Exponential Stabilization of Transverse Vibration and Trajectory Tracking for General In-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods

Abstract

In this paper, exponential stabilization of vibration of a flexible beam together with its general in-plane trajectory tracking is presented. Coupled beam dynamics including beam vibration (flexible/fast subsystem) and its rigid in-plane motion (rigid/slow subsystem) takes place in two different time domains. Therefore, to have the beam track a desired trajectory while suppressing its vibration by an exponential rate of decay, a composite control scheme is elaborated by two-time scale (TTS) control theory. This control law has two parts: one is a tracking controller designed for the rigid subsystem based on inverse dynamic law, and the other one is an exponential stabilizing controller for the flexible subsystem based on boundary control (BC) laws. Exponential stabilization is proved by using a metric containing kinetic and potential energies of the fast subsystem and by feedback of the rate of deflection and the slope at one end of the beam. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and together with the slow inverse controller is able to provide very good trajectory tracking with acceptable actuating forces/moments. Also, they illustrate that tracking errors and the vibration amplitude are decreased versus time by the fast exponential stabilizing control law compared with an asymptotic stabilizing control law.

Robust Control of Collaborative Manipulators - Flexible Object System

“…Hence, from (18), (19) and (20) the beam dynamics of the rigid motion with respect to Cartesian coordinates   T 0 0…”

“…The beam has rigid as well as flexible parameters and are coupled together in rf M , rf C , rf  and also in R. These flexible parameters have to be uncoupled from the above matrices and vectors by using the singular perturbation technique. This technique also accounts for the neglected high frequency characteristics when the beam undergoes vibration [20].…”

“…Using a perturbation parameter, say 2 ,  the order of the system dynamics can be changed and this small parameter depends upon the system variable. Keeping that in mind, the term EI /  in (36), which has large magnitude compared to other coefficients [20], can be redefined as,…”

“…In these schemes, the strain signal is used instead of the measurements of flexible coordinates and their time derivatives. Quite recently, [20] and [21] also considered the PDE in their controller designs. In the present study the dynamics for the flexible object moved by two planar rigid manipulators will be described by the PDE, and the controller will be designed without using the assumed mode method.…”

“…The proposed robust controller has the following features: 1) it does not need the information of the modes; 2) exact knowledge of the manipulators parameters or the beam are not required; and 3) it satisfies Tikhnov's theorem. It should be noted that the singular perturbation technique has also been applied to the singlelink flexible beams, say, for example, in [20] and [21], where a slow system and fast subsystems were identified as well. Comparing with the case of the manipulators handling a flexible beam, the major difference lies in the so-called slow system, where it couples with manipulator dynamics.…”