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same as LMA after an elapse of
M
−
1 time sample, a nice convergence result
can be expected from RLMA. Hence
ε
P
≥
J
1
≥
J
2
≥···≥
J
p
(11)
According to (6), RMPC is convergent.
4 Case Study
The RMPC described above is applied to the Permanent Magnet Synchronous
Motor (PMSM) current control to verify its effectiveness. The dynamics of the
PMSM are modeled in the
dq
reference frame [14]:
i
d
(
k
+1)=
i
d
(
k
)+
L
d
(
Ri
d
(
k
)+
n
p
ωL
q
i
q
(
k
)) +
L
d
u
d
(
k
)
−
(12)
i
q
(
k
+1)=
i
q
(
k
)+
L
q
(
n
p
ωφ
)+
L
q
u
q
(
k
)
−
Ri
q
(
k
)
−
n
p
ωL
d
i
d
(
k
)
−
where
i
d
,
i
q
are
d
and
q
components of the stator current,
u
d
,
u
q
are
d
and
q
components of the stator voltage. The parameters of (12) are listed in Table.1.
It should be noted that
T
in (12) is the sample interval, which is required to be
around 0
.
2 ms to obtain good performance for PMSM[15].
Tabl e 1.
Specifications of the PMSM
Symbols
Values
Units
L
d
0
.
000334
H
L
q
0
.
000334
H
R
0
.
4578
Ω
n
p
4
φ
0
.
171
Wb
ω
20
rad/s
T
0
.
2
ms
The RMPC controller is designed with cost function as shown in (2), where
y
(
k
+
i
)=[
i
d
(
k
+
i
)
i
q
(
k
+
i
)]
T
, initial value
y
0
(
k
+
i
)=[3 0]
T
,
y
r
(
k
+
i
)=
[10 5]
T
. An input sequence is acceptable if
J<
2.Designparameter
Q
is set
as diag[1 1], learning filter
L
in ILC is diag[10 10]. The initial value of
η
is 1,
but
η
is updated in every predict step to guarantee that every step is effective
in decreasing
J
.
α
in (3) is 0
.
6.The simulation is based on Matlab/Simulink.
When
P
is 20 and
M
is 10, the values of the cost function in a prediction
horizon of RMPC are shown in Fig.2. It can be seen from Fig.2 that, all the
CVs are within limits. However,
J
converged before 10, but diverged after that.
It is because the RMPC only utilize the information of present step. All the
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