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, p. 446-471

This paper is the second part of a series of papers dealing with realization theory of switched systems. The current Part II addresses realization theory of bilinear switched systems. In Part I [Petreczky, ESAIM: COCV, DOI: 10.1051/cocv/2010014] we presented realization theory of linear switched systems. More precisely, in Part II we present necessary and sufficient conditions for a family of input-output maps to be realizable by a bilinear switched system, together with a characterization of minimal realizations. Similarly to Part I, the paper deals with 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 : https://doi.org/10.1051/cocv/2010015
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_446_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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {2},
     year = {2011},
     pages = {446-471},
     doi = {10.1051/cocv/2010015},
     zbl = {1233.93021},
     zbl = {1233.93020},
     mrnumber = {2801327},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_2_446_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. 446-471. doi : 10.1051/cocv/2010015. http://www.numdam.org/item/COCV_2011__17_2_446_0/

[1] P. D'Alessandro, A. Isidori and A. Ruberti, Realization and structure theory of bilinear dynamical systems. SIAM J. Control 12 (1974) 517-535. | MR 424307 | Zbl 0254.93008

[2] A. Isidori, Nonlinear Control Systems. Springer-Verlag (1989). | MR 1015932 | Zbl 0569.93034

[3] D. Liberzon, Switching in Systems and Control. Birkhäuser, Boston (2003). | MR 1987806 | Zbl 1036.93001

[4] M. Petreczky, Realization theory for bilinear switched systems, in Proceedings of 44th IEEE Conference on Decision and Control (2005). [CD-ROM only.] | Zbl 1155.93335

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

[6] M. Petreczky, Realization theory linear and bilinear switched systems: A formal power series approach - Part I: Realization theory of linear switched systems. ESAIM: COCV (2010) DOI: 10.1051/cocv/2010014. | Numdam | MR 2801326 | Zbl 1233.93020

[7] E.D. Sontag, Realization theory of discrete-time nonlinear systems: Part I - The bounded case. IEEE Trans. Circuits Syst. 26 (1979) 342-359. | MR 529666 | Zbl 0409.93014

[8] Y. Wang and E. Sontag, Algebraic differential equations and rational control systems. SIAM J. Control Optim. 30 (1992) 1126-1149. | MR 1178655 | Zbl 0762.93015