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Abstract

This paper provides some new techniques to construct a Lyapunov-Krasovskii functional for time varying delay systems. The construction is based on a partitioning scheme of the time-varying delay leading to a new type of Lyapunov-Krasovskii functional. This functional is depending on an augmented state and also on an integral quadratic constraint added to reduce the conservatism of the proposed methodology. This approach is then extended to the robust case. Finally, some examples support our approach.

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Résumé

De nombreux travaux de recherche ont montré les liens existants entre le contrôle de congestion dans les réseaux de communication et la théorie de la commande en Automatique. Nous présentons dans cet article une méthode pour la synthèse d'un AQM (Active Queue Management) assurant le contrôle de congestion d'un routeur. Le modèle utilisé est une représentation fluide linéarisée du comportement de TCP (Transmission Control Protocol). La synthèse de l'AQM est alors transformée en un problème de commande par retour d'état pour les systèmes à retards. Enfin, un exemple numérique extrait de la littérature et des simulations via le simulateur de réseaux NS-2 complètent notre étude.

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Mots-Clés / Keywords

Abstract

This paper deals with the stability analysis of linear time-delay sys- tems. The time-delay is assumed to be a time-varying C1 function be- longing to an interval and has a bounded derivative. Considering stable delay-free system, the quadratic separation framework is used to assess the maximal allowable value of the delay that preserves stability. To take into account the time-varying nature of the delay, the quadratic separation principle has been extended to cope with the general case of uncertain op- erators (instead of just uncertain matrices). Then, separation conditions lead to linear matrix inequality (LMI) which can be efficiently solved with Semi Definite Programming solvers in Matlab. The paper concludes with an illustrative academic example.

Abstract

Recent research has shown the link between congestion control in communication networks and feedback control system. In this paper, the design of an active queue management (AQM) which can be viewed as a controller, is considered. Based on a state space representation of a linearized fluid flow model of TCP, the AQM design is converted to a state feedback synthesis problem for time delay systems. Finally, an example extracted from the literature and simulations via a network simulator NS (under cross traffic conditions) support our study.

Abstract

Stability analysis of linear systems with timevarying delay is investigated. 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 formulated in terms of linear matrix inequalities (LMI). Finally, an example shows the effectiveness of the proposed methodology.

Abstract

For the last few years, we assist to a growing interest of designing AQM (Active Queue Management) using control theory. In this paper, we focus on the synthesis of an AQM based on the Lyapunov theory for time delay systems. With the help of a recently developed Lyapunov-Krasovskii functional and using a state space representation of a linearized fluid model of TCP, two robust AQMs stabilizing the TCP model are constructed. Notice that our results are constructive and the synthesis problem is reduced to a convex optimization scheme expressed in terms of linear matrix inequalities (LMIs). Finally, an example extracted from the literature and simulations via NS simulator [1] support our study.