State space realization of 2D finite dimensional behaviours

This paper deals with the state space realization of autonomous autoregressive two-dimensional (2-D) systems in the context of the behavioural approach. An arbitrary autoregressive 2-D system $\Sigma $ can be viewed as the sum of an externally controllable subsystem $\Sigma ^c $ with an autonomous one $\Sigma ^a $, so that a state space realization of $\Sigma $ can be obtained by separately realizing $\Sigma ^c $ and $\Sigma ^a $. Since a procedure for realizing externally controllable systems in state/driving-variable form is already available in the literature, the general realization problem is easily reduced to the autonomous case. Here, some properties of finite-dimensional autonomous systems are discussed, allowing for a realization procedure that uses the Gröbner bases theory.