Controllability and reachability of 2-D positive system: A graph theoretic approach

Abstract—When dealing with two-dimensional (2-D) discrete state-space models, controllability properties 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 2-D positive systems by means of a graph theoretic approach. For all these properties, necessary and sufficient conditions, which refer to the structure of the digraph, are provided. While the global reachability index is bounded by the system dimension, the local reachability index may far exceed the system dimension. Upper bounds on the local reachability index for some special classes of positive systems are finally derived. Index Terms—Controllability, finite memory systems, influence digraph, reachability, strong connectedness, two-dimensional (2-D) positive systems, zero controllability. I