IFAC Journal of Systems andkrishnaj/Publications_files/papers/...A part of this work appeared in...

IFAC Journal of Systems and Control 2 (2017) 18–32

Contents lists available at ScienceDirect

IFAC Journal of Systems and Control

journal homepage: www.elsevier.com/locate/ifacsc

The modified optimal velocity model: stability analyses and design

guidelines

�

Gopal Krishna Kamath

∗, Krishna Jagannathan , Gaurav Raina

Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

a r t i c l e i n f o

Article history:

Received 14 August 2017

Revised 14 November 2017

Accepted 15 November 2017

Available online 16 November 2017

Keywords:

Transportation networks

Car-following models

Time delays

Stability

Convergence

Hopf bifurcation

a b s t r a c t

Reaction delays are important in determining the qualitative dynamical properties of a platoon of vehicles

traveling on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics

of the Modified Optimal Velocity Model (MOVM). Specifically, we analyze the MOVM in three regimes –

no delay, small delay and arbitrary delay. In the absence of reaction delays, we show that the MOVM is

locally stable. For small delays, we then derive a sufficient condition for the MOVM to be locally stable.

Next, for an arbitrary delay, we derive the necessary and sufficient condition for the local stability of the

MOVM. We show that the traffic flow transits from the locally stable to the locally unstable regime via

a Hopf bifurcation. We also derive the necessary and sufficient condition for non-oscillatory convergence

and characterize the rate of convergence of the MOVM. These conditions help ensure smooth traffic flow,

good ride quality and quick equilibration to the uniform flow. Further, since a Hopf bifurcation results in

the emergence of limit cycles, we provide an analytical framework to characterize the type of the Hopf

bifurcation and the asymptotic orbital stability of the resulting non-linear oscillations. Finally, we corrob-

orate our analyses using stability charts, bifurcation diagrams, numerical computations and simulations

conducted using MATLAB.

© 2017 Elsevier Ltd. All rights reserved.

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1. Introduction

Intelligent transportation systems constitute a substantial

theme of discussion on futuristic smart cities. In this context,

self-driven vehicles are a prospective solution to address traf-

fic issues such as resource utilization and commute delays;

(see Rajamani, 2012 , Section 5.2, van den Berg and Verhoef, 2016;

Greengard, 2015; Vahidi and Eskandarian, 2003 and references

therein). To ensure that these objectives are met, in addition to en-

suring human safety, the design of control algorithms for these ve-

hicles becomes important. To that end, it is imperative to have an

in-depth understanding of human behavior and vehicular dynam-

ics. This has led to the development and study of a class of dynam-

ical models known as the car-following models ( Bando, Hasebe,

Nakanishi, & Nakayama, 1998; Chowdhury, Santen, & Schadschnei-

der, 20 0 0; Gazis, Herman, & Rothery, 1961; Helbing, 2001; Kamath,

Jagannathan, & Raina, 2015; Orosz & Stépán, 2006 ).

� A part of this work appeared in Proceedings of the 53rd Annual Allerton

Conference on Communication, Control and Computing, pp. 538–545, 2015. DOI:

10.1109/ALLERTON.2015.7447051 ∗ Corresponding author.

E-mail addresses: [email protected] (G.K. Kamath), [email protected]

(K. Jagannathan), [email protected] (G. Raina).

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https://doi.org/10.1016/j.ifacsc.2017.11.003

2468-6018/© 2017 Elsevier Ltd. All rights reserved.

Feedback delays play an important role in determining the

ualitative behavior of dynamical systems ( Hale & Lunel, 2011 ). In

articular, these delays are known to destabilize the system and in-

uce oscillatory behavior ( Kamath et al., 2015; Sipahi & Niculescu,

006 ). In the context of human-driven vehicles, predominant com-

onents of the reaction delay are psychological and mechanical

n nature ( Sipahi & Niculescu, 2006 ). In contrast, delays in self-

riven vehicles arise due to sensing, communication, signal pro-

essing and actuation, and are envisioned to be smaller than hu-

an reaction delays ( Kesting & Treiber, 2008 ).

In this paper, we investigate the impact of delayed feedback on

he qualitative dynamical properties of a platoon of vehicles trav-

ling on a straight road. Specifically, we consider each vehicle’s

ynamics to be modeled by the Modified Optimal Velocity Model

MOVM) ( Kamath et al., 2015 ). Motivated by the wide range of val-

es assumed by reaction delays in various scenarios, we analyze

he MOVM in three regimes; namely, ( i ) no delay, ( ii ) small de-

ay and ( iii ) arbitrary delay. In the absence of delays, we show that

he MOVM is locally stable. When the delays are rather small, as in

he case of self-driven vehicles, we derive a sufficient condition for

he local stability of the MOVM using a suitable approximation. For

he arbitrary-delay regime, we analytically characterize the region

f local stability for the MOVM.

In the context of transportation networks, two additional prop-

rties are of practical importance; namely, ride quality (lack of


http://www.ScienceDirect.com

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https://doi.org/10.1109/ALLERTON.2015.7447051

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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 19

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erky vehicular motion) and the time taken by the platoon to attain

he desired equilibrium when perturbed. Mathematically, these

ranslate to studying the non-oscillatory property of the MOVM’s

olutions and the rate of their convergence to the desired equilib-

ium. In this paper, we also characterize these properties for the

OVM.

In the context of human-driven vehicles, model parameters

enerally correspond to human behavior, and hence cannot be

tuned” or “controlled.” However, our work enhances phenomeno-

ogical insight into the emergence and evolution of traffic conges-

ion. For example, a peculiar phenomenon known as the “phan-

om jam” is observed on highways ( Chowdhury et al., 20 0 0; Hel-

ing, 2001 ). Therein, a congestion wave emerges seemingly out of

owhere and propagates up the highway from the point of its ori-

in. Such an oscillatory behavior in the traffic flow has typically

een attributed to a change in the driver’s sensitivity, such as a

udden deceleration; for details, see Chowdhury et al. (20 0 0) and

elbing (2001) . In general, feedback delays are known to induce

scillations in state variables of dynamical systems ( Kamath et al.,

015; Sipahi & Niculescu, 2006 ). Since the MOVM explicitly incor-

orates feedback delays, and relative velocities and headways con-

titute state variables of the MOVM, our work provides a theoret-

cal basis for understanding the emergence and evolution of oscil-

atory phenomena such as “phantom jams.” In particular, our work

erves to highlight the possible role of reaction delays in the emer-

ence of oscillatory phenomena in traffic flows. More generally,

ur results reveal an important observation: the traffic flow may

ransit into instability due to an appropriate variation in any sub-

et of model parameters. To capture this complex dependence of

tability on various parameters, we introduce an exogenous, non-

imensional parameter in our dynamical model. We then analyze

he behavior of the resulting system as the exogenous parameter

s pushed just beyond the stability boundary. We show that non-

inear oscillations, termed limit cycles , emerge in the traffic flow

ue to a Hopf bifurcation .

In the context of self-driven vehicles, reaction delays are ex-

ected to be smaller than their human counterparts ( Kesting &

reiber, 2008 ). Hence, it would be realistically possible to achieve

maller equilibrium headways (Rajamani, 2012, Section 5.2) . This

ould, in turn, vastly improve resource utilization without com-

romising safety ( Greengard, 2015 ). In this paper, based on our

heoretical analyses, we provide some design guidelines to appro-

riately tune the parameters of the so-called “upper longitudinal

ontrol algorithm” (Rajamani, 2012, Section 5.2) . Mathematically,

ur analytical findings highlight the quantitative impact of delayed

eedback on the design of control algorithms for self-driven vehi-

les. Specifically, our design guidelines take into consideration var-

ous aspects of the longitudinal control algorithm such as stability,

ood ride quality and fast convergence of the traffic to the uni-

orm flow. In the event that the traffic flow does lose stability, our

esign guidelines help tune the model parameters with an aim of

educing the amplitude and angular velocity of the resultant limit

ycles.

.1. Related work on car-following models

The motivation for our paper comes from the key idea be-

ind the Optimal Velocity Model (OVM) proposed by Bando et al.

n Bando, Hasebe, Nakayama, Shibata, and Sukiyama (1995) for a

latoon of vehicles on a circular loop. However, the model con-

idered therein was devoid of reaction delays. Thus, a new model

as proposed in Bando et al. (1998) to account for the drivers’

elays. Therein, the authors also claimed that these delays were

ot central to capturing the dynamics of the system. In response,

avis showed via numerical computations that reaction delays in-

eed play an important part in determining the qualitative behav-

or of the OVM Davis (2002) . This led to a further modification to

he OVM in Davis (2003) . However, this too did not account for

he delay arising due to a vehicle’s own velocity. It was shown

n Gasser, Sirito, and Werner (2004) that the OVM without delays

oses local stability via a Hopf bifurcation. For the OVM with de-

ays, Orosz, Krauskopf, and Wilson (2005) performed an initial nu-

erical study of the bifurcation phenomenon before supplying an

nalytical proof in Orosz and Stépán (2006) .

While a control-theoretic treatment of car-following models

as been widely studied (see Bekey, Burnham, and Seo, 1977;

ey et al., 2016; Li et al., 2017 and references therein), the the-

atic issue on “Traffic jams: dynamics and control” ( Orosz, Wil-

on, & Stépán, 2010 ) highlights the growing interest in a syner-

ized control-theoretic and dynamical systems viewpoint of trans-

ortation networks. A recent exposition of linear stability analysis

n the context of car-following models can be found in Wilson and

ard (2011) .

From a vehicular dynamics perspective, most upper longitudinal

ontrollers in the literature assume the lower controller’s dynam-

cs to be well modeled by a first-order control system, in order

o capture the delay lag (Rajamani, 2012, Section 5.3) . The upper

ongitudinal controllers are then designed to maintain either con-

tant velocity, spacing or time gap; for details, see Rajamani and

hu (2002) and the references therein. Specifically, it was shown

n Rajamani and Zhu (2002) that synchronization with the lead

ehicle is possible by using information only from the vehicle di-

ectly ahead. This reduces implementation complexity, and does

ot mandate vehicles to be installed with communication devices.

However, in the context of autonomous vehicles, communi-

ation systems are required to exchange various system states

equired for the control action. This information is used ei-

her for distributed control ( Rajamani & Zhu, 2002 ) or coordi-

ated control ( Qu, Wang, & Hull, 2008 ) of vehicles. Formation

ontrol ( Anderson, Sun, Sugie, Azuma, & Sakurama, 2017; Cha-

an, Belur, Chakraborty, & Manjunath, 2015 ) and platoon stabili-

ies ( Summers, Yu, Dasgupta, & Anderson, 2011 ) have also been

tudied considering information flow among the vehicles. How-

ver, these works do not consider the effect of delays in relay-

ng the required information. In contrast, when latency increases

ue to randomness in the communication environment, strategies

ave been developed to make use of only on-board sensors with

inimal degradation in performance ( Ploeg, Semsar-Kazerooni, Li-

ster, van de Wouw, & Nijmeijer, 2015 ). For an extensive review,

ee Dey et al. (2016) . Usage of communication systems is also

nown to mitigate phantom jams ( Won, Park, & Son, 2016 ). It may

e noted that, for our scenario of straight road with a single lane,

he formation control problem subsumes the problem of stabilizing

platoon. Thus, our work can also be thought of as a formation

ontrol problem in the presence of reaction delays and using only

n-board sensors.

