State Estimation for the discretized LWR PDE using explicit polyhedral representations of the Godunov scheme

This article investigates the problem of estimating the state of discretized hyperbolic scalar partial differential equations. It uses a Godunov scheme to discretize the so-called Lighthill-Whitham-Richards equation with a triangular flux function, and proves that the resulting nonlinear dynamical system can be decomposed in a piecewise affine manner. Using this explicit representation, the system is written as a switching dynamical system (hybrid system), with an exponential number of modes. The estimation problem is posed using Kalman filtering in each of the linear mode, and the approach becomes computationally tractable by tracking the mode evolution as the estimation is performed at each time step. Numerical results are presented using the Mobile Millennium data set, and compared to results obtained using ensemble Kalman filtering, which is used for estimation in traffic monitoring.