基于PDE模型的空间柔性曲梁无穷维Kalman滤波器设计
刘福寿，金栋平，文浩
摘要:状态观测器设计是大型空间结构主动控制设计中的一项重要内容。由于大型空间结构频率低、模态密集且阻尼小，传统的基于模态截断模型的观测器设计方法容易导致观测溢出问题。直接基于结构的偏微分方程(PDE)模型进行状态观测器设计可以避免上述问题，目前正成为该领域的研究热点。针对具有弱阻尼的空间柔性曲梁，对曲梁在悬臂状态下的面内振动进行状态估计。首先建立考虑剪切变形和转动惯量的曲梁面内振动PDE模型，并通过变量替换将PDE模型转化为Hilbert空间中的抽象状态方程，然后基于Kalman滤波理论构建曲梁的无穷维状态观测器模型，并采用频域下的谱分解方法得到无穷维Kalman滤波器增益函数的解析解。最后通过数值算例对无穷维Kalman滤波器的观测效果进行了验证，并揭示了该方法在防止观测溢出方面的有效性。
关键词：柔性曲梁,PDE模型,状态估计,观测溢出
Infinite-dimensional Kalman filter design for flexible curved beams based on PDE models
Fushou Liu, Dongping Jin, Hao Wen
Abstract：State observer design is a crucial step in active vibration control of large flexible space structures (LFSSs). Since the LFSSs often have lower natural frequencies, high modal density and low damping, the traditional observer design methods which use the modal truncation model are easily leading to the problem of observation spillover. Instead, the approaches directly using the partial differential equation (PDE) models of flexible structures can avoid the spillover problem, and are becoming the hotspot of research in this field. The presented research aimed at the flexible curved beams which are often used as support components in space antennas, and designed the infinite dimensional state observer for the in-plane vibration of cantilevered curved beams with weakly internal damping based on the theory of Kalman filtering. Firstly, a second order PDEs model for the in-plane vibration of the curved beam is established which considers the shear deformation and rotary inertia of the curved beam simultaneously, then by using the variable substitution method, the second order PDEs model is reduced to first order and transformed to an abstract state space model (SSM) in Hilbert space. The infinite-dimensional Kalman filter for the curved beam is designed using the SSM, and the explicit formula for the feedback gain operators of the Kalman filter are obtained based on the spectral factorization method in frequency domain. Finally, a numerical example is carried out to validate the effect of the presented infinite dimensional Kalman filter design method, and to reveal the effectiveness of the method on preventing the observation spillover.
Keyword：flexible curved beams, PDE models, state estimation, observation spillover
原文链接：http://engine.scichina.com/publisher/scp/journal/SSPMA/47/10/10.1360/SSPMA2017-00097?slug=fulltext
