Networked Control for a Group of Trains ◾ 207
0 200 400 600800
1000
0 200 400 600800
1000
0 200 400 600800
1000
−1.0
0.0
1.0
2.0
2.5
−0.5
0.5
1.5
2.5
3.5
−2.0
−1.0
0.0
1.0
2.0
2.5
Time slot (k)
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
Figure 9.13 Performances of T3 using the Sv scheme,
TP
k
i
======
0.3s,( 0)
γγ
P
k
i
(0
)0.1
θθ==== .
0 200 400 600 800
1000
0 200 400 600800
1000
0 200 400 600800
1000
−1.0
0.0
1.0
2.0
2.5
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
−0.5
0.5
1.5
2.5
3.5
−2.0
−1.0
0.0
1.0
2.0
2.5
Time slot (k)
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
Figure9.14 Performances of T3 using the Lv_d scheme,
TP
k
i
=
=====
0.3s,( 0)γγ
P
k
i
(0
)0.1
θθ==== .
208 ◾ Advances in Communications-Based Train Control Systems
e performances of T
3
using dierent control schemes are illustrated in
Figure9.16. It is shown that the Lv_f scheme has a much better applied force and
velocity performances at the cost of the distance performance compared with the
Sv scheme, which means less energy consumption and better riding comfortability
at the expense of line capacity. e Lv_d scheme outperforms the Sv scheme in all
kinds of performance measures.
e comparisons of the total cost between the proposed Lv_f/Lv_d scheme and
the Sv scheme with dierent coecients are given in Figure 9.17. It can be found
that both the Lv_f and Lv_d schemes outperform the Sv scheme.
9.7 Conclusion
CBTC systems use wireless networks to transmit trains’ status and control data. In
this chapter, the issues related to trains’ control in CBTC systems with lossy wire-
less networks were studied. e system was modeled as an NCS with packet drops
in both uplink and downlink transmissions. We studied packet drops introduced
by random transmission errors and handovers. e packet drop rate was formulated
and related to handovers in CBTC. e analytical results on the rate of packet
0 200 400 600800
1000
0 200 400 600800
1000
0 200 400 600800
1000
−1.0
0.0
1.0
2.0
2.5
T3 w
T3 w/o
T3 w
T3 w/o
T3 w
T3 w/o
−0.5
0.5
1.5
2.5
3.5
−2.0
−1.0
0.0
1.0
2.0
2.5
Time slot (k)
δd
k
i
(m)
δv
k
i
(m/s)
δf
k
i
(m/s
2
)
Figure 9.15 Performances of T3 using the Lv_f scheme,
TP
k
i
=
=====
0.3s,( 0)γγ
P
k
i
(0
)0.1
θθ==== .
Networked Control for a Group of Trains ◾ 209
0 500 1000
0
20
40
60
80
100
120
Time slot (k)
0500 1000
Time slot (k)
0500
1000
Time slot (k)
Cost of divagation of distance deviation ( J
N,δd
)
Cost of divagation of velocity deviation (
J
N,δv
)
Cost of deviation of applied force deviation (
J
N,δf
)
Lv_f, 0.1
Sv, 0.1
Lv_d, 0.1
Lv_f, 0.01
Sv, 0.01
Lv_d, 0.01
0
10
20
30
40
50
60
70
80
90
100
Sv, 0.1
Lv_d, 0.1
Lv_f, 0.1
Sv, 0.01
Lv_f, 0.01
Lv_d, 0.01
0
50
100
150
200
250
Sv, 0.1
Lv_d, 0.1
Lv_f, 0.1
Sv, 0.01
Lv_d, 0.01
Lv_f, 0.01
Figure 9.16 Cost of T3 using the Sv, Lv_f, and Lv_d schemes, respectively,
T ==
0.
3s,
PP
k
i
k
i
(0)( 0) 0.1,
0.01
γγθθ======== .
0
500
1000
1500
2000
2500
3000
3500
4000
Lv_f and Sv scheme
Sv, α
f
= 10
Sv, α
f
= 1
Sv, α
f
= 5
Lv_f, α
f
= 10
Lv_f, α
f
= 1
Lv_f, α
f
= 5
Sv, α
d
= 10
Sv, α
d
= 1
Sv, α
d
= 5
Lv_d, α
d
= 10
Lv_d, α
d
= 1
Lv_d, α
d
= 5
0 200 400 600 800 100
0
0
200
400
600
800
1000
1200
1400
1600
1800
Time slot (k)
0 200 400 600 800 1000
Time slot (k)
Cost ( J
N
)
Cost ( J
N
)
Lv_d and Sv scheme
Figure 9.17 Total cost of T3 using the Sv, Lv_f, and Lv_d schemes, respectively,
T == 0.3s
,
PP
k
i
k
i
(0)( 0) 0.
1
γγθθ======== .
210 ◾ Advances in Communications-Based Train Control Systems
drops introduced by handover were proved to be in line with the eld test results
obtained from business operating CBTC lines. e eects of packet drops on the
stability and performances of trains’ control systems were analyzed.
We proposed two novel schemes to improve performances of trains’ control
system under packet drops. Each train uses dierent combinations of the status or
estimated status of foregoing trains to generate control commands through select-
ing dierent closed-loop gains to minimize divagation of the applied force or dis-
tance deviations from the optimal values. Extensive simulation results of systems
with currently adopted and proposed schemes under dierent packet drop rates
were presented. e results showed that the proposed schemes outperform the cur-
rently used scheme in CBTC systems. ey can provide less energy consumption,
better riding comfortability, and higher line capacity.
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