Global optimizasyon metodu kullanılarak bir doğru akım motorunun hız kontrolunun gerçekleştirilmesi

Abstract

Mühendislikte bir problemin en uygun çözümünü bulmak çok önemlidir. Amaç kullanılan sistemin hem isteklere tam yanıt vermesi hemde olabildiğince ekonomik olmasıdır. Teoride tüm gereksinimlere karşılık verebilen fiziksel sistemler oluşturmak olasıdır. Ancak, bu üstünlüklerin getirdiği büyük ekonomik yükler göz ardı edilemez. Bu yüzden en iyi çözüm aranırken, kullanım alanına ve beklentilere göre bir amaç ölçütü belirlenir ve bulunabilecek en uygun çözüm elde edilmeye çalışılır. Bu çalışmada, yukarıdaki açıklamalar doğrultusunda global bir optimizasyon metodu yardımıyla »fiziksel bir sistemin tanımlanan performans büyüklüğü maksimize edilmeye çalışılmıştır. Uygulanan arama metodu, fiziksel sistemin kontrolü için önemli olan çıkış büyüklüklerini veri olarak alıp global aramayı gerçekleştirdiğinden, hem doğrusal hemde doğrusal olmıyan tüm sistemler için kulanılabilir. Yapılan işlem ; nasıl bir sistem olursa olsun, sistem çıkış verilerini bilgisayara almak, tanımlanan perfor mansa göre kontrol katsayılarını belirleyici aramayı yapmak ve bir sonraki adımda yeni katsayılara göre oluşturulmuş kontrol işaretini fiziksel sisteme tekrar uygulamak şeklinde açıklanabilir. Çalışmamızda kullandığımız fiziksel sistem, bir doğru akım motorudur ve hızının kontrolü yapılmaya çalışılmıştır. Doğal olarak böyle bir sistemde kontrol amaçları bellidir. Bunlar ; sistemin verilen hız referansına en kısa sürede, olabilecek en az aşımla ve kararlı hal hatası kalmayacak şekilde oturması, biçiminde özetlenebilir. Tezin bundan sonraki aşamalarında, izlenilen yollar ve alınan sonuçlar ayrıntılarıyla açıklanmaya çalışılacaktır.

Abstract : In this study, a dc motor's speed had been controlled by using a digital controller. The controller structure was in one case classical PID controller and in another case, it will be a nonlinear state feedback controller. The best values of the controller parameters had been tried to find by using a new global optimization method. Keywords *. Digital controller, PID controller, global optimization, performance index. 1. INTRODUCTION To find the best solution of a problem is very important for all the fields of engineering and science. If we define the problem in terms of mathema tical representation, such that the measure of the per formance is given by f, which is a real valued, non-li near function of n parameters, x= (xl, x2,..., xn), then the problem becomes minimizing or maximizing the function f. There is no real difference between mini mizing or maximizing since the maximum of f(x) is the minimum of -f(x). When the best solution of a system was found, we can say that the performance of the system is maximum. In our study, the best values of the transient state such as the percent overshoot, the risetime and the steady state error had been found by using the new global search method. 2. GLOBAL OPTIMIZATION METHOD In this study, a new global search method was applied to the system. The method uses W.L. PRICE'S Controlled Random Search, B. E. STUCKMAN ' s Global Optimization Using Brownian Motion For Multivariable Functions, and Grid Point Search. This method had been developed by Olcay BOZ. In Price's method, a search domain is defined, N trial points are chosen and stored in the array A. At each iteration a new trial point is found and if the function value at P is greater than fm, (M is the point which has the lowest value in the points in A ) then M is replaced with P in the array. Iteration goes on in this manner. In Stuckman's method, the function is evaluated at the second power of 2 vertices and line segments are constructed between each two points for all the line segments. The probability that the unknown function will exceed *fm(x), the largest value of the function yet found after m guesses by some pozitive constant Km is maximized, then the location becomes the next guess. Line segments joining the new point to the other points are added and returned back. In the Boz's method, given a function n variables, an initial search domain is defined by using the limits of each variable. The search domain is divided into grids and N sample points are found from the grids. The func tion is evaluated at each trial point and the position and the function value corresponding to each point are stored in an array A. Standart deviation of the function values of the points stored in A are found and this is used for finding Km. Line segments which connect each set of two points are constructed. For each line segment the point that maximizes the probability that the function will exceed *fm(x), by Km is compared with the point (L) of lowest function value in the array. If fp > fl, then L is replaced with P. It goes in this manner until all the line segments are processed. If the function values of the points with the greatest and lowest function value in A are identical to a predeter mined accuracy we stop. If not standart deviation of the function values of the points stored in A are found. Li ne segments which connect each set of two points in the array are constructed and iterations goes on this manner. vx 3. DESING METHOD Boz ' s hybrid method which was written in the pre vious section was tested on the speed control system of a dc motor. The principle circuit of the design equipment had been shown in Figure 3-1. Conventional optimal control techniques optimize the performance of a system by minimizing a quadratic index of performance. A common definition of a quadratic performance index or cost J is : /"« J = l/2 J (XT(t)QX(t) + UT(t)RU(t) )dt (3.1) o Where X is the state matrix and U is the system input matrix. Matrices R and Q are used for weighting the parameters in the index. This index is chosen to simplify the mathematics involved to allow a method of solution, i.e., the Ricatti equation for linear systems and Hamilton-Jacobi-Bellman equation for non-linear systems. The performance for our system is to be a weighted sum of the percent overshoot, the rise time and the steady state error. G(Q,R)= -cl.PO - c2.Tr -c3.e.. (3.2) The method has been presented of determining the optimal Q and R for the best design of an optimal cont rol system. The advantage of this technique is that it yields a design which is simultaneusly optimal in the quadratic sense as well as in terms of a seperate, more meaningful performance criterion developed by the disigner. The formulation of the problem is such that this meaningful performance criterion can be any realvalued function. Vll Controller System Uoot = R-*BTK Uo p t x(t) = Ax(t) + Buopt(t) X(0) = Xo Measurement y(t) y(t) = Cx(t) x(t) Figure 3-1 Design equipment principle circuit ( Block Diagram ) vm CONCLUSION In this thesis, a global optimization method was applied to a physical system such as separately excited dc motor to find the optimal parameters of the controller unit. The method doesn't need the mathematical model of the system. Because, it takes the real values of the system from the state variables which is chosen by ourself. Therefore, we are free to choose a relation ship that is meaningful to the application regardless of the system complexity. However, the main disadvantage of this way is many repetitions of the same experiment must be performed to find the best performance of the system. In spite of this disadvantage, it can be used all the systems whether linear or non-linear. And this is really good advantage for the method.