Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

TL;DR

This paper develops a robust feedback stabilization method for a liquid-filled tank with wall friction, ensuring spill-free control even with uncertain friction parameters, using Lyapunov functionals.

Contribution

It introduces a novel CLF-based feedback law that guarantees stabilization despite wall friction uncertainties, extending results to both constant and variable friction coefficients.

Findings

Achieved spill-free exponential stabilization with robustness to wall friction

Designed feedback law without requiring exact friction coefficient knowledge

Extended stabilization results to cases with variable friction coefficients

Abstract

We solve the feedback stabilization problem for a tank, with friction, containing a liquid modeled by the viscous Saint-Venant system of Partial Differential Equations. A spill-free exponential stabilization is achieved, with robustness to the wall friction forces. A Control Lyapunov Functional (CLF) methodology with two different Lyapunov functionals is employed. These functionals determine specific parameterized sets which approximate the state space. The feedback law is designed based only on one of the two functionals (which is the CLF) while the other functional is used for the derivation of estimates of the sup-norm of the velocity. The feedback law does not require the knowledge of the exact relation of the friction coefficient. Two main results are provided: the first deals with the special case of a velocity-independent friction coefficient, while the second deals with the general case. The obtained results are new even in the frictionless case.