At a microscopic level, Chen et al. proposed a behavioral car-

ollowing model based on empirical data that captures phan-

om jams ( Chen, Laval, Zheng, & Ahn, 2012 ). Therein, the authors

howed statistical correlation in drivers’ behavior before and dur-

ng traffic oscillations. However, no suggestions to avoid phantom

ams were offered. To that end, Nishi et al. developed a frame-

ork for “jam-absorbing” driving in Nishi, Tomoeda, Shimura, and

ishinari (2013) . A “jam-absorbing vehicle” appropriately varies its

eadway with the aim of mitigating phantom jams. This work was

xtended by Taniguchi, Nishi, Ezaki, and Nishinari (2015) to in-

lude car-following behavior. Therein, the authors also numerically

onstructed the region in parameter space that avoids formation of

econdary jams.

In the context of platoon stability, it has been shown that well-

laced, communicating autonomous vehicles may be used to sta-

ilize platoons of human-driven vehicles ( Orosz, 2016 ). More gen-

20 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32

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erally, the platooning problem has been studied as a consensus

problem with delays ( Bernardo, Salvi, & Santini, 2015 ). Such an ap-

proach aids the design of coupling protocols between interacting

agents (in this context, vehicles). In contrast, we provide design

guidelines to appropriately choose protocol parameters, for a given

coupling protocol. Additionally, the effect of communication delays

has been studied in the literature, both when the delays are de-

terministic ( Ge & Orosz, 2017 ) and random ( Qin, Gomez, & Orosz,

2017 ). It may be noted that, our work differs from these at a funda-

mental level; these models assume vehicles to be traversing a cir-

cular loop, thus yielding a periodic boundary condition. In contrast,

our work studies the effect of (deterministic) reaction delays on

the qualitative dynamics of a platoon of vehicles using the MOVM

on a straight road. Further, in addition to characterizing the region

for local stability, we study two practically relevant properties –

non-oscillatory convergence and the rate of convergence. More im-

portantly, our analysis goes beyond that of the linearized system

by making use of bifurcation theory to take into account non-linear

terms. For a treatment of bifurcations in non-delayed systems,

the reader is referred to the classical text by Guckenheimer and

Holmes (1983) ; for Hopf bifurcations in systems with time delays,

the reader may refer to the excellent texts by Hassard, Kazarinoff,

and Wan (1981) or Marsden and McCracken (1976) .

1.2. Our contributions

Our contributions are as follows.

(1) We derive a variant of the OVM for an infinitely-long road –

called the MOVM – and analyze it in three regimes; namely,

( i ) no delay, ( ii ) small delay and ( iii ) arbitrary delay. We

prove that the ideal case of instantaneously-reacting drivers

is locally stable for all practically significant parameter val-

ues. We then derive a stability condition for the small-delay

regime by conducting a linearization on the time variable.

(2) For the case of an arbitrary delay, we derive the necessary

and sufficient condition for the local stability of the MOVM.

We then prove that, upon violation of this condition, the

MOVM loses local stability via a Hopf bifurcation.

(3) We provide an analytical framework to characterize the type

of the Hopf bifurcation and the asymptotic orbital stability

of the emergent limit cycles using Poincaré normal forms

and the center manifold theory.

(4) In the case of human-driven vehicles, our work enhances

phenomenological insight into the emergence and evo-

lution of traffic congestion. For example, the Hopf bi-

furcation analysis provides a mathematical framework to

offer a possible explanation for the observed “phantom

jams” Kamath et al. (2015) . In the case of self-driven vehi-

cles, our work offers suggestions for their design guidelines.

(5) We derive a necessary and sufficient condition for non-

oscillatory convergence of the MOVM. This is useful in the

context of a transportation network since oscillations lead

to jerky vehicular movements, thereby degrading ride qual-

ity and possibly causing collisions.

(6) We characterize the rate of convergence of the MOVM,

thereby gaining insight into the time required for the pla-

toon to equilibrate, when perturbed. Such perturbations oc-

cur, for instance, when a vehicle departs from a platoon.

Therein, we also bring forth the trade-off between the

rate of convergence and non-oscillatory convergence of the

MOVM.

(7) We corroborate the analytical results with the aid of stabil-

ity charts, bifurcation diagrams, numerical computations and

simulations performed using MATLAB.

The remainder of this paper is organized as follows. In

ection 2 , we summarize the OVM and derive the MOVM. In

ections 3 –5 , we characterize the stable regions for the MOVM

n no-delay, small-delay and arbitrary-delay regimes respectively.

e then derive the necessary and sufficient condition for non-

scillatory convergence of the MOVM in Section 6 , and characterize

ts rate of convergence in Section 7 . In Section 8 , we present the

ocal Hopf bifurcation analysis for the MOVM. In Section 9 , we cor-

oborate our analyses using MATLAB simulations before concluding

he paper in Section 10 .

. Models

In this section, we first provide an overview of the setting of

ur work. We then briefly explain the OVM, before ending the sec-

ion by deriving the MOVM.

.1. The setting

We consider N + 1 idealistic vehicles (with 0 length) traveling

n an infinitely long, single-lane road with no overtaking. The lead

ehicle is indexed with 0, the vehicle following it with 1, and so

n. The acceleration of each vehicle is updated based on a com-

ination of its position, velocity and acceleration as well as those

orresponding to the vehicle directly ahead. We use x i ( t ), ˙ x i (t) and

¨ i (t) to denote the position, velocity and acceleration of the ve-

icle indexed i at time t respectively. We also assume that the

ead vehicle’s acceleration and velocity profiles are known. Specifi-

ally, we only consider leader profiles that converge to x 0 = 0 and

< ˙ x 0 < ∞ in finite time; that is, there exists T 0 < ∞ such that

¨ 0 (t) = 0 , ˙ x 0 (t) = ˙ x 0 > 0 , ∀ t ≥ T 0 . We also use the terms “driver”

nd “vehicle” interchangeably throughout. Further, we make use of

I units throughout.

.2. The Optimal Velocity Model (OVM)

The OVM, proposed by Bando et al. in Bando et al. (1995) , is

ased on the key idea that each vehicle in a platoon tries to at-

ain an “optimal” velocity, which a function of its headway. Hence,

ach vehicle updates its acceleration proportional to the difference

etween this optimal velocity and its own velocity. This was mod-

fied in Bando et al. (1998) to account for the reaction delay. For N

ehicles traveling on a circular loop of length L units, the dynamics

s captured by Bando et al. (1998)

¨ 1 (t) = a ( V (x N (t − τ ) − x 1 (t − τ )) − ˙ x 1 (t − τ ) ) ,

x i (t) = a ( V (x i −1 (t − τ ) − x i (t − τ )) − ˙ x i (t − τ ) ) , (1)

or i ∈ {2, ��, N }. Here, a > 0 is the drivers’ sensitivity coefficient,

is the common reaction delay and V : R + → R + is called the

ptimal Velocity Function (OVF). As pointed out in Batista and

wrdy (2010) , an OVF satisfies:

(i) Monotonic increase,

(ii) Bounded above, and,

(iii) Continuous differentiability.

Let V max = lim y →∞

V (y ) . The limit exists as a consequence of (i)

nd (ii) above. Also, (iii) ensures that an OVF will be invertible.

.3. The Modified Optimal Velocity Model (MOVM)

Next, we derive a version of the OVM for the infinite highway

etting. To that end, we begin by re-writing system (1) as

¨ 1 (t) = a ( V (x 0 (t − τ1 ) − x 1 (t − τ1 )) − ˙ x 1 (t − τ1 ) ) ,

x i (t) = a ( V (x i −1 (t − τi ) − x i (t − τi )) − ˙ x i (t − τi ) ) , (2)


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V , y ≥ y .

here x 0 ( t ) is the position of the lead vehicle at time t . To capture

eality better, we have accounted for heterogeneity in reaction de-

ays. Notice that, in contrast to (1) , system (2) no longer possesses

he circular structure resulting from the periodic boundary condi-

ion. Indeed, the second vehicle (with index 1) now follows the

ead vehicle rather than the vehicle with index N . Further, each

ehicle requires external information from the vehicle preceding it

nly. Hence, on a technological level, on-board sensors suffice to

mplement our strategy.

From (2) , it may be noted that x i ( t ) → ∞ as t → ∞ for each

. To apply tools from non-linear dynamics, we require bounded

tate variables. To that end, we use the change of variables y i ( t )

x i −1 (t) − x i (t) and v i ( t ) = ˙ y i (t) = ˙ x i −1 (t) − ˙ x i (t) . Here, y i ( t ) and

i ( t ) represent the relative distance (headway) and relative velocity

etween the vehicles i and i − 1 at time t respectively. Substituting

hese in (2) , we obtain the following system after some algebraic

anipulations

˙ 1 (t) = x 0 (t) + a ( x 0 (t − τ1 ) − V (y 1 (t − τ1 )) − v 1 (t − τ1 ) ) ,

˙ v k (t) = a ( V (y k −1 (t − τk −1 )) − V (y k (t − τk )) − v k (t − τk ) ) ,

˙ y i (t) = v i (t) , (3)

or i ∈ {1, 2, ��, N } and for k ∈ {2, 3, ��, N }. We refer to sys-

em (3) as the Modified Optimal Velocity Model (MOVM). We em-

hasize that, given the absolute variables { x i } N i =1 , the relative vari-

bles { y i } N i =1 are uniquely determined, and vice versa (when the

nitial positions are known). Hence, systems (2) and (3) are equiv-

lent, i.e., they are representations of the same system in different

ariables.

