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Prediction-Based Control of Linear Systems by Compensating Input-Dependent Input Delay of Integral-Type

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Recent Results on Nonlinear Delay Control Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 4))

Abstract

This study addresses the problem of delay compensation via a predictor-based output feedback for a class of linear systems subject to input delay which itself depends on the input. The equation defining the delay is implicit and involves past values of the input through an integral relation, the kernel of which is a polynomial function of the input. This modeling represents systems where transport phenomena take place at the inlet of a system involving a nonlinearity, which frequently occurs in the processing industry. The conditions of asymptotic stabilization require the magnitude of the feedback gain to comply with the initial conditions. Arguments for the proof of this novel result include general Halanay inequalities for delay differential equations and take advantage of recent advances in backstepping techniques for uncertain or varying delay systems.

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Notes

  1. 1.

    The delay is positive and, besides, its derivative can be expressed as \(\dot{D}(t) = 1- \varphi (u(t))/(\varphi (u(t-D))) < 1\) which guarantees strict causality.

  2. 2.

    This is the case if \(\dot{D} < 1\).

  3. 3.

    Interestingly, a similar condition is often stated in Linear Matrix Inequality approaches, such as [28] for example, where the delay is also assumed to be time-differentiable.

  4. 4.

    More precisely, in [11], this result is stated for \(a > b >0\).

  5. 5.

    The case when x is identically 0 is trivial. The continuity (and even more) is obtained by assuming \(\psi \) is smooth enough.

  6. 6.

    \(\gamma \ge 0\) is the unique solution on \([0,\infty [\) of \( a - \gamma = b \mathrm{exp}(\gamma \overline{D})\).

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Author information

Authors and Affiliations

  1. CNRS at GIPSA-lab, Control Department, 11 Rue des Mathématiques, 38000, Grenoble, France

    Delphine Bresch-Pietri

  2. Centre Automatique et Systèmes at MINES ParisTech, 60 Boulevard Saint-Michel, 75006, Paris, France

    Nicolas Petit

Authors
  1. Delphine Bresch-Pietri
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  2. Nicolas Petit
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Correspondence to Delphine Bresch-Pietri .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Iasson Karafyllis

  2. Department of Mathematics, Louisiana State University (LSU), Baton Rouge, Louisiana, USA

    Michael Malisoff

  3. CNRS-CentraleSupélec-Université Paris Sud, EPI Inria “DISCO”, Laboratory of Signals and Systems (L2S, UMR CNRS 8506), Gif-sur-Yvette, France

    Frederic Mazenc

  4. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L’Aquila, Italy

    Pierdomenico Pepe

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Bresch-Pietri, D., Petit, N. (2016). Prediction-Based Control of Linear Systems by Compensating Input-Dependent Input Delay of Integral-Type. In: Karafyllis, I., Malisoff, M., Mazenc, F., Pepe, P. (eds) Recent Results on Nonlinear Delay Control Systems. Advances in Delays and Dynamics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-18072-4_4

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  • DOI: https://doi.org/10.1007/978-3-319-18072-4_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18071-7

  • Online ISBN: 978-3-319-18072-4

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