Stabilization of the linearized water tank system

TL;DR

This paper investigates the controllability and stabilization of linearized water tank systems around non-uniform steady states, demonstrating controllability in Sobolev spaces and designing explicit feedback laws for exponential stabilization.

Contribution

It proves controllability of the linearized water tank system around non-uniform steady states and develops explicit feedback laws for exponential stabilization.

Findings

Controllability in Sobolev spaces for non-uniform steady states.

Explicit Proportional Integral feedback law for stabilization.

Stability achieved with arbitrarily large decay rate.

Abstract

In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.