Stop-and-Go Suppression in Two-Class Congested Traffic
This paper develops boundary feedback control laws in order to damp out traffic oscillations in the congested regime of the linearized two-class Aw-Rascle (AR) traffic model. The macroscopic second-order two-class AR traffic model consists of four hyperbolic partial differential equations (PDEs) describing the dynamics of densities and velocities on freeway. The concept of area occupancy is used to express the traffic pressure and equilibrium speed relationship yielding a coupling between the two classes of vehicles. Each vehicle class is characterized by its own vehicle size and driver’s behavior. The considered equilibrium profiles of the model represent evenly distributed traffic with constant densities and velocities of both classes along the investigated track section. After linearizing the model equations around those equilibrium profiles, it is observed that in the congested traffic one of the four characteristic speeds is negative, whereas the remaining three are positive. Backstepping control design is employed to stabilize the heterodirectional hyperbolic PDEs. The control input actuates the traffic flow at outlet of the investigated track section and is realized by a ramp metering. A full-state feedback is designed to achieve finite time convergence of the density and velocity perturbations to the equilibrium at zero. This result is then combined with an anti-collocated observer design in order to construct an output feedback control law that damps out stop-and-go waves in finite time by measuring the velocities and densities of both vehicle classes at the inlet of the investigated track section. The performance of the developed controllers is verified by simulation.
Stuttgart]Mark Burkhardt, UCSD,*]Huan Yu and UCSD]Miroslav Krstic
Institute for System Dynamics, University of Stuttgart, Waldburgstr. 17/19, Stuttgart, 70563, Germany
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, 92093, USA
Key words: Multi-class traffic model, PDE control, Backstepping, Output feedback controller.
Nowadays, more and more people own a car leading to crowded highways and congested traffic during rush hours in many countries of the world. Stop-and-go traffic is common to appear in congested traffic. This phenonemon is characterized by traffic density and velocity perturbations, causing higher fuel consumption and a higher risk of accidents. Traffic management systems like ramp metering or variable speed limits exist in order to regulate traffic. Moreover, they can be used to damp out those traffic oscillations and distribute traffic evenly on the highway. Many recent efforts are focused on boundary control of stop-and-go traffic including , ,  and . However, the results presented in this paper assume heterogeneous vehicle sizes and drivers’ behavior. Thus, the overall challenge addressed in this work is the design of a ramp metering traffic management system to reduce traffic oscillations in the congested regime while distinguishing two different vehicle classes.
In general, traffic models are categorized in micro-, meso- and macroscopic models. Macroscopic models describe the traffic as a continuum and are thus more suitable to investigate traffic jams and disturbances in traffic flow. Typically, their model equations are PDEs. The first representative of this model category is the Lighthill-Whitham-Richards (LWR) model,  and , that is given by a single PDE conserving the traffic flow. Although capturing a good amount of realistic traffic phenomena, the LWR model fails to model important phenomena like platoon dispersion  or stop-and-go waves and is faulty under light traffic conditions. Thus, a second order extension is provided independently by  and . This second order extension is denoted as the PW model and introduces a second PDE modeling the velocity dynamics of the vehicles in addition to the conversation law. However, the critique in  points out that the PW model fails to portray that personalities of vehicles remain unchanged and drivers react more likely to events in front of them. To overcome these issues,  and  presented the Aw-Rascle-Zhang (ARZ) model which is also a second-order model describing the interaction between the vehicles by a traffic pressure function.
Macroscopic multi-class models were proposed after the publication of the ARZ model, including the first order models , , , , , , ,  and . The extended LWR model, introduced by , is the first macroscopic multi-class extension. Instead of adjusting their velocity only depending on the density of their own class, vehicles are affected by the densities of all other classes. Thus, the assumed speed-density relationship is formulated with respect to the sum of all densities, the total density. Furthermore, the -populations model  extends this idea of coupling regarding the average length of the vehicle classes by denoting the speed-density relationship in dependence of the mean free space between the vehicles. In  and , a phenomenon called creeping is introduced. This behavior occurs in reality and corresponds to the scenario where one vehicle class, for instance trucks, are jammed and stopped due to congestion and a second vehicle class, for instance motorcycles, still moves in the gap between the trucks.
