Synthesis of optimal control of technological processes based on a multialternative parametric description of the final state
DOI:
https://doi.org/10.15587/1729-4061.2017.105294Keywords:
optimal control of technological processes, Pontryagin’s maximum principle, speed, final state line, multialternative description of the final state, ridge analysisAbstract
A method is proposed to establish control of technological processes that would be optimal in terms of speed and final state on the basis of analyzing the solution of a system of stochastic differential equations (SDEs), which is a mathematical model of a controlled process. The results of the numerical modeling have proved that, being sufficiently simple, the proposed method helps obtain solutions that are completely consistent with the results obtained using the Pontryagin maximum principle for the speed problem. It has been shown that such an approach to the search for optimal control of technological processes opens up additional opportunities in solving the task of retaining the parameters of the technological process within a given area. Two alternatives of the control implementation are proposed and justified, differing in the principle of selecting control switching times.
It has been shown that the determining factor for the choice of optimal control is the initial state of the system, described by the position of the phase space point characterizing the actual initial state relative to the final state line. If the final state is described by the equation of the straight line, it is proposed to reduce it to its normal form and to calculate the corresponding deviation of the point of the preceding state from this straight line, which uniquely determines the sign of control. It has been proved that the problem of finding the optimal control of technological processes must be preceded by the problem of obtaining a mathematical description of the final state, based on the construction of regression equations in which the output variable can be the quality of the finished technological product.
It is proposed to obtain a multialternative parametric description of the final state for the search for optimal control of the technological process using a ridge analysis. It has been shown that each of the alternatives represents a set of suboptimal values of the output variable, which provides optimal values of the output variable describing the quality of the finished technological product in the chosen sense. Due to this approach, it is possible to synthesize the optimal control in terms of the speed and final state of technological processes in conditions of a multialternative description of the final state of the technological systemReferences
- Lutsenko, I. (2014). A practical approach to selecting optimal control criteria. Technology audit and production reserves, 2 (1 (16)), 32–35. doi: 10.15587/2312-8372.2014.23432
- Trufanov, I. D., Chumakov, K. I., Bondarenko, A. A. (2005). Obshcheteoreticheskie aspekty razrabotki stohasticheskoy sistemy avtomatizirovannoy ehkspertnoy ocenki dinamicheskogo kachestva proizvodstvennyh situaciy ehlektrostaleplavleniya. Eastern-European Journal of Enterprise Technologies, 6 (2 (18)), 52–58.
- Trufanov, I. D., Metel'skiy, V. P., Chumakov, K. I., Lozinskiy, O. Yu., Paranchuk, Ya. S. (2008). Ehnergosberegayushchee upravlenie ehlektrotekhnologicheskim kompleksom kak baza povysheniya ehnergoehffektivnosti metallurgii stali. Eastern-European Journal of Enterprise Technologies, 6 (1 (36)), 22–29.
- Lutsenko, I., Fomovskaya, E. (2015). Identification of target system operations. The practice of determining the optimal control. Eastern-European Journal of Enterprise Technologies, 6 (2 (78)), 30–36. doi: 10.15587/1729-4061.2015.54432
- Lutsenko, I., Vihrova, E., Fomovskaya, E., Serdiuk, O. (2016). Development of the method for testing of efficiency criterion of models of simple target operations. Eastern-European Journal of Enterprise Technologies, 2 (4 (80)), 42–50. doi: 10.15587/1729-4061.2016.66307
- Lutsenko, I. (2015). Identification of target system operations. Development of global efficiency criterion of target operations. Eastern-European Journal of Enterprise Technologies, 2 (2 (74)), 35–40. doi: 10.15587/1729-4061.2015.38963
- Lutsenko, I., Fomovskaya, E. (2015). Synthesis of cybernetic structure of optimal spooler. Metallurgical and Mining Industry, 9, 297–301.
- Diligenskiy, N. V., Dymova, L. G., Sevast'yanov, P. V. (2004). Nechetkoe modelirovanie i mnogokriterial'naya optimizaciya proizvodstvennyh sistem v usloviyah neopredelennosti: tekhnologiya, ehkonomika, ehkologiya. Moscow: Mashinostroenie-1, 397.
- Hong, D. H., Lee, S., Do, H. Y. (2001). Fuzzy linear regression analysis for fuzzy input-output data using shape-preserving operations. Fuzzy Sets and Systems, 122 (3), 513–526. doi: 10.1016/s0165-0114(00)00003-8
- Yang, M.-S., Lin, T.-S. (2002). Fuzzy least-squares linear regression analysis for fuzzy input-output data. Fuzzy Sets and Systems, 126 (3), 389–399. doi: 10.1016/s0165-0114(01)00066-5
- Seraya, O. V., Demin, D. A. (2012). Linear Regression Analysis of a Small Sample of Fuzzy Input Data. Journal of Automation and Information Sciences, 44 (7), 34–48. doi: 10.1615/jautomatinfscien.v44.i7.40
- Tseng, Y.-T., Ward, J. D. (2017). Comparison of objective functions for batch crystallization using a simple process model and Pontryagin’s minimum principle. Computers & Chemical Engineering, 99, 271–279. doi: 10.1016/j.compchemeng.2017.01.017
- Demin, D. A. (2012). Synthesis process control elektrodugovoy smelting iron. Eastern-European Journal of Enterprise Technologies, 2 (10 (56)), 4–9. Available at: http://journals.uran.ua/eejet/article/view/3881/3557
- Demin, D. A. (2012). Synthesis of optimal temperature regulator of electroarc holding furnace bath. Scientific Bulletin of National Mining University, 6, 52–58.
