Stabilization of one nonlinear system with coefficients depending on the condition of the control object
DOI:
https://doi.org/10.26577/JMMCS-2019-1-588Keywords:
effective management accounting, three- sector economic cluster, Lagrange multipliers method, nonlinear systems, quadratic functionalAbstract
For the mathematical model of a three-sector economic cluster, an optimal control problem is posed on an infinite time interval. An optimal stabilization problem is considered for a single class of nonlinear systems with coefficients depending on the state of the control object with constraints on control. A non-linear stabilizing control has been found taking into account constraints to the control, which depends on the state of the system and the current point in time. The results obtained for a nonlinear system are used in the construction of control parameters for a three-sector economic cluster over an infinite time interval. For the considering example, the optimal distribution of labor and investment resources has been determined, which satisfy the balance ratios. The given example illustrates the use of the proposing control method of a nonlinear system.
References
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[23] Murzabekov Z.N., "Dostatochnoye usloviya optimal’nosti dinamicheskikh system upravleniya s zakreplennymi kontsami [Sufficient conditions for optimality of dynamic control systems with fixed ends] " , Matematicheskiy zhurnal [Mathematical Journal], vol.4, no 3(12)-(2012): 52-59.
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[2] Bellman R., Kalaba R., Dinamicheskoye programmirovaniye i sovremennaya teoriya upravleniya [Dynamic programming and modern control theory] , (M.: Nauka, 1968)
[3] Krotov V.F., Gurman V.I., Metody i zadachi optimal’nogo upravleniya [Methods and objectives of optimal control] , (M.: Nauka, 1973)
[4] V.F. Krotov [i dr.]; pod red. V.F. Krotova.[V.F. Krotov [et al.]; by ed. V.F. Krotov], Osnovy teorii optimal’nogo upravleniya [Fundamentals of the theory of optimal control] , (M.: Vysshaya shkola, 1990)
[5] Krotov V.F., Metody i zadachi optimal’nogo upravleniya [Methods and tasks of optimal control] ,(M.: Nauka, 1973)
[6] Kurzhanskiy A.B., "Differentsial’nyye uravneniya v zadachakh sinteza upravleniy. I. Obyknovennyye sistemy [Differential equations in control synthesis problems. I. Ordinary systems]" , Differents. uravneniya, vol:41, no 1, (2005): 12-22.
[7] Kalman R.E., "Contributions to the Theory of Optimal Control" , Bol. Soc. Mat. Mexicana, vol:5 no 1 (1960): 102-119.
[8] K.A. Pupkova,N.D. Yegupova Metody klassicheskoy i sovremennoy teorii avtomaticheskogo upravleniya [Methods of classical and modern theory of automatic control], "The optimal control problem with fixed-end trajectories for a threesector economic model of a cluster." , (M.:MGTU Baumana, 2004).
[9] Afanasyev V.N., Kolmanovskiy V.B., Nosov V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya [Mathematical theory of designing control systems] , (M.: Vyssh.shk, 2003)
[10] Brayson A., Kho YU-shi., Prikladnaya teoriya optimal’nogo upravleniya [Applied theory of optimal control] , (M.: Mir, 1972)
[11] Roytenberg YA.N., Avtomaticheskoye upravleniye [Automatic control] , - Uchebnoye posobiye. - izd.2-ye, pererab.idopoln. Glavnaya redaktsiya fiziko- matematicheskoy literatury [Tutorial. ed.2-e, pererab.dopoln. Main edition of the physical and
mathematical literature], (M.: Nauka, 2015)
[12] Afanas’yev A.P., Dzyuba S.M., Yemel’yanova I.I., "Optimal’noye upravleniye s obratnoy svyaz’yu odnim classom nelineynykh sistem po kvadratichnomu kriteriyu [Optimal feedback control of one class of nonlinear systems by quadratic criterion]" , Vestnik TGU. vol.5, no 20(2015). : 1024-1033.
[13] Lobanov S.M., Zatylkin V. V., Malonga O. SH., "Postroyeniye optimal’nogo upravleniya odnim klassom nelineynykh sistem po kvadratichnomu kriteriyu [Construction of optimal control of one class of nonlinear systems by quadratic criterion] " , Vestnik VGUIT no 3 (2012): 54-48.
[14] Dmitriyev M. G., Makarov D. A., "Gladkiy nelineynyy regulyator v slabo nelineynoy sisteme upravleniya s koeffitsiyentami, zavisyashchimi ot sostoyaniya [Smooth non-linear controller in a weakly non-linear control system with state-dependent coefficients]" , Trudy ISA RAN, vol.4, no 64(2014): 53-58.
[15] Afanas’yev V. N., Orlov P. V., "Suboptimal’noye upravleniye nelineynym ob’yektom, linearizuyemym obratnoy svyaz’yu [Suboptimal control of a nonlinear object linearized by feedback]" , Izvestiya RAN. Teoriya i sistemy upravleniya (Theory and Control Systems), no 3 (2011): 13-22.
[16] Kolemayev V.A., Economic-mathematical modeling (in Russian) , (Moscow:UNITY, 2005).
[17] Aseev S.M., Besov K.O., Kryazhimskii A.V., "Infinite-horizon optimal control problems in economics" , Russian Math. Surveys no 67(2) : 195-253 (2012).
[18] Murzabekov Z., Milosz M. and Tussupova K, "Modeling and optimization of the production cluster" , Proceedings of 36th International Conference on Information Systems and Architecture and Technology- ISAT-2015, Advances in Intelligent Systems and Computing. - Karpacz, vol:2, (2014): 99-108.
[19] Klamka J., "Constrained controllability of dynamics systems" , International Journal of Applied Mathematics and Computer Science, no 9(2): 231-244 (1999).
[20] Aipanov, Sh., Murzabekov, Z., "Analytical solution of a linear quadratic optimal control problem with control value constraints" , Computer and Systems Sciences International, vol:1, no 53 (2014): 84-91.
[21] Murzabekov Z., Milosz M. and Tussupova K., "The optimal control problem with fixed-end trajectories for a three-sector economic model of a cluster." , Intelligent Information and Database Systems, ACIIDS, (2018). : 382-391.
[22] Murzabekov Z.N., Avtomaticheskoye upravleniyeOptimizatsiya upravlyayemykh sistem [Optimization of the control system] , (Almaty:ATU, 2009)
[23] Murzabekov Z.N., "Dostatochnoye usloviya optimal’nosti dinamicheskikh system upravleniya s zakreplennymi kontsami [Sufficient conditions for optimality of dynamic control systems with fixed ends] " , Matematicheskiy zhurnal [Mathematical Journal], vol.4, no 3(12)-(2012): 52-59.
[24] Mracek C.P., Cloutier J.R., "Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method" , International Journal of robust and nonlinear control, vol.8, no 4-5 (1998): 401-433.
[25] Cimen T., "Sdependent Riccati Equation (SDRE) control: A Survey" , Proceedings of the 17th World Congress The International Federation of Automatic Control. Seoul, Korea, (July 6-11. 2008): P.3761-3775.
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How to Cite
Murzabekov, Z. N., & Mirzakhmedova, G. A. (2019). Stabilization of one nonlinear system with coefficients depending on the condition of the control object. Journal of Mathematics, Mechanics and Computer Science, 101(1), 76–86. https://doi.org/10.26577/JMMCS-2019-1-588
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