On Analytic Nonlinear Input-output Systems: Expanded Global Convergence and System Interconnections

Functional series representations of nonlinear systems first appeared in engineering in the early 1950’s. One common representation of a nonlinear input-output system are Chen-Fliess series or Fliess operators. Such operators are described by functional series indexed by words over a noncommutative alphabet. They can be viewed as a noncommutative generalization of a Taylor series. A Fliess operator is said to be globally convergent when its radius of convergence is infinite, in other words, when there is no a priori upper bound on both the L1-norm of an admissible input and the length of time over which the corresponding output is well defined. If such bounds are required to ensure convergence, then the Fliess operator is said to be locally convergent with a finite radius of convergence. However, in the literature, a Fliess operator is classified as locally convergent or globally convergent based solely on the growth rate of the coefficients in its generating series. The existing growth rate bounds provide sufficient conditions for global convergence which are very conservative. Therefore, the first main goal of this dissertation is to develop a more exact relationship between the coefficient growth rate and the nature of convergence of the corresponding Fliess operator. This first goal is accomplished by introducing a new topological space of formal power series which renders a Fréchet space instead of the more commonly used ultrametric space. Then, a direct relationship is developed between the nature of convergence of a Fliess operator and its generating series. The second main goal of this dissertation is to show that the global convergence of Fliess operators is preserved under the nonrecursive interconnections, namely the parallel sum and product connections and the cascade connection. This fact had only been understood previously in a narrow sense based on the more conservative tests for global convergence