Analysis and Control of a Non-Standard Hyperbolic PDE Traffic Flow Model
For traffic flow modelers and control engineers, this work provides a theoretically sound model with minimal measurement requirements for stabilization, but the approach is incremental as it extends existing PDE control techniques to a specific non-standard model.
The paper introduces a non-standard hyperbolic PDE traffic flow model with anisotropic properties and forward-only information propagation, proves global existence and uniqueness of classical solutions, and demonstrates global stabilization via a boundary feedback law requiring only inlet velocity measurements.
The paper provides results for a non-standard, hyperbolic, 1-D, nonlinear traffic flow model on a bounded domain. The model consists of two first-order PDEs with a dynamic boundary condition that involves the time derivative of the velocity. The proposed model has features that are important from a traffic-theoretic point of view: is completely anisotropic and information travels forward exactly at the same speed as traffic. It is shown that, for all physically meaningful initial conditions, the model admits a globally defined, unique, classical solution that remains positive and bounded for all times. Moreover, it is shown that global stabilization can be achieved for arbitrary equilibria by means of an explicit boundary feedback law. The stabilizing feedback law depends only on the inlet velocity and consequently, the measurement requirements for the implementation of the proposed boundary feedback law are minimal. The efficiency of the proposed boundary feedback law is demonstrated by means of a numerical example.