The paper extends the concept of the Lie derivative of the vector field, used in the study of the continuous-time dynamical systems, for the discrete-time case. In the continuous-time case the Lie derivative of a vector field (1-form or scalar function) with respect to the system dynamics is defined as its rate of change in time. In the discrete-time case we introduce the algebraic definition of the Lie derivative, using the concepts of forward and backward shifts. The definitions of discrete-time forward and backward shifts of the vector field are based on the concepts of already known forward and backward shifts of the 1-forms and on the scalar product of 1-form and vector field. Further we show that the interpretation of the discrete-time Lie derivative agrees with its interpretation as the rate of change in the continuous-time case. Finally, the geometric property of the discrete-time Lie derivative is also examined and shown to mimic the respective property in the continuous-time case.
1. Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups. Springer, 1983.
2. Bohner, M. and Peterson, A. Dynamic Equations on Time Scales. Birkhäuser, Boston-Basel-Berlin, 2001.
3. Fliess, M. Reversible linear and nonlinear discrete-time dynamics. IEEE Trans. Autom. Control, 1992, 37, 1144–1153.
4. Nam, K. Linearization of discrete-time nonlinear systems and canonical structure. IEEE Trans. Autom. Control, 1989, 34, 119–122.
5. Lee, W. and Nam, K. Observer design for nonlinear discrete-time systems. In Proceedings of the 29th CDC, Honolulu, Hawaii, December, 1990, 768–769.
6. Aranda-Bricaire, E., Kotta, Ü., and Moog, C. H. Linearization of discrete-time systems. SIAM J. Control. Optim., 1996, 34, 1999–2023.
7. Verriest, E. I. and Gray, W. S. Discrete time nonlinear balancing. In Proceedings of the 5th IFAC Symposium of NOLCOS, St. Petersburg, Russia, July 2001, 515–520.
8. Rieger, K., Schlacher, K., and Holl, J. On the observability of discrete-time dynamic systems – a geometric approach. Automatica, 2008, 44, 2057–2062.
9. Mullen, P. et al. Discrete Lie advection of differential forms. Found. Comput. Math., 2011, 11, 131–149.
10. Hirani, A. N. Discrete Exterior Calculus: Applications in Mechanics and Computer Science. Ph.D. Thesis, Caltech, 2003.
11. Nijmeijer, H. and van der Schaft, A. J. Nonlinear Dynamic Control Systems. Springer Verlag, New York, 1991.
12. Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E., and Wyrwas, M. Algebraic formalism of differential one-forms for nonlinear control systems on time scales. Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 264–282.
13. Isidori, A. Nonlinear Control Systems. Springer Verlag, London, 1995.
14. Califano, C., Monaco, S., and Normand-Cyrot, D. On the problem of feedback linearization. Systems Contr. Lett., 1999, 36, 61–67.
15. Rahula, M. The New Problems in Differential Geometry. World Scientific Publishing Co. Pte. Ltd., 1993.
16. Mullari, T., Kotta, Ü., Bartosiewicz, Z., and Pawłuszewicz, E. Discrete-time Lie derivative with respect to the system dynamics. In Proceedings of the 18th IFAC World Congress: Milano, Italy, 2011, 11000–11005.
17. Van der Schaft, A. J. Transformations and representations of nonlinear systems. In Perspectives in Control Theory (Jakubczyk, B., Malanowski, K., and Respondek, W., eds), Birkhäuser, Boston, 1990, 297–314.
18. Delaleau, E. and Respondek, W. Lowering the orders of derivatives of controls in deneralized state space systems. J. Math. Syst. Estim. Control, 1995, 5, 1–27.
19. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Springer Verlag, London, 2007.
20. Van der Schaft, A. J. On realization of nonlinear systems described by higher order differential equations. Math. Syst. Theory, 1987, 19, 239–275.