1832 D. Chen et al.
optimal velocity (OV) function which contains time-
varying delay and headway. τ(t) is the time-varying
delay which satisfies:
0 ≤ τ(t) ≤ h, ˙τ(t) ≤ θ<1. (3)
where h is the upper boundary of varying-time delay.
u
n
(t) is the feedback control term to be designed.
A nonlinear OV function is proposed as [23]:
F(y
n
(t)) =
v
max
n
2
[
tanh(y
n
(t) − h
c
) + tanh(h
c
)
]
. (4)
where v
max
n
is the maximum velocity and h
c
is the neu-
tral distance. It is assumed that the velocity of head
vehicle v
0
is less than the maximum velocity:
v
0
≤ v
max
n
. (5)
It is easy to prove that the nonlinear OV function (4)
satisfies the following equation according to Lipschitz
continuous theory:
F(y
2
(t)) − F(y
1
(t))
≤ κ
y
2
(t) − y
1
(t)
. (6)
where constant κ<∞ exists.
When headway of vehicle n is smaller than the still
distance y
min
n
, the vehicle adopts the dead-stop imple-
ment to guarantee that moving vehicles cannot occur
the reverse or collision:
if y
n
(t)<y
min
n
, then x
n
(t) is invariant,v
n
(t) = 0.
(7)
Vehicle dynamics (2) is defined under the equilibrium
state:
[y
∗
n
(t), v
∗
n
(t)]
T
=[F
−1
(v
0
), v
0
]
T
. (8)
which means that all the vehicles move orderly with
constant velocity v
0
and headway F
−1
(v
0
) in the steady
state.
The feedback control of ACC vehicles was designed
as [48,49]:
u
n,ACC
(t) = k
1
(y
n−1
(t)−y
n
(t))+k
2
(v
n−1
(t)−v
n
(t)).
(9)
where k
1
, k
2
are control gains.
n-1nn+1n+iN
Flow direction
Connected
vehicle
RSU
V2V V2I
ACC vehicle
LIDAR sensor
Fig. 3 Information flow network: a ACC vehicles formation, b
Connected vehicles formation
According to control signal Eq. (9), information flow
network of ACC vehicles adopts preceding-following
topology shown in Fig. 3a. Although this control term
works to some extent, it is not suitable for the devel-
opment of intelligent transportation systems, particu-
larly connected vehicles. In connect framework, each
vehicle obtains state information from other vehicles
by V2V communication and traffic signal informa-
tion from road side unit ( RSU) by V2I communication
shown in Fig. 3b.
Therefore, a new feedback control signal u
n
(t) for
connected eco-driving system is designed as follows:
u
n,connect
(t) =
M
m=1
k
1,m
(y
n−m
(t) − y
n−m+1
(t))
+
M
m=1
k
2,m
(v
n−m
(t) − v
n−m+1
(t)),
(10)
where k
1,m
, k
2,m
are the feedback gains to be deter-
mined and M is the number of considered ahead vehi-
cles.
For the vicinity of traffic signals, the external dis-
turbances (e.g., irregular geometrical surfaces, bad
weather, emergencies) lead to terrible driving condi-
tions. It is easy to induce traffic congestion and increase
fuel consumption. Thus, we introduce the external dis-
turbances into the closed-loop system (2):
˙v
n
(t) = a
n
(F(y
n
(t − τ(t))) − v
n
(t))
+ u
n,connect
(t) + b
n
w
n
(t),
˙y
n
(t) = v
n−1
(t) − v
n
(t). (11)
In order to structure error dynamics considering the
equilibrium state, let ˜y
n
(t) = y
n
(t) − y
∗
n
, ˜v
n
(t) =
v
n
(t) − v
∗
n
. Considering Eqs. (8) and (11), the error
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