# Output Feedback Control of Two-lane Traffic Congestion

###### Abstract

This paper develops output feedback boundary control to mitigate traffic congestion of a unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw-Rascle-Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with nonlinear coupled hyperbolic PDEs. Based on driver’s preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSL) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved for both the controllers and observer designs. Numerical simulations validate our design in two different traffic scenarios.

*]Huan Yu and

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, 92093, United States

Key words: ARZ traffic model; Output feedback; Boundary observer: PDE backstepping.

^{1}

^{1}footnotetext: Corresponding author.

## 1 Introduction

Traffic congestion on freeways been investigated intensively over the past decades. Motivations behind are to understand the formation of traffic congestion, and further to prevent or suppress instabilities of traffic flow. Macroscopic modeling of traffic dynamics is to describe evolution of aggregated traffic state values including traffic density and velocity. Traffic dynamics are governed by hyperbolic PDEs, including the first-order model by Ligthill, Whitham and Richards (LWR), the second-order Payne-Whitham model, and the second-order Aw-Rascle-Zhang (ARZ) model [3] [23]. The LWR model is a conservation law of traffic density. It is simple yet powerful in understanding the formation and propagation of traffic shockwaves on freeway. But it fails to describe stop-and-go traffic, also known as ”jamiton” [9]. The oscillations of densities and velocities travel with traffic stream, causing unsafe driving conditions, increased consumptions of fuel and delay of travel time.

In order to describe this common phenomenon, the second-order traffic models are developed, consisting of nonlinear hyperbolic PDEs of traffic density and velocity. The ARZ model advances the PW model in correctly prediction of propagation of traffic velocity. Validation of the ARZ model with empirical traffic data is conducted in [10]. [6] discusses the heterodirectional propagations of characteristics waves for congested traffic by the ARZ model. Previous work of authors [20] [21] discuss the linear stability of uniform steady states of the nonlinear hyperbolic ARZ model. Instabilities appear in the congested regime of the ARZ model when drivers on the road are aggressive. To tackle the traffic congestion, traffic management infrastructures like VSL and ramp metering provide opportunities for boundary actuation of traffic velocity and flow. Many recent efforts [5] [19] [20] [21] [24] [25] [26] are focused on boundary control of traffic PDE model. Among these results, [20] firstly applied the backstepping control design to stabilize the stop-and-go traffic of the one-lane ARZ model with ramp metering.

The above-mentioned models treat multi-lane freeway traffic cumulatively as a single lane by assuming averaged velocity and density over cross section of all lanes. The individual dynamics of each lane and inter-lane interactions are neglected. However, distinct density and velocity equilibrium exist in multi-lane problems. The differences of velocities give rise to lane-changing interactions and further lead to traffic congestion [15]. To address the phenomenon, a number of macroscopic multi-lane models [12] [13] [16] [17] [18] have been developed from microscopic, then kinetic to macroscopic descriptions. In this paper, we adopt the multi-lane ARZ traffic model proposed by [17] [13] to describe a two-lane freeway traffic with lane-changing between the two lanes. Lane interactions appear as interchanging source terms in the system, leading to more involved couplings and a higher order of PDEs. The complexity of the multi-lane model is greatly increased compared to the one-lane problem.

Feedback boundary control design for a general class of hyperbolic PDEs using backstepping method are studied in [1] [2] [7] [8] [11] [14] [20]. In [8], stabilization of a counter convecting hyperbolic PDEs is achieved with a single boundary. [14] presents a solution to output feedback of a fully general case of heterodirectional first-order linear coupled hyperbolic PDEs. Actuation of all the PDEs from the same boundary is required to stabilize the system in finite time. A shorter convergence time is further obtained in [2] by modifying the target system structure.

In this problem, a two-lane ARZ model of a freeway segment presents heterodirectional coupled nonlinear hyperbolic PDEs, governing the traffic densities and velocities of the fast and slow lanes. We aim to stabilize the oscillations in the two-lane traffic using the PDE backstepping method, based on the stabilization results in [14]. Actuation of traffic velocities at the outlet boundary are realized by two VSLs.

