WP2: Control/estimation of the Moskowitz model.

The Moskowitz Hamilton-Jacobi (HJ) Partial Differential Equation (PDE) is a well-know continuum model of highway traffic flow. It is the main goal of WP2 to develop boundary feedback laws as well as boundary observers for a class of (potentially viscous) HJ PDEs. The activities of WP2 are organized as follows.

Task 2.1: Design of a boundary feedback controller for the Moskowitz model.

Undoubtedly, maximizing the throughput at bottleneck locations is one of the most important goals of any traffic flow control algorithm. For control purposes, accurate traffic flow models may distinguish the traffic dynamics at bottlenecks from the dynamics of the rest of traffic. Typically, traffic flow control at a bottleneck area may be achieved manipulating the flow at a location upstream of the bottleneck. This location may be either fixed, such as, e.g., in the case of ramp metering or moving, such as, e.g., in the case where Variable Speed Limits (VSLs) are applied, activated at a location closer to or farther from the bottleneck location (depending, potentially, on the flow/density at the bottleneck or the VSL-application areas). Such traffic flow models may consist, in addition of a PDE state, of an ODE state that describes the traffic dynamics at the bottleneck area. Motivated by the need for control at bottleneck areas and capitalizing on the close relation of the cumulative number-of-vehicles equation, in particular, the Moskowitz model, to the conservation-of-vehicles equation, in particular, the Lighthill-Whitham-Richards (LWR) model (the cumulative number of vehicles may be derived from the density of vehicles via a certain integral transformation and, respectively, the density of vehicles may be obtained from the cumulative number of vehicles via differentiation), the following activities are performed.

A1. A methodology for stabilization of general nonlinear systems with actuator dynamics governed by general, quasilinear, first-order hyperbolic PDEs is presented. Since for such PDE-ODE cascades the speed of propagation depends on the PDE state itself (which implies that the prediction horizon cannot be a priori known analytically), the key design challenge is the determination of the predictor state. This challenge is resolved and a PDE predictor-feedback control law that compensates the transport actuator dynamics is introduced. Since it may be intriguing as to what is the exact relation of the cascade to a system with input delay, the fact that the considered PDE-ODE cascade gives rise to a system with input delay, with a delay that depends on past input values (defined implicitly via a nonlinear equation), is highlighted.

  • N. Bekiaris-Liberis and M. Krstic, Compensation of actuator dynamics governed by quasilinear hyperbolic PDEs,Automatica, vol. 92, pp. 29--40, 2018.
  • N. Bekiaris-Liberis and M. Krstic, Control of nonlinear systems with actuator dynamics governed by quasilinear first-order hyperbolic PDEs, European Control Conference, 2018.

    • A2. The stabilization problem of a transport PDE/nonlinear ODE cascade, in which the PDE state evolves on a domain whose length depends on the boundary values of the PDE state itself, is introduced and solved. In particular, a predictor-feedback control design, which compensates such transport PDE dynamics, is developed. The relation of the PDE-ODE cascade to a nonlinear system with input delay that depends on past input values is also highlighted and the predictor-feedback control design for this representation is presented as well.

  • N. Bekiaris-Liberis and M. Krstic, Compensation of transport actuator dynamics with input-dependent moving controlled boundary, IEEE Transactions on Automatic Control, vol. 63, pp. 3889--3896, 2018.
  • N. Bekiaris-Liberis and M. Krstic, Compensation of transport actuator dynamics with input-dependent moving controlled boundary, European Control Conference, 2018.

    • A3. An optimal, robust delay-compensating feedback law is developed for multi-input linear systems with distinct input delays. The introduced control design methodology provides a systematic tool for optimal and robust coordinated ramp metering-based feedback control of traffic flow at distant bottlenecks, when several on-ramps may be actuated (located at different distances from the bottleneck area).

  • X. Cai, N. Bekiaris-Liberis, and M. Krstic, Input-to-state stability and inverse optimality of predictor feedbackfor multi-input linear systems, Automatica, vol. 103, pp. 549--557, 2019.

    • A4. Utilizing the conservation-of-vehicles equation, in its discrete time and discrete space form, a novel methodology for integrated lane-changing and ramp metering control, which exploits the presence of connected and partly automated vehicles (capable of receiving and executing specific control commands, such as lane-changing actions), is developed. The proposed control approach aims at robustly maximizing throughput at motorway bottlenecks employing a Linear Quadratic Integral (LQI) regulator in combination with a certain anti-windup scheme.

