PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models

Abstract

This paper investigates the proportional–integral (PI) boundary feedback control for the linear hyperbolic systems of balance laws which control and output measures are located at the boundaries. We address the issue of feedback stabilization by means of PI boundary controllers. By constructing a new weighted Lyapunov function, the sufficient conditions in terms of matrix inequalities are developed for the exponential stability of closed-loop systems. These results are illustrated by the linearized Aw–Rascle–Zhang (ARZ) traffic flow model. We design a PI boundary controller to stabilize the oscillations of the traffic parameters of a freeway segment and evaluate the performance with numerical simulations.