Please use this identifier to cite or link to this item: `http://hdl.handle.net/11527/16497`
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dc.contributor.authorÜstündağ, Berktr_TR
dc.date2000tr_TR
dc.date.accessioned2018-07-18T08:31:14Z-
dc.date.available2018-07-18T08:31:14Z-
dc.date.issued2000tr_TR
dc.identifier.urihttp://hdl.handle.net/11527/16497-
dc.descriptionTez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2000tr_TR
dc.descriptionThesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 2000en_US
dc.description.abstractLet X be a compact set called feasible region and f be an objective function such that X0, f(u(k))-»r (2) or XVI fi»-r I ->0 =>u=x, f(x)=r (3) k-x» where * represents the optimal solution and r is the reference value. The output of the controller u yields the solution of the equation f(x)-r=0 as it can be seen from the equation (3). If reference sign r is particularly set to zero then the steady state value of u will be equal to one of the roots of f(x). This control theory approach was previously used for the calculation of the step length in numerical solution of differential equations [2]. Roots of a function which will be evaluated in obtaining solution of the global optimization problem is determined by the same approach. A single or multi- variable objective function is placed instead of f(.) and a fuzzy rule based controller takes the place of G in the proposed method. The closed loop control system shown in figure 2 finds the solution of the equation f(x)-r=0 where k represents the number of steps and controller output is considered as the solution. Some problems arise in application of classical linear controllers with control theory approach for the solution of equations instead of well known iterative numerical methods. For example, when a PID type controller is used for root seeking, stability problems are met in reaching the solution. Large valued derivatives of the function around the roots to be solved in an equation, not only causes oscillations, and it may also be the reason of missing the root. d(k)=±1 1 -JrOx e(k) £ 51 |z-1| V- CE ce(k) &. -m- J FUZZY LOGIC du,-J CONTROLLER -^U du=Gc(E,CE) LL. Figure 2 Block diagram of the system used for solution of the single variable equation f(x)-r=0. In this new method, the fuzzy logic controller structure offered by Mamdani is used in the block diagram given in figure 2 [24]. e is actual error, ce is change of actual error, E is scaled error, CE is scaled change of error, See is scaling factor of the error, Scce is scaling factor of the change of error, du is output of the fuzzy logic controller, u is discretely integrated controller output and u* represents the value of calculated root in figure 2. Output of this kind of fuzzy controller saturates over a certain limit depending on the error and change of the error. The integrator at the output of the fuzzy controller is only the memory element in the closed loop system used for finding of the roots. xvn Unsuitable selection of the parameters (See, Scce) causes only limited amplitude oscillation around the root since new method generates bounded output for bounded input if at least fuzzy decision table has been chosen properly. Let the upper bound of the search region be US (upper bound) and starting point of search be AS (lower bound) of the region then u(0) must be set to AS for searching in the interval (AS,US). This numerical assignment provides search to be started from a predetermined point. There exist many methods for optimizing or finding the roots of a unimodal function. All of these methods may fail in non-unimodal functions. Complexity of the function does not affect the achievement in reaching the solution, since only the fuzzy decision table and previously trial points are evaluated starting from a given point. This property will be used in solution of global optimization problem. There are several possible approaches for global optimization with fuzzy logic controller. The proposed optimization algorithm is based on scanning of the feasible region in a closed loop control system with fuzzy logic controller. Search direction may be from the upper bound to lower bound or from lower bound to upper bound or it even can be applied in sub-intervals covering the feasible region. Optimization algorithm accepts the lower bound as the starting point and finds the maximum value of a given function f(.). Roots of the equation f'(u)=0 is searched through running towards the right hand side for r=0 and the first found root is considered to be new global maximum value if the slope of the function is positive in the starting point of search. Otherwise root of f is searched through running towards the right hand side for r=f(u) and the new reached point determined by the solution of the equation f '(u)=0 is considered to be a new global maximum value. This procedure is repeated with solution of f(u)=r by taking r is equal to the global maximum value in both cases. The lastly determined root of f '(u)=0 will be the global maximum value, if controller output (u) exceeds the upper bound US (saturation). This search procedure is illustrated with an example function shown in figure 3. The bold line represents the route of the search algorithm in finding the solution. f(k)-f(k-l) value can be evaluated by adding a shift operator (z"1) and a subtracter on the feedback line for evaluating the derivative f '(u) however this operator decreases the performance in finding the correct solution since step lengths are different in each iteration. The numerical equivalent of derivative such as f(x+h)-f(x))/h has to be preferred because these operations are handled in computer environment. xviu Ref2 Refl Figure 3 Search of global maximum value of a single variable function There is no need target resolution to be a precise value of global optimization as the ending criteria in the sub operation f(x)=0 because global extremal points are only reached after the solution of f (x)=0. For example, if the demanded upper limit of the error is Em=0.000001, the ending criteria for root search is set as Em=0.01 and stopping criteria of the derivative loops is set as Em=0.000001 then the same correct global maximum value will be found by lesser number of evaluations. In order to fit in restrictions on the function optimization parameters penalty components can be added to the objective function. Finding the correct solution is guaranteed ifa Lipchitz value can be given for f(x) in the proposed method of global optimization. Although theoretically none of the solution methods can guarantee finding the correct global maximum value, the proposed method is successful in solution of standard test functions. Previously found global maxima is set as new reference for other variables after a search for a variable in multi variable optimization problems Otherwise, the number of evaluations would be increased exponentially in proportion to the number of variables. When the problem is solved by proposed method on a network, each computer shares a sub interval and new global maximum points are sent to each other after all of the derivative loops. This feature provides decreasing the amount of iterations for the machines those evaluate some intervals far from real global maxima region. en_US
dc.languageturtr_TR
dc.publisherFen Bilimleri Enstitüsütr_TR
dc.publisherInstitute of Science and Technologyen_US
dc.rightsKurumsal arşive yüklenen tüm eserler telif hakkı ile korunmaktadır. Bunlar, bu kaynak üzerinden herhangi bir amaçla görüntülenebilir, ancak yazılı izin alınmadan herhangi bir biçimde yeniden oluşturulması veya dağıtılması yasaklanmıştır.tr_TR
dc.rightsAll works uploaded to the institutional repository are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission.en_US
dc.subjectBulanık denetimtr_TR
dc.subjectOptimizasyontr_TR
dc.subjectOptimizasyon modelleritr_TR
dc.subjectFuzzy controlen_US
dc.subjectOptimizationen_US
dc.subjectOptimization modelsen_US
dc.titleBulanık kontrolör ile yeni bir global eniyileme yöntemitr_TR
dc.title.alternativeA New method of global optimization by using fuzzy controlleren_US
dc.typeThesisen_US
dc.typeTeztr_TR
dc.contributor.authorID100797tr_TR
dc.contributor.departmentKontrol ve Otomasyon Mühendisliğitr_TR
dc.contributor.departmentControl and Automation Engineeringen_US
dc.description.degreeDoktoratr_TR
dc.description.degreePh.D.en_US
Appears in Collections:Kontrol ve Otomasyon Mühendisliği Lisansüstü Programı - Doktora

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