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

Tadeusz Kaczorek

International Journal of Applied Mathematics and Computer Science (2010)

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

Abstract

top
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.

How to cite

top

Tadeusz 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
  1. Ansaklis, P.J. and Michel, N. (1997). Linear Systems, McGrowHill, New York, NY. 
  2. 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
  3. 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
  4. 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
  5. 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
  6. Fornasini, E. and Marchesini, G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties, Mathematical System Theory 12: 59-72. Zbl0392.93034
  7. Kaczorek, T. (1992). Linear Control Systems, Vol. 2, Wiley, New York, NY. Zbl0784.93002
  8. Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London. Zbl1005.68175
  9. Kaczorek, T. (2007). Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer-Verlag, London. Zbl1114.15019
  10. Kailath, T. (1980). Linear Systems, Prentice Hall, Englewood Cliffs, NJ. Zbl0454.93001
  11. 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. 
  12. 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
  13. 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
  14. 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
  15. Ntogramatzis, L. (2010). A geometric theory for 2-D systems, Multidimensional Systems and Signal Processing, (submitted). 
  16. 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
  17. Wonham, W.M. (1979). Linear Multivariable Control: A Geometric Approach, Springer, New York, NY. Zbl0424.93001
  18. Żak S. (2003). Systems and Control, Oxford University Press, New York, NY. 

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.