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

• 成果类型: Article
• 署名作者: Lhachemi, Hugo; Prieur, Christophe; Trelat, Emmanuel
• 署名单位: University College Dublin; Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); Inria; Universite Paris Cite; Sorbonne Universite
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.2996598
• 发表日期: 2021
• 页码: 1573-1587
• 关键词: Mathematical model; DELAYS; Perturbation methods; Feedback control; stability analysis; Heating systems; Numerical stability; delay boundary control; Neumann trace; partial differential equations (PDEs); proportional– integral (PI) regulation control; 1-D reaction– diffusion equation

摘要

The general context of this article 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 article 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.