eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
On the properties of forward and backward shifts of vector fields; pp. 314–325

Arvo Kaldmäe, Vadim Kaparin ORCID Icon, Ülle Kotta ORCID Icon, Tanel Mullari, Ewa Pawluszewicz ORCID Icon

The paper investigates some properties of recently defined forward and backward shifts of vector fields. The main purpose of the paper is to show that the forward and backward shifts of vector fields commute with the Lie bracket operator and with some commonly used system transformations. The latter include, for example, classical and parametrized state transformations as well as static and dynamic state feedbacks. These properties become important when studying control problems involving such transformations.


1. Aranda-Bricaire, E., Kotta, Ü . and Moog, C. H. Linearization of discrete-time systems. SIAM J. Control. Optim., 1996, 34(6), 1999–2023.

2. Belikov, J., Kaldmäe, A. and Kotta, Ü. Global linearization approach to nonlinear control systems: a brief tutorial overview. Proc. Estonian Acad. Sci., 2017, 66(3), 243–263.

3. Boothby, W. M. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, Orlando, 1986.

4. Califano, C., Monaco, S. and Normand-Cyrot, D. Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time. Automatica, 2009, 45(11), 2483–2490.

5. Conte, G., Moog, C. H. and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Springer, London, 2007.

6. Fliess, M., Levine, J., Martin, P., Ollivier, F., and Rouchon, P. A remark on nonlinear accessibility conditions and infinite prolongations. Syst. Control Lett., 1997, 31, 77–83.

7. Isidori, A. Nonlinear Control Systems. 3rd edSpringer–Verlag, London, 1995.

8. Kaparin, V. and Kotta, Ü . Transformation of nonlinear discrete-time system into the extended observer form. Int. J. Control, 2018, 91(4), 848–858.

9. Kaparin, V. and Kotta, Ü. Transformation of nonlinear MIMO discrete-time systems into the extended observer form. Asian J. Control, 2019, 21(5), 2208–2217.

10. Kolar, B., Schöberl, M. and Diwold, J. Differential-geometric decomposition of flat nonlinear discrete-time systems. Automatica, 2021, 132, 109828.

11. Levine, J. Analysis and Control of Nonlinear Systems: a Flatness-based Approach. Springer, Berlin, 2009.

12. Mullari, T. and Kotta, Ü. Linearization of discrete-time control system by state transformation. Proc. Estonian Acad. Sci., 2021, 70(1), 62–79.

13. Mullari, T. and Kotta, Ü. Transformation of nonlinear discrete-time state equations into the observer form: extension to non-reversible case. Proc. Estonian Acad. Sci., 2021, 70(3), 235–247.

14. Mullari, T., Kotta, Ü., Bartosiewicz, Z., Pawluszewicz, E. and Moog, C. H. Forward and backward shifts of vector fields: towards the dual algebraic framework. IEEE Trans. Automat. Contr., 2017, 62(6), 3029–3033.

15. Nijmeijer, H. and van der Schaft, A. J. Nonlinear Dynamical Control Systems. Springer, New York, 1990.

16. Rieger, K. and Schlacher, K. Implicit discrete-time systems and accessibility. Automatica, 2011, 47(9), 1849–1859.

17. Rieger, K., Schlacher, K. and Holl, J. On the observability of discrete-time dynamic systems – a geometric approach. Automat- ica, 2008, 44(8), 2057–2062.

18. Spivak, M. A Comprehensive Introduction to Differential Geometry. 3rd. ed. Publish or Perish, Houston, 1999. 

Back to Issue