Linear Quadratic Regulation for First Order Hyperbolic PDEs

TL;DR

This paper develops an LQR-based control approach for first order hyperbolic PDEs, deriving a Riccati PDE and optimal feedback, demonstrated on chemical reactor and traffic flow models.

Contribution

It introduces a novel LQR framework for hyperbolic PDEs, deriving the Riccati PDE and explicit feedback control, with applications to chemical and traffic models.

Findings

Derived Riccati PDE for hyperbolic PDEs

Obtained explicit optimal feedback control

Validated approach on chemical reactor and traffic models

Abstract

We consider transport processes that are modeled by first order hyperbolic partial differential equations. Our goal is to find a full state feedback that makes a given reference profile locally asymptotically stable. To accomplish this we employ Linear Quadratic Regulation (LQR) with finite dimensional patch or point control actuation. We derive the Riccati partial differential equation whose solution is the kernel of the optimal cost. The optimal state feedback is also found. The derivation is accomplished by elementary techniques such as integration by parts and completing the square. We apply this theory to two examples that have appeared in the literature and that were solved by a modification of LQR. The first example deals with a model of a fixed-bed chemical reactor and the second example deals with traffic congestion on a stretch of freeway.