Freeway ramp metering: An LPV set theoretical analysis

The paper contributes to the analysis of freeway traffic flow dynamics by set theoretic methods. First, the macroscopic, non-linear and second-order model of freeway traffic flow dynamics is transformed to an equivalent and quasi Linear Parameter Varying (LPV) representation by steady-state centering and state variable factorization. Second, a polytopic LPV model form is obtained from the quasi model reformulation. The latter polytopic LPV form is then used as a basis for the computation and analysis of disturbance invariant sets. This framework is able to characterize constrained sets of states which can be reached by pure ramp metering control input signals. Furthermore, these sets become invariant to other measured and unmeasured disturbance inputs. The application of disturbance invariant set theory provides an analytical tool for constrained freeway ramp metering describing the set of states being invariant under the system dynamics, measured disturbance and other physical constraints regardless to the value unmeasured disturbance signal. The proposed idea is fully based on the analysis of the (transformed) non-linear macroscopic system and aims at filling the gap between the traffic modeling and quantitative freeway ramp metering.