Real-time estimation of distributed parameters systems: application to traffic monitoring

The coupling of the physical world with information technology promises to help meet increasing demands for efficient, sustainable, and secure management of our built infrastructure and natural environment. A mathematical abstraction of the physical environment can be achieved the form of distributed parameters systems described by partial differential equations. Yet, initial and boundary conditions, and other model parameters necessary for complete characterization of the model are often unknown, driving the need for distributed sensing of the physical environment. Because of the nonlinearities and distributed nature inherent to these processes, efficient estimation algorithms to reconcile modeling and measurement errors in real-time remains an open challenge for many applications.

This work investigates the problem of real-time estimation of distributed parameters systems in the context of monitoring traffic. The recent explosion of cell phones with Internet connectivity and GPS are rapidly increasing sensor coverage on roadways – with a catch. GPS velocity measurements, further degraded to preserve user anonymity, cannot be easily iterated into the well-known density-based network of partial differential equations typically used to describe traffic. This challenge is circumvented by transforming the density-based traffic model into a new but equivalent evolution equation for velocity, which retains the nonlinearity and nondifferentiability of traffic due to shocks. Because of this transformation, GPS velocity measurements become a direct observation of the state. The resulting state estimation problem is then solved in real-time for large road networks with an ensemble Kalman filtering algorithm. This approach has been implemented on the Northern California highway network, which receives several million measurements per day, and it has served as the core real-time traffic engine inside the Mobile Millennium traffic monitoring system at UC Berkeley for two years.