Advertisement

Differential Equations

, Volume 54, Issue 4, pp 497–508 | Cite as

Covering Method for Trajectory Generation and Orbital Decomposition of Systems

  • Yu. S. Belinskaya
  • V. N. Chetverikov
Control Theory
  • 13 Downloads

Abstract

Based on previously stated approaches, we propose a method for solving point-topoint steering problems in the case where a 2n-parametric family of solutions of a nonlinear system is known and for Liouville systems. Two examples of the helicopter motions in the vertical plane and the Kapitsa pendulum are considered to demonstrate the efficiency of the proposed method.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belinskaya, Yu.S. and Chetverikov, V.N., Covering method for terminal control with regard to constraints, Differ. Equations, 2014, vol. 50, no. 12, pp. 1632–1642.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Ed. by Vinogradov, A.M. and Krasil’shchik, I.S., in Trans. Math. Monogr., Vol. 182, 1999.Google Scholar
  3. 3.
    Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., Flatness and defect of nonlinear systems: Introductory theory and examples, Internat. J. Control, 1995, vol. 61, no. 6, pp. 1327–1361.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control, 1999, vol. 44, no. 5, pp. 922–937.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Belinskaya, Yu.S. and Chetverikov, V.N., Symmetries, coverings, and decomposition of systems and trajectory generation, Differ. Equations, 2016, vol. 52, no. 11, pp. 1423–1435.CrossRefzbMATHGoogle Scholar
  6. 6.
    Belinskaya, Yu.S. and Chetverikov, V.N., Covering method for point-to-point control of constrained flat systems, IFAC-Papers OnLine, 2015, vol. 48, no. 11, pp. 924–929.CrossRefGoogle Scholar
  7. 7.
    Sokolov, V.V., Pseudosymmetries and differential substitutions, Funct. Anal. Appl., 1988, vol. 22, no. 2, pp. 121–129.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Respondek, W., On decomposition of nonlinear control systems, Systems Control Lett., 1982, vol. 1, pp. 301–308.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chetverikov, V.N., On the structure of integrable C-fields, Differential Geom. Appl., 1991, vol. 1, pp. 309–325.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sira-Ramirez, H., Castro-Linares, R., and Liceaga-Castro, E., A Liouvillian systems approach for the trajectory planning-based control of helicopter models, Internat. J. Robust Nonlinear Control, 2000, vol. 10, pp. 301–320.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chelouah, A., Extensions of differential flat fields and Liouvillian systems, in Proc. of the IEEE Conference on Decision and Control, San Diego, 1997, pp. 4268–4273.CrossRefGoogle Scholar
  12. 12.
    Chetverikov, V.N., Liouville systems and symmetries, Differ. Equations, 2012, vol. 48, no. 12, pp. 1639–1651.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chetverikov, V.N., Coverings and integrable pseudosymmetries of differential equations, Differ. Equations, 2017, vol. 53, no. 11, pp. 1428–1439.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Bauman Moscow State Technical UniversityMoscowRussia

Personalised recommendations