State-space descriptions and observability properties of 2D finite-dimensional autonomous behaviors

Finite-dimensional autonomous behaviors (SIAM J. Control Optimiz. 31 (1993) 1502; IEEE Trans. Circuits Systems Part I, CAS1-47 3 (2000) 290) are the sets of solutions of certain two-dimensional (2D) di8erence equations, endowed with the property of constituting #nite-dimensional vector spaces. Among all possible types of representations for this class of behaviors, we will be speci#cally interested in two of them: kernel descriptions (corresponding to right factor prime polynomial matrix operators) or 2D state-space descriptions (associated with a pair of nonsingular commuting system matrices). In this paper, we explore the observability property of these state-space representations, and analyze the algebraic connections between the properties of the kernel descriptions and the properties of the corresponding state-space realizations.