Set‐theoretic analysis of the isolated ramp metering problem

Abstract

Analysis of ramp metering, as an effective freeway traffic control solution, is focused in the paper. The studied traffic control problem is discussed in a set-theoretic context in order to quantitatively characterize its effectivenes. The notions of maximal robust controlled invariant set, as well as t-step robust controllable set are defined and used for analyzing the ramp metering problem independently of the control policy applied. Algorithms are developed to compute these sets, with special attention to their practical interpretations. Numerical examples with graphical representation of the proposed methodology are given to examine local ramp metering and conclude implications to control strategies. 1247The ramp metering control problem has been investigated in [5] and [10] by using optimal control considerations. The first-order macroscopic freeway model of and Richards [12] was used for representing dynamics, and the total time spent (TTS) objective is minimized subject to the dynamical equations. The mathematical analysis of the constrained optimization showed that it is not beneficial to meter ramps when the freeway is either uniformly congested or uniformly uncongested. At the same time, under mixed traffic conditions, ramp metering is proved to improve network performance. Based on a first-order description, Gomes and Horowitz [13,14] provided a similar analysis of the optimal control problem. Freeway dynamics were represented by the asymmetric cell transmission model (CTM) [15], and numerical conditions were derived for the equivalent linear formulation of the non-linear optimization problem. As a consequence, a nearglobal optimal ramp metering law has also been established. The characterization of the equilibrium solutions of the CTM and its implication for ramp metering have been carried out in [16]. By exploring the equilibrium of the model, it has been shown that discharge flow can be increased by applying ramp metering. In the case of an initially congested freeway, the existence of a ramp metering law, which dissolves congestion and improves network efficiency, has been derived. Furthermore, it has been shown that the TTS objective can be minimized by controlling the on-ramp volume even when the congestion cannot be resolved completely. Recently, Wang et al. [17] provided a control theoretic analysis of the local ramp metering problem. Specific ramp control laws of ALINEA and PI-ALINEA have been considered and analyzed by using linear and non-linear stability techniques. Design conditions for the feedback gains were concluded from the results of the analysis.Within this line of research, this paper contributes to the mathematical analysis of the ramp metering problem. The presence of controlled and uncontrolled variables with physical bounds implies a set-theoretic setup for investigating the effect of ramp metering. For this purpose, a systematic, settheoretic framework is proposed and developed throughout the paper. First, the non-linear model is embedded into a linear parameter varying (LP...

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Editorial for the Special Issue on recent trends in traffic modelling and control

“…It is also worth noting that the modern vision of traffic control encompasses a strong integration between freeway traffic control systems and urban traffic management systems, in order to design efficient ‘smart mobility’ systems. For this reason, this Special Issue includes contributions relevant to both freeway traffic systems and urban traffic networks . These contributions are able to provide a significant scientific, technological, social and economic impact, with immediate effects on quality of life and environmental health, especially in terms of congestion reduction, efficient use of energy resources, reduction of costs associated with fuel consumption and wear of vehicles.…”

“…One of the most common and effective freeway traffic control actions, that is, ramp metering, is analysed in . The considered traffic control problem is discussed in a set‐theoretic context.…”