The Choice of the Forms of Lyapunov Functions for a Positive 2D Roesser Model

Kaczorek, Tadeusz

doi:10.2478/v10006-007-0039-7

Abstract

The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^T PA - P is negative definite. The theoretical deliberations will be illustrated by numerical examples.

References

Benvenuti L. and Farina L. (2004): A tutorial on the positive realization problem.IEEE Transactions on Automatic Control, Vol. 49, No. 5, pp. 651-664.10.1109/TAC.2004.826715
Search in Google Scholar
Bose N. K. (1985): Multidimensional Systems Theory Progress, Directions and Open Problems, Dordrecht: D. Reidel Publishing Co.10.1007/978-94-009-5225-6_8
Search in Google Scholar
Farina L. and Rinaldi S. (2000): Positive Linear Systems. Theory and Applications. New York: Wiley.10.1002/9781118033029
Search in Google Scholar
Fornasini E. and Marchesini G. (1978): Double indexed dynamical systems.Mathematical Systems Theory, Vol. 12, pp. 59-72.10.1007/BF01776566
Search in Google Scholar
Fornasini E. and Marchesini G. (1976): State-space realization theory of two- dimensional filters.IEEE Transactions on Automatic Control, Vol. AC-21, pp. 484-491.10.1109/TAC.1976.1101305
Search in Google Scholar
Fornasini E. and Valcher M.E. (1996): On the spectral and combinatorial structure of 2D positive systems.Linear Algebra and Its Applications, Vol. 245, pp. 223-258.10.1016/0024-3795(94)00242-8
Search in Google Scholar
Fornasini E. and Valcher M.E. (1997): Recent developments in 2D positive systems theory.International Journal of Applied Mathematics and Computer Science, Vol. 7, No. 4, pp. 101-123.
Search in Google Scholar
Gałkowski K. (1997): Elementary operation approach to state space realization of 2D systems.IEEE Transaction on Circuits and Systems, Vol. 44, No. 2, pp. 120-129.10.1109/81.554329
Search in Google Scholar
Kaczorek T. (1999): Externally positive 2D linear systems.Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol. 47, No. 3, pp. 227-234.
Search in Google Scholar
Kaczorek T. (1996): Reachability and controllability of non-negative 2D Roesser type models.Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol. 44, No. 4, pp. 405-410.
Search in Google Scholar
Kaczorek T. (2000): Positive 1D and 2D Systems. London: Springer.
Search in Google Scholar
Kaczorek T. (2002): When the equilibrium of positive 2D Roesser model are strictly positive.Bulletin of the Polish Academy of Sciences: Technical Sciences, Vol. 50, No. 3, pp. 221-227.
Search in Google Scholar
Kaczorek T. (1985): Two-Dimensional Linear Systems. Berlin: Springer.10.1007/BFb0005617
Search in Google Scholar
Klamka J. (1999): Controllability of 2D linear systems, In: Advances in Control Highlights of ECC 1999 (P.M. Frank, Ed.), Berlin: Springer, pp. 319-326.
Search in Google Scholar
Klamka J. (1991): Controllability of dynamical systems. Dordrecht: Kluwer.
Search in Google Scholar
Kurek J. (1985): The general state-space model for a two-dimensional linear digital systems.IEEE Transactions on Automatic Control, Vol. -30, No. 2, pp. 600-602.10.1109/TAC.1985.1103998
Search in Google Scholar
Kurek J. (2002): Stability of positive 2D systems described by the Roesser model.IEEE Transactions on Circuits and Systems I, Vol. 49, No. 4, pp. 531-533.10.1109/81.995672
Search in Google Scholar
Roesser R.P. (1975) A discrete state-space model for linear image processing.IEEE Transactions on Automatic Control, Vol. AC-20, No. 1, pp. 1-10.10.1109/TAC.1975.1100844
Search in Google Scholar

The Choice of the Forms of Lyapunov Functions for a Positive 2D Roesser Model

Abstract

Paradigm

My account