Synthesis of Time-Optimal Control for One Fourth-Order Linear System

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

A linear fourth-order control system is studied, describing in the first approximation the dynamics of an inverted pendulum with an active dynamic damper. Based on Pontryagin’s maximum principle and the method proposed by A.A. Feldbaum for constructing sets on which control switching occurs, the problem of synthesizing optimal control that brings the system to a state of rest in minimal time is solved. The properties of the system under consideration make it possible to reduce the solution of the optimal response problem to the solution of a similar problem for a system of lower dimension.

Full Text

Restricted Access

About the authors

I. M. Ananievsky

Ishlinsky Institute for Problems in Mechanics of the RAS

Author for correspondence.
Email: anan@ipmnet.ru
Russian Federation, Moscow

References

  1. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The Mathematical Theory of Optimal Processes. N.Y.: Wiley, 1962. 360 p.
  2. Milyutin A.A., Dmitruk A.V., Osmolovskiy N.P. The Maximum Principle in Optimal Control. Moscow. MSU. Pub., 2004. 168 p. (in Russian)
  3. Feldbaum A.A. On synthesis of optimal systems with the help of phase space // Avtomat. i Telemeh., 1955, vol. 16, no. 2, pp. 129–149. (in Russian)
  4. Formalskii A.M. Stabilisation and Motion Control of Unstable Objects. Vol. 33. De Gruyter Studies in Mathematical Physics. Berlin;München;Boston: De Gruyter, 2016. https://doi.org/10.1515/9783110375893
  5. Lavrovsky E.K. On quick action in the problem of controlling the vertical position of a pendulum by the movement of its base // J. Comput. Syst. Sci. Int., 2021, vol. 60, no. 1, pp. 39–47.
  6. Reshmin S.A., Chernous’ko F.L. Time-optimal control of an inverted pendulum in the feedback form // J. Comput. Syst. Sci. Int., 2006, vol. 45, no. 3, pp. 383–394.
  7. Chernousko F.L., Akulenko L.D., Sokolov B.N. Control of Oscillations. Moscow: Nauka, 1980. 384 p. (in Russian)
  8. Chernousko F.L., Ananievski I.M., Reshmin S.A. Control of Nonlinear Dynamical Systems. Methods and Applications. Berlin;Heidelberg: Springer, 2008. 396 p.
  9. Anan’yevskii I.M. Control of a fourth-order linear system with mixed constraints // PMM, 2000, vol. 64, iss. 6, pp. 863–870.
  10. Ananievskii I.M., Dunaev I.A. The fastest damping of a linear inverted pendulum using a dynamic absorber // J. Comput. Syst. Sci. Int., 2024, vol. 63, no. 3, pp. 390–402.
  11. Kalman R.E., Falb P.L., Arbib M.A. Topics in Mathematical System Theory. N.Y.: McGraw-Hill, 1969. 358 p.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1

Download (10KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences