Adaptive Boundary Observers for Hyperbolic PDEs With Application to Traffic Flow Estimation

This article studies the problem of adaptive boundary observer design for a class of linear hyperbolic partial differential equations (PDEs) subject to in-domain and boundary parameter uncertainties. Based on the swapping transformation technique, a Luenberger-type boundary observer with the least squares parameter estimation law is designed, which relies only on the measurements at boundaries of the system. By employing the Lyapunov function method, we prove that the exponential convergence of the proposed boundary observer design scheme is guaranteed when the observer gains satisfy a set of matrix inequality conditions. Finally, we illustrate the effectiveness of the adaptive estimation scheme by applying it to the linearized Aw-Rascle-Zhang traffic flow model involving the in-domain relaxation time and boundary flux fluctuation uncertainties.