182 ◾ Advances in Communications-Based Train Control Systems
roughout this chapter, unless otherwise stated, the superscript “
i
” denotes the
i
th train, and the subscript “
k
” indicates the
k
th period (time slot).
e result of uplink transmission of the status from the
i
th train to ZC is
denoted as
γ
k
i
, where
γ
k
i
=
0
means that the uplink transmitted packet is dropped
and
γ
k
i
=1
indicates that the packet is successfully received by ZC. Due to packet
drops in uplink transmissions, ZC can only get the estimated status of trains,
ˆˆ
.sv
k
i
k
i
,
e LMAs of the leading train are
ˆ
s
k
0
and
ˆ
v
k
0
, and
c
k
i
is the LMA of the
i
th train.
e result of downlink transmission from ZC to the
i
th train is denoted as
θ
k
i
. Due
to packet drops in downlink transmissions, trains can only get the estimated LMA
ˆ
c
k
i
to calculate the speed/distance prole.
ere are two kinds of uncertainties in train–ground transmissions: transmis-
sion delay and packet drop. Due to the uncertainties in train–ground communi-
cations, retransmissions are needed. Random transmission errors lead to delays if
packets are successfully received before reaching the retry limits. Otherwise, pack-
ets are dropped.
In WLANs, the station (STA) must break o the link with the original access
point (AP) before associating with the objective AP, which introduces packet drops.
e relationship between packet drops and handovers will be studied in Section 9.4.
ZC
Wireless
network
(downlink)
Wireless
network
(uplink)
Controller
Actuator Sensor
Plant
Traction
brake
Train
1
VOBC
Odometer
speedometer
Traction
brake
Train
2
VOBC
Odometer
speedometer
Traction
brake
Train
n
VOBC
Odometer
speedometer
θ
k
i
γ
k
i
ˆ
c
k
i
ˆ ˆ
i−1
c
k
= (s
k
, v
k
)
i
i−1
s
k
, v
k
, i = 1, … , n
ii
ˆˆ
0
(s
k
, v
k
)
0
ˆˆ
i
(s
k
, v
k
)
i
Figure9.3 Model of system to control a group of trains in CBTC.
Networked Control for a Group of Trains ◾ 183
9.3.2.2 Equivalent NCS
In Figure 9.4, we model the control system as an NCS with packet drops in both
uplink and downlink transmissions. e group of trains composes the plant. e
sensors are odometers and velocimeters of all the trains. All the vehicle onboard
components (VOBCs) constitute the controller of the system. e actuator is all the
traction and brake equipment of running trains.
Both uplink and downlink transmissions are diverse among trains due to vari-
ous delays and packet drops. Some handover strategies are related to packet drops.
A very large maximum number of retransmission times impair handover perfor-
mance, because the STA may keep associating with the original AP even under a
poor link quality. For such reason, a small maximum number of retry limits are
usually adopted in CBTC systems, which means that delays introduced by random
errors are very small. At the same time, trains and ZC are strictly time synchro-
nized. On receiving the status of its preceding train, the train knows the exact
sensor–controller delays, which can be estimated and compensated.
e main objectives of trains’ control system are as follows:
◾ To keep distances between trains to the scheduled headway to maximize line
capacity
◾ To control trains’ velocities close to the maximum allowed speed to minimize
the average travel time
◾ To avoid unnecessary traction and brake to decrease energy consumption and
make passengers comfortable
We dene the system state as the distance and velocity deviations of all the trains,
and dene the output of the controller as the excess applied force. e status of
trains is transmitted from odometers/speedometers to ZC and from ZC to VOBCs
Controller
Sensor
Uplink
wireless
Actuator
ZC
i = 1,...,
ni
= 1,...,n
Downlink
wireless
ˆ
X
k
~
X
k
~
X
k
X
k
θ
k
Plant
s
k
, v
k
i
i
γ
k
i
i
Figure9.4 Equivalent networked control system.
184 ◾ Advances in Communications-Based Train Control Systems
because each train can directly get location and velocity of its own from onboard
sensor. After receiving the status of the preceding train, the train generates control
based on its distance and speed deviations. e uplink and downlink transmissions
are equivalent to the transmission of system state.
As illustrated in Figure 9.4,
X
k
is the system state and γ
k
i
is the result of uplink
transmission of packet from the
i
th train to ZC. Due to packet drops in uplink
transmission, ZC can only get the estimated state,
X
k
. ZC issues LMA,
X
k
.
