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![Page 1: IFAC Journal of Systems andkrishnaj/Publications_files/papers/...A part of this work appeared in theProceedings of the 53rd Annual Allerton Conference on Communication, Control theand](https://reader035.vdocuments.site/reader035/viewer/2022071404/60f9284747533a60af7079ea/html5/thumbnails/1.jpg)
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
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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-
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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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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 21
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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
0nd 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
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22 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32
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.
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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 23
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)
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24 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32
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.
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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 25
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 ).
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26 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32
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 )
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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 27
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
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28 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32
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.
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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 29
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 20his, 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
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30 G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32
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
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G.K. Kamath et al. / IFAC Journal of Systems and Control 2 (2017) 18–32 31
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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