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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">novtexmech</journal-id><journal-title-group><journal-title xml:lang="ru">Мехатроника, автоматизация, управление</journal-title><trans-title-group xml:lang="en"><trans-title>Mekhatronika, Avtomatizatsiya, Upravlenie</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">1684-6427</issn><issn pub-type="epub">2619-1253</issn><publisher><publisher-name>Commercial Publisher «New Technologies»</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17587/mau.20.367-375</article-id><article-id custom-type="elpub" pub-id-type="custom">novtexmech-648</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>ДИНАМИКА, БАЛЛИСТИКА И УПРАВЛЕНИЕ ДВИЖЕНИЕМ ЛЕТАТЕЛЬНЫХ АППАРАТОВ</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>DYNAMICS, BALLISTICS AND CONTROL OF AIRCRAFT</subject></subj-group></article-categories><title-group><article-title>Сравнение прямого метода и принципа максимума в задаче формирования программного управления летательным аппаратом</article-title><trans-title-group xml:lang="en"><trans-title>Comparison of the Direct Method and the Maximum Principle in the Problem of the Aircraft Program Control Design</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Корсун</surname><given-names>О. Н.</given-names></name><name name-style="western" xml:lang="en"><surname>Korsun</surname><given-names>O. N.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Д-р техн. наук, проф.</p><p>г. Москва.</p></bio><bio xml:lang="en"><p>D. Sc., Professor.</p><p>Moscow.</p></bio><email xlink:type="simple">marmotto@rambler.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Стуловский</surname><given-names>А. В.</given-names></name><name name-style="western" xml:lang="en"><surname>Stulovskii</surname><given-names>A. V.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Инженер.</p><p>г. Москва.</p></bio><bio xml:lang="en"/><email xlink:type="simple">avstlv2@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru">Государственный научно-исследовательский институт авиационных систем.<country>Россия</country></aff><aff xml:lang="en">State Research Institute of Aviation Systems.<country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2019</year></pub-date><pub-date pub-type="epub"><day>05</day><month>06</month><year>2019</year></pub-date><volume>20</volume><issue>6</issue><fpage>367</fpage><lpage>375</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Commercial Publisher «New Technologies», 2019</copyright-statement><copyright-year>2019</copyright-year><copyright-holder xml:lang="ru">Commercial Publisher «New Technologies»</copyright-holder><copyright-holder xml:lang="en">Commercial Publisher «New Technologies»</copyright-holder><license xlink:href="https://mech.novtex.ru/jour/about/submissions#copyrightNotice" xlink:type="simple"><license-p>https://mech.novtex.ru/jour/about/submissions#copyrightNotice</license-p></license></permissions><self-uri xlink:href="https://mech.novtex.ru/jour/article/view/648">https://mech.novtex.ru/jour/article/view/648</self-uri><abstract><p>Обсуждается решение задачи формирования программного управления для объекта, задаваемого нелинейной системой дифференциальных уравнений. Известные методы оптимального управления требуют решения двухточечной краевой задачи, что в общем случае связано с принципиальными сложностями. Поэтому предлагается методика, использующая прямой метод решения, в котором оптимизация осуществляется посредством популяционного алгоритма. На примере движения летательного аппарата проводится сравнение решений, полученных прямым методом и на основе классической теории оптимального управления, в первую очередь, принципа максимума Понтрягина. Приводятся результаты, показывающие высокую степень совпадения полученных управлений при различных способах выбора целевого функционала.</p></abstract><trans-abstract xml:lang="en"><p>The article deals with the problem of program control design for a dynamic object defined by a nonlinear system of differential equations. Known methods of optimal control require the two-point boundary value problem solution, which in general is coupled with fundamental difficulties. Therefore, this paper proposes a technique that uses the direct method, in which the functional is minimized directly using a population-based algorithm. The use of direct methods is based on the assumption that control signals may be defined by a finite set of parameters. Then a scalar functional is formed, the numerical value of which measures the quality of the obtained solutions. In this case, the search for optimal control is reduced to the problem of single-criterion multi-parameter optimization. The practical importance of this approach is that it eliminates the need to solve a two-point boundary value problem. However, this results in another difficulty, since the approximation of control, in general, requires a large number of parameters. It is known that in this case, the effectiveness of conventional gradient numerical optimization methods decreases markedly. Therefore, it is proposed to take the next step and apply genetic or population-based optimization algorithms that have confirmed their performance in solving this class of problems. For this purpose the paper uses one of the modifications of the particle swarm algorithm. The technique is applied to a test problem describing the spatial movement of a maneuverable aircraft. The direct method is compared with two classical solutions based on the condition that the partial control derivatives of the Hamilton function are equal to zero and with the condition of Hamilton function maximum over controls (Pontryagin’s maximum principle). The presented results show the high degree of similarity between obtained controls for all considered methods of selecting the target functional. At the same time, the accuracy of classical algorithms turns out to be slightly worse, and they show a higher sensitivity to the quality of the initial approximation. Thus, the obtained results confirm the approximate equivalence of the direct method and the classical methods of program control design, at least for the class of problems under consideration. The practical significance of this research is that the use of the direct method is much simpler than solving a two-point boundary value problem necessary for classical algorithms.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>оптимальное программное управление</kwd><kwd>двухточечная краевая задача</kwd><kwd>принцип максимума</kwd><kwd>прямые методы</kwd><kwd>популяционный алгоритм численной оптимизации</kwd></kwd-group><kwd-group xml:lang="en"><kwd>optimal program control</kwd><kwd>two-point boundary value problem</kwd><kwd>Pontryagin’s maximum principle</kwd><kwd>populationbased numerical optimization algorithm</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Справочник по теории автоматического управления / Под ред. А. А. Красовского. М.: Наука, 1987. 712 с.</mixed-citation><mixed-citation xml:lang="en">Krasovskij A. A. ed. 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