# Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

International Journal of Applied Mathematics and Computer Science (2010)

- Volume: 20, Issue: 3, page 507-512
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topTadeusz Kaczorek. "Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems." International Journal of Applied Mathematics and Computer Science 20.3 (2010): 507-512. <http://eudml.org/doc/208003>.

@article{TadeuszKaczorek2010,

abstract = {The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.},

author = {Tadeusz Kaczorek},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {common canonical form; similarity transformation; 2D linear system; state feedback},

language = {eng},

number = {3},

pages = {507-512},

title = {Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems},

url = {http://eudml.org/doc/208003},

volume = {20},

year = {2010},

}

TY - JOUR

AU - Tadeusz Kaczorek

TI - Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

JO - International Journal of Applied Mathematics and Computer Science

PY - 2010

VL - 20

IS - 3

SP - 507

EP - 512

AB - The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.

LA - eng

KW - common canonical form; similarity transformation; 2D linear system; state feedback

UR - http://eudml.org/doc/208003

ER -

## References

top- Ansaklis, P.J. and Michel, N. (1997). Linear Systems, McGrowHill, New York, NY.
- Basile, G. and Marro, G. (1969). Controlled and conditioned invariant subspaces in linear system theory, Journal of Optimization Theory and Applications 3(5): 306-315. Zbl0172.12501
- Basile, G. and Marro, G. (1982). Self-bounded controlled invariant subspaces: A straightforward approach to constrained controllability, Journal of Optimization Theory and Applications 38(1): 71-81. Zbl0471.93008
- Conte, G. and Perdon, A. (1988). A geometric approach to the theory of 2-D systems, IEEE Transactions on Automatics Control AC-33(10): 946-950. Zbl0658.93021
- Conte, G., Perdon, A. and Kaczorek, T. (1991). Geometric methods in the theory of singular 2D linear systems, Kybernetika 27(3): 262-270. Zbl0746.93021
- Fornasini, E. and Marchesini, G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties, Mathematical System Theory 12: 59-72. Zbl0392.93034
- Kaczorek, T. (1992). Linear Control Systems, Vol. 2, Wiley, New York, NY. Zbl0784.93002
- Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. Zbl1005.68175
- Kaczorek, T. (2007). Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer-Verlag, London. Zbl1114.15019
- Kailath, T. (1980). Linear Systems, Prentice Hall, Englewood Cliffs, NJ. Zbl0454.93001
- Karmanciolu, A. and Lewis, F.L. (1990). A geometric approach to 2-D implicit systems, Proceedings of the 29th Conference on Decision and Control, Honolulu, HI, USA.
- Karmanciolu, A. and Lewis, F.L. (1992). Geometric theory for the singular Roesser model, IEEE Transactions on Automatics Control AC-37(6): 801-806. Zbl0755.93014
- Kurek, J. (1985). The general state-space model for a twodimensional linear digital systems, IEEE Transactions on Automatics Control AC-30(6): 600-602. Zbl0561.93034
- Malabre, M., Martínez-García, J. and Del-Muro-Cuéllar, B. (1997). On the fixed poles for disturbance rejection, Automatica 33(6): 1209-1211. Zbl0879.93007
- Ntogramatzis, L. (2010). A geometric theory for 2-D systems, Multidimensional Systems and Signal Processing, (submitted).
- Roesser, R.P. (1975). A discrete state-space model for linear image processing, IEEE Transactions on Automatic Control AC-20(1): 1-10. Zbl0304.68099
- Wonham, W.M. (1979). Linear Multivariable Control: A Geometric Approach, Springer, New York, NY. Zbl0424.93001
- Żak S. (2003). Systems and Control, Oxford University Press, New York, NY.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.