Backstepping stabilization of the linearized Saint-Venant–Exner model

Abstract

a b s t r a c tUsing 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 closedloop 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.

A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope

“…Later on, this result was generalized in [27] to n × n quasilinear hyperbolic systems. In particular, in the context of the Saint-Venant equations, the backstepping method has been used to achieve exponential stabilization of the linearized Saint-Venant Exner equations with arbitrary slope or friction in [20], both in subcritical and supercritical regime. It has also been used in [21] to stabilize a linearized bilayered Saint-Venant model, a 4 × 4 system of two Saint-Venant systems interacting with each other.…”

Disturbance rejection in general heterodirectional 1-D linear hyperbolic systems using collocated sensing and control

“…Systems of this type have lately been subject to extensive research due to the vast amount of physical systems that can be modeled this way. Relevant systems are heat exchangers (Xu & Sallet, 2010), transmission lines (Curró, Fusco & Manganaro, 2011), road traffic (Amin, Hante & Bayen, 2008), oil wells (Landet, Pavlov & Aamo, 2013) and multiphase flow (Di Meglio, 2011;Diagne, Diagne & Tang, in press), to mention a few.…”

Boundary control of linear stochastic reaction‐diffusion systems

“…, then, under the observer-based boundary controller (19), the disturbed linear SRDSs (18) can achieve the finite horizon mean-square H ∞ performance.…”

“…In the work of Pan et al, a boundary control was used to achieve the mean‐square asymptotical stability for SRDSs with Neumann boundary conditions, but the result in the aforementioned work is conservative because only one single point's state was used to design the boundary controller. As a classical method for studying boundary control of deterministic systems, the backstepping method, until recently, can only be used to study deterministic linear systems . In the works of Liu and Diagne et al, the backstepping method is applied to scalar parabolic systems.…”

“…As a classical method for studying boundary control of deterministic systems, the backstepping method, until recently, can only be used to study deterministic linear systems. [17][18][19][20][21] In the works of Liu 17 and Diagne et al, 18 the backstepping method is applied to scalar parabolic systems. In other works, [19][20][21] the backstepping design of multivariable systems is considered.…”