Modeling, Estimation and Control of Distributed Parameter Systems: Application to Transportation Networks
The research presented in this dissertation is motivated by the need for well-posed mathematical models of traffic flow for data assimilation of measurements from heterogeneous sensors and flow control on the road network.
A new 2 X 2 partial differential equation (PDE) model of traffic with phase transitions is proposed. The system of PDEs constitutes an extension to the Lighthill-Whitham-Richards model accounting for variability around the empirical fundamental diagram in the congestion phase. A Riemann solver is constructed and a variation on the classical Godunov scheme, required due to the non-convexity of the state-space, is implemented. The model is validated against experimental vehicle trajectories recorded at high resolution, and shown to capture complex traffic phenomena such as forward-moving discontinuities in the congestion phase, which is not possible with scalar hyperbolic models of traffic flow. A corresponding mesoscopic interpretation of these phenomena in terms of drivers behavior is proposed.
The structure of the uncertainty distribution resulting from the propagation of initial uncertainty in weak entropy solutions to first order scalar hyperbolic conservation laws is characterized in the case of a Riemann problem. It is shown that at shock waves, the uncertainty is a mixture of the uncertainty on the left and right initial condition, and the consequences of this specific class of uncertainty on estimation accuracy is assessed in the case of the extended Kalman filter and the ensemble Kalman filter. This sets the basis for filtering-based traffic estimation and traffic forecast with appropriate treatment of the specific type of uncertainty arising due to the mathematical structure of the model used, which is of critical importance for road networks with sparse measurements.
As a first step towards controlling general distributed models of traffic, a benchmark problem is investigated, in the form of a first order scalar hyperbolic conservation law. The weak entropy solution to the conservation law is stabilized around a uniform solution using boundary actuation. The control is designed to be compatible with the proper weak boundary conditions, which given specific assumptions guarantees that the corresponding initial-boundary value problem is well-posed. A semi-analytic boundary control is proposed and shown to stabilize the solution to the scalar conservation law. The benefits of introducing discontinuities in the solution are discussed. For traffic applications, this method allows us to pose the problem of ramp metering on freeways for congestion control and reduction of the amplitude of the capacity drop, as well as the problem of vehicular guidance for phantom jam stabilization on road networks, in a proper mathematical framework.