The contribution of this paper is the following. This is the first result dealing with control problem of multi-lane traffic PDE model. The dynamics of two-lane traffic are studied from control perspectives. Theoretical result of output feedback control of the general class of heterodirectional linear hyperbolic PDE systems is developed in [2] [14], but has never been applied in traffic application. Being the first paper to adopt the methodology, our result opens the door for solving related multi-lane traffic problems with PDE control techniques. Furthermore, we advance the theoretical results in [2] [14] by proposing a collocated boundary observer and controller design. The output feedback controllers in both papers are constructed with full-state feedback controllers and an anti-collocated observer. In implementation, collocated boundary observer and controllers are more practically applicable. We bridge this gap by developing a observer with sensing at outlet, which is also a more challenging problem in the design for the system of this paper.

The paper is organized as follows: in Section 2 we introduce the two-lane ARZ traffic model and then linearize the nonlinear hyperbolic PDEs around uniform steady states. In Section 3 backstepping transformation is derived for the linearized model in Riemann coordinates. We present full-state feedback control laws to actuate outlet boundary velocities. In Section 4, we design collocated boundary observers and then obtain output feedback control laws. In Section 5, control design in two different traffic scenarios are discussed and tested with numerical simulation.

## 2 Problem Statement

In this section, two-lane traffic ARZ model is introduced. We derive lane-specific uniform steady states according to the drivers’ overall preference for the lanes and then linearize the nonlinear system around the steady states. The linearized system is transformed to Riemann variables. A coupled first-order hetero-directional hyperbolic system is obtained for control design.

### 2.1 Two-lane traffic ARZ model

The two-lane traffic on unidirectional roads is described with the following two-lane traffic ARZ model by [17] [13]. The diagram in Fig.1 is shown with the faster lane on the left and slower lane on the right. The two-lane traffic ARZ model is given by

(1) | ||||

(2) | ||||

(3) | ||||

(4) |

The traffic density and velocity are defined in , , where is the length of the freeway segment. The above nonlinear hyperbolic PDEs consist of two subsystems of second-order nonlinear hyperbolic PDEs, each describing one-lane traffic dynamics. Lane-changing interactions and drivers’ behavior adapting to the traffic appear as source terms on the right hand side of PDEs.

The variable is defined as the traffic density pressure

(5) |

which is an increasing function of density . is the maximum traffic velocity, is the maximum traffic density and the constant coefficient reflects the aggressiveness of drivers on road. The parameter is defined as relaxation time that reflects driver’s behavior adapting to the traffic equilibrium velocity in the lane . The parameter describes the driver’s preference for remaining in lane , which relates to the both lanes’ density and velocity. We consider them to be constant coefficients in this paper.

The equilibrium velocity-density relationship is given in the form of the Greenshield’s model,

(6) |

We choose the Greenshield’s model for due to its simplicity but the control design presented later is not limited by this choice. Note that the equilibrium velocity-density model (6) is for cumulative single lane traffic. Distinct velocity equilibrium does exist in each of the two lanes [18]. The lane-specific steady traffic velocities will be discussed in the following section.

### 2.2 Driver’s preference for two lanes

We consider to linearize the nonlinear hyperbolic system around uniform steady states . We obtain the following equations

(7) | ||||

(8) | ||||

(9) |

The steady state density-velocity relations are defined based on (6). Thus the steady states need to satisfy

(10) | ||||

(11) | ||||

(12) |

where and differ from single-lane . The ratio coefficients and are defined as

(13) | ||||

(14) |

The parameter defines driver’s preference for the fast lane over slow lane according to (7),(10),

(15) |

Compared with the single-lane Greenshield’s model in (6), the relations of steady state traffic velocities and densities depend on the drivers lane-changing preference parameter .

Assuming that overall drivers prefer fast lane over slow lane, we use Fig.2 to show the equilibrium velocity-density relation and fundamental diagram of the single-lane, the fast and slow lane. represents the equivalent maximum density of the single-lane. The actual maximum density in the fast lane and in the fast lane are related to by

(16) | ||||

(17) |

If drivers prefer the fast lane, the decrease of velocity gets steeper in the slow lane and less steep in the fast lane. At the same density, the fast lane traffic is ”more tolerant to risk” of high density than in the single-lane case, and the slow lane traffic is ”less tolerant to risk” than in the single-lane case. As a result, the traffic flux of fast lane is higher than the slow lane at the same density in the fundamental diagram shown in fig.2.

