Trafficflowstabilityinducedbyconstanttimeheadway ...lixxx099/papers/LiShri02.pdfTrafficflowstabilityinducedbyconstanttimeheadway policyforadaptivecruisecontrolvehiclesq PerryY.Li*,AnkurShrivastava

Traffic flow stability induced by constant time headwaypolicy for adaptive cruise control vehicles q

Perry Y. Li *, Ankur Shrivastava

Department of Mechanical Engineering, University of Minnesota, 111 Church Street SE, Minneapolis, MN 55455, USA

Received 19 January 2000; accepted 17 January 2002

Abstract

This paper is concerned with the traffic flow stability/instability induced by a particular adaptive cruisecontrol (ACC) policy, known as the ‘‘constant time headway (CTH) policy’’. The control policy is analyzedfor a circular highway using three different traffic models, namely a microscopic model, a spatially discretemodel, and a spatially continuous model. It is shown that these three different modeling paradigms canresult in different traffic stability properties unless the control policy and traffic dynamics are consistentlyabstracted for the different paradigms. The traffic dynamics will, however, be qualitatively consistent acrossthe three modeling paradigms if a consistent biasing strategy is used to adapt the CTH policy to the variousmodeling frameworks. The biasing strategy determines whether the feedback quantity for use in the con-trol, is taken colocatedly, downstream or upstream to the vehicle/section/highway location. For ACC ve-hicles equipped with forward looking sensors, the downstream biasing strategy should be used. In this case,the CTH policy induces exponentially stable traffic flow on a circular highway in all three modelingframeworks. It is also shown that traffic flow stability will be preserved for an open stretch highway if theentry and exit conditions are made to observe the downstream biasing strategy.� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Traffic flow; Stability; Adaptive cruise control; Macroscopic model; Headway; Biasing strategy

1. Introduction

The ability to stabilize traffic to a desired pattern is important to prevent congestion and tofully utilize the capacity of the highway. For fully automated highway systems (AHS) (Varaiya,

Transportation Research Part C 10 (2002) 275–301www.elsevier.com/locate/trc

qResearch supported by Minnesota Department of Transportation and the Center for Transportation Studies (CTS)

and CAMDAC at the University of Minnesota.* Corresponding author. Tel.: +1-612-626-7815; fax: +1-612-625-9395.

E-mail addresses: [email protected] (P.Y. Li), [email protected] (A. Shrivastava).

0968-090X/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0968-090X(02)00004-9

mail to: [email protected]

1993), traffic control systems known as link layer controllers, have previously been developed(Alvarez et al., 1999; Li et al., 1997a; Peng, 1995) to stabilize traffic to desired patterns specified bya pair of density and velocity profiles. Unfortunately, these controllers require the highway systembe fully automated, meaning that all vehicles must be automated and can communicate with eachother and with the roadside infrastructure. This poses a significant impediment to the adoption offully automated AHS due to the large initial cost layout that is necessary. Traffic flow controlsystems that can be implemented in a distributed and incremental fashion would be more practicaland more likely to be adopted. Adaptive cruise control (ACC) vehicles, also called autonomousintelligent cruise control (AICC) or simply intelligent cruise control (ICC) vehicles, which areexpected to gradually appear on the market soon, hold the potential to realize this goal. ACCvehicles can sense intervehicle spacings in addition to being able to control vehicle speeds like theexisting cruise control technology. ACC was originally developed as an enhancement for safetyand comfort. However, our research goal is to exploit the sensing and control capabilities of ACCvehicles to achieve an implementation of a traffic flow control system which is distributed and canbe incrementally implemented without requiring a fully functional fully automated highwaysystem.Control policies of ACC vehicles are characterized by the speed-spacing relationships that the

vehicle control system aims to observe. A popular control policy is the ‘‘constant time headway’’(CTH) policy which states that the desired speed of the vehicle should be proportional to theintervehicle spacing, with the specified headway being the reciprocal of the constant of propor-tionality. CTH is appealing because it observes the normal driving intuition that vehicles shouldslow down as the intervehicle spacings decrease. Since headway and flowrate are roughly inverselyrelated, regulating headway has the effect of determining the flowrate on the highway. We do not,in this paper, consider the vehicle control issue such as that of ensuring that the headway policy isobserved or of maintaining safety (see Girault and Yovine (1999), Liang and Peng (1998), Li et al.(1997b), Ioannou and Chien (1993), etc. for these aspects). Rather, we are interested in how theCTH policy affects the global traffic flow pattern, in terms of density profiles in some sense, for thehighway as a whole. As such, we assume that each vehicle faithfully observes the specified ACC-CTH policy. We analyze the stability property of equilibrium traffic flow patterns on circularhighways populated by ACC vehicles that obey the CTH policy. Analysis is performed usinghighway models of three types of abstraction:

1. the microscopic model in which vehicles are individually modeled,2. a family of spatially discrete models in which the highway is partitioned into sections and the

aggregated density and speed associated with each highway section are accounted for and3. a family of spatially continuous models in which the traffic flow is analogous to fluid flow, and

the spatial distribution of the aggregated density and speed at every location on the highway areaccounted for.

The last two sets of models are macroscopic models. Macroscopic models are abstract modelsof the actual system. They are generally less computationally burdensome than microscopicmodels. This is because they consider average or aggregate information only instead of keepingtrack of behaviors of individual vehicles. The models that we study here are quite standard in theliterature. For example, the spatially continuous model is basically the celebrated Lighthill,

276 P.Y. Li, A. Shrivastava / Transportation Research Part C 10 (2002) 275–301

Whitham, Richards (LWR) model (Daganzo, 1997). The spatially discrete model is similar to thatin Payne (1971). This model is subsequently modified in many other works such as Karaslaanet al. (1990) and Papageorgiou et al. (1990), by adding on extra dynamics of the traffic velocity.Recently, Broucke and Varaiya (1996) developed a new mesoscale model based on vehicle ac-tivities and space-time resources of the highway. We do not study the abstraction of the ACC-CTH policy in this new framework.A recent paper by Swaroop and Rajagopal (Dharba and Rajagopal, 1999) investigated

the traffic flow stability on an open stretch of highway with entries and exits populatedby ACC vehicles under the same ‘‘constant time headway’’ policy that we are considering.Their conclusion, obtained based on a spatially discrete model of the highway is that the re-sulting traffic flow is unstable. They attributed the instability to a traveling wave that travelsupstream.In the present paper, instead of studying an open stretch of highway as in Dharba and Ra-

