In this paper, general in-plane trajectory tracking problem of a flexible beam is studied. To obtain the dynamic equations of motion of the beam, Hamiltonian dynamics is used and then Lagrange’s equations of beam dynamics and corresponding expressions for boundary conditions are derived. Resulting equations show that the coupled beam dynamics including beam vibration and its rigid in-plane 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 the mass center of the beam track a desired trajectory while suppressing its 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 (BC) method does not require any information about vibration along the beam except at the end points, nor requires discretizing the partial differential equation of beam vibration to a set of ordinary differential equations. So, the method avoids the need for instruments to measure data from vibration of any point along the beam or designing an observer for estimating this information. Also, the proposed method prevents control spillover due to discretization. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and the slow controller provides very good trajectory tracking with acceptable actuating forces/moments.

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The boundary stabilization of a coupled fluid-structure system consisting of a vibrating parachute dam in contact with a fluid is studied in this paper. The parachute dam dynamics is presented by nonlinear partial differential equations. The fluid is assumed to be Newtonian, barotropic, and compressible. For the stability analysis of the coupled system, the boundary control method is used; a boundary feedback is constructed to stabilize the vibrations of the dam and the fluid simultaneously. The control force consists of the feedback from dam tension at its end. Moreover, the exponential stabilization of the parachute dam is achieved using a Lyapunov functional and boundary feedback.

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.