Controllability and reachability of 2D positive systems: a graph theoretic approach

When dealing with two-dimensional (2D) discrete state-space models, reachability, controllability and zero-controllability are introduced in two different forms: a local form, which refers to single local states, and a global form, which instead pertains the infinite set of local states lying on a separation set. In this paper, these concepts are investigated in the context of 2D positive systems. Their combinatorial nature suggests a graph theoretic approach to their analysis, as, indeed, to every 2D positive state-space model of dimension n with m inputs one can associate a 2D influence digraph with n vertices and m sources. For all these properties, necessary and sufficient conditions, which refer to the structure of the digraph, are provided. It turns out that while the global reachability index is bounded by the system dimension n, the local reachability index may far exceed the system dimension and even reach n^2/4.