DOI: https://doi.org/10.32515/2414-3820.2018.48.35-44
The Problem of Selection of the Optimal Strategy of Minimax Control by Objects in Agricultural Production with Distributed Parameters
About the Authors
Oleksiy Lobok, Associate Professor, Doctor of Physical and Mathematical Sciences, National University of Food Technologies, Kyiv, Ukraine
Boris Goncharenko, Professor, Doctor in Technics (Doctor of Technics Sciences), National University of Food Technologies, Kyiv, Ukraine, e-mail: goncharenkobn@i.ua
Larisa Vihrova, Professor, PhD in Technics (Candidate of Technics Sciences), Central Ukrainian National Technical University, Kropyvnytskyi, Ukraine, e-mail: vihrovalg@ukr.net
Marina Sych, PhD in Technics (Candidate of Technics Sciences), National University of Bioresources and Nature Management of Ukraine, Kyiv, Ukraine, e-mail: sm.nuft@gmail.com
Abstract
Тhe problem of minimax control synthesis for objects in agricultural production that are described by a two-dimensional heat conduction equation of parabolic type is solved. It is assumed that the control object functions under uncertainty conditions, and the perturbations acting on the object belong to some given hyperelipsoid. The problem of constructing a regulator in the state of an object for cases of point and mobile limit control is considered in accordance with the integral-quadratic quality criterion. With the help of numerical optimization methods, the problem of determining the optimal location of concentrated regulators at the boundary of a rectangular region and the problem of finding the optimal law of motion of a mobile limit regulator is solved. The problem is posed and solved in the minimax formulation when there is an optimal control on the state of the object functioning under uncertainty conditions so that the regulator minimizes the maximum control error from a set of possible values, taking into account the most unfavorable perturbations that can act on the object or system. In this case, the perturbations of the object belong to a given limited region. The results of computational experiments illustrating the effectiveness of the constructed limiting concentrated and moving regulators are presented. The obtained results indicate that the controls found in the work are indeed optimal and ensure minimum errors (deviations from the given state) of the functioning of the system and energy costs for the implementation of control under given conditions and in the absence of any information on external action other than the region of permissible perturbations.
In the work, for the first time, a minimax approach was used to control the objects described by the two-dimensional parabolic type thermal conductivity equation; the theoretical positions of synthesis of minimax regulators for cases of lumped boundary (point) and moving regulators are considered; algorithmic software is developed that allows to simulate the dynamics of the constructed minimax-regulators and to investigate the corresponding transients.
Keywords
minimax control, regulators, distributed parameter systems, optimization, gradient projection method, point and mobile limit controls
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References
1. Butkovskij, A.G. (1975). Metodi upravlenija sistemami s raspredelennymi parametrami [Control Systems for Distributed Parameters]. Moskow: Nauka [in Russian].
2. Butkovskij, A.G., Darinskij, Ju.V., Pustyl'nikov, L.M. (1976). Podvizhnoe upravlenie sistemami s raspredelennymi parametrami [Movable control systems with distributed parameters]. Avtomatika i telemehanika, 2, 15-25 [in Russian].
3. Butkovskij, A.G., Pustyl'nikov, L.M. (1980). Teorija podvizhnogo upravlenija sisistemami s raspredelennymi parametrami [Theory of mobile control systems with distributed parameters]. Moskow: Nauka [in Russian].
4. Lions Zh.-L. (1972). Optimal'noe upravlenie sistemami, opisyvajushhimisja uravnenijami s chastnymi proizvodnymi [Optimal control of systems described by partial differential equations]. Moskow: Mir [in Russian].
5. Ladyzhenskaja, O.A., Ural'ceva, N.N., Solonnikov, V.A. (1967). Linejnye i kvazilinejnye uravnenija parabolicheskogo tipa [Linear and quasilinear parabolic equations]. Moskow: Nauka [in Russian].
6. Lobok, O.P., Honcharenko, B.M., Savits'ka, N.M. (2015). Minimaksne upravlinnia v linijnykh dynamichnykh systemakh iz rozpodilenymy parametramy [Minimax control in linear dynamic systems with distributed parameters ]. Zhurnal «Naukovi pratsi NUKhT», Vol. 21, 6, 16 ̶ 26. [in Ukrainian].
7. Kirichenko, N.F. (1982). Minimaksnoe upravlenie i ocenivanie v dinamicheskih sistemah [Minimax control and estimation in dynamic systems]. Avtomatika i telemehanika, 1, 32-39 [in Russian].
8. Lobok, A.P. (1983). Minimaksnye reguljatory v sistemah s raspredelennymi parametrami [Minimax control and estimation in dynamic systems]. Modelirovanie i optimizacija slozhnyh sistem., Vol.2, 62-67[in Russian].
9. Vasil'ev, F.P. (1981). Metody reshenija jekstremal'nyh zadach [Methods for solving extremal problems]. Moskow: Nauka [in Russian].
GOST Style Citations
- Бутковский А.Г. Методи управления системами с распределенными параметрами. Москва: Наука, 1975. 568.
- Бутковский А.Г., Даринский Ю.В., Пустыльников Л.М. Подвижное управление системами с распределенными параметрами. Автоматика и телемеханика, 1976. №2. С. 15-25
- Бутковский А.Г., Пустыльников Л.М. Теория подвижного управления сисистемами с распределенными параметрами. Москва: Наука. 1980. 397 с.
- Лионс Ж.-Л. Оптимальное управление системами, описывающимися уравнениями с частными производными. Москва: Мир, 1972. 414 с.
- Ладыженская О.А., Уральцева Н.Н., Солонников В.А. Линейные и квазилинейные уравнения параболического типа. Москва: Наука, 1967. 736 с.
