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In the literature, networks are usually assumed to operate at their steady states in static traffic assignment models. Given a set of demand management and supply regulation strategies, whether the vehicular traffic on a network would approach or not to a steady state under certain assumptions of behavioral realism is of importance and great interest in practice. Moreover, one primary reason for understanding the long-term traffic evolution dynamics is to control or influence the system trajectory in an efficient manner. To address these problems, we focus on a dynamic feedback control design and its subsequent steady-state analysis for a class of day-to-day (DTD) disequilibrium process so that the network traffic would evolve towards the desired stable traffic equilibrium. We propose an interconnected dynamic system framework for the stability and convergence analysis of DTD traffic adjustment process under the dynamic feedback control. Distinguished from the existing studies, we introduce a notion of output stability as well as the Lyapunov-like conditions to carry out the investigation noting that the steady states of the dynamic tolls that correspond to the Lagrange multipliers of the traffic equilibrium problem can be non-unique. The proposed method gives rise to a new approach to selecting pricing strategy. The proposed method involves only two adjustment parameters with well-identified physical meanings while the requirement on adjustment parameters for convergence and stability is mild. Finally, several numerical examples are presented to illustrate the insightful findings. It is found that, with dynamic feedback control, the network traffic state can converge to the desired equilibrium in a much faster manner than the case without control. The dynamic feedback controller can be implemented as demand management and supply regulation strategies such as road pricing and signal control.

The Macroscopic Fundamental Diagram (MFD) framework has been widely utilized to describe traffic dynamics in urban networks as well as to design perimeter flow control strategies under stationary (constant) demand and deterministic settings. In real world, both the MFD and demand however suffer from various intrinsic uncertainties while travel demand is of time-varying nature. Hence, robust control for traffic networks with uncertain MFDs and demand is much appealing and of greater interest in practice. In literature, there would be a lack of robust control strategies for the problem. One major hurdle is of requirement on model linearization that is actually a basis of most existing results. The main objective of this paper is to explore a new robust perimeter control framework for dynamic traffic networks with parameter uncertainty (on the MFD) and exogenous disturbance induced by travel demand. The disturbance in question is in general time-varying and stochastic. Our main contribution focuses on developing a control-Lyapunov function (CLF) based approach to establishing a couple of universal control laws, one is almost smooth and the other is Bang-bang like, for different implementation scenarios. Moreover, it is indicated that the almost smooth control is more suited for road pricing while the Bang-bang like control for signal timing. In sharp contrast to existing methods, in which adjusting extensive design parameters are usually needed, the proposed methods can determine the control in an automatic manner. Furthermore, numerical results demonstrate that the control can drive the system dynamics towards a desired equilibrium under various scenarios with uncertain MFDs and travel demand. Both stability and robustness can be substantially observed. As a major consequence, the proposed methods achieve not only global asymptotic stability but also appealing robustness for the closed-loop traffic system.

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.