In addition to first-order macroscopic multi-class models, second-order multi-class models are introduced in , , , ,  and . While first-order models assume that the velocities of all vehicles equal to their equilibrium velocities at every time, second-order models provide PDEs describing the velocity dynamics for each class. The denoted second-order models differ in the terms that occur in the velocity dynamics and in the principles which are used to deduce them. For instance,  presents an originally microscopic model that is transformed to the macroscopic one, whereas the model in  is based on the macroscopic second-order model of . The main focus of this paper is the macroscopic multi-class model . Considering this extension for the case of two different classes yields four coupled nonlinear hyperbolic PDEs which are denoted as the two-class AR traffic model in the following. In order to consider vehicle sizes, the vehicles are assumed to adjust their speed according to a measure called area occupancy,  and , which needs to be distinguished from occupancy .
Typically, traffic management systems act on the boundary of the investigated track section yielding a boundary control problem. In the literature, different techniques are proposed that achieve convergence of the states of hyperbolic coupled PDEs to a constant equilibrium with boundary control. The main focus of this paper is on the backstepping stabilization technique. While , , , , ,  provide results in case of or coupled hyperbolic systems, there also exists literature if an arbitrary amount of linear coupled PDEs is considered , , , , . In fact, the presented output feedback control of this work corresponds to the special case of the theoretical result in  for and . The elimination of traffic oscillations by applying the backstepping technique is achieved for the ARZ model in  and in . Moreover,  also presents adaptive control results.
Main contribution of this paper: this work presents the first result on boundary control design for traffic congestion consisting of two different vehicle classes. On one hand, this work contributes to traffic modeling in the sense of deducing a macroscopic multi-class traffic model in its characteristic form and investigating the obtained characteristic speeds. On the other hand, a connection between the theoretical control design method backstepping and an up-to-date extension of the AR traffic model for two classes is created by designing a full-state feedback controller and output feedback controller as ramp metering signal. Moreover, the results are a first step for control problem of more than two vehicle classes or the combination of multiple classes with multiple lanes.
The rest of this paper is structured as follows: Section introduces the two-class AR traffic model, the parameters characterizing the two classes and where they occur as well as the assumed boundary conditions. Section includes the preparation of the linearized model for the control design and the formulation of the control design model. Furthermore, the full-state feedback controller result is presented in section and the following section presents the anti-collocated observer design. Section provides the combination of the results to derive an the output feedback controller and section verifies the performance of the presented controllers with simulation results. Some future work is discussed in section .
2 Problem statement
The two-class AR traffic model is presented and important model parameters are explained. Afterwards, the model equations are linearized around a constant equilibrium, followed by a discussion of the free-flow or congested regimes.
2.1 Two-class AR traffic model
The Extended AR model for heterogeneous traffic presented in  is investigated in case of two classes. This two-class AR traffic model is given by
where each vehicle class is described by traffic density and velocity with . The parameter is the length of the investigated track section. The traffic density is defined as vehicles per unit length. The higher the traffic density, the more crowded is the traffic of class vehicles at a specific spatial point. In addition, the velocity describes the velocity of class vehicles at a specified spatial point along the investigated track section. The traffic density and velocity correspond to the first vehicle class and the traffic density and velocity correspond to the second vehicle class. The non-zero terms on the right hand side represent the adaption of the vehicles to their desired velocities, where is the adaptation time.
where is the occupied surface per vehicle class and the width of the investigated track. Assuming that the traffic densities are and along the entire considered highway section, the area occupancy is the percentage of occupied road space by any class. It holds that . The area occupancy depends on both densities since the occupied road surface is influenced by the vehicles of both classes.
