Fractional Fliess operators: Two approaches

Winter-Arboleda, Irina M.; Gray, W. Steven; Espinosa, Luis A. Duffaut

doi:10.1109/ciss.2015.7086831

Cited by 18 publications

(8 citation statements)

References 14 publications

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“…The notion of Gevrey series also have applications in the context of the theory of Fliess operators (see [23,24]). These operators, as input-output map of a nonlinear control system, can be applied in biology, for example to the modeling of homeostasis.…”

mentioning

confidence: 99%

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Mégret¹,

Demongeot²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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mentioning

confidence: 99%

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Mégret¹,

Demongeot²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…The set of all locally convergent generating series is denoted by R ℓ LC X . The supremum of the set of all max{R, T } for which a Fliess operator F c is a well-defined mapping from Winter-Arboleda(2019), Winter-Arboleda, et al(2015)]. The infimum of all the γ ∈ [0, 1[ such that (4) is satisfied for a series c ∈ R ℓ X is called the Gevrey order of the series c. A series c ∈ R ℓ X obeying the growth condition in (4) is called a globally convergent series.…”

Section: Fliess Operatorsmentioning

confidence: 99%

“…Thus, the space S ∞ is separable. A countable dense topological subspace is constructed in [Winter-Arboleda(2019)]. Hence, the limit space S ∞ is better behaved than the spaces S ∞ (R) from which it is constructed.…”

Section: Interconnections Of Fliess Operatorsmentioning

confidence: 99%

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Formal Power Series Approach to Nonlinear Systems with Additive Static Feedback

Venkatesh¹,

Gray²

2021

Preprint

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The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen-Fliess series and an additive static output feedback is applied. The first step is to consider the so called Wiener-Fliess connection consisting of a Chen-Fliess series followed by a memoryless function. Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. To explicitly compute the generating series, two Hopf algebras are needed, the existing output feedback Hopf algebra used to describe dynamic output feedback, and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener-Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener-Fliess composition product and the additive static feedback product are completely characterized.

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“…If there exist real numbers K c , M c > 0 and r ∈ [0, 1) such that the coefficients of the generating series c = η∈X * (c, η)η ∈ R ℓ X satisfying the growth bound [20] for details. In this case, the operator is said to be globally convergent, and the set of all series satisfying (13) is designated by R ℓ GC X .…”

Section: Chen-fliess Seriesmentioning

confidence: 99%

Evaluating Generating Functions for Periodic Multiple Polylogarithms via Rational Chen–Fliess Series

Ebrahimi‐Fard

Gray

Manchon

2020

Springer Proceedings in Mathematics &Amp; Statistics

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The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.

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Fractional Fliess operators: Two approaches

Cited by 18 publications

References 14 publications

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Formal Power Series Approach to Nonlinear Systems with Additive Static Feedback

Evaluating Generating Functions for Periodic Multiple Polylogarithms via Rational Chen–Fliess Series

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