Two-Dimensional Proper Rational Matrices and Causal Input/Output Representations of Two-Dimensional Behavioral Systems

The concept of proper rational matrix is strictly connected with the representation of causal transfer matrices. In the two-dimensional (2D) case there is much freedom in defining proper rational matrices. This freedom is connected to the fact that past and future in the 2D case can be determined by a 2D cone. In this way the concept of rational matrix which is proper with respect to a cone can be introduced. Moreover, an algorithm that checks the properness of a rational matrix is proposed. Finally, this algorithm is used for determining all possible causal input/output (I/O) representations of a behavior given by a kernel representation.