Doubly-Rational Series: Recognizability and Realizations

The goal of this paper is to introduce the notion of a doubly-rational formal power series and two equivalent characterizations, one in terms of an analogue of recognizability of a series, and the other in the context of state space realizations of nonlinear input-output systems represented by Chen-Fliess series. As an application, it is shown how to model a bilinear system with output saturation in this context.