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3M inR su s
Theorem 1.
Consider the time-delay discrete-time systems (2) is asymptoti-
cally stable, if there is an appropriate dimension of the positive definite symmet-
ric matrix
P>
0
and
S>
0
, at the same time satisfy the following LMI
⎡
⎤
−
P
1
+
P
2
∗∗
0
−P
2
∗
P
1
G
0
P
1
G
1
−P
1
⎣
⎦
<
0
(3)
Proof.
Define the Lyapunov function as
k−
1
V
(
x
(
k
)) =
x
T
(
k
)
Px
(
k
)+
x
T
(
i
)
Sx
(
i
)
t
=
k−h
The full differential form of
ΔV
(
x
(
k
)), along the trajectories of
V
(
k
)isgivenby
ΔV
(
x
(
k
)) =
V
(
x
(
k
+1))
−V
(
x
(
k
))
=
x
T
(
k
+1)
P
1
x
(
k
+1)
x
T
(
k
)
P
1
x
(
k
)
−
k
k−
1
x
T
(
i
)
P
2
x
(
i
)
x
T
(
i
)
P
2
x
(
i
)
+
−
i
=
k
+1
−h
i
=
k−h
h
)
Ω
x
(
k
)
x
(
k
=
x
T
(
k
)
x
T
(
k
−
−
h
)
where
Ω
=
G
0
PG
0
−
P
1
+
P
2
G
0
P
1
G
1
G
1
P
1
G
0
G
1
P
1
G
1
−
P
2
Hence when
ΔV <
0, we can obtain
Ω<
0, It can be proved.
Theorem 2.
Consider the uncertain time-delay discrete-time systems (2) with
additive controller gain perturbations (4), if there is an appropriate dimension
of the positive definite symmetric matrix
X>
0
and
W>
0
and constant value
ε
i
>
0
,
(
i
=0
,
1
,
2)
, at the same time satisfy the following LMI
⎡
⎣
⎤
⎦
MG
0
X
+
H
0
YG
1
X
0
0
0
0
X
T
E
0
X
T
E
2
∗−
X
+
W
0
∗
∗
−
W
0
0
XE
1
<
0
(4)
∗
∗
∗ −
ε
0
I
0
0
∗
∗
∗ ∗ −
ε
2
I
0
∗
∗
∗ ∗
∗ −
ε
1
I
then time-delay closed-loop system is asymptotically stable, at this time, there is
passive controller in the uncertain time-delay system
K
=
YX
−
1
.
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