Realization and Modelling in System Theory pp 325-331 | Cite as

# Computation of the fundamental matrix sequence and the Cayley-Hamilton theorem in singular 2-D systems

## Abstract

The study of 2-D singular or generalized systems was the objective of a number of papers recently. The 2-D generalized state-space models may provide a more suitable structure for a number of applications, such as processing of images, and in heat flow exchangers, long trasmission lines, neural networks, multivariable networks and interconnected 2-D systems. Recently we have defined (with F.L. Lewis) the fundamental matrix of the 2-D generalized Roesser model and demonstrated a number of properties and its fundamental importance in the analysis of 2-D generalized systems. This paper refers to a computational technique for the calculation of the fundamental matrix sequence. Using a three dimension recursion algorithm, and exploing the Toeplitz and Hessenberg forms of certain auxiliary matrices, each matrix of the fundamental matrix sequence is expressed via an ARMA model. Moreover the generalized 2-D Caley-Hamilton theorem is presented, in terms of the fundamental matrix sequence. The presented algebraic results provide useful tools for the analysis and design of 2-D generalized state-space systems.

### Keywords

Lution Acoustics## Preview

Unable to display preview. Download preview PDF.

### References

- [1]N.K. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, 1982.Google Scholar
- [2]D.F. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, Englewood Cliffs, N.J., 1984.Google Scholar
- [3]A.N. Venetsanopoulos, “Digital image processing and analysis”, in Signal Processing (Eds. T.S. Durrani and J.L. Lacoume), North-Holland, 1886.Google Scholar
- [4]J.F. Delansky, “Some synthesis methods for adjustable networks using multivariable techniques”, IEEE Trans. Circuit Theory, vol. 7, pp. 251–260, 1960.Google Scholar
- [5]D.C. Youla, J.D. Phodes and P.C. Marston, “Recent deve opments in the synthesis of a class of lumped-distributed filters, Int. J. Control., vol. 1, pp. 59–80, 1973.Google Scholar
- [6]W. Marszalek, “On modelling of distributed proceses with two-dimensional discrete linear equations”, Rozprawy Elektrotechniczne, vol. 33, pp. 627–640, 1987.Google Scholar
- [7]W. Marszalek, “On solving some heat exchangers problems via image processing equations”, Archiwum Termodynamiki, vol. 8, pp. 55–71, 1987.Google Scholar
- [8]G. Garibotto and R. Molpen, “Two-dimensional recursive filtering in seismic signal processing”, Proc. European Signal Processing Conference, pp. 95–96, Lausanne, Switzerland, Sept. 1980.Google Scholar
- [9]E.A. Robinson and S. Treitel, Geophysical Signal Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1980.Google Scholar
- [10]F.L. Lewis, “A survey linear singular systems”, Circuits, Systems and Signal Processing, vol. 5, No. 1, pp. 3–36, 1986.CrossRefGoogle Scholar
- [11]Special issue on Recent Advances in Singular Systems, Circuits, Systems and Signal Processing, (Eds. F.L. Lewis and B.G. Mertzios), vol. 8, 1989, in press.Google Scholar
- [12]R.P. Roesser, “A discrete state-space model for linear image processing”, IEEE Trans. Automat. Control., vol. AC-20, No. 1, pp. 1–10, Feb. 1975.CrossRefGoogle Scholar
- [13]S.-Y. Kung, B.C. Lévy, M. Morf, and T. Kailath, “New results in 2-D systems theory, Part II: 2-D state-space models, realization and the notions of controllability, observability and minimality, Proc. IEEE, vol. 65, No. 6. pp. 945–961, June 1977.CrossRefGoogle Scholar
- [14]D.G. Luenberger, “Dynamic equations in descriptor form, IEEE Trans. Automat. Control, vol. AC-22, No. 3, pp. 312–321, June 1977.CrossRefGoogle Scholar
- [15]F.L. Lewis and B.G. Mertzios, “On the analysis of two-dimensional discrete singular systems”. Submitted for publication.Google Scholar
- [16]B.G. Mertzios and F.L. Lewis, “An algorithm for the computation of the transfer function matrix of generalized 2-D systems”, Circuits, Systems, and Signal Proc., vol. 7, no. 4, pp. 459–466, 1988.CrossRefGoogle Scholar
- [17]B.W. Dickinson, “Efficient solution of linear equations with banded Toeplitz matrices”, IEEE Trans. Acoustics, Speech, Signal Proc., vol. ASSP-27, pp. 421–423, Aug. 1979.CrossRefGoogle Scholar
- [18]B.G. Mertzios and F.L. Lewis, “Fundamental matrix of discrete singular systems”, Circuits, Systems, and Signal Proc., 1989, in press.Google Scholar
- [19]F.L. Lewis, “Further remarks on the Cayley-Hamilton theorem and Leverrier method for the matrix pencil (sE-A)”, IEEE Trans. Automat. Control. vol. AC-31, pp. 869–870, Sept. 1986.CrossRefGoogle Scholar
- [20]B.G. Mertzios and M.A. Christodoulou, “On the generalized Cayley-Hamilton theorem”, IEEE Trans. Automat. Control, vol. AC-31, pp. 156–157, Feb. 1986.CrossRefGoogle Scholar
- [21]T. Ciftsibasi and O. Yuksel, “On the Ckyley-Hamilton theorem for two-dimensional systems”, IEEE Trans. Automat. Control, vol. AC-27, No. 1, pp. 193–194, Feb. 1982.CrossRefGoogle Scholar