jagopal (1999), we study a circular highway, so as to highlight the intrinsic property of the CTHpolicy itself. Various types of entry and exit conditions, if present, can potentially alter the sta-bility property of the overall highway system given the same ACC policy. Afterall, entry and exitconditions are results of interconnections of roadways in a network. It is well known that in-terconnections of stable systems can result in an overall system that is unstable. By focusing on acircular highway, the possibly confounding effect of the entry and exit conditions which exist foran open stretch of highway can thus be avoided, because potentially prejudicial choices of entryand exit conditions do not need to be made. To generalize the result to open stretches of highwayspopulated by ACC vehicles after understanding the intrinsic stability property of the ACC policy,our approach is to investigate what types of entry and exit conditions are appropriate or con-sistent. Our assumption is that entry and exit conditions are to some extent, modifiable by design(e.g. using traffic control devices such as ramp metering). In this paper, flow stability induced byACC-CTH policy is studied using various highway modeling paradigms. This is to ensure that ourconclusions do not fall victim to the modeling abstraction process. Our analysis shows that on anyfinite length circular highway, the traffic flow induced by the CTH policy is in fact exponentiallystable. Moreover, it is found that the manner in which the CTH policy is abstracted forthe various highway models must be performed consistently and with care; otherwise, errone-ous stability properties will be arrived at. A key consideration for consistent abstraction is thebiasing strategy which determines whether the feedback is taken from downstream, upstreamor colocatedly. The notion of biasing strategy also provides useful insight as to what entryand exit conditions can preserve traffic flow stability under the CTH policy on an open stretch ofhighway.The rest of the paper is organized as follows: In Section 2, a microscopic model is used to show

that the CTH policy induces exponentially stable traffic flow. In Section 3, we study the sameCTH policy under three different modeling abstractions and show that inconsistent results can beobtained. In Section 4, we propose the concept of biasing strategy and analyze the varioushighway models after incorporating this concept. Simulation results that demonstrate that thehighway model abstractions are consistent with each other when biasing strategy is considered isshown in Section 5. How the results obtained in this paper can be applied to an open stretch ofhighway is discussed in Section 6. General discussion of the results is given in Section 7. Section 8contains some concluding remarks.

P.Y. Li, A. Shrivastava / Transportation Research Part C 10 (2002) 275–301 277

2. Stability of CTH policy: microscopic analysis

The vehicle’s headway is the time taken for the vehicle to traverse the intervehicle spacing aheadof it. Thus, if the vehicle is traveling at speed vi, and the intervehicle spacing is Dxi, then, theheadway is given by

hw ¼ Dxi=vi:

Therefore, ACC vehicles under the CTH policy observe a speed-spacing relation given by

vi ¼1

hwDxi; ð1Þ

where hw is the specified headway.Consider now a single lane circular highway of length L (Fig. 1). The highway’s circle topology

can be parameterized by ½0; LÞ � R so that x ¼ 0 is identified with x ¼ L. Assume that the highwayis populated by N ordered, ACC vehicles located at xiðtÞ 2 ½0; LÞ, i ¼ 1; . . . ;N , at time t. Theordering of the vehicles are such that vehicle i is the j–ith vehicle ahead of vehicle j (i.e. fromdownstream to upstream, and from leader to followers). For simplicity, arithmetics of the indicesare taken to be of modulo n (e.g. i ¼ 1 ) i� 1 ¼ N ). Let Lv be the length of the vehicles. Theintervehicle spacing ahead of the ith vehicle is Dxi ¼ ðLþ xi�1 � xi � LvÞ modL where x=L denotesthe remainder of x=L. It is assumed that Dxi P 0.In the microscopic description of the highway, vehicles are individually modeled. We can use

Dxi, i ¼ 1; . . . ;N as the state of the traffic. The dynamics are given by

d

dtDxi ¼ _xxi�1 � _xxi; ð2Þ

where _xxi would be given by (1) if the vehicles observe the CTH policy. Thus,

d

dtDxi ¼ fi Dx1;Dx2; . . . ;DxNð Þ ð3Þ

for some functions fi: Rn ! R.

Fig. 1. A microscopic traffic flow model.


Given an ACC policy, let Dxi;1, i ¼ 1; . . . ;N be the set of equilibrium intervehicle spacings, i.e.for each i ¼ 1; . . . ;N

fi Dx1;1;Dx2;1; . . . ;DxN ;1ð Þ ¼ 0:

We define the traffic flow stability in the sense of Lyapunov. Let ~DDxiðt ¼ 0Þ ¼ Dxiðt ¼ 0Þ�Dxi;1, i ¼ 1; . . . ;N be a set of initial perturbations of the intervehicle spacings, and ~DDx :¼½~DDx1; ~DDx2; . . . ; ~DDxN �T. The traffic flow at Dxi;1, i ¼ 1; . . . ;N is stable in the sense of Lyapunov if givenany � > 0, there exists a d > 0 so that

k~DDxð0Þk6 d ) k~DDxðtÞk6 � 8 tP 0:

It is asymptotically stable if in addition to being stable, limt!1 ~DDxðtÞ ¼ 0.We now analyze the traffic flow stability under the CTH policy. The traffic dynamics (3) and (1)

can be combined and represented in vector form:

d

dt

Dx1Dx2...

DxN�1DxN

2666664

3777775 ¼ 1

hw

�1 0 0 . . . 11 �1 0 . . . 0

..

. ... ..

. ... ..

.

0 . . . 1 �1 00 . . . 0 1 �1

266664

377775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Amicro

Dx1Dx2...

DxN�1DxN

2666664

3777775: ð4Þ

From (4), the equilibrium intervehicle spacings are of the form

Dx1;1 ¼ Dx2;1 ¼ � � � ¼ DxN ;1;

i.e. the vehicles are equally spaced. Because the highway length is L, the equilibrium spacings mustbe L=N � Lv where Lv is the length of each vehicle.

Theorem 1. Consider the microscopic traffic model given by (2), with the N vehicles under the controlof the constant time headway (CTH) ACC policy given by (1). The equilibrium traffic pattern withthe intervehicle spacings Dxi;1 ¼ L=N � Lv is asymptotically stable with an exponential convergencerate.

Proof. Since the traffic dynamics in (4) is clearly linear, the stability of the equilibrium trafficpattern is determined by the Hurwitzness of the matrix Amicro in (4), which in turn, can be eval-uated by considering its Gershgorin disks (Demmel, 1997). Gershgorin’s disks of a matrix aredisks with centers at the diagonal elements, and with radii being the absolute sums of the offdiagonal elements in the corresponding rows. The eigenvalues of the matrix lie in the regiondefined by the union of all the Gershgorin disks. In our case, the disks are centered at �1=hw andhave radii of 1=hw. Hence all the eigenvalues lie in the closed left half plane (LHP). Furthermore,by expanding the expression for detðsI � AmicroÞ, it can be shown that a1 6¼ 0 in

detðsI � AmicroÞ ¼ sn þ an�1sn�1 þ � � � þ a1s:

Thus, Amicro has only one eigenvalue at the origin. The corresponding eigenvector is½1; 1; 1; . . . ; 1�T. A simple Jordan form argument then establishes that the state vector must ex-ponentially converge to the one dimensional subspace fDx1;1 ¼ Dx2;1 ¼ � � � ¼ DxN ;1g. Since the


Jordan decomposition of the state vector ½Dx1;Dx2; . . . ;DxN �T at any time in the direction of½1; 1; . . . ; 1� is invariant (since the associated eigenvalue is 0), it must be ðL=N � LvÞ � ½1; 1; . . . ; 1�T,the desired equilibrium state. Thus, the state vector converges to the equilibrium exponen-tially. �

It is instructive to investigate the factors that affect the rate of convergence, which is determinedby the magnitude of the real part of the non-zero eigenvalues closest to the imaginary axis. From

Fig. 2. Magnitude of the real part of the dominant eigenvalue of Amicro versus the number of vehicles for hw ¼ 1. Top:

uniform scale. Bottom: log scale.


