Transverse Vibration Control of Axially Moving Membranes by Regulation of Axial Velocity

In this brief, a novel control algorithm that suppresses the transverse vibrations of an axially moving membrane system is presented. The proposed control method is to regulate the axial transport velocity of the membrane so as to track a desired profile according to which the vibration energy of the membrane at the end of transport decays most quickly. An optimal control problem that generates the desired profile of the axial transport velocity is solved by the conjugate gradient method. The Galerkin method is applied in order to reduce the partial differential equations describing the dynamics of the axially moving membrane into two sets of ordinary differential equations (ODEs) representing longitudinal/lateral and transverse displacements. For control design purposes, these ODEs are rewritten into state-space equations. The vibration energy of the axially moving membrane is represented by a quadratic form of the state variables. In the optimal control problem, the cost function modified from the vibration energy function is subject to the constraints on the state variables, and the axial transport velocity is considered as a control input. The effectiveness of the proposed control method is illustrated via numerical simulations.