- Лобок О.П., Гончаренко Б.М., Савіцька Н.М. Мінімаксне управління в лінійних динамічних системах із розподіленими параметрами. Журнал «Наукові праці НУХТ». Київ: НУХТ, 2015. Т. 21, № 6. С.16 ̶ 26.
- Кириченко Н.Ф. Минимаксное управление и оценивание в динамических системах. Автоматика и телемеханика, 1982. №1. С. 32-39
- Лобок А.П. Минимаксные регуляторы в системах с распределенными параметрами. Вестн. Киев. универ. Моделирование и оптимизация сложных систем, 1983. Вып.2. С. 62-67.
- Васильев Ф.П. Методы решения экстремальных задач. Москва: Наука, 1981. 400 с.
Copyright (c) 2018 Oleksiy Lobok, Boris Goncharenko, Larisa Vihrova, Marina Sych
The Problem of Selection of the Optimal Strategy of Minimax Control by Objects in Agricultural Production with Distributed Parameters
About the Authors
Oleksiy Lobok, Associate Professor, Doctor of Physical and Mathematical Sciences, National University of Food Technologies, Kyiv, Ukraine
Boris Goncharenko, Professor, Doctor in Technics (Doctor of Technics Sciences), National University of Food Technologies, Kyiv, Ukraine, e-mail: goncharenkobn@i.ua
Larisa Vihrova, Professor, PhD in Technics (Candidate of Technics Sciences), Central Ukrainian National Technical University, Kropyvnytskyi, Ukraine, e-mail: vihrovalg@ukr.net
Marina Sych, PhD in Technics (Candidate of Technics Sciences), National University of Bioresources and Nature Management of Ukraine, Kyiv, Ukraine, e-mail: sm.nuft@gmail.com
Abstract
Keywords
Full Text:
PDFReferences
1. Butkovskij, A.G. (1975). Metodi upravlenija sistemami s raspredelennymi parametrami [Control Systems for Distributed Parameters]. Moskow: Nauka [in Russian].
2. Butkovskij, A.G., Darinskij, Ju.V., Pustyl'nikov, L.M. (1976). Podvizhnoe upravlenie sistemami s raspredelennymi parametrami [Movable control systems with distributed parameters]. Avtomatika i telemehanika, 2, 15-25 [in Russian].
3. Butkovskij, A.G., Pustyl'nikov, L.M. (1980). Teorija podvizhnogo upravlenija sisistemami s raspredelennymi parametrami [Theory of mobile control systems with distributed parameters]. Moskow: Nauka [in Russian].
4. Lions Zh.-L. (1972). Optimal'noe upravlenie sistemami, opisyvajushhimisja uravnenijami s chastnymi proizvodnymi [Optimal control of systems described by partial differential equations]. Moskow: Mir [in Russian].
5. Ladyzhenskaja, O.A., Ural'ceva, N.N., Solonnikov, V.A. (1967). Linejnye i kvazilinejnye uravnenija parabolicheskogo tipa [Linear and quasilinear parabolic equations]. Moskow: Nauka [in Russian].
6. Lobok, O.P., Honcharenko, B.M., Savits'ka, N.M. (2015). Minimaksne upravlinnia v linijnykh dynamichnykh systemakh iz rozpodilenymy parametramy [Minimax control in linear dynamic systems with distributed parameters ]. Zhurnal «Naukovi pratsi NUKhT», Vol. 21, 6, 16 ̶ 26. [in Ukrainian].
7. Kirichenko, N.F. (1982). Minimaksnoe upravlenie i ocenivanie v dinamicheskih sistemah [Minimax control and estimation in dynamic systems]. Avtomatika i telemehanika, 1, 32-39 [in Russian].
8. Lobok, A.P. (1983). Minimaksnye reguljatory v sistemah s raspredelennymi parametrami [Minimax control and estimation in dynamic systems]. Modelirovanie i optimizacija slozhnyh sistem., Vol.2, 62-67[in Russian].
9. Vasil'ev, F.P. (1981). Metody reshenija jekstremal'nyh zadach [Methods for solving extremal problems]. Moskow: Nauka [in Russian].
GOST Style Citations
- Бутковский А.Г. Методи управления системами с распределенными параметрами. Москва: Наука, 1975. 568.
- Бутковский А.Г., Даринский Ю.В., Пустыльников Л.М. Подвижное управление системами с распределенными параметрами. Автоматика и телемеханика, 1976. №2. С. 15-25
- Бутковский А.Г., Пустыльников Л.М. Теория подвижного управления сисистемами с распределенными параметрами. Москва: Наука. 1980. 397 с.
- Лионс Ж.-Л. Оптимальное управление системами, описывающимися уравнениями с частными производными. Москва: Мир, 1972. 414 с.
- Ладыженская О.А., Уральцева Н.Н., Солонников В.А. Линейные и квазилинейные уравнения параболического типа. Москва: Наука, 1967. 736 с.
- Лобок О.П., Гончаренко Б.М., Савіцька Н.М. Мінімаксне управління в лінійних динамічних системах із розподіленими параметрами. Журнал «Наукові праці НУХТ». Київ: НУХТ, 2015. Т. 21, № 6. С.16 ̶ 26.
- Кириченко Н.Ф. Минимаксное управление и оценивание в динамических системах. Автоматика и телемеханика, 1982. №1. С. 32-39
- Лобок А.П. Минимаксные регуляторы в системах с распределенными параметрами. Вестн. Киев. универ. Моделирование и оптимизация сложных систем, 1983. Вып.2. С. 62-67.
- Васильев Ф.П. Методы решения экстремальных задач. Москва: Наука, 1981. 400 с.