Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs

TL;DR

This paper develops a PDE backstepping control method for stabilizing and controlling large systems of coupled hyperbolic PDEs with arbitrary numbers and directions of transport, using only boundary actuation.

Contribution

It extends control techniques to fully general coupled hyperbolic PDE systems with actuation at a single boundary, including homodirectional and heterodirectional cases.

Findings

Successfully stabilizes systems with arbitrary numbers of PDEs in both directions.

Provides boundary control laws for full-state and output feedback stabilization.

Enables trajectory planning and tracking for coupled hyperbolic PDEs.

Abstract

Research on stabilization of coupled hyperbolic PDEs has been dominated by the focus on pairs of counter-convecting ("heterodirectional") transport PDEs with distributed local coupling and with controls at one or both boundaries. A recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport PDEs that convect at different speeds against the direction of the PDE whose boundary is actuated. In this paper we present a solution to the fully general case, in which the number of PDEs in either direction is arbitrary, and where actuation is applied on only one boundary (to all the PDEs that convect downstream from that boundary). To solve this general problem, we solve, as a special case, the problem of control of coupled "homodirectional" hyperbolic linear PDEs, where multiple transport PDEs convect in the same direction with arbitrary local coupling. Our approach is based on PDE backstepping and yields solutions to stabilization, by both full-state and observer-based output feedback, trajectory planning, and trajectory tracking problems.