State Estimation for Polyhedral Hybrid Systems and Applications to the Godunov Scheme for Highway Traffic Estimation

This paper 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, with a state space partitioned into an exponential number of polyhedra in which one mode is active. We propose a feasible approach based on the interactive multiple model (IMM) which is a widely used algorithm for estimation of hybrid systems in the scientific community. The number of modes is reduced based on the geometric properties of the polyhedral partition. The k-means algorithm is also applied on historical data to partition modes into clusters. The performance of these algorithms are compared to the extended Kalman filter and the ensemble Kalman filter in the context of Highway Traffic State Estimation. In particular, we use sparse measurements from loop detectors along a section of the I-880 to estimate the state density for our numerical experiments.