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This paper studies the feedback stabilization problem of the motion of a tankthat contains an incompressible, Newtonian, viscous liquid. The control inputis the force applied on the tank and the overall system consists of twononlinear Partial Differential Equations and two Ordinary DifferentialEquations. Moreover, a spill-free condition is required to hold. By applyingthe Control Lyapunov Functional methodology, a set of initial conditions (statespace) is determined for which spill-free motion of the liquid is possible byapplying an appropriate control input. Semi-global stabilization of the liquidand the tank by means of a simple feedback law is achieved, in the sense thatfor every closed subset of the state space, it is possible to find appropriatecontroller gains, so that every solution of the closed-loop system initiatedfrom the given closed subset satisfies specific stability estimates. Theclosed-loop system exhibits an exponential convergence rate to the desiredequilibrium point. The proposed stabilizing feedback law does not requiremeasurement of the liquid level and velocity profiles inside the tank andsimply requires measurements of: (i) the tank position error and tank velocity,(ii) the total momentum of the liquid, and (iii) the liquid levels at the tankwalls. The obtained results allow an algorithmic solution of the problem of thespill-free movement and slosh-free settlement of a liquid in a vessel oflimited height (such as water in a glass) by a robot to a pre-specifiedposition, no matter how full the vessel is.

Spill-Free Transfer and Stabilization of Viscous Liquid