PI Control for the Cascade Channels Modeled by General Saint-Venant Equations

The input-to-state stability of the nonhorizontal cascade channels with different arbitrary cross section, slope, and friction modeled by Saint-Venant equations is addressed in this article. The control input and measured output are both on the collocated boundary. The proportional-integral (PI) control is proposed to study both the exponential stability and the output regulation of closed-loop systems with the aid of the Lyapunov approach. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the nonuniform steady-states of different channels is constructed. We show that by a suitable choice of the boundary feedback controls, the local exponential stability and the input-to-state stability of the nonlinear Saint-Venant equations for the H² norm are guaranteed, then validated with numerical simulations. Meanwhile, the output regulation and the rejection of constant disturbances are realized as well.