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Shuffle-Rational Series: Recognizability and Realizations

The notion of a shuffle-rational formal power series is introduced. Then two equivalent characterizations are presented, one in terms of an analogue of Schützenberger’s recognizability of a series, and the other in the context of state space realizations of nonlinear input-output systems represented by Chen-Fliess series. An underlying computational framework is also described which employs the Hopf algebra associated with the shuffle group. As an application, it is shown how to model a bilinear system with output saturation in this context.