Reduction of Band Matrices in Large Dynamic Systems Controllability and Observability

Abstract

An approach is proposed for linear stationary dynamical system with controllability and observability band matrices making it possible to simplify procedures for evaluating controllability and observability of this system. The obtained results are based on the fact that the controllability and observability criteria of a dynamic system are equivalent due to their required and sufficient properties; therefore, any transformations of one criterion not violating the conditions of necessity and sufficiency could be reduced to transformations in a sense equivalent to the initial transformations. The Popov --- Belevich --- Hautus transformations of the controllability and observability criteria were taken as a basis, and then results of such transformations were correctly extended to the band criteria. It was proved that the controllability and observability analysis of a linear stationary system with a large number of the state dimensions was reduced to studying the matrices rank of a much smaller size. The proposed approach is based on the existence condition for a numerical matrix of a certain rank of the nondegenerate matrices that satisfy certain transformations. The corresponding controllability and observability theorems for the stationary dynamical systems were provided. It was shown that for systems with one input and one output, the controllability and observability analysis was reduced to the analysis of scalars

Please cite this article as:

Zubov N.E., Ryabchenko V.N. Reduction of band matrices in large dynamic systems controllability and observability. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 4 (103), pp. 39--49. DOI: https://doi.org/10.18698/1812-3368-2022-4-39-49

References

[1] Kailath T. Linear systems. Prentice Hall, 1980.

[2] Bernstein D. Matrix mathematics. Princeton Univ. Press, 2005.

[3] Zubov N.E., Mikrin E.A., Ryabchenko V.N. Matrichnye metody v teorii i praktike sistem avtomaticheskogo upravleniya letatel’nykh apparatov [Matrix methods in theory and practice of automated control systems for aircraft]. Moscow, BMSTU Publ., 2016.

[4] Gadzhiev M.G., Misrikhanov M.Sh., Ryabchenko V.N., et al. Matrichnye metody analiza i upravleniya perekhodnymi protsessami v elektroenergeticheskikh sistemakh [Matrix control and analysis methods for transient processes in electrical power systems]. Moscow, MPEI Publ., 2019.

[5] Wonham W.M. Linear multivariable control: a geometric approach. In: Stochastic Modelling and Applied Probability. New York, NY, Springer, 1985. DOI: https://doi.org/10.1007/978-1-4612-1082-5

[6] Zybin E.Yu., Misrikhanov M.Sh., Ryabchenko V.N. Minimal’naya parametrizatsiya resheniy lineynykh matrichnykh uravneniy. V kn.: Sovremennye metody upravleniya mnogosvyaznymi sistemami. Vyp. 2 [Minimum parametrization of solutions for linear systems of matrix equations. In: Modern Control Methods for Multicoupling Systems. Iss. 2]. Moscow, Energoatomizdat Publ., 2003, pp. 191--202 (in Russ.).

[7] Taufer J. Randwertprobleme für Systeme von Linearen Differentialgleichungen. Praha, Academia, 1973.

[8] Barinov V.A., Sovalov S.A. Rezhimy energosistem. Metody analiza i upravleniya [Regimes of power systems. Analysis and control methods]. Moscow, Energoatomizdat Publ., 1990.

[9] Gusseynov F.G. Uproshchenie raschetnykh skhem elektricheskikh system [Computational scheme simplification for electrical systems]. Moscow, Energiya Publ., 1978.

[10] Christensen G.S., El-Hawary M.E., Soliman S.A. Optimal control applications electric power systems. Series Mathematical Concepts and Methods in Science and Engineering. New York, NY, Springer, 1987. DOI: https://doi.org/10.1007/978-1-4899-2085-0

[11] Kundur P. Power system stability and control. McGraw-Hill, 1994.

[12] Zybin E.Yu., Misrikhanov M.Sh., Ryabchenko V.N. Recursive controllability and observability tests for large dynamic systems. Autom. Remote Control, 2006, vol. 67, no. 5, pp. 783--795. DOI: https://doi.org/10.1134/S0005117906050109

[13] Gantmakher F.R. Teoriya matrits [Matrix theory]. Moscow, Nauka Publ., 1968.

[14] Misrikhanov M.Sh., Ryabchenko V.N. Reduction of state dimensions at analysis of controllability and observability of linear models for power systems. Izvestiya TRTU, 2005, no. 11, pp. 42--54 (in Russ.).

[15] Mikrin E.A., Zubov N.E., Lapin A.V., et al. Analytical formula of calculating a controller for linear SIMO-system. Differentsial’nye uravneniya i protsessy upravleniya [Differential Equations and Control Processes], 2020, no. 1 (in Russ.). Available at: https://diffjournal.spbu.ru/pdf/mikrin1.pdf

[16] Mikrin E.A., Ryabchenko V.N., Zubov N.E., et al. Analysis and synthesis of dynamic MIMO-system based on band matrices of special type. Differentsial’nye uravneniya i protsessy upravleniya [Differential Equations and Control Processes], 2020, no. 2 (in Russ.). Available at: https://diffjournal.spbu.ru/RU/numbers/2020.2/article.1.1.html

[17] Zubov N.E., Mikrin E.A., Misrikhanov M.Sh., et al. Output control of the spectrum of a descriptor dynamical system. Dokl. Math., 2016, vol. 93, pp. 259--261. DOI: https://doi.org/10.1134/S106456241603008X

[18] Zubov N.E., Lapin A.V., Mikrin E.A., et al. Output control of the spectrum of a linear dynamic system in terms of the Van der Woude method. Dokl. Math., 2017, vol. 96, pp. 457--460. DOI: https://doi.org/10.1134/S1064562417050179

[19] Mikrin E.A., Zubov N.E., Efanov D.E., et al. Superfast iterative solvers for linear matrix equations. Dokl. Math., vol. 98, pp. 444--447. DOI: https://doi.org/10.1134/S1064562418060145