Canonical forms of singular 1D and 2D linear systems

Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science (2003)

  • Volume: 13, Issue: 1, page 61-72
  • ISSN: 1641-876X

Abstract

top
The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.

How to cite

top

Kaczorek, Tadeusz. "Canonical forms of singular 1D and 2D linear systems." International Journal of Applied Mathematics and Computer Science 13.1 (2003): 61-72. <http://eudml.org/doc/207624>.

@article{Kaczorek2003,
abstract = {The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.},
author = {Kaczorek, Tadeusz},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {canonical form; transformation; singular; 2D Roesser model; 1D system; singular systems; 1D systems; discrete-time linear systems; realizations},
language = {eng},
number = {1},
pages = {61-72},
title = {Canonical forms of singular 1D and 2D linear systems},
url = {http://eudml.org/doc/207624},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Kaczorek, Tadeusz
TI - Canonical forms of singular 1D and 2D linear systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 1
SP - 61
EP - 72
AB - The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.
LA - eng
KW - canonical form; transformation; singular; 2D Roesser model; 1D system; singular systems; 1D systems; discrete-time linear systems; realizations
UR - http://eudml.org/doc/207624
ER -

References

top
  1. Aplevich J.D. (1985): Minimal representations of implicit linear systems. - Automatica, Vol. 21, No. 3, pp. 259-269. Zbl0565.93012
  2. Brunovsky P.A. (1970): A classification of linear controllable systems. - Kybernetika, Vol. 6, No. 3, pp. 173-187. Zbl0199.48202
  3. Cobb D. (1984): Controllability, observability and duality in singular systems. - IEEE Trans. Automat. Contr., Vol. A-26, No. 12, pp. 1076-1082. 
  4. Dai L. (1989): Singular Control Systems. - New York: Springer. Zbl0669.93034
  5. Eising R. (1978): Realization and stabilization of 2-D systems. - IEEE Trans. Automat. Contr., Vol. AC-23, No. 5, pp. 795-799. Zbl0397.93022
  6. Fornasini E. and Marchesini G. (1976): State-space realization theory of two-dimensional filters. - IEEE Trans. Automat. Contr., Vol. AC-21, No. 4, pp. 484-491. Zbl0332.93072
  7. Fornasini E. and Marchesini G. (1978): Doubly-indexed dynamical systems: State-space models and structural properties. - Math. Syst. Theory, Vol. 12, pp. 59-72. Zbl0392.93034
  8. Gałkowski K. (1981): The state-space realization of an n-dimensional transfer function. - Int. J. Circ. Theory Appl., Vol. 9, pp. 189-197. Zbl0458.94050
  9. Gałkowski K. (1992): Transformation of the transfer function variables of the singular n-dimensional Roesser model. - Int. J. Circ. Theory Appl., Vol. 20, pp. 63-74. Zbl0743.94026
  10. Gałkowski K. (1997): Elementary operation approach to state-space realizations of 2-D systems. - IEEE Trans. Circ. Syst. Fund. Theory Appl., Vol. 44, No. 2, pp. 120-129. Zbl0874.93028
  11. Hayton G.E., Walker A.B. and Pugh A.C. (1988): Matrix pencil equivalents of a general polynomial matrix. - Int. J. Contr., Vol. 49, No. 6, pp. 1979-1987. Zbl0683.93015
  12. Hinamoto T. and Fairman F.W. (1984): Realisation of the Attasi state space model for 2-D filters. - Int. J. Syst. Sci., Vol. 15, No. 2, pp. 215-228. Zbl0539.93013
  13. Kaczorek T. (1985): Two-Dimensional Linear Systems. - Berlin: Springer. Zbl0593.93031
