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On the Codimension of the Singularity at the Origin for Networked Delay Systems Read chapter Read first chapter

2016 | OriginalPaper | Chapter

7. Output Feedback Event-Triggered Control

Authors : Mahmoud Abdelrahim, Romain Postoyan, Jamal Daafouz, Dragan Nešić

Published in: Delays and Networked Control Systems

Publisher: Springer International Publishing

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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.

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previous chapter On the Stability Analysis of Sampled-Data Systems with Delays
next chapter Stabilization by Quantized Delayed State Feedback
Footnotes
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
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
The symbol \(\displaystyle \star \) denotes symmetric blocks while \(\displaystyle \varSigma (.)\) stands for \(\displaystyle (.) + (.)^{T}\).
 
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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Metadata
Title
Output Feedback Event-Triggered Control
Authors
Mahmoud Abdelrahim
Romain Postoyan
Jamal Daafouz
Dragan Nešić
Copyright Year
2016
Book
Delays and Networked Control Systems

Print ISBN: 978-3-319-32371-8

Electronic ISBN: 978-3-319-32372-5

Copyright Year: 2016

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