Stabilization of the linearized water tank system
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.
Type
Publication
In Archive for Rational Mechanics and Analysis
Authors
Christophe Zhang
(he/him)
researcher
I am currently in secondment at Inria Paris, as a researcher in the CAGE team. I work on control, stabilization and optimal control problems in infinite dimension (PDEs and evolution equation), with a particular focus on constrained control.
I have also dabbled in data-driven modelling and control, in particular methods involving the Koopman operator.
More recently, I have taken an interest in reachability analysis. I co-supervised the PhD thesis of Ivan Hasenohr with Camille Pouchol and Yannick Privat, during which we developed computer-assisted proofs of reachability and non-reachability.