Transformation of degenerate indirect control systems in the vicinity of a program manifold
DOI:
https://doi.org/10.70474/wcda3590Keywords:
program manifold, degenerate systems, equivalence of systems, indirect control automatic systems, Lyapunov transformation, canonical formsAbstract
We consider one of the classes of implicit differential systems, systems of ordinary differential equations that are not resolved with respect to the highest derivative. Such equations are often found in everyday life in mechanics, physics, economics, biology, etc. The problems of constructing automatic control systems according to a given smooth program manifold also come down to such equations. This is the case when the dimension of the system of equations under construction is greater than the dimension of the program manifold. Then systems of algebraic equations with a rectangular matrix arise. We consider a system with a square matrix,
the discriminant of which is zero. The general problem of constructing systems of differential equations for a given manifold is considered.
The necessary and sufficient condition is drawn that the manifold is integral to the system of equations. The Yerugin function is linear with respect to the manifold. Then an indirect control system is built, taking into account that a given manifold is integral to it under certain conditions. In general, the Jacobi matrix is rectangular. The case is investigated when the matrix is quadratic and has zero roots. The manifold is assumed to be linear with respect to the desired variable. A degenerate indirect control system is obtained, unresolved with respect to the highest derivative. equivalent to a certain system is established, the matrices of which are constant and have a special structure. Lyapunov transformation matrices are found. It is shown that the considered control systems can be reduced to a central canonical form. A brief overview is provided.
Downloads
References
Samoilenko A.M, Yacobets V.P. On a reducibility of degenerate linear systems , to central canonical form, Dopov. AN Ukraine, 4 (1993), 10–15.
Yacobets V.P. Principal properties of degenerate linear systems, Ukraine mathematical journal, 49:9 (1997), 1278–1296. DOI: https://doi.org/10.1007/BF02487351
Mazanic S.A. On the linear differential systems equivalent with respect to Lyapunov’s generalized transformation, Differential equations, 22:9 (1986), 1619–1622.
Zhumatov S.S. Central canonical form and stability of degenerate control systems, Mathematical Journal, 3:3 (2003), 48–54.
Erugin N.P. Construction all the set of systems of differential equations, possessing by given integral curve, Prikladnaya Matematika i Mechanika, 6 (1952), 659–670.
Galiullin A.S., Mukhametzyanov I.A., Mukharlyamov R.G.and other, Conztruction program motion’s system, Nauka, Moskow (1971).
Galiullin A.S., Mukhametzyanov I.A., Mukharlyamov R.G. A survey of investigating on analytic construction of program motion’s systems, Vestnik RUDN. 1 (1994), 5–21 (In Russian).
Llibre J., Ramirez R. Inverse Problems in Ordinary Differential Equations and Applications, Springer International Publishing Switzerland, 2016. DOI: https://doi.org/10.1007/978-3-319-26339-7
Zhumatov S.S. On the stability of a program manifold of control systems with variable coefficients, Ukrainian Mathematical Journal, 71:8 (2020), 1202–1213. DOI: 10.1007/s11253-019-01707-7 DOI: https://doi.org/10.1007/s11253-019-01707-7
Maygarin B.G. Stability and quality of process of nonlinear automatic control system, Alma-Ata: Nauka, 1981 (In Russian).
Zhumatov S.S. Stability of a program manifold of control systems with locally quadratic relations, Ukrainian Mathematical Journal, 61:3 (2009), 500–509. DOI: 10.1007/s 11253-008-0224-y. DOI: https://doi.org/10.1007/s11253-009-0224-y
Byarov T.N. Theory of motion’s stability on a limit interval time, Gylym, Almaty (2003).
Tleubergenov M.I. On the inverse stochastic reconstruction problem, Differential Equations, 50:2 (2014), 274–278. DOI: 10.1134/s 0012266 11402 0165. DOI: https://doi.org/10.1134/S0012266114020165
Mukharlyamov R.G. Controlling the dynamics of a system with differential connections, News of the Russian Academy of Sciences. Theory and control systems. 3 (2019), 22–33.
Kaspirovich I.E., Mukharlyamov R.G. On methods for constructing dynamic equations taking into account the stabilization of connections News of the Russian Academy of Sciences. MTT. 3 (2019), 123–134.
Tleubergenov M.I., Ibraeva G.T. On the restoration problem with degenerated diffusion, TWMS Journal of Pure and Applied Mathematics, 6:1 (2015), 93–99.
Vassilina G.K., Tleubergenov M.I. Solution of the problem of stochastic stability of an integral manifold by the second Lyapunov method, Ukrainian Mathematical Journal, 68:1 (2016), 14–28. DOI: 10.1007/s 11253-016-1205-6. DOI: https://doi.org/10.1007/s11253-016-1205-6
Zhumatov S.S., Vasilina G.K. The Absolute Stability of Program Manifold of Control Systems with Rigid and Tachometric Feedbaks, Lobachevskii Journal of Mathematics, 43:11 (2022), 3344–3351. Pleiadis Publiching, Ltd, 2022. DOI: https://doi.org/10.1134/S1995080222140402
Zhumatov S.S., Mynbayeva S.T. Stability of the program manifold of automatic indirect control systems taking into account the external loads Advances in the Theory of Nonlinear Analysis and its Applications, 7:2 (2023), 405–412. DOI: https://doi.org/10.31197/atnaa.1200890
Additional Files
Published
Issue
Section
License
Copyright (c) 2025 Kazakh Mathematical Journal
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
One can find the license terms "CC Attribution-NonCommercial-NoDerivatives 4.0" here.
