Summary
Theory of systems on homogeneous time scales unifies theories of continuous-time and discrete-time systems. The characterizations of external dynamical equivalence known for continuous-time and discrete-time systems are extended here to systems on time scales. Under assumption of uniform observability, it is shown that two analytic control systems with output are externally dynamically equivalent if and only if their delta universes are isomorphic. The delta operator associated to the system on a time scale is a generalization of the differential operator associated to a continuous-time system and of the difference operator associated to a discrete-time system.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
. Bartosiewicz Z, Johnson J (1994) Systems on universe spaces. Acta Applicandae Mathematicae 34.
Bartosiewicz Z, Jakubczyk B, Pawluszewicz E (1994) Dynamic feedback equiv alence of nonlinear discrete-time systems. In: Proceedings of First International Symposium on Mathematical Models in Automation and Robotics, Sept. 1–3, 1994, Miedzyzdroje, Poland, Tech. Univ. of Szczecin Press, Szczecin.
. Bartosiewicz Z, Pawluszewicz E (1998) External equivalence of unobservable discrete-time systems. In: Proceedings of NOLCOS’98Enschede, Netherlands, July 1998.
Bartosiewicz Z, Pawluszewicz E (2004) Unification of continuous-time and discrete-time systems: the linear case. In: Proceedings of Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004) Katholieke Universiteit Leuven, Belgium July 5–9, 2004, Leuven.
. Bartosiewicz Z, Pawluszewicz E (2005) Dynamic feedback equivalence of non linear systems on time scales. In: Proceedings of 16th IFAC Congress, Prague (CD-ROM).
. Bartosiewicz Z, Pawluszewicz E (2006) Realizations of linear control systems on time scales. Control & Cybernetics, 35(4).
Bohner M, Peterson A (2001) Dynamic Equation on Time Scales. Birkhäuser, Boston Basel Berlin.
Charlet B, Levine J, Marino R (1991) Sufficient conditions for dynamic state feedback linearization. SIAM J. Control and Optimization 29:38–57.
Fausett LV, Murty KN (2004) Controllability, observability and realizability criteria on time scale dynamical systems. Nonlinear Stud. 11:627–638.
Fliess M, Lévine J, Martin Ph, Rouchon P (1992) Sur les systèmes non linéaires différentiellement plats. C. R. Acad. Sci. Paris Sér. I Math., 315(5):619–624.
. Goodwin GC, Graebe SF, Salgado ME (2001) Control System Design. Prentice Hall International.
Gürses M, Guseinov GSh, Silindir B (2005) Integrable equations on time scales. Journal of Mathematical Physics 46:113–510.
. Hilger S (1988) Ein Maßkettenkalkülmit Anwendung auf Zentrumsmannig faltigkeiten. Ph.D. thesis, Universität Würzburg, Germany External Dynamical Equivalence 69.
Jakubczyk B (1992) Remarks on equivalence and linearization of nonlinear systems. In: Proceedings of the 2nd IFAC NOLCOS Symposium, 1992, Bordeaux, France, 393–397.
. Jakubczyk B (1992) Dynamic feedback equivalence of nonlinear control systems. Preprint.
. Johnson J (1986) A generalized global differential calculus I. Cahiers Top. et Geom. Diff. XXVII.
. Martin Ph, Murray RM, Rouchon P (1997) In: Bastin G, Gevers M (Eds) Flat systems. Plenary Lectures and Mini-Courses, European Control Conference ECC’97, Brussels.
Middleton RH, Goodwin GC (1990) Digital Control and Estimation: A Unified Approach. Englewood Cliffs, NJ: Prentice Hall.
Pawluszewicz E (1996) Dynamic linearization of input-output discrete-time systems. In: Proceedings of the International Conference UKACC Control’96, Exeter, UK.
Pawluszewicz E, Bartosiewicz Z (1999) External Dynamic Feedback Equivalence of Observable Discrete-time Control Systems. Proc. of Symposia in Pure Mathematics, vol. 64, American Mathematical Society, Providence, Rhode Island.
Pawluszewicz E, Bartosiewicz Z (2003) Differential universes in external dynamic linearization. Proceedings of European Control Conference ECC’03, Cambridge, UK.
Pomet J-B (1995) A differential geometric setting for dynamic equivalence and dynamic linearization, in: Geometry in Nonlinear Control and Differential Inclusions, Banach Center Publications, vol. 32, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bartosiewicz, Z., Pawłuszewicz, E. (2008). External Dynamical Equivalence of Analytic Control Systems. In: Sarychev, A., Shiryaev, A., Guerra, M., Grossinho, M.d.R. (eds) Mathematical Control Theory and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69532-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-69532-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69531-8
Online ISBN: 978-3-540-69532-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)