Исследование управляемости динамических систем при наличии ограничения на управления с применением интервальной математики
DOI:
https://doi.org/10.26577/JMMCS-2019-2-27Ключевые слова:
критерий, управляемость, управление, интервальная математика, динамические системы, интервал, интервальный вектор, дифференциальное уравнениеАннотация
Статья посвящена актуальной проблеме математической теории управляемости. В ней исследована математическая модель управления, описываемая обыкновенными дифференциальными уравнениями, учитывающая ограничения на управление. Как известно, проблема нахождения управляемости динамических систем с фазовыми ограничениями и ограничениями на управление до сих пор остается актуальной. Существует множество подходов к решению названной задачи. Классическая теория управления сегодня модифицируется и находит новые методы решения задач управляемости, оптимального управления и устойчивости, полученных решений. В ходе исследования управляемости динамической системы авторы применили интервальную математику, которая позволила получить эффективный критерий управляемости динамический систем с фазовыми ограничениями и ограничениями на управления. Данный метод применим для определенного класса задач, в которых данные описываются нормальным законом распределения.
Конструктивность предложенного критерия демонстрируется на двух примерах. Первый – модельная задача, описываемая уравнениями 2-го порядка. Второй – электромеханическая следящая система автоматического манипулятора, описываемая уравнениями 3-го порядка. Таким образом для динамических систем получили достаточное условие управляемости.
Библиографические ссылки
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[14] Kaucher E., "Interval analysis in the extended interval space IR," Computing Supplement 2 (1980): 33-49.
[15] Hodko S.Т., Proektirovanie sistem upravleniya s nestabilnymi parametrami [Designing control systems with unstable parameters] (L.: Engineering, 1987).
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[17] Ivlev R.S., Postroenie i issledovanie svoistv mnogomernyh sistem upravleniya intervalno-zadanymi obektami
[Construction and study of the properties of multidimensional control systems of interval-specified objects] (Almaty: Diss. Cand. those. Sciences, 1999).
[18] Ivlev R.S. and Sokolova S.P., "Postroenie vektornogo upravleniya mnogomernym intervalno-zadanym
obektom"[Construction of vector control multidimensional interval specified object], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 4 (1999): 3-13.
[19] Haritonov V.L., "Ob assimtoticheskoi ustoichivosti polozheniya ravnovesiya semeistva sistem lineinyh diferencialnyh uravneni"[On the asymptotic stability of the equilibrium position of a family of systems of linear differential equations], Differents. the Equations 14 (1978): 2086-88.
[20] Sharyi S.P., Intervalnye algebraicheskie zadachi i ih chislennoe reshenie [Interval algebraic problems and their numerical solution] (Novosibirsk: Diss. Dr. Phys.-Mat. Sciences, 2000).
[21] Shashihin V.N., "Zadacha robastnogo razmesheniya polusov v intervalnyh krupnomashtabnyh sistemah"[The task of robust pole placement in large-scale interval systems], Automation and Remote Control 2 (2002): 34-43.
[22] Hlebalin N.А., Sintez intervalnyh regulyatorov v zadache modalnogo upravleniya [Synthesis of interval regulators in the modal control problem] (Sarat.: Analytical methods for regulator synthesis: Interst. scientific Sat Saratov, 1988).
[23] Aschepkov L.T and. Dolgy D.V., "The universal solutions of interval systems of linear algebraical equations," Int. J. of Software Eng. and Knowledge Eng. 3 (1993): 477-85.
[24] Shashihin V.N., "Sintez robastnogo upravleniya dlya intervalnyh krupnomashtabnyh sistem s posledeistviem"[Synthesis of robust control for interval large-scale systems with aftereffect], Automation and Remote Control 12 (1997): 164-74.
[25] Li E.B. and Markus L., Osnovy teorii optimalnogo upravleniya [Fundamentals of optimal control theory] (М.: Nauka, 1972).
[26] Bialas S.A., "A necessary and sufficient conditions for stability of interval matrices," Int.J.Contr. 37 (1983): 717-22.
