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Abstract

In this article, we are interested in analysing the stability of systems that incorporate time-varying delays in their dynamic. The Lyapunov-Krasovskii approach is definitely the most popular method to address this issue and many results have proposed new functionals and enhanced techniques for deriving less conservative stability conditions. In the present work, we propose an original approach: the quadratic separation. To this end, the delay operator properties are exploited to provide delay range stability conditions. In particular, L2-norm of delay-dependent operators are computed so as to reduce the conservatism of the approach. Moreover, the main result is able to assess the stability of non-small delay systems, i.e, it can detect a stability interval for systems that are unstable without any delay. Several examples illustrate the benefit of our methodology.

Abstract

This thesis investigates the existing links between the control theory and the communication networks supporting the well-known communication protocol TCP. The key idea consists in using the tools from control theory for the network traffic stabilization. First, we have addressed the stability analysis of time-varying delay systems via two temporal approaches. On one hand, we consider the well-known Lyapunov-Krasovskii method in which new functionals are built according to original modelings of the system (delay fractionning , system derivative). On the other hand, the stability is also assessed with an input-output approach, borrowing then tools from the robust control framework. The time delay system is then rewritten as the interconnection of a linear application with a matrix consisting in a set of operators that defines the former system. Thus, having revisited the quadratic separation principle, additional auxiliary operators are proposed in order to provide an enhanced modeling of the delayed dynamic. In a second part, the developed methodology is used to cope with the congestion phenomenon in a router supporting TCP communications. This end-to-end protocol is sensitive to packet losses and adjusts thereof its sending rate with respect to the AIMD (Additive-Increase Multiplicative-Decrease) algorithm. Based on the Active Queue Management (AQM) principle, we design a controller embedded into the router that monitors the packet losses. A such mechanism allows to stabilize the network traffic and to control the congestion phenomenon in spite of the delays induced by the network. All theoretical results are tested through experiments under the network simulator NS-2.

Résumé

Cette thèse s'inscrit dans une thématique pluridisciplinaire explorant les liens existants entre la théorie de la commande et les réseaux informatiques. L'idée consiste à appliquer les outils de l'Automatique pour la stabilisation du trafic dans les réseaux de communication. Premièrement, nous nous sommes intéressés à l'analyse de stabilité des systèmes à retards variables au travers de deux approches temporelles. D'une part, nous avons considéré la méthode de Lyapunov-Krasovskii dans laquelle nous avons élaboré des fonctionnelles en adéquation avec de nouvelles modélisations du système (segmentation du retard, dérivée temporelle). D'autre part, la stabilité a également été abordée avec une approche entrée-sortie, empruntant alors les outils de l'analyse robuste. Le système à retard est alors réécrit comme l'interconnexion d'une application linéaire avec une matrice constituée d'opérateurs définissant le système original. Après avoir revisité le principe de séparation quadratique, nous développons des opérateurs auxiliaires afin de caractériser au mieux la dynamique retardée et proposer des critères moins pessimistes. Deuxièmement, la méthodologie développée est ensuite utilisée pour le problème de contrôle de congestion d'un routeur lors de communications TCP. Ce protocole de bout en bout est sensible à la perte de paquet et modifie en conséquence son taux d'émission selon l'algorithme du AIMD. Il s'agit alors de commander le taux de perte par l'intermédiaire d'un mécanisme d'Active Queue Management situé au niveau du routeur afin de réguler le trafic. Les résultats théoriques sont ensuite évalués à l'aide du simulateur de réseaux NS-2.

Mots-Clés / Keywords

Automatique; Systèmes à retards; Lyapunov; Séparation quadratique; LMI; Réseau; TCP; Contrôle de congestion; Active queue management; Automatic control; Time-varying delay systems; Quadratic separation; Network; Congestion congestion;

Abstract

Stability analysis of linear systems with time-varying delay is inves- tigated. In order to highlight the relations between the variation of the delay and the states, redundant equations are introduced to construct a new modeling of the delay system. New types of Lyapunov Krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion. Delay dependent stability conditions are then formu- lated in terms of linear matrix inequalities (LMI). Finally, several exam- ples show the effectiveness of the proposed methodology.

Mots-Clés / Keywords

Time-delay systems stability; Lyapunov-Krasovskii functional; LMI;

Mots-Clés / Keywords

Active queue management; Congestion control; Control theory; Multiple time delay system; Stability; TCP/IP;

Abstract

This paper investigates the overload problem of a single congested router in TCP (Transmission Control Protocol) networks. To cope with the congestion phenomenon, we design a feedback control based on a multiple time-delays model of the set TCP/AQM (Active Queue Management ). Indeed, using robust control tools, especially in the quadratic separation framework, the TCP/AQM model is rewritten as an intercon- nected system and a structured state feedback is constructed to stabilize the network variables. Finally, we illustrate the proposed methodology with a numerical example and simulations using NS-2 simulator.

Mots-Clés / Keywords

Time-delay systems control; Congestion control; AQM; TCP networks;