(4), it is clear that the rate of convergence would be inversely proportional to the specifiedheadway, hw. Moreover, Fig. 2 shows that for large N (number of vehicles), the rate of con-vergence decreases with (1=N 2). A detailed analysis shows that the real part of the dominanteigenvalue is

ð�1þ cosð2p=NÞÞ=hw � 2p2=ðN 2hwÞ:The above stability result seems to be in conflict with the result in Dharba and Rajagopal (1999)which indicates that the CTH induced traffic flow is not asymptotically stable. Their results wereestablished for an open stretch of highway (with entries and exits), via a spatially continuousmodel (which indicates limitedly stable traffic flow) (Fig. 3); 1 and via a spatially discrete highwaymodel. Our result, on the other hand, is established for a circular highway (i.e. no entry or exit),using a microscopic model. A microscopic model in which each vehicle is modeled should be moreaccurate than either a spatially continuous model or a spatially discrete model since no ab-straction is employed.In the sequel, we show that the abstraction process must be performed with care in order to

have consistent stability property.

3. Two inconsistent model abstractions

3.1. Spatially continuous model

Consider now a macroscopic framework in which the traffic flow is modeled using a com-pressible fluid flow analogy. Let qðx; tÞ and vðx; tÞ denote the traffic density and speed respectivelyat location x 2 ½0; LÞ and at time t. vðx; tÞ is considered to be the control variable. The trafficdynamics which can be derived using a conservation of mass (continuity) argument are given by

oqot

ðx; tÞ ¼ � o

oxqðx; tÞ � vðx; tÞ½ �: ð5Þ

To prescribe the traffic dynamics under the spatially continuous framework when the vehiclesare supposed to be under the control of the CTH policy (1), we must define what the feedback lawfor the traffic speed vðx; tÞ is. Because the intervehicle spacing is Dxi at the location of the vehicle i,the local density would be 1=ðDxi þ LvÞ, it appears reasonable that a translation of the individualvehicle’s control law (1) is

Fig. 3. Spatially continuous traffic flow model.

1 Their definition of traffic stability in Dharba and Rajagopal (1999) is stronger than commonly used, and requires

convergence.


vðx; tÞ ¼ 1

hw

1

qðx; tÞ

�� Lv

: ð6Þ

This translation of the control law is also utilized in Dharba and Rajagopal (1999). The abstractedcontrol law (6) will be referred to as neutrally biased or colocatedly biased since the feedbackinformation for location x is taken from the same location.Substituting (6) into (5), we have

oqot

ðx; tÞ ¼ Lvhw

oqox

ðx; tÞ: ð7Þ

Eq. (7) is a linear wave equation. The desired equilibrium traffic pattern is given by qðx; tÞ ¼ N=L,a constant where N is the number of vehicles on the circular highway. To analyze its stability,let ~qqðx; tÞ :¼ qðx; tÞ � N=L, and define the Lyapunov functional to be related to the L2 norm of~qqð�; tÞ,

V ðtÞ ¼ 1

2

Z L

0

~qqðx; tÞ2 dx ¼ 1

2~qqð�; tÞ�� 2

2: ð8Þ

Thus, since

o~qqot

ðx; tÞ ¼ Lvhw

o~qqox

ðx; tÞ;

_VV ðtÞ ¼ Lvhw

Z L

0

~qqðx; tÞ o~qqox

ðx; tÞdx

¼ Lv2hw

Z L

0

o

ox~qq2ðx; tÞdx

¼ Lv2hw

~qq2ðL; tÞh

� ~qq2ð0; tÞi¼ 0:

ð9Þ

The last equality is due to the fact that the highway is circular so that x ¼ 0 and x ¼ L are the samepoint. This analysis demonstrates that the equilibrium traffic pattern qðx; tÞ ¼ N=L is only mar-ginally L2 Lyapunov stable in the following sense: given any � > 0, we can choose d6 � so thatwhenever the L2 norm of the initial perturbation, k~qq2ð�; t ¼ 0Þk26 d, the subsequent perturbationat any time tP 0, k~qq2ðx; tÞk26 �. The stability is only marginal because the size of the perturbationdoes not abate. In particular, unlike the microscopic model in Section 2, the density profile qðx; tÞdoes not converge to the desired equilibrium N=L.

3.2. Spatially discrete model

Consider now a spatially discrete model of the circular highway (Fig. 4). The highway, of lengthL, is partitioned into n equal length sections, and the density in the ith section is denoted by qi

(defined as the ratio of the number of vehicles in the section to the section length). It is assumed


that there is a single representative speed for each section, denoted by vi, i ¼ 1; . . . ; n, which arethe control variables. Readers are alerted to the possible confusion due to the indexing conven-tions that we have adopted. For the spatially discrete highway, the section numbers increase as wego downstream. This is done to be consistent with the literature such as Dharba and Rajagopal(1999), Chien et al. (1994), Karaslaan et al. (1990) and Papageorgiou et al. (1990). The conventionadopted in the microscopic model in Section 3.1 is that vehicle numbers increase from the leader(downstream) to the followers (upstream). This is also done to be consistent with the literature. Inthe literature, because microscopic analysis is generally concerned with car following, such anordering is reasonable (see e.g. Swaroop et al., 1994; Varaiya, 1993). In Karaslaan et al. (1990), amixed convention is also adopted).The dynamics of the section densities are given by the following model, which is also used in

Dharba and Rajagopal (1999), Karaslaan et al. (1990), Chien et al. (1994) and others. Fori ¼ 1; . . . ; n,

_qqi ¼nL½ � qi þ qi�1�; ð10Þ

where qi is the flowrate out of section i. 2 Notice that (10) implies that vehicles are conserved sothat

Xni¼1

qiðtÞ ¼ constant:

Each qi in (10) is defined to be a convex combination of the two ideal upstream and downstreamflowrates:

qi ¼ aiqivi þ ð1� aiÞqiþ1viþ1: ð11Þ

In (11), 06 ai 6 1, i ¼ 1; . . . ; n are the mixing coefficients that weigh the relative importance of theupstream and downstream data. In traffic modeling literature, ai’s are chosen to match experi-mental data.

Fig. 4. Spatially discrete (sectioned) highway traffic flow model.

2 Modular n arithmetic is assumed for the indices (i.e. i� 1 ¼ n when i ¼ 1).


To analyze the stability of the traffic flow induced by CTH policy for the individual vehicles (1)in this framework, we must choose, similar to the case of the spatially continuous model, rea-sonable control laws vi’s for the section to mimic the CTH control law. A reasonable choice is

vi ¼1

hw

1

qi

�� Lv

: ð12Þ

Referring to Fig. 4, a reasonable choice at first sight for the mixing coefficients ai in (11) are ai ¼ 1if we assume uniform spatial density profile of qðx; tÞ ¼ qiðtÞ whenever x is within section i. Thedesired equilibrium density pattern is given by q1 ¼ q2 ¼ � � � ¼ qn ¼ N=L where N is the totalnumber of vehicles on the highway.Let ~qqiðtÞ :¼ qiðtÞ � N=L, the traffic dynamics density errors are given by

d

dt

~qq1

~qq2

..

