Backstepping stabilization of the linearized Saint-Venant–Exner model

Abstract

Using backstepping design, exponential stabilization of the linearized Saint-Venant–Exner (SVE) model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water–sediment interaction, is achieved. The linearized SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control strategy with actuation located only at the downstream gate is employed. A full state feedback controller is designed which guarantees exponential stability of the desired setpoint of the resulting closed-loop system. Using the reconstruction of the distributed state through a backstepping observer, an output feedback controller is established, resulting in the exponential stability of the closed-loop system at the desired setpoint. The proposed state and output feedback controllers can deal with both subcritical and supercritical flow regimes without any restrictive conditions.