Shuffle algorithm for two-dimensional singular systems with a Fornasini-Marchesini model

The condition for existence of solution for 2-D singular systems is usually stated as the nonsingularity of a matrix pencil in two complex variables. The determinant of this matrix pencil equals the characteristic polynomial of the system. Although many authors assume this regularity condition to be satisfied, there has been no apparent effort towards developing computational methods to evaluate such a condition. In this paper we present several, easy to evaluate, sufficient conditions to determine whether the regularity condition is satisfied. The main contribution of the paper is a new 2-D shuffle algorithm which is a natural extension of that of Luenberger [5]. The algorithm is a useful tool to test for regularity and should also contribute to the study of the geometric structure of 2-D singular systems.