Controllable and autonomous nD linear systems

Description/Abstract

The theory of multidimensional systems suffers in certain areas from a lack of development of fundamental concepts. Using the behavioural approach, the study of linear shift-invariant nD systems can be encompassed within the well-established framework of commutative algebra, as previously shown by Oberst. We consider here the discrete case. In this paper, we take two basic properties of discrete nD systems, controllability and autonomy, and show that they have simple algebraic characterizations. We make several non-trivial generalizations of previous results for the 2D case. In particular we analyse the controllable-autonomous decomposition and the controllable subsystem of autoregressive systems. We also show that a controllable nD subsystem of $(k^q)^{(\Ints^n)}$ is precisely one which is minimal in its transfer class.