Backstepping Stabilization of the Linearized Saint-Venant-Exner Model

TL;DR

This paper develops backstepping boundary controllers for the linearized Saint-Venant-Exner PDE model of water-sediment flow, ensuring exponential stabilization in both subcritical and supercritical regimes without restrictive conditions.

Contribution

It introduces a novel backstepping control and observer design for the coupled SVE PDEs, handling arbitrary parameters and flow regimes with guaranteed exponential stability.

Findings

Achieves exponential stabilization of the SVE model.

Designs effective boundary control strategies for different flow regimes.

Provides stability guarantees without restrictive assumptions.

Abstract

Using the backstepping design, we achieve exponential stabilization of the coupled Saint-Venant-Exner (SVE) PDE model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water-sediment interaction under subcritical or supercritical flow regime. The studied SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control (with actuation located only at downstream) strategy is adopted. A full state feedback controller is firstly designed, which guarantees the exponential stability of the closed-loop control system. Then, an output feedback controller is designed based on the reconstruction of the distributed state with a backstepping observer. It also guarantees the exponential stability of the closed-loop control system. The flow regime depends on the dimensionless Froude number Fr, and both our controllers can deal with the subcritical (Fr < 1) and supercritical (Fr > 1) flow regime. They achieve the exponential stability results without any restrictive conditions in contrast to existing results.