PI controllers for 1-D nonlinear transport equation

Jean-Michel Coron (1, 2) , Amaury Hayat (1, 2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry

In this paper, we introduce a method to get necessary and sufficient stability conditions for systems governed by 1-D nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified.

Mots clés: Distributed parameter system; Boundary control; Lyapunov functions; Partial differential equation; Exponential stability; Hyperbolic Equation

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Jean-Michel Coron, Amaury Hayat. PI controllers for 1-D nonlinear transport equation. IEEE Transactions on Automatic Control, 2019, 64 (11), pp.4570-4582. ⟨hal-01766261⟩