Algebraic structure of discrete-time polynomial systems

In this paper we present some new results on controllability and realization of discrete-time, finite-dimensional, constant, polynomial systems. We study the realization problem of both systems linear-in-state and systems quadratic-in-state. Explicit existence criteria for span-canonical realizations as well as space isomorphism theorems are obtained. Combining the methods used for realization of systems linear-in-state and systems quadratic-in-state we can realize any polynomial systems which may be realized by construction of a generalized Hankel matrix. A sufficient condition as well as a necessary and sufficient condition for controllability of polynomial systems are given.