Finite-Dimensional Boundary Control of the Linear Kuramoto-Sivashinsky Equation Under Point Measurement With Guaranteed L2-Gain

• 成果类型: Article
• 署名作者: Katz, Rami; Fridman, Emilia
• 署名单位: Tel Aviv University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3121234
• 发表日期: 2022
• 页码: 5570-5577
• 关键词: Boundary control; LMI; modal decomposition; observer-based control; parabolic PDEs

摘要

Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the one dimensional (1D) heat equation, under the assumption that one of the observation or control operators is bounded. This article suggests a constructive method for such controllers for 1D parabolic partial differential equations (PDEs) with both observation and control operators being unbounded. We consider the Kuramoto-Sivashinsky equation under either boundary or in-domain point measurement and boundary actuation in the presence of disturbances in the PDE and measurement. We employ a modal decomposition approach via dynamic extension, using eigenfunctions of a Sturm-Liouville operator. The controller dimension is defined by the number of unstable modes, whereas the observer dimension N may be larger. We suggest a direct Lyapunov approach to the full-order closed-loop system, which results in a linear matrix inequality (LMI), for input-to-state stabilization (ISS) and guaranteed L-2-gain, whose elements and dimension depend on N. The value of N and the decay rate are obtained from the LMI. We prove that the LMI is always feasible provided N and the L-2 or ISS gains are large enough, thereby obtaining guarantees for our approach. Moreover, for the case of stabilization, we show that feasibility of the LMI for some N implies its feasibility for N + 1. Numerical examples demonstrate the efficiency of the method.