On Structural Properties of Optimal Feedback Control of Traffic Flow under the Cell Transmission Model

In this paper, we investigate the structure of the finite-time optimal feedback control for a linear freeway traffic network modeled by the Cell Transmission Model. Piecewise affine supply and demand functions are considered and optimization with respect to a general linear objective function is studied. Using the framework of multi-parametric linear programming, we show that the optimal feedback control can be represented in a closed-form by a piecewise affine function on polyhedra of the network traffic density. The resulting optimal feedback control law, however, has a centralized structure and requires instantaneous access to the state of the entire network that may lead to prohibitive communication requirements in large-scale complex networks. We subsequently examine the design of a decentralized optimal feedback controller with a one-hop information structure, wherein the optimum outflow rate from each segment of the network depends only on the density of that segment and the density of the segment immediately downstream. The proposed control strategy has a simple closed-form representation and is scalable to arbitrarily large networks. Moreover, we demonstrate that, with respect to certain meaningful linear performance indexes, the centralized optimal controller has a decentralized realization.