具有外部干扰的不稳定剪切梁的输入–状态稳定

本文针对一端受到范德华力的不稳定剪切梁方程,考虑其输入–状态稳定性问题.通过可逆变换把方程等价地变成一个具有反馈循环的2×2的一阶运输方程与常微分方程的耦合系统.通过自抗扰控制方法,给出具有时变增益的扩张状态观测器来估计干扰.应用Backstepping变换和干扰估计量,设计系统的反馈控制来补偿系统本身的不稳定以及消除匹配干扰.通过C₀–半群方法证明闭环系统的适定性,以及Lyapunov方法证明闭环系统的输入–状态稳定性.数值仿真验证理论结果的正确性.

Input-to-state stabilization of a destabilized shear beam with external disturbances

In this paper, the input-to-state stabilization of an unstable shear beam with van der Waals forces at one end is considered. Through an invertible transformation, the equation is transformed into a 2 × 2 system of first-order transport equations, which convects in opposite directions cascaded with an ordinary differential equation (ODE). Using the active disturbance rejection control (ADRC) method, an extended state observer with the time-varying gain is given to estimate the disturbance. Applying the backstepping transformation and the disturbance estimation, the feedback control of the closed-loop system is proposed to compensate for the instability of the system itself and cancel the matched disturbance. By the C0-semigroup method and the Lyapunov method, the well-posedness and the input-to-state stability (ISS) of theclosed-loop systems are proved, respectively. The validity of the theoretical results is verified by numerical simulations.