Bölünmez sınır koşullu optimizasyon problemlerinde geri beslemeli bir yapının bilgisayarda realizasyonu

Abstract

Bu çalışmada, bölünmez sınır koşullu optimal kontrol problemlerinde geri beslemeli (feedback) bir yapı oluşturularak bilgisayarda bunun realizasyonu üzerinde çalışılmıştır. Konu üç bölümde sunulmuştur. Birinci bölümde parametreli optimizasyon problemlerinin tanımı ve çözümü üzerinde durulmuştur. ikinci bölümde dinamik sistemler için optimizasyon problemleri ele alınmıştır. Tek adımlı ve çok adımlı sistemler hakkında bilgi verilmiştir. Üçüncü bölümde çalışmanın ana teması incelenmiştir. Bu bölümde bölünmez sınır koşullu optimizasyon problemlerinde geri beslemeli bir yapı oluşturulmuştur. Kullanılan algoritmada \(i) (i = 0,1,... İV) Lagrange çarpanları için bir kabul yapılıp bu kabulün doğru olduğu görülmüştür. Ayrıca aynı probleme EK- A da başka bir çözüm ile yaklaşılmıştır. İncelenen problemde elde edilen her iki çözüm içinde bilgisayarda programlar yazılarak Ekler bölümünde verilmiştir. Bu programlar kullanılarak yapılan örnek problemlerin çözüm çıktıları ve elde edilen sonuçlarla ilgili değerlendirme çalışma içinde görülmektedir.

On this work an algorithm for finding solution of the optimization prob lems with unseperated two-point boundary conditions is proposed and the related computer codes are given. 1. Statement of The Problem The controlled process under consideration is described by the equations x(i + 1) = F(i)x(i) + G(i)u(i) + V{i), (1) the boundary conditions fax(0) + fax(N) = q (2) and the function N-l 2 1 1 N_1 J = t-xT(N)Sx(N) + £ V xT(i)Q(i)x(i) +uT(i)R(i)u(i) +2xT(i)L(i)u{i) (3) that has to be minimized. Here u is m-dimensional vector of the controls, x is the n-dimensoinal vector of the phase coordinates. Coefficient matrices Q(i) = QT(i), R(i) = RT(i), S = ST are nxn,mxm and nxn dimensional respectively (i = 0, 1,... N - 1) and q is fe-dimensional vector [1-3]. For the solution of the problem we have two approaches. 2. The Method 1 One can show that solution of the problem (1), (2), (3) reduces to the solution of the system of equations x(i + 1) = M(i)x(i) + K(i)X(i + 1) + V{i) (4) A(t) = P(i)x(i) + MT(i)X(i + 1) (5) with boundary conditions (2) and A(0) = fxu (6) A(JV) = Sx(N) - (0) \ ri6s 5(0) F(0) -4TJ\ v J~\ -wT\_( A(0) >I; \Sx(N)-X(N) Assume that, non-singular matrix N with appropriate dimension is defined with the following equation t )u=\ c.,.f7\n_ vah (19) VI lT wlÎH-vo That is, 0 /" ~ V JV21 JV22 7 lV 0 JW^' %Wn)m+(iy»)-(})m}. Seperatiug JV to its approj>riate subparts, 'Nn JV12 JV =. İV21 iV22 we have A^1A(0) + JV12Sx(JV)-JV12A(JV) = z/ (k equations) (20) iV2iA(0) + iV22Sx(JV)-JV22A(JV) = 0 (2n-k equations) (21) Equations (2) and (21) can put together as follows -fa 0 \fx(N)\_(-fa 0 \(*(0)\ (q\ N-xS -Ni2)\X{N)J ~ V 0 -Nn)\X(0)J + V0; - v v -^^^ ©1 02 9 denoting *>-($). solution of the above equation can be written neatly as -(OHM"1 {«*(#)-«.}. (22) Now we want to write equations (4) and (5) also in the matrix form as, / - Jv(0 \ / x(i + l)\_( M(i) 0 \ / x(i) \ ( V(İ) \ 0 ^(ojiAii+ijj-^-pw i)\\(i)j + \ 0 ; Pi(i) P-2(i) Vi(«) Briefly it is P^zii + 1) = P2(*M0 + Vi(») (23) This recursive equation after using JV times beginning with i = 0 gives z(N) and it can be written as z{N) = Pz(Q) + V (24) vii where the known matrices P and V, can be shown as following: F = n£olP(0 (25) _ ( Z17 i^llıPU^Vİİ) i£jN-l 4. Conclusion The computer codes are given for the both approaches to the solution. The flow charts of the programs can in a short form be given in the written form as follows. For the first approach: The z(i), i = 0, 1,..., N) values are computed by use of the given datas. ( x(i)\ Because z(i) =, ;.; ] this means that we have all the values of x(i) and \X(i)J X(i). Then using these, u(i) 's are computed. For the second approach: Because in this systematic we have X(i) = S(i)x(i) + N(i)v + W(i), first matrices S(i), N(i) and W(i) are calculated. Then in -fax(N) = NT(i)x(i) + n(i)u + w(i) matrices n(i) and w(i) are computed. The next step is to solve the following system of equations 5(0) and obtain z(Q) and u. cjn -Ar (0) -7i(0) ^ (<0)\ _ fq + w{0)\ Now we know the starting values of x and A as x(0) and A(0). Solving the recursive equations x(i + 1) = M(i)x(i) + K(i)X(i + 1) + V(i) X(i) = S(i)x(i) +N(İ)u + W(i) all x(i) and X{i) {i = 0, 1,..., N - 1) become known. viii The first approach doesn't work for all the possible values of n and k. This because the matrix [«^j in the equation (A.22) as a result of its construction, when n ^ k is always singular. For the n = k case solution can be found under the assumption of the non-singularness of the matrices R(i) and F(i). On the other hand, the second approach can produce solutions under the only assumption that the matrices R(i) and F(i) are invertible. To decide the correction of the codes, the relative errors are calculated considering the equations z(* + l) = /*(*(*), u(0) q = (x(0),x(N)) and using the obtained values of x(i) and u(i). In general the second ap proach is better in this test with respect to the first.