Corentin Briat

The stability analysis and control of uncertain LTI and LPV time-delay systems are studied. The approach is based on the use of a delay-fragmented parameter-dependent Lyapunov-Krasovskii functional (LKF) and leads to stability conditions expressed in terms of parameter-dependent LMIs. However, the obtained conditions cannot be directly used for control due to the inherent structure of the LMIs. To overcome this difficulty, a dilated version of the LMI is provided in order to shape its structure and adapt it for control purposes. Constructive sufficient conditions for the existence of memoryless controllers and controllers with exact-memory are then obtained. The approach is finally illustrated through several examples of the literature.

@article{Briat:aut10,
author = "C.~Briat and O.~Sename and J-F.~Lafay",
title = "Robust $L_2$ Control of Uncertain LTI/LPV Time-Delay Systems - A Delay-Fragmented Lyapunov-Krasovskii Functional Approach",
year = "----",
journal = "Submitted to Automatica"
}

The stability analysis of systems with varying sampling period is analyzed in the framework of dynamic equations on time-scales. Lyapunov theory is used, with sample-period-dependent and independent Lyapunov functions, to obtain stability conditions expressed in terms of parameter dependent matrix inequalities. The examples illustrate the efficiency of the approach which is able to recover, for some systems, the theoretical results for the constant sampling case even in the varying case. It is also shown that some systems may have admissible varying sampling periods located in disjoint sets. Finally, stabilization results are provided. Two types of switched controllers are considered, namely robust-controllers and sampling-period-dependent controllers. It is shown that the latter ones, using the information on the sampling period, can improve stability properties. Stabilization examples illustrate the effectiveness of the approach.

@article{Briat:aut10,
author = "C.~Briat and U.T.~{J"{o}nsson}",
title = "Stability analysis and Stabilization of sampled-data systems - A time-scale approach",
year = "2010",
journal = "Submitted to Automatica"
}

The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Its conservatism is studied through the use of the Gr"{u}ss Inequality, in both the continuous and discrete cases, with particular focus on problems arising in stability analysis of time-delay and sampled-data systems. It has been noted in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results. We propose to prove this in the general setting and we show that the gap can be made arbitrarily small provided that the order of fragmentation is chosen sufficiently large. The convergence is also shown to be faster when the involved functions are differentiable. Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided. It is shown that the other bounds are equivalent to Jensen's together with a higher computational complexity. However, these bounds exhibit important well-posedness and linearity properties. This demonstrates their importance and suggests that Jensen's might be the best convex bound we could expect both in terms of tightness and computational complexity.

@article{Briat:tac10,
author = "C.~Briat",
title = "Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems",
year = "2010",
journal = "Submitted to IEEE Transactions on Automatic Control"
}

The aim of this paper is to propose a model-based feedback control strategy for indoor temperature regulation in buildings equipped with underfloor air distribution (UFAD). Supposing distributed sensing and actuation capabilities, a 0D model of the ventilation process is derived, based on the thermodynamics properties of the flow. A state-space description of the process is then inferred, including discrete events and nonlinear components. The use of a wireless sensor network (WSN) and the resulting communication constraints with the IEEE 802.15.4 standard are discussed. Both synchronous and asynchronous transmissions are considered. Based on the linear part of the model, different H¥ robust multiple-input multiple-output (MIMO) controllers are designed, first with a standard mixed-sensitivity approach and then by taking into account the network-induced delay explicitly. The impact of the communication constraints and the relative performances of the controllers are discussed based on simulation results.

@article{WiMaPaBri:ima10,
author = "E.~Witrant and P.~Di~Marco and P.~Park and C.~Briat",
title = "Limitations and Performances of Robust Control over WSN: UFAD Control in Intelligent Buildings",
year = "2010",
journal = "Accepted at IMA Journal of Mathematical Control and Information"
}

The stabilization of uncertain LTI/LPV time delay systems with time varying delays by state-feedback controllers is addressed. At the difference of other works in the literature, the proposed approach allows for the synthesis of resilient controllers with respect to uncertainties on the implemented delay. It is emphasized that such controllers unify memoryless and exact-memory controllers usually considered in the literature. The solutions to the stability and stabilization problems are expressed in terms of LMIs which allow to check the stability of the closed-loop system for a given bound on the knowledge error and even optimize the uncertainty radius under some performance constraints; in this paper, the $\mathcal{H}_\infty$ performance measure is considered. The interest of the approach is finally illustrated through several examples.

@article{Briat:aut09,
author = "C.~Briat and O.~Sename and J-F.~Lafay",
title = "Memory Resilient Gain-scheduled State-Feedback Control of Uncertain {LTI/LPV} Time-Delay Systems with Time-Varying Delays",
year = "2010",
journal = "Systems \& Control Letters"
volume = "59"
pages = "451--459"
}

This paper introduces a new model for disease outbreaks. This model describes the disease evolution through a system of nonlinear differential equations with distributed-delay. The main difference between classical SIR-model resides in the fact that the recovery rate of the population is expressed as a distributed-delay term modeling the time spent being sick by infected people. This model is identified to fit realistic measurements which shows the effectiveness of the model. Finally, we develop an optimal campaign vaccination strategy based on recent results on the impulsive control of time-delay systems.

@article{Briat:bspc09,
author = "C.~Briat and E.~I.~Verriest",
title = "A New Delay-SIR Model for Pulse Vaccination",
year = "2009",
journal = "Biomedical Signal Processing and Control",
volume = "4",
pages = "272-277",
}

This paper deals with delay-scheduled control of linear systems with time-varying delays. First, we introduce a new model transformation which allows us to provide an unified approach in the framework of linear parameter varying systems, to analyze stability and design state-feedback controllers. Second, we extend this method to develop a new type of controller where the state-feedback is time-varying and scheduled by the value of the delay (delay-scheduled controllers). All the results are provided in terms of Linear Matrix Inequalities (LMIs).

@article{Briat:tac09,
author = "C.~Briat and O.~Sename and J.F.~Lafay",
title = "$\mathcal{H}_\infty$ delay-scheduled control of linear systems with time-varying delays",
year = "2009",
journal = "{IEEE} Transactions on Automatic Control",
volume = "42(8)",
pages = "2255-2260",
}

This paper is concerned in the synthesis of delay-scheduled state-feedback stabilizing linear systems with time-varying delay when the delay can be approximatively known in real-time. First we introduce a new model transformation turning the time-delay system into an uncertain LPV system. Using this reformulation we elaborate delay-dependent stability test based on the so-called full block S-procedure and derive from it a delay-dependent stabilization lemma. Our results are then relaxed using a new relaxation lemma which is shown to have good properties and provide then LMI based theorems, well-known for their tractability. Our results tackle error measurement on the delay. We show the efficiency of the method through an example.

@article{Briat:scl09,
author = "C.~Briat and O.~Sename and J.F.~Lafay",
title = "Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays - A LPV Approach",
year = "2009",
journal = "Systems \& Control Letters", volume = "58(9)",
pages = "664-671",

}