The MOVM is described by a system of Delay Differential Equa-

ions (DDEs). Since such systems are hard to analyze, we obtain

onditions for their local stability by analyzing them in the neigh-

orhood of their equilibria. Such an analysis technique is called lo-

al stability analysis . To obtain the equilibrium for the MOVM, we

rst equate the Right Hand Sides (RHSs) corresponding to ˙ y i (t) to

ero, thus yielding v ∗i

= 0 for each i . Next, we equate the RHSs cor-

esponding to ˙ v k (t) to zero, for k ∈ {2, 3, ��, N }. Using the equi-

ibria for the relative velocities, we obtain V (y ∗i ) = V (y ∗

j ) , ∀ i, j .

quating the RHS of the very first differential equation to zero,

e obtain V (y ∗1 ) = ˙ x 0 . Combining these, and using the properties

f the OVF, we obtain y ∗i

= V −1 ( x 0 ) for each i . Therefore, v ∗i

= 0 ,

∗i

= V −1 ( x 0 ) , i = 1 , 2 , · · · , N represents the unique equilibrium of

he MOVM. Therefore, to linearize (3) about this equilibrium, we

rst consider a small perturbation u i ( t ) about the equilibrium of

he relative spacing pertaining to vehicle indexed i . That is, u i ( t ) = i ( t ) - y ∗

i . Next, we consider the Taylor’s series expansion of u i ( t ),

nd set the leader’s profile to zero, to obtain the linearized model,

iven by

˙ 1 (t) = − du 1 (t − τ1 ) − a v 1 (t − τ1 ) ,

˙ v k (t) = du k −1 (t − τk −1 ) − du k (t − τk ) − a v k (t − τk ) ,

˙ u i (t) = v i (t) , (4)

or i ∈ {1, 2, ��, N } and for k ∈ {2, 3, ��, N }. Here, d = aV ′ (V −1 ( x 0 ))

s the equilibrium coefficient, where the prime indicates differ-

ntiation with respect to a state variable. Henceforth, we denote˜ = V

′ (V −1 ( x 0 )) . Therefore, d = a d .

The MOVM is completely specified by the relative velocities v i ’s

nd the headways y i ’s. Therefore, the state of the MOVM at time

t ” is given by S (t) = [ v 1 (t) v 2 (t ) · · · v N (t ) u 1 (t ) u 2 (t ) · · · u N (t )] T ∈

2 N . Thus, system (4) can be succinctly written in matrix form as

˙ (t) =

N ∑

k =0

A k S (t − τk ) . (5)

his is the evolution equation of the MOVM in the standard state-

pace representation. Here, τ is introduced for notational brevity
0
nd set to zero. Also, the matrices A k ∈ R

2 N×2 N for each k are the

ynamics matrices , which capture the dependence of the derivative

n the state variable delayed by the k th reaction delay. For instance,

hen N = 2 , the evolution equations are

˙ 1 (t) = − du 1 (t − τ1 ) − a v 1 (t − τ1 ) ,

˙ 2 (t) = du 1 (t − τ1 ) − du 2 (t − τ2 ) − a v 2 (t − τ2 ) ,

˙ 1 (t) = v 1 (t) ,

˙ 2 (t) = v 2 (t) .

he above equations can be re-written in the matrix form as

˙ v 1 (t) ˙ v 2 (t) ˙ y 1 (t) ˙ y 2 (t)

⎤

⎥ ⎦

︷︷︸ ˙ S (t)

=

⎡

⎢ ⎣

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

⎤

⎥ ⎦

︸︷︷︸ A 0

⎡

⎢ ⎣

v 1 (t) v 2 (t) y 1 (t) y 2 (t)

⎤

⎥ ⎦

︸︷︷︸ S(t)

+

⎡

⎢ ⎣

−a 0 −d 0

0 0 d 0

0 0 0 0

0 0 0 0

⎤

⎥ ⎦

︸︷︷︸ A 1

⎡

⎢ ⎣

v 1 (t − τ1 ) v 2 (t − τ1 ) y 1 (t − τ1 ) y 2 (t − τ1 )

⎤

⎥ ⎦

︸︷︷︸ S(t−τ1 )

+

⎡

⎢ ⎣

0 0 0 0

0 −a 0 −d 0 0 0 0

0 0 0 0

⎤

⎥ ⎦

︸︷︷︸ A 2

⎡

⎢ ⎣

v 1 (t − τ2 ) v 2 (t − τ2 ) y 1 (t − τ2 ) y 2 (t − τ2 )

⎤

⎥ ⎦

︸︷︷︸ S(t−τ2 )

.

For an arbitrary N , the matrices A k , k = 1 , 2 · · · , N, are defined

s follows.

0 =

[0 N×N 0 N×N

I N×N 0 N×N

].

ere, 0 N × N and I N × N denote zero and identity matrices of order

× N respectively. For 1 ≤ k ≤ N − 1 , we have

(A k ) i j =

⎧ ⎪ ⎨

⎪ ⎩

−a, i = j = k,

−d, i = k, j = N + k,

d, i = k + 1 , j = k,

0 , elsewhere ,

nd

(A N ) i j =

{ −a, i = j = N,

−d, i = N, j = 2 N,

0 , elsewhere .

.4. Optimal Velocity Functions (OVFs)

There are several functions that satisfy the properties

entioned in Section 2.2 . We mention four widely-used

VFs Batista and Twrdy (2010) , obtained by fixing a functional

orm for V ( · ).

(a) Underwood OVF:

V 1 (y ) = V 0 e − 2 y m

y .

(b) Bando OVF:

V 2 (y ) = V 0

(tanh

(y − y m

˜ y

)+ tanh

(y m

˜ y

)).

(c) Trigonometric OVF:

V 3 (y ) = V 0

(tan

−1 (

y − y m

˜ y

)+ tan

−1 (

y m

˜ y

)).

(d) Hyperbolic OVF:

V 4 (y ) =

{0 , y ≤ y 0 , (

(y −y 0 ) n )

0 ( y ) n +(y −y 0 ) n 0


w

a

B

w

W

v

e

C

w

a

m

w

T

i

m

t

O

F

s

r

w

T

λ

W

Here, V 0 , y 0 , y m

, ˜ y and n are model parameters.

As captured by Kamath, Jagannathan, and Raina (2016 , Fig. 1),

the aforementioned OVFs behave similarly with varying headway.

The following are noteworthy: ( i ) The values attained by these

OVFs, in the vicinity of the equilibrium, are almost the same, ( ii )

their slopes, evaluated at the equilibrium, are different. The lin-

earized version of the MOVM, captured by system (5) , brings forth

the dependence on the slope via the variable ˜ d , and ( iii ) we make

use of the Bando OVF throughout this paper, except in Section 8 .

Therein, we consider both the Bando OVF and the Underwood OVF,

consistent with Kamath et al. (2015) .

We now proceed to understand the dynamical behavior of a

platoon of cars running the MOVM.

3. The no-delay regime

We first consider the idealistic case of instantaneously-reacting

drivers. This results in zero reactions delays. Therefore, the model

described by system (5) boils down to the following system of Or-

dinary Differential Equations (ODEs):

˙ S (t) =

(

N ∑

k =0

A k

)

S (t) . (6)

We denote by A , the sum of matrices A k , which is known as the

dynamics matrix . To characterize the stability of system (6) , we re-

quire the eigenvalues of A to be negative (Györi & Ladas, 1991, The-

orem 5.1.1) . To that end, we compute its characteristic function as

f (λ) = det (λI 2 N×2 N − A ) = det

([(λ + a ) I N×N

ˆ D

I N×N λI N×N

])= 0 ,

where ˆ D is derived from the dynamics matrix A . The diagonal en-

tries of ˆ D are all d , while its sub-diagonal entries are −d. Fur-

ther, the diagonal matrices of the above block matrix are invertible,

and the off-diagonal matrices commute with each other. Hence,

from Silvester (20 0 0 , Theorem 3), the characteristic equation can

be simplified to (λ2 + aλ + d) N = 0 , which holds true if and only

if

λ2 + aλ + d = 0 . (7)

Solving the above quadratic, we notice that the poles correspond-

ing to system (6) will be negative if a > 0 and

˜ d = V ′ (V −1 ( x 0 )) > 0 .

We note that, from physical constraints, a > 0. Also, since V ( · ) is

an OVF, it is monotonically increasing. Therefore, ˜ d > 0 . Hence, for

all physically relevant values of the parameters, the corresponding

poles will lie in the open left-half of the Argand plane. Thus, the

MOVM is locally stable for all physically relevant values of the pa-

rameters, in the absence of delays.

4. The small-delay regime

Having studied the MOVM in the absence of reaction delays, we

now analyze it in the small-delay regime. A way to obtain insight

for the case of small delays is to conduct a linearization on time.

This would yield a system of ODEs, which serves as an approxima-

tion to the original infinite-dimensional system (5) , valid for small

delays. We derive the criterion for such a system of ODEs to be

stable, thereby emphasizing the design trade-off inherent among

various system parameters and the reaction delay.

We begin by applying the Taylor’s series approximation to

the time-delayed state variables thus: v i (t − τi ) ≈ v i (t) − τi v i (t) ,

and u i (t − τi ) ≈ u i (t) − τi ˙ u i (t) . Using this approximation for terms

in (4) , substituting v i ( t ) for ˙ u i (t) and re-arranging the resulting

equations, we obtain the matrix equation

B

S (t) = A S (t) . (8)

here the matrix A is the dynamics matrix, as defined in Section 3 ,

nd B is a block matrix of the form

=

[B s 0 N×N

0 N×N I N×N ,

],

here

(B s ) i j =

{1 − aτi , i = j, 0 , elsewhere .

e note that, since B s is a diagonal matrix, so is B . Also, B is in-

ertible if and only if a τ i � = 1, for each i . Thus, when a τ i � = 1, for

ach i , we define ˜ C = B −1 A, which is of the form

˜ =

[˜ C s ˜ C c

I N×N 0 N×N ,

],

here

(C s ) i j =

⎧ ⎨

⎩

−a + dτi

1 −aτi , i = j,

−dτ j

1 −aτi , j = i − 1 ,

0 , elsewhere ,

nd

(C c ) i j =

⎧ ⎨

⎩

−d 1 −aτi

, i = j, d

1 −aτi , j = i − 1 ,

0 , elsewhere .

For system (8) to be stable, the real part of eigenvalues of C

ust be negative (Györi & Ladas, 1991, Theorem 5.1.1) . To that end,

e compute its characteristic function as

f (λ) = det (λI 2 N×2 N − ˜ C ) = det

([λI N×N − ˜ C s − ˜ C c

−I N×N λI N×N

])= 0 .

he diagonal matrices of the aforementioned block matrix are

nvertible, and the matrices in the second row therein com-

ute with each other. Hence, the characteristic equation simplifies

o (Silvester, 20 0 0, Theorem 3)

f (λ) = det (λ(λI N×N − ˜ C s ) − ˜ C c

)= 0 .

n further simplification, this yields

f (λ) =

N ∏

i =1

((1 − aτi ) λ

2 + (a − dτi ) λ + d )

= 0 .

or multiple terms in the above product to equal zero, their re-

pective reaction delays must be equal. Such a possibility is not

ealistic, hence we ignore it. Therefore, for some i ∈ {1, 2, ��, N },

e have

(1 − aτi ) λ2 + (a − dτi ) λ + d = 0 .

he roots of this quadratic equation are given by

1 , 2 =

−(a − dτi ) ±√

(a − dτi ) 2 − 4 d(1 − aτi )

2(1 − aτi ) .

e now consider the following (exhaustive) cases.

(1) Let a τ i > 1. Since d > 0, it follows that 4 d(1 − aτi ) < 0 . Then,

the eigenvalues are real. Further, one of these eigenvalues

will be positive and the other negative. Hence, we require

a τ i < 1 for system (8) to be stable.

(2) Let (a − d τi ) 2 ≥ 4 d (1 − aτi ) . Then, the eigenvalues are real.

They are negative if and only if a − dτi > 0 , i.e ., ˜ d τi < 1 .

Hence, we require ˜ d τi < 1 for system (8) to be stable.

(3) Let (a − d τi ) 2 < 4 d (1 − aτi ) . Then, the eigenvalues are com-

plex. The real part of the eigenvalues will be negative if and

only if a − dτi > 0 , i.e ., ˜ d τi < 1 . Hence, we require ˜ d τi < 1 for

system (8) to be stable.


o

m

f

t

c

s

5

r

W

s

c

c

5

a

g

t

t

a

c

b

n

x

i

x

w

h

s

t

i

o

s

S

a

s

i

v

f

t

i

S

w

c

t

T

λ

w

w

l

m

t

S

e

F

s

r

w

λ

o

L

p

A

o

κ

κ

S

S

ω

S

r

ω

F

t

f

ω

S

κ

T

s

t

κ

t

L

R

From the above cases, it is clear that system (8) is stable if and

nly if

ax (a, ˜ d ) τi < 1 , (9)

or each i ∈ {1, 2, ��, N }. Recall that we obtained system (8) by

runcating the Taylor’s series to first order. Hence, (9) is a suffi-

ient condition for the local stability of the MOVM described by

ystem (3) , valid for small values of the reaction delay.

. The arbitrary-delay regime

Having studied system (3) in the no-delay and the small-delay

egimes, in this section, we focus on the arbitrary-delay regime.

e first derive the necessary and sufficient condition for the local

tability of the MOVM. We then show that, upon violation of this

ondition, the corresponding traffic flow transits via a Hopf bifur-

ation to the locally unstable regime.

.1. Transversality condition

Hopf bifurcation is a phenomenon wherein, on appropriate vari-

tion of system parameters, a dynamical system either loses or re-

ains stability because of a pair of conjugate eigenvalues crossing

he imaginary axis in the Argand plane (Hale & Lunel, 2011, Chap-

er 11, Theorem 1.1) . Mathematically, Hopf bifurcation analysis is

rigorous way of proving the emergence of limit cycles (isolated

losed trajectory in state space) in non-linear dynamical systems.

To determine if system (3) undergoes a stability loss via a Hopf

ifurcation, we follow Raina (2005) and introduce an exogenous,

on-dimensional parameter κ > 0. A general system of DDEs

˙ (t) = f (x (t) , x (t − τ1 ) , · · · , x (t − τn )) , (10)

s modified to

˙ (t) = κ f (x (t) , x (t − τ1 ) , · · · , x (t − τn )) , (11)

ith the introduction of the exogenous parameter. Note that ( i ) κas no effect on the equilibrium of system (10) , and ( ii ) we obtain

ystem (10) by setting κ = 1 in system (11) . We first linearize sys-

em (11) about its non-trivial equilibrium and derive its character-

stic equation. We then search for a pair of conjugate eigenvalues

n the imaginary axis in the Argand plane. This yields the neces-

ary and sufficient condition for the local stability of system (11) .

etting the exogenous parameter to unity then yields the necessary

nd sufficient condition for system (10) . The exogenous parameter

o introduced helps simplify the requisite algebra and capture any

nterdependence among the system parameters.

For the MOVM, introducing κ in (3) yields

˙ 1 (t) = x 0 (t) + κa ( x 0 (t − τ1 ) − V (y 1 (t − τ1 )) − v 1 (t − τ1 ) ) ,

˙ v k (t) = κa ( V (y k −1 (t − τk −1 )) − V (y k (t − τk )) − v k (t − τk ) ) ,

˙ y i (t) = κv i (t) , (12)

or i ∈ {1, 2, ��, N } and for k ∈ {2, 3, ��, N }. We linearize this about

he equilibrium v ∗i

= 0 , y ∗i

= V −1 ( x 0 ) , i = 1 , 2 , · · · , N, and write it

n matrix form to obtain

˙ (t) =

N ∑

k =0

˜ A k S (t − τk ) , (13)

here the matrices ˜ A k = κA k , for k = 0 , 1 , · · · , N, where the matri-

es A k are as defined in Section 2 .

The characteristic equation corresponding to system (13) is ob-

ained as (Györi & Ladas, 1991, Section 5.1)

f (λ) = det

(

λI 2 N×2 N −N ∑

k =0

e −λτk ˜ A k

)

= 0 .

he matrix in consideration is a block matrix of the form

I 2 N×2 N −N ∑

k =0

e −λτk ˜ A k =

[˜ A

˜ B

˜ C ˜ D

],

here ˜ C = −κ I N×N and

˜ D = λI N×N . Further, ˜ A is a diagonal matrix

ith the i th diagonal entry being λ + κae −λτi , and

˜ B is a sparse

ower-triangular matrix. Clearly, ˜ A and

˜ D are invertible, and

˜ C com-

utes with

˜ D . Therefore, the characteristic equation simplifies to

he form (Silvester, 20 0 0, Theorem 3)

f (λ) = det

([˜ A

˜ B

˜ C ˜ D

])= det

(˜ A

D − ˜ B

C )

= 0 .

implifying the above expression, we obtain the characteristic

quation pertaining to (13) as

f (λ) =

N ∏

i =1

(λ2 + κaλe −λτi + κ2 de −λτi ) = 0 . (14)

or multiple terms in the above product to equal zero, their re-

pective reaction delays must be equal. Such a possibility is not

ealistic, hence we ignore it. Therefore, for some i ∈ {1, 2, ��, N },

e have

2 + κaλe −λτi + κ2 de −λτi = 0 . (15)

System (12) will be locally stable if and only if all the roots

f (15) lie in the open left-half of the Argand plane (Györi &

adas, 1991, Theorem 5.1.1) . Therefore, we search for a conjugate

air of eigenvalues of (15) that crosses the imaginary axis in the

rgand plane. To that end, we substitute λ = jω in (15) , with

j =

√ −1 . We then equate the real and imaginary parts to zero and

btain

aω sin (ωτi ) + κ2 d cos (ωτi ) = ω

2 , (16)

aω cos (ωτi ) − κ2 d sin (ωτi ) = 0 . (17)

quaring and adding (16) and (17) yields ω

4 − κ2 a 2 ω

2 − κ4 d 2 = 0 .

olving for ω

2 , we obtain

2 1 , 2 = κ2

(a 2 ± √

a 4 + 4 d 2

2

).

ince we are searching for a positive root, we discard the negative

oot. The positive root of the above expression is given by

= κ

√

a (a +

√

a 2 + 4

d 2 )

2

. (18)

or convenience, we write the above equation as ω = κχ. Notice

hat, on re-arranging (17) , we obtain κ ˜ d tan (ωτi ) = ω. Substituting

or ω in the above equation and simplifying yields

0 =

1

τi

tan

−1 (χ

˜ d

). (19)

ubstituting ω 0 in (17) and simplifying, we obtain

cr =

1

τi χtan

−1 (χ

˜ d

). (20)

hus, (19) and (20) yield the angular frequency of the oscillatory

olution and the value of κ at which such a solution exists respec-

ively.

We now show that the MOVM undergoes a Hopf bifurcation at

= κcr . To that end, we need to prove the transversality condi-

ion of the Hopf spectrum. That is, we must show that (Hale &

unel, 2011, Chapter 11, Theorem 1.1)

eal

(d λ

d κ

)κ= κcr

� = 0 . (21)


Fig. 1. Stability chart: Illustrates the necessary and sufficient condition

(N&SC) (23) and the sufficient condition (SC) (9) for the MOVM, for small

delays. The plot serves to validate our analysis presented in Section 4 .

6

M

t

n

m

j

n

n

t

g

t

b

To that end, we differentiate (15) with respect to κ and simplify it,

to obtain

Real

( (d λ

d κ

)−1 )

κ= κcr

=

κcr ω

2 0 τi (κ

2 cr

˜ d cos (ω 0 τi ) + ω

2 0 )

(κ2 cr

˜ d cos (ω 0 τi ) + ω 0 ) 2 + (κ2 cr

˜ d sin (ω 0 τi )) 2 > 0 . (22)

The positivity in (22) follows because cos ( ω 0 τ i ) = κcr ˜ d / (κ2

cr ˜ d 2 +

ω

2 0 ) is positive. This expression follows from (17) using trigono-

metric manipulations. Also, Real( z ) > 0 if and only if Real(1/ z ) > 0 ∀z ∈ C . Hence, from (22) we have

Real

(d λ

d κ

)κ= κcr

> 0 .

This proves the transversality of the Hopf spectrum. Therefore,

the MOVM transits from the locally stable to the locally unstable

regime via a Hopf bifurcation at κ = κcr . It can be shown that for

sufficiently small values of κ , system (12) is locally stable. Ad-

ditionally, the above strict inequality implies that the eigenvalues

move from left to right in the Argand plane as κ is increased in

the neighborhood of κcr . Therefore, κ < κcr is the necessary and

sufficient condition for local stability of system (12) .

5.2. Discussion

A few comments are in order.

(1) Note that the characteristic Eq. (15) is transcendental, hence

there exist infinitely many roots. However, system (12) loses

local stability when the first conjugate pair of eigenvalues

crosses the imaginary axis as the exogenous parameter is

varied. Due to the positivity of the derivative in (22) , system

stability cannot be restored by increasing κ .

(2) The equation of the stability boundary pertaining to sys-

tem (12) is κ = κcr . It is also called the Hopf boundary of the

said system. To obtain the Hopf boundary corresponding to

the MOVM described by system (3) , we tune the system pa-

rameters such that κcr = 1 in (20) . In particular, the MOVM

is locally stable if and only if, for each i ∈ {1, 2, ��, N }, we

have

τi <

1

χtan

−1 (χ

˜ d

). (23)

It is clear from (23) that when the reaction delay increases,

the MOVM loses local stability via a Hopf bifurcation. Also

note that when τ = 0 , (23) is trivially satisfied for all phys-

ically relevant parameter values. This is in agreement with

the result derived in Section 3 . To validate the analysis pre-

sented in Section 4 , we plot the RHSs of (9) and (23) for

small values of the reaction delay in Fig. 1 . Clearly, we notice

from Fig. 1 that (9) indeed represents a sufficient condition

for the local stability of the MOVM for small delays.

(3) Loss of local stability via a Hopf bifurcation results in the

emergence of limit cycles. Since the dynamical variables for

the MOVM correspond to relative velocities and headways,

these non-linear oscillations physically manifest as back-

propagating congestion wave on a highway. Thus, as men-

tioned in the Introduction, our analysis provides a mathe-

matical basis to the commonly-observed “phantom jam.”

(4) Note that the non-dimensional parameter κ is not a model

parameter; rather, it is an exogenous mathematical entity

introduced to aid the analysis and capture any interdepen-

dence among model parameters. It also serves to simplify

the algebra required to obtain the necessary and sufficient

condition for local stability of the MOVM. Further, since sub-

stituting κ = 1 yields the MOVM, it is useful in a neighbor-

hood around 1, i.e., near the stability boundary.

(5) Gain parameters are known to destabilize feedback sys-

tems (Rajamani, 2012, Section 3.7) . Thus, we need to ver-

ify that the bifurcation phenomenon proved in this section

is not an artefact of the exogenous parameter. To that end,

we need to verify that the MOVM also undergoes a Hopf bi-

furcation when one of the model parameters is chosen as

the bifurcation parameter. It was shown in Manjunath and

Raina (2014) that the transversality condition of the Hopf

spectrum holds true for the characteristic equation of the

form (15) (with κ = 1 ) when τ is used as the bifurcation pa-

rameter, although in a different context.

(6) Note that the non-dimensional parameter κ can also be in-

terpreted as a time-scale change for the case of the MOVM.

This can be seen from (12) by multiplying both sides by 1/ κ ,

and making the change of variable ˜ t = κt. Then, the “rela-

tive importance” of the reaction delays to the system time

scale would be κτi / t , for each i . Thus, in this new time scale,

an increase in κ can be interpreted as a uniform (multi-

plicative) increase in all the reaction delays. Thus, the afore-

mentioned viewpoint may also be useful in interpreting the

single-parameter bifurcation analysis presented in this pa-

per.

. Non-oscillatory convergence

In the previous three sections, we derived conditions for the

OVM to be locally stable in three different regimes. In the next

wo sections, we explore two important properties of the MOVM;

amely, non-oscillatory convergence and the rate of convergence.

In the context of transportation networks, ride quality is of ut-

ost importance. This, in turn, mandates that the vehicles avoid

erky motion. Since relative velocities and headways constitute dy-

amical variables for the MOVM, it boils down to studying the

on-oscillatory property of its solutions. In particular, we derive

he necessary and sufficient condition for non-oscillatory conver-

ence of the MOVM. Mathematically, this amounts to ensuring that

he eigenvalues corresponding to system (5) are negative real num-

ers.


o

t

κ

w

t

λ

a

a

S

T

0

o

T

o

s

y

e

A

t

ω

T

a

i

S

σ

N

t

a

c

e

N

n

m

t

τ

w

o

r

c

m

o

τ

Fig. 2. Illustration of the region of non-oscillatory convergence for the MOVM.

Here, τ cr and τ noc represent the boundaries of the locally stable region and the

region of non-oscillatory convergence of the MOVM respectively. Notice the strin-

gent requirements on the reaction delay for the solutions of the MOVM to be non

oscillatory, for a given sensitivity coefficient.

f

τ

o

t

t

s

T

r

p

t

f

e

v

t

t

C

o

t

t

t

k

a

e

(

r

N

p

c

a

w

d

To derive the necessary and sufficient condition for non-

scillatory convergence of the MOVM, we begin with the charac-

eristic equation corresponding to system (3) , obtained by setting

= 1 in (15) . We also drop the subscript “i ” for convenience. Thus,

e obtain

f (λ) = λ2 + (aλ + d) e −λτ = 0 . (24)

To ensure non-oscillatory convergence of the MOVM, we require

he roots of (24) to be real and negative. To that end, we substitute

= σ + jω in (24) , where j =

√ −1 . This yields

ω sin (ωτ ) + (aσ + d) cos (ωτ ) = (ω

2 − σ 2 ) e στ , and (25)

ω sin (ωτ ) + (aσ + d) cos (ωτ ) = (−2 σω) e στ . (26)

quaring and adding (25) and (26) , we obtain

(aω) 2 + (aσ + d) 2 = (ω

2 + σ 2 ) 2 e 2 στ . (27)

o ensure that the roots are real, we require a condition for ω = to be the only solution of (27) . Substituting ω = 0 in (27) , we

btain

(aσ + d) 2 = σ 4 e 2 στ . (28)

hus, the above condition is necessary for ω = 0 to be a solution

f (27) . To check whether it is also a sufficient condition, we first

eparate the terms containing ω in (27) from those without it. This

ields

2 στω

4 + (2 σ 2 e 2 στ − a 2 ) ω

2 = (aσ + d) 2 − σ 4 e 2 στ .

ssuming (aσ + d) 2 = σ 4 e 2 στ , we solve the above quadratic in ω

2

o obtain

2 = 0 or ω

2 =

a 2 − 2 σ 2 e 2 στ

e 2 στ.

hus, for ω = 0 to be the unique solution of (27) , we require

2 = 2 σ 2 e 2 στ in addition to the condition mentioned in (28) . That

s, (24) has real eigenvalues if and only if

(aσ + d) 2 = σ 4 e 2 στ , and a 2 = 2 σ 2 e 2 στ . (29)

olving the above two equations for the eigenvalue, we obtain

=

˜ d m ±, with m ± = −2 ±√

2 . (30)

otice from the foregoing analysis that the eigenvalues are guaran-

eed to be negative if they are real. Substituting (30) in (24) and re-

rranging, we obtain the boundary for the region of non-oscillatory

onvergence as

− ˜ d τm ± =

−m

2 ± ˜ d

a (m + 1) .

otice that the Left Hand Side (LHS) in the above equation is a

on-negative quantity. The RHS is non-negative for m − but not for

+ . Hence, we set m = m − in the above equation, and re-arrange

o obtain

noc =

1

m

d ln

(−a (m + 1)

m

2 ˜ d

), (31)

here τ noc represents the boundary for the region of non-

scillatory convergence in the τ -domain. Therefore, τ < τ noc rep-

esents the necessary and sufficient condition for non-oscillatory

onvergence of the MOVM. We note that the following inequalities

ust be satisfied: 0 < τ noc < τ cr , where τ cr is the RHS of (23) .

In summary, the necessary and sufficient condition for non-

scillatory convergence of the MOVM is

i <

1

m

d ln

(−a (m + 1)

m

2 ˜ d

), (32)

t

or each i ∈ {1, 2, ��, N }, when the RHS is positive and less than

cr .

We now illustrate the boundary for the region of non-

scillatory convergence of the MOVM described by (31) . In order

o better-understand the stringent constraints on system parame-

ers to achieve non-oscillatory convergence, we also plot the neces-

ary and sufficient condition for local stability (23) of the MOVM.

o that end, we make use of the Bando OVF. We let the equilib-

ium velocity of the lead vehicle to be ˙ x 0 = 5 m/s, and the model

arameters as y ∗ = 2 m, ˜ y = 5 m and y m

= 1 m. We then compute

he corresponding V 0 and

˜ d . We vary the sensitivity coefficient a

rom 1 and 5, and compute the requisite boundaries using the sci-

ntific computation software MATLAB.

Fig. 2 portrays regions of local stability and non-oscillatory con-

ergence for the MOVM in the ( a, τ )-space. For a fixed a , the reac-

ion delay must not exceed τ cr (respectively, τ noc ) for the MOVM

o be locally stable (respectively, possess non-oscillatory solutions).

learly, the values of τ need to be much smaller for the solutions

f the MOVM to be non oscillatory as opposed to the stability of

he MOVM, for a fixed value of a . In fact, as the sensitivity parame-

er a increases, the corresponding value of reaction delays required

o ensure non-oscillatory convergence decreases rapidly.

We end this section with two remarks. ( i ) To the best of our

nowledge, the analysis presented in this section is the first to

ddress non-oscillatory convergence of systems with characteristic

quations of the form (24) using spectral-domain techniques, and

ii ) we can obtain ω = 0 as the only solution to (27) by a geomet-

ical method as follows. Re-arranging (27) yields

(ω

2 + σ 2 ) 2 = (a 2 e −2 στ ) ω

2 + (aσ + d) 2 e −2 στ .

otice that the LHS and the RHS of the above equation represent a

arabola and a line in ω

2 respectively. Since a parabola is strictly

onvex, the tangent to a parabola at any point will intersect it only

t that point. It can be shown that the RHS of the above equation

ill be the tangent to the LHS at ω

2 = 0 if and only if the con-

itions in (29) hold. Details of this approach can be found in the

echnical report ( Kamath, Jagannathan, & Raina, 2017 ).


t

t

r

w

T

I

p

p

l

8

g

e

v

a

w

t

g

i

c

�

f

a

e

t

b

v

t

v

7. Rate of convergence

In this section, we characterize the time required to attain

the uniform traffic flow, once the traffic flow is perturbed (by

events such as the departure of a vehicle from the platoon). Math-

ematically, it is related to the rate of convergence of solutions

of the MOVM to the desired equilibrium. To that end, we fol-

low Chong, Lee, and Kang (2001) and first characterize the rate of

convergence of the MOVM. Then, using the notion of settling time,

we derive an expression for the time a platoon takes to attain the

desired equilibrium following a perturbation.

We begin by recalling the characteristic equation pertaining to

system (5) from Section 5.1 . Dropping the subscript “i ” for ease of

exposition, and setting κ = 1 in (15) , we obtain

λ2 + aλe −λτ + de −λτ = 0 .

Using the change of variables z = λτ, the above equation results

in

z 2 e z + a ∗z + d ∗ = 0 , (33)

where a ∗ = aτ and d ∗ = dτ 2 . Notice that (33) has the

same form as (Chong et al., 2001 , Eq. (22)). Hence, follow-

ing Chong et al. (2001) , we substitute z = ψ − σ, where σ is

non-negative and real, in (33) to obtain

(ψ

2 − 2 σψ + σ 2 ) e ψ + a ∗e σψ + (d ∗ − a ∗σ ) e σ = 0 .

The characteristic equation corresponding to the above system

is obtained by substituting ψ = τλ as

λ2 +

(−2 σ

τ

)λ + (ae σ ) λe −λτ +

(d − aσ

τ

)e σ e −λτ +

(σ 2

τ 2

)= 0 .

(34)

The rate of convergence is the largest σ ≥ 0 such that the root

of (34) with the largest real part is negative Chong et al. (2001) . As

pointed out in Chong et al. (2001) , finding such a σ analytically is

intractable. Hence, we illustrate the variation of the rate of conver-

gence numerically with respect to both the sensitivity parameter a

and the reaction delay τ , using the scientific computation software

MATLAB.

We consider the Bando OVF, and set the following parameters:

y m

= 1 m, ˜ y = 5 m, y ∗ = 2 m and ˙ x 0 = 5 m/s. We then compute

the corresponding values of V 0 and

˜ d . Next, we vary the sensitiv-

ity coefficient a in the range [1, 5], and for each of its values, we

compute the critical value of the reaction delay τ cr using (23) . We

then vary the reaction delay τ in the range [0, τ cr ], for each a .

For every pair ( a, τ ) in this range, σ is increased from 0, till the

root of (34) with the largest real part crosses the imaginary axis in

the Argand plane. Since the resulting plot would be three dimen-

sional, we present the corresponding contour plots in Fig. 3 . For

clarity in presentation, the contour plots are segregated as follows:

Fig. 3 a is for low to medium values of the rate of convergence,

whereas Fig. 3 b is for high values. It can be seen from Fig. 3 a that

small changes in a or τ causes the rate of convergence to change

from 0.3 to 0.9. However, it would require relatively larger changes

in a or τ for the rate of convergence to change from 0.1 to 0.3.

That is, the gradient of the rate of convergence increases rather

rapidly with an increase in the rate of convergence. Also, for low

values of the rate of convergence, non-oscillatory convergence can

be guaranteed. In contrast, Fig. 3 b brings forth the trade-off be-

tween the rate of convergence and non-oscillatory convergence;

very high rates of convergence cannot be achieved if the solutions

are to be non oscillatory.

The rate of convergence determines the time taken by a platoon

to reach an equilibrium (denoted by T e MOV M

). To characterize T e MOV M

,

we first define the time taken by the i th pair of vehicles in the

platoon following the standard control-theoretic notion of “settling

ime.” That is, by t e i (ε) , we denote the minimum time taken by the

ime-domain trajectory of the MOVM to enter and subsequently

emain within the ε-band around the equilibrium. For simplicity,

e drop the explicit dependence on ε. Then,

e MOV M

=

N ∑

i =1

t e i . (35)

t is clear that (35) is an upper bound on the time taken by the

latoon to equilibrate. However, the equality holds since the i th

air cannot equilibrate till the (i − 1) th pair has reached its equi-

ibrium.

. Hopf bifurcation analysis

In the previous sections, we have characterized the stable re-

ion for the MOVM, and studied two of its most important prop-

rties; namely, non-oscillatory convergence and the rate of con-

ergence. We have also proved that system (3) loses stability via

Hopf bifurcation, thus resulting in limit cycles. In this section,

e provide an analytical framework to characterize the type of

he bifurcation and the asymptotic orbital stability of the emer-

ent limit cycles. We closely follow the style of analysis presented

n Hassard et al. (1981) , which uses Poincaré normal forms and the

enter manifold theory.

We begin by denoting the RHS of (12) as f i . That is, for i ∈ {1, 2,

��, N },

f i � aκ( V (y i −1 (t − τi −1 )) − V (y i (t − τi )) − v k (t − τi ) ) . (36)

Define μ = κ − κcr . Notice that the system undergoes a Hopf bi-

urcation at μ = 0 , where κ = κcr . Henceforth, we shall consider μs the bifurcation parameter. Also, it is clear that when μ> 0, the

xogenous parameter κ changes from κcr to κcr + μ, thus pushing

he system into an unstable regime.

We now provide a step-by-step overview of the detailed local

ifurcation analysis, before delving into its technical details.

Step 1 : We begin by applying Taylor’s series expansion to the

RHS of (36) . Next, we separate the linear terms from their

non-linear counterparts. This allows us to cast the resulting

equation into the standard form of an Operator Differential

Equation (OpDE).

Step 2 : When μ = 0 , the system has exactly one pair of purely

imaginary eigenvalues with non-zero angular velocity, as

seen from (22) . We call the linear space spanned by the

corresponding eigenvectors as the critical eigenspace. For

the purpose of our analysis, we also require a locally in-

variant manifold that is a tangent to the critical eigenspace

at the system’s equilibrium. The center manifold theo-

rem Hassard et al. (1981) guarantees the existence of such

a manifold.

Step 3 : Next, we project the system onto its critical eigenspace

and its complement when μ = 0 . Thus, we may write the

dynamics of the original system on the center manifold as

an ODE in a single complex variable.

Step 4 : Finally, using Poincaré normal forms, we evaluate the

Lyapunov coefficient and the Floquet exponent. These, in

turn, help characterize the type of the Hopf bifurcation and

the asymptotic orbital stability of the emergent limit cycles.

We begin the analysis by expanding (12) about the equilibrium

∗i

= 0 , y ∗i

= V −1 ( x 0 ) , i = 1 , 2 , · · · , N, using Taylor’s series, to ob-

ain

˙ i (t) = (−κa ) v i,t (−τi ) + (−κaV

′ (y ∗i )) y i,t (−τi )

+ (−κaV

′′ (y ∗i )) y

2 i,t (−τi ) + (−κaV

′′′ (y ∗i )) y

3 i,t (−τi )

+ ζ (1) i

y (i −1) ,t (−τi −1 ) + ζ (2) i

y 2 (i −1) ,t (−τi −1 )


Fig. 3. Contour plots: Contour lines of the rate of convergence overlaying the boundaries of the locally stable region and the region of non-oscillatory convergence of the

MOVM. While ( a ) is for low to medium values of rate of convergence, ( b ) is for high values. From ( a ), observe: ( i ) the rapid change in the gradient of the rate of convergence,

and ( ii ) for lower values of the rate of convergence, non-oscillatory convergence is also guaranteed. In contrast, ( b ) shows that very high rates of convergence cannot be

achieved if the solutions are to be non oscillatory.

y

w

v

t

t

ζ

s

a

e

c

f

w

S

H

c

C

F

d

T

O

s

S

d

o

v

e

C

L

I

L

d

w

A

a

R

H

t

p

t

T

p

[

a

+ ζ (3) i

y 3 (i −1) ,t (−τi −1 ) + higher order terms ,

˙ i (t) = κx i (t) (37)

here we use the shorthand v i,t (−τi ) and y i,t (−τi ) to represent

i (t − τi ) and y i (t − τi ) respectively. Also, V ′ , V

′′ and V

′′′ denote

he first, second and third derivatives of the OVF with respect to

he state variable respectively. Additionally, the coefficients ζ (1) i

,

(2) i

and ζ (3) i

represent −κaV ′ (y ∗

i ) , −κaV

′′ (y ∗

i ) and −κaV

′′′ (y ∗

i ) re-

pectively for i > 1, and are zero for i = 1 .

In the following, we use C k ( A ; B ) to denote the linear space of

ll functions from A to B which are k times differentiable, with

ach derivative being continuous. Also, we use C to denote C 0 , for

onvenience.

With the concatenated state S (t) , note that (12) is of the

orm:

d S (t)

d t = L μS t (θ ) + F( S t (θ ) , μ) , (38)

here t > 0, μ ∈ R , and where for τ = max i

τi > 0 ,

t (θ ) = S (t + θ ) , S : [ −τ, 0] −→ R

2 N , θ ∈ [ −τ, 0] .

ere, L μ : C ([ −τ, 0] ; R

2 N )

−→ R

2 N is a one-parameter family of

ontinuous, bounded linear functionals, whereas the operator F :

([ −τ, 0] ; R

2 N )

−→ R

2 N is an aggregation of the non-linear terms.

urther, we assume that F( S t , μ) is analytic, and that F and L μ

epend analytically on the bifurcation parameter μ, for small | μ|.

he objective now is to cast (38) in the standard form of an

pDE:

d S t

d t = A (μ) S t + R S t , (39)

ince the dependence here is on S t alone rather than both S t and

(t) . To that end, we begin by transforming the linear problem

S (t) / d t = L μS t (θ ) . We note that, by the Riesz representation the-

rem (Rudin, 1987, Theorem 6.19) , there exists a 2 N × 2 N matrix-

alued measure η(·, μ) : B

(C ([ −τ, 0] ; R

2 N ))

−→ R

2 N×2 N , wherein

ach component of η( · ) has bounded variation, and for all φ ∈

([ −τ, 0] ; R

2 N ), we have

μφ =

∫ 0

−τd η(θ, μ) φ(θ ) . (40)

n particular,

μS t =

∫ 0

−τd η(θ, μ) S (t + θ ) .

Motivated by the linearized system (13) , we define

η =

[˜ A

˜ B

˜ C ˜ D

]d θ,

here

( A ) i j =

{ −κdδ(θ + τi ) , i = j, κdδ(θ + τ j ) , j = i − 1 , i > 1 ,

0 , otherwise,

( B ) i j =

{−κaδ(θ + τi ) , i = j, 0 , otherwise,

˜ C = κ I N×N and

˜ D = 0 N×N .

For φ ∈ C 1 ([ −τ, 0] ; C

2 N ), we define

(μ) φ(θ ) =

{

d φ(θ ) d θ

, θ ∈ [ −τ, 0) , ∫ 0 −τ d η(s, μ) φ(s ) ≡ L μ, θ = 0 ,

(41)

nd

φ(θ ) =

{0 , θ ∈ [ −τ, 0) , F(φ, μ) , θ = 0 .

With the above definitions, we observe that d S t / d θ ≡ d S t / d t.

ence, we have successfully cast (38) in the form of (39) . To ob-

ain the required coefficients, it is sufficient to evaluate various ex-

ressions for μ = 0 , which we use henceforth. We start by finding

he eigenvector of the operator A (0) with eigenvalue λ(0) = jω 0 .

hat is, we want an 2 N × 1 vector (to be denoted by q ( θ )) with the

roperty that A (0) q (θ ) = jω 0 q (θ ) . We assume the form: q (θ ) =1 φ1 φ2 · · · φ2 N−1 ]

T e jω 0 θ , and solve the eigenvalue equations. We

lso assume the following:

(i)

− jω 0 e jω 0 τ1 + κd

2 =

−1 + e jω 0 τ

2 ,

κ a ω

0


z

T

w

w

E

r

a

m

c

a

z

T

z

T

e

O

e

g

N

t

w

T

w

w

H

N

i

p

l

−

−

W

S

k

t

(ii) For each i ∈ { 1 , 2 , · · · N − 1 } , the following matrix is invert-

ible: [κd e − jω 0 τi +1 + j ω 0 κae − jω 0 τi +1

κβ − jω 0

],

where β = j(−1 + je − jω 0 τ ) /ω 0 . Then, for i ∈ { 1 , 2 , · · · N − 1 } ,

φN =

κβ

jω 0

, φi = − j κω 0 d e − jω 0 τi

�M i

, and φN+ i = −βκ2 de − jω 0 τi

�M i

,

where �M i = ω

2 0

− κdω 0 sin (ω 0 τi +1 ) − κ2 βa cos (ω 0 τi +1 ) +j(κ2 βa sin (ω 0 τi +1 ) − κdω 0 cos (ω 0 τi +1 )) .

We define the adjoint operator as follows:

A

∗(0) φ(θ ) =

{

− d φ(θ ) d θ

, θ ∈ (0 , τ ] , ∫ 0 −τ d ηT (s, 0) φ(−s ) , θ = 0 ,

where d ηT is the transpose of d η. We note that the domains

of A and A

∗ are C 1 ([ −τ, 0] ; C

2 N )

and C 1 ([0 , τ ] ; C

2 N )

respec-

tively. Therefore, if j ω 0 is an eigenvalue of A , then − jω 0 is

an eigenvalue of A

∗. Hence, to find the eigenvector of A

∗(0)

corresponding to − jω 0 (to be denoted by p ( θ )), we assume

the form: p(θ ) = B [ ψ 2 N−1 ψ 2 N−2 ψ 2 N−3 · · · 1] T e jω 0 θ , and solve

A

∗(0) p(θ ) = − jω 0 p(θ ) . We also assume the following:

(i)

κd − jω 0 e jω 0 τN

κ2 a =

−1 + e jω 0 τ

ω

2 0

,

(ii) For each i ∈ { 1 , 2 , · · · N − 1 } , the following matrix is invert-

ible: [jω 0 −κae − jω 0 τi

κβ κd e − jω 0 τi − j ω 0

].

Then, for i ∈ { 1 , 2 , · · · N − 1 } , we obtain

ψ N =

jω 0 e jω 0 τN

κa , ψ N+ i =

j ω 0 κd ψ N+ i −1 e − jω 0 τN−i

� ˜ M i

,

and ψ i =

κ2 adψ N+ i −1 e − jω 0 τN−i

� ˜ M i

,

where � ˜ M i = ω

2 0

+ κdω 0 sin (ω 0 τN−i ) + κ2 βa cos (ω 0 τN−i ) +j (κd ω 0 cos (ω 0 τN−i ) − κ2 βa sin (ω 0 τN−i )) .

The normalization condition for Hopf bifurcation requires that

〈 p, q 〉 = 1, thus yielding an expression for B .

For any q ∈ C ([ −τ, 0] ; C

2 N )

and p ∈ C ([0 , τ ] ; C

2 N ), the inner

product is defined as

〈 p, q 〉 � p · q −∫ 0

θ= −τ

∫ θ

ζ=0

p T (ζ − θ ) d ηq (ζ ) d ζ , (42)

where the overbar represents the complex conjugate and the “ · ′ ′ represents the regular dot product. The value of B such that the

inner product between the eigenvectors of A and A

∗ is unity can

be shown to be

B =

1

ζ1 + ζ2 + ζ3 + ζ4

,

where

ζ1 =

(2 e jω 0 τ − e j2 ω 0 τ − 1

2

)N−1 ∑

i =0

κψ N−i −1 φi ,

ζ2 =

N−1 ∑

i =0

(e jω 0 τi +1 − e j2 ω 0 τi +1

jω 0

)κψ 2 N−1 −i (a φi + d φN+ i ) ,

ζ3 =

N−2 ∑

i =0

(e j2 ω 0 τi +1 − e jω 0 τi +1

jω 0

)κd φi ψ 2 N−2 −i ,

t

and, ζ4 =

2 N−1 ∑

i =0

ψ 2 N−1 −i φi .

For S t , a solution of (39) at μ = 0, we define

(t) = 〈 p(θ ) , S t 〉 , and w (t, θ ) = S t (θ ) − 2 Real (z(t) q (θ )) .

hen, on the center manifold C 0 , we have w (t, θ ) = (z(t) , z (t) , θ ) , where

(z(t) , z (t) , θ ) = w 20 (θ ) z 2

2

+ w 02 (θ ) z 2

2

+ w 11 (θ ) z z + · · · . (43)

ffectively, z and z are the local coordinates for C 0 in C in the di-

ections of p and p respectively. We note that w is real if S t is real,

nd we deal only with real solutions. The existence of the center

anifold C 0 enables the reduction of (39) to an ODE in a single

omplex variable on C 0 . At μ = 0, the said ODE can be described

s

˙ (t) = 〈 p, A S t + R S t 〉 ,

= jω 0 z(t) + p (0) . F ( w (z, z , θ ) + 2 Real (z(t) q (θ )) ) ,

= jω 0 z(t) + p (0) . F 0 (z, z ) . (44)

his is written in abbreviated form as

˙ (t) = jω 0 z(t) + g(z, z ) . (45)

he objective now is to expand g in powers of z and z . How-

ver, this requires w i j (θ ) ’s from (43) . Once these are evaluated, the

DE (44) for z would be explicit (as given by (45) ), where g can be

xpanded in terms of z and z as

(z, z ) = p (0) . F 0 (z, z ) = g 20 z 2

2

+ g 02 z 2

2

+ g 11 z z + g 21 z 2 z

2

+ · · · .

(46)

ext, we write ˙ w =

˙ S t − ˙ z q − ˙ z q . Using (39) and (45) , we then ob-

ain the following ODE:

˙ =

{A w − 2 Real ( p (0) . F 0 q (θ )) , θ ∈ [ −τ, 0) , A w − 2 Real ( p (0) . F 0 q (0)) + F 0 , θ = 0 .

his can be re-written using (43) as

˙ = A w + H(z, z , θ ) , (47)

here H can be expanded as

(z, z , θ ) = H 20 (θ ) z 2

2

+ H 02 (θ ) z 2

2

+ H 11 (θ ) z z + H 21 (θ ) z 2 z

2

+ · · · .

(48)

ear the origin, on the manifold C 0 , we have ˙ w = w z z + w z z. Us-

ng (43) and (45) to replace w z z (and their conjugates, by their

ower series expansion) and equating with (47) , we obtain the fol-

owing operator equations:

(2 jω 0 − A ) w 20 (θ ) = H 20 (θ ) , (49)

A w 11 = H 11 (θ ) , (50)

(2 jω 0 + A ) w 02 (θ ) = H 02 (θ ) . (51)

e start by observing that

t (θ ) = w 20 (θ ) z 2

2

+ w 02 (θ ) z 2

2

+ w 11 (θ ) z z + zq (θ ) + z q (θ ) + · · · .

From the Hopf bifurcation analysis ( Hassard et al., 1981 ), we

now that the coefficients of z 2 , z 2 , z 2 z , and z z terms are used

o approximate the system dynamics. Hence, we only retain these

erms in the expansions.


m

s

w

F

w

F

F

a

T

e

−

g

w

p

g

w

c

w

∈H

w

H

H

F

w

w

S

w

w

f

[

y

t

e[U

e

w

d

e

s

f

s

t

F

q

i

n

i

c

B

D

E

s

s

κ

fi

v

v

o

W

o

t

w

O

N

fi

3

t

u

τ

To obtain the effect of non-linearities, we substitute the afore-

entioned terms appropriately in the non-linear terms of (37) and

eparate the terms as required. Therefore, for each i ∈ {1, 2, ��, 2 N },

e have the non-linearity term to be

i = F 20 i

z 2

2

+ F 02 i

z 2

2

+ F 11 i z z + F 21 i

z 2 z

2

, (52)

here, for i ∈ {1, 2, ��, N }, the coefficients are given by

20 i = �(1) i

w 20 i (−τi ) + ζ (1) i −1

w 20(i −1) (−τi −1 ) ,

02 i = �(1) i

w 02 i (−τi ) + ζ (1) i −1

w 02(i −1) (−τi −1 ) ,

F 11 i = �(1) i

w 11 i (−τi ) + ζ (1) i −1

w 11(i −1) (−τi −1 ) ,

F 21 i = 2�(2) i

(w 20 i (−τi ) e jω 0 τi + 2 w 11 i (−τi ) e

− jω 0 τi )

+ 2 ζ (2) i −1

(w 20(i −1) (−τi −1 ) e jω 0 τi −1 + 2 w 11(i −1) (−τi −1 ) e

− jω 0 τi −1 ) ,

nd for i ∈ { N + 1 , N + 2 , · · · , 2 N} , each of these coefficients is zero.

his is so, since last N states correspond to the headways who

volution equations are linear. Here, �(1) i

= −κaV ′ (y ∗

i ) , �(2)

i =

κaV ′′ (y ∗

i ) , and �(3)

i = −κaV

′′′ (y ∗

i ) .

Next, we compute g in (45) as

(z, z ) = p (0) . F 0 = B

2 N ∑

l=1

ψ 2 N−l F l , (53)

here F 0 = [ F 1 F 2 · · · F 2 N ] T . Substituting (52) in (53) , and com-

aring with (46) , we obtain

x = B

2 N ∑

l=1

ψ 2 N−l F xl , (54)

here x ∈ {20, 02, 11, 21}. Using (54) , the corresponding coefficients

an be computed. However, computing g 21 requires w 20 (θ ) and

11 (θ ) . Hence, we perform the requisite computation next. For θ [ −τ, 0) , H can be simplified as

(z, z , θ ) = −Real ( p (0) . F 0 q (θ ) ) ,

= −(

g 20 z 2

2

+ g 02 z 2

2

+ g 11 z z + · · ·)

q (θ )

−(

g 20 z 2

2

+ g 02 z 2

2

+ g 11 z z + · · ·)

q (θ ) ,

hich, when compared with (48) , yields

20 (θ ) = −g 20 q (θ ) − g 20 q (θ ) , (55)

11 (θ ) = −g 11 q (θ ) − g 11 q (θ ) . (56)

rom (41) , (49) and (50) , we obtain the following ODEs:

˙ 20 (θ ) = 2 jω 0 w 20 (θ ) + g 20 q (θ ) + g 02 q (θ ) , (57)

˙ 11 (θ ) = g 11 q (θ ) + g 11 q (θ ) . (58)

olving (57) and (58) , we obtain

20 (θ ) = − g 20

jω 0

q (0) e jω 0 θ − g 02

3 jω 0

q (0) e − jω 0 θ + e e 2 jωθ , (59)

11 (θ ) =

g 11

jω 0

q (0) e jω 0 θ − g 11

jω 0

q (0) e − jω 0 θ + f , (60)

or some vectors e and f , to be determined.

To that end, we begin by defining the following vector: ˜ F 20 � F 201 F 202 · · · F 20(2 N) ]

T . Equating (49) and (55) , and simplifying,

ields the operator equation: 2 jω e − A ( e e 2 jω 0 θ ) =

˜ F . To solve
0 20
his, we assume that the following matrices to be invertible for

ach i ∈ {1, 2, ��, N },

2 jω o + κ(a + d) e − jω 0 τi κ(a + d) e − jω 0 τi

−κτ 2 jω o

].

nder this condition, we obtain for i ∈ {1, 2, ��, N },

i =

2 jω 0 F 20 i

�M

∗i

, and, e N+ i =

κτF 20 i

�M

∗i

, (61)

here �M

∗i

= −4 ω

2 0

+ 2 ω 0 κ(a + d) sin (ω 0 τi ) + τκ2 (a +) cos (ω 0 τi ) + j(2 ω 0 κ(a + d) cos (ω 0 τi ) − τκ2 (a + d) sin (ω 0 τi )) .

Next, equating (50) and (56) , and simplifying, we obtain the op-

rator equation A f = − ˜ F 11 , with

˜ F 11 � [ F 111 F 112 · · · F 11(2 N) ] T . On

olving this equation, we obtain for i ∈ {1, 2, ��, N },

i = 0 , and, f N+ i =

F 11 i

κτi (a + d) . (62)

Substituting for e and f from (61) and (62) in (59) and (60) re-

pectively, we obtain w 20 (θ ) and w 11 (θ ) . This, in turn, facilitates

he computation of g 21 . We can then compute

c 1 (0) =

j

2 ω 0

(g 20 g 11 − 2 | g 11 | 2 − 1

3

| g 02 | 2 )

+

g 21

2

,

α′ (0) = Real

(d λ

d κ

)κ= κcr

, μ2 = −Real (c 1 (0))

α′ (0) ,

and β2 = 2 Real (c 1 (0)) .

Here, c 1 (0) is known as the Lyapunov coefficient and β2 is the

loquet exponent. It is known from Hassard et al. (1981) that these

uantities are useful since

( i ) If μ2 > 0, then the bifurcation is supercritical , whereas if

μ2 < 0, then the bifurcation is subcritical .

( ii ) If β2 > 0, then the limit cycle is asymptotically orbitally un-

stable , whereas if β2 < 0, then the limit cycle is asymptoti-

cally orbitally stable .

Some of the details pertaining to the derivation can be found

n the technical report Kamath et al. (2017) . We now present

umerically-constructed bifurcation diagrams to gain some insight

nto the effect of various parameters on the amplitude of the limit

ycle.

ifurcation diagrams

To obtain bifurcation diagrams, we make use of

DE-BIFTOOL Engelborghs, Luzyanina, and Roose (2002) ,

ngelborghs, Luzyanina, and Samaey (2001) . We first input

ystem (12) and their first-order derivatives with respect to the

tate and delayed state variables to DDE-BIFTOOL. We then set

= 1 and initialize the model parameters appropriately. We also

x a range of variation for the bifurcation parameter. DDE-BIFTOOL

aries the bifurcation parameter accordingly and finds its critical

alue. We then increase the value of κ and record the amplitude

f the resulting limit cycle, thus obtaining the bifurcation diagram.

e use the SI units throughout; time will be expressed in “sec-

nds,” distance in “meters,” velocity in “meters per second” and

he sensitivity coefficient in “inverse second.” For our comparison,

e consider two optimal velocity functions; namely, the Bando

VF and the Underwood OVF.

For the Bando OVF, we initialize the parameters as follows:

= 4 , a = 1 . 2 , τ1 = 0 . 2 , τ2 = 0 . 2 , τ3 = 0 . 3911 and τ4 = 0 . 2 . We

x y m

= 2 and ˜ y = 5 , and compute V 0 for each of y ∗i

= 1 , 2 and

. The vehicle indexed 3 is considered to undergo a Hopf bifurca-

ion. For the case of the Underwood OVF, we set the following val-

es for the parameters. N = 3 , a = 1 . 2 , τ1 = 0 . 1 , τ2 = 0 . 11885 and

3 = 0 . 1 . We fix y m

= 2 , and compute V 0 for each of y ∗i

= 1 , 2 and


Fig. 4. Bifurcation diagrams: Amplitude of the emergent limit cycles in relative velocity variable as a function of the exogenous parameter κ . ( a ) is for the Bando OVF, while

( b ) is for the Underwood OVF. For a fixed κ ∈ [1, 1.1], the Underwood OVF results in limit cycles of smaller relative velocity than its Bando counterpart.

Fig. 5. Simulation results: Shows the variations in relative velocity and headway around their respective equilibria. ( a ) portrays the limit cycles predicted by (22) , while ( b )

presents an instance of non-oscillatory behavior when parameters are chosen appropriately satisfying (32) .

t

c

T

B

l

t

y

a

r

f

a

v

u

τ

h

T

y

b

i

f

t

2

3. The vehicle indexed 2 is then considered to undergo a Hopf bi-

furcation. We choose the equilibrium velocity of the lead vehicle,

˙ x 0 = 5 .

The bifurcation diagrams are shown in Fig. 4 . As seen from the

figure, the amplitude of the relative velocity increases with an in-

crease in κ . However, for a fixed value of the exogenous parameter,

the Underwood OVF yields limit cycles with smaller relative veloc-

ity than its Bando counterpart, which is desirable. Also, notice that

the amplitude of the emergent limit cycles increases with an in-

crease in the equilibrium headway. This is intuitive because larger

equilibrium headways offer more space for the resulting limit cy-

cles to oscillate in.

9. Simulations

Thus far, we have analyzed the MOVM in no-delay, small-delay

and arbitrary-delay regimes. We also studied two of its important

properties – non-oscillatory convergence and the rate of conver-

gence. In the previous section, we presented an analytical frame-

work to characterize the type of Hopf bifurcation and the asymp-

totic orbital stability of the limit cycles that emerge when the sta-

bility conditions are marginally violated.

In this section, we present the simulation results of the MOVM

that serve to corroborate our analytical findings. We make use of

he scientific computation software MATLAB to implement a dis-

rete version of system (3) , thus simulating the MOVM. We use

s = 10 −4 s as the update time. Throughout, we use SI units.

To corroborate the insight from Section 5 , we make use of the

ando OVF. We consider a platoon of four vehicles following a

ead vehicle on an infinite highway, i.e., N = 4 . Further, we assume

hat the lead vehicle’s velocity profile is given by 5(1 − e 10 t ) , thus

ielding an equilibrium velocity for the leader as ˙ x 0 = 5 . We also

ssume that the 3 rd vehicle undergoes a Hopf bifurcation, while

emaining vehicles are locally stable. The remaining parameters

or various vehicles are chosen as follows. a = 1 . 2 , ˜ y = 5 , y m

= 1

nd y ∗i

= 3 for i = 1 , 2 , 3 , 4 . We then compute the corresponding

alue of V 0 using the functional form for the Bando OVF and τ cr

sing (23) . Further, we set τ1 = τcr / 10 , τ2 = τcr / 3 , τ3 = τcr and

4 = τcr / 2 . We plot the variation of the relative velocity and the

eadway about their respective equilibria for the vehicle indexed 3.

hat is, we plot ˜ v 3 (t) = v 3 (t) − v ∗3

= v 3 (t ) and ˜ y 3 (t ) = y 3 (t) − y ∗3

= 3 (t) − 3 . Fig. 5 a shows the emergence of limit cycles, as predicted

y the transversality condition of the Hopf spectrum (22) .

Next, we corroborate the analysis presented in Section 6 us-

ng the Bando OVF. Again, we consider a platoon of four vehicles

ollowing a lead vehicle on an infinite highway, i.e., N = 4 . Fur-

her, we assume that the lead vehicle’s velocity profile is given by

5(1 − e 10 t ) , thus yielding an equilibrium velocity for the leader


a

s

W

t

W

a

a

v

n

1

b

t

a

r

t

f

t

a

l

b

m

p

j

c

o

j

fl

c

t

t

M

t

l

l

t

c

t

A

t

v

H

i

v

g

a

A

o

c

t

M

R

A

B

B

B

B

v

B

C

C

C

C

D

D

D

E

E

G

G

G

G

G

G

H

H

H

K

K

K

K

L

M

M

s ˙ x 0 = 25 . The remaining parameters for various vehicles are cho-

en as follows. a = 2 , ˜ y = 25 , y m

= 15 and y ∗i

= 15 for i = 1 , 2 , 3 , 4 .

e then compute the corresponding value of V 0 using the func-

ional form for the Bando OVF. We then compute τ noc using (31) .

e set the reaction delays as τ1 = τnoc / 10 , τ2 = τnoc / 3 , τ3 = τnoc / 2

nd τ4 = τnoc / 5 . Fig. 5 b shows an instance of the relative velocity

nd headway variations around their respective equilibria for the

ehicle indexed 3. The headway and relative velocities possess the

on-oscillatory behavior, as predicted by the analysis in Section 6 .

0. Concluding remarks

In this paper, we highlighted the importance of delayed feed-

ack in determining the qualitative dynamical properties of a pla-

oon of vehicles traveling on a straight road. Specifically, we an-

lyzed the Modified Optimal Velocity Model (MOVM) in three

egimes – no delay, small delay and arbitrary delay. We proved

hat, in the absence of reaction delays, the MOVM is locally stable

or all practically relevant values of model parameters. We then ob-

ained a sufficient condition for the local stability of the MOVM by

nalyzing it in the small-delay regime. We also characterized the

ocal stability region of the MOVM in the arbitrary-delay regime.

We then proved that the MOVM undergoes a loss of local sta-

ility via a Hopf bifurcation. The resulting limit cycles physically

anifest as a back-propagating congestion wave. Thus, our work

rovides a mathematical basis to explain the observed “phantom

ams.” For the said analysis, we used an exogenous parameter that

aptures any interdependence among the model parameters.

We then derived the necessary and sufficient condition for non-

scillatory convergence of the MOVM, with the aim of avoiding

erky vehicular motions. This, in turn, guarantees smooth traffic

ow and improves ride quality. Next, we characterized the rate of

onvergence of the MOVM, which affects the time taken by a pla-

oon to equilibrate. We also brought forth the trade-off between

he rate of convergence and non-oscillatory convergence of the

OVM.

Finally, we provided an analytical framework to characterize the

ype of Hopf bifurcation and the asymptotic orbital stability of the

imit cycles which emerge when the stability conditions are vio-

ated. Therein, we made use of Poincaré normal forms and the cen-

er manifold theory. We corroborated our analyses using stability

harts, bifurcation diagrams, numerical computations and simula-

ions conducted using MATLAB.

venues for further research

There are numerous avenues that merit further investigation. In

his work, we have derived the conditions for pairwise stability of

ehicles in a platoon, whose dynamics are captured by the MOVM.

owever, the string stability of such a platoon remains to be stud-

ed.

From a practical standpoint, the parameters of the MOVM may

ary, for varied reasons. Hence, it becomes imperative that the lon-

itudinal control algorithm be robust to such parameter variations,

nd to unmodeled dynamics.

cknowledgments

This work is undertaken as a part of an Information Technol-

gy Research Academy (ITRA), Media Lab Asia, project titled “De-

ongesting India’s transportation networks.” The authors are also

hankful to Debayani Ghosh, Rakshith Jagannath and Sreelakshmi

anjunath for many helpful discussions.

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