Trajectory Tracking and Vibration Suppression for General Spatial Motion of a Flexible Plate via Two-time Scale and Boundary Control Methods

Abstract

In this paper, general spatial trajectory tracking problem of a flexible plate is studied. To obtain dynamic equations of motion of the plate, Hamiltonian dynamics is used and then Lagrange’s equations of the plate dynamics and the corresponding expressions for boundary conditions are derived. Resulting equations show that the coupled plate dynamics including plate vibration and its rigid spatial motion take place in two different time domains. By using two-time scale (TTS) control theory, a control scheme is elaborated that makes the orientation and position of selected points of the plate track a desired trajectory while suppressing the plate’s vibration. TTS composite controller has two parts: one is a tracking controller designed for the slow (rigid) subsystem, and the other one is a stabilizing controller for the fast (flexible) subsystem. For the fast subsystem, the proposed boundary control method does not require any information about vibration of the interior points of the plate; nor requires discretizing the partial differential equation of the plate vibration to a set of ordinary differential equations. This part of the control commands consists of feedback of the velocities at the boundary points of the plate. So, the method avoids the need for instruments to measure data from vibration of any points inside the plate or designing an observer for estimating this information. Also the proposed method prevents control spillover due to discretization and resorting to truncation of the model. Simulation results show that the fast stabilizing control commands at the boundary are able to remove undesirable vibration of the flexible plate and the slow controller provides very good trajectory tracking with acceptable actuating forces/moments.