The traffic pressure function is formulated as
where corresponds to the free-flow velocity, to the traffic pressure exponent and to the maximum area occupancy. The traffic pressure is the experienced traffic pressure by class vehicles and depends on the area occupancy. The higher the area occupancy, the higher the experienced traffic pressure. For instance, if a vehicle suddenly decelerates, then the following vehicle experiences a high traffic pressure forcing another deceleration. Thereby, the free-flow velocity represents the desired velocity of a driver, if no other vehicles of any class are present. The pressure exponent is a parameter that models the experience of the traffic pressure. Higher traffic pressure exponents lead to less experienced pressure. However, the maximum experienced pressure remains the same and is given by the free-flow velocity. The maximum area occupancy describes the percentage of occupied road surface for which the corresponding vehicle class is jammed. To obtain physically meaningful results, holds. For instance, means that if of the highway are covered by vehicles of any class, then the class vehicles are jammed and therefore their desired velocity is zero. Finally, the equilibrium speed- relationship is
according to the model of Greenshield , and represents the desired velocity of the class vehicles. It depends on the area occupancy since a very crowded road implies a lower desired speed in contrast to a nearly empty road. If the area occupancy is at the maximum , then the corresponding equilibrium speed- relationship value is . In order to show the qualitative behavior of the traffic pressure function (6) and the equilibrium speed- relationship (7), both functions are plotted in Figure 1 using an example parameter set. It is illustrated, that a more crowded highway, corresponding to a higher area occupancy, implies a higher experienced traffic pressure and a lower equilibrium speed.
2.2 Linearized two-class AR traffic model
Thus, the equilibrium velocities are determined by the equilibrium densities and . The perturbations of the distributed variables and are defined as
for each class and the linearized model equations are given by
where the introduced Jacobian matrices are
and the abbreviations
are introduced with . The abbreviations represent the derivative of the class traffic pressure function with respect to class traffic density. The boundary conditions are assumed to be
where (19) and (20) assume that the same total traffic flow enters and leaves the track section which is given by the sum of the class and class equilibrium flows and . The traffic flow of class is defined as
Boundary conditions (17) and (18) indicate that the traffic densities of the incoming traffic flow are equivalent to the equilibrium densities. Thus, not only the entering traffic flow is constant, in fact the densities of both classes in this traffic flow are assumed to be constant. The linearization of the introduced boundary conditions (17) to (20) is
2.3 Free/congested regime analysis
In general, two different regimes of traffic are distinguished: the free-flow regime and the congested regime. The free-flow regime is characterized by the fact that the total information of the system travels downstream along with the vehicles. In that case, the model equations correspond to four homodirectional hyperbolic PDEs. On the other hand, a partial upstream propagation of information characterizes the traffic flow in the congested regime. The corresponding heterodirectional behavior causes the development of stop-and-go traffic which implies increased fuel consumption and risk of accidents. Therefore, it is reasonable to investigate which choices of equilibrium densities and parameters lead to heterodirectional information propagation. Therefore, the characteristic speeds are computed and their signs are considered in the following. First, the linearized model equations (12) need to be decoupled in time leading to
with the new Jacobian matrices
Then, the characteristic speeds are given by the eigenvalues of which are
For model validity, the equilibrium velocities of both vehicle classes are chosen to be positive, i.e. and and all vehicles travel downstream. Thus, the first two characteristic speeds (29) and (30) are positive. In addition, it is shown in  that
holds. Because and , (35) implies that is positive as well. Hence, the only characteristic speed that may have a negative sign is . Therefore, traffic is defined to be in the free regime if the equilibrium densities and parameters satisfy
and in the congested regime if they meet
Since a controller dealing with congested traffic is designed later on, it is assumed that the equilibrium densities and parameters are chosen such that (37) holds throughout the rest of this paper. In fact, this result allows to formulate a traffic froude number as it is presented in  for the ARZ model. Besides, in , a physical interpretation of the characteristic speeds (29) to (32) is given. While and correspond to the flow of class vehicles and class vehicles, is related to the fact that the faster vehicle class overtakes the slower one. For that reason, it is reasonable to obtain , if is assumed.