Drivers prefer the slow lane. The decrease of velocity is steeper in the fast lane than that of slow lane at the same density. The slow lane is more tolerant to high density and the traffic flux is higher in the slow lane.

In general, the activities of lane changing segregate the drivers into the more ”risk-tolerant” ones in the fast lane and the more ”risk-averse” in the slow lane. The risk-tolerant drivers prefer to drive with a faster speed at the same density, compared with risk-averse drivers.

### 2.3 Linearized two-lane ARZ model

Before linearizing the nonlinear system (2.1)-(4) to steady states (10)-(12), we consider the following boundary conditions of -system. We assume constant traffic flux entering from the inlet boundary of the two lanes.

(18) |

Two VSLs implemented at the outlet and actuate the traffic velocity variations for the fast and slow lanes respectively.

(19) | ||||

(20) | ||||

(21) | ||||

(22) |

Then we linearize the above nonlinear hyperbolic system around steady states that satisfy (10)-(12). The deviations from the steady states are defined as

(23) | ||||

(24) |

The linearized hyperbolic system is obtained

(25) | ||||

(26) | ||||

(27) | ||||

(28) |

with the linearized boundary conditions

(29) | ||||

(30) | ||||

(31) | ||||

(32) |

In order to diagonalize the spatial derivatives on the left hand side of the equations, we write the above linearized hyperbolic system in the Riemann coordinates as

(33) | ||||

(34) |

We consider the congested regime in [20] where steady state traffic density disturbances convect downstream and the velocity disturbances travel upstream. Therefore the following conditions hold for the characteristic speeds of ,

(35) |

We obtain a coupled first-order hetero-directional hyperbolic system in ,

(36) | ||||

(37) | ||||

(38) | ||||

(39) | ||||

(40) | ||||

(41) | ||||

(42) | ||||

(43) |

where the constant boundary coefficients are defined as

(44) |

and the constant parameter block matrix is denoted by

(45) |

The elements of sub-matrices of are defined as,

(46) | ||||

(47) | ||||

(48) | ||||

(49) | ||||

(50) | ||||

(51) | ||||

(52) | ||||

(53) |

The flow diagram of -system is shown in Fig.3. The first-order hyperbolic system is composed of two coupled second-order heterodirectional hyperbolic systems. States convect downstream while states propagate upstream. We use two VSLs to damp out the oscillations to zero from the outlet.

## 3 Full-state Feedback Control Design with VSLs

To apply the backstepping approach and to design boundary control for the system in (36)-(43), we scale the state variables and in space to cancel the diagonal terms in their equations. The Riemann variables and remain to be the same. The scaled variables and are defined as

(54) | ||||

(55) |

Then we obtain the scaled system:

(56) | ||||

(57) | ||||

(58) | ||||

(59) | ||||

(60) | ||||

(61) | ||||

(62) | ||||

(63) |

We denote the transports speeds as

(64) | ||||

(65) |

Note that the steady velocity of the fast lane is larger than that of the slow lane, the constant transport speeds satisfy the following inequalities,

(66) |

where the constant coefficients and are defined as

(67) |

The new in-domain coefficient matrix is given by

(68) |

where the sub-matrices are obtained as

(69) | ||||

(70) | ||||

(71) | ||||

(72) |

Among the transformed sub-matrices, the elements of are constant and the elements of , and are spatially-varying coefficients. We summarize the transformation between and from (33), (34) and (54), (55) as follows:

(73) | ||||

(74) | ||||

(75) | ||||

(76) |

Then we introduce the backstepping transformation to the scaled -system in (56)-(63),

(77) | ||||

(78) |

where the kernel matrices are denoted as

(79) |

The kernel variables and evolve in the triangular domain . Taking derivative with respect to time and space on both sides of (77)-(78) along the solution of a target system given later, we obtain the following kernel equations. The kernels and are governed by

(80) | |||

(81) | |||

(82) | |||

(83) | |||