- Ozatay, E., Ozguner, U., Filev, D. (2017). Velocity profile optimization of on road vehicles: Pontryagin's Maximum Principle based approach. Control Engineering Practice, 61, 244–254. doi: 10.1016/j.conengprac.2016.09.006
- Saerens, B., Van den Bulck, E. (2013). Calculation of the minimum-fuel driving control based on Pontryagin’s maximum principle. Transportation Research Part D: Transport and Environment, 24, 89–97. doi: 10.1016/j.trd.2013.05.004
- Bauer, S., Suchaneck, A., Leon, F. P. (2014). Thermal and energy battery management optimization in electric vehicles using Pontryagin's maximum principle. Journal of Power Sources, 246, 808–818. doi: 10.1016/j.jpowsour.2013.08.020
- Onori, S., Tribioli, L. (2015). Adaptive Pontryagin’s Minimum Principle supervisory controller design for the plug-in hybrid GM Chevrolet Volt. Applied Energy, 147, 224–234. doi: 10.1016/j.apenergy.2015.01.021
- Fang, H., Wie, X., Zhao, F. (2015). Structural optimization of double-tube once-through steam generator using Pontryagin's Maximum Principle. Progress in Nuclear Energy, 78, 318–329. doi: 10.1016/j.pnucene.2014.09.008
- Candido, J. J., Justino, P. A. P. S. (2011). Modelling, control and Pontryagin Maximum Principle for a two-body wave energy device. Renewable Energy, 36 (5), 1545–1557. doi: 10.1016/j.renene.2010.11.013
- Krasovskiy, A. A., Taras'ev, A. M. (2007). Dinamicheskaya optimizaciya investiciy v modelyah ehkonomicheskogo rosta. Avtomatika i telemekhanika, 10, 38–52.
- Ohsawa, T. (2015). Contact geometry of the Pontryagin maximum principle. Automatica, 55, 1–5. doi: 10.1016/j.automatica.2015.02.015
- Blot, J., Kone, M. I. (2016). Pontryagin principle for a Mayer problem governed by a delay functional differential equation. Journal of Mathematical Analysis and Applications, 444 (1), 192–209. doi: 10.1016/j.jmaa.2016.06.027
- Pereira, F. L., Silva, G. N. (2011). A Maximum Principle for Constrained Infinite Horizon Dynamic Control Systems. IFAC Proceedings Volumes, 44 (1), 10207–10212. doi: 10.3182/20110828-6-it-1002.03622
- Stecha, J., Rathousky, J. (2011). Stochastic maximum principle. IFAC Proceedings Volumes, 44 (1), 4714–4720. doi: 10.3182/20110828-6-it-1002.01501
- Arutyunov, A. V., Karamzin, D. Yu., Pereira, F. (2012). Pontryagin’s maximum principle for constrained impulsive control problems. Nonlinear Analysis: Theory, Methods & Applications, 75 (3), 1045–1057. doi: 10.1016/j.na.2011.04.047
- Khlopin, D. V. (2016). On the Hamiltonian in infinite horizon control problems. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 22 (4), 295–310. doi: 10.21538/0134-4889-2016-22-4-295-310
- Ballestra, L. V. (2016). The spatial AK model and the Pontryagin maximum principle. Journal of Mathematical Economics, 67, 87–94. doi: 10.1016/j.jmateco.2016.09.012
- Tregub, V. G., Chorna, Yu. O. (2010). Optimal control of batch processes with interphase transitions. Eastern-European Journal of Enterprise Technologies, 6 (4 (48)), 10–12. Available at: http://journals.uran.ua/eejet/article/view/3270/3072
- Kocur, M. P. (2016). Mathematical modeling and optimization of transiant thermoelectric cooling process. Technology audit and production reserves, 1 (2 (27)), 29–34. doi: 10.15587/2312-8372.2016.59320
- Levchuk, I. L. (2015). Managing the process of catalytic reforming by the optimal distribution of temperature at the reactor block inlets. Technology audit and production reserves, 2 (4 (22)), 56–60. doi: 10.15587/2312-8372.2015.40592
- Musaev, A. (2002). Intelligent Control Systems for Refinery Technological Processes. Proceedings of conf. ICPI’02 (Intelligent computing for the petroleum industry). Mexico, 2, 6–17.
- Musaev, A. A. (2003). Virtual'nye analizatory: koncepciya postroeniya i primeneniya v zadachah upravleniya nepreryvnymi tekhnologicheskimi processami. Avtomatizaciya v promyshlennosti, 8, 28–33.
