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Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization

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Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 267))

Abstract

A detailed expose of the Hopf algebra approach to interconnected input-output systems in nonlinear control theory is presented. The focus is on input-output systems that can be represented in terms of Chen–Fliess functional expansions or Fliess operators. This provides a starting point for a discrete-time version of this theory. In particular, the notion of a discrete-time Fliess operator is given and a class of parallel interconnections is described in terms of the quasi-shuffle algebra.

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Notes

  1. 1.

    For notational convenience, \(p=(p,\emptyset )\emptyset \in {\mathbb R}\langle X \rangle \) is often abbreviated as \(p=(p,\emptyset )\).

  2. 2.

    The superscript \(\ell \) will be dropped when \(\ell =1\).

  3. 3.

    The same symbol will be used for composition on \({\mathbb R}^m\langle \langle X \rangle \rangle \), \({\mathbb R}^m\langle \langle X_\delta \rangle \rangle \), and \({\mathbb R}^{\ell } \langle \langle X \rangle \rangle \times {\mathbb R}^m\langle \langle X_\delta \rangle \rangle \). It will always be clear which product is being used since the arguments of these products have a distinct notation, namely, c versus \(c_\delta \).

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Acknowledgements

The third author was supported by grant SEV-2011-0087 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. This research was also supported by a grant from the BBVA Foundation.

Author information

Authors and Affiliations

  1. Department of Electrical and Biomedical Engineering, University of Vermont, Burlington, VT, 05405, USA

    Luis A. Duffaut Espinosa

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology – NTNU, NO-7491, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, VA, 23529, USA

    W. Steven Gray

Authors
  1. Luis A. Duffaut Espinosa
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  2. Kurusch Ebrahimi-Fard
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  3. W. Steven Gray
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Correspondence to W. Steven Gray .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology—NTNU, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  2. Institute for the Mathematical Sciences (ICMAT), Madrid, Spain

    María Barbero Liñán

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Duffaut Espinosa, L.A., Ebrahimi-Fard, K., Gray, W.S. (2018). Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_5

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