202 ◾ Advances in Communications-Based Train Control Systems
9.6.2 Design of the Closed-Loop Control Systems
A control system of three trains is designed. Train
1
(T1) is the leading train. Train
3
(T3) is the last train. Train
2
(T2) is in the middle. e main parameters used in
our simulations are listed in Table 9.6.
We design the closed-loop control systems based on the following control
schemes.
“Short view” scheme (Sv). e following train uses the speed and velocity infor-
mation of its own and its previous train to generate control commands.
“Long view and minimized distance” scheme (Lv_d). Each train uses dierent
combinations of the status or estimated status of foregoing trains to generate con-
trol commands by selecting its closed-loop gain to minimize the divagation of its
distance deviations from the optimal values.
“Long view and minimized force” scheme (Lv_f). Each train selects its closed-loop
gain to minimize the divagation of its applied force deviations from the optimal
values.
e Sv scheme is based on the currently adopted scheme in CBTC systems. e
Lv_d and Lv_f schemes are based on our proposed methods.
e system of Equation 9.28 is controllable and observable. e poles place-
ment method is used to design the closed-loop gains [40,41]. e leading train has
only one closed-loop gain.
G
1,1
0.0820000.6845
00
=−
[]
Train
2
has two alternatives.
G
2,1
0 1.2026 000.9016 0=−
[]
G
2,2
0.0449 0.4326 0 0.07157 0.7707 0=− −−
[]
Table 9.6 Simulation Parameters
Parameters Value α
8
Length of train (m) 118
Mass of train 1
Optimal speed (km/h) 90
Maximum normal brake rate (m/s
2
) 1.5
Emergency brake rate (m/s
2
) 2
Maximum acceleration rate (m/s
2
) 2
Maximum input distance error (m) 15
Networked Control for a Group of Trains ◾ 203
Train
3
has four options.
G
3,1
00 1.5023000.8032=−
[]
G
3,2
0.0382 0.0380 0.3850=−
[
−−
−−
]
0.07331 0.0783 0.7048
G
3,3
0.0460 0 0.4629 0.0733300.7048=− −−
[]
G
3,4
0 0.1035 1.0360 0 0.07333 0.7048=− −−
[]
G
s
are combinations of closed-loop gains of all the trains. For the control system of
three trains, there are eight selections.
GGGG
GGGG
GGGG
1
1,12,1 3,1
2
1,12,1 3,2
3
1,12,2
=
=
=
′′′
′′ ′
′′
33,4
′
For comparison of performances, control systems that use
G
1
–
G
8
as the closed-loop
gains have the same distance, velocity, and applied force deviations in steady state.
9.6.3 Simulation Results of Trains’ Control
System Impacted by Packet Drops
e stability of trains’ control system in current CBTC systems using the short
view strategy is analyzed using eorem 1 under dierent packet drop rates. With
regard to the short view scheme,
Φ
k
in Equation 9.28 depends on
Θ
k
.
Θ
kkkk
kk
n
k
n
=
−
Diag
γθ γθ γθ
01 12 1
For the control system of three trains,
Θ
k
has eight alternatives. Accordingly, there
are eight options for
Φ
k
,
ΦΦ
ks
s
∈
,=1,
,8
.
r
s
is the occurrence rate of
Φ
s
; thus,
r
s
s
=
=
∑
1
1
8
. For simplicity, we assume that the uplink and downlink packet trans-
missions are independent and identically Bernoulli distributed random variables.
rpp
s
mm
=−−
−
−
1(1) (1 )
2
3
2
where:
p
is the packet drop rate
m
is the number of trains that have successfully received LMA
204 ◾ Advances in Communications-Based Train Control Systems
e admissible set of
α
s
is
0.1 1, 1,2, ,7
12
.
