Date of Award

Summer 2021

Document Type


Degree Name

Doctor of Philosophy (PhD)


Electrical & Computer Engineering


Electrical Engineering

Committee Director

W. Steven Gray

Committee Member

Oscar R. Gonzalez

Committee Member

Luis A. Duffaut Espinosa

Committee Member

Alexander Schmeding


The problem statement for this dissertation is two-fold. The first problem considered is when does a Chen-Fliess series in an additive static feedback connection with a formal static map yield a closed-loop system with a Chen-Fliess series expansion? This work proves that such a closed-loop system always has a Chen-Fliess series representation. Furthermore, an algorithm based on the Hopf algebras for the shuffle group and the dynamic output feedback group is designed to compute the generating series of the closed-loop system. It is proved that the additive static feedback connection preserves local convergence and relative degree, but a counterexample shows that the additive static feedback does not preserve global convergence in general. This dissertation then pivots to the second problem considered, the shuffle rationality problem. The notion of shuffle rationality and shuffle recognizability are first defined, akin to the traditional notion of rational series in bilinear systems theory. It is proved that shuffle rationality and shuffle recognizability coincide, similar to Schutzenberger’s theorem. An equivalent characterization of shuffle rational series is provided in terms of a canonical state space realization. Specifically, it is shown that a shuffle rational series corresponds to a realization of a nilpotent bilinear system cascaded with a static rational map.


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