Fliess Operators in Cascade and Feedback Systems

Given two nonlinear input-output systems represented as Fliess operators, this paper considers properties of the cascade and feedback interconnected systems. The cascade system is characterized via the composition product for formal power series. In the multivariable setting this product is shown to be generally well defined and continuous in its first argument. Then a condition is introduced under which the composition product preserves rationality and local convergence. In preparation for the feedback analysis, it is also shown that the composition product produces a contractive mapping on the set of all formal power series using the familiar ultrametric. Next, the feedback connection is considered in the special case where inputs are supplied from an exosystem which is itself a Fliess operator. In particular, a sufficient condition is given under which a unique solution to the feedback equation is known to exist. Then the closed-loop system is characterized as the output of new Fliess operator when a certain series factorization property is available. This leads to an implicit characterization of the feedback product for formal power series.