On the stability of traffic perimeter control in two‐region urban cities

In this paper, stability analysis of traffic control for two‐region urban cities is treated. It is known in control theory that optimality does not imply stability. If the optimal control is applied in a heavily congested system with high demand, traffic conditions might not change or the network might still lead to gridlock. A city partitioned in two regions with a Macroscopic Fundamental Diagram (MFD) for each of the regions is considered. Under the assumption of triangular MFDs, the two‐region MFDs system is modeled as a piecewise second‐order system. Necessary and sufficient conditions are derived for stable equilibrium accumulations in the undersaturated regimes for both MFDs. Moreover, the traffic perimeter control problem for the two‐region MFDs system is formulated. Phase portraits and stability analysis are conducted, and a new algorithm is proposed to derive the boundaries of the stable and unstable regions. Based on these regions, a state‐feedback control strategy is derived. Trapezoidal shape of MFDs are also addressed with numerical solutions.