Feedback Stabilization of Tank-Liquid System with Robustness to Surface Tension

TL;DR

This paper develops a robust feedback control method for stabilizing a viscous liquid in a tank described by PDEs, ensuring spill-free exponential convergence despite surface tension effects.

Contribution

It introduces a Control Lyapunov Functional-based feedback law that guarantees robust, exponential stabilization of the tank-liquid system with surface tension effects.

Findings

Achieves spill-free exponential stabilization.

Provides robustness to surface tension variations.

Constructs a parameterized family of stabilizing sets.

Abstract

We construct a robust stabilizing feedback law for the viscous Saint-Venant system of Partial Differential Equations (PDEs) with surface tension and without wall friction. The Saint-Venant system describes the movement of a tank which contains a viscous liquid. We assume constant contact angles between the liquid and the walls of the tank and we achieve a spill-free exponential stabilization with robustness to surface tension by using a Control Lyapunov Functional (CLF). The proposed CLF provides a parameterized family of sets which approximate the state space from the interior. Based on the CLF, we construct a nonlinear stabilizing feedback law which ensures that the closed-loop system converges exponentially to the desired equilibrium point in the sense of an appropriate norm.