Mathematical model of a digital control system with background controllers of the Neuman type for complex multicirculated objects

E. V. Larkin ; A. N. Privalov ; T. A. Akimenko ; I. N. Larioshkin

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 129-141.

Résumé

In the work, a mathematical model of digital control of multi-circuit objects is built, taking into 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 and are widely used in engineering practice, however, in the overwhelming majority, they involve the formation of models that do not take into account the presence of time intervals between transactions in a Von Neumann type computer. To solve the problem, a typical block diagram of complex multi-loop control systems with digital controllers of the Von Neumann type has been developed, which takes into account the random nature of the processed data and real time delays between transactions. It is proposed, taking into account the randomness of the time interval between transactions and the stochastic nature of switching to conjugate operators, to consider a semi-Markov process as 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 parameters of time intervals between transactions in cyclic control algorithms, which makes it possible to evaluate the characteristics of the system at the design stage, and therefore is the key to the rational design of digital control systems for multi-circuit objects with control algorithms of almost any complexity. An example of mathematical modeling of a two-circuit system with digital control is presented.

Détail

Keywords: semi-Markov process, wandering time, digital control system, controller, digital control system control algorithm, transaction.

Bibliographie

[1] Malin Löfving M., Säfsten K., Winroth M., “Manufacturing strategy formulation, leadership style and organizational culture in small and medium-sized enterprises”, IJMTM, 30:5 (2016), 306–325 | DOI

[2] Zhou M. C., “Modeling, Simulation, and Control of Flexible Manufacturing Systems: A Petri Net Approach”, Wspc 1999, 428 pp.

[3] Landau I. D., Zito G., Digital Control Systems, Design, Identification and Implementation, Springer, 2006, 484 pp.

[4] Aström J., Wittenmark B., Computer Controlled Systems: Theory and Design, Tsinghua University Press. Prentice Hall, 2002, 557 pp.

[5] 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), 124–129

[6] Fadali M. S., Visioli A., Digital control engineering: Analysis and design, Elsevier Inc, 2013, 239–272

[7] Auslander D. M., Ridgely J. R., Jones J. C., “Real-time software for implementation of feedback control”, The control handbook. Control system fundamentals, ed. W.S. Levine, CRC Press. Taylor and Francis Group, US, 2017, 16–32 | MR

[8] 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 pp.

[9] Tzafestas S. G., Introduction to Mobile Robot Control, Elsevier, 2014, 750 pp.

[10] Babishin V., Taghipour S., “Optimal maintenance policy for multicomponent systems with periodic and opportunistic inspections and preventive replacements”, Applied Mathematical Modelling, 40:23 (2016), 10480–10505 | DOI | MR | Zbl

[11] Arnold K. A., “Timing analysis in embedded systems”, Embedded hardware by J. Ganssler, K. Arnold et all, Elsevier Inc, USA, 2008, 239–272

[12] Balsamo S., Harrison P. G., Marin A., “Methodological construction of product-form stochastic Petri nets for performance evaluation”, Journal of Systems and Software, 85:7 (2012), 1520–1539 | DOI

[13] Hamann A., Racu R., Ernst R., “Multi-dimensional robustness optimization in heterogeneous distributed embedded systems”, Proceedings of the 13th IEEE Real Time and Embedded Technology and Applications Symposium, RTAS '07, IEEE Computer Society, Washington, DC, USA, 2007, 269–280

[14] Schiff J. L., The Laplace transform: Theory and applications, Undergraduate Texts in Mathematics, 199, Springer Verlag, USA, NY, 233 pp. | MR

[15] Larkin E. V., Bogomolov A. V., Privalov A. N., “A Method for Estimating the Time Intervals between Transactions in Speech-Compression Algorithms”, Automatic Documentation and Mathematical Linguistics, 51:5 (2017), 214–219 | DOI | MR

[16] Bielecki T. R., Jakubowski J., Niewȩglowski M., “Conditional Markov chains: Properties, construction and structured dependence”, Stochastic Processes and their Applications, 127:4 (2017), 1125–1170 | DOI | MR | Zbl

[17] Ching W. K., Huang X., Ng M. K., Siu T. K., Markov Chains: Models, Algorithms and Applications, International Series in Operations Research Management Science, 189, Springer Science + Business Media, NY, 2013, 241 pp. | DOI | MR | Zbl

[18] Howard R. A., Dynamic Probabilistic Systems, v. 1, Markov Models, Courier Corporation, 2012, 205 pp.; v. II, Semi-Markov and Decision Processes

[19] Janssen J., Manca R., Applied Semi-Markov processes, Springer US, 2006, 310 pp. | MR | Zbl

[20] Kobayashi H., Marl B. L., Turin W., Probability, Random Processes and Statistical Analysis, Cambridge University Press, 2012, 812 pp. | MR | Zbl

[21] Petersen P., Linear algebra, Springer-Verlag, NY, 2012, 427 pp. | Zbl