8
≤≤ =≤≤αα
s
s
and
en we use the LMI toolbox of MATLAB to verify the feasibility of LMIs in
eorem1. Genetic algorithm (GA) is used to nd the maximum decay rate. e
tness function of GA is to nd the maximum
∏
ss
r
s
=1
8
α
throughout the admissible
set of
α
s
that makes all the LMIs feasible.
e exponential stability of system with transmission period
T 0.3s=
under
packet drop rate
0.1
–
0.7
is veried. e maximum decay rate
α
0.05
and the
corresponding
α
s
are given in Table 9.7. It can be seen that the current train’s
control system in CBTC keeps stable even at a very high packet drop rate,
PP
k
i
k
i
(0)( 0)
0.7
γθ== == . As the drop rate increases, the decay rate of the system
decreases.
e performances of T
1
and T
2
using the Sv scheme under packet drop
rate
PP
k
i
k
i
(0)( 0)
0.1
γθ== == are illustrated in Figure 9.9. e statuses of T
1
and T
2
are used as the input to T
3
’s controller. e distance, velocity, and applied
force deviations of T
3
using Sv, Lv_d, or Lv_f scheme under a low packet drop rate
PP
k
i
k
i
(0)( 0)
0.01
γθ== == are depicted in Figures 9.10 through 9.12, respectively.
e deviations of T
3
under a much higher packet drop rate
PP
k
i
k
i
(0)( 0)
0.1
γθ== ==
are given in Figures 9.13 through 9.15.
It can be found that the Lv_f scheme has the smallest uctuation of the applied
force deviations and the smoothest velocity deviations around the optimal values
with a little bit higher distance deviation compared with the Sv and Lv_d schemes.
e Lv_d scheme outperforms the Sv scheme in distance, velocity, and applied force
deviations.
Table 9.7 LMI Feasibility of the System Using Current Control Scheme
(T= 0.3 s)
Drop
Rate
Decay
Rate α
1
α
2
α
3
α
4
α
5
α
6
α
7
α
8
0.01 1.0331 0.9842 0.9997 0.9927 0.9999 0.9848 0.9993 0.9997 1.9973
0.1 1.0185 0.9938 0.9944 0.9949 0.9999 0.9972 0.9999 0.9972 1.9979
0.2 1.0091 0.9958 0.9999 0.9979 0.9980 0.9968 0.9999 0.9997 1.9984
0.3 1.0040 0.9972 0.9961 0.9998 0.9998 0.9965 0.9998 0.9998 1.9968
0.4 1.0015 0.9997 0.9979 0.9987 0.9998 0.9983 0.9990 0.9999 1.9717
0.5 1.0005 0.9984 0.9997 0.9996 0.9928 0.9995 0.9971 0.9946 1.9927
0.6 1.00008 0.9985 0.9985 0.9999 0.9999 0.9987 0.9955 0.9999 1.9981
0.7 1.00001 0.9999 0.9997 0.9988 0.9933 0.9999 0.9977 0.9986 1.9956
Networked Control for a Group of Trains ◾ 205
−2
0
2
4
6
8
10
12
14
16
−0.5
0.0
0.5
1.0
1.5
2.0
0 500
1000
−1.0
−0.5
0.0
0.5
1.0
1.5
Time slot (k)
0500 1000
Time slot (k)
0 500 1000
Time slot (k)
T1
T2
T1
T2
T1
T2
Acceleration rate, δf
k
i
(m/s
2
)
Velocity deviations, δv
k
i
(m/s)
Distance deviations, δd
k
i
(m)
Figure 9.9 Performances of T1 and T2 using the Sv scheme,
T == 0.3s
,
PP
k
i
k
i
(0)( 0)
0.01
γγθθ======== .
−1
0
1
2
0
1
2
0 100 200 300 400 500 600 700 800 900
1000
0 100 200 300 400 500600 700800 90
01
00
0
0 100 200 300 400 500600 700800 90
01
00
0
−1
0
1
Time slot (k)
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
Figure 9.10 Performances of T3 using the Sv scheme,
TP
k
i
=
=====
0.3s,( 0)γγ
P
k
i
(0
)0
.0
1
θθ==== .
206 ◾ Advances in Communications-Based Train Control Systems
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
0 200 400 600800
1000
0 200 400 600800
1000
0 200 400 600800
1000
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
Time slot (k)
0
1
2
−1
0
1
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
Figure9.11 Performances of T3 using the Lv_d scheme,
TP
k
i
=
====
=0.3s,( 0)γγ
P
k
i
(0
)0
.0
1
θθ==== .
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
0 200 400 600800
1000
0 200 400 600800
1000
0 200 400 600800
1000
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
Time slot (k)
0
1
2
−1
0
1
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
Figure 9.12 Performances of T3 using the Lv_f scheme,
TP
k
i
=
=====
0.3s,( 0)γγ
P
k
i
(0)0.0
1
θθ==== .
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