Generating series for nonlinear cascade and feedback systems

Given two analytic nonlinear input-output systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection and feedback connection. In each case, the corresponding generating series is produced, when one exists, and conditions for convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series is generalized to the multivariable setting, and its set of known properties is expanded. In addition, the notion of a feedback product for formal power series is introduced and characterized.