Realization theory for linear and bilinear switched systems: A formal power series approach
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 410-445.

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.

DOI: 10.1051/cocv/2010014
Classification: 93B15, 93B20, 93B25, 93C99
Keywords: hybrid systems switched linear systems, switched bilinear systems, realization theory, formal power series, minimal realization
@article{COCV_2011__17_2_410_0,
     author = {Petreczky, Mih\'aly},
     title = {Realization theory for linear and bilinear switched systems: {A} formal power series approach},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {410--445},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {2},
     year = {2011},
     doi = {10.1051/cocv/2010014},
     mrnumber = {2801326},
     zbl = {1233.93020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2010014/}
}
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
PY  - 2011
SP  - 410
EP  - 445
VL  - 17
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2010014/
DO  - 10.1051/cocv/2010014
LA  - en
ID  - COCV_2011__17_2_410_0
ER  - 
%0 Journal Article
%A Petreczky, Mihály
%T Realization theory for linear and bilinear switched systems: A formal power series approach
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 410-445
%V 17
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2010014/
%R 10.1051/cocv/2010014
%G en
%F COCV_2011__17_2_410_0
Petreczky, Mihály. Realization theory for linear and bilinear switched systems: A formal power series approach. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 410-445. doi : 10.1051/cocv/2010014. http://archive.numdam.org/articles/10.1051/cocv/2010014/

[1] J. Berstel and C. Reutenauer, Rational series and their languages, EATCS Monographs on Theoretical Computer Science. Springer-Verlag (1984). | MR | Zbl

[2] M.F. Callier and A.C. Desoer, Linear System Theory. Springer-Verlag (1991). | MR | Zbl

[3] P. D'Alessandro, A. Isidori and A. Ruberti, Realization and structure theory of bilinear dynamical systems. SIAM J. Control 12 (1974) 517-535. | MR | Zbl

[4] S. Eilenberg, Automata, Languages and Machines. Academic Press, New York-London (1974). | MR | Zbl

[5] M. Fliess, Matrices de Hankel. J. Math. Pures Appl. 53 (1974) 197-222. | MR | Zbl

[6] M. Fliess, Realizations of nonlinear systems and abstract transitive Lie algebras. Bull. Amer. Math. Soc. 2 (1980) 444-446. | MR | Zbl

[7] M. Fliess, Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3-40. | Numdam | MR | Zbl

[8] F. Gécseg and I. Peák, Algebraic theory of automata. Akadémiai Kiadó, Budapest (1972). | MR | Zbl

[9] A. Isidori, Direct construction of minimal bilinear realizations from nonlinear input-output maps. IEEE Trans. Automat. Contr. AC-18 (1973) 626-631. | MR | Zbl

[10] A. Isidori, Nonlinear Control Systems. Springer-Verlag (1989). | Zbl

[11] N. Jacobson, Lectures in Abstract Algebra, Vol. II: Linear algebra. D. van Nostrand Company, Inc., New York (1953). | MR | Zbl

[12] B. Jakubczyk, Existence and uniqueness of realizations of nonlinear systems. SIAM J. Control Optim. 18 (1980) 455-471. | MR | Zbl

[13] 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. | MR | Zbl

[14] W. Kuich and A. Salomaa, Semirings, Automata, Languages, in EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1986). | MR | Zbl

[15] D. Liberzon, Switching in Systems and Control. Birkhäuser, Boston (2003). | MR | Zbl

[16] 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.] | Zbl

[17] M. Petreczky, Realization theory for bilinear hybrid systems, in 11th IEEE Conference on Methods and Models in Automation and Robotics (2005). [CD-ROM only.]

[18] M. Petreczky, Realization theory for bilinear switched systems, in Proceedings of 44th IEEE Conference on Decision and Control (2005). [CD-ROM only.] | Zbl

[19] 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).

[20] M. Petreczky, Realization Theory of Hybrid Systems. Ph.D. Thesis, Vrije Universiteit, Amsterdam (2006). [Available online at: http://www.cwi.nl/~mpetrec.]

[21] M. Petreczky, Realization theory for linear switched systems: Formal power series approach. Syst. Control Lett. 56 (2007) 588-595. | MR | Zbl

[22] 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. | MR

[23] M.-P. Schtzenberger, On the definition of a family of automata. Inf. Control 4 (1961) 245-270. | MR | Zbl

[24] E.D. Sontag, Polynomial Response Maps, Lecture Notes in Control and Information Sciences 13. Springer Verlag (1979). | MR | Zbl

[25] E.D. Sontag, Realization theory of discrete-time nonlinear systems: Part I - The bounded case. IEEE Trans. Circuits Syst. 26 (1979) 342-356. | MR | Zbl

[26] Z. Sun, S.S. Ge and T.H. Lee, Controllability and reachability criteria for switched linear systems. Automatica 38 (2002) 115-786. | MR | Zbl

[27] H. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory 10 (1977) 263-284. | MR | Zbl

[28] Y. Wang and E. Sontag, Algebraic differential equations and rational control systems. SIAM J. Control Optim. 30 (1992) 1126-1149. | MR | Zbl

Cited by Sources: