Infinite-dimensional Kalman filter design for flexible curved beams based on PDE models

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.