We solve the boundary output-feedback tracking problem for a moving rigid body-liquid system described by the linearized Saint-Venant PDEs. The control input is an external force acting on the rigid body. The control law is a boundary state-feedback type augmented with a Luenberger boundary observer, leading to a dynamic feedback law. The main complexity lies in the fact that we can only measure the position and velocity of the rigid body. To get around this limitation, the observer estimates the liquid levels at the cavity walls. We utilize a control Lyapunov functional methodology (with two different functionals for the observer and the dynamic feedback law) and achieve a slosh-free asymptotic convergence to the prespecified equilibrium point (in the sense of an appropriate norm). Finally, through a simulation, we demonstrate the feasibility of the boundary output-feedback controller for the (linear and nonlinear) system.
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