PI controllers for the general Saint-Venant equations

Amaury Hayat

doi:10.5802/jep.210

PI controllers for the general Saint-Venant equations
[Contrôles PI pour les équations de Saint-Venant générales]

Amaury Hayat ¹

¹ Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique, École des Ponts ParisTech 6-8 avenue Blaise Pascal, Cité Descartes – Champs-sur-Marne, 77455 Marne-la-Vallée, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1431-1472.

Résumé
Abstract

Nous étudions la stabilité exponentielle en norme $H^{2}$ des équations de Saint-Venant non-linéaires avec un frottement arbitraire et une pente. Le système est régulé avec un unique contrôle proportionnel-intégral (PI) à une extrémité du canal. En utilisant une fonction de Lyapunov adéquate, nous trouvons une condition simple et explicite sur le contrôle PI pour assurer la stabilité exponentielle de tous les états stationnaires. Cette condition est indépendante de la pente, du coefficient de friction, de la longueur de la rivière, ou encore de la perturbation du débit entrant. Plus surprenant : elle peut être rendue indépendante de l’état stationnaire considéré. Lorsque la perturbation du débit entrant dépend du temps et qu’il n’existe pas d’état stationnaire, nous pouvons quand même montrer l’« input-to-state stability » (ISS) du système. Par ailleurs, une légère modification du contrôle PI permet de retrouver la stabilité exponentielle des trajectoires à variation lente.

We study the exponential stability in the $H^{2}$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain of the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the input-to-state stability (ISS) of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.

Reçu le : 2021-08-05
Accepté le : 2022-08-28
Publié le : 2022-10-05

DOI : 10.5802/jep.210

Classification : 93D15, 93D23, 93D25, 35B35, 35F30
Keywords: Saint-Venant equations, proportional integral control, exponential stability, input-to-state stability, nonlinear systems, partial differential equations
Mot clés : Équations de Saint-Venant, contrôle proportionnel intégral, stabilité exponentielle, stabilité de l’entrée à l’état, systèmes non-linéaires, équations aux dérivées partielles

Affiliations des auteurs :

Amaury Hayat ¹

¹ Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique, École des Ponts ParisTech 6-8 avenue Blaise Pascal, Cité Descartes – Champs-sur-Marne, 77455 Marne-la-Vallée, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Amaury Hayat. PI controllers for the general Saint-Venant equations. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1431-1472. doi : 10.5802/jep.210. https://jep.centre-mersenne.org/articles/10.5802/jep.210/

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