NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS

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1 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana Mustafna and Vladmr Vaytev Bashkr State Unversty, Address: 3, Valdy Str., Ufa, Russa E-Mal: ABSTRACT In the artcle based on the method of varatons n the space of controls the algorthm s developed and program was mplemented to determne the optmal control problems wth free rght end. As an llustraton method, presents the results of numercal soluton of the three examples wth constrants on the control and phase varables. The advantage of ths algorthm s the lack of requrements for the selecton of the ntal approxmaton control parameter and phase varables. The algorthm has good convergence and can be used to solve a large class of applcatons n varous branches of the economy. By usng the developed algorthm determned the optmum trajectory and the numercal values of the control parameter for the test problems. A comparatve analyss of the results of the numercal soluton of the examples for dfferent values of ntal approxmaton control and precson. Keywords: method of varatons, optmal control, phase varables. 1. INTRODUCTION Methods of the optmum control theory are ntensvely used n varous applcaton areas. Control theory s applcaton-orented mathematcs that deals wth the basc prncples underlyng the analyss and desgn of (control) systems. Systems can be engneerng systems, economc systems, bologcal systems and so on. To control means that one has to nfluence the behavour of the system n a desrable way: for example, n the case of an ar condtoner, the am s to control the temperature of a room and mantan t at a desred level, whle n the case of an arcraft, we wsh to control ts alttude at each pont of tme so that t follows a desred trajectory. As a result, more and more people wll beneft greatly by learnng to solve the optmal control problems numercally. Ths work s devoted to an actual problem - development of effcent and unversal algorthms of numercal problem solvng of optmal control.. PROBLEM STATEMENT Consder the followng optmal control mnnmze I u Subject to G x t (1) dx f ( t, x( t), u( t)), t t 0, T, x(0) x0, () dt u 0 (3) Where u t R s the functon characterzng the n operatng nfluence, x t R s functon descrbng a condton of process and t s tme. Let s consder varous algorthms for problem solvng of optmal control..1 Performance crteron A performance crteron (also called cost functonal or smply cost) must be specfed for evaluatng the performance of a system quanttatvely. By analogy to the problems of the calculus of varatons, the cost functonal I : U[ t0, t1] R may be defned n the socalled Lagrange form: t1 I ( u) f ( t, x( t), u( t)) dt. (4) t THE ALGORITHM OF THE METHOD OF VARIATIONS The algorthm conssts of 9 steps: 1. Guess an ntal approxmaton of control U 0.. Break nterval [ t 0, t k ] to n parts, consttutng a unform system of unts. 3. Select startng node t 0, whch wll be a varaton of controls. 4. Compute U ( t 0 ) U. 5. Compute x (t), u(t) by solvng (3). 6. Calculate I (u) accordng to (4). 7. Go to t 1 and go to step 4 for all remanng ponts t. 8. Determne the mnmum value of the crteron calculated for all ponts t and defne a new control U1 corresponds to the lowest value crteron. 9. Set U U. Then, wth the control U 1, go to step 3 untl wll not fnd varaton n whch the performance crteron wll not be mproved. 4. DISCUSSIONS The software for the numercal calculatons presented below n ths artcle was developed n Borland 30

2 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Network (ARPN). All rghts reserved. Delph envronment. For each of the followng cases, we wll compute the Eucldean norm of the soluton error: x x ( t ), x x ( t ), x1 u 1 u u ( t). 1 x Example-1: Consder the followng optmal x u( t), x(0) 0, x(1) 0.5, x R, u R, t 0,1. (5) 1 I u t x t dt mn. (6) 0 The optmal control problem s to fnd a control law u whch mnmzes cost functonal (6). The analytcal soluton of ths problem s presented n []. Fgure-1 show the comparson between numercal soluton and approxmate soluton for u Table-1 presents smulaton results for dfferent ntal guess and accuracy of ths problem. The performance measure s: Fgure-1. Comparson between numercal soluton and approxmate soluton, Example 1. Table-1. Smulaton results for dfferent ntal guess and accuracy, Example 1.. u0 Accuracy Elapsed tme, s. u 1 0 0,1 0,36 1,4 0,93 0 0,01 0,74 0,99 0, ,001 1, 0,083 0, ,6 0,001 1,4 0,004 0,00 5-0,9 0,0001 4,75 0,0004 0, ,1 0, ,43 0, ,00013 x Example-: Consder the followng optmal x 1t x t, x t u t 0.5 u t ; x 0 0, 0 0, 1 x 0 t 1, 0 u 1. (7) 31

