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

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 2, page 410-445
- ISSN: 1292-8119

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topPetreczky, Mihály. "Realization theory for linear and bilinear switched systems: A formal power series approach." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 410-445. <http://eudml.org/doc/276332>.

@article{Petreczky2011,

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.
},

author = {Petreczky, Mihály},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hybrid systems switched linear systems; switched bilinear systems; realization theory; formal power series; minimal realization; hybrid systems switched linear systems},

language = {eng},

month = {5},

number = {2},

pages = {410-445},

publisher = {EDP Sciences},

title = {Realization theory for linear and bilinear switched systems: A formal power series approach},

url = {http://eudml.org/doc/276332},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Petreczky, Mihály

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2011/5//

PB - EDP Sciences

VL - 17

IS - 2

SP - 410

EP - 445

AB -
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.

LA - eng

KW - Hybrid systems switched linear systems; switched bilinear systems; realization theory; formal power series; minimal realization; hybrid systems switched linear systems

UR - http://eudml.org/doc/276332

ER -

## References

top- 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).
- P. D'Alessandro, A. Isidori and A. Ruberti, Realization and structure theory of bilinear dynamical systems. SIAM J. Control12 (1974) 517–535.
- S. Eilenberg, Automata, Languages and Machines. Academic Press, New York-London (1974).
- M. Fliess, Matrices de Hankel. J. Math. Pures Appl.53 (1974) 197–222.
- M. Fliess, Realizations of nonlinear systems and abstract transitive Lie algebras. Bull. Amer. Math. Soc.2 (1980) 444–446.
- M. Fliess, Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France109 (1981) 3–40.
- 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).
- B. Jakubczyk, Existence and uniqueness of realizations of nonlinear systems. SIAM J. Control Optim.18 (1980) 455–471.
- 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 .] URIhttp://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: .] URIhttp://www.cwi.nl/~mpetrec
- M. Petreczky, Realization theory for linear switched systems: Formal power series approach. Syst. Control Lett.56 (2007) 588–595.
- 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.
- M.-P. Schtzenberger, On the definition of a family of automata. Inf. Control4 (1961) 245–270.
- E.D. Sontag, Polynomial Response Maps, Lecture Notes in Control and Information Sciences13. Springer Verlag (1979).
- E.D. Sontag, Realization theory of discrete-time nonlinear systems: Part I – The bounded case. IEEE Trans. Circuits Syst.26 (1979) 342–356.
- Z. Sun, S.S. Ge and T.H. Lee, Controllability and reachability criteria for switched linear systems. Automatica38 (2002) 115–786.
- H. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory10 (1977) 263–284.
- Y. Wang and E. Sontag, Algebraic differential equations and rational control systems. SIAM J. Control Optim.30 (1992) 1126–1149.

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