chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2020-21-3-129-141cheb-855Research ArticleСтатьиArticleМатематическая модель цифровой системы управления с контроллерами фон Неймановского типа сложными многоконтурными объектамиMathematical model of a digital control system with background controllers of the Neuman type for complex multicirculated objectsЛаркинЕвгений ВасильевичLarkinEvgenii Vasil’evich

доктор технических наук, профессор

Doctor of technical sciences, Professor

elarkin@mail.ruПриваловАлександр НиколаевичPrivalovAlexander Nikolaevich

доктор технических наук, профессор

Doctor of technical sciences, Professor

privalov.61@mail.ruАкименкоТатьяна АлексеевнаAkimenkoTatyana Alekseevna

кандидат технических наук, доцент

Candidate of Technical Sciences, Associate Professor

tantan72@mail.ruЛариошкинИван НиколаевичLarioshkinIvan Nikolaevich

аспирант

postgraduate student

tantan72@mail.ruТульский государственный университетРоссияTula State UniversityRussian FederationТульский государственный педагогический университет им. Л. Н. ТолстогоРоссияTula State Pedagogical UniversityRussian Federation202028122020213129141Copyright © Ларкин Е.В., Привалов А.Н., Акименко Т.А., Лариошкин И.Н., 20202020Ларкин Е.В., Привалов А.Н., Акименко Т.А., Лариошкин И.Н.Larkin E.V., Privalov A.N., Akimenko T.A., Larioshkin I.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/855

В работе построена математическая модель цифрового управления многоконтурнымиобъектами, учитывающая реальные характеристики цифрового контроллера, как элементасистемы управления. Сформулирована проблема, заключающаяся в том, что методы мо-делирования цифровых систем управления известны и широко применяются в практикеинженерной деятельности, однако в подавляющем большинстве они предполагают форми-рование моделей, не учитывающих наличие временных интервалов между транзакциямив ЭВМ фон Неймановского типа.Для решения задачи разработана типовая структурная схема сложных многоконтур-ных систем управления с цифровыми контроллерами фон Неймановского типа, котораяучитывает случайный характер обрабатываемых данных и реальные временных задержкимежду транзакциями.Предложено с учетом случайности временного интервала между транзакциями и стоха-стического характера переключения в сопряженные операторы считать адекватной моде-лью алгоритма функционирования цифровых систем управления считать полумарковскийпроцесс.На основе полумарковских процессов предложен метод оценки параметров временныхинтервалов между транзакциями в циклических алгоритмах управления, который позво-ляет оценить характеристики системы на этапе ее проектирования, а следовательно явля-ется ключом к рациональному проектированию цифровых систем управления многокон-турными объектами с алгоритмами управления практически любой сложности. Представ-лен пример математического моделирования двухконтурной системы с цифровым управ-лением.

In the work, a mathematical model of digital control of multi-circuit objects is built, takinginto account the real characteristics of a digital controller as an element of a control system.The problem is formulated that the methods of modeling digital control systems are known andare widely used in engineering practice, however, in the overwhelming majority, they involvethe formation of models that do not take into account the presence of time intervals betweentransactions in a Von Neumann type computer.To solve the problem, a typical block diagram of complex multi-loop control systems withdigital controllers of the Von Neumann type has been developed, which takes into account therandom nature of the processed data and real time delays between transactions.It is proposed, taking into account the randomness of the time interval between transactionsand the stochastic nature of switching to conjugate operators, to consider a semi-Markov processas an adequate model of the algorithm for the functioning of digital control systems.On the basis of semi-Markov processes, a method is proposed for estimating the parametersof time intervals between transactions in cyclic control algorithms, which makes it possible toevaluate the characteristics of the system at the design stage, and therefore is the key to therational design of digital control systems for multi-circuit objects with control algorithms ofalmost any complexity. An example of mathematical modeling of a two-circuit system withdigital control is presented.

