An algorithm is developed for the computation of the transfer function matrix of a two-dimensional system, which is given in its state-space form, without inverting a polynomial matrix. A new transformation has been considered so that the well known Fadeeva's algorithm for regular systems can be used as the basisfor the derivation of the present algorithm. The above transformation can be generally used in the reduction of many two-dimensional problems to the corresponding one-dimensional ones. The algorithm presented is well-suited for computer use.
I. Zntvoduction
Two-dimensional
(2-D) systems have drawn considerable attention in recent years, since they provide the mathematical framework for the study of 2-D digital filters which find numerous crucial applications in image processing, medical imaging and processing of geophysical and seismic data (l-3).
This paper establishes a new algorithm for the direct computation of the transfer function matrix of a linear, shift-invariant, discrete, multi-variable 2-D system, from its state-space description, without inverting a polynomial matrix in two variables. This problem has been considered by Koo and Chen (4) who extended Fadeeva's algorithm (5) for the inversion of the resolvent matrix (d-A) in the 2-D case. Moreover, a formula has been presented which allows the determination of the transfer function matrix of a multivariable 2-D system in terms of the state transition matrix and the characteristic equation ( ). The proposed algorithm in the present paper uses the simple, well known 1-D Fadeeva's algorithm (also called Leverrier's algorithm) and does not involve the 2-D nature of the system and the relevant complicated notation. The algorithm reduces computational cost and is well suited for computer use. It is useful in the analysis, synthesis and control of 2-D and multidimensional systems, since it can be extended to more dimensions.
ZZ. Algorithm
Consider the linear