.

~qqn�1~qqn

0BBBBBB@

1CCCCCCA ¼ nLv

Lhw

1 0 0 . . . 0 0 �1�1 1 0 . . . 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 . . . 0 �1 1 0

0 0 . . . 0 0 �1 1

2666664

3777775

~qq1

~qq2

..

.

~qqn�1~qqn

0BBBBBB@

1CCCCCCA: ð13Þ

Using a Gershgorin disk argument, it is easy to see that the matrix in (13) is not Hurwitz (the disksare centered at nLv=ðLhwÞ and have radii nLv=ðLhwÞ, and there is a singleton of zero eigenvalue).Thus, the traffic dynamics at the equilibrium are exponentially unstable and there are initialperturbations that will grow exponentially. In fact, as we shall see later on, the dynamics wouldremain unstable whenever a > 0:5.

3.3. Summary

Thus, we have the interesting situation in which three models: a microscopic model given inSection 2, a spatially continuous model given by (5) and (6), and a spatially discrete model givenby (10)–(12), which were developed to model the same physical process, are nevertheless incon-sistent with each other and exhibit completely different stability properties. The microscopicmodel indicates that traffic flow induced by CTH policy is exponentially stable, the spatiallycontinuous model indicates that it is marginally stable, and the spatially discrete model indicatesthat it is exponentially unstable. It should be noted, however, that the microscopic model shouldbe the most reliable since it does not make further assumption once the CTH policy has beenspecified.

4. Biasing strategies

We suggest that the discrepancy noted above exists because in the process of abstracting thetraffic dynamics for the spatially continuous or the spatially discrete highway models, whether thefeedback information is taken from the upstream or the downstream has not been taken intoaccount with care. We refer to this modeling aspect as the biasing strategy. For example, in themicroscopic model in Section 2, vehicles sense and utilize for feedback, intervehicle spacings


ahead of them. We call this downstream biased. In the spatially continuous model, feedback in-formation is obtained at the same location as for the control. We call this neutrally biased orcolocatedly biased. In the spatially discrete model, biasing is determined by the mixing coefficientsai’s in (11). It is upstream biased if ai > 0:5 (as in the analysis above), and it is downstream biased ifa < 0:5. It is neutrally biased if ai ¼ 0:5.Our assertion is that consistent stability properties would be obtained if the same biasing

strategy is employed in each of the three modeling paradigms. To show that this is indeed the case,we incorporate the three biasing strategies into the three models, and investigate their stabilityproperties.

4.1. Microscopic model

For the microscopic model in (2), the CTH control policy for the three biasing strategies are asfollows. The readers are again reminded that vehicle indices increase from the leader (down-stream) to followers (upstream).

• Downstream biasing. The speed of vehicle i is given by

vi ¼1

hw½Dxi�; ð14Þ

where Dxi is the intervehicle spacing ahead of the vehicle i. This is the same as (1) and would be theactual CTH control policy for ACC vehicles equipped with forward looking sensors.• Neutral biasing. The speed of vehicle i is given by

vi ¼1

2hw½Dxi þ Dxiþ1�: ð15Þ

Thus, it uses the mean of the intervehicle spacings both ahead and behind for feedback.• Upstream biasing. The speed of vehicle i is

vi ¼1

hw½Dxiþ1�: ð16Þ

Thus, it uses the intervehicle spacings behind the vehicle for feedback. This scheme requires arearward looking sensor and would probably not be a sensible approach for actual implemen-tation.

We present the traffic flow stability results for the three biasing strategies.

Theorem 2. For the microscopic traffic model given by (2), let N be the number of vehicles on thehighway, and L be the highway length, and Lv be the vehicle length. The equilibrium intervehiclespacings Dxi ¼ L=N � Lv have the following properties:

1. it is exponentially stable under the downstream biasing strategy (14);2. it is marginally stable under the neutral biasing strategy (15);3. it is exponentially unstable under the upstream biasing strategy (16).


Proof. The downstream biasing case has already been analyzed in Section 2.For the other two cases, define ~DDxiðtÞ :¼ DxiðtÞ � ðL=N � LvÞ to be the spacing errors.Under the neutral biasing strategy, the dynamics of the system are given by

d

dt

~DDx1~DDx2...

~DDxN�1~DDxN

2666664

3777775 ¼ 1

hw

0 �1 0 . . . 11 0 �1 . . . 0

..

. ... ..

. ... ..

.

0 . . . 1 0 �1�1 0 . . . 1 0

266664

377775

~DDx1~DDx2...

~DDxN�1~DDxN

2666664

3777775: ð17Þ

Consider the Lyapunov function,

V ¼ 1

2

XNi¼1

~DDx2i :

Therefore, _VV ¼ 0 since the matrix in (17) is skew symmetric. This shows that the equilibriumtraffic pattern is marginally Lyapunov stable. Any perturbation will neither subside nor grow.Under the upstream biasing case, the dynamics are given by

d

dt

Dx1Dx2...

DxN�1DxN

2666664

3777775 ¼ 1

hw

1 �1 0 . . . 00 1 �1 . . . 0

..

. ... ..

. ... ..

.

0 . . . 0 1 �1�1 0 . . . 0 1

266664

377775

Dx1Dx2...

DxN�1DxN

2666664

3777775: ð18Þ

Since the matrix in (18) is just the negative of that in (4), the traffic dynamics are exponentiallyunstable. �

4.2. Spatially discrete model

The spatially discrete model (Fig. 4) is given by Eqs. (10)–(12). Recall that the flowrate out of asection, defined in (11), is a convex combination of the ideal flowrates of the upstream and thedownstream sections. The mixing coefficients ai’s in (11) determine this combination. Three classes(not exhaustive) of biasing strategies are:

• downstream biasing for each i ¼ 1; . . . ; n, ai < 0:5;• neutral biasing a1 ¼ a2 ¼ � � � ¼ an ¼ 0:5;• upstream biasing for each i ¼ 1; . . . ; n, ai > 0:5.

In the analysis for the spatially discrete model put forth in Section 3, ai’s were chosen to be 1.So an upstream biasing strategy was adopted.Traffic flow dynamics for an n-sectioned circular highway, can be written as

_qq ¼ A � q; ð19Þwhere q ¼ ½q1; . . . ;qn�

Tand A 2 Rn�n


A ¼ nLvLhw

�ð1� an � a1Þ ð1� a1Þ 0 . . . 0 0 �an

�a1 �ð1� a1 � a2Þ ð1� a2Þ . . . 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 . . . 0 �an�2 �ð1� an�2 � an�1Þ ð1� an�1Þð1� anÞ 0 . . . 0 0 �an�1 �ð1� an�1 � anÞ

2666664

3777775:

ð20Þ

Theorem 3. Consider the spatially discrete highway model given by Eqs. (10) and (11), and with theCTH control law given by (12). Assume that the number of vehicles on the highway is N, and the totallength of the highway is L. The equilibrium traffic pattern, given by

qiðtÞ ¼NL; i ¼ 1; . . . ; n

has the following stability properties.

