Spill-Free Transfer and Stabilization of Viscous Liquid

Abstract

This article studies the feedback stabilization problem of the motion of a tank that contains an incompressible, Newtonian, viscous liquid. The control input is the force applied on the tank and the overall system consists of two nonlinear partial differential equations and two ordinary differential equations. Moreover, a spill-free condition is required to hold. By applying the control Lyapunov functional methodology, a set of initial conditions (state space) is determined for which spill-free motion of the liquid is possible by applying an appropriate control input. Semi-global stabilization of the liquid and the tank by means of a simple feedback law is achieved, in the sense that for every closed subset of the state space, it is possible to find appropriate controller gains, so that every solution of the closed-loop system initiated from the given closed subset satisfies specific stability estimates. The closed-loop system exhibits an exponential convergence rate to the desired equilibrium point. The proposed stabilizing feedback law does not require measurement of the liquid level and velocity profiles inside the tank and simply requires measurements of 1) the tank position error and tank velocity, 2) the total momentum of the liquid, and 3) the liquid levels at the tank walls. The obtained results allow an algorithmic solution of the problem of the spill-free movement and slosh-free settlement of a liquid in a vessel of limited height (such as water in a glass) by a robot to a prespecified position, no matter how full the vessel is.