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Lhachemi, Hugo
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Lhachemi, Hugo
Official Name
Lhachemi, Hugo
Research Output
Now showing 1 - 10 of 15
Publication
Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays
2021-01, Lhachemi, Hugo, Prieur, Christophe, Shorten, Robert
This paper discusses the in-domain feedback stabilization of reaction–diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a -semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent.
Publication
On the derivation of stability properties for time-delay systems without constraint on the time-derivative of the initial condition
2021-11, Lhachemi, Hugo, Shorten, Robert
Stability of retarded differential equations is closely related to the existence of Lyapunov-Krasovskii functionals. Even if a number of converse results have been reported regarding the existence of such functionals, there is a lack of constructive methods for their selection. For certain classes of time-delay systems for which such constructive methods are lacking, it was shown that Lyapunov-Krasovskii functionals that are also allowed to depend on the time-derivative of the state-trajectory are efficient tools for the study of the stability properties. However, in such an approach the initial condition needs to be assumed absolutely continuous with a square integrable weak derivative. In addition, the stability results hold for initial conditions that are evaluated based on the magnitude of both the initial condition and its time-derivative. The main objective of this paper is to show that, for certain classes of time-delay systems, the aforementioned stability results can actually be extended to initial conditions that are only assumed continuous and that are evaluated in uniform norm.
Publication
ISS Property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems
2019-11, Lhachemi, Hugo, Shorten, Robert
This paper deals with the establishment of Input-to-State Stability (ISS) estimates for infinite dimen-sional systems with respect to both boundary and distributed disturbances. First, a new approach isdeveloped for the establishment of ISS estimates for a class of Riesz-spectral boundary control systemssatisfying certain eigenvalue constraints. Second, a concept of weak solutions is introduced in order torelax the disturbances regularity assumptions required to ensure the existence of classical solutions.The proposed concept of weak solutions, that applies to a large class of boundary control systemswhich is not limited to the Riesz-spectral ones, provides a natural extension of the concept of bothclassical and mild solutions. Assuming that an ISS estimate holds true for classical solutions, we showthe existence, the uniqueness, and the ISS property of the weak solutions.
Publication
Boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and state-delay
2020-06, Lhachemi, Hugo, Shorten, Robert
This paper discusses the boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we show that the closed-loop system is exponentially input-to-state stable with fading memory.
Publication
Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control
2020-04, Lhachemi, Hugo, Shorten, Robert, Prieur, Christophe
This paper deals with the exponential input-to-state stabilization with respect to boundary disturbances of a class of diagonal infinite-dimensional systems via delay boundary control. The considered input delays are uncertain and time-varying. The proposed control strategy consists of a constant-delay predictor feedback controller designed on a truncated finite-dimensional model capturing the unstable modes of the original infinite-dimensional system. We show that the resulting closed-loop system is exponentially input-to-state stable with fading memory of both additive boundary input perturbations and disturbances in the computation of the predictor feedback.
Publication
Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances
2020-03, Lhachemi, Hugo, Saussié, David, Zhu, Guchuan, Shorten, Robert
This note establishes the exponential input-to-state stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.
Publication
ISS of a Clamped-Free Damped String for the Configurations Associated with the Loss of the Riesz-Spectral Properties
2019-06-28, Lhachemi, Hugo, Shorten, Robert
This paper deals with the Input-to-State Stability (ISS) of a clamped-free damped string with respect to boundary disturbances for the configurations associated with the loss of the Riesz-spectral properties. Specifically, for most of the values of the physical parameters (namely the stiffness parameter and the damping coefficient), the ISS property of the clamped free damped string can be established based on the fact that the underlying disturbance-free operator is a Riesz-spectral operator. However, such a Riesz-spectral property does not hold true for certain configurations of the physical parameters of the damped string. This paper specifically investigates the establishment of an ISS estimate for these configurations. The proposed strategy relies on the projection of the original system trajectories in a Riesz basis obtained by adequately completing the set of eigenvectors of the disturbance-free operator.
Publication
On Design for Additive Manufacturing: Review of Challenges and Opportunities utilising Visualisation Technologies
2019-06-19, Newell, Anthony, George, Abraham, Papakostas, Nikolaos, Lhachemi, Hugo, Malik, Ammar, Shorten, Robert
Design for additive manufacturing poses new challenges and opportunities for manufacturers to produce highly customised parts while reducing cost, production time and improving quality. Manufacturing constraints of conventional manufacturing methods, such as geometric complexity limitations and workpiece handling, have shaped the landscape of computer-aided design tools, which are therefore not suitably adapted to design for additive manufacturing. Furthermore, computer-aided design tools require a high level of training to produce appropriate models. Augmented reality and feedback technologies pose an interesting opportunity for design for additive manufacturing, whereby the interaction with 3D models in an augmented or virtual design space can provide intuitive feedback to engineers and designers, providing fast validation of designs, parametric modelling and opportunities for training and use in both professional and amateur designer communities. This paper will explore and review the opportunities this exciting new technology provides.
Publication
Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems With Delay Boundary Control
2021-01, Lhachemi, Hugo, Prieur, Christophe
This article studies the boundary feedback stabilization of a class of diagonal infinite-dimensional boundary control systems. In the studied setting, the boundary control input is subject to a constant delay while the open-loop system might exhibit a finite number of unstable modes. The proposed control design strategy consists of two main steps. First, a finite-dimensional subsystem is obtained by truncation of the original infinitedimensional system (IDS) via modal decomposition. It includes the unstable components of the IDS and allows the design of a finite-dimensional delay controller by means of the Artstein transformation and the pole-shifting theorem. Second, it is shown via the selection of an adequate Lyapunov function that: 1) the finite-dimensional delay controller successfully stabilizes the original IDS and 2) the closed-loop system is exponentially input-to-state stable (ISS) with respect to distributed disturbances. Finally, the obtained ISS property is used to derive a small gain condition ensuring the stability of an IDS-ODE interconnection.
Publication
PI Regulation of a Reaction-Diffusion Equation with Delayed Boundary Control
2020-05-22, Lhachemi, Hugo, Prieur, Christophe, Trélat, Emmanuel
The general context of this work is the feedback control of an infinite-dimensional system so that the closed loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction diffusion equation might be either open-loop stable or unstable.
The proposed control strategy goes as follows. First, a finite dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system is obtained via spectral
decomposition. The truncated model is then augmented by an integral component on the tracking error of the left Neumann trace. After resorting to the Artstein transformation to handle the control input delay, the PI controller is designed by pole shifting. Stability of the resulting closed-loop infinite-dimensional system,
consisting of the original reaction-diffusion equation with the PI controller, is then established thanks to an adequate Lyapunov function. In the case of a time-varying reference input and a time-varying distributed disturbance, our stability result takes the form of an exponential Input-to-State Stability (ISS) estimate
with fading memory. Finally, another exponential ISS estimate with fading memory is established for the tracking performance of the reference signal by the system output. In particular, these results assess the setpoint regulation of the left Neumann trace in the presence of distributed perturbations that converge to
a steady-state value and with a time-derivative that converges to zero. Numerical simulations are carried out to illustrate the efficiency of our control strategy.