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.
@article{bwmeta1.element.bwnjournal-article-amcv13i1p61bwm,
author = {Kaczorek, Tadeusz},
title = {Canonical forms of singular 1D and 2D linear systems},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {13},
year = {2003},
pages = {61-72},
zbl = {1051.93021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv13i1p61bwm}
}
Kaczorek, Tadeusz. Canonical forms of singular 1D and 2D linear systems. International Journal of Applied Mathematics and Computer Science, Tome 13 (2003) pp. 61-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv13i1p61bwm/
[000] Aplevich J.D. (1985): Minimal representations of implicit linear systems. - Automatica, Vol. 21, No. 3, pp. 259-269. | Zbl 0565.93012
[001] Brunovsky P.A. (1970): A classification of linear controllable systems. - Kybernetika, Vol. 6, No. 3, pp. 173-187. | Zbl 0199.48202
[002] Cobb D. (1984): Controllability, observability and duality in singular systems. - IEEE Trans. Automat. Contr., Vol. A-26, No. 12, pp. 1076-1082.
[003] Dai L. (1989): Singular Control Systems. - New York: Springer. | Zbl 0669.93034
[004] Eising R. (1978): Realization and stabilization of 2-D systems. - IEEE Trans. Automat. Contr., Vol. AC-23, No. 5, pp. 795-799. | Zbl 0397.93022
[005] 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. | Zbl 0332.93072
[006] Fornasini E. and Marchesini G. (1978): Doubly-indexed dynamical systems: State-space models and structural properties. - Math. Syst. Theory, Vol. 12, pp. 59-72. | Zbl 0392.93034
[007] Gałkowski K. (1981): The state-space realization of an n-dimensional transfer function. - Int. J. Circ. Theory Appl., Vol. 9, pp. 189-197. | Zbl 0458.94050
[008] 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. | Zbl 0743.94026
[009] 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. | Zbl 0874.93028
[010] 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. | Zbl 0683.93015
[011] 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. | Zbl 0539.93013
[012] Kaczorek T. (1985): Two-Dimensional Linear Systems. - Berlin: Springer. | Zbl 0593.93031
[013] 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. | Zbl 0667.93018
[014] 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. | Zbl 0655.93046
[015] Kaczorek T. (1992): Linear Control Systems, Vols. 1 and 2. - New York: Wiley. | Zbl 0784.93002
[016] 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.
[017] 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. | Zbl 0888.93009
[018] 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. | Zbl 1065.93523
[019] Kaczorek T. (1997b): Positive realization in canonical form of the 2D Roesser type model. - Proc. Control and Decision Conf., San Diego, pp. 335-336.
[020] Kaczorek T. (1997c): Realisation problem for positive 2D Roesser model. -Bull. Pol. Acad. Sci. Techn. Sci., Vol. 45, No. 4, pp. 607-619. | Zbl 0959.93010
[021] Kaczorek T. (1998): Realisation problem for singular 2D linear systems.- Bull. Pol. Acad. Sci. Techn. Sci., Vol. 46, No. 3, pp. 317-330. | Zbl 0923.93012
[022] Kaczorek T. (2000): Determination of realisations in canonical forms for singular linear. - Proc. Polish-Ukrainian School-Seminar, Solina, Poland, pp. 47-51.
[023] Klamka J. (1991): Controllability of Dynamical Systems. -Dordrecht: Kluwer. | Zbl 0732.93008
[024] 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. | Zbl 0561.93034
[025] Lewis F.L. (1984): Descriptor systems: Decomposition into forward and backward subsystems. - IEEE Trans. Automat. Contr., Vol. AC-29, No. 2, pp. 167-170. | Zbl 0534.93013
[026] Lewis F.L. (1986): A survey of linear singular systems. - Circ. Syst. Signal Process., Vol. 5, No. 1, pp. 1-36. | Zbl 0613.93029
[027] 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.
[028] Luenberger D.G. (1967): Canonical forms for linear multivariable systems. - IEEE Trans. Automat. Contr., Vol. AC-12, No. 63, pp. 290-293.
[029] Luenberger D.G. (1978): Time-invariant descriptor systems. - Automatica Vol. 14, pp. 473-480. | Zbl 0398.93040
[030] Ozcaldiran K. and Lewis F.L. (1989): Generalized reachability subspaces for singular systems. - SIAM J. Contr. Optim., Vol. 26, No. 3, pp. 495-510. | Zbl 0689.93004
[031] 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. | Zbl 0304.68099
[032] 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.
[033] 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. | Zbl 0891.93046
[034] 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.