Networked Control for a Group of Trains ◾ 197
e system state, estimated system state, parameter matrix, closed-loop gain, and
control vector are dened as follows:
YX
XA
AA
YX
XB
kk ky
kk k
n
y
=
′′
′
=
[]
()
=
′
=
′′
,
,
1
BlkDiag
B
ˆˆ
ll kDiag
BlkDiag
BB
GGGU
UU
kkkk
kk
[]
()
=
[]
()
=
′′
[]
′
ˆ
ˆ
,
e system model of a group of trains’ control with variant uplink and downlink
packet drops can be expressed as
YA
YBGY
AY BG CY EY FY HW
kykykk
yk yk kk kk kk k
+
−−
=−
=− +++
1
110
(
ˆ
ˆ
ˆˆ
ˆˆ
ˆˆ
))
ˆ
ˆ
ˆ
UG
Y
kkk
=−
(9.26)
9.5.3 Effects of Packet Drops on the Stability
of the Trains’ Control System
To analyze the eects of packet drops on the stability of the trains’ control system,
an augmented state is dened as
ZY
YYW
kkkk
=
′
′
′
′
−−110
ˆ
(9.27)
e augmented closed-loop system is
ZZ
kk
k+1
=
Φ
(9.28)
Φ
k
yykk ykkykk yk k
nnnn
nn
n
ABGC BGEBGF BGH
I
=
−−
−−
ˆ
ˆˆˆ
22,2 2,
22
,2
000
ˆˆ
ˆˆˆ
CE
FH
I
kk
kk
nn nn nn n
000
2,22,2 2,
22
where:
Φ
k
is a time variable matrix depending on
ϒ
k
j
and
Θ
k
j
198 ◾ Advances in Communications-Based Train Control Systems
If we assume the uplink and downlink transmissions are independent,
Θϒ
k
j
k
j
has
2
n
alternatives. Accordingly, there are
21
21n
−
+
options for
Φ
k
,
ΦΦ
ks
n
s∈= +
{}
−
,1,,
21
21
. e occurrence rate of
Φ
s
is designated as
r
s
; thus,
r
s
s
n
=
−
+
∑
1
=1
2
21
1
.
e system of Equation 9.28 falls under the class of asynchronous dynamical
systems (ADSs) [23]. e stability of ADS is given by the following theorem:
eorem 1: For the asynchronous dynamical system
ZZ
kk
k+
=
1
Φ
where:
ΦΦ
ks
sN
∈=
{}
,1,, is the coecient matrix at the
kth
period.
Φ
s
occurs with
rate
r
s
,
r
s
s
N
=
=
∑
1
1
.
e system is exponentially stable with decay rate of
α
0.05
, if there exists a Lyapunov
function
Vx()
,
RR
n
→
+
:
ββ
1
2
2
2
() xV
xx
≤≤
where
ββ
12
,0>
and
V
satises the following conditions:
◾ ere exists
α
s
sN>=
0, 1, ,
, such that
Vx Vx Vx
kk
sk
()()
(1
)( )
1
0.1
+
−
−≤−α
◾ The
α
s
satisfies
αα
s
r
s
s
N
>>
∏
1
=1
.
e method to prove eorem 1 is similar to that in [22]. We use a smaller expo-
nent of
α
s
in the denition of the Lyapunov function, because it makes the linear
matrix inequality (LMIs) more feasible under high packet drop rate.
9.5.4 Effects of Packet Drops on the Performances
of the Trains’ Control System
It is assumed that deviations of distance, velocity, and applied force without trans-
mission errors are the optimal values. e total cost, cost of divagation of distance,
velocity, and applied force from the optimal values are dened as follows:
JJ
JJ
NdNd vNvf
Nf
=++ααα
δδδ,,,
(9.29)
Networked Control for a Group of Trains ◾ 199
Jd
d
Nd
k
N
i
n
k
i
k
i
,
=1 =1
2
()
δ
δδ=−
∑∑
Jv
v
Nv
k
N
i
n
k
i
k
i
,
=1 =1
2
()
δ
δδ=−
∑∑
Jf
f
Nf
k
N
i
n
k
i
k
i
,
=1 =1
2
()
δ
δδ=−
∑∑
where the optimal distance, velocity, and applied force deviations are designated as
δ
d
k
i
,
δ
v
k
i
, and δ
f
k
i
, respectively. e total cost is dened as J
N
, J
Nd
,δ
is the cost of
divagation of distance deviation from
δ
d
k
i
,
J
Nv
,δ
is the cost of divagation of velocity
deviation from
δ
v
k
i
, and
J
Nf
,δ
is the cost of divagation of applied force deviation
from
δ
f
k
i
. e weight coecients are dened as
α
d
,
α
v
, and
α
f
, respectively.
e power of a train is proportional to the applied force at any given speed. e
applied forces without packet drops are the most energy-saving ones, because they only
overcome the basic resistance of trains in steady state. To reduce energy consumption
of trains’ control in CBTC with packet drops is to decrease unnecessary traction and
brake and keep the applied forces of trains close to the optimal curves. erefore, a
smaller
J
Nf
,δ
means less energy consumption. e line capacity is inversely propor-
tional to the headway. As
J
Nd
,δ
indicates the uctuation of train’s distance from the
one in steady state, a shorter headway is needed to eliminate the interaction between
adjacent trains for a smaller
J
Nd
,δ
, which implies a higher line capacity.