- Demin, D. A. (2011). Methodology of forming functional in the optimal control electric smelting. Technology audit and production reserves, 1 (1 (1)), 15–24. Available at: http://journals.uran.ua/tarp/article/view/4082/3748
- Musaev, A. A., Nikitin, V. A. (2007). Optimal'noe upravlenie processom smesheniya tovarnogo topliva v potoke. Pribory i sistemy, 4, 5–11.
- Alekseeva, L. B.; Chernov, S. S. (Ed.) (2012). Struktura vzaimodeystviya virtual'nogo monitoringa s sistemoy upravleniya nepreryvnym tekhnologicheskim processom. Novosibirsk: Izd-vo NGTU, 114–118.
- Afanas'ev, V. N., Kolmanovskiy, V. B., Nosov, V. R. (1989). Matematicheskaya teoriya konstruirovaniya sistem upravleniya. Moscow: Vysha Shkola, 447.
- Demin, D. A. (2012). Synthesis process control elektrodugovoy smelting iron. Eastern-European Journal of Enterprise Technologies, 2 (10 (56)), 4–9. Available at: http://journals.uran.ua/eejet/article/view/3881/3557
- Demin, D. A. (2013). Adaptive modeling in problems of optimal control search termovremennoy cast iron. Eastern-European Journal of Enterprise Technologies, 6 (4 (66)), 31–37. Available at: http://journals.uran.ua/eejet/article/view/19453/17110
- Demin, D. A. (2014). Mathematical description typification in the problems of synthesis of optimal controller of foundry technological parameters. Eastern-European Journal of Enterprise Technologies, 1 (4 (67)), 43–56. doi: 10.15587/1729-4061.2014.21203
- Horbiychuk, M. I. (1997). Sposib vidboru kryteriyiv optymal'nosti pry adaptyvnomu upravlinni protsesom burinnya. Rozvidka i rozrobka naftovykh i hazovykh rodovyshch. Seriya: Tekhnichna kibernetyka ta elektryfikatsiya ob’yektiv palyvno-enerhetychnoho kompleksu, 34 (5), 18–23.
- Horbiychuk, M. I. (1998). Adaptyvne upravlinnya protsesom pohlyblennya sverdlovyn. Rozvidka i rozrobka naftovykh i hazovykh rodovyshch. Seriya: Tekhnichna kibernetyka ta elektryfikatsiya ob'yektiv palyvno-enerhetychnoho kompleksu, 35 (6), 3–9
- Suzdal', V. S., Epifanov, Yu. M., Sobolev, A. V., Tavrovskiy, I. I. (2009). Parametricheskaya identifikaciya Varmax modeley processa kristallizacii krupnogabaritnyh monokristallov. Naukovyi visnyk KUEITU, 4 (26), 23–29.
- Suzdal', V. S. (2011). Model reduction at synthesis of controllers for crystallization control. Eastern-European Journal of Enterprise Technologies, 2 (3 (50)), 31–34. Available at: http://journals.uran.ua/eejet/article/view/1745/1642
- Suzdal', V. S. (2011). Optimization of synthesis control problem for crystallization processes. Eastern-European Journal of Enterprise Technologies, 6 (3 (54)), 41–44. Available at: http://journals.uran.ua/eejet/article/view/2247/2051
- Zyelyk, Y. I., Lychak, M. M., Shevchenko, V. N. (2003). Simulation and Identification of Controlled Objects with the Use of the Interval-Set Analysis MATLAB Toolbox. Journal of Automation and Information Sciences, 35 (3), 31–44. doi: 10.1615/jautomatinfscien.v35.i3.40
- Zyelyk, Y. I. (2000). Convergence of a matrix gradient algorithm of solution of extremal problem under constraints. Journal of Automation and Information Sciences, 32 (9), 34–41.
- Zyelyk, Y. I. (2000). Convergence of a Matrix Gradient Control Algorithm with Feedback Under Constraints. Journal of Automation and Information Sciences, 32 (10), 35–45. doi: 10.1615/jautomatinfscien.v32.i10.50
- Kachanov, P. A. (2000). Optimal'noe upravlenie sostoyaniem dinamicheskih sistem v usloviyah neopredelennosti. Kharkiv: KhGPU, 209.
- Raskin, L. G., Seraya, O. V. (2008). Nechetkaya matematika. Kharkiv: Parus, 352.
- Hartman, K., Leckiy, E., Shefer, V. et. al. (1977). Planirovanie ehksperimenta v issledovanii tekhnologicheskih processov. Moscow: Mir, 552.
- Demin, D. A., Pelikh, V. F., Ponomarenko, O. I. (1998). Complex alloying of grey cast iron. Liteynoe Proizvodstvo, 10, 18–19.
- Demin, D. A., Pelikh, V. F., Ponomarenko, O. I. (1995). Optimization of the method of adjustment of chemical composition of flake graphite iron. Liteynoe Proizvodstvo, 7-8, 42–43.
- Mohanad, M. K., Kostyk, V., Demin, D., Kostyk, K. (2016). Modeling of the case depth and surface hardness of steel during ion nitriding. Eastern-European Journal of Enterprise Technologies, 2 (5 (80)), 45–49. doi: 10.15587/1729-4061.2016.65454
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