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Therefore, we obtain
2
lim
Δ
U
k
+
1
R
k
→∞
(
)
(
)
()
(
(
)
(
)
)
()
(17)
≤
lim
e
U
+
e
U
⋅
e
U
−
e
U
k
ILC k
,
+
1
k
ILC k
,
+
1
Q
Q
k
→∞
Q
Q
(
)
()
(
)
≤⋅
M lim
e
U
−
e
U
k
ILC k
,
+
1
Q
k
→∞
Q
()
(
)
where
M
is the upper bound of
e
U
+
e
U
, namely
k
ILC k
,
+
1
Q
Q
(
)
()
e
U
+
e
U
≤
M
.
k
ILC k
,
+
1
Q
Q
Thus, we have the conclusion that
convergences to 0, namely
Δ
U
conver-
Δ
U
k
k
R
gences to 0. This completes the proof. Q.E.D.
3.2
Tracking Performance Analysis
2
()
*
is defined as the minimum of
⋅
Q
(the ideal value of
*
is zero). Moti-
M
e
M
vated by our previous work [21], the definition of bounded tracking and zero tracking
of the integrated learning optimization control system are defined as
()
Definition 1: Bounded-tracking.
If there exists a
for every
ε >
0
and
δδε
=
>
0
2
( )
such that the inequality
e
U
−<
M
*
ε
holds when
U=U
+
U
k
SNPCk
,
ILCk
,
k
Q
for every
kk
>
.
r
+
<
δ
1
0
0
δ >
0
Definition 2: Zero-tracking.
If it is bounded-tracking and there exists
and
2
( )
*
U=U
+
U
such that the equality
lim
e
U
−=
M
0
holds when
k
+
1
SNPCk
,
+
1
ILCk
,
+
1
k
+
1
Q
k
→∞
.
r
+
<
δ
1
0
( )
k
Theorem 2
The tracking error
e
U
of the proposed integrated optimization prob-
lem is bounded-tracking for arbitrary initial control profiles
.
U=U
+
U
k
SNPCk
,
ILCk
,
0
0
0
2
()
Moreover, if the function of
U=U U
) is derivable and the
optimization solution is not in the boundary, it is zero-tracking for arbitrary initial
control profiles
eU
Q
(
+
k
k
SNPCk
,
ILCk
,
.
U=U
+
U
k
SNPCk
,
ILCk
,
0
0
0
()
Proof:
It is easy to know that for every
ε > , there exists
0
such that the
δδε
=
>
0
(
)
2
*
optimal solution in
k
+ -th batch satisfies
1
e
U
−<
M
ε
when
.
r
+
<
δ
0
ILC k
,
+
1
1
0
Q
0
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