3 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Network (ARPN). All rghts reserved. The performance measure s: x x x 1 max. I (8) 1, 1 The optmal control problem s to fnd a control law u whch mnmzes cost functonal (8). The analytcal soluton of ths problem s presented n [3]. Fgure- shows the comparson between numercal soluton and approxmate soluton for u Table- presents smulaton results for dfferent ntal guess and accuracy of ths problem. At the same tme the estmated value of the control parameter n the range 0 t 1 has a constant value equal to 1. Fgure-. The suboptmal states. Table-. Smulaton results for dfferent ntal guess and accuracy, Example.. u0 Accuracy Elapsed tme, s. u x 1 1 0,6 0,1,3 1,06 1,105 0,85 0,6 0,01 6,43 1,009 0,04 0, ,6 0,001 9,45 1 0,001 0, ,8 0,001 11,58 1 0,003 0, ,8 0, ,3 1 0,0033 0, ,6 0, ,85 1 0,0000 0,00007 x Example-3: Consder the followng optmal x 1t x t, x t x1 t u t ; x 0 0, 0 0, 1 x 0 t, u 1. The performance measure s: x x x mn. 1, (9) I (10) The optmal control problem s to fnd a control law u whch mnmzes cost functonal (10). 1, t 0.5, The exact soluton are: u ( t) 0, 0.5 t, 1, t.5. Fgure-3, Fgure-4 shows the comparson between numercal soluton and approxmate soluton for u Table 3 presents smulaton results for dfferent ntal guess and accuracy of ths problem. 3

4 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Network (ARPN). All rghts reserved. Fgure-3. The suboptmal states. Fgure-4. The suboptmal control. Table-3. Smulaton results for dfferent ntal guess and accuracy, Example 3.. u0 Accuracy Elapsed tme, s. u x ,1,06 3,06 1,11 1,1 0 0,01,85,99 0,14 0, ,001 4,1,987 0,018 0, ,6 0,001 3,94,854 0,019 0, ,9 0,0001 1,06,004 0,1086 0, ,1 0, ,35,003 0,1093 0,1088 x 33

5 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Network (ARPN). All rghts reserved. 5. CONCLUSIONS For many optmal control problems, the method of varatons s the best opton we have. The advantage of ths algorthm s that t does not have requrements wth ntal guess. The algorthm has good convergence and can be used to solve a large class of applcatons n varous felds of natonal economy. REFERENCES Igor Grgoryev, Svetlana Mustafna, Oleg Larn «Numercal soluton of optmal control problems by the method of successve approxmatons». Internatonal Journal of Pure and Appled Mathematcs. 111(4): do:10.173/jpam.v Igor Grgoryev, Svetlana Mustafna «Global optmzaton of functons of several varables usng parallel technologes». Internatonal Journal of Pure and Appled Mathematcs. 106(1): do: /jpam.v Gulnaz Shangareeva, Igor Grgoryev, Svetlana Mustafna «Comparatve Analyss of Numercal Soluton of Optmal Control Problems». Internatonal Journal of Pure and Appled Mathematcs. 110(4): do: /j-pam.v Igor Grgoryev, Eldar Mftakhov, Svetlana Mustafna «Mathematcal Modellng of the Copolymerzaton of Styrene wth Malec Anhydrde n a Homogeneous Envronment». Internatonal Journal of Chemcal Scences. 14(1): Mustafna S., Mftakhov E., Mkhalova T «Solvng the drect problem of butadene-styrene copolymerzaton». Internatonal Journal of Chemcal Scences. 1():