The notion of the transfer function of the discrete-time nonlinear control system is defined. The definition is based on a non-commutative twisted polynomial ring, which can be by the Ore condition extended into its quotient ring (field of fractions). Some properties of the transfer function, related to accessibility and observability of the system, are studied and the transfer functions of different composite systems (series, parallel, and feedback connections) are given. The resulting theory is, in principle, similar to that in the linear case, except that the polynomial description relates now the differentials of inputs and outputs, and the resulting polynomial ring is non-commutative.
1. Blomberg, H. and Ylinen, R. Algebraic Theory for Multivariable Linear Systems. Academic Press, London, 1983.
2. Zheng, Y., Willems, J. and Zhang, C. A polynomial approach to nonlinear system controllability. IEEE Trans. Automat. Control, 2001, 46, 1782–1788.
https://doi.org/10.1109/9.964691
3. Conte, G., Moog, C. H. and Perdon, A. M. Nonlinear Control Systems: An Algebraic Setting. Springer-Verlag, London, 1999.
4. Zheng, Y. and Cao, L. Transfer function description for nonlinear systems. J. East China Normal Univ. (Nat. Sci.), 1995, 2, 15–26.
5. Halás, M. An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica (accepted).
6. Halás, M. and Huba, M. Symbolic computation for nonlinear systems using quotients over skew polynomial ring. In 14th Mediterranean Conference on Control and Automation, Ancona, Italy. 2006.
https://doi.org/10.1109/MED.2006.328725
7. Zhang, C. and Zheng, Y. A polynomial approach to discrete-time nonlinear system controllability. Int. J. Control, 2004, 77, 491–497.
https://doi.org/10.1080/00207170410001682506
8. Bartosiewicz, Z., Kotta, Ü., Nõmm, S. and Pawluszewicz, E. Input-output equivalence transformations for discrete-time nonlinear systems. In Proceedings of the 2nd IFAC Symposium on Systems Structure and Control, Oaxaca, Mexico. 2004, 706–711.
9. Kotta, Ü. Irreducibility conditions for nonlinear input-output difference equations. In Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia. 2000, 3404–3408.
10. Kotta, Ü. and Tõnso, M. Irreducibility conditions for discrete-time nonlinear multiinput multi-output systems. In Proceedings of the 6th IFAC Symposium on Nonlinear Control (NOLCOS), Stuttgart, Germany. 2004, 269–273.
11. Grizzle, J. W. A linear algebraic framework for the analysis of discrete-time nonlinear systems. SIAM J. Control Optim., 1993, 31, 1026–1044.
https://doi.org/10.1137/0331046
12. Aranda-Bricaire, E., Kotta, Ü. and Moog, C. H. Linearization of discrete-time systems. SIAM J. Control Optim., 1996, 34, 1999–2023.
https://doi.org/10.1137/S0363012994267315
13. Kotta, Ü. Decomposition of discrete-time nonlinear control systems. Proc. Estonian Acad. Sci. Phys. Math., 2005, 54, 154–161.
14. Kotta, Ü., Zinober, A. S. I. and Liu, P. Transfer equivalence and realization of nonlinear higher order input-output difference equations. Automatica, 2001, 37, 1771–1778.
https://doi.org/10.1016/S0005-1098(01)00144-3
15. Farb, B. and Dennis, R. Noncommutative Algebra. Springer-Verlag, New York, 1993.
https://doi.org/10.1007/978-1-4612-0889-1
16. Bronstein, M. and Petkovšek, M. An introduction to pseudo-linear algebra. Theor. Computer Sci., 1996, 157, 3–33.
https://doi.org/10.1016/0304-3975(95)00173-5
17. Ore, O. Linear equations in non-commutative fields. Annals Math., 1931, 32, 463–477.
https://doi.org/10.2307/1968245
18. Ore, O. Theory of non-commutative polynomials. Annals Math., 1933, 34, 480–508.
https://doi.org/10.2307/1968173