9.5.5 Two Proposed Novel Control Schemes
Previously, we modeled the control of a group of trains in CBTC system as an NCS
with packet drops. e packet drop rate in CBTC systems is formulated and related
to handovers. e impact of packet drops on the estimated system state is analyzed.
After that, packet drops are introduced into the NCS model and cost functions
are dened to analyze the impact of packet drops on the stability and perfor-
mances of the trains’ control system in CBTC. Our previous work has claried
the two unclear questions we mentioned in Section 9.1. e QoS of train–ground
communication should satisfy the stability and performance requirement of trains’
control system. Also, the eect of packet drops on the trains’ control system can be
analyzed through our method. In the next, we will use the NCS model to design
the control scheme. By doing so, the design of trains’ control is integrated with the
train–ground communication to improve the performances of trains’ control under
packet drops in CBTC systems.
200 ◾ Advances in Communications-Based Train Control Systems
For the current control scheme, only the status of the previous train is used for
the following train to create control command. In this section, we propose two
control schemes to improve the performances of trains’ control system under packet
drops. As ZC has status of all the running trains, we propose to use all or some
of the forgoing trains’ status for the following train to generate control command.
erefore, it can respond more rapidly to the status change of foregoing trains. Due
to packet drops, the following train may receive the real or estimated status of the
forgoing trains; it can select to use all or some of the status to keep its performances
close to the optimal values. e proposed schemes will be veried in Section 9.6 by
simulation results. e proposed schemes are as follows:
◾ From the beginning of each period, ZC waits to receive the status of all the
trains before sending LMAs. ZC’s waiting time is less than a predened
uplink reception interval. e equivalent uplink transmission delay is uni-
form among trains.
◾ If a train’s status is successfully received by ZC, it will be inserted into the
LMA for the following trains. Otherwise, the estimated status of the train
will be used.
◾ ZC sends the status or estimated status of all the foregoing trains together in
one packet to a specic train to generate control commands.
◾ e controller waits for a predened duration from the beginning of each
period to generate and output control commands to the actuators. e length
of the duration
τ
is no less than
2τ
m
. Here,
τ
m
is the delay introduced by the
maximum number of retransmissions.
◾ If the LMA containing the status of all the foregoing trains is available when the
controller generates and sends controls to the actuator, it will be used to calculate
the control commands. Otherwise, an estimated LMA will be used instead.
◾ Based on the status or estimated status of the foregoing trains, each train
selects the closed-loop gain to minimize the uctuation of applied force or
distance deviations around the optimal values. Each train selects its closed-
loop gain by the following two criteria:
– Minimized force criterion. To minimize its cost of divagation of the
applied force deviations from the optimal values.
G
s
is the combination
of the closed-loop gains of all the trains.
min
i
n
k
i
k
i
ff
=
∑
−
1
2
()δδ
(9.30)
– Minimized distance criterion. To minimize its cost of divagation of the
distance deviations from the optimal values.
min
i
n
k
i
k
i
dd
=1
2
()
∑
−
δδ
(9.31)
Networked Control for a Group of Trains ◾ 201
G
k
of
ˆ
G
k
in (9.26) is time variable. It is a combination of closed-loop gains of all
the trains.
G
k
has multiple options,
GGsM
ks
∈=
{, 1,
,}
.
9.6 Field Test and Simulation Results
In this section, we rst give eld test results on the rate of packet drops introduced
by handovers in a real CBTC system. en, we design a control system of three
trains, and the two novel control schemes are evaluated. Finally, simulation results
are presented, and the eects of packet drops on the stability and performances of
trains’ control system are discussed.
9.6.1 Field Test Results on the Packet Drop Rate
We measure the packet drop rate on Beijing Yizhuang Subway Line. Yizhuang Line
is a typical CBTC line that began its commercial operation at the end of 2010. It is
23 km
in length. One hundred and eight APs are installed in the main line.
We use self-developed software, mentioned in Section 9.4, to obtain the packet
drop rate introduced by handover. Field test results at
60 km/h
speed are listed in
Table 9.5.
In order to get enough samples, we transmit packets with a much shorter period,
T =
16 ms. e packet drop rate is between 0.003 and 0.005, which is close to the
analytical result we can obtain using the method given in Section 9.4,
T =
16 ms
,
P
ho
= 0.0074
. e eld test results prove the validity of our proposed analytical
model of packet drop rate in CBTC. en, we use the analytical model to get the
packet drop rate at a very low speed (
30 km/h
), P
ho
(30) ≈ 0.01, and at a much higher
speed (
200 km/h
), P
ho
(200) ≈ 0.1. According to the two drop rates, random packet
drops will be generated to analyze the performances of dierent control schemes in
the simulations described in Sections 9.6.2 and 9.6.3.
Table 9.5 Field Test Results on the Rate of Packet Drops Introduced
byHandovers
Test Duration (s) Dropped Packets Total Packets Drop Rate α
8
2448 517 145975 0.003542
2378 466 133615 0.003488
2748 510 162783 0.003133
4749 1098 222766 0.004929
1719 399 102208 0.003904
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