PI Regulation of a Reaction-Diffusion Equation with Delayed Boundary Control

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dc.contributor.authorLhachemi, Hugo-
dc.contributor.authorPrieur, Christophe-
dc.contributor.authorTrélat, Emmanuel-
dc.date.copyright2020 IEEEen_US
dc.identifier.citationIEEE Transactions on Automatic Controlen_US
dc.description.abstractThe 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.en_US
dc.description.sponsorshipEuropean Commission - European Regional Development Funden_US
dc.description.sponsorshipScience Foundation Irelanden_US
dc.rights© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.en_US
dc.subject1-D reaction-diffusion equationen_US
dc.subjectPI regulation controlen_US
dc.subjectNeumann traceen_US
dc.subjectDelay boundary controlen_US
dc.subjectPartial Differential Equations (PDEs)en_US
dc.titlePI Regulation of a Reaction-Diffusion Equation with Delayed Boundary Controlen_US
dc.typeJournal Articleen_US
dc.statusPeer revieweden_US
dc.description.othersponsorshipI-Form industry partnersen_US
dc.description.adminUpdate citation details during checkdate report - ACen_US
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