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u
j
+1
(
k
)=
u
j
(
k
)+
Le
j
(
k
+1)
(4)
where subscript j represent the iteration index. is a parameter matrix in P-type
ILC.
It should be noted that, (4) works under unconstrained condition. Although
constrained ILC [11, 12] can be used here conveniently, it does not need to
do so for three reasons. Firstly, it has been proved that model (1) using input
derived from (4)can converge to the setpoint [10], so it can satisfy the output
constraint. Secondly, choosing a suitable
α
in reference trajectory (3) can avoid
too aggressive MV moves. Thirdly, unconstrained ILC need less computational
time.
As the first MV has made the predictive trajectory close to the final solution,
it is relatively simple to obtain other MVs
u
(
k
+
i
)
i
=1
...M
−
1usingRLMA[6]:
u
(
k
+
i
+1)=
u
(
k
+
i
)+
N
(
k
+
i
)
Ψ
(
k
+
i
)
e
(
k
+
i
)
N
(
k
+
i
)=
N
(
k
+
i −
1)
− N
(
k
+
i −
1)
Ψ
∗
(
k
+
i
)
S
(
k
+
i
)
−
1
Ψ
∗
(
k
+
i
)
T
N
(
k
+
i −
1)
S
(
k
+
i
)=
Ψ
∗
(
k
+
i
)
T
N
(
k
+
i −
1)
(
−
1)
Ψ
∗
(
k
+
i
)+
Λ
∗
(
k
+
i
)
(5)
∂u
(
k
)
,Ψ
∗
(
k
+
i
)=
Ψ
(
k
+
i
)
T
T
,Λ
∗
(
k
+
i
)
−
1
=
10
0
η
∂J
Ψ
(
k
)=
0
···
1
···
0
Ψ
∗
(
k
+
i
)isa
M
by 2 matrix, the second column of
Ψ
∗
(
k
+
i
) is designed to
deal with the damped term in LMA. The 1 in the second column of
Ψ
∗
(
k
+
i
)is
placed at (
tmod
(
M
−
1)) + 1.
η
is the damping factor. After an elapse of
M
−
1
time units, (5) is virtually the same as LMA.
The flowchart of RMPC in each predictive horizon is shown in Fig.1.
Suppose the dimension of
u
(
k
+
i
)is
d
. It can be observed from (4) and (5) that
RMPC requires the computation complexity of Ø(
d
2
) to solve an optimization
problem recursively. In contrast, the computation complexity of conventional
Read the initial
state
Improve the first
state using ILC
Obtain the latter
terms using RLMA
The value of cost function
minimum enough?
Output the first term
of input sequence
Fig. 1.
Flowchart of RMPC in one predictive horizon
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