Feedforward boundary control of $2 \times 2$ nonlinear hyperbolic systems with application to Saint-Venant equations

TL;DR

This paper develops a causal, stable feedforward control method for 2x2 hyperbolic systems, extending from linear to nonlinear cases, with applications to Saint-Venant equations and cascade pools, enhancing disturbance rejection.

Contribution

It introduces a novel approach to design stable, causal feedforward controllers for nonlinear hyperbolic systems, extending linear theory to practical, complex applications.

Findings

Effective feedforward control for Saint-Venant equations demonstrated

Modified control law prevents oscillations in cascade pools

Method applicable to nonlinear hyperbolic systems with disturbance rejection

Abstract

Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of $2 \times 2$ hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint- Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.