Adaptive vibration control for stabilisation of the Euler–Bernoulli beam with an unknown payload

In this paper, an adaptive boundary controller is designed to solve the boundary control problem of Euler–Bernoulli beams with an unknown payload. Therefore, a boundary controller with an adaptive law is designed to compensate for the parameter uncertainty of the system. Partial Differential Equations (PDEs) are used to describe the Euler–Bernoulli beam system. The well-posedness of the system under the action of an adaptive boundary controller is proved by using the linear operator semigroup method. Meanwhile, the asymptotic stability of the closed-loop system is derived from the extended Krasovskii–LaSalle invariance principle. The effectiveness and superiority of the proposed method are illustrated by simulation in comparison with the existing results.

本文针对带未知负载的欧拉-伯努利梁（Euler–Bernoulli beam）的边界控制问题，设计了自适应边界控制器。为此，本文设计了带有自适应律的边界控制器，用以补偿系统的参数不确定性。本文采用偏微分方程（Partial Differential Equations, PDEs）对欧拉-伯努利梁系统进行建模，通过线性算子半群方法证明了自适应边界控制器作用下系统的适定性。同时，基于推广的克拉索夫斯基-拉萨尔不变性原理，推导得到闭环系统的渐近稳定性。最后通过仿真实验，将所提方法与现有研究成果进行对比，验证了所提方法的有效性与优越性。