Abstract
The paper describes an algebraic construction of the inversive differential ring, associated with a nonlinear control system, defined on a nonhomogeneous but regular time scale. The ring of meromorphic functions in system variables is constructed under the assumption that the system is submersive, and equipped with three operators (delta- and nabla-derivatives, and the forward shift operator) whose properties are studied. The formalism developed unifies the existing theories for continuous- and discrete-time nonlinear systems, and accommodates also the case of non-uniformly sampled systems. Compared with the homogeneous case the main difficulties are noncommutativity of delta (nabla) derivative and shift operators and the fact that the additional time variable t appears in the definition of the differential ring. The latter yields that the new variables of the inversive closure, depending on t, have to be chosen to be smooth at each dense point t of the time scale.
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Acknowledgments
The work of Z. Bartosiewicz and M. Wyrwas is supported by the Białystok University of Technology grant No. S/WI/2/2011. The work of E. Pawłuszewicz was partly done during her stay in Aveiro University (Department of Mathematics). Her work is additionally supported by the Białystok University of Technology grant No. S/WM/2/08. The work of Ü. Kotta is supported by the Estonian Science Foundation Grant No. 6922.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bartosiewicz, Z., Kotta, Ü., Pawłuszewicz, E. et al. Control systems on regular time scales and their differential rings. Math. Control Signals Syst. 22, 185–201 (2011). https://doi.org/10.1007/s00498-011-0058-7
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DOI: https://doi.org/10.1007/s00498-011-0058-7