полумарковский процессвремя блужданияцифровая система управ- ленияконтроллералгоритм управления цифровой системой управлениятранзакцияsemi-Markov processwandering timedigital control systemcontrollerdigital control system control algorithmtransaction.Исследование выполнено при финансовой поддержке РФФИ по проекту 19-47-710004_р_аReferencesMalin Löfving M., Säfsten K.; Winroth M. Manufacturing strategy formulation, leadership style and organizational culture in small and medium-sized enterprises // IJMTM. 2016. Vol. 30. no. 5, pp. 306-325.Malin Löfving M., Säfsten K.; Winroth M., 2016, Manufacturing strategy formulation, leadership style and organizational culture in small and medium-sized enterprises, IJMTM. Vol. 30. no. 5, pp. 306 - 325.Zhou M.C. Modeling, Simulation, and Control of Flexible Manufacturing Systems: A Petri Net Approach // Wspc. 1999, 428 р.Zhou M.C., 1999, Modeling, Simulation, and Control of Flexible Manufacturing Systems: A Petri Net Approach. Wspc, 428 р.Landau I.D., Zito G., Digital Control Systems, Design, Identification and Implementation // Springer. 2006, 484 p.Landau I.D., Zito G., 2006, Digital Control Systems, Design, Identification and Implementation. Springer, 484 p.Aström J., Wittenmark B., Computer Controlled Systems: Theory and Design // Tsinghua University Press. Prentice Hall. 2002, 557 p.Aström J., Wittenmark B., 2002, Computer Controlled Systems: Theory and Design. Tsinghua University Press. Prentice Hall, 557 p.Larkin E., Bogomolov A., Privalov A., Estimation of Events Flow Generated with Ergodic Semi-Markov Processes // 2nd International Ural Conference on Measurements (URALCON), South Ural State University, Chelyabinsk, Russia, 2017, pp. 124-129.Larkin E., Bogomolov A., Privalov A., 2017, Estimation of Events Flow Generated with Ergodic Semi-Markov Processes // 2nd International Ural Conference on Measurements (URALCON), South Ural State University, Chelyabinsk, Russia, pp. 124-129.Fadali M.S., Visioli A., Digital control engineering: Analysis and design // Elsevier Inc. 2013, pp. 239 - 272.Fadali M.S., Visioli A., 2013, Digital control engineering: Analysis and design. Elsevier Inc., pp. 239 - 272.Auslander D.M., Ridgely J.R. Jones J.C., Real-time software for implementation of feedback control // The control handbook. Control system fundamentals. Ed. by W.S. Levine. CRC Press. Taylor and Francis Group, US. 2017, pp.16--32.Auslander D.M., Ridgely J.R. Jones J.C., 2017, Real-time software for implementation of feedback control // The control handbook. Control system fundamentals. Ed. by W.S. Levine. CRC Press. Taylor and Francis Group, US. pp.16--32.Karnopp D.C., Margolis D.L., Rosenberg R.C., System dynamics: Modeling, simulation and control of mechatronic systems // John Willey & Sons. New Jersey. 2012, 636 p.Karnopp D.C., Margolis D.L., Rosenberg R.C., 2012, System dynamics: Modeling, simulation and control of mechatronic systems. John Willey \& Sons. New Jersey, 636 p.Tzafestas S.G. Introduction to Mobile Robot Control. Elsevier, 2014. 750 p.Tzafestas S.G. Introduction to Mobile Robot Control. Elsevier, 2014. 750 p.Babishin V., Taghipour S. Optimal maintenance policy for multicomponent systems with periodic and opportunistic inspections and preventive replacements // Applied Mathematical Modelling 2016, V. 40. no. 23, pp. 10480-10505.Babishin V., Taghipour S.. 2016, Optimal maintenance policy for multicomponent systems with periodic and opportunistic inspections and preventive replacements // Applied Mathematical Modelling, V. 40. no. 23, pp. 10480 -10505.Arnold K. A. “Timing analysis in embedded systems”, In embedded hardware by J. Ganssler, K. Arnold