192 ◾ Advances in Communications-Based Train Control Systems
are
n
APs with
n −1
overlapping coverage areas. e running time of a train
atspeed
v
m
is
t
dndn
v
r
m
=
()
ap ov
⋅− ⋅−
1
(9.17)
where:
d
ap
is the coverage range of an AP
To make the problem tractable, we make the following simplications:
◾ For business operation speed, handovers introduce the average handover time
t
ah
.
◾ ere is no ping-pong handover. STA makes only one handover in the over-
lapping coverage area.
◾ e number of packet drops caused by each handover is
n
t
T
ho
ah
=
where:
t
ah
is the average handover time
t
T
ah
is the minimum integer that is greater than or equal to
t
T
ah
0 50 100 150 200 250 300 350 40
0
0
10
20
30
40
50
60
70
80
90
100
Handover time (ms)
Number of samples
Figure9.8 Field test results on handover time.
Networked Control for a Group of Trains ◾ 193
e number of total transmitted packets is
n
t
T
r
to
=
. e rate of packet drops
introduced by handovers is
P
nn
n
n
ho
ho
to
lim
=
−
→∞
(1)
(9.18)
Based on the eld test results on handover time,
t
ah
ms=
67 . Taking parameters
from Table 9.4, we can obtain the rate of packet drops introduced by handovers
at a train speed of
60
km/h
, P
ho
(60)
≈0.027. For future CBTC applications with
a speed of
200 km/h
, we assume that handovers always introduce the maximum
communication interruption time,
t
mh
ms=
180 . It has
P
ho
(200
)
≈0.1.
9.5 Trains’ Control in CBTC with Packet Drops
In this section, packet drops are rst introduced into the trains’ control system in
CBTC. en, the eects of packet drops on state transmission and estimation are
studied. eir impacts on stability and performances of trains’ control system are
also analyzed. Finally, we propose two novel control schemes to improve the per-
formances of trains’ control system under packet drops.
9.5.1 Currently Used Control Scheme in CBTC Systems
In current CBTC systems, a train uses the status of its own and the previous train
to generate control commands. Each train can directly obtain its status through
onboard sensors without impairment from the train–ground communication.
If the packet containing the status of the preceding train is lost, the following
train uses the real status of its own and the estimated status of the previous train
Table 9.4 Parameters to Analyze the Rate
ofPacket Drops Introduced by Handovers
Parameters Value α
8
d
ap
(m) 200
d
ov
(m) 20
t
mh
(ms) 180
t
ah
(ms) 67
T (ms) 300
v
m
(km/h) 200
194 ◾ Advances in Communications-Based Train Control Systems
to generate control commands. For safety, the following train assumes that the
preceding train keeps still at the last received position.
9.5.2 States Estimation under Packet Drops
e handover of onboard STA may aect either or both the uplink and downlink
transmissions. We assume that
γ
k
i
and
θ
k
j
obey independent and identical Bernoulli
distribution.
PPp
k
i
k
j
γθ
=
()
==
()
=
11
PP p
k
i
k
j
γθ=
()
==
()
=−
00
1
e
j
th train’s estimated status of the
i
th train depends on the results of the uplink
transmission from the
i
th train to ZC and the downlink transmission from ZC to
the
j
th train.
ˆˆ
sss
k
ij
k
ij
k
i
k
ij
k
ij
,, ,
1
,
=
ϑϑ
+
−
(9.19)
ˆˆ
vvv
k
ij
k
ij
k
i
k
ij
k
ij
v
,, ,
1
,
()
=+ +
−
ϑϑ ∆ (9.20)
where:
ˆ
s
k
ij
,
and
ˆ
v
i
ij
,
are the
j
th train estimated location and the speed of the
i
th train,
respectively
∆
v
is the maximum increase of speed a train may experience in one period
T
ϑ
k
i
j
,
indicates that if the
j
th train received the status of the
i
th train
Here, we only consider the case that the following train uses the status of the fore-
going trains,
ij<
.
ϑ
γθ
ϑ
k
ij
k
i
k
j
k
ii
ij
ij
,
,
,
=1,
=
<
≥
ϑϑ
k
ij
k
ij,,
1=−
(9.21)
For the currently adopted control scheme in CBTC systems, the status of the
i
th
train is only sent to the
(1)
i
+
th train to generate control,
ji=+1
.
