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Erschienen in:
Introduction to LPV Systems Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

5. Stability Analysis of Time-Delay Systems

verfasst von : Corentin Briat

Erschienen in: Linear Parameter-Varying and Time-Delay Systems

Verlag: Springer Berlin Heidelberg

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Abstract

This chapter presents the main stability and instability results for general time-delay systems. These results are further adapted to the analysis of linear time-delay system using the extensions of Lyapunov theory, namely the Lyapunov-Krasovskii and Lyapunov-Razumikhin theorems, and input-output analysis techniques such small-gain techniques, integral quadratic constraints and quadratic separation. Theoretical results regarding the conservatism of model-transformations and bounding techniques are also derived. The different approaches are compared with each other based on their corresponding stability criteria. Some discussions about complexity reduction are also provided. As for LPV systems, all the obtained stability criteria take the form of matrix inequalities.

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Vorheriges Kapitel Introduction to Time-Delay Systems
Nächstes Kapitel Introduction to LPV Time-Delay Systems
Fußnoten
1
It will be shown later that, in the case of linear systems, Lyapunov-Krasovskii functionals can be connected to robust stability analysis in the \(L_2\)-norm, and Lyapunov-Razumikhin functions to robust stability analysis in the \(L_\infty \)-norm: see also [65, 66].
 
2
Let \(X\) be symmetric positive definite with positive definite square root \(X^{1/2}\), then \((X^{1/2}x+X^{-1/2}y)^\text {T}(X^{1/2}x+X^{-1/2}y)\ge 0\) implies that \(-2x^\text {T}y\le x^\text {T}Xx+y^\text {T}X^{-1}y\) holds. A more thorough discussion on this type of bounds is provided in Sects. 5.6.2 and 5.6.4.
 
3
Sometimes called discretized Lyapunov-Krasovskii functional.
 
4
Note that usually they are defined using the “essential supremum” operator. We assume here that the considered signals are bounded on all intervals of measure zero.
 
5
Additional IQCs can be found in [60, 126].
 
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Metadaten
Titel
Stability Analysis of Time-Delay Systems
verfasst von
Corentin Briat
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Buch
Linear Parameter-Varying and Time-Delay Systems

Print ISBN: 978-3-662-44049-0

Electronic ISBN: 978-3-662-44050-6

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-662-44050-6_5

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