et all, USA, Elsevier Inc. 2008. pp. 239 - 272.Arnold K. A. “Timing analysis in embedded systems”, In embedded hardware by J. Ganssler, K. Arnold et all, USA, Elsevier Inc. 2008. pp. 239 - 272.Balsamo S., Harrison P.G., Marin A., “Methodological construction of product-form stochastic Petri nets for performance evaluation” // In Journal of Systems and Software. Elsevier Inc., 2012, Vol. 85, no. 7. pp. 1520–1539.Balsamo S., Harrison P.G., Marin A., 2012, “Methodological construction of product-form stochastic Petri nets for performance evaluation”, In Journal of Systems and Software. Elsevier Inc., Vol. 85, no. 7. pp. 1520–1539.Hamann A., Racu, R., Ernst R., “Multi-dimensional robustness optimization in heterogeneous distributed embedded systems” // In Proceedings of the 13th IEEE Real Time and Embedded Technology and Applications Symposium, RTAS ’07, IEEE Computer Society, Washington, DC, USA. 2007, pp. 269–280.Hamann A., Racu, R., Ernst R., 2007, “Multi-dimensional robustness optimization in hete\-rogeneous distributed embedded systems”, In Proceedings of the 13th IEEE Real Time and Embedded Technology and Applications Symposium, RTAS ’07, IEEE Computer Society, Washington, DC, USA. pp. 269–280.Schiff J.L., The Laplace transform: Theory and applications, USA, NY Springer Verlag, 199, 233 p.Schiff J.L., 199, The Laplace transform: Theory and applications, USA, NY Springer Verlag, 233 p.Larkin E.V., Bogomolov A.V., Privalov A.N., “A Method for Estimating the Time Intervals between Transactions in Speech-Compression Algorithms”, In: Automatic Documentation and Mathematical Linguistics. 2017, Vol. 51. Iss. 5. pp. 214-219.Larkin E.V., Bogomolov A.V., Privalov A.N., 2017, “A Method for Estimating the Time Intervals between Transactions in Speech-Compression Algorithms”, In: Automatic Docu\-men\-tation and Mathematical Linguistics, Vol. 51. Iss. 5. pp. 214-219.Bielecki T.R., Jakubowski J., Niewc{e}glowski M., “Conditional Markov chains: Properties, construction and structured dependence”, In Stochastic Processes and their Applications. 2017, V. 127, no. 4. pp. 1125–1170.Bielecki T.R., Jakubowski J., Niew\c{e}glowski M., 2017, “Conditional Markov chains: Properties, construction and structured dependence”, In Stochastic Processes and their Applications, V. 127, no. 4. pp. 1125–1170.Ching W.K., Huang X., Ng M.K., Siu T.K., “Markov Chains: Models, Algorithms and Applications”, In International Series in Operations Research & Management Science, V. 189. Springer Science + Business Media NY, 2013, 241 p.Ching W.K., Huang X., Ng M.K., Siu T.K., 2013, “Markov Chains: Models, Algorithms and Applications”, In International Series in Operations Research \& Management Science, V. 189. Springer Science + Business Media NY,. 241 p.Howard R.A., Dynamic Probabilistic Systems. Vol. 1: Markov Models. Vol. II: Semi-Markov and Decision Processes // Courier Corporation. 2012, 205p.Howard R.A., 2012, Dynamic Probabilistic Systems. Vol. 1: Markov Models. Vol. II: Semi-Markov and Decision Processes, Courier Corporation, 205p.Janssen J., Manca R., Applied Semi-Markov processes. Springer US. 2006, 310 p.Janssen J., Manca R., 2006, Applied Semi-Markov processes. Springer US, 310 p.Kobayashi H., Marl B.L., Turin W., Probability, Random Processes and Statistical Analysis: Cambridge University Press. 2012, 812 p.Kobayashi H., Marl B.L., Turin W., 2012, Probability, Random Processes and Statistical Analysis: Cambridge University Press.. 812 p.Petersen P.,Linear algebra. Springer-Verlag. NY. 2012, 427 p.Petersen P., 2012, Linear algebra. Springer-Verlag. NY. 427 p.

The authors declare that there are no conflicts of interest present.