State Space Realization of 2-D 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 Σ can be viewed as the sum of an externally controllable subsystem Σ ᶜ with an autonomous one Σ ᵃ, so that a state space realization of Σ can be obtained by separately realizing Σ ᶜ and Σ 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.