[27] Liu M., "Interval Analysis of Dynamic Response of Structures," International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) (2015): 810-13.
[28] Chen S. H. and Zhang, X. M., "Dynamic response of closedloop system with uncertain parameters using interval finite element method," Journal of Engineering Mechanics 132 (2006): 830-40.
[29] Qiu Z. P. and Wang X. J., "Parameter perturbation method for dynamic responses of structures with uncertain-butbounded parameters based on interval analysis," International Journal of Solids and Structures 42 (2005): 4958-70.
[30] Wang Z and Tian Q and Hu H, "Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters,"Springer Sci 84 (2015): 527–48.
[31] Wu, J.L. and Zhang, Y.Q., "The dynamic analysis of multibody systems with uncertain parameters using interval method,"Appl. Mech. Mater. (2012): 1555–61.
[32] Yunlong Li and Xiaojun Wang and Ren Huang and Zhiping Qiu, "Balanced-based model reduction of uncertain systems with interval parameters," Journal of Vibration and Control 22 (2016): 2958.
[2] Moore R.E. and Kearfott R.B. and Cloud V.J., Introduction to interval analysis (Philadelphia: SIAM Press, 2009), 6-10.
[3] Mazakov T.Zh. and Jomartova Sh. A., "Primenenie intervalnogo analiza v prakticheskih vychisleniyah"[Application of interval analysis in practical calculations], Computational technologies 7 (2002): 230-34.
[4] Jomartova Sh. A., "“Prakticheskie” intervalnye vychisleniya"[“Hands-on” interval calculations], Bulletin of NAS RK. 2 (2002): 41-6.
[5] Kolmanovskiy V. V. and Nosov V. R., Ustoichivost i periodicheskie rezhimy reguliruemyh sistem s posledeistviem [Stability and periodic regimes of controlled systems with aftereffect] (M.: Nauka, 1981).
[6] Kurzhanskiy А.B., Upravlenie i nabludenie v usloviyah neopredelennosti [Management and supervision in the conditions of uncertainty] (М.: Nauka, 1977).
[7] Аfanasiev V.N. and Kolmanovskiy V. V. and Nosov V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya [Mathematical theory of designing control systems] (М.: Vyshay shkola, 1998).
[8] Alefeld G. and Herzberger J., Introduction to Interval Computations, (York: Academic Press, 1983), 15-25.
[9] Yedavalli R.K., "Stability analysis of interval matrices: another sufficient condition," Int.J.Contr. 43 (1986): 767-72.
[10] Voevoda А.А. and Plohotnikov V.V., O metodike sinteza regulyatorov dlya obektov s intervalnymi parametrami [On the method of synthesis of regulators for objects with interval parameters] (Collection of scientific papers of NSTU. Novosibirsk, 1998).
[11] Davydov D.V., "Localnaya stabilizaciya intervalno nabludaemoi sistemy s neopredelennymi parametrami"[Local stabilization of an interval-observable system with uncertain parameters], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 8 (2003): 44-51.
[12] Zaharov А.V. and Shokin U.I., "Sintez sistem upravleniya pri intervalnoi neopredelennosti parametrov ih matematicheskih modelei"[Synthesis of control systems with interval uncertainty parameters of their mathematical models], DAN USSR 299 (1988).
[13] Moiseev А.N., Modalnoe upravlenie mnogomernoi dinamicheskoi sistemoi s parametricheskimi neopredelennostyami intervalnogo tipa [Modal control of a multidimensional dynamic system with parametric uncertainties of interval type] (Tomsk: Diss. Cand. those. Sciences, 1997).
[14] Kaucher E., "Interval analysis in the extended interval space IR," Computing Supplement 2 (1980): 33-49.
[15] Hodko S.Т., Proektirovanie sistem upravleniya s nestabilnymi parametrami [Designing control systems with unstable parameters] (L.: Engineering, 1987).
