Leontev Input-Output Balance Model as a Dynamic System Control Problem

The purpose of the study was to determine the problem of control of a dynamic system of higher dimension. Relying on Leontev input-output balance, we formalized the dynamic system and synthesized its control. Within the research, we developed a mathematical model that combines different working objects that consume and release various resources. The value of the penalty for all nodes and objects is introduced into the matrix representation of the problem, taking into account various options for their interaction, i.e., the observation problem. A matrix representation of the planning task at each working object is formed. For the formed system, a control loop is created; the influencing parameters of the external environment are indicated. We calculated the system operational mode, taking into account the interaction of the nodes of objects with each other when the parameters of the external environment influence them. Findings of research show that in achieving a complex result, the system is inefficient without optimal planning and accounting for the matrix of penalties for the interaction of nodes and objects of the dynamic system with each other. In a specific example, for a dynamic system with a dimension of 4.8 million parameters, we estimated the control taking into account the penalty matrix, which made it possible to increase the inflow of additional resources from the outside by 2.4 times from 130 billion conv. units up to 310 conv. units in 5 years. Taking into account the maximum optimization of control in the nodes, an increase of 3.66 times in the inflow of additional resources was ensured --- from 200.46 to 726.62 billion rubles

References

[1] Wiener N. Cybernetics or сontrol and communication in the animal and the machine. New York, MIT Press, 1961.

[2] Krasovskiy A.A. Istoricheskiy ocherk razvitiya i sostoyaniya teorii upravleniya. Sovremennaya prikladnaya teoriya upravleniya. Ch. I [Historical development outline and state of control theory. Modern theory of applied control]. Taganrog, TRTU Publ., 2000.

[3] Tsypkin Ya.Z. Adaptatsiya i obuchenie v avtomaticheskikh sistemakh [Adaptation and training in automatic systems]. Moscow, Nauka Publ., 1968.

[4] Ressler O.E. Chemical turbulence: chaos in a small reaction-diffusion system. Z. Naturforsch. A., 1976, vol. 31, no. 10, pp. 1168--1172. DOI: https://doi.org/10.1515/zna-1976-1006

[5] Tyukin I. Adaptation in dynamical systems. Cambridge University Press, 2011.

[6] Lorenz E.N. Deterministic nonperiodic flow. J. Atoms. Sc., 1963, vol. 20, no. 2, pp. 130--141. DOI: https://doi.org/10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2

[7] Birkhoff G.D. Dynamical systems. New York, American Mathematical Society, 1927.

[8] Guckenheimer J., Holmes P. Nonlinear оscillations, dynamical systems and bifurcations of vector fields. New York, Springer, 1983.

[9] Palis J., De Melo W.Jr. Geometric theory of dynamical systems. New York, Springer, 1982.

[10] Golestani M., Mohammadzaman I., Yazdanpanah M.J. Robust finite-time stabilization of uncertain nonlinear systems based on partial stability. Nonlinear Dyn., 2016, vol. 85, no. 1, pp. 87--96. DOI: https://doi.org/10.1007/s11071-016-2669-5

[11] Haddad W.M., L’Afflitto A. Finite-time partial stability and stabilization and optimal feedback control. J. Franklin Inst., 2015, vol. 352, no. 6, pp. 2329--2357. DOI: https://doi.org/10.1016/j.jfranklin.2015.03.022

[12] Jammazi C., Abichou A. Controllability of linearized systems implies local finite-time stabilizability: applications to finite-time attitude control. IMA J. Math. Control. Inf., 2018, vol. 35, no. 1, pp. 249--277 DOI: https://doi.org/10.1093/imamci/dnw047

[13] Jenkins M., Larrain F., Esquivel G. Export processing zones in Latin America. Harvard Institute for International Development Working, 1998, no. 646. DOI: https://dx.doi.org/10.2139/ssrn.168174

[14] Kumar A., Shankar R., Choudhary A., et al. A big data MapReduce framework for fault diagnosis in cloud-based manufacturing. Int. J. Prod. Res., 2016, vol. 54, no. 23, pp. 7060--7073. DOI: https://doi.org/10.1080/00207543.2016.1153166

[15] L’Afflitto A. Differential games, finite-time partial-state stabilization of nonlinear dynamical systems, and optimal robust control. Int. J. Control, 2017, vol. 90, no. 9, pp. 1861--1878. DOI: https://doi.org/10.1080/00207179.2016.1226518

[16] Leontief W.W. The structure of American Economy, 1919--1939. Cambridge, Harvard University Press, 1941.

[17] Krotov V.F. Osnovy teorii optimal’nogo upravleniya [The basics of optimal management]. Moscow, Vysshaya shkola Publ., 1990.

[18] Schminke А., Van Biesebroeck J. Using export market performance to evaluate regional preferential policies in China. Rev. World Econ., 2013, vol. 149, no. 2, pp. 343--367. DOI: https://doi.org/10.1007/s10290-012-0145-y

[19] Masaev S.N. Metodika kompleksnoy otsenki upravlencheskikh resheniy v proizvodstvennykh sistemakh s primeneniem korrelyatsionnoy adaptometrii. Dis. kand. tekh. nauk [Methodology for integrated assessment of managerial decisions in production systems using correlation adaptometry. Cand. Sc. (Eng.). Diss.]. Krasnoyarsk, SFU, 2011 (in Russ.).