• it is exponentially stable if the traffic model is downstream biased,• it is exponentially unstable if the model is upstream biased,• it is marginally stable, if the model is neutrally biased.

Proof. Let ~qqiðtÞ :¼ qiðtÞ � ðN=LÞ and ~qq ¼ ~qq1 . . . ~qqnð Þ. Introduce a new coordinate ½�qqT?; �qq0�

T, where

�qq? 2 Rn�1, which is related to ~qq via:

~qq ¼

1

I ...

10 � � � 0 1

0BB@

1CCA

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}T

j�qq?j�qq0

0BB@

1CCA: ð21Þ

Notice that �qq0ðtÞ is the mean density error since it is the deviation of the total number of vehiclesfrom N, multiplied by n=L. It can easily be shown that �qq0ðtÞ is invariant.Define the Lyapunov function,

V ¼ 1

2

Xni¼1

~qq2i ¼

1

2~qqT~qq ¼ 1

2�qqT? �qq0

� � 1

In�1�n�1...

1

1 . . . 1 n

0BBBB@

1CCCCA

�qq?

�qq0

!:

Then,

_VV ¼ ~qqTA~qq ¼ ~qqT Aþ AT

2

� |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

As

~qq ¼ �qqT? �qq0

� � 0

�AAs...

0

0 . . . 0 0

0BBBB@

1CCCCA

�qq?

�qq0

!¼ �qqT

?�AAs�qq?:


Explicitly,

As ¼nLvLhw

an þ a1 � 1 12� a1 0 . . . 0 0 1

2� an

12� a1 a1 þ a2 � 1 1

2� a2 . . . 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 . . . 0 12� an�2 an�2 þ an�1 � 1 1

2� an�1

12� a1 0 . . . 0 0 1

2� an�1 an�1 þ an � 1

26666664

37777775

and

�AAs ¼nLvLhw

an þ a1 � 1 12� a1 0 . . . 0 0 0

12� a1 a1 þ a2 � 1 1

2� a2 . . . 0 0 0

..

. ... ..

. ... ..

. ...

12� an�2

0 0 . . . 0 0 12� an�2 an�2 þ an�1 � 1

26666664

37777775:

The stability of the system is determined by the signs of the eigenvalues (which are real) of thesymmetric matrix �AAs. These can be evaluated using Gershgorin disks. First, we remark that �AAs isnon-singular when the highway model is either downstream biased, or upstream biased. Thisresult can be obtained by evaluating the determinant of �AAs recursively in terms of its principleminors.For the case of downstream biasing, each ai < 0:5. The union of the n� 1 Gershgorin disks of

�AAs is contained in the union of the disks with centers at ai þ ai�1 � 1 < 0 and with radiij1� ai � ai�1j, for i ¼ 1; . . . ; n� 1 (modulo n index calculation assumed). Therefore, the disks andthe eigenvalues lie on the closed LHP. Thus, V ðtÞ is non-increasing. By LaSalle’s theorem, andsince As is non-singular, �qq?ðtÞ ! 0, i.e. there exists some rðtÞ so

~qq1ðtÞ...

~qqn�1ðtÞ

0B@

1CA! rðtÞ

1...

1

0@

1A:

Since we can repeat the above analysis with the transformation T in (21) replaced by a permu-tation of the columns, such as

T ¼

1 0 � � � 01...

I

1

0BB@

1CCA

then, we would have found that

~qq2ðtÞ~qq3ðtÞ...

~qqnðtÞ

0BBB@

1CCCA! r�ðtÞ

1...

1

0@

1A:


Thus, we must have limt!1 rðtÞ ¼ r�ðtÞ, or limt!1 ~qqiðtÞ ¼ ~qqjðtÞ for each i; j ¼ 1; . . . ; n. SincePni¼1 ~qqiðtÞ ¼ 0, ~qqiðtÞ ! 0. Finally, because (19) is a linear system and for linear systems, asymp-

totic convergence implies exponential convergence, �qq?ðtÞ ! 0 exponentially.For the case of upstream biasing, each ai > 0:5. All the Greshgorin disks lie on the closed

RHP. Because �AAs is non-singular, the eigenvalues are strictly positive. The reversed timesystem

_~qq~qq ¼ �A~qq;

would therefore be exponentially stable as in the downstream biasing case. This shows that theoriginal system (19) is exponentially unstable.For the case of neutral biasing, ai ¼ 0:5, �AAs ¼ 0, so that _VV ¼ 0. This indicates that the equi-

librium is stable but not asymptotically stable. Perturbations will neither be attenuated nor am-plified. �

It is instructive to investigate numerically the eigenvalues of A in (19) since the convergence rateis determined by the magnitude of the real part of the eigenvalue of A with the smallest real part.For simplicity, we computed the case when a1 ¼ a2 ¼ � � � ¼ an ¼ a. In Fig. 5, we see that for each06 a6 1, the n eigenvalues of A lie on an ellipsoidal configuration which touches the origin. Asexpected, when a < 0:5, the ellipsoids lie on the closed LHP (stable); when a ¼ 0:5, the eigenvalueslie on the imaginary axis (marginally stable); when a > 0:5, the ellipsoids lie on the closed RHP(unstable). When a ¼ 0, the ellipsoid is a circle, centered at �nLv=ðLhwÞ, with radius nLv=ðLhwÞ.The ellipsoids for a > 0 can be thought of as copies of this circle scaled by ð1� 2aÞ along the realaxis.

Fig. 5. Locus of eigenvalues of A for the spatially discrete highway for various mixing coefficients a, with n ¼ 6,

Lv=ðhwLÞ ¼ 1. Filled locations are eigenvalues at a ¼ 0, 0.2, 0.4, 0.6, 0.8, 1.


This observation can be confirmed from the estimation of the convergence rate using eigen-values of the matrix �AAs. This is because when a1 ¼ � � � ¼ an ¼ a, �AAs and it eigenvalues are pro-portional to ð0:5� aÞ. Therefore, as a varies, the real part of the dominant eigenvalues (in fact alleigenvalues) of A should also be proportional to ð0:5� aÞ. Fig. 6 shows that this is indeed thecase.The effect of the number of sections, n, is investigated next. As shown in Fig. 7, the magnitude

of the real part of the dominant eigenvalue of A, which determines the exponential convergencerate, decreases with the number of sections n. Moreover, the plot in log scale (Fig. 7) reveals thatfor sufficiently large n, the convergence rate is inversely proportional to n.In fact, it can be shown that when a < 0:5, the magnitude of the real part of the dominant

eigenvalue of A is given by

Lvð1� 2aÞnLhw

1

�� cos

2pn

� ��:

4.3. Spatially continuous model

In this abstraction, the traffic flow is modeled as compressible fluid flow (Fig. 3). The evolutionof the density profile qð�; tÞ : ½0;LÞ ! R is described by Eq. (5) where vðx; tÞ is the velocity profileof the flow which is assumed to be the manipulated quantities. The readers are reminded that xincreases from upstream to downstream. The three classes of biasing strategies for the CTH policyare defined according to whether the density information is obtained downstream, colocatedly, orupstream to the location where control is applied. They are

Fig. 6. Magnitude of the real part of dominant eigenvalues of A for the spatially discrete highway as a varies from

0 ! 1, n ¼ 5, Lv=ðLhwÞ ¼ 1.