We dene the
j
th train estimated system state as
ˆˆˆ
XDV
k
j
k
j
k
j
=
′′
(9.22)
Networked Control for a Group of Trains ◾ 195
where:
ˆ
D
k
j
and
ˆ
V
k
j
are
n
-dimensional vectors of deviations of the estimated distances
and velocities
ˆ
ˆˆˆ
ˆ
ˆˆˆ
Dd
dd
Vv vv
k
j
k
j
k
ij
k
nj
k
j
k
j
k
ij
k
n
=
=
δδδ
δδδ
1,
,,
1, ,
,, j
where:
δ
ˆ
d
k
ij
,
and δ
ˆ
v
k
ij
,
are deviations of the
j
th train estimated distance and velocity of
the
i
th train, respectively
If the status of the
i
th train is successfully received by the
j
th train, it will be used to
get
δ
ˆ
d
k
ij
,
and
δ
ˆ
v
k
ij
,
. Otherwise, the
j
th train uses the last estimated distance and speed
of the
i
th train, and considers the worst case to update the deviation of the estimated
distance and velocity.
δ
ˆ
d
k
ij
,
and
δ
ˆ
v
k
ij
,
depend on
γ
k
i−
1
,
γ
k
i
, and
θ
k
j
, and can be expressed as
δϑ ϑδ
ϑϑ
ˆ
ˆ
ˆ
ds
sd
k
ij
k
ij
k
ij
k
ii
k
ij
k
ij
d
k
ij
k
,,1, ,
1
,
,
()
()
=−−+ −
=
−
−
∆∆
iij
k
i
k
ij
k
i
k
ij
k
ij
k
i
d
T
v
T
vv
−
−
−
−
−
−+
+
1,
1,
,
1,
1
2
2
(2)
(
δ
ϑ
δ
ϑ
ϑ
δ
11)
,1,
1
,,
−−
−
−
ϑϑ δϑ
k
ij
k
ij
k
ij
k
ij
d
d
ˆ
∆
(9.23)
δϑδϑδ
ˆˆ
vvv
k
ij
k
ij
k
i
k
ij
k
ij
v
,, ,
1
,
()
=+ +
−
∆ (9.24)
where:
Δ
d
is
the maximum distance a train may travel during one period
T
Δ
v
is
the maximum increase of speed a train may experience during one time slot
We omit the item
ϑϑ
k
ij
k
ij
vT
,
−
1,
, because
vT
is very small compared with Δ
i
. It can be
included in Δ
i
for safety. For algebraic operation, we dene the following diagonal
matrix to indicate the uplink and downlink transmissions:
Θ
k
j
k
j
k
nj
=
()
Diag
ϑϑ
1, ,
Θ
k
j
k
j
k
nj
=
1, ,
Diag
ϑϑ
()
ϒ
k
j
k
j
k
nj
=
()
−
Diag
ϑϑ
0, 1,
196 ◾ Advances in Communications-Based Train Control Systems
ϒ
k
j
k
j
k
nj
=
0, 1,
Diag
ϑϑ
−
()
where:
Θ
k
j
, Θ
k
j
, ϒ
k
j
, and ϒ
k
j
nn
R∈
×
Diag
X
()
generates a matrix, the main diagonal of which are components of
vector X
From Equations 9.23 and 9.24, the
jth train estimated system state can be
expressed as
ˆ
ˆ
ˆˆ
ˆˆ
XCXEXF
XH
W
k
j
k
j
k
k
j
k
k
j
k
j
k
jj
=
+++
−
−
1
1
0
(9.25)
ˆ
ˆ
C
T
E
T
k
j
k
j
k
j
k
j
k
j
nn
k
j
k
j
nn
k
j
k
j
nn
=
−
=
−Θϒ Θϒ
Θ
Θϒ
2
0
,
0
2
0
,
,
,
00
,nn
ˆˆ
F
I
H
k
j
n
k
j
k
j
nn
nn
k
j
k
j
k
j
nn
nn
k
j
=
−
Θϒ
Θ
Θ
Θ
0
0
,=
0
0
,
,
,
,
W
j
dd
vv
0
=− −
[]
′
∆∆
∆∆
For a group of trains’ analysis in CBTC systems, we dene the following matrices:
ˆˆ
ˆ
ˆˆ
ˆ
ˆ
CC
C
EE
E
F
kk
k
n
kk
k
n
k
=
(
)
=
(
)
=
BlkDiag
BlkDiag
Bl
1
1
kkDiag
BlkDiag
ˆˆ
ˆˆ
ˆ
FF
HH
H
WW W
kk
n
kk
k
n
1
1
00
1
0
(
)
=
(
)
=
′′
nn
()
where:
BlkDiag
([ ])
constructs block diagonal matrix from input arguments
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