PI controllers for the general Saint-Venant equations

TL;DR

This paper demonstrates that a simple PI controller can ensure exponential stability of the nonlinear Saint-Venant equations in a river channel, regardless of physical parameters or steady-state conditions, using Lyapunov functions.

Contribution

It provides explicit, parameter-independent conditions for PI control gains to stabilize the nonlinear shallow water equations in various scenarios.

Findings

Explicit stability condition on PI gain independent of physical parameters

Exponential stability achieved for steady-states and slowly varying trajectories

Input-to-State stability established for time-dependent inflow disturbances

Abstract

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 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 of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.