Optimal Control Problem by Linear Dynamic System of the Second Order

Мирослав Михайлович Копець

Abstract


Background. For controlled object which behavior is described by the system of two linear differential equations and with criteria of quality in the form of quadratic functional the optimal control problem is considered. In contrast to the general used methods for investigation of this problem (Pontryagin’s maximum principle, Bellman’s method of dynamic programming) the Lagrange’s multipliers method is proposed by author. Such approach provided the possibility to more effectively obtain the solution of the investigated optimization problem in comparison with the above-mentioned methods.

Objective. The purpose of the research is to obtain the formulas for calculation of optimal control and minimal value of the cost functional.

Methods. In the paper the methods of calculus of variations were used.

Results. In the paper the necessary optimality conditions were obtained and the uniqueness of optimal control was proved. On the basis of these conditions the system of differential Riccati equations is deduced.

Conclusions. The solution of this system permits to write the closed formula for optimal control and formula for calculation of minimal value of the performance functional. The results obtained in the paper may be used for investigation of the process of landing of airplane.


Keywords


Quadratic functional; Method of Lagrange multipliers; Necessary conditions of optimality; Optimal control; System of differential Riccati equations

References


J.N. Andreev, Control by Finite Dimensional Linear Objects. Moscow, USSR: Nauka, 1976, 424 p. (in Russian).

B.N. Bublic and N.F. Kirichenko, Fundamentals of the Control Theory. Kyiv, USSR: Vyshcha Shkola, 1975, 328 p. (in Russian).

J.N. Rojtenberg, Automatic Control. Moscow, USSR: Nauka, 1978, 551 p. (in Russian).

A.N. Vasiljev, Mathematica. A practical Course with Examples of the Solution of Applied Poblems. Kyiv, Ukraine: Vek+, St. Petersburg, Russia: Korona.Vek, 2008, 448 p. (in Russian).

A.M. Letov, The Dynamics of Flight and Control. Moscow, USSR: Nauka, 1969, 360 p. (in Russian).

M.M. Kopets, “Optimal control problem by process of a string vibration”, Teoriya Optymal'nykh Rishen', pp. 32–38, 2014 (in Russian).

A.A. Chikrii, Conflict Controlled Processes. Кyiv, Ukraine: Naykova Dumka, 1992, 384 p. (in Russian).

A.A. Chikrii and S.D. Eidel'man, “Game control problem for quasi-linear systems with fractional derivatives of Riemann–Liouville”, Cybernetics and Systems Analysis, vol. 36, no. 6, pp. 66–99, 2012.

S.D. Eidel'man and A.A. Chikrii, “Dynamic game approach problem for equations of fractional order”, Ukrayins'kyy Matematychnyy Zhurnal, vol. 52, no. 11, pp. 1566–1583, 2000.


GOST Style Citations


  1. Андреев Ю.Н. Управление конечномерными линейными объектами. – М.: Наука, 1976. – 424 с.
     
  2. Бублик Б.Н., Кириченко Н.Ф. Основы теории управления. – К.: Вища школа, 1975. – 328 с.
     
  3. Ройтенберг Я.Н. Автоматическое управление. – М.: Наука, 1978. – 551 с.
     
  4. Васильев А.Н. Mathematica. Практический курс с примерами решения прикладных задач. – К.: Век+, СПб: Корона.Век, 2008. – 448 с.
     
  5. Летов А.М. Динамика полета и управление. – М.: Наука, 1969. – 360 с.
     
  6. Копец М.М. Задача оптимального управления процессом колебания струны // Теорія оптимальних рішень. – 2014. – С. 32–38.
     
  7. Чикрий А.А. Конфликтно управляемые процессы. – К.: Наук. думка, 1992. – 384 с.
     
  8. Chikrii A.A., Eidel'man S.D. Game control problem for quasi-linear systems with fractional derivatives of Riemann–Liouville // Cybernetics and Systems Analysis. – 2012. – 35, № 6. – P. 66–99.
     
  9. Eidel'man S.D., Chikrii A.A. Dynamic game approach problem for equations of fractional order // Український мат. журнал. – 2000. – 52, № 11. – С. 1566–1583.




DOI: https://doi.org/10.20535/1810-0546.2015.6.72816

Refbacks

  • There are currently no refbacks.




Copyright (c) 2017 NTUU KPI