A universal feedback control policy for arbitrary networks of signalized intersections

The paper presents a universal feedback control policy (UCP) for an arbitrary road network of signalized intersections. At each time and at each intersection, the control simultaneously activates one set of compatible phases or streams. The control is a function of the current state of the road network, the vector of all queue lengths in the network. The activation of a phase permits vehicles to move from the input link of the phase to its output link at a pre-specified random saturation rate. The stochastic demand is modeled by vehicles that enter the network according to a stationary Bernoulli process. Vehicles then make turns randomly with pre-specified routing (turn) probabilities, and they leave the network upon reaching an exit link. A control policy is said to stabilize a particular demand if the mean queue length at each intersection stays bounded. UCP is universal because it stabilizes a demand whenever there exists any stabilizing control policy for that demand. UCP is local: the control at an intersection depends only on the queues adjacent to the intersection. Remarkably, UCP does not require knowledge of the demand, although it does require knowledge of the turn probabilities. An adaptive UCP (AUCP) is also presented. AUCP estimates the turn probabilities on-line, hence it requires no prior knowledge of the demand or turn probabilities. The properties of UCP are proved using a (stochastic) Lyapunov function construction, which was first formulated in the context of multi-hop wireless networks. The proof makes use of a linear program previously developed for the design of a fixed-time control that maximizes reserve capacity. It is also shown that if the average demand is known, and there exists a stabilizing policy, there is also a fixed-time stabilizing control. The feedback function of UCP can be interpreted as a queue-minimizing control. The interpretation suggests other stabilizing feedback control policies.