• Downstream biasing:

vðx; tÞ ¼ 1

hw

1

qðxþ Dðx; tÞ; tÞ

�� Lv

; Dðx; tÞ > 0: ð22Þ

• Neutral biasing:

vðx; tÞ ¼ 1

hw

1

qðx; tÞ

�� Lv

: ð23Þ

Fig. 7. Magnitude of the real part of dominant eigenvalues of A for the spatially discrete highway versus the number of

highway sections n, with a ¼ 0:1, Lv=ðLhwÞ ¼ 1.


• Upstream biasing:

vðx; tÞ ¼ 1

hw

1

qðx� Dðx; tÞ; tÞ

�� Lv

; Dðx; tÞ > 0: ð24Þ

Similar classes of biasing strategies can also be defined. For example, one can use instead ofpointwise feedback, use feedback based on integrated information downstream and upstream. Inthis case, downstream biasing would rely more heavily on downstream information, and upstreambiasing would rely more on upstream information. We used the pointwise feedback definitionsabove because they can be analyzed relatively simply. Our goal for the spatially continuous modelis to illustrate that consistent stability result can be recovered when consistent biasing strategiesare used. Because of this focus, we prefer definition of the biasing strategies which can be analyzedin a relatively straight forward manner.Assuming that there are N vehicles on the highway. Under the CTH policy, the desired equi-

librium density profile is

qðx; tÞ ¼ N=L:

The stability property of the equilibrium profile when the neutral biased case (marginallystable) has already been established in Section 3. Before we analyze the downstream and upstreambiased cases, we first perform a coordinate transformation for Eq. (5) and define a new statevariable so that the closed loop dynamics become linear. We are motivated by the observationthat both the microscopic and spatially discrete highway models exhibit linear closed loop dy-namics.Assume that qðx; tÞ > 0 and is sufficiently smooth for all x 2 ½0;LÞ and tP 0. 3 We define a new

time varying spatial coordinate �xxðx; tÞ such that:

o�xxox

ðx; tÞ ¼ qðx; tÞ; o�xxot

ð0; tÞ ¼ �qð0; tÞvð0; tÞ: ð25Þ

Using the continuity Eq. (5), and the fact that o2�xx=otox ¼ o2�xx=oxot, this definition results in

o�xxot

ðx; tÞ ¼ �qðx; tÞvðx; tÞ: ð26Þ

In the new coordinates, the highway length is LðtÞ :¼ �xxðL; tÞ � �xxð0; tÞ and the boundaries areidentified: �xxðL; tÞ ¼ �xxð0; tÞ. Since qðx; tÞ > 0 by assumption, the inverse mapping of �xxð�; tÞ exists.We denote it by dð�; tÞ : ½0; LðtÞÞ ! ½0; LÞ so that dð�xxðx; tÞ; tÞ ¼ x. The new coordinate system is atranslation and a scaling of the original system (5). Its physical significance is that the density

3 These assumptions are made for technical purposes. Notice that if qðx; tÞ is smooth, the control policies that weconsider are also smooth with respect to x. It is reasonable to assume that qð�; �Þ 6¼ 0, which would imply that on a

circular highway (which is compact), qð�; tÞ will be strictly positive (physical reasoning precludes that it is negative).

qð�; �Þ must be non-zero because if qðx; tÞ ¼ 0 at some x, ðoq=oxÞðx; tÞ must also be 0 (since qðx; tÞP 0). Therefore,

according to (5), any point at which qðx; tÞ ¼ 0 must remain 0. Convergence to the desired density will be impossible.

For logistical reasons therefore, the case qðx; tÞ ¼ 0 at some x should be excluded in the spatially continuous highway

model paradiagm.


becomes uniform in this coordinate system. To see this, notice that if �xx1 ¼ �xxðx1ðtÞ; tÞ and�xx2 ¼ �xxðx2ðtÞ; tÞ (or equivalently, x1ðtÞ ¼ dð�xx1; tÞ and x2ðtÞ ¼ dð�xx2; tÞ), we have

�xx2 � �xx1 ¼Z x2ðtÞ

x1ðtÞqðx; tÞdx;

where the right hand side represents the number of vehicles in the interval demarcated by �xx1 and�xx2. In essence, the new coordinate system is constructed so that it is attached to the moving ve-hicles. Hence the original Eulerian framework has been converted into a Lagrangian framework.Let us define the spacing profile in the original coordinates to be,

gðx; tÞ :¼ 1

qðx; tÞ :

In the new coordinates, it is

f ð�xx; tÞ :¼ gðdð�xx; tÞ; tÞ :¼ 1

qðx ¼ dð�xx; tÞ; tÞ : ð27Þ

We now show that the traffic dynamics, described in terms of the spacing profile in the new co-ordinates, f ð�xx; tÞ, is linear with respect to the speed control policy. Differentiating the expressiondð�xxðx; tÞ; tÞ ¼ x with respect to x and t, we have

odo�xx

ð�xxðx; tÞ; tÞ ¼ o�xxox

ðx; tÞ !�1

¼ qðx; tÞ;

odot

ð�xxðx; tÞ; tÞ �xx

¼ � odo�xx

ð�xxðx; tÞ; tÞ o�xxot

ðx; tÞ ¼ �q�1ðx; tÞ o�xxot

ðx; tÞ:

Making use of (26), we have

ofot

ð�xx; tÞ ¼ o�vvo�xx

ð�xx; tÞ; ð28Þ

where �vvð�xx; tÞ is the control policy in the new coordinate system:

�vvð�xx; tÞ ¼ vðdð�xx; tÞ; tÞ ¼ vðx; tÞ: ð29Þ

Given Dðx; tÞ > 0, because the coordinate transformation �xxð�; tÞ is monotonic, there existDupð�xx; tÞ > 0 and Ddownð�xx; tÞ > 0 so that the CTH policy with the downstream biasing (22) andupstream biasing (24) can be rewritten as

• Downstream biasing:

�vvð�xx; tÞ ¼ 1

hwf ð�xxþ Ddownð�xx; tÞ; tÞ: ð30Þ

• Upstream biasing:

�vvð�xx; tÞ ¼ 1

hwf ð�xx� Dupð�xx; tÞ; tÞ: ð31Þ


Let us define a Lyapunov function:

