Robust boundary stabilization of a vibrating rectangular plate using disturbance adaptation

Summary This paper considers robust boundary control with disturbance adaptation to stabilize the vibration of a rectangular plate under disturbances with unknown upper‐bounds. Disturbances are considered to be distributed both over the plate interior domain and along the boundary (in‐domain and boundary distributed disturbances). Applying Hamilton's principle, the dynamics of the plate is represented in the form of a fourth‐order partial differential equation subject to static and dynamic boundary conditions. The proposed model considers the membrane effect of axial force and the effect of actuator mass dynamics on the plate vibration. A robust boundary control is established that stabilizes the plate in presence of both in‐domain and boundary disturbances. A rigorous Lyapunov stability analysis shows that the vibration of the plate is uniformly ultimately bounded and converges to the vicinity of zero by proper selection of control gains. For the vanishing in‐domain disturbances, it is seen that exponential stability is achieved by the proposed control. Next, a disturbance adaptation law is introduced to stabilize the plate vibration in response to disturbances with unknown bounds, and the stability of the robust boundary control with disturbance adaptation is studied using Lyapunov. Simulation results verify the efficiency of the suggested control.