2. Main