[16] Smagin Е.М. and Moiseev А.N. and Moiseeva S.P., "Nekotorye metody vychisleniya koefficientov IHP intervalnyh matric"[Some methods for calculating the ICP coefficients of interval matrices], Computational technologies, Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 2 (1997): 52-61.
[17] Ivlev R.S., Postroenie i issledovanie svoistv mnogomernyh sistem upravleniya intervalno-zadanymi obektami
[Construction and study of the properties of multidimensional control systems of interval-specified objects] (Almaty: Diss. Cand. those. Sciences, 1999).
[18] Ivlev R.S. and Sokolova S.P., "Postroenie vektornogo upravleniya mnogomernym intervalno-zadanym
obektom"[Construction of vector control multidimensional interval specified object], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 4 (1999): 3-13.
[19] Haritonov V.L., "Ob assimtoticheskoi ustoichivosti polozheniya ravnovesiya semeistva sistem lineinyh diferencialnyh uravneni"[On the asymptotic stability of the equilibrium position of a family of systems of linear differential equations], Differents. the Equations 14 (1978): 2086-88.
[20] Sharyi S.P., Intervalnye algebraicheskie zadachi i ih chislennoe reshenie [Interval algebraic problems and their numerical solution] (Novosibirsk: Diss. Dr. Phys.-Mat. Sciences, 2000).
[21] Shashihin V.N., "Zadacha robastnogo razmesheniya polusov v intervalnyh krupnomashtabnyh sistemah"[The task of robust pole placement in large-scale interval systems], Automation and Remote Control 2 (2002): 34-43.
[22] Hlebalin N.А., Sintez intervalnyh regulyatorov v zadache modalnogo upravleniya [Synthesis of interval regulators in the modal control problem] (Sarat.: Analytical methods for regulator synthesis: Interst. scientific Sat Saratov, 1988).
[23] Aschepkov L.T and. Dolgy D.V., "The universal solutions of interval systems of linear algebraical equations," Int. J. of Software Eng. and Knowledge Eng. 3 (1993): 477-85.
[24] Shashihin V.N., "Sintez robastnogo upravleniya dlya intervalnyh krupnomashtabnyh sistem s posledeistviem"[Synthesis of robust control for interval large-scale systems with aftereffect], Automation and Remote Control 12 (1997): 164-74.
[25] Li E.B. and Markus L., Osnovy teorii optimalnogo upravleniya [Fundamentals of optimal control theory] (М.: Nauka, 1972).
[26] Bialas S.A., "A necessary and sufficient conditions for stability of interval matrices," Int.J.Contr. 37 (1983): 717-22.
[27] Liu M., "Interval Analysis of Dynamic Response of Structures," International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) (2015): 810-13.
[28] Chen S. H. and Zhang, X. M., "Dynamic response of closedloop system with uncertain parameters using interval finite element method," Journal of Engineering Mechanics 132 (2006): 830-40.
[29] Qiu Z. P. and Wang X. J., "Parameter perturbation method for dynamic responses of structures with uncertain-butbounded parameters based on interval analysis," International Journal of Solids and Structures 42 (2005): 4958-70.
[30] Wang Z and Tian Q and Hu H, "Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters,"Springer Sci 84 (2015): 527–48.
[31] Wu, J.L. and Zhang, Y.Q., "The dynamic analysis of multibody systems with uncertain parameters using interval method,"Appl. Mech. Mater. (2012): 1555–61.
[32] Yunlong Li and Xiaojun Wang and Ren Huang and Zhiping Qiu, "Balanced-based model reduction of uncertain systems with interval parameters," Journal of Vibration and Control 22 (2016): 2958.
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Как цитировать
Jomartova, S. A., Nikulin, V. V., & Karymsakova, N. T. (2019). Исследование управляемости динамических систем при наличии ограничения на управления с применением интервальной математики. Вестник КазНУ. Серия математика, механика, информатика, 102(2), 69–80. https://doi.org/10.26577/JMMCS-2019-2-27
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Прикладная математика
