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 repres entation of causal transfer matrices. In the two-dimensional (2D) case ther e is much freedom in defining proper rational matrices. This freedom is con nected 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 th e properness of a rational matrix is proposed. Finally, this algorithm is u sed for determining all possible causal input/output (I/O) representations of a behavior given by a kernel representation.