arXiv subjects

Frédéric Gouaisbaut

Publications and source records attributed to Frédéric Gouaisbaut.

18 recordsLinked to original sources

arXiv2025

Reduced-order Smith predictor for state feedback control with guaranteed stability

This article deals with the implementation of the Smith Predictor for state feedback control in state space representation. The desired control law, obtained using partial differential equations and backstepping control, contains an integral term that has to be approximated for implementation. In this article, we propose a new way to implement this control law using a dynamic controller. The control law is composed of a state feedback term and a dynamic term that approaches the integral term that has to be estimated for implementation. Using a Lyapunov functional, we provide sufficient conditions, in terms of a linear matrix inequality, to guarantee that the closed-loop system is stable when the proposed control law is applied. We use three examples, taken from the literature, to show the benefits of the proposed approach.

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arXiv2022

Systems with both constant and time-varying delays: a switched systems approach and application to observer-controller co-design

In this paper, we study the application of switched systems stability criteria to derive delay-dependent conditions for systems affected by both a constant and a time-varying delay. The main novelty of our approach lies on the use of path-complete Lyapunov techniques along with the proposition of a new modified functional to obtain convex analysis conditions while avoiding the need of computing a dwell time for each mode in a switched system representation, as usual in the \textit{switched approach} for time-delay systems. Furthermore, we leverage the developed analysis to obtain LMIs for the closed-loop stabilization of systems with time-varying sensor delays by means of an observer-based compensator. A numerical example illustrates the proposed methods.

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arXiv2022

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov-Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel-Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

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arXiv2022

Necessary and sufficient stability condition for time-delay systems arising from Legendre approximation

Recently, sufficient conditions of stability or instability for time-delay systems have been proven to be necessary. In this way, a remarkable necessary and sufficient condition has then been developed by Gomez et al. It is presented as a simple test of positive definiteness of a matrix issued from the Lyapunov matrix. In this paper, an extension of this result is presented. Without going into details, the uniform discretization of the state has been replaced by projections on the first Legendre polynomials. Like Gomez et al., based on convergence arguments, the necessity is obtained in finite order, which can be calculated analytically. Compared to them, by relying on the fast convergence rate of Legendre approximation, the required order to ensure stability has been reduced. Thanks to this major modification, as shown in the example section, it is possible the find stable regions for low orders and unstable ones for even smaller orders.

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arXiv2021

On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming

Stability and stabilization for linear state feedback control systems in the presence of sensor quantization are studied. As the closed-loop system is described by a discontinuous right-hand side differential equation, Krasovskii solutions (to the closed-loop system) are considered. Sufficient conditions in the form of matrix inequalities are proposed to characterize uniform global asymptotic stability of a compact set containing the origin. Such conditions are shown to be always feasible whenever the quantization-free closed-loop system is asymptotically stable. Building on the obtained conditions, computationally affordable algorithms for the solution to the considered problems are illustrated. The effectiveness of the proposed methodology is shown in three examples.

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arXiv2020

$\mathcal{L}_2$ State Estimation with Guaranteed Convergence Speed in the Presence of Sporadic Measurements

This paper deals with the problem of estimating the state of a linear time-invariant system in the presence of sporadically available measurements and external perturbations. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error at each sampling time. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve global exponential stability with a given decay rate to a set wherein the estimation error is equal to zero. Robustness with respect to external perturbations and $\mathcal{L}_2$-external stability from the plant perturbation to a given performance output are considered. Moreover, computationally efficient algorithms based on the solution to linear matrix inequalities are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in three examples.

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arXiv2020

Practical Stability Analysis of a Drilling Pipe under Friction with a PI-Controller

This paper deals with the exponential stability of a drilling pipe controlled by a PI controller. The model used leads to a coupled ODE / PDE and is consequently of infinite dimension. Using recent advances in time-delay systems, we derive a new Lyapunov functional based on an state extension made up of projections of the Riemann coordinates. Two cases will be considered. First, we will provide an exponential stability result expressed using the LMI framework. This result is dedicated to a linear version of the torsional dynamic. On a second hand, the nonlinear terms in the initial model, that generates the well-known stick-slip phenomenon is captured through a new stability theorem. Numerical simulations show the effectiveness of the method and that the stick-slip oscillations cannot be weaken using a PI controller.

