Date of Award

Spring 2004

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Electrical/Computer Engineering

Committee Director

W. Steven Gray

Committee Member

Oscar R. Gonzalez

Committee Member

Matthias Kawski

Committee Member

Brett Newman

Committee Member

John J. Swetits

Abstract

Formal power series methods provide effective tools for nonlinear system analysis. For a broad range of analytic nonlinear systems, their input-output mapping can be described by a Fliess operator associated with a formal power series. In this dissertation, the inter connection of two Fliess operators is characterized by the generating series of the composite system. In addition, the formal Laplace-Borel transform of a Fliess operator is defined and its fundamental properties are presented. The formal Laplace-Borel transform produces an elegant description of system interconnections in a purely algebraic context.

Specifically, four basic interconnections of Fliess operators are addressed: the parallel, product, cascade and feedback connections. For each interconnection, the generating series of the overall system is given, and a growth condition is developed, which guarantees the convergence property of the output of the corresponding Fliess operator.

Motivated by the relationship between operations on formal power series and system interconnections, and following the idea of the classical integral Laplace-Borel transform, a new formal Laplace-Borel transform of a Fliess operator is proposed. The properties of this Laplace-Borel transform are provided, and in particular, a fundamental semigroup isomorphism is identified between the set of all locally convergent power series and the set of all well-defined Fliess operators.

A software package was produced in Maple based on the ACE package developed by the ACE group in Université de Marne-la Vallée led by Sébastien Veigneau. The ACE package provided the binary operations of addition, concatenation and shuffle product on the free monoid of formal polynomials. In this dissertation, the operations of composition, modified composition, chronological products and the evaluation of Fliess operators are implemented in software. The package was used to demonstrate various aspects of the new interconnection theory.

DOI

10.25777/rt2a-5498

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