A polynomial matrix approach to the structural properties of 2D positive systems

Reachability and observability of two-dimensional (2D) discrete state-space models are introduced in two different forms: a local form, which refers to single local states, and a global form, which pertains to the infinite set of local states lying on a separation set [M. Bisiacco, State and output feedback stabilizability of 2D systems, IEEE Trans. Circ. Syst., CAS-32 (1985) 1246–1249; E. Fornasini, G. Marchesini, Global properties and duality in 2-D systems, Syst. Control Lett. 2 (1) (1982) 30–38]. While local reachability and observability can be naturally characterized by resorting to classical state space techniques, their global counterparts are better addressed by means of polynomial techniques. In this paper, reachability and observability are introduced in the context of 2D positive systems and their global versions investigated via a polynomial approach. Necessary and sufficient conditions for the existence of these properties are provided and, in particular, polynomial canonical forms for globally reachable/observable positive systems with scalar inputs/scalar outputs are provided.