Mathematical Problems in Engineering
thepredictedvalueofthestateiscalculated.Aergetting
the prediction error by comparing predicted value of control
signal with the true value at time +1,thestatevalue
isupdatedspontaneously.Inaddition,totracethestate
trajectory of system, we use certain parameters of the object
model () to predict and calculate the referential value of
control signal. e prediction model can be described as
follows:
(
+1
)
=
(
)
+
(
)
,
(
)
=
(
)
,
()
where , ,andrefer to ()and()and
is the referential
value of control signal.
e predicted error of control signal produced by the
model is shown as follows:
(
)
=
(
)
−
(
)
.
()
2.3. e Description of Scheduling Policy. With the rapid
development of computer technology, sensors sampling fre-
quency and the processing speed of the controller are being
improved continually; network conict is becoming more
and more serious at actuators side because of the limited
channels of network during the transmission of informa-
tion. So it is important to introduce reasonable scheduling
policy to reduce the network conict at actuator node.
Here we introduce the restrained condition of transmission
as |
𝑖
()| ≤
𝑖
(
𝑖
represents scheduling threshold, =
[1,2,...,],andis the dimension of the control signal).
Controller will not send the control signal
𝑖
() taken as
unimportant information to actuator and the actuator keeps
thevalueofcontrolsignalattime−1if the restrained
condition is satised, which helps to reduce the transmission
frequency of unimportant information at actuator node.
According to the description above, piecewise function as
followsisintroduced:
𝑖
(
)
=
𝑖
(
−1
)
,
𝑖
(
)
≤
𝑖
,
𝑖
(
)
,
𝑖
(
)
>
𝑖
.
()
Moreover, we introduce
𝑖
(
)
=
0,
𝑖
(
)
≤
𝑖
,
1,
𝑖
(
)
>
𝑖
,
=
[
1,2,...,
]
, ()
where
𝑖
=0represents that
𝑖
should not be transmitted,
while
𝑖
=1represents that
𝑖
should be transmitted. We
dene Φ=diag(
1
,
2
,...,
𝑚
).Obviously,equality()is
equivalent to
(
)
=Φ
𝑗
,
𝑚×𝑚
−Φ
𝑗
𝑇
(
)
,
𝑇
(
−1
)
𝑇
.
()
Remark 2. According to the description about Φabove, the
total number of cases that Φcould appear should be 2
𝑚
in the
wholeschedulingprocess;thatistosay,
Φ=Φ
𝑗
∈Φ
1
,Φ
2
,...,Φ
2
𝑚
, =1,2,...,2
𝑚
.
()
Remark 3. Obviously, dierent from the previous method,
such as [], in which referential value of scheduling policy
based on deadline of message is a xed value. e referential
value in Section . of scheduling policy based on predicted
error obtained by using certain parameters of the object
model () to predict the value of system is varied with the
change of the system state trajectory. In this way, a bigger
chanceforlossesofunimportantinformationinNCSis
oered than the previous method as []; that is to say,
scheduling policy based on predicted error is more eective
to avoid the network conict and save energy of nodes.
2.4. Augmented System Model of NCS under Scheduling
Policy Based on Predicted Error. Based on the description of
equalities ()and()andRemark , it can be obtained that
(
)
=Φ
𝑗
(
)
+−Φ
𝑗
(
−1
)
. ()
e augmented matrix is dened as
=
𝑇
(
)
,
𝑇
(
−1
)
,...,
𝑇
(
−−1
)
,
𝑇
(
−
)
,
𝑇
(
−−1
)
𝑇
.
()
erefore, the augmented NCS model becomes
(
+1
)
=
_
(
)
+
_
(
)
,
(
)
=
_
(
)
,
()
where
_
=
0⋅⋅⋅00Φ
𝑗
(−Φ
𝑗
)
0⋅⋅⋅00 0 0
0⋅⋅⋅00 0 0
.
.
.
.
.
. d
.
.
.
.
.
.
.
.
.
.
.
.
00⋅⋅⋅0 0 0
00⋅⋅⋅0 0 0
00⋅⋅⋅00Φ
𝑗
−Φ
𝑗
(𝑑+2)×(𝑑+2)
,
_
=
𝑇
0⋅⋅⋅0000
𝑇
,
_
=
0⋅⋅⋅0000
,
=1,2,...,2
𝑚
.
()
Obviously, () is a switching model; the number of switching
modes is 2
𝑚
.