Advertisement

On Linearizability Conditions for Non-autonomous Control Systems

Conference paper
  • 369 Downloads
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)

Abstract

The paper deals with the problem of mappability of nonlinear non-autonomous control systems to linear non-autonomous systems with analytic matrices. We study the existence of non-local linearizing map of class \(C^2\) for nonlinear systems of class \(C^1\). In the paper K. Sklyar, On mappability of control systems to linear systems with analytic matrices. Systems Control Lett. 134 (2019), 104572, linearizability conditions were obtained under the additional requirement concerning existence of a non-local driftless form of the system. The goal of the present paper is to reduce this requirement.

Keywords

Linearizability problem Non-autonomous systems Driftless form Non-local first integrals 

References

  1. 1.
    Korobov, V.I.: Controllability, stability of some nonlinear systems. Differ. Uravnenija 9, 614–619 (1973). (in Russian)Google Scholar
  2. 2.
    Krener, A.: On the equivalence of control systems and the linearization of non-linear systems. SIAM J. Control 11, 670–676 (1973).  https://doi.org/10.1137/0311051MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Celikovsky, S.: Global linearization of nonlinear systems – a survey. In: Geom. in Nonlin. Control and Diff. Incl., Warszawa, pp. 123–137 (1995).  https://doi.org/10.4064/-32-1-123-137
  4. 4.
    Celikovsky, S., Nijmeijer, H.: Equivalence of nonlinear systems to triangular form: the singular case. Syst. Control Lett. 27, 135–144 (1996).  https://doi.org/10.1016/0167-6911(95)00059-3MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Korobov, V.I., Pavlichkov, S.S.: Global properties of the triangular systems in the singular case. J. Math. Anal. Appl. 342, 1426–1439 (2008).  https://doi.org/10.1016/j.jmaa.2007.12.070MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Korobov, V.I., Sklyar, K.V., Skoryk, V.O.: Stepwise synthesis of constrained controls for single input nonlinear systems of special form. Nonlinear Differ. Equ. Appl. 23–31 (2016).  https://doi.org/10.1007/s00030-016-0385-y
  7. 7.
    Brockett, R.W.: Feedback invariance for nonlinear systems. In: Proceedings of the Seventh World Congress IFAC, Helsinki, pp. 1115–1120 (1978)Google Scholar
  8. 8.
    Jakubczyk, B., Respondek, W.: On linearization of control systems. Bull. Acad. Sci. Polonaise Ser. Sci. Math. 28, 517–522 (1980)Google Scholar
  9. 9.
    Su, R.: On the linear equivalents of nonlinear systems. Syst. Control Lett. 2, 48–52 (1982).  https://doi.org/10.1016/S0167-6911(82)80042-XMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Respondek, W.: Geometric methods in linearization of control systems. In: Mathematical Control Theory, vol. 14, pp. 453–467. Banach Center Publication, PWN, Warsaw (1985)Google Scholar
  11. 11.
    Respondek, W.: Linearization, feedback and Lie brackets. In: Scientific Papers of the Institute of Technical Cybernetics of the Technical University of Wroclaw, no. 70, Conf. 29, pp. 131–166 (1985)Google Scholar
  12. 12.
    Nicolau, F., Respondek, W.: Flatness of multi-input control-affine systems linearizable via one-fold prolongation. SIAM J. Control Optim. 55(5), 3171–3203 (2017).  https://doi.org/10.1137/140999463MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, S., Moog, C.H., Respondek, W.: Maximal feedback linearization and its internal dynamics with applications to mechanical systems on \(\mathbb{R}^4\). Int. J. Robust Nonlinear Control 29(9), 2639–2659 (2019).  https://doi.org/10.1002/rnc.4507MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sklyar, G.M., Sklyar, K.V., Ignatovich, S.Y.: On the extension of the Korobov’s class of linearizable triangular systems by nonlinear control systems of the class \(C^1\). Syst. Control Lett. 54, 1097–1108 (2005).  https://doi.org/10.1016/j.sysconle.2005.04.002MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sklyar, K.V., Ignatovich, S.Y., Skoryk, V.O.: Conditions of linearizability for multi-control systems of the class \(C^1\). Commun. Math. Anal. 17, 359–365 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Sklyar, K.V., Ignatovich, S.Y.: Linearizability of systems of the class \(C^1\) with multi-dimensional control. Syst. Control Lett. 94, 92–96 (2016).  https://doi.org/10.1016/j.sysconle.2016.05.016MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sklyar, K.V., Ignatovich, S.Y., Sklyar, G.M.: Verification of feedback linearizability conditions for control systems of the class \(C^1\). In: 25th Mediterranean Conference on Control and Automation, MED, pp. 163–168 (2017). 7984112Google Scholar
  18. 18.
    Sklyar, K.V., Sklyar, G.M., Ignatovich, S.Y.: Linearizability of multi-control systems of the class \(C^1\) by additive change of controls. In: André, C., Bastos, M., Karlovich, A., Silbermann, B., Zaballa, I. (eds.) Operator Theory, Operator Algebras, and Matrix Theory. Operator Theory: Advances and Applications, vol. 267, pp. 359–370. Birkhauser/Springer, Cham (2018)CrossRefGoogle Scholar
  19. 19.
    Sklyar, K.: On mappability of control systems to linear systems with analytic matrices. Syst. Control Lett. 134, 104572 (2019).  https://doi.org/10.1016/j.sysconle.2019.104572MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Arnol’d, V.I.: Ordinary Differential Equations. Springer-Verlag, Heidelberg (1992)zbMATHGoogle Scholar
  21. 21.
    Hartman, P.: Ordinary Differential Equations, vol. XIV. Wiley, New York, London, Sydney (1964)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of SzczecinSzczecinPoland
  2. 2.V. N. Karazin Kharkiv National UniversityKharkivUkraine

Personalised recommendations