A note on some characterization of invariant zeros in singular systems and algebraic criteria of nondegeneracy

Jerzy Tokarzewski

International Journal of Applied Mathematics and Computer Science (2004)

  • Volume: 14, Issue: 2, page 149-159
  • ISSN: 1641-876X

Abstract

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The question how the classical definition of the Smith zeros of an LTI continuous-time singular control system can be generalized and related to state-space methods is discussed. The zeros are defined as those complex numbers for which there exists a zero direction with a nonzero state-zero direction. Such a definition allows an infinite number of zeros (then the system is called degenerate). A sufficient and necessary condition for nondegeneracy is formulated. Moreover, some characterization of invariant zeros, based on the Weierstrass-Kronecker canonical form of the system and the first nonzero Markov parameter, is obtained.

How to cite

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Tokarzewski, Jerzy. "A note on some characterization of invariant zeros in singular systems and algebraic criteria of nondegeneracy." International Journal of Applied Mathematics and Computer Science 14.2 (2004): 149-159. <http://eudml.org/doc/207686>.

@article{Tokarzewski2004,
abstract = {The question how the classical definition of the Smith zeros of an LTI continuous-time singular control system can be generalized and related to state-space methods is discussed. The zeros are defined as those complex numbers for which there exists a zero direction with a nonzero state-zero direction. Such a definition allows an infinite number of zeros (then the system is called degenerate). A sufficient and necessary condition for nondegeneracy is formulated. Moreover, some characterization of invariant zeros, based on the Weierstrass-Kronecker canonical form of the system and the first nonzero Markov parameter, is obtained.},
author = {Tokarzewski, Jerzy},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Markov parameters; singular control systems; state-space methods; multivariable zeros},
language = {eng},
number = {2},
pages = {149-159},
title = {A note on some characterization of invariant zeros in singular systems and algebraic criteria of nondegeneracy},
url = {http://eudml.org/doc/207686},
volume = {14},
year = {2004},
}

TY - JOUR
AU - Tokarzewski, Jerzy
TI - A note on some characterization of invariant zeros in singular systems and algebraic criteria of nondegeneracy
JO - International Journal of Applied Mathematics and Computer Science
PY - 2004
VL - 14
IS - 2
SP - 149
EP - 159
AB - The question how the classical definition of the Smith zeros of an LTI continuous-time singular control system can be generalized and related to state-space methods is discussed. The zeros are defined as those complex numbers for which there exists a zero direction with a nonzero state-zero direction. Such a definition allows an infinite number of zeros (then the system is called degenerate). A sufficient and necessary condition for nondegeneracy is formulated. Moreover, some characterization of invariant zeros, based on the Weierstrass-Kronecker canonical form of the system and the first nonzero Markov parameter, is obtained.
LA - eng
KW - Markov parameters; singular control systems; state-space methods; multivariable zeros
UR - http://eudml.org/doc/207686
ER -

References

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  2. Ben-Israel A. and Greville T.N.E. (2002): Generalized Inverses: Theory and Applications, 2nd Ed. - New York: Wiley. Zbl0451.15004
  3. Dai L. (1989): Singular Control Systems. - Berlin: Springer. Zbl0669.93034
  4. Gantmacher F.R. (1988): Theory of Matrices. - Moscow: Nauka (in Russian). 
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  6. Kaczorek T. (1999): Control and Systems Theory. - Warsaw: Polish Scientific Publishers (in Polish). Zbl0997.93501
  7. Kaczorek T. (2000): Positive One- and Two-Dimensional Systems. -Warsaw: University of Technology Press (in Polish). 
  8. Kaczorek T. (2003): Decomposition of singular linear systems. - Przegląd Elektrotechniczny, Vol. LXXIX, No. 2, pp. 53-58. 
  9. Misra P., Van Dooren P. and Varga A. (1994): Computation of structural invariants of generalized state-space systems. - Automatica, Vol. 30, No. 12, pp. 1921-1936. Zbl0816.93040
  10. Tokarzewski J. (1998): On some characterization of invariant and decoupling zeros in singular systems. - Arch. Contr. Sci., Vol. 5, No. 3-4, pp. 145-159. Zbl1193.93106
  11. Tokarzewski J. (2002a): Zeros in Linear Systems: A Geometric Approach.- Warsaw: University of Technology Press. Zbl1044.93036
  12. Tokarzewski J. (2002b): Relationship between Smith zeros and invariant zeros in linear singular systems. - Proc. 8th IEEE Int. Conf. Methods and Models in Automation and Robotics, MMAR 2002, Szczecin, Poland, Vol. I, pp. 71-74. 
  13. Tokarzewski J. (2003): A characterization of invariant zeros in singular systems via the first nonzero Markov parameter and algebraic criterions of nondegeneracy. - Proc. 9th IEEE Int. Conf. Methods and Models in Automation and Robotics, MMAR 2003, Międzyzdroje, Poland, Vol. I, pp. 437-442. 

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