Research Interests




Stability of time-delay systems

Stability of linear time-delay systems has been intensively studied since several decades (see [Fridman et al 2006], [Richard 2003], [Gu et al 2003] and references therein). A such success can be explained by their applied aspect. Indeed, many processes include dead-time phenomena such as biology, chemistry, economics, as well as population dynamics [Kolmanovskii et al 1999] [Niculescu 2001] [Sipahi 2011]. Processing time and propagation time in actuators and sensors generally induce also such delays, especially if some devices are physically distant. That is the challenge of networked controlled systems [Bushnell 2001] [Tarbouriech et al 2005].

In our work, two approaches are considered: the well-known Lyapunov-Krasovskii method [Ariba et al 2007,2008] and the quadratic separation method [Ariba et al 2008,2009,2010]. This latter is an original approach based on robust control tools that provides a fruitful framework to address the stability analysis of nonlinear and uncertain systems [Iwasaki et al 1998] [Peaucelle et al 2007]. We then adapted this framework to the stability analysis of time-delay systems. The key idea lies in rewording the system as a feedback interconnection where the delay dynamic is embedded into an uncertain operator. The original feature of our contribution is to design a set of additional auxiliary operators that enhance the system modelling and reduce the conservatism of the analysis. Our stability conditions are expressed in terms of linear matrix inequality conditions which can be efficiently solved with available semi-definite programming algorithms.

The quadratic separation principle provides a unified framework to study systems with constant/varying/distributed delays, as well as uncertainties on the model. The choice of the set of operators leads to different types of stability condition (regarding the nature of the delay):
  • Delay-dependent stability condition for time-varying delay systems - Theorem 3 in [Ariba et al 2009].
  • Delay-dependent robust stability condition for time-varying delay systems with uncertain matrices (model uncertainty, neglected dynamics...) - Theorem 4 in [Ariba et al 2009].
  • Delay-range stability condition for time-varying delay systems - Theorem 2 in [Ariba et al 2010].
  • Delay-dependent stability condition for distributed time-delay systems where the distributed kernel is a polynomial function - Theorem 2 in [Gouaisbaut et al 2009].
  • Delay-range stability condition for distributed time-delay systems where the distributed kernel is a polynomial function - Theorem 4 in [Gouaisbaut et al 2009].
The corresponding MATLAB function codes can be found here.


My Ph.D. thesis


Title:

On the stability of time-varying delay systems:
 theory and application  to congestion control of a router.
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Abstract:

This thesis investigates the existing links between the control theory and the communication networks supporting the well-known communication protocol TCP (Transmission Control Protocol). The key idea consists in using the tools from control theory for the network traffic stabilization. Hence, this manuscript has naturally been turned towards two research lines: the design of stability and stabilization conditions for delay systems and the congestion control issue in IP network (Internet Protocol).

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. To this end, we proceed to a model augmentation through delay fractionning or with the use of the system derivative. On the other hand, the stability is also assessed with an input-output approach, borrowing then tools from the robust control framework. More precisely, the time delay system is rewritten as the interconnection of a linear application with a uncertain 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 of the system. In both cases, we aim at taking into account relevant informations on the system to reduce the conservatism of the stability analysis. All the stability criteria are expressed as linear matrix inequality conditions.

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 nonlinear simulations in Matlab as well as some experiments under the network simulator NS-2.

The manuscript can be found here (in french) and the slides of my defense are here (in french).