According to the definitions in (36) and (37), the boundary between the two regimes is defined as . In case of the two-class AR traffic model, this boundary is a line which can be drawn in the --plane. Compared to the single class consideration, the boundary is equivalent to a single density which is the critical density. The numerically computed boundary between the two regimes is plotted as a contour plot for an example parameter set in Figure 2. The figure also indicates that small values for both equilibrium densities and correspond to a positive value of and therefore homodirectional behavior. On the other hand, large values of densities lead to a negative value of indicating heterodirectional behavior. Hence, smaller equilibrium densities correspond to free-flow regime and large equilibrium densities correspond to congested regime.
3 Boundary control design model
In a next step, the control objective and input is explained and afterwards the introduced linearized two-class AR traffic model, (12), (22) to (25), is prepared for the control design. The preparation is done using two transformations, the transformation to Riemann coordinates and a second transformation to further simplify the equations expressed in Riemann coordinates resulting in the control design model.
3.1 Control input and objective
The overall goal is to damp out stop-and-go traffic in the congested regime and achieve convergence to the equilibrium states in a finite time. The term of stop-and-go traffic refers to oscillations of the density and velocity perturbations around their constant equilibrium values along the highway. Moreover, a ramp metering is considered to be installed at the outlet of the investigated track section regulating the traffic outflow. In this work, the ramp metering is used to damp out the introduced stop-and-go waves. Thus, the boundary condition (25) becomes
Compared to the application of multi-phase flow in oil pipelines, , a ramp metering works as a valve at the end of the pipe to control the outgoing flow.
3.2 Transformation to Riemann coordinates
The system is transformed to Riemann coordinates to accomplish a decoupling of the spatial derivatives. The Riemann variables are defined as new coordinates. The linear state transformation is given by
where the constant invertible transformation matrix satisfies
and therefore diagonalizes the Jacobian . The entries of are denoted as
The matrix is straightforward to obtain but omitted in this paper due to its complexity and length. Inserting the transformation in (12) yields the model equations in Riemann coordinates
and the entries of the Jacobian are denoted by
Since the coefficient matrix of the spatial derivatives is now diagonal, a decoupling in spatial derivatives is achieved. The characteristic speeds (29) to (32) form the diagonal because they are the eigenvalues of . In addition, the same transformation is applied to the boundary conditions (22), (23), (24) and (38) yielding
The matrices are given by
and are obtained by formulating the linearized boundary conditions in matrix form, inserting the transformation law to Riemann coordinates and decoupling afterwards. Besides, in (47) and (48), the abbreviations
are inserted. Furthermore, the input transformation, used in (46), is
Overall, the model equations in Riemann coordinates are given by (42), (45) and (46). Because the inverse linear state transformation exists, the system expressed in Riemann coordinates shares the same stability properties with the original system. In a next step, a second transformation is performed in order to complete the preparation for the control design and obtain the control design model. The resulting coordinates are . The second transformation achieves zero elements on the diagonal of in (42) and sorts the positive characteristic speeds (29) to (31) in ascending order on the diagonal of the coefficient matrix of the spatial derivatives. However, an ascending order is only defined uniquely, if it is known whether or holds, according to (35). In the following, it is assumed that class vehicles represent small and fast average vehicles whereas class describes big trucks which are large and slow. Thus, for the equilibrium velocities holds and therefore the ascending order of positive characteristic speeds is . Hence, the transformation law
is applied to (42) yielding the transformed PDEs
The abbreviations for the coefficients of the source term, are:
The diagonal elements of are sorted in an ascending order and the relations , (29), and , (30), are inserted. In addition, , and are depending on the spatial coordinate. Their entries are bounded and either positive or negative on the whole domain, depending on the sign of the corresponding . Applying the transformation (51) to (54) on the boundary conditions (45) and (46) yields
In addition, the input given in (63) is defined as
All in all, the control design model is given by (55), (56), (62) and (63). In Figure 3, the qualitative behavior of the control design model is illustrated. According to the sign of the characteristic speeds, the propagation direction for each state is drawn in Figure 3. It shows that the control input acts at the outlet of system, first propagating upstream and, after it is carried through the boundary condition at the inlet of the investigated track section, affecting downstream traffic.
The summary of the two transformations is
The overall transformation law depends on the spatial coordinate. In addition, both input transformations combined are
and the inversion is given by