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.
For notational convenience, \(p=(p,\emptyset )\emptyset \in {\mathbb R}\langle X \rangle \) is often abbreviated as \(p=(p,\emptyset )\).
- 2.
The superscript \(\ell \) will be dropped when \(\ell =1\).
- 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.
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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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