Integer invariants of 2D autonomous systems

For a linear system of ordinary differential/difference equations (1D systems), some integer invariants, such as the number of inputs, the number of outputs, and the dimension of the state space in a minimal state space representation, play an important role in systems and control theory. Algorithms for calculating these invariants of 1D systems are also available in the literature. However, a complete description of the structure indices is unavailable for linear systems of partial differential/difference equations (nD systems). Some of the works available in the literature are those of generalizing minimal lag descriptions to discrete nD systems (Wood et al., 1998), and generalization of structure indices for discrete nD systems (Wood et al., 2000) by Wood and co-authors. One of the assumptions in these works is that the domain is the n-dimensional grid of natural numbers. We revisit this problem of defining integer invariants for 2D systems, whose domain is the 2-dimensional integer grid, by considering the restriction of 2D systems to sublattices of rank one.