Ayrık sistemler için bazı optimal kontrol problemlerinin dinamik programlama metodu ile incelenmesi
Research of some optimal control problems using dynamic programming method for discrete systems
- Tez No: 83091
- Danışmanlar: PROF. DR. SEYİDALİ S. AKHİEV
- Tez Türü: Yüksek Lisans
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Ayrık sistemler, Bellman denklemi, Dinamik programlama, Optimum denetim, Discrate systems, Bellman equation, Dynamic programming, Optimum control
- Yıl: 1999
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Mühendislik Bilimleri Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
AYRIK SİSTEMLER İÇİN BAZI OPTIMAL KONTROL PROBLEMLERİNİN DİNAMİK PROGRAMLAMA METODU İLE İNCELENMESİ ÖZET Tez çalışmasında, R. Bellman' in dinamik programlama metodunun bir parametreli ayrık sistemlerle ilgili olan bazı optimal kontrol problemlerine uygulanması verilmiştir. x(t + l) = f(t,x(t),u(t))> t = 0,...,N-\; (1.1) u(t)eV(t), t = 0,...,N-l; (1.2) N-l J(x,u) - £g('»*(0»M(0) + ç>(x(N)) -» min (1.3) (=0 şeklinde ifade edilen optimal kontrol problemi incelenmiştir. Tez çalışması altı bölümden oluşur. İlk bölümde çalışmanın kısa özeti verilmiştir. İkinci bölümde ise bir parametreli ayrık sistemler için optimal kontrol probleminin sunuluşu verilmiştir. Üçüncü bölümde, genel şekilde verilmiş olan (1.4)- (1.7) problemi için dinamik problemler oluşturulmuştur. x(* + 1) = f(t,x{t),u(t)\ t = 0,...,N-l; (1.4) jc(0) = jc°, (1.5) AM J(x,u) = YuSİUx{t),u{t)) + ç(x(N)) -> min (1.6) u(t)eV(t), t = 0,...,N-\; (1.7) Dördüncü bölümde ise bu problem için optimallik prensibi verilmiş ve ispat edilmiştir. Beşinci bölümde, (1.4)-(1.7) problemi için Bellman denklemi verilmiştir. Altıncı bölümde ise Bellman denkleminin çözümü verilmiş ve bir örnek ele alınmıştır.
Özet (Çeviri)
RESEARCH OF SOME OPTIMAL CONTROL PROBLEMS USING DYNAMIC PROGRAMMING METHOD FOR DISCRETE SYSTEMS SUMMARY The theory of optimal control came out in the second half of the twentieth century. The Maximum Principle of L.S.Pontryagin has an important role in coming out this theory. The second important method of this theory is The Dynamic Prograrnming Method of R.Bellman. The Dynamic Programming method is used to solve many optimal prosesses approximately. In this study, application of R.Bellman's dynamic programming method to some optimal control problems related with one parametered discrete sistems is given. Discrete system held here can be considered as an approach of continuous optimal control problems expressed as first order differential equations. In this study, the optimal control problem for the discrete system held in this study, can be written shortly as shown below; x(t + l) = f(t,x(t),u(t)), t = 0,...,N-\; (1.1) u(t)eV(t), t = 0,...,N-l; (1.2) JV-l J(x,u) = Y,S{Ux{t),u{t)) + (p(x{N)) -» min (1.3) <=0 Here, x(t) = (xl(t),...,xn(t)) is the state vector, u(t) = (ul(t)i...iur(t)) IS the control vector; f = {fx,--,fn) is the given n-dimensional vector function; g and <p are the given functions, V(t) is the sub cell of Rr. One of the basic characteristics of the optimal control problem described in (1.1)-(1.3) is that the maximum principle of L.S.Pontryagin is not generally valid for such problems. In other words, for optimal control problems described in (1.1)-(1.3), L.S.Pontryagin' s Maximum Principal is valid for only special cases. For this reason, the problem held in this thesis is researched by using Bellman's dynamic programming method. The thesis contains six parts. In the first part, short summary of the study is given. In the second part, the presantation of the optimal control problem for one parametered discrete system is given. This presantation can be written shortly as below. viAssuming t = 0,...,N is the discrete time, let's show the state of the system at /time as x(t ) = (xx (/),... xn (/)) n dimensioned vector and the control vector applied to this system at /time as u(t) = (ul(t),..Mr(t)) r dimensioned vector. Also let's assume that the state of the system at t + 1 time is defined as below, regarding to t,x(t) and u(t). x(t + 1) = f{t,x{t),u(t)\ t = 0,..., N-l; (1.4) Here, f(t,x,u) = {fl(t,x,u),...,fn(t,x,u)) is the n dimensioned continous vector function that is given for t = 0,..., N - 1 values of discrete /time and is defined in R“ x Rr according to (x,u) vector argument for every / time. The system defined in (1.1) will only be controlled at / = 0,...,iV - 1 time. This function is controlled by means of u(0),...,u(N - 1) vectors. Because of this, u(Q),...,u(N - 1) vectors fit a definite control program. So the values of u(t) vector at / = 0,..., İV - 1 time form a definite control program. Definition 1.1 : Let's say that, u{t) vector function is defined for / = 0,..., N - 1 points and the u(t) value obtained at / time is a definite element of V(f) cell which was given before. u(t)eV(t), t = 0,...,N-l (1.5) Here, V(f) is a definite sub cell of Rr space for arbitrary t = 0,...,N - 1 times. In this condition, u(f) vector is called as possible vector function. Let's show possible control vectors cell with U. It is clear that we can define the U cell as below. U = {u(t) = (Ul(t),...,ur(t))\ u(t)eV(t), t = 0,...,N-l } Now let's handle a possible u(t) control vector and place it on the right side of the system (1.4). Thus (1.4) system x(0) = x° (1.6)will have the unique definite x(t) t = 0,1,...,^, solution that supports the beginning condition. We will call this solution as the trajectory valid for u(t) control of (1.4)-(1.6) problem. Optimal control problem for (1.4) discrete system can be presented as below: 1. Let's assume that x° ={xl,...,xn) is the given beginning state, (p(x) is the continous function defined at R”. Under these circumstances, find such u(t), t = 0,..., N - 1, control vector that supports (1.5) condition and gives the possible minimum value for J(u) = (p(x{N)) -> min (1.7) functional with the x(t), t = 0,...,N, solution of (1.5), (1.6) problem. Optimal control problem can be presented as below when x beginning point is not certain. 2. Find such u(t), t = 0,...,N -1 control and such x(t), t = 0,...,N that supports (1.5) condition and (w(/),x(/)) pairs supports the equation (1.4), thus x(N) value lets the J{u)=(p{x{N))-*rmn (1.8) functional have the minimum value. In the third part, dynamic problems have been formed for given (2.5)-(2.8) problem as general. In the forth part, proof of the optimization principle has been given for this problem. In the fifth part, Bellman equation has been given for this problem. In the sixth part, solution for the Bellman equation has been given and an example has been handled. vin
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