分布参数柔性梁的建模与振动边界控制

Modeling and Boundary Vibration Control of a Distributed-Parameter Flexible Beam

研究了一类具有未知扰动和变张力柔性梁的二维振动控制问题。柔性梁式结构属于典型的无穷维分布参数系统,其动力学模型由一组偏微分方程(PDEs)和一组用常微分方程(ODEs)混合构成。为了避免控制溢出和实现二维振动控制,基于柔性梁原始无穷维分布参数模型,结合边界控制技术和Lyapunov直接法,设计了纵向和横向二维PD(Proportional Derivative)控制器用以抑制柔性梁的振动,设计的PD控制器简单可行且独立于系统参数,因此具有较好的实时性和鲁棒性。其后利用经典的Lyapunov直接法对柔性梁系统的稳定性和一致有界性进行了证明。最后对所设计控制方法的有效性进行了仿真验证。

A boundary control in two-dimensional is proposed for a distributed-parameter flexible beam with unknown disturbance and varying tension to minimize the beam vibrations. Flexible beam is a typical infinite-dimensional distributed parameter systems, and its hybrid dynamic model is described in terms of partial differential equations （PDEs） and ordinary differential equations （ODEs）. To avoid con- trol spillover and achieve vibration control in two-dimensional, the PD （Proportional Derivative） bound- ary controllers in longitudinal and lateral direction are designed respectively based on the original infinite- dimensional PDEs model and Lyapunov＇s direct method to reduce the flexible vibrations. With the proposed PD boundary control, the real-time and robustness of control system are ensured because the proposed controller is simple and independent of system parameters. The uniform boundedness and closed- looped stability can be achieved by Lyapunov＇s direct method. Simulation results illustrate the effectiveness of the proposed boundary control.