Boundary conditions and behavior of the macroscopic fundamental diagram based network traffic dynamics: A control systems perspective

Macroscopic fundamental diagram (MFD), establishing a mapping from the network flow accumulation to the trip completion rate, has been widely used for aggregate modeling of urban traffic network dynamics. Based on the MFD framework, extensive research has been dedicated to devising perimeter control strategies to protect the network from gridlock. Recent research has revealed that the stochasticity and time-varying nature of travel demand can introduce significant scattering in the MFD, thus reducing the definition of the MFD dynamics. However, this type of demand effect on the behavior of the MFD dynamics has not been well studied. In this article, we investigate such effect and propose some appropriate boundary conditions to ensure that the MFD dynamics are well-defined. These boundary conditions can be regarded as travel demand adjustment in traffic rationing. For perimeter control design, a set of sufficient conditions that guarantee the controllability, an important but yet untouched issue, are derived for general multi-region MFD systems. The stability of the network equilibrium and convergence of the network dynamics are then analyzed in the sense of Lyapunov. Both theoretical and numerical results indicate that the network traffic converges to the desired uncongested equilibrium under proper boundary conditions in conjunction with proper control measures. The results are consistent with some existing studies and offer a control systems perspective regarding the demand-oriented behavior analysis of MFD-based network traffic dynamics. A surprising finding is that if the control purpose is to regulate the traffic to a desired level of service, the perimeter control gain can be simply chosen as its desired steady state, that is, the control gain is a constant and can be implemented as proportional control. This property sheds light on the road pricing design based on the MFD framework by minimizing the gap between the actual traffic state and the desired traffic state.