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Nonlinear Sampled-Data Stabilization with Delays

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Delays and Interconnections: Methodology, Algorithms and Applications

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 10))

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Abstract

In this work, how sampling can be instrumental for stabilizing nonlinear dynamics with delays is discussed through several approaches developed by the authors in a comparative perspective with respect to the existing literature. Performances and computational aspects are illustrated through academic examples.

L2S (CNRS and Université Paris-Sud) and DIAG (Università di Roma ‘La Sapienza’) with mobility support from the Université Franco-Italienne/Università Italo-Francese (UFI/UIF).

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Notes

  1. 1.

    Assuming the delay free dynamics forward complete ensures that the delayed one (1) is too: \(\forall x_0\) and \(u\in M^{[-\tau , \infty )}_U\) the solution x(t) of (1) with initial condition \(x(0) = x_0 \in \mathbb {R}^n\) exists \(\forall t\ge 0\).

  2. 2.

    \(L_f\) denotes the Lie derivative operator, \(L_f = \sum _{i= 1}^n f_i(\cdot )\frac{\partial }{\partial x_i}\). \(e^{L_f} \)(or \(e^f\), when no confusion arises) denotes the associated Lie series operator, \(e^{\mathrm {L}_f} := 1 + \sum _{i \ge 1}\frac{L_f^i}{i!}\).

  3. 3.

    A function \(R(x,\delta )= O(\delta ^p)\) is said of order \(\delta ^p; p \ge 1\) if whenever it is defined it can be written as \(R(x, \delta ) = \delta ^{p-1}\tilde{R}(x, \delta )\) and there exist a function \({\theta \in \mathcal {K}_{\infty }}\) and \(\delta ^* >0\) s.t. \(\forall \delta \le \delta ^*\), \(| \tilde{R} (x, \delta )| \le \theta (\delta )\).

  4. 4.

    \(\circ \) denotes the composition of operators and functions.

  5. 5.

    Mappings and dynamics are parameterized by \(\delta \) as indicated with superscript \((\cdot )^{\delta }\).

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Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Informatica, Automatica e Gestionale, DIAG Università di Roma “La Sapienza”, via Ariosto 25, 00185, Roma, Italy

    Salvatore Monaco

  2. Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506) CNRS; 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France

    Dorothée Normand-Cyrot

  3. DIAG (Università di Roma ‘La Sapienza’), Rome, Italy

    Mattia Mattioni

Authors
  1. Salvatore Monaco
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  2. Dorothée Normand-Cyrot
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  3. Mattia Mattioni
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Corresponding author

Correspondence to Dorothée Normand-Cyrot .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Rifat Sipahi

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Monaco, S., Normand-Cyrot, D., Mattioni, M. (2019). Nonlinear Sampled-Data Stabilization with Delays. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_19

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