Generating Series for Interconnected Nonlinear Systems and the Formal Laplace-Borel Transform

GENERATING SERIES FOR INTERGONNECTED NONLINEAR SYSTEMS AND THE FORMAL LAPLACE-BOREL TRANSFORM Yaqin Li Old Dominion University, 2004 Director: Dr. W. Steven Gray 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 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.