  We solve the output-feedback stabilization problem for a tank with a liquid
modeled by the viscous Saint-Venant PDE system. The control input is the
acceleration of the tank and a Control Lyapunov Functional methodology is used.
The measurements are the tank position and the liquid level at the tank walls.
The control scheme is a combination of a state feedback law with functional
observers for the tank velocity and the liquid momentum. Four different types
of output feedback stabilizers are proposed. A full-order observer and a
reduced-order observer are used in order to estimate the tank velocity while
the unmeasured liquid momentum is either estimated by using an appropriate
scalar filter or is ignored. The reduced order observer differs from the full
order observer because it omits the estimation of the measured tank position.
Exponential convergence of the closed-loop system to the desired equilibrium
point is achieved in each case. An algorithm is provided that guarantees that a
robotic arm can move a glass of water to a pre-specified position no matter how
full the glass is, without spilling water out of the glass, without residual
end point sloshing and without measuring the water momentum and the glass
velocity. Finally, the efficiency of the proposed output feedback laws is
validated by numerical examples, obtained by using a simple finite-difference
numerical scheme. The properties of the proposed, explicit, finite-difference
scheme are determined.
