  Ramp metering, which regulates the flow entering the freeway, is one of the
most effective freeway traffic control methods. This paper introduces an
output-feedback adaptive approach to ramp metering that combines model
predictive control (MPC) with set-membership parameter and state estimation.
The set-membership estimator is designed based on a mixed-monotone embedding of
underlying traffic dynamics. The embedding is also used as the modeling basis
for MPC optimization. For a line freeway network with unknown parameters and
partial measurement on the freeway mainline, we provide sufficient conditions
on the control horizon, cost functions, terminal sets of MPC, and inflow demand
at the ramps such that the queue lengths in the closed-loop system remain
bounded. The sufficient condition on the demand matches the necessary
condition, thereby proving maximal throughput under the proposed controller.
The result is strengthened to input-to-state stability when model parameters
and demand are known. The samples generated by the proposed controller are
shown to facilitate finite time estimation of free-flow model parameters, i.e.,
free-flow speed and turning ratios. Simulation results illustrate stability of
the closed-loop system under the proposed controller even under few mainline
measurements, for which the system becomes unstable under existing approaches.
