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Delays and Networked Control Systems pp 113–131Cite as

Output Feedback Event-Triggered Control

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Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 6))

Abstract

Event-triggered control has been proposed as an alternative implementation to conventional time-triggered approach in order to reduce the amount of transmissions. The idea is to adapt transmissions to the state of the plant such that the loop is closed only when it is needed according to the stability or/and the performance requirements. Most of the existing event-triggered control strategies assume that the full state measurement is available. Unfortunately, this assumption is often not satisfied in practice. There is therefore a strong need for appropriate tools in the context of output feedback control. Most existing works on this topic focus on linear systems. The objective of this chapter is to first summarize our recent results on the case where the plant dynamics is nonlinear. The approach we follow is emulation as we first design a stabilizing output feedback law in the absence of sampling; then we consider the network and we synthesize the event-triggering condition. The latter combines techniques from event-triggered and time-triggered control. The results are then proved to be applicable to linear time-invariant (LTI) systems as a particular case. We then use these results as a starting point to elaborate a co-design method, which allows us to jointly construct the feedback law and the triggering condition for LTI systems where the problem is formulated in terms of linear matrix inequalities (LMI). We then exploit the flexibility of the method to maximize the guaranteed minimum amount of time between two transmissions. The results are illustrated on physical and numerical examples.

R. Postoyan—This work is partially supported by the ANR under the grant COMPACS (ANR-13-BS03-0004-02).

D. Nešić—This work is also supported by the Australian Research Council under the Discovery Projects.

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Notes

  1. 1.

    A continuous function \(\displaystyle \gamma : \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is of class \(\displaystyle \mathscr {K}\) if it is zero at zero, strictly increasing, and it is of class \(\displaystyle \mathscr {K}_{\infty }\) if in addition \(\displaystyle \gamma (s) \rightarrow \infty \) as \(\displaystyle s \rightarrow \infty \).

  2. 2.

    A continuous function \(\displaystyle \gamma : \mathbb {R}_{\ge 0} \times \mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is of class \(\displaystyle \mathscr {KL}\) if for each \(\displaystyle t \in \mathbb {R}_{\ge 0}\), \(\displaystyle \gamma (.,t)\) is of class \(\displaystyle \mathscr {K}\), and, for each \(\displaystyle s \in \mathbb {R}_{\ge 0}\), \(\displaystyle \gamma (s,.)\) is decreasing to zero.

  3. 3.

    The symbol \(\displaystyle \star \) denotes symmetric blocks while \(\displaystyle \varSigma (.)\) stands for \(\displaystyle (.) + (.)^{T}\).

  4. 4.

    In view of the Schur complement of LMI (7.30), we deduce that \(\displaystyle \left[ \begin{array}{cc}\varvec{Y} &{} \mathbb {I}_{n_{p}} \\ \mathbb {I}_{n_{p}} &{} \varvec{X} \end{array}\right] >0\) which implies that \(\displaystyle \varvec{X} - \varvec{Y}^{-1}>0\) and thus, \(\displaystyle \mathbb {I}_{n_{p}} - \varvec{X}\varvec{Y}\) is nonsingular. Hence, the existence of nonsingular matrices U, V is always ensured.

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Authors and Affiliations

  1. Université de Lorraine, CRAN, UMR 7039 and CNRS, CRAN, UMR 7039, Vandoeuvre-les-Nancy, France

    Mahmoud Abdelrahim, Romain Postoyan & Jamal Daafouz

  2. Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, VIC, 3010, Australia

    Dragan Nešić

Authors
  1. Mahmoud Abdelrahim
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  2. Romain Postoyan
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  3. Jamal Daafouz
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  4. Dragan Nešić
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Corresponding author

Correspondence to Mahmoud Abdelrahim .

Editor information

Editors and Affiliations

  1. LAAS—UPR CNRS, Université de Toulouse, Toulouse, France

    Alexandre Seuret

  2. CRISTAL—UMR CNRS, École centrale de Lille, Villeneuve d'Ascq Cedex, France

    Laurentiu Hetel

  3. Université de Lorraine, CRAN—UMR CNRS, INPL,, Vandoeuvre-lès-Nancy Cedex, France

    Jamal Daafouz

  4. School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden

    Karl H. Johansson

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Abdelrahim, M., Postoyan, R., Daafouz, J., Nešić, D. (2016). Output Feedback Event-Triggered Control. In: Seuret, A., Hetel, L., Daafouz, J., Johansson, K. (eds) Delays and Networked Control Systems . Advances in Delays and Dynamics, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-32372-5_7

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  • DOI: https://doi.org/10.1007/978-3-319-32372-5_7

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