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

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.