V ¼ 1

2

If ð�xx; tÞh

� L=Ni2d�xx; ð32Þ

where the integration is taken over the domain of the circular highway, �xx 2 ½�xxð0; tÞ;�xxðL; tÞÞ.Notice that V ! 0 means that f ð�xx; tÞ ! L=N and qðx; tÞ ! N=L. Taking time derivative of V inEq. (32),

_VV ¼I

ðf ð�xx; tÞ � L=NÞ ofot

ð�xx; tÞd�xx: ð33Þ

Inserting Eq. (28) into (33), and using the biased control policy (30) and (31), and recognizing thatHðof =o�xxÞd�xx ¼ 0, we get

_VV ¼Hf ð�xx; tÞ of ð�xxþDdownÞ

o�xx d�xx downstream biasing;Hf ð�xx; tÞ of ð�xx�DupÞ

o�xx d�xx upstream biasing:

(

By shifting the reference frame by Ddown and �Dup respectively, we get

_VV ¼Hf ð�xx� Ddown; tÞ of

o�xx ð�xx; tÞd�xx ¼H

f ð�xx� Ddown; tÞ � f ð�xx; tÞh i

ofo�xx ð�xx; tÞd�xx downstream biasing;H

f ð�xxþ Dup; tÞ ofo�xx ð�xx; tÞd�xx ¼

Hf ð�xxþ Dup; tÞ � f ð�xx; tÞh i

ofo�xx ð�xx; tÞd�xx upstream biasing:

8<:

The last equality is becauseIf ð�xx; tÞ of

o�xxð�xx; tÞd�xx ¼

I1

2

o

oxf 2ð�xx; tÞd�xx ¼ 0:

Since for small jDj, D � ðof =o�xxÞð�xx; tÞ and f ð�xxþ DÞ � f ð�xxÞ share the same signs, therefore, forsufficiently small positive Ddownð�xx; tÞ and Dupð�xx; tÞ, we have _VV 6 0 when the downstream biasingstrategy is employed, and _VV P 0 when the upstream biasing strategy is employed. This means thaterror in the spacing profile, and also the density profile, would decay when a downstream biasingstrategy is used, and would grow when an upstream strategy is used. If further technical condi-tions such as uniform smoothness, boundedness of signals etc. are met, the above analysis wouldsuggest that when a downstream biasing strategy is used, ðof =o�xxÞð�xx; tÞ ! 0. This in turn impliesthat qðx; tÞ ! N=L, the desired equilibrium density.

4.4. Summary

In this section, we have analyzed the microscopic model, the spatially discrete highway model,and the spatially continuous model when biasing strategies are accounted for. These results aresummarized in Table 1. We found that if consistent biasing strategies are applied to each of themodels, consistent stability properties can be regained. In particular, for the CTH policy, ifdownstream biasing strategies are applied, then for each of the three modeling frameworks, thedesired uniform traffic flow is stable, and any initial perturbations in the density or spacingprofiles will decay and dissipate.


5. Simulations

To illustrate ideas, simulations are performed. The circular highway is L ¼ 2000 m long. It ispopulated by N ¼ 100 vehicles, each has length Lv ¼ 5 m. The highway is modeled in the mi-croscopic, spatially discrete and spatially continuous frameworks. A CTH policy with hw ¼ 1 s isassumed. The spatially discrete model has n ¼ 20 sections, and the mixing coefficients ai’s i ¼1; . . . ; 20 are chosen to be ð1� ðnL=LvN 2ÞÞ ¼ 0:1. This choice is made so that the convergencerates for the spatially discrete and microscopic models should be compatible. For the spatiallycontinuous model, the downstream biasing parameter is arbitrarily chosen to be D ¼ 20 m. Aninitial density or intervehicle spacing perturbation for each framework is set to be sinusoidal, andthey are compatible between models.As shown in Fig. 8 the evolutions of the density profiles for the three modeling frameworks are

very similar in shape. As expected, all three models exhibit responses that converge to the uniformdensity profile. Notice that in the case of the microscopic model, the intervehicle spacings areconverted to density profiles for easy comparisons. From Fig. 9, we see that the convergence ratesof the errors for the spatially discrete model and for the microscopic model are also similar. Theseresults show that the macroscopic models are reasonably good abstractions of the microscopicsystem, as long as the a consistent downstream biasing strategy is used.

6. Open stretch highway

The result in Section 2 is that the CTH policy induces exponentially stable traffic flow patternson a circular highway. Similar results for both the spatially discrete and the spatially continuousmacroscopic circular highway models when a downstream biasing strategy is used are also ob-tained in Section 4. On the other hand, the authors in Dharba and Rajagopal (1999) studied theCTH policy for an open stretch of highway and concluded from the analysis of a spatially con-tinuous model and a spatially discrete model, that the resulting traffic dynamics are unstable.How should one explain this difference? On the one hand, one observes that the traffic flow

induced by ACC-CTH is stable when there are no entries or exits, but becomes unstable when theentry/exit conditions in Dharba and Rajagopal (1999) are imposed. This seems to suggest that theACC-CTH policy has some intrinsically stability property which can be destroyed by some (im-proper) inlet and outlet conditions. On the other hand, one may also propose an explanationbased on the observation that on a circular highway (but not on an openstretch of highway),information can travel around the highway back to the same location. Therefore, the circular

Table 1

Traffic stability properties for various modeling paradigms and biasing strategies

Biasing Highway models

Spatially continuous Spatially discrete Microscopic

Downstream Stable Stable Stable

Neutral Marginally stable Marginally stable Marginally stable

Upstream Unstable Unstable Unstable


Fig. 8. Evolutions of density profiles for the same circular highway using three modeling paradigms. Top: microscopic

model; middle: spatially discrete model; bottom: spatially continuous model.


feedback path provides the mechanism for stabilizating an ACC-CTH traffic flow which isotherwise intrinsically unstable. We believe that the former explanation (i.e. entry/exit conditionscontribute to instability) is more plausible. Our opinion is formed based on the fact (see below)that it is possible to have stable traffic flow on an openstretch of highway (which does not have thecircular feedback path) using ACC-CTH policy, when suitable entry and exit conditions areimposed.To illustrate, consider the inlet and outlet conditions used in Dharba and Rajagopal (1999)

which are that (a) the inlet flowrate q0ðtÞ is independent of the traffic condition, and (b) the outletflow is given by qnðtÞ ¼ qnðtÞvnðtÞ. Notice that both the inlet and outlet conditions are defined suchthat the flow rates do not depend on the densities in the downstream sections. In fact, arbitraryinlet flow into a highway section is allowed even if it is already completely jammed.In this situation, the matrix A in (20) that describes the density error dynamics becomes

(Dharba and Rajagopal, 1999):

A2 ¼nLvLhw

a1 ð1� a1Þ 0 . . . 0 0 0�a1 �ð1� a1 � a2Þ ð1� a2Þ . . . 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 . . . 0 �an�2 �ð1� an�2 � an�1Þ ð1� an�1Þ0 0 . . . 0 0 �an�1 an�1

2666664

3777775:

ð34Þ

Notice that A2 in (34) differs from A in (20) only in the first and last rows, thus verifying that thedifference between A2 in (34) and A in (20) are caused that the entry and exit conditions. A2 in (34)can be numerically verified to have eigenvalues with positive real parts (hence unstable) for any ai.Therefore, as concluded in Dharba and Rajagopal (1999) the combination of an open stretch of

Fig. 9. RMS density errors for the spatially discrete and microscopic model versus time.


highway populated by ACC-CTH vehicles, and with these input and output conditions, producesunstable traffic flow.We now ask the question whether there exist entry and exit conditions under which an open

stretch of highway populated by ACC-CTH vehicles can be stable. The answer is yes. Let usconsider a different set of inlet and outlet conditions below, which are motivated by our under-standing of the role that biasing strategies play in traffic stability for the circular highway. Todefine the new entry and exit conditions, we assume that there are two fictitious sections, oneupstream to the first section (the source), and one downstream to the last section (the sink).Now, define the inlet flowrate to be

q0 ¼ � Lhw

ð1� a0Þq1 þ r0; ð35Þ

where r0 is some exogenous input signal. This inlet flow condition would have been generated by(11) if the density of the source section q0ðtÞ satisfies:

r0 :¼1

hw1½ � Lvanq0ðtÞ�:

Similarly, we define the outlet condition qn according to (11) assuming that qn drains into thesink section with density qnþ1 � 0, i.e.

qn ¼ anðqnvnÞ þ ð1� anÞðqnþ1vnþ1Þ

¼ 1

hw� Lvhw

ðanqn þ ð1� anÞqnþ1Þ

¼ 1

hw� Lvhw

anqn:

ð36Þ

Notice that these new entry/exit conditions are dependent on the densities in the downstreamsections for the inlet and outlet flows. Under these conditions, the matrix that determines thetraffic flow stability becomes:

A3 ¼nLvLhw

�ð1� a0 � a1Þ ð1� a1Þ 0 . . . 0 0�a1 �ð1� a1 � a2Þ ð1� a2Þ . . . 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 . . . 0 �an�2 �ð1� an�2 � an�1Þ ð1� an�1Þ0 0 . . . 0 0 �an�1 �ð1� an�1 � anÞ

2666664

3777775:

ð37Þ

Notice that if we choose the mixing parameters for the inlet and the outlet to be completelyupstream biased, i.e. a0 ¼ an ¼ 1, then the system matrix A3 in (37) reduces to A2 in (34). Thisindicates that the inlet and outlet conditions used to generate the (unstable) matrix A2 in (34) areequivalent to upstream biasing strategies.What happens if we choose the mixing parameters to be a0 < 0:5 in (35) and an < 0:5 in (36) so

that the inlet and outlet conditions correspond to downstream biasing strategies? In this case, allthe eigenvalues of A3 in (37) have again non-positive real parts. To see this, notice that A3 þ AT

3

is negative semidefinite when ai < 0:5, i ¼ 0; . . . ; n. The Hurwitzness of A3 can be obtained by


defining the Lyapunov function as in the proof of Theorem 3. Hence, the system becomes stablewhen the new inlet and outlet conditions defined using downstream biasing strategies are imposed.What we have shown is in this section is that with some appropriate inlet and outlet conditions,

an open stretch of highway populated with ACC-CTH vehicles will exhibit stable traffic flowpattern. Notice that stability does not require the use of the circular feedback path. Whether theseinlet and outlet conditions can be realistically achieved or can naturally occur remain to be seen.This discussion will be beyond the scope of the present paper. Intuitively, ramp metering or othertraffic devices may be helpful to impose these inlet/outlet conditions in the event that they do notalready naturally occur.

7. Discussion

The fact that the CTH-ACC policy induces stable traffic on a circular highway and openstretchhighways with entry and exit conditions defined to obey downstream biasing suggests that theACC-CTH policy can remain a viable option for ACC vehicles to accomplish the objective ofstable traffic flow control.Because CTH policy is intrinsically stable on circular highways or highways with suitably

defined inlet/outlet conditions, an intriguing possibility that we are currently exploring is to utilizeon-ramp metering strategies to define suitable inlet and output conditions for non-circularhighways. One idea is to use ramp metering to construct ‘‘logically circular’’ highways from anopen stretch of highway. Since the convergence rate generally decreases with highway length,number of sections, or the number of vehicles, it is expected that the convergence rates wouldimprove if a long highway (circular or not) is partitioned into several short logically circularhighways.In our analysis, we demonstrated that the biasing strategy is an important factor that deter-

mines traffic flow stability. Specifically, as feedback information is taken from a location furtherand further downstream, the convergence rate of the traffic flow is expected to improve. ACCpolicy generally utilizes the intervehicle spacing ahead of the vehicle for feedback. An interestingquestion to explore is whether or not any benefits can be obtained by defining ACC speed spacingpolicies using intervehicle spacings that are one or more vehicles ahead.It is interesting to note that the so called ‘‘greedy rule’’ traffic management plan proposed using

the activity framework in Broucke and Varaiya (1996) in which the speed of an upstream sectionis specified to occupy the available gap in a downstream section, can be considered a downstreambiased strategy. This traffic plan has also been shown to be stable.In this paper, we have assumed that vehicles observe faithfully the speed-spacing policy. It is

natural to wonder what would happen to the traffic flow stability when the dynamics associatedwith the vehicle control system is considered. Intuitively, since CTH policy is exponentially stable,we should expect that the stability should have some amount of robustness to factors such as thedynamics of the vehicle control system. A more intriguing opportunity is that when vehicle dy-namics are considered, one can and must consider, instead of the speed/spacing relationship, atleast the speed/spacing/range-rate relationship. Notice that when the desired speed-spacing isassumed to be faithfully executed, if the vehicle’s speed is made to depend on the speed of itsdownstream neighbor, the formulation of the ACC policy must include an extra non-autonomous


term corresponding to the speed of a leader (in the case of an open stretch highway), or a biasingdesign factor (in the case of a circular highway). Such a non-autonomous formulation is some-what unnatural. It can be avoided if vehicle dynamics are included.The CTH policy considered in this paper is an idealization because it assumes that an infinite

vehicle speed can be achieved. In reality, vehicle speed is limited. Since one might like to operatethe system at maximum speed (so as to maximize level of service, LOS), it would also beworthwhile to investigate the effect of velocity saturation on the traffic flow stability.

8. Conclusions

In this paper, we studied the traffic flow stability induced by the CTH-ACC policy on a circularhighway using three modeling abstractions: a microscopic model, a spatially discrete model, and aspatially continuous model. It is found that CTH is exponentially stable. However, this propertycan be easily missed if the analysis is performed in the abstract macroscopic frameworks withoutusing a consistent biasing strategy. If the biasing strategy is consistently applied, it can be shownthat the traffic flow would be stable (unstable) in all the three modeling abstractions when adownstream (upstream) biasing strategy is used. In addition, the stability result can be extended tonon-circular openstretch highways, as long as entry and exit conditions are defined to obey thedownstream biasing strategy. These results suggest that CTH may yet remain a viable speed-spacing policy candidate for ACC vehicles.

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