Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-15T12:06:13.137Z Has data issue: false hasContentIssue false
  • English
  • Français

Realization theory for linear and bilinear switched systems: A formal power series approach

Part I: Realization theory of linear switched systems*

Published online by Cambridge University Press:  31 March 2010

Mihály Petreczky
Mihály Petreczky*
Affiliation:
Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. M.Petreczky@maastrichtuniversity.nl
Get access

Abstract

The paper represents the first part of a series of papers on realization theory of switched systems. Part I presents realization theory of linear switched systems, Part II presents realization theory of bilinear switched systems. More precisely, in Part I necessary and sufficient conditions are formulated for a family of input-output maps to be realizable by a linear switched system and a characterization of minimal realizations is presented. The paper treats two types of switched systems. The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences is admissible, but within this restricted set the switching times are arbitrary. The paper uses the theory of formal power series to derive the results on realization theory.

Keywords

Hybrid systems switched linear systemsswitched bilinear systemsrealization theoryformal power seriesminimal realization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 410 - 445
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. Berstel and C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science. Springer-Verlag (1984).
M.F. Callier and A.C. Desoer, Linear System Theory. Springer-Verlag (1991).
D'Alessandro, P., Isidori, A. and Ruberti, A., Realization and structure theory of bilinear dynamical systems. SIAM J. Control 12 (1974) 517–535.
S. Eilenberg, Automata, Languages and Machines. Academic Press, New York-London (1974).
Fliess, M., Matrices de Hankel. J. Math. Pures Appl. 53 (1974) 197–222.
Fliess, M., Realizations of nonlinear systems and abstract transitive Lie algebras. Bull. Amer. Math. Soc. 2 (1980) 444–446. CrossRef
Fliess, M., Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3–40. CrossRef
F. Gécseg and I. Peák, Algebraic theory of automata. Akadémiai Kiadó, Budapest (1972).
A. Isidori, Direct construction of minimal bilinear realizations from nonlinear input-output maps. IEEE Trans. Automat. Contr. AC-18 (1973) 626–631.
A. Isidori, Nonlinear Control Systems. Springer-Verlag (1989).
N. Jacobson, Lectures in Abstract Algebra, Vol. II: Linear algebra. D. van Nostrand Company, Inc., New York (1953).
Jakubczyk, B., Existence and uniqueness of realizations of nonlinear systems. SIAM J. Control Optim. 18 (1980) 455–471. CrossRef
B. Jakubczyk, Realization theory for nonlinear systems, three approaches, in Algebraic and Geometric Methods in Nonlinear Control Theory, M. Fliess and M. Hazewinkel Eds., D. Reidel Publishing Company (1986) 3–32.
W. Kuich and A. Salomaa, Semirings, Automata, Languages, in EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1986).
D. Liberzon, Switching in Systems and Control. Birkhäuser, Boston (2003).
M. Petreczky, Realization theory for linear switched systems, in Proceedings of the Sixteenth International Symposium on Mathematical Theory of Networks and Systems (2004). [ Draft available at http://www.cwi.nl/~mpetrec.]
M. Petreczky, Realization theory for bilinear hybrid systems, in 11th IEEE Conference on Methods and Models in Automation and Robotics (2005). [CD-ROM only.]
M. Petreczky, Realization theory for bilinear switched systems, in Proceedings of 44th IEEE Conference on Decision and Control (2005). [CD-ROM only.]
M. Petreczky, Hybrid formal power series and their application to realization theory of hybrid systems, in 17th International Symposium on Mathematical Networks and Systems (2006).
M. Petreczky, Realization Theory of Hybrid Systems. Ph.D. Thesis, Vrije Universiteit, Amsterdam (2006). [Available online at: http://www.cwi.nl/~mpetrec.]
Petreczky, M., Realization theory for linear switched systems: Formal power series approach. Syst. Control Lett. 56 (2007) 588–595. CrossRef
C. Reutenauer, The local realization of generating series of finite lie-rank, in Algebraic and Geometric Methods in Nonlinear Control Theory, M. Fliess and M. Hazewinkel Eds., D. Reidel Publishing Company (1986) 33–43.
Schtzenberger, M.-P., On the definition of a family of automata. Inf. Control 4 (1961) 245–270.
E.D. Sontag, Polynomial Response Maps, Lecture Notes in Control and Information Sciences 13. Springer Verlag (1979).
Sontag, E.D., Realization theory of discrete-time nonlinear systems: Part I – The bounded case. IEEE Trans. Circuits Syst. 26 (1979) 342–356. CrossRef
Sun, Z., Ge, S.S. and Lee, T.H., Controllability and reachability criteria for switched linear systems. Automatica 38 (2002) 115–786.
Sussmann, H., Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory 10 (1977) 263–284. CrossRef
Wang, Y. and Sontag, E., Algebraic differential equations and rational control systems. SIAM J. Control Optim. 30 (1992) 1126–1149. CrossRef