  • F. Tajdari, C. Roncoli, N. Bekiaris-Liberis, and M. Papageorgiou, Integrated ramp metering and lane-changing feedback control at motorway bottlenecks, European Control Conference, 2019.

    • Task 2.2: Observer design for estimation of the PDE state assuming boundary measurements.

    A. The availability of real-time traffic state estimates is a prerequisite for successful application of traffic management and control strategies. In particular, lane-specific highway traffic management and control has considerable potential in traffic flow optimization. However, the effectiveness of lane-based traffic management and control strategies largely depends on the quality and accuracy of traffic monitoring at a lane level. A reliable and cost-effective traffic monitoring solution should amply rely on the presence of connected vehicles, which are capable of providing accurate position and speed information (they can act as mobile sensors), whereas it should rely on the minimum number (e.g., at highway boundaries) of (costly) fixed detectors. For these reasons, we address the problem of per-lane density estimation as well as ramp flow estimation in highways, via the development of a model-based estimation approach, which relies largely on the presence of connected vehicles.

    Capitalizing on the close relation of the cumulative number-of-vehicles equation (in particular, the Moskowitz model) to the conservation-of-vehicles equation (in particular, the state of the cumulative number-of-vehicles model may be obtained from the state of the conservation law model) and in order to obtain an estimation methodology that is as ready as possible for real data testing and even for potential actual implementation, we develop a methodology based on the following three basic ingredients: (1) a data-driven version of the conservation-of-vehicles equation (in its time- and space-discretized form); (2) the utilization of position and speed information from connected vehicles’ reports, as well as total flow measurements obtained from a minimum number (sufficient for the observability of the model) of fixed detectors, such as, for example, at the main entry and exit of a given highway stretch; and (3) the employment of a standard Kalman filter.

  • N. Bekiaris-Liberis, C. Roncoli, and M. Papageorgiou, Highway traffic state estimation per lane in the presence of connected vehicles, Transportation Research Part B, vol. 106, pp. 1--28, 2017.
  • 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.

    • Task 2.3: Combination of the full-state feedback controller with the observer in a dynamic output-feedback control design.

    A. Arguably, it is desirable that any real-time traffic flow control/estimation algorithm, besides being able to efficiently exploit the capabilities of the available actuators and sensors, potentially, by also capitalizing on appropriate coordination and integration of different strategies, can be also made robust with respect to sensor or actuator errors and with respect to noisy measurements (which may deteriorate the performance of real-time traffic flow control).

    We are concerned with the coordinated, fault-tolerant, and noise-robust highway traffic flow planning, control, and estimation. The control strategies, which may be employed via ramp metering or variable speed limits (or, via combination of both), utilize efficiently the typical minimum actuation capabilities, as they are constructed considering ``bilateral” boundary actuation, i.e., at the two boundary ends of a given highway stretch. The ``dual” estimation strategies, which may be employed combining measurements from fixed detectors and probe vehicles, exploit in an efficient way the typically available measurement information from fixed detectors, as the basis for their design is a bilateral sensing framework, in which measurements from the two highway boundaries are appropriately combined. The control and estimation schemes are developed for a class of viscous Hamilton-Jacobi PDEs, which include the Moskowitz model with quadratic Hamiltonian as special case, and are capable of capturing certain realistic ``smoothing” effects in traffic flow due to drivers’ look-ahead ability.

    The output-feedback control design consists of three ingredients. The first, is the design of the feedforward actions at both boundaries, which provide the desired trajectory for the traffic state. This is achieved via solving the nonlinear trajectory generation problem for this type of PDEs. The second ingredient is the development of full-state feedback laws for the two boundary ends of the highway, which guarantee tracking of the desired trajectory of the traffic flow with an arbitrary decay rate. The third element is the construction of a nonlinear observer for estimation of the traffic state via utilization of measurements from both highway boundaries. The three different elements are then combined together in a complete observer-based output-feedback traffic flow control strategy. All of the designs are constructed interlacing PDE feedback linearization with PDE backstepping.

  • N. Bekiaris-Liberis and R. Vazquez, Nonlinear bilateraloutput-feedback control for a class of viscous Hamilton-Jacobi PDEs, Automatica, vol. 101, pp. 223--231, 2019.
  • N. Bekiaris-Liberis and R. Vazquez, Nonlinear bilateral full-state feedback trajectory tracking for a class of viscous Hamilton-Jacobi PDEs, IEEE Conference on Decision and Control, 2018.