WP3 - Description

WP3: Control/estimation of the Aw-Rascle-Zhang model.

The second-order Aw-Rascle-Zhang (ARZ) system of first-order hyperbolic Partial Differential Equation (PDE) is a celebrated continuum model of highway traffic flow. The main goal of WP3 is to develop feedforward as well as feedback boundary control laws for a class of 2x2 systems of first-order hyperbolic PDEs. The activities of WP3 are organized as follows.

Task 3.1: Feedforward control design generating the desired reference trajectory.

Each of the control laws developed for the considered ARZ-type models consists of a feedback (see Task 3.2) and a feedforward part. The latter is designed such that, a given traffic system, operates at a desired operating point. In particular, the following feedforward laws are constructed.

A1. For a specific traffic flow model (see Task 3.2.A1 for details) the operating point may be chosen either as constant or varying. In the former case, the set-point may be chosen such that traffic flow is regulated at a uniform profile, which maximizes the total flow at a considered highway stretch. In the latter case, the operating point is constructed such that the flow at the outlet of a given highway stretch follows a desired trajectory, which may vary, depending, for example, on the time of the day or the season, etc., aiming at maximization of the outflow under different traffic conditions.

I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, Feedback control of nonlinear hyperbolic PDE systems inspired by traffic flow models, IEEE Transactions on Automatic Control, vol. 64, pp. 3647--3662, 2019.

I. Papamichail, N. Bekiaris-Liberis, A. I. Delis, D. Manolis, K.-S. Mountakis, I. K. Nikolos, C. Roncoli, & M. Papageorgiou, Motorway traffic flow modelling, estimation and control with vehicle automation and communication systems, Annual Reviews in Control, to appear, 2019.

I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, Traffic flow inspired analysis and boundary control for a class of 2x2 hyperbolic systems, European Control Conference, 2018.

A2. For an ARZ-type model for mixed traffic, i.e., traffic consisting of both manual and Adaptive Cruise Control-equipped (ACC-equipped) vehicles (see Task 3.2.A3 for details), a feedforward control law is presented. The feedforward action guarantees a constant point of operation for a given highway stretch. The set-point may be such that the speed profile of the considered stretch is maximized. More complicated reference trajectories, guaranteeing, for instance, maximization of the speed profile of the whole stretch, may be also constructed, exploiting the distributed nature of the manipulated variable.

N. Bekiaris-Liberis and A. Delis, PDE-based feedback control of freeway traffic flow via time-gap manipulation of ACC-equipped vehicles, IEEE Transactions on Control Systems Technology, provisionally accepted, 2019.

N. Bekiaris-Liberis and A. Delis, Feedback control of freeway traffic flow via time-gap manipulation of ACC-equipped vehicles: A PDE-based approach, IFAC Workshop on Control of Transportation Systems, 2019.

Task 3.2: Design of the PI controller forcing the system to the desired profile, and compensating disturbances and uncertainties on the system.

A1. Continuum second-order traffic flow models, such as, for example, the Aw-Rascle-Zhang model, consist of two PDE states, where, typically, one describes the density dynamics and the other describes the speed evolution. The study of the properties of such second-order traffic flow models is of significant importance because they may capture phenomena, such as, for example, ``stop-and-go" phenomena, which cannot be captured by first-order models (i.e., models that incorporate one PDE state, which describes the density dynamics). Moreover, the problem of regulation of the density and speed profiles to any desired equilibrium point with minimum actuation/measurement instrumentation requirements (e.g., only at the inlet of a highway stretch) is by far a non-trivial problem.

A novel, hyperbolic, nonlinear, 1-D traffic flow model on a bounded domain for relatively crowded roads is proposed. It consists of two first order PDEs with a dynamic boundary condition that involves the time derivative of the velocity. The proposed model has all of the features that are important from a traffic-theoretic point of view, namely: (i) it includes the vehicle conservation equation, (ii) it admits bounded solutions which predict positive values for both density and velocity, (iii) it obeys to the anisotropy principle (i.e., the fact that a vehicle is influenced only by the traffic dynamics ahead of it), and (iv) it allows waves traveling forward exactly in the same speed as traffic. Moreover, a simple (yet, nonlinear) explicit boundary feedback law, which achieves global stabilization for arbitrary equilibria is developed. The stabilizing feedback law adjusts the inlet flow (ramp metering) and depends only on the inlet velocity (i.e., sensor and actuator are collocated); consequently, the measurement/actuation requirements for implementation of the proposed boundary feedback law are minimal.

I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, Traffic flow inspired analysis and boundary control for a class of 2x2 hyperbolic systems, European Control Conference, 2018.

A2. Traffic waves, such as, for example, ``stop-and-go“ waves, not only may jeopardize drivers’ safety and comfort, but they also result in increased fuel consumption. Adaptive Cruise Control (ACC) has been proved to be useful in mitigating the effect of such waves in traffic. Yet, the presence of delays, due to, for example, actuators and sensors, may deteriorate the performance of nominal ACC laws. For this reason, we introduce an ACC design methodology aiming at the compensation of the negative effects of such delays in traffic flow, via utilization of a simple microscopic version of continuum second-order traffic flow models, which may reflect the effect in traffic flow behavior due to the presence of ACC-equipped vehicles.

Specifically, we develop a predictor-based adaptive cruise control design with integral action (based on a nominal constant time-headway policy) for the compensation of large actuator and sensor delays in vehicular systems utilizing measurements of the relative spacing as well as of the speed and the short-term history of the desired acceleration of each ``ego” vehicle.

N. Bekiaris-Liberis, C. Roncoli, and M. Papageorgiou, Predictor-based adaptive cruise control design with integral action, IFAC Symposium on Control in Transportation Systems, 2018.

A3. Although traffic congestion may be unavoidable nowadays, due to the continuous increase in the number of vehicles and in the traffic demand, some of its ramifications may be alleviated employing real-time traffic control strategies. Among other reasons, certain traffic flow instability phenomena, such as, for example, stop-and-go waves, are some of the causes of traffic congestion’s negative consequences on fuel consumption, total travel time, drivers’ comfort, and safety. One promising avenue to traffic flow stabilization is the development of control design tools that exploit the capabilities of automated and connected vehicles, while retaining the distributed nature of traffic flow dynamics. Specifically, it may be beneficial for traffic flow, to appropriately manipulate in real time the ACC (Adaptive Cruise Control) settings of vehicles already equipped with an ACC feature (as it has been also observed in microscopic simulation).

A feedback control strategy is developed for stabilization of traffic flow in congested regime, manipulating the time-gap setting of vehicles equipped with ACC and utilizing a control-oriented, ARZ-type model with ACC (which is shown to possess certain important traffic flow-theoretic properties). Due to the presence (on average) of a certain penetration rate of ACC-equipped vehicles in a given freeway stretch, the traffic flow control problem is recast to the problem of stabilization of a 2x2 linear system of first-order, heterodirectional hyperbolic PDEs with in-domain actuation.