The problem of tracking an unknown input action $u(\cdot)$ in a system of nonlinear ordinary differential equations is considered. Its essence consists in the construction of an algorithm for calculating some function that approximates~$u(\cdot)$ in the mean square. The algorithm in question should implement the tracking process in real time, i.e., it should calculate an approximation of the input action realized up to a time moment $t$, not later than this time. The input data to the algorithm are the results of inaccurate measurements of the system's phase state at discrete times. As a consequence of this feature of the problem, the exact tracking of $u(\cdot)$ is impossible. Therefore, we construct an algorithm of approximate tracking based on a controlled model. The model control obtained by the feedback principle taking into account current phase states is formed on the basis of an appropriate modification of the dynamic discrepancy method well-known in the theory of ill-posed problems.
Keywords: dynamic discrepancy method, input action tracking
Received December 23, 2024
Revised January 27, 2025
Accepted January 27, 2025
Published online March 20, 2025
Vyacheslav Ivanovich Maksimov, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: maksimov@imm.uran.ru
REFERENCES
1. Polyak B.T., Shcherbakov P.S. Robastnaya ustoychivost’ i upravleniye [Robust stability and control]. Moscow, Nauka Publ., 2002, 303 p. ISBN: 5-02-002561-5 .
2. Boulite S., Idrissi A., Ould Maaloum A. Robust multivariable PI-controllers for linear systems in Banach state spaces. J. Math. Anal. Appl., 2009, vol. 349, no. 1, pp. 90–99. https://doi.org/10.1016/j.jmaa.2008.08.039
3. Chen W.H., Yang J., Guo L., Li H. Disturbance-observer-based control and related methods: an overview. IEEE Trans. Ind. Electron., 2015, vol. 63, no. 2, pp. 1083–1095. https://doi.org/10.1109/TIE.2015.2478397
4. Guo B.-Z., Wu Z.-H., Zhou H.-C. Active disturbance rejection control approach to output-feedback stabilization of a class of uncertain nonlinear systems subject to stochastic disturbance. IEEE Trans. Autom. Control, 2015, vol. 61, no. 6, pp. 1613–1618. https://doi.org/10.1109/TAC.2015.2471815
5. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text was published in Krasovskii N. N., Subbotin A. I. Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.
6. Krasovskiy N.N. Upravleniye dinamicheskoy sistemoy. Zadacha o minimume garantirovannogo rezul’tata [Dynamic system control. Minimum guaranteed result problem], Moscow, Nauka Publ., 1985, 520 p.
7. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under the conditions of uncertainty], Moscow, Nauka Publ., 1977, 392 p.
8. Kurzhanski A., Valui I. Ellipsoidal calculus for estimation and control. Boston, Birkhäuser, 1996, 321 p. ISBN-10: 0817636994 .
9. Anan’ev B.I., Gusev M.I., Filippova T.F. Upravleniye i otsenivaniye sostoyaniy dinamicheskikh sistem s neopredelennost’yu [Control and estimation of states of dynamic systems with uncertainty]. Novosibirsk, Izd-vo SO RAN, 2018. 193 p. ISBN: 978-5-7692-1624-4 .
10. Osipov Yu.S., Kryazhimskii A.V. Inverse problems for ordinary differential equations: dynamical solutions, Basel, Gordon and Breach, 1995, 625 p. ISBN: 978-2881249440 .
11. Osipov Yu.S., Kryazhimskii A.V., Maksimov V.I. Metody dinamicheskogo vosstanovleniya vkhodov upravlyaemykh sistem [Methods for dynamic reconstruction of inputs of control systems], Yekaterinburg, Ural Branch of RAS Publ., 2011, 292 p.
12. Maksimov V.I. The methods of dynamical reconstruction of an input in a system of ordinary differential equations. J. Inverse Ill-Posed Probl., 2021, vol. 29, no. 1, pp. 125–156. https://doi.org/10.1515/jiip-2020-0040
13. Vasil’ev F.P. Metody resheniya ekstremal’nykh zadach [Methods for solving extremal problems]. Moscow, Nauka Publ., 1981, 400 p.
14. Osipov Yu.S., Kryazhimskii A.V. On methods of positional modeling of control in dynamic systems. In the book: Kachestvennyye voprosy teorii differentsial’nykh uravneniy i upravlyayemykh sistem [Qualitative issues of the theory of differential equations and control systems]. Sverdlovsk, UNTS Publ., 1988, pp. 34–44.
15. Blizorukova M.S. Input modeling in time-delayed systems. Comput. Math. Model., 2001, vol. 12, no. 2, pp. 174–185. https://doi.org/10.1023/A:1012518317520
16. Blizorukova M.S., Maksimov V.I. Dynamic discrepancy method in the problem of reconstructing the input of a system with time delay control. Comput. Math. Math. Phys., 2021, vol. 61, no. 3, pp. 359–367. https://doi.org/10.1134/S0965542521030040
17. Surkov P.G. Application of the residual method in the right hand side reconstruction problem for a system of fractional order. Comput. Math. Math. Phys., 2019, vol. 59, no. 11, pp. 1781–1790. https://doi.org/10.1134/S0965542519110113
18. Maksimov V.I. Some dynamical inverse problems for hyperbolic systems. Control and Cybernetics, 1996, vol. 25, no. 3, pp. 464–482.
19. Maksimov V.I. Positional modeling of controls and initial functions for Volterra systems. Differentsial’nye Uravneniya, 1987, vol. 23, no. 4, pp. 618–629 (in Russian).
20. Maksimov V.I. The dynamical decoupled method in the input reconstruction problem. Comput. Math. Math. Phys., 2004, vol. 44, no. 2, pp. 278–288.
21. Vasil’eva E.V. The dynamic discrepancy method for a differential equation with memory. Comput. Math. Model., 1999, vol. 10, no. 1, pp. 55–60. https://doi.org/10.1007/BF02358922
22. Mart’yanov A.S. The dynamic residual method in the problem of reconstructing inputs with incomplete information. Comput. Math. Math. Phys., 2005, vol. 45, no. 2, pp. 213–221.
23. Ioffe A.D., Tikhomirov V.M. Theory of extremal problems. Elsevier Sci. Publ., 2009, 459 p. ISBN: 9780080875279 . Original Russian text published in Ioffe A. D., Tikhomirov V. M. Teoriya ekstremal’nykh zadach, Moscow, Nauka Publ., 1974, 480 p.
Cite this article as: V.I. Maksimov. On an algorithm of tracking an input action in a system of differential equations.Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 141–154
