Novel Communications-Based Train Control System ◾ 125
train control model. In other words, other more sophisticated train control models
can be used in our algorithm as well.
We consider the linear quadratic cost as our performance measure in this
chapter. Specically, the controller described in Equation 7.3 is designed to
minimize
JzkzkQzk zk ukRu k
k
LQ
=
→∞
∑
−
′
−
+
′
() () () () ()
()
(7.4)
where:
zk
Cxk()
()=
o
is the observed train states
zk
()
is the destination train states
e weight matrix
Q
is positive semidenite
R
is positive denite
We can tune the system performance by choosing dierent
Q
and
R
. e rst
part in Equation 7.4 indicates the minimization of state tracking error, and the
second part considers the control magnitude. Note that the square root of this per-
formance measure in Equation 7.4 is equivalent to the
H
2
norm for systems with
perfect feedback [9]. We will refer to the performance measure as the
H
2
norm due
to this equivalence.
e optimal control can be realized only if the MA is updated when the train
moves forward. e service brake started by the ATP subsystem due to commu-
nication latency will make the train travel o the optimized guidance trajectory,
which degrades the system control performance.
7.3.2 Communication Channel Model
In this chapter, we use nite-sate Markov channel (FSMC) models in CBTC sys-
tems. FSMC models have been widely accepted in the literature as an eective
approach to characterize the correlation structure of the fading process, includ-
ing satellite channels [16], indoor channels [17], Rayleigh fading channels [18,19],
Rician fading channels [20], and Nakagami fading channels [21,22]. Considering
FSMC models may enable substantial performance improvement over the schemes
with memoryless channel models [23].
In FSMC models, the range of the received SNR can be partitioned into dis-
crete levels. Each level corresponds to a state in the Markov chain. Assume that
there are
L
states in the model. Let
i
and γ denote the instantaneous channel state
and SNR, respectively. When the channel is in state
i
, the corresponding SNR is
γ
i
. en we have
γγγ
ii
<<
1
+
,
11≤≤ −iL
. e probability of transition from state
i
to state
j
in the Markov model is denoted by P
ij
,
. In real systems, the values of
the above transition probability can be obtained from the history observation of
the CBTC system.
126 ◾ Advances in Communications-Based Train Control Systems
7.4 Communication LatencyinCBTCSystems
with CoMP
In this section, we derive the communication latency in CBTC systems with CoMP.
We rst derive the outage capacity and bit error rate (BER). en, we obtain the
communication latency.
7.4.1 Coordinated Multipoint Transmission and Reception
In cellular networks, interference exists between intercells, which aects spectral
eciency, especially in urban cellular systems. CoMP, which was originally pro-
posed to overcome this limitation, can signicantly improve average spectral e-
ciency and also increase the cell edge and average data rates.
In our system, CoMP is used among the active base stations to improve train–
ground communication performance. In the downlink, the neighboring active
base stations transmit cooperatively, and thus their coverage increases. Inaddition,
diversity gain can be realized when the MT on the train combines the received
signals from multiple base stations. In the uplink, multiple active base stations
receive in coordination, which eectively reduces the requirement of received sig-
nal power at each individual base station. Together, CoMP can provide coverage
for train MT in nearby cells that are under deep fading and improve the system
availability.
In order to achieve the optimal performance, all of the base stations in the net-
work should cooperate with each other in each transmission or reception process.
However, the introduced complexity is not acceptable in real systems. erefore,
standards have specied the maximal number of base stations that may cooper-
ate with each other [24]. In this chapter, we assume that there are only two base
stations in each CoMP cooperation cluster. is is because the base stations are
linearly deployed in CBTC train–ground communication systems. e base sta-
tions in the cluster can be switched on in a combination set
Θ={}
1,2 , and each
combination θ is an element of set
Θ
.
7.4.2 Data Transmission Rate and BER
We rst derive the system data transmission rate. For an arbitrary element
θ∈Θ
,
the uplink sum capacity for the cluster
C()θ
can be calculated as follows [25]:
CH IPHH(, )( )
2
||
θ
θ
=+
+
log
det
o
(7.5)
where:
I
||
θ
denotes a
||||θθ×
identity matrix
P
o
denotes the transmission power of each user terminal
HC
∈
×||
||θθ
denotes the channel matrix
Novel Communications-Based Train Control System ◾ 127
Once a cluster has been selected, the data rate allocation can be performed. Weselect
the modulation and coding scheme for the train MT. e corresponding data rate
can be dened as
Rate
=⋅βθCH
(, )
(7.6)
where:
β can be chosen from the interval [0,1]
Equation 7.5 works only if the channel matrix can be obtained with perfect accu-
racy by the central scheduler, which is dicult to be realized in most wireless sys-
tems. is is because CoMP systems suer from constraints that are imposed by
the backhaul infrastructure, which is required to communicate with the central
scheduler and exchange channel information for joint processing.
In addition to the train control information, passenger information system,
public security monitor system, and station-to-train audio dispatch system are all
integrated into a single backhaul network. In a backhaul network with so much
trac, the communication congestion and latency are unavoidable in the backhaul
network. is is why the CoMP suers from constraints that are imposed by the
backhaul infrastructure of CBTC systems.