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arXiv2018

Stability analysis of a system coupled to a heat equation

As a first approach to the study of systems coupling finite and infinite dimensional natures, this article addresses the stability of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov functional technique. Inspired from recent developments in the area of time delay systems, a new methodology to study the stability of such a class of distributed parameter systems is presented here. The idea is to use a polynomial approximation of the infinite dimensional state of the heat equation in order to build an enriched energy functional. A well known efficient integral inequality (Bessel inequality) will allow to obtain stability conditions expressed in terms of linear matrix inequalities. We will eventually test our approach on academic examples in order to illustrate the efficiency of our theoretical results.

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arXiv2018

Nonlinear control for an uncertain electromagnetic actuator

This paper presents the design of a nonlinear control law for a typical electromagnetic actuator system. Electromagnetic actuators are widely implemented in industrial applications, and especially as linear positioning system. In this work, we aim at taking into account a magnetic phenomenon that is usually neglected: flux fringing. This issue is addressed with an uncertain modeling approach. The proposed control law consists of two steps, a backstepping control regulates the mechanical part and a sliding mode approach controls the coil current and the magnetic force implicitly. An illustrative example shows the effectiveness of the presented approach.

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arXiv2018

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.

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arXiv2018

Stabilization of an unstable wave equation using an infinite dimensional dynamic controller

This paper deals with the stabilization of an anti-stable string equation with Dirichlet actuation where the instability appears because of the uncontrolled boundary condition. Then, infinitely many unstable poles are generated and an infinite dimensional control law is therefore proposed to exponentially stabilize the system. The idea behind the choice of the controller is to extend the domain of the PDE so that the anti-damping term is compensated by a damping at the other boundary condition. Additionally, notice that the system can then be exponentially stabilized with a chosen decay-rate and is robust to uncertainties on the wave speed and the anti-damped coefficient of the wave equation, with the only use of a point-wise boundary measurement. The efficiency of this new control strategy is then compared to the backstepping approach.

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arXiv2018

Exponential Lyapunov Stability Analysis of a Drilling Mechanism

This article deals with the stability analysis of a drilling system which is modelled as a coupled ordinary differential equation / string equation. The string is damped at the two boundaries but leading to a stable open-loop system. The aim is to derive a linear matrix inequality ensuring the exponential stability with a guaranteed decay-rate of this interconnected system. A strictly proper dynamic controller based on boundary measurements is proposed to accelerate the system dynamics and its effects are investigated through the stability theorem and simulations. It results in an efficient finite dimension controller which subsequently improves the system performances.

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arXiv2018

Static State and Output Feedback Synthesis for Time-Delay Systems

In this paper, the design of a static feedback gain for a linear system subject to an input delay is studied. This synthesis is based on a stability analysis conducted using Lyapunov-Krasovskii theorem and Bessel-Legendre inequalities expressed in terms of LMIs. Some bilinear non-convex matrix inequalities are obtained to go from analysis to synthesis. They are then difficult to solve and an iterative LMI procedure is given which takes advantage of the elimination lemma. Naturally, slack variables are introduced and then, following an optimization process, values for them are proposed to reduce the conservatism. The two main corollaries discuss the static state and output feedback synthesis. Finally, a comparison is proposed and shows that this formulation introduces small conservatism.

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arXiv2018

Lyapunov stability analysis of a string equation coupled with an ordinary differential system

This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the classical Lyapunov functional proposed in the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality together with Legendre polynomials. Numerical examples illustrate the potential of our approach through three scenari: a stable ODE perturbed by the PDE, an unstable open-loop ODE stabilized by the PDE and an unstable closed-loop ODE stabilized by the PDE.

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arXiv2009

Multiple time-delays system modeling and control for router management

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.

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arXiv2009

Robust control tools for traffic monitoring in TCP/AQM 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.

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arXiv2009

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 {\it NS simulator} support our study.

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arXiv2009

Design and performance evaluation of a state-space based AQM

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.

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