  14. Kaczorek T. (1987): Realization problem for general model of two-dimensional linear systems. - Bull. Pol. Acad. Sci. Techn. Sci., Vol. 35, No. 11-12, pp. 633-637. Zbl0667.93018
  15. Kaczorek T. (1988): Singular general model of 2-D systems and its solution. - IEEE Trans. Automat. Contr., Vol. AC-33, No. 11, pp. 1060-1061. Zbl0655.93046
  16. Kaczorek T. (1992): Linear Control Systems, Vols. 1 and 2. - New York: Wiley. Zbl0784.93002
  17. Kaczorek T. (1995): Singular 2-D continuous-discrete linear systems. Dynamics of continuous, discrete and impulse systems. - Adv. Syst. Sci. Appl., Vol. 1, No. 1, pp. 103-108. 
  18. Kaczorek T. (1996): Reachability and controllability of non-negative 2-D Roesser type models. - Bull. Pol. Acad. Sci. Techn. Sci., Vol. 44, No. 4, pp. 405-410. Zbl0888.93009
  19. Kaczorek T. (1997a): Positive realisations of improper transfer matrices of discrete-time linear systems. - Bull. Pol. Acad. Techn. Sci., Vol. 45, No. 2, pp. 277-286. Zbl1065.93523
  20. Kaczorek T. (1997b): Positive realization in canonical form of the 2D Roesser type model. - Proc. Control and Decision Conf., San Diego, pp. 335-336. 
  21. Kaczorek T. (1997c): Realisation problem for positive 2D Roesser model. -Bull. Pol. Acad. Sci. Techn. Sci., Vol. 45, No. 4, pp. 607-619. Zbl0959.93010
  22. Kaczorek T. (1998): Realisation problem for singular 2D linear systems.- Bull. Pol. Acad. Sci. Techn. Sci., Vol. 46, No. 3, pp. 317-330. Zbl0923.93012
  23. Kaczorek T. (2000): Determination of realisations in canonical forms for singular linear. - Proc. Polish-Ukrainian School-Seminar, Solina, Poland, pp. 47-51. 
  24. Klamka J. (1991): Controllability of Dynamical Systems. -Dordrecht: Kluwer. Zbl0732.93008
  25. Kurek J. (1985): The general state-space model for a two-dimensional linear digital system. - IEEE Trans. Automat. Contr., Vol. AC-30, No. 6, pp. 600-602. Zbl0561.93034
  26. Lewis F.L. (1984): Descriptor systems: Decomposition into forward and backward subsystems. - IEEE Trans. Automat. Contr., Vol. AC-29, No. 2, pp. 167-170. Zbl0534.93013
  27. Lewis F.L. (1986): A survey of linear singular systems. - Circ. Syst. Signal Process., Vol. 5, No. 1, pp. 1-36. Zbl0613.93029
  28. Lewis F.L. and Mertzios B.G. (1989): On the analysis of discrete linear time-invariant singular systems. - IEEE Trans. Automat. Contr., Vol. AC-34. 
  29. Luenberger D.G. (1967): Canonical forms for linear multivariable systems. - IEEE Trans. Automat. Contr., Vol. AC-12, No. 63, pp. 290-293. 
  30. Luenberger D.G. (1978): Time-invariant descriptor systems. - Automatica Vol. 14, pp. 473-480. Zbl0398.93040
  31. Ozcaldiran K. and Lewis F.L. (1989): Generalized reachability subspaces for singular systems. - SIAM J. Contr. Optim., Vol. 26, No. 3, pp. 495-510. Zbl0689.93004
  32. Roesser R.B. (1975): A discrete state space model for linear image processing. - IEEE Trans. Automat. Contr., Vol. AC-20, No. 1, pp. 1-10. Zbl0304.68099
  33. Silverman L.M. (1966): Transformation of time- variable systems to canonical (phase-variable) form. - IEEE Trans. Automat. Contr., Vol. AC-11, No. 2, pp. 300-303. 
  34. Valcher M.E. (1997): On the internal stability and asymptotic behavior of 2-D positive systems. - IEEE Trans. Circ. Syst. - I, Vol. 44, No. 7, pp. 602-613. Zbl0891.93046
  35. Żak S.H., Lee E.B. and Lu W.-S. (1986): Realizations of 2-D filters and time delay systems. - IEEE Trans. Circ. Syst., Vol. CAS-33, No. 12, pp. 1241-1244. 

NotesEmbed ?

top

You must be logged in to post comments.