On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently

On the Radius of Convergence of Interconnected Analytic Nonlinear Systems

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and unity feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions on the coefficients of the generating series for the component systems, the radius of convergence of each interconnected system is computed assuming the component systems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the unity feedback connection, it is shown definitively that local convergence is preserved under unity feedback. This had been an open question in the literature

SISO Output Affine Feedback Transformation Group and Its Faa di Bruno Hopf Algebra

The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen-Fliess functional expansions or Fliess operators and are independent of the existence of any state space models. This interconnection, called an output affine feedback connection, is distinguished from conventional output feedback by the presence of a multiplier in an outer loop. Once this transformation group is established, three basic questions are addressed. How can this transformation group be used to provide an explicit Fliess operator representation of such a closed-loop system? Is it possible to use this feedback scheme to do system inversion purely in an input-output setting? In particular, can feedback input-output linearization be posed and solved entirely in this framework, i.e., without the need for any state space realization? Lastly, what can be said about feedback invariants under this transformation group? A final objective of the paper is to describe the Lie algebra of infinitesimal characters associated with the group in terms of a pre-Lie product.Comment: revised manuscript; title and abstract changed; new material adde

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

Entropy of Generating Series for Nonlinear Input-Output Systems and their Interconnections

This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions