Ieeepro techno solutions 2013 ieee embedded project model predictive control of vehicles on urban roads for improved fuel economy

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 3, MAY 2013 831
Model Predictive Control of Vehicles on Urban
Roads for Improved Fuel Economy
Md. Abdus Samad Kamal, Member, IEEE, Masakazu Mukai, Member, IEEE,
Junichi Murata, Member, IEEE, and Taketoshi Kawabe, Member, IEEE
Abstract—Energy consumption of a vehicle is greatly
influenced by its driving behavior in highly interacting urban
traffic. Strategies for fuel efficient driving have been studied and
experimented with in various conceptual frameworks. This paper
presents a novel control system to drive a vehicle efficiently
on roads containing varying traffic and signals at intersections
for improved fuel economy. The system measures the relevant
information of the current road and traffic, predicts the future
states of the preceding vehicle, and computes the optimal
vehicle control input using model predictive control (MPC).
A typical control objective is chosen to maximize fuel economy by
regulating a safe head-distance or cruising at the optimal velocity
under bounded driving torque condition. The proposed vehicle
control system is evaluated in urban traffic containing thousands
of diverse vehicles using the microscopic traffic simulator
AIMSUN. Simulation results show that the vehicles controlled
by the proposed MPC method significantly improve their fuel
economy.
Index Terms—Automotive control, fuel economy, model
predictive control (MPC).
I. INTRODUCTION
ENVIRONMENTALLY friendly vehicles are highly
demanding for efficient utilization of energy resources
and reduction of exhaust gas emissions in road networks.
Energy consumption of a vehicle depends on various physical
factors, such as characteristics of its engine, power train
system, structure of the vehicle against aerodynamic drag,
road surface conditions, and weather. Besides these factors,
driving behavior also has a great influence on fuel consumption
of a vehicle. Ecological (eco)-driving that addresses strate-
gies for fuel efficient maneuvering of a vehicle has received
great attention recently. In realizing fuel efficient driving, a
driver has to anticipate the road traffic situations properly
and pose perfect knowledge of the engine dynamics of his
car, which is hardly possible for a human driver. Therefore, a
driver can be technologically assisted to drive in such a fuel
efficient style.
Manuscript received September 28, 2011; revised February 15, 2012;
accepted April 16, 2012. Manuscript received in final form May 2, 2012.
Date of publication June 8, 2012; date of current version April 17, 2013.
Recommended by Associate Editor A. Alessandri.
M. A. S. Kamal is with the Institute of Industrial Science, The University
of Tokyo, Tokyo 153-8505, Japan (e-mail: maskamal@ieee.org).
M. Mukai, J. Murata, and T. Kawabe are with the Graduate School of
Information Science and Electrical Engineering, Kyushu University, Fukuoka
819-0395, Japan (e-mail: mukai@cig.ees.kyushu-u.ac.jp; murata@ees.
kyushu-u.ac.jp; kawabe@ees.kyushu-u.ac.jp).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCST.2012.2198478
Research related to the idea of controlling a vehicle for
improving its fuel economy has a long track. Some early
studies were conducted to determine the optimal cruising
velocity of a vehicle based on its internal operating charac-
teristics [1], [2]. In a study, the optimal values of acceleration
for starting up and cruising speed over hilly terrains or flat
roads were determined for fuel efficient driving [3]. In a
similar research, an algorithm was proposed to derive the
optimum speeding strategies for an entire drive cycle [4].
Utilizing information about the future road slopes, model
predictive control (MPC) using discrete dynamic programming
was used for driving heavy trucks with the aim of reducing
fuel consumption over a route without increasing the total
travel time [5], [6]. Some dynamic programming approaches
are usually used to obtain the globally optimum strategies over
an entire driving cycle. While useful in identifying the globally
optimum strategies, these methods are not suited to real-time
implementation as they require knowledge of the driving cycle
before the trip, which is clearly infeasible [7]. MPC is used to
derive the optimal vehicle control input on free hilly roads for
ecological driving, by obtaining the road terrain information
from a digital road map. Noticeable reduction in fuel consump-
tion of a vehicle was observed by simulation over a road of
Fukuoka city, Japan [8]. Freightliner new innovation truck uses
satellite-based route previews with predictive cruise control to
reduce fuel consumption by adapting the preset cruise control
speed without reducing the average speed [9]. Since influences
of the preceding traffic are not considered, these methods are
not appropriate for urban roads where a vehicle often follows
slower traffic and frequently stops at red signals. Periodically
obtained information of density and average velocity of traffic
is utilized to determine the optimal engine torque for improv-
ing fuel economy of a vehicle, using MPC [10]. The system
is designed to be deactivated when a car is detected at a close
distance ahead. Therefore, it is also not appropriate for most
urban roads.
The concept of maximizing fuel efficiency of a vehicle con-
sidering influences of the preceding traffic has received further
attention recently. Various studies and experiments reveal that
the car following behavior has a great influence on emissions
and energy consumption of a vehicle [11], [12]. Fuel-efficient
or eco-driving strategies avoid unnecessary acceleration and
braking by anticipating the preceding traffic, cruising at a
steady velocity, and braking slowly at stops [13], [14]. In
some recently manufactured cars, there is an indicator on the
display panel that shows a green “ECO” mark to its driver
1063-6536/$31.00 © 2012 IEEE

832 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 3, MAY 2013
when the car consumes little or no fuel. The driver would
find his driving as ecological only when he maintains a steady
velocity at a reasonable level or decelerates the car by releasing
the accelerator pedal. Some car service providers have recently
launched an off-board eco-driving support service for their
member users, in which after driving records are sent to
a telemetric data center for off-line analysis, and advice is
sent to the driver for improving his driving style at the
next time. Based on past performance, they have proposed
an on-board assistance system to motivate a driver for eco-
driving by showing his comparative driving efficiency, his
position in the fuel composition ranking, etc. [15]. Another
driving assistance system uses information of traffic signal,
jams, road gradient, and distance between cars, and advice
is given in a very rough form, such as “keep driving” or
“reduce pressure on pedal,” depending on motivation of the
driver [16]. An investigation of dynamic eco-driving, based
on advice given in real-time to drivers with changing traf-
fic conditions in the vehicles’ vicinity, reveals 10%–20%
improved fuel consumption and lower emissions [17]. Some
other studies show that the fuel economy of a vehicle on
urban roads can be improved, by using intelligent technologies
with traffic flow information at less cost than the cost of a
hybrid vehicle [18]. Utilizing the signal transition timing of
the upcoming intersection, fuel consumption of a vehicle can
also be reduced by avoiding aggressive braking [19], [20].
These systems need information of signal timing in advance,
and they may not provide any advantages on crowded roads
where further speeding up to avoid a red signal is not possible.
An ecological driver assistance system was proposed in [21],
aiming at steady driving depending on traffic flow conditions.
A modified eco-driving system for urban roads was presented
focusing on the optimum fuel economy with a simplified
traffic prediction model [22]. Both of the above approaches
lack perfect prediction of the preceding traffic due to their
model simplification. Therefore, in some situations, imperfect
prediction of the preceding traffic may deviate driving from the
optimal track.
Most of the above eco-driving approaches are very super-
ficial as they only provide speeding assistance periodically,
without analyzing road-traffic situations in a perfect way.
Toward a comprehensive eco-driving system, it is necessary to
develop an intensive vehicle control method that measures and
analyzes the situations and computes the optimal control input
at an interval of a fraction of second. The existing sensing
technologies used in a vehicle have limitations in measuring
information of the preceding vehicles, as they often change the
lane due to overtaking, exiting, turning or for some unknown
reasons. Inter-vehicle communication (IVC) using wireless
technology has recently been regarded as an effective way for
transmitting traffic-related information for realization of safe,
efficient, and comfortable transportation systems [23], [24].
Through IVC, precise information of the local vehicles from
the preceding traffic can easily be obtained with a negligible
delay, and trends of local traffic flow can be estimated more
accurately.
Assuming relevant traffic information can be obtained
through an advanced IVC, a model predictive vehicle control
Fig. 1. Concept of the proposed model predictive vehicle control system on
an urban road.
system for improved fuel economy is presented in this paper.
With regards to the developments found in the above lit-
eratures, contributions of this paper are as follows. Instead
of empirical driving rules, tips, or assistance provided peri-
odically without analyzing the current traffic situations, the
proposed system generates the optimal control input, at a
short sampling interval, by predicting the road traffic situations
through rigorous computation. The simplified models found
in previous developments, in [21] and [22], are replaced by
enhanced dynamic models of the vehicles in which influences
of resistances and driving forces, effect of road gradient, and
presence of the preceding vehicle and signals at the end of
sections are included. Driving performance depends on precise
anticipation of the preceding vehicle, which is realized in
this development by incorporating a prediction model for-
mulated by analyzing experimental driving data. Therefore,
the system’s ability to handle changing situations, such as an
overtaking, or appearance of a red signal, is strengthened. The
control objective is chosen to maximize the fuel economy by
regulating a safe head distance, while following a slow car or
cruising at the optimal velocity under bounded driving torque
condition, instead of considering only the velocity preference.
The fundamental concept of the proposed vehicle control
system is shown in Fig. 1. The system receives information of
a vehicle, called the host, its preceding vehicle, road gradient,
and traffic signal, and computes the optimal vehicle control
inputs using MPC. For simplicity, in this paper, the computed
optimal control input is directly fed to drive the vehicle,
assuming an adaptive cruise control (ACC) system requests
the torque and the engine exactly follows it. The proposed
vehicle control system is simulated in AIMSUN microscopic
traffic simulator on a pseudo realistic urban road network
containing thousands of vehicles and traffic signals at inter-
sections. Performance of the proposed system, including its
computation time and various controlling and fuel saving
aspects, is evaluated from one hundred independent tests using
different vehicles. The statistics obtained from simulation
results show significant improvement in fuel economy of the
vehicles controlled by the proposed MPC.
The rest of this paper is organized as follows. The vehicle
control system, including problem formulation, operational
assumptions, prediction models, and algorithm using nonlinear
MPC is presented in Section II. Section III describes
simulation settings and results, followed by discussion in

KAMAL et al.: MPC OF VEHICLES ON URBAN ROADS 833
Section IV. Finally, conclusions and future work are included
in Section V.
II. VEHICLE CONTROL SYSTEM
A. Problem Formulation
This section describes the formulation of the vehicle control
problem, its operational assumptions, and technological
settings to make the system feasible. Lateral movement of a
vehicle within a lane is strictly limited by the shape of the
road. The task of steering control of a vehicle for both lane
keeping and lane changing is assumed to be perfectly handled
by its driver. Only the longitudinal motion of a vehicle, called
host vehicle, needs to be controlled for improving its fuel
consumption. In general, the state equation of a nonlinear
control system at time t can be represented by
˙x(t) = f (x(t), u(t), q(t)) (1)
where x ∈ RNx, u ∈ RNu, and q are the state vector, the
input vector, and a time dependent or external parameter,
respectively. It is assumed that the velocity of a vehicle is
determined by the state of its immediate preceding vehicle,
and the vehicles behind have no influences on it. Since a
safe gap must be maintained behind the preceding vehicle,
its state can be considered as a dynamic reference in the
vehicle control problem. Therefore, it is necessary to include
the model of the reference or preceding vehicle together with
the host for designing an effective control system. Here, the
state vector of size Nx = 4 is define as x = [xh, vh, xp, vp]T,
where xh and vh are the position and velocity of the host
vehicle, and xp and vp are the position and velocity of the
preceding vehicle, respectively. In this paper, by “velocity”
it is meant that the motion is in the direction that the car
is heading. The control input uh ∈ u is only applied to the
host vehicle. The acceleration of the preceding vehicle ap is
assumed to be measurable and it is represented by the time
varying external parameter q(t). The suffix h and p represent
the host and the preceding vehicle, respectively.
The velocity of the host at t is subject to the total forces
acting on the vehicle, and it is expressed by
Mh
dvh(t)
dt
= FT (t) − FR(t) (2)
where Mh, FT (t), and FR(t) are the equivalent mass of the
vehicle and its rotating parts, the driving force, and the sum
of all motion resistance forces, respectively. The resistance
forces, including aerodynamic, rolling, and gradient forces can
be represented by
FR =
1
2
CDρa Avv2
h +μMh gcosθ(xh)+ Mhgsinθ(xh) (3)
where CD, ρa, Av, μ, and θ(xh) are the drag coefficient, the
air density, the frontal area of the vehicle, the rolling resistance
coefficient, and the road gradient angle as a function of xh,
respectively. The driving force is given by the mass of the
vehicle and the control input as FT (t) = Mhuh(t). The road
gradient angle θ is usually very small, and therefore for com-
putational simplicity it can be approximated as sin(θ) ≈ θ,
and cos(θ) ≈ 1.0. The state equation (1) can be rewritten as
f (x, u, q) =
⎡
⎢
⎢
⎣
vh
− 1
2Mh
CDρa Avv2
h − μg − gθ(xh) + uh
vp
ap
⎤
⎥
⎥
⎦. (4)
Here in (4), the parameter q = ap denotes acceleration of the
preceding vehicle. The control input uh is related to the driving
or braking torque of the host vehicle, which is applied through
its throttle or brake, respectively. A very high magnitude of
input or its rate is not desirable considering the fuel efficiency
and driving comfort. Therefore, the input and its rate are
bounded symmetrically as −umax ≤ uh ≤ umax and |˙uh| ≤ α.
It is assumed that these limits are sufficient for efficient and
comfortable driving, in usual driving conditions. Handling of
abnormal situations like approaching a collision is not the main
interest of this paper. However, in real implementation, before
executing the computed control input, a lower level check
or a simple collision avoidance system can be used to avoid
any violation of a minimum head distance from the preceding
vehicle, or other abnormal situations.
The purpose of controlling the host vehicle is to improve
its fuel economy by regulating a safe distance, according to
the state of the preceding vehicle and the signal at the next
intersection. More accurate and precise control performance
may be achieved by considering a few more vehicles and
remaining time of the current green or red signal in the
model. However, such a formulation would make the sensing
system very complex and hard to realize using the existing
technology. With the above assumptions related to the dynamic
equation of (4), it is necessary to measure velocity, position,
and acceleration of the preceding vehicle, and in addition,
distance to the next intersection with a red signal. Collecting
information from the preceding vehicles at real-time t is the
key issue for realizing the proposed system. It is assumed
that necessary information of the preceding vehicle from its
sensory system and status of the next traffic signal can be
obtained or estimated precisely through an IVC system based
on wireless technology, or by some advance sensing system
on the vehicle. It is also assumed that the optimal control
input can be fed directly to the vehicle using an advanced
ACC system that can operate in stop-and-go traffic. Therefore,
it is possible to make the proposed vehicle control system
feasible using existing technology. Such an ACC system in a
car automatically detects the preceding vehicle, and adjusts
the closing distance by using the throttle and brake, after
being activated by a driver [25]. With the above technological
and operational assumptions, it can be stated that the vehicle
control problem formulated here is feasible. The next section
describes the way of treating a red signal and anticipating the
preceding vehicle as a dynamic reference to control the host
vehicle and relevant assumptions.
B. Prediction of the Preceding Vehicle
It is assumed that the position xp(t), velocity vp(t), acceler-
ation ap(t) of the preceding vehicle, and status and position of
the red signal xred(t) can be measured or known at real-time t.

For implementing MPC, it is necessary to estimate the states
of the preceding vehicle by predicting its acceleration ap(τ)
for τ > t in a given horizon. A simple choice is to consider
that the acceleration ap remains constant in the horizon,
ap(τ)|τ>t = ap(t) , if the next signal is green. However, such
a choice in not realistic since it may lead to a very high or a
negative predicted velocity of the preceding vehicle at the end
of a long horizon. Therefore, at any time t, the acceleration
of the preceding vehicle at τ > t in the prediction horizon is
obtained as
ap(τ) = f1ap (vp(τ), ap(t))
=
ap(t)
1 + e−β1(vp(τ)−γ1) 1 + eβ2(vp(τ)−γ2)
. (5)
The parameters γ1 and γ2 of the sigmoid functions in (5) define
an approximate range of velocities. The continuous function
(5) states that ap(τ) ≈ ap(t) if vp(τ) is within that range,
otherwise ap(τ) ≈ 0, and β1 and β2 express the sharpness
of the sigmoid functions. Which means the acceleration of the
preceding vehicle approaches zero when it reaches a maximum
velocity or stops completely and it never moves backward. It is
assumed that the current signal status will remain the same
in the prediction horizon since the signal changing time is
assumed to be unknown in advance. If there is no preceding
vehicle within the next stopping point at a red signal, it is
considered that a dummy vehicle is idling there by suitably
replacing xp = xred, vp = 0, and ap = 0. By this way,
the signaling system is introduced in the model without any
change in the problem structure. Based on the above stated
context, the time-dependent parameter q for τ > t of (1) can
be obtained as
q(τ) = ap(τ) = f1ap (vp(τ), ap(t)). (6)
In the case, the preceding vehicle is approaching a red signal
from far without dropping its velocity, prediction using (5)
is not appropriate, since it is certain that it must stop at
the end of the section after a known distance of (l(t) =
xred(t) − xh(t)). For this special case, a typical predic-
tion model of the preceding vehicle based on experimental
driving data has been proposed [26]. The prediction model
has been formulated by analyzing various stopping patterns
from real driving data of some drivers of different skills,
which were recorded experimentally on the national route
129 in Kanagawa prefecture, Japan. The model is formed
based on a reference stopping pattern, which is approximated
as the mean braking velocity curve v∗
b(l) for l < 180 m
as follows:
v∗
b(l) = 5.635 × 10−10
l5
− 3.446 × 10−7
l4
+ 7.925
×10−5
l3
− 8.519 × 10−3
l2
+ 0.4805. (7)
When a vehicle exactly follows the stopping pattern v∗
b(l)
(7), its braking rate (deceleration) is obtained as
a∗
b(l) =
dv∗
b(l)
dl
v∗
b(l). (8)
A red signal may appear when vehicles are at different
velocities and distances from the intersections. When a vehicle
has a velocity v(l) > v∗
b(l), it must stop at the distance l with
a higher braking rate than (8). Similarly, when a vehicle has a
velocity v(l) < v∗
b(l), it may stop at the same distance l with
a lower braking rate than (8). It is assumed that the braking
rate of any vehicle would be symmetric to (8) as the stopping
distances are the same, which can be obtained by multiplying
a factor to a∗
b(l) as
ab(l) = a∗
b(l)
v(l)
v∗
b(l)
2
. (9)
The above formulation provides a relationship of the braking
rate with respect to both velocity and stopping distance.
Details of this modeling and validation are illustrated in [26].
In this paper, (9) is used to predict the stopping behavior
of the preceding vehicle when it approaches a red signal
from far. Therefore, in this case the time-dependent parameter
q(t), simply defined by a function f2ap , is obtained for
t < τ as
q(τ) = f2ap (vp(τ), xp(τ), xred) = −ab(l(τ)). (10)
C. MPC
It is necessary to define a safe strategy of the host vehi-
cle in following a preceding vehicle. A common way to
define a desired separation Sd(t) at time t with a velocity
vh(t) is
Sd(t) = S0 + thdvh(t) (11)
where thd represents a safe headway time while following the
preceding vehicle, and S0 is the minimum separation between
the vehicles. In this paper, the above preference of maintaining
a velocity-dependent safe distance is taken into account in the
form of a soft constraint, which is described in the setting of
the performance index.
For computational simplicity, the inequality constraint relat-
ing the control input uh is converted into an equivalent equality
constraint using a dummy input ud as
C(x, u) =
1
2
u2
h + u2
d − u2
max = 0. (12)
Therefore, in this optimization problem, the control vector
u of size Nu = 2 is defined as u = [uh, ud]T . The
rate constraint is not directly included in this problem of
optimization. Instead, a saturation function is used to limit
the computed input rate before applying it to the vehi-
cle. State constraint is not considered for simplicity, assum-
ing the driver can handle any abnormal state situations.
Such simplifications are to ensure the optimization problem
solved at each step is tractable and can be executed in
real-time.
Subject to the dynamic equation (4) and constraint (12), the
following optimal control problem is solved at each time t
with the current state x(t) used as the initial state:
min
u
J =
t+T
t
L (x(τ), u(τ), q(τ))dτ (13)
where T is the horizon over which the optimal control inputs
are determined. The cost function L is chosen in this paper to

have the following form:
L =
1
2
(x − xd)T
Q(x − xd) +
1
2
(u − ur )T
R(u − ur )
+ we Feco(x) + ws(x)(Serr)2
(14)
where xd and ur are the desired state and input vector,
respectively, and Q and R are their corresponding
weighting matrices. The cost related to the fuel consumption
Feco(x) is multiplied by a weight of we. The cost due to
separation error or deviation from desired headway, Serr(t) =
Sd(t) − (xp(t) − xh(t)), is multiplied by a weighting function
ws(x). The weighting function ws(x) provides a large value
at the closing separation, similar to a barrier function, and a
small or negligible values when the separation is safe, which is
defined as
ws(x) = ρe
−σ
(x p1 (t)−xh(t)−S0)
vh(t)+δv
−thd
. (15)
For deriving the condition for optimal solution, the
Hamiltonian function is formed using (4), (12), and (14) as
follows:
H(x, λ, u, μ, q) = L(x, u, q)+λT
f (x, u, q)+μT
C(x, u)
(16)
where the vector λ denotes costates, and μ denotes Lagrange
multiplier associated with the constraint. Since the optimiza-
tion problem does have convex structure, its solution can be
local optimal.
To solve the optimal control problem numerically, the hori-
zon T is discretized into M steps. The necessary conditions
for the optimality are obtained in the following form:
F (U(t), x(t), t) := 0 (17)
where
F (U(t), x(t), t)
:=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
HT
u x∗
0 (t), λ∗
1(t), u∗
0(t), μ∗
0(t), q∗
0 (t)
C x∗
0(t), u∗
0(t)
...
HT
u x∗
M−1(t), λ∗
M (t), u∗
M−1(t), μ∗
M−1(t), q∗
M−1(t)
C x∗
M−1(0), u∗
M(0)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(18)
and
U(t) := u∗T
0 (t), μ∗T
0 (t), . . . , u∗T
M−1(t), μ∗T
M−1(t)
T
. (19)
Here Hu is the Jacobian and {x∗
i (t)}M−1
i=0 , {λ∗
i (t)}M
i=1,
{u∗
i (t)}M−1
i=0 , {μ∗
i (t)}M−1
i=0 , and {q∗
i (t)}M−1
i=0 denote correspond-
ing discretized values at the ith step. Which means both
Hu = 0 and C = 0 need to be satisfied at every step in
the horizon. Therefore, the size of the optimization problem
becomes as follows: the number of variables corresponding to
discrete states, inputs, costates, and Lagrange multipliers are
Nx M, Nu M, Nx M, and M, respectively. For a given vector
U(t) and step size in the horizon δ = T/M, discretized
sequences of the states are obtained as
x∗
0 (t) = x(t) (20)
a∗
p 0(t) = ap(t) (21)
x∗
i+1(t) = x∗
i (t) + f (x∗
i (t), u∗
i (t), q∗
i (t))δ (22)
and
a∗
p i+1(t) = f1ap (v∗
p i (t), a∗
p 0(t)) (23)
or
a∗
p i+1(t) = f2ap (v∗
p i(t), x∗
p i(t), xred(t)). (24)
Here, ap i+1 is computed using (24) when the preceding
vehicle approaches a red signal, otherwise (23) is used.
The sequence of costates is determined recursively using the
Jacobians Hx and the terminal costate λ∗
M(t) = 0 as
λ∗
i (t) = λ∗
i+1(t) + HT
x (x∗
i (t), λ∗
i+1(t), u∗
i (t)μ∗
i (t), q∗
i (t))δ.
(25)
The above discretized nonlinear algebraic equations determine
the optimal control sequence of the problem as an implicit
function of the current state and time.
In determining the optimal control input over the horizon,
the nonlinear equation F(U(t), x(t), t) = 0 needs to be solved
with a given U(t) and x(t) at each time t. Instead of a costly
iterative algorithm, the solution of the above optimal control
problem can be obtained efficiently using Continuation and
generalized minimum residual (C/GMRES) method [27]. It is
based on the idea that F can identically be zero if the following
conditions hold:
d
dt
F(U(t), x(t), t) = −ζ F(U(t), x(t), t) (26)
F(U(0), x(0), 0) = 0 (27)
where ζ is a positive constant to stabilize F = 0. For
nonsingular FU , (26) can be rewritten using ˙U(t), and
Jacobians Fx , FU, and Ft as
˙U = F−1
U (−Fx ˙x − Ft − ζ F). (28)
With a coefficient matrix FU the above (28) can also be
treated as a linear algebraic equation to determine ˙U for given
U, x, ˙x, and t. This can be solved efficiently by a linear
solver like the GMRES method. The GMRES method is a
kind of Krylov subspace method to solve nonsymmetric linear
systems represented in the form of Ax = b by successively
generating orthogonal vectors and combining these through
a least-squares solve and update, in minimizing the residual
b − Ax [27], [28]. The solution U(t) can be traced by
integrating ˙U of (28) in real-time using the continuation
method, which is a nonlinear problem solving method used for
large-scale system engineering problems [29]. In continuation
method, the derivative of the control sequence with respect to
time is determined according to the corresponding derivative
of the state with respect to time. The details of the C/GMRES
method, its error analysis and proof can be found in [27].
Since C/GMRES method does not require iterative searches,
it is much faster than any iterative method, and it can
be implemented for an on-board vehicle control system
in real-time.
C/GMRES involves several approximations, and the error
F is not necessarily expected to converge to zero due to

Algorithm 1 Vehicle Control Using C/GMRES Method
1) Initialize δ = t, T , M = T/δ, n := 0; measure x(t),
and q(t); Initialize the control input {unδ(t )}
t =(n+M−1)δ
t =nδ
analytically or numerically.
2) Set n := n + 1.
3) Measure x(t) and q(t) at t = nδ.
4) Using x(t), q(t), initial guess of input
{u(n−1)δ(t )}t =(n+M−1)δ
t =nδ , obtain the predicted state
{ˆxnδ(t )}
t =(n+M)δ
t =(n+1)δ , co-state {λnδ(t )}
t =(n+M)δ
t =(n+1)δ , weight
{w4nδ(t )}t =(n+M)δ
t =nδ , and compute {˙unδ(t )}t =(n+M−1)δ
t =nδ
using GMRES method.
5) Compute
{unδ(t ) = u(n−1)δ(t ) + αtanh(˙unδ(t )/α)δ}t =nδ and
{unδ(t ) = u(n−1)δ(t ) + ˙unδ(t )δ}
t =(n+M−1)δ
t =(n+1)δ .
6) Set and execute the vehicle control input unδ(t) =
unδ(nδ), for the period nδ ≤ t ≤ (n + 1)δ.
7) Go back to Step 2.
persistent approximation error. If FU , Fx , and Ft , which are
related to open-loop characteristics of the system, are Lipschitz
continuous with respect to (U, x, t) and FU , F−1
U , Fx
Ft , and Ft are bounded by a positive constant for all
(U, x, t), the boundedness of the error F is guaranteed
in C/GMRES [27]. Since the control input is constrained in
the proposed vehicle control system, if there is a feasible
solution of the optimal control problem, the speed of the
host vehicle does not diverge. However, in the proposed
system the host vehicle is controlled based on the state of
the preceding vehicle, and considering the inherent variations
and randomness in traffic flows, such as the sudden appearance
of red signals or change in the preceding vehicle, a feasible
solution is not always guaranteed. Such abnormal cases can
easily be detected from computation and handled, as stated
earlier in the problem formulation, and the control algorithm
can be reset later in normal conditions.
The overall algorithm of the proposed model predictive
vehicle control system using C/GMRES method can be sum-
marized as in Algorithm 1.
At each sampling step, only the immediate vehicle control
input is applied through a saturation function with a maximum
value of α. The set of control inputs is used as the initial
guess of the solution of the optimization problem for the
next sampling step, for newly measured states of the vehicles.
Repeating the whole process and renewing the control input at
each sampling time are necessary to overcome the influences
of varying traffic and unmodeled disturbances in computation.
III. EVALUATION OF THE VEHICLE CONTROL SYSTEM
A. Settings for Numerical Simulation
A host vehicle is chosen with the following model parame-
ters: Mh = 1200 kg, CD = 0.32, ρa = 1.184 kg/m3, Av =
2.5 m2, and μ = 0.015. The proposed vehicle control system
has been simulated considering preferences of umax = 1.50,
thd = 1.8 s, and S0 = 5.0 m. A suitable horizon T = 60 s is
chosen to cover prediction of the vehicle up to an intersection
with red signal from a reasonable distance. The prediction
horizon T is split into M = 120 steps of size δ = 0.5 s.
The value of α is chosen by analyzing the rate of acceleration
obtained from experimental driving data of three drivers over
a total of 32-km driving. Values of acceleration at an interval
of 0.5 s are obtained, and only the values above 0.3 m/s2
are taken into account. More than 99% of the counted cases,
the rates of acceleration were found below 1.5. Assuming this
figure reflects usual comfortable driving, the value of α is set
at 1.5 in simulation. The prediction model of the preceding
vehicle (5) is set with β1 = 2.5, β2 = 1.0, γ1 = 1.0, and
γ2 = 16. For estimating fuel consumption of a vehicle, an
approximate and differentiable function of velocity and control
input is taken here [8]. For a typical vehicle, fuel consumption
in ml/s is estimated as
fV = fcruise + faccel (29)
where fcruise = b0 + b1v + b2v2 + b3v3 represents
fuel consumed per sec at a steady velocity of v, and
faccel = â(c0 + c1v + c2v2) is additional consumption due
to presence of acceleration at the velocity v. The equivalent
acceleration of the vehicle considering the effect of road
gradient is defined as (â = −(1)/(2M)CDρa Avv2
h −μg +uh).
It is assumed that during braking from a high velocity when
uh < 0, no fuel is consumed since the engine is rotated by
the kinetic energy of the vehicle. At idling condition, the fuel
supply resumes automatically to keep the engine rotating. The
consumption parameters, which were approximated using the
data obtained from an engine torque-speed-efficiency map
of a typical vehicle, are b0 = 0.1569, b1 = 2.450 × 10−2,
b2 = −7.415 × 10−4, b3 = 5.975 × 10−5, c0 = 0.07224,
c1 = 9.681 × 10−2, and c2 = 1.075 × 10−3. Details of
the formation and determination of parameters of this fuel
consumption model are described in [8].
The performance index is set as xd = [0, 13.89, 0, 0]T,
Q = diag[0, 0.31, 0, 0], ur = [FR − gθ(xh), 0]T
, R =
diag[22.2, 0], and we = 220. The desired velocity of the
vehicle is chosen at the same value of the velocity limit of
the road, and the other desired states are ignored by choosing
the coefficient matrix Q. The weighting parameters of wS
is set at ρ = 0.017 and σ = 2.954. The cost related to
the fuel economy is defined as Feco = fcruise/(vh + δv).
A small positive number δv = 0.1 is added with vh to avoid
singularity at vh = 0. Instead of total fuel consumption, the
cruising consumption rate is chosen in the performance index
so that the vehicle at the steady condition runs at a velocity
that maximizes distance per unit of fuel, i.e., gas mileage.
Calibration of the weighting parameters is an important issue
for attaining safe and efficient behavior in complex traffic flow.
Weighting parameters are tuned manually by observing the
driving performance in two steps: 1) Q, R, and we are tuned
for maximum fuel economy by simulating a vehicle in the
absence of a preceding vehicle and 2) wS, ρ, σ, and also R
are tuned in such a way that the vehicle can avoid rear-ends
collision by simulating it on crowded roads.
A test route of about 4.0 km on a network consists of 14 sec-
tions (S1, . . . , S14) and 13 intersections has been constructed
in AIMSUN simulator, Fig. 2. In the network, the consecutive
sections are connected through traffic control signals, which

S1 S14S13
Test Route 4 km
S2
Fig. 2. Image of the test route, road network in AIMSUN used in simulation.
0
100
200
300
400
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14
Length[m]
Length of the section
0
1
2
3
4
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14
Lanes
Number of Lanes of the section
0
600
1200
1800
2400
3000
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14
Road Section
Trafficflow[Veh/h]
Vehicles per hour on the section
(a)
(b)
(c)
Fig. 3. Settings of section S1–S14 in the test route. (a) Length of the section.
(b) Number of lanes. (c) Traffic flow per hour.
are synchronously set at 90 s cycle, including 52 s for green
and 2 s for yellow signal. Vehicles can enter and leave the
network only at intersections. The recommended velocity limit
of the roads is 50 km/h. The length, number of lanes, and
traffic flow rate of each road section with respect to the test
route are shown in Fig. 3. With the above settings of the
network, actual traffic flow rates observed on the sections
are as follows. A total of 2715 vehicles enter the network
in one hour through various points, among which only 1261
vehicles enter through the beginning of S1 and the rest of
1291 vehicles enter through the intersections. The number
of vehicles leave the end of S14 is 1424, and the rest
1454 vehicles leave through intersections. The traffic contains
72.19% cars, 3.13% trucks, 8.95% taxis, and 15.73% vans.
The default car following model implemented in AIMSUN
is based on Gipps model [30], [31]. The lane changing task
on a multilane road is also controlled by a parameterized
lane change model in AIMSUN. Various parameters of the
Gipps model and lane changing model among vehicles are
set stochastically using typical normal distributions. Therefore,
driving characteristics of each vehicles differ from any other
vehicles, and such diverse vehicles create a pseudo realistic
traffic environment. An extension of AIMSUN simulator was
created through application program interface (API) to collect
traffic data and control a vehicle from the outside of AIMSUN
for evaluating the developed control system.
B. Simulation Results
An arbitrary car from the traffic on the first section S1
is selected as the host vehicle, and then it is controlled
through API until it exits the last section S14, after traveling
a distance of about 4.0 km. For the purpose of comparison,
the same vehicle with the same initial conditions is controlled
by the proposed MPC and Gipps model-based driving in two
independent tests. For simplicity in representation, a notation
“MPC-vehicle” is used to mean a vehicle that is controlled by
the proposed algorithm, the same applies to “Gipps-vehicle”
which is fully controlled by AIMSUN as per its default
settings.
Figs. 4 and 5 show driving scenarios with respect to
simulation time using various plots of an MPC-vehicle and
a Gipps-vehicle, respectively. The following plots, shown in
each figure, illustrate the traffic environment: (a) the sequence
of the synchronized traffic signals [green (G), yellow, and red
(R)] at the intersections; (b) the current road section of the
host vehicle (each of 14 sections is represented by a step);
(c) the number of vehicles on the corresponding section of the
host vehicle; (d) the velocity of the preceding vehicle; (e) the
head distance or range clearance; and (f) the points at which
the preceding vehicle is replace by another vehicle due to a
lange change, overtaking, or turning event. These plots show
complexities and uncertainties in the traffic flow in which the
host vehicle is controlled. The performance of the host vehicle
is shown by the following plots: (g) the velocity of the host
vehicle; (h) the control input of the host vehicle; and (i) the
cumulative fuel consumption curve.
Although initial conditions in both cases of an MPC-vehicle
and Gipps-vehicle are the same, due to different control
methods and inherent variations in traffic flows, they encounter
different situations on the route. It is observed that the input
of the MPC-vehicle is kept limited within a moderate range,
whereas the Gipps-vehicle is controlled aggressively. During
speeding up, the MPC-vehicle gradually reduces input as its
velocity approaches a high value. By this behavior, it avoids
very high fuel consumption rate at high velocity. When it
approaches an intersection with a red signal, the MPC vehicle
begins to decelerate much earlier than the deceleration of the
Gipps-vehicle, which enables reuse of the kinetic energy of
the vehicle. In traveling a distance of about 4.0 km, the MPC-
vehicle consumes fuel of 225.11 ml, whereas the Gipps-vehicle
consumes 262.91 ml. Sudden changes in the preceding vehicle
cause the input of the MPC-vehicle to fluctuate slightly, since
such changes affect anticipation of the future states of the
preceding vehicle. Acceleration of the preceding vehicle is
estimated using its velocities measured at the last two sampling
steps, which often slightly differs from its actual values in
the next step. Therefore, prediction using slightly different
acceleration of the preceding vehicle also causes the control
input to sometimes oscillate slightly. In spite of such little
noisy control input, the MPC-vehicle ultimately reduces the
total fuel consumption. However, the real mechanical actuators

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 4. Performance of an MPC-vehicle and situations in test environment in
terms of traffic, road, and signals. (a) Traffic signal, (b) current road section of
the host vehicle, (c) number of vehicles on the current section, (d) velocity of
the preceding vehicle, (e) range clearance, (f) change in the preceding vehicle,
(g) velocity of the host vehicle, (h) control input, and (i) fuel consumption.
have delays that would make the effective input smoother and
slightly better results can be expected.
In driving on crowded urban roads, a vehicle encounters
unique traffic situations since driving behavior and preference
among the vehicles are not the same. For a statistically
meaningful comparison, fuel consumptions of the MPC and
Gipps vehicles in 100 independent cases are observed at
different simulation time. At a certain time, a vehicle is
chosen randomly from the traffic at section S1, and with the
same initial conditions, it is driven on the same route using
MPC and Gipps methods in two separate tests. In the next
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 5. Performance of a typical Gipps vehicle and situations in test
environment in terms of traffic, road, and signals. (a) Traffic signal, (b) current
road section of the host vehicle, (c) number of vehicles on the current section,
(d) velocity of the preceding vehicle, (e) range clearance, (f) change in the
preceding vehicle, (g) velocity of the host vehicle, (h) control input, and
(i) fuel consumption.
observation, at a later time another vehicle is chosen from the
traffic and driven in the same way using both methods. In
this way, fuel consumption obtained from 100 independent
vehicles are plotted in Fig. 6, in the sequence of their
observations. The solid lines show the corresponding average
fuel consumptions of these vehicles. Table I summarizes the
relative merits of the MPC-vehicles over the Gipps-vehicles.
The MPC-vehicles reduce fuel consumption by 13.21% and
improve fuel economy (km/l) by 14.96%, compared with
that of the Gipps vehicles. The mean value and standard

Fig. 6. Fuel consumption of MPC and Gipps vehicles in 100 randomly
chosen independent cases.
TABLE I
COMPARISON OF MPC VEHICLES AND GIPPS VEHICLES
MPC Gipps Comparison
Fuel [ml] 221.69 255.43 −13.21%
Mileage [km/l] 18.18 15.82 +14.95%
Total time [s] 529.73 454.91 +16.44 %
Idling time [s] 95.80 116.60 −17.84 %
deviation of fuel economy of the MPC-vehicles are 18.18
and 0.5742 km/l, respectively. Whereas, the mean values
and standard deviation of fuel economy of Gipps-vehicles
are 15.83 and 0.9585 km/l, respectively. Although the MPC-
vehicles have the same driving preferences, variations in their
fuel consumptions are due to influences of the other vehicles
and timing of appearance of red signals in the network.
Although, average idling time of MPC-vehicles at red
signals is 17.88% less than that of Gipps-vehicles, they take
16.44% extra time to travel over the same route. Instead of any
direct cost for trip time, cost for fuel economy (consumption
divided by velocity) is used in the performance index, which
influences an MPC-vehicle to avoid both very low and very
high velocities. The reason of taking longer time by an
MPC-vehicle can be understood from its velocity and control
characteristics shown in Fig. 4. The MPC-vehicles start with
limited acceleration and attain the rated or a steady velocity
at a longer time, comparatively, and they do not exceed the
recommended or desired velocity limit. Therefore, they take
longer time to pass a section and are more likely to face
red signals than the average traffic. In separate simulation
on a typical traffic flow condition, stopping frequencies of
30 vehicles controlled by MPC and Gipps methods have been
observed. The MPC-vehicles stopped at red signals 242 times,
whereas Gipps-vehicles stopped 213 times. Accelerating char-
acteristics and stopping at red signals are the main reasons that
make the MPC-vehicle more time consuming for traveling the
same distance than a Gipps-vehicle. By considering the cost
of traveling time and advance information of signal timing
may help an MPC-vehicle to achieve a balanced performance
by trading off the fuel consumption with travel time. Such
formulation and development can be investigated in the future.
A number of comparative fuel saving aspects observed in
the evaluation are summarized here.
1) MPC-vehicles find and run at the optimal cruising
velocity (when preceding vehicle is far), but the cruising
velocities among Gipps-vehicles vary stochastically
according to the drivers’ preferences.
2) MPC-vehicles apply comparatively lower value of
acceleration to smoothly reach the optimal cruising
velocity or a steady velocity than that of Gipps-vehicles.
3) MPC-vehicles usually keep higher range clearance with
respect to the preceding vehicle, which helps avoiding
sudden or aggressive braking. Therefore, it is a safer
approach than the Gipps model.
4) At a red signal MPC-vehicles start decelerating much
earlier and slowly that ensures utilization of kinetic
energy of the vehicle (except for sudden appearance of
red signal or being overtaken by a vehicle).
The above features illustrate the significance of the proposed
MPC-based vehicle control system. Introducing such an on
board fuel efficient driving system may make the transporta-
tion systems more environmentally friendly.
IV. DISCUSSION
It is important that the model be very simple to keep
the optimization problem computationally tractable for real-
time execution. AIMSUN simulator facilitates an option of
interactive simulation in real-time, and the proposed algorithm
is found fast enough to run a vehicle without causing any
delay. A computation time of 6.43 milliseconds per sampling
step was observed on a typical PC. Therefore, the proposed
MPC for vehicle driving can be executed in real-time for
realizing an ACC system.
In simulation, the proposed MPC system is activated after
the vehicle starts from standstill by the default driving model
like activation of an ACC system and deactivated when it stops
at a red signal. Fuel consumed in a vehicle during idling to
keep its engine running can only be reduced by stopping the
engine. In the case of MPC vehicles, idling fuel consumption
of 15.05 ml is included in the total consumption of 221.69
ml. If an automatic switching on–off system can be used for
idling stop and restart, the fuel economy can be improved
further.
Performance of the proposed MPC system is compared
with the Gipps model-based driving in the simulator. Gipps
model is a parameterized car following model that represents
different human driving behavior depending on the values
of its parameters. In AIMSUN, parameters of Gipps model
among vehicles are randomly chosen by normal distributions.
Therefore, each vehicle exhibits distinct behavior on the road,
and a pseudo realistic traffic environment is created. In this
paper, such one hundred from thousands of vehicles on the net-
work are chosen randomly and compared to obtain statistically
meaningful results, assuming Gipps vehicles represent usual
human driving behavior. However, depending on the distance
between the intersections, number of lanes, signal cycle, and
traffic flow, the fuel consumption may vary in a similar way
for both driving systems.

Apparently the proposed MPC method is similar to a car
following model if there is a preceding car at a short distance.
However, in some aspects it differs from the conventional car
following models. An MPC vehicle predicts the preceding
vehicle and uses the signal status to choose its acceleration.
It closely follows a slower car at steady conditions like a
simple car following model, but it keeps a distance when there
is velocity fluctuations in the preceding vehicle and stops at a
red signal by smooth deceleration.
The approximate fuel consumption model presented here is
valid for a gasoline engine. By replacing its parameters and
choosing suitable weights of the cost function, the proposed
system can be applied to other vehicles. Any alternative fuel
consumption models using continuous functions may also be
used by replacing the presented model. For a hybrid electric
or electric vehicle, a similar MPC-driving system can be for-
mulated considering the energy consumption model including
the effect of regenerative braking and charging-discharging
characteristics of the battery.
Any reliable sensing system that can collect the relevant
information of the preceding vehicle and signal can be used for
implementing the proposed control method. On-board camera,
laser, or other sensors have limitations in precise detection and
measurement of the state of a preceding vehicle, especially
on the curved or hilly roads, during turning or overtaking
and at intersections. IVC system may provide information
in such situations with a high accuracy. IVC is an emerg-
ing technology currently under experimentation in intelligent
transportation systems targeting at the enhancement of safety
in road transportation. A widespread implementation of IVC
requires further studies and development in the research area
of wireless communication to solve various issues, such as
frequency band, detection and establishment of links, delay
and so on. Such technical issues of IVC have not been
investigated in this paper.
V. CONCLUSION
A MPC system for improved fuel economy of a vehicle
has been proposed in this paper. The system predicts the pre-
ceding vehicle and considers the signal status of the upcoming
intersections to compute the optimal vehicle control input. The
fuel economy of the vehicle was maximized by regulating a
safe head-distance or cruising at the optimal velocity under a
bounded driving torque condition. The proposed system has
been evaluated in AIMSUN traffic simulator on a typically
crowded urban road network. The computation time was
found to be fast enough to implement the proposed system
in real-time. The vehicle has been controlled safely in spite of
changing and uncertain phenomena in the traffic environment.
Simulation results reveal significant improvement in overall
fuel consumption for traveling a given distance compared with
conventional driving. The proposed system can be used to
develop an ACC or driving assistance system for the next
generation vehicles through further technological advancement
in intelligent transportation systems.
In the proposed method, sometimes use of hard braking
cannot be avoided if a red signal appears when the vehicle
is close to an intersection. If the remaining duration of
the traffic signal is known in advance, by decelerating the
vehicle optimally further improvement in fuel economy can
be realized. It would be worthy to include such features and
enhance the proposed vehicle control system in the future.
ACKNOWLEDGMENT
The authors would like to thank Nissan Motors Company
Ltd., Japan, for providing experimental driving data and valu-
able comments on this development.
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Md. Abdus Samad Kamal (M’01) received the
B.Sc. Eng. degree from the Department of Elec-
trical and Electronic Engineering, Khulna Univer-
sity of Engineering and Technology, Phul Barigate,
Bangladesh, in 1997, and the Master’s and Ph.D.
degrees from the Graduate School of Information
Science and Electrical Engineering, Kyushu Uni-
versity, Fukuoka, Japan, in 2003 and 2006, respec-
tively.
He served as a Lecturer with the Khulna Uni-
versity of Engineering and Technology until 2000.
He served as a Post-Doctoral Fellow with Kyushu University, an Assistant
Professor with International Islamic University Malaysia, Malaysia, and a
Researcher with the Fukuoka Industry, Science, and Technology Foundation.
He is currently a Researcher with the Institute of Industrial Science, The
University of Tokyo, Tokyo, Japan. His current research interests include
intelligent transportation systems, DAS, and applications of model predictive
control.
Dr. Kamal is a member of the Society of Instrument and Control Engineers.
Masakazu Mukai (M’06) received the B.E., M.E.,
and Dr.Eng. degrees in electrical engineering from
Kanazawa University, Kanazawa, Japan, in 2000,
2002, and 2005, respectively.
He is currently with the Graduate School of Infor-
mation Science and Electrical Engineering, Kyushu
University, Fukuoka, Japan. His current research
interests include receding horizon control and its
applications.
Dr. Mukai is a member of the Society of
Instrument and Control Engineers, the International
Student Committee on Industrial Ecology, and the Institute of Electrical
Engineers of Japan.
Junichi Murata (M’97) received the Master’s and
Dr.Eng. degrees from Kyushu University, Fukuoka,
Japan, in 1983 and 1986, respectively.
He then became a Research Associate, an Asso-
ciate Professor, and a Professor with the Graduate
School of Information Science and Electrical Engi-
neering, Kyushu University. His current research
interests include neural networks, self-organizing
systems, and their applications to control and iden-
tification.
Prof. Murata is a member of the Society of Instru-
ment and Control Engineers, the International Student Committee on Industrial
Ecology, and the Institute of Electrical Engineers of Japan.
Taketoshi Kawabe (M’98) received the B.Sc. and
M.Sc. degrees in pure and applied physics from the
Department of Applied Physics, School of Science
and Engineering, Waseda University, Tokyo, Japan,
in 1981 and 1984, respectively, and the Ph.D. degree
from Tokyo University, Tokyo, in 1994.
He was with Nissan Research Center, Nissan
Motor Co., Ltd., Japan, from 1984 to 2005. He
was a Research Student with the Department of
Mathematical Engineering and Information Physics,
Tokyo University, Tokyo, in 1992 and 1993. He has
been a Professor with the Faculty of Information Science and Electrical
Engineering, Kyushu University, Fukuoka, Japan, since April 2005. His
current research interests include motion control, vibration control, and related
automotive control technology.
Dr. Kawabe is a member of the Society of Instrument and Control Engi-
neers, the Japan Society of Mechanical Engineers, the Institute of Electrical
Engineers of Japan, and the Society of Automotive Engineering of Japan.

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