Given the Markov channel model and the current observed channel state
H
,
the channel state will transition to
ˆ
H
with probability
PR HH
T
d
(,
)
ˆ
after an approx-
imately backhaul infrastructure latency
T
d
.
PR HH
T
d
(,
)
ˆ
is the channel transition
probability, which can be obtained from eld tests. e system capacity
CH
(,
)
θ
ˆ
under channel state
ˆ
H
can be calculated as described in Equation 7.5.
If the system capacity
CH
(,
)
θ
ˆ
is less than the allocated data rate, there is an
outrage and the resulted data rate is 0. Otherwise the resulted data rate
Rate
is
equal to the allocated data rate
Rate
. erefore, we have
Rate
Rate
=
=
0, (, )<
,
if Rate
Rate others
CHθ
ˆ
(7.7)
e outrage probability
P
ou
t
for the observed channel state
H
and allocated data
rate Rate can be approximately derived as
PH PR HH CH
H
T
d
out
Rate if Rate(, )(,),(,)<=
∑
ˆ
ˆˆ
θ
(7.8)
As a result, the ultimate data rate can be calculated as
Rate
=⋅−
[]
Rate Rate
out
1(
,)
PH
(7.9)
Next, we start to derive the BER. According to [26], when CoMP is not used in our
proposed system, that is, the selected cluster is a single base station, communication
128 ◾ Advances in Communications-Based Train Control Systems
over the Rayleigh channel from source to destination without diversity gain has the
BER performance:
BER
1
1
2
1
1
=−
+
γ
γ
(7.10)
where:
γ
is the received signal SNR, and it can be obtained from the transmission power
and channel state
When neighboring active base stations transmit or receive cooperatively to obtain
diversity gain, that is, the selected cluster has two base stations, we consider the
maximum ratio combining (MRC) of two diversity branches to derive the system
BER as [26]
BER
212
12
2
2
1
1
(, )
1
2
1
1
1(1)
1
(1 )
γγ
γγ
γ
γ
γ
γ
=+
−
+
−
+
/
/
(7.11)
where:
γ
1
and
γ
2
are the received signal SNRs for the two diversity branches, and they
can be obtained from the transmission power and channel state
Similar to the circumstance regarding data transmission rate, when CoMP is
used in the system, given the current observed channel states
H
1
and
H
2
for the
two diversity branches, the channel state will transition to
ˆ
H
1
and
ˆ
H
2
with prob-
ability
PR HV
T
d
(,)
11
ˆ
and PR
HH
T
d
(,
)
22
ˆ
after latency
T
d
, respectively. erefore,
given the current observed channel states
H
1
and H
2
, the real system BER canbe
calculated as
ˆˆ
ˆˆ
ˆ
ˆˆ
BE
RB
ER
2
12
11 11 21122
,=
()
×
()
×
()
∑∑
HH
T
d
T
d
PR HH PR HH
HH
γγ (7.12)
7.4.3 Communication Latency
In order to improve system reliability, an automatic repeat request (ARQ) scheme
is needed. As selective-repeat (SR)-ARQ has been proven to outperform other
basic forms of ARQ schemes (such as stop-and-wait ARQ and go-back-N ARQ)
[27], we use SR-ARQ in this study. According to an SR-ARQ protocol, the aver-
age number of transmissions needed for one packet to be successfully accepted by
the destination is
12(1 )3 (1 )
1
2
⋅+⋅−+⋅ −+=ppppp
p
ccccc
c
(7.13)
Novel Communications-Based Train Control System ◾ 129
where
p
c
is the probability that a packet is successfully received, and it can be
calculated as
p
L
c
packet
BER=(
1)
− (7.14)
where:
BER
is described in Equations 7.10 and 7.12
L
packet
is the number of bits contained in one packet
With the average number of transmissions needed for one packet to be successfully
accepted
1/
p
c
, the communication latency can be approximately calculated as
T
p
LL
Rate
la
c
packet ack
=
1
⋅
+
ˆ
(7.15)
where:
L
ac
k
is the ACK length
ˆ
Ra
te is the data transmission rate
7.5 Control Performance OptimizationinCBTC
Systems with CoMP
As described in Section 7.2.1, communication latency in train–ground commu-
nication systems could make the train travel away from the guidance trajectory
and severely aect CBTC system performance. In order to mitigate the impacts of
communication latency on system performance, we jointly consider CoMP cluster
selection and hando decision issues in CBTC systems. Particularly, the optimal
scheme is composed of two parts. For the rst part, we propose an SMDP-based
optimization algorithm, with the optimization objective to minimize the linear
quadratic cost of train control system. In addition, when the train controller nds
its speed deviates from a preset value due to communication latency, it will recal-
culate the optimal guidance trajectory again. e train will travel along the new
guidance trajectory after the recalculation.
In this section, we rst describe our SMDP-based model. e online value itera-
tion algorithm via stochastic approximation to solve the SMDP model is introduced
next. Finally, the optimal guidance trajectory calculation approach is described.
7.5.1 SMDP-Based CoMP Cluster Selection
and Handoff Decision Model
With the objective to minimize the linear quadratic cost function, we model the
train–ground communication optimization problem as an SMDP. We rst present
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