Because
of packet drops in downlink transmission, indicated by
θ
k
i
, the controller can only
get estimated LMA,
ˆ
X
k
.
9.3.3 Analytical Formulation of CBTC
To set up the analytical formulation of CBTC, we rst consider the ideal case with-
out packet drops. In CBTC systems, trains and ZC are strictly time synchronized
and sampling with a very short period. We can consider that the system is discrete
and linear time invariant.
XA
XB
U
kkk
+
+
1
=
UG
X
kk
=
−
(9.1)
where:
XR
k
n
∈
×
21
is the augmented system state
UR
k
n
∈
×
1
is the output of the controller
AR
nn
∈
×
22
and
BR
nn
∈
×2
are real constant system matrices
e augmented system state includes distance and speed deviations of all the trains.
X
D
V
k
k
k
=
(9.2)
Dd d
kk k
n
=
1
δδ
Vv v
kk k
n
=
1
δδ
Uf
f
kk k
n
=
1
δδ
where:
D
k
,
V
k
, and
U
k
are
n
-dimensional vectors
n
is the number of trains
D
k
is the deviation of distances between trains from the scheduled headways
V
k
is the speed deviation from the optimal speed
U
k
is the deviation of the applied forces from the basic resistance of trains
Networked Control for a Group of Trains ◾ 185
For every distance deviation in
D
k
, it has
ds s
k
i
k
i
k
i
=
1
−
− (9.3)
δdd
in
k
i
k
ii
=,=1
,,
−∆ (9.4)
where:
d
k
i
is the distance between the
i
th and
(1)
i
−
th trains
δ
d
k
i
is the distance deviation of the
i
th train
e optimal distance of the
i
th train is dened as Δ
i
, which is a constant depending
on the parameters of the
i
th train, such as emergency brake performance, length of
the train, and safe margin.
e velocity deviation is dened as
δvvv
k
i
k
i
= − (9.5)
where:
v
k
i
is the train speed
We assume that the optimal speed
v
is a constant. To achieve the shortest travel
time,
v
is just below the maximum speed with a reserved margin for safety.
For every deviation of applied force in
U
k
, it has
δff
gm
a
k
i
k
i
k
ii
k
i
==
− (9.6)
where:
δf
k
i
is the deviation of the applied force to the train from the basic resistance
of the train
f
k
i
is the applied force to the
i
th train
g
k
i
is the basic resistance of the
i
th train
Based on kinematic equations, it has
δδvvaT
k
i
k
i
k
i
+
=+
1
(9.7)
ssvT aT
k
i
k
i
k
i
k
i
+
=+ +
1
2
1
2
(9.8)
δδδδddvvTaaT
k
i
k
i
k
i
k
i
k
i
k
i
+
−−
+−
+−
1
11
2
=( )
1
2
()
(9.9)
where:
a
k
i
is the acceleration of the
i
th train
T
is the sampling period of the sensor in the system
Based on Equations 9.7 through 9.9, we get matrices
A
and
B
in Equation 9.1.
186 ◾ Advances in Communications-Based Train Control Systems
9.4 Packet Drops in Train–Ground Communications
In this section, we rst analyze packet drops introduced by random transmission
errors. en, we study the packet drops caused by handovers, which have a drop
rate depending on the handover time, the AP coverage range, and the overlapping
coverage area between APs.
9.4.1 Packet Drops due to Random Transmission Errors
ere is an automatic repeat request (ARQ) scheme with carrier sense multiple
access/collision avoidance (CSMA/CA) at the media access control (MAC) layer of
IEEE 802.11. Usually, a small number of retransmission times are used in CBTC
systems. e maximum number of retransmission times of Beijing Yizhuang and
Changping Lines is less than
10
.
We use MS Visual C++ 8.0 and IT++ 4.03 to build a link-level simulator of
WLANs based on [32]. e frame error rate (FER) at dierent train speeds is given
in Figure 9.5. e data rate is
6Mbits/s
and the packet size is
200
byte
s
. e simu-
lated channel is at fading channel with predened train speeds. From Figure9.5,
we can see that the FER increases signicantly with the growth of train speed.
Because of the random backo scheme in CSMA/CA, the time interval between
retransmissions (expect for the rst two retransmissions) is much bigger than the channel
10 15 20 25 30 35
40
SNR (dB)
200 km/h
160 km/h
120 km/h
90 km/h
60 km/h
3 km/h
10
−4
10
−3
10
−2
10
−1
10
0
FER
WLAN in flat fading channel with different train speeds
Figure9.5 FER at certain train speeds.
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