LQ Control of Traffic Flow Models via Variable Speed Limits

In this paper, an extension of a linear control design for hyperbolic linear partial differential equations is presented for a first-order traffic flow model. Starting from the Lighthill-Whitham-Richards (LWR) model, variable speed limit control (VSL) is applied through a modification of Greenshield's equilibrium flow model. Then, an optimal linear quadratic (LQ) controller is designed on the linear LWR model. The LQ state feedback function is found via the solution of a Riccati differential equation. Unlike previous studies, the control input is the rate of change of the input, not the input itself. The proposed controller is then verified on both the linear and nonlinear models. In both cases, the controller is able to drive the system to a desired density profile. In the nonlinear application, a higher control gain is needed to achieve similar results as in the linear case.

I. Introduction

Congestion is a growing concern in major cities as it leads to increased energy usage and pollution [1]. In 2022, the average commuter in the Unites States spent around 73 extra hours in traffic. This is up over 40% from 2016 [2], [3]. The modeling of traffic flow and its control offer opportunities to mitigate congestion, improve travel time, and reduce overall energy usage associated with the transportation sector.