Frédéric Gouaisbaut

An augmented model for robust stability analysis of time-varying delay systems

Stability analysis of linear systems with time-varying 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 modelling 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. Finally, several examples show the effectiveness of the proposed methodology.

Delay-dependent stability analysis of linear systems with time-varying delay

Stability analysis of linear systems with time- varying 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.

Stability interval for time-varying delay systems

We investigate the stability analysis of linear time-delay systems. The time-delay is assumed to be a time-varying continuous function belonging to an interval (possibly excluding zero) with a bound on its derivative. To this end, we propose to use the quadratic separation framework to assess the intervals on the delay that preserves the stability. Nevertheless, to take the time-varying nature of the delay into account, the quadratic separation principle has to be extended to cope with the general case of time-varying operators. The key idea lies in rewording the delay system as a feedback interconnection consisting of operators that characterize it. The original feature of this contribution is to design a set of additional auxiliary operators that enhance the system modelling and reduce the conservatism of the methodology. Then, separation conditions lead to linear matrix inequality conditions which can be efficiently solved with available semi-definite programming algorithms. The paper concludes with illustrative academic examples.

Feedback control for router management and TCP/IP network stability

Several works have established links between congestion control in communication networks and feedback control theory. In this paper, following this paradigm, the design of an AQM (active queue management) ensuring the stability of the congestion phenomenon at a router is proposed. To this end, a modified fluid flow model of TCP (transmission control protocol) that takes into account all delays of the topology is introduced. Then, appropriate tools from control theory are used to address the stability issue and to cope with the time-varying nature of the multiple delays. More precisely, the design of the AQM is formulated as a structured state feedback for multiple time delay systems through the quadratic separation framework. The objective of this mechanism is to ensure the regulation of the queue size of the congested router as well as flow rates to a prescribed level. Furthermore, the proposed methodology allows to set arbitrarily the QoS (quality of service) of the communications following through the controlled router. Finally, a numerical example and some simulations support the exposed theory.

An unknown input sliding observer for anomaly detection in TCP/IP networks

This paper deals with the issue of anomaly detection in TCP/IP networks based on a control theory approach. Using a previously developed sliding mode observer, an improvement of the anomaly detection and reconstruction is proposed. More specifically, the ability of distinguishing false/true positives and false/true negatives in a prescribed finite time is ensured thanks to the design of an unknown input observer combined to some low pass filters. A high quality of service (QoS) is thus guaranteed to the network. To elucidate the proposed method, a network topology is then tested via Simulink as well as via the network simulator NS-2. Finally, detailed results analysis confirm the enhancement brought to the detection of an anomaly flowing through the network.

On designing Lyapunov-Krasovskii based AQM for routers supporting TCP flows

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 [Fall, K., et al., www.isi.edu/nsnam/ns/] support our study.

Robust control tools for traffic monitoring in TCP networks

Several studies have considered control theory tools for traffic control in communication networks, as for example the congestion control issue in IP (Internet Protocol) routers. In this paper, we propose to design a linear observer for time-delay systems to address the traffic monitoring issue in TCP/AQM (Transmission Control Protocol/Active Queue Management) networks. Due to several propagation delays and the queueing delay, the set TCP/AQM is modeled as a multiple delayed system of a particular form. Hence, appropriate robust control tools as quadratic separation are adopted to construct a delay dependent observer for TCP flows estimation. Note that, the developed mechanism enables also the anomaly detection issue for a class of DoS (Denial of Service) attacks. At last, simulations via the network simulator NS-2 and an emulation experiment validate the proposed methodology.

Integral Inequality for Time-Varying Delay Systems and Its Application to Output-Feedback Control

This chapter considers the stability and stabilization of time-varying delay systems. We develop some new integral inequalities which are proved to encompass the celebrated Jensen’s inequality. These technical tools allow to construct simple Lyapunov-Krasovskii functionals very efficient in practice. Notice that our procedure is coupled with the use of the reciprocal convexity result in order to reduce the conservatism induced by the LMI optimisation setup. The effectiveness of the proposed results is illustrated by an example extracted from the dynamics of machines chatter.

L2 stability for quantized linear systems with saturations

Abstract This paper deals with ultimate bounded stability analysis and stabilization conditions for systems involving input saturation and quantized control law, which corresponds to the state quantization case. The state feedback control design problem is then addressed. Theoretical results to ensure the ultimate boundedness and the L 2 stability of the closed-loop system are presented both in local as well as global contexts. The saturation and quantized nonlinearities are tackled through the use of some modified sector conditions. The proposed conditions are then cast in convex optimization problems aiming at maximizing the region of attraction of the closed-loop system and minimizing the set in which the closed-loop trajectories are ultimately bounded, maximizing the bound of admissible L 2 disturbances or maximizing the L 2 -gain from the disturbance to the regulated output.

Input / Output Stability of a Damped String Equation coupled with Ordinary Differential System

The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived using pole locations. Then, based on the Small-Gain theorem and on the Quadratic Separation framework, some robust stability criteria are provided. The latter follows from a projection of the infinite dimensional state on an orthogonal basis of Legendre polynomials. Numerical examples comparing these results with the ones in the literature are proposed and a comparison of its efficiency is made.