzy relation the least relation is if- X t Y Theorem 4. : For given Y and R biggest X is given by Xt' suprx (R t Y) Meanwhile the design aspects of the fuzzy controller can be given by 1. Completeness of control rules By completeness, we mean that the controller can generate control for any input fuzzy state X. A set of control rules If Xi, then U1, i = 1, 2,..., N is complete if VxexlistiK Xâ(x) > 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 XIFor 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 = <?i I The problem is to maximize J subject to the model of the system. G is the set of goal. Gj equals to the goal for output at instant i. The dynamic state feedback set is gathered by an algorithm, widely given in the text, so that the control signal is realized as v* - **. ** jreXxU In this study a new approach is made for the solution of the optimization problem in fuzzy systems. The simulation studies give perfect solutions. However, it must be noted that the speed of the microprocessors of our age is not fast enough to make this much calculations in a sampling period. Therefore, realization of this algorithm is strongly recommended bu a set of fuzzy chips. The solutions to optimal control problems in fuzzy systems can be reached by different methods. One of them is given by Pedrycz in ”Fuzzy Sets and Systems", but however the point of wiev is precise. This study XIIstands as a step to the realization of other control problems. Any question about the study have an answer in the text. XIII
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