This study focuses on the observer and observer-based output feedback control design for a one-dimensional wave equation with van der Pol type nonlinear boundary condition and Dirichlet boundary control. The contribution of this work is divided into two parts. Firstly, the anti-collocated backstepping observer design for the nonlinear wave system is investigated, and the well-posedness and the asymptotical stability analysis of the observer error are studied. Secondly, an output feedback observer-based controller is set up by using boundary displacement measurement only to stabilise the nonlinear wave system; also the asymptotical stability of the closed-loop system is shown. The validity of the theoretical results is demonstrated by numerical simulations.

本研究侧重于具有 van der Pol 型非线性边界条件和 Dirichlet 边界控制的一维波动方程的观测器和基于观测器的输出反馈控制设计。这项工作的贡献分为两部分。首先研究了非线性波系统的反并置反步观测器设计，研究了观测器误差的适定性和渐近稳定性分析。其次，建立了基于输出反馈观测器的控制器，仅利用边界位移测量来稳定非线性波系统；还显示了闭环系统的渐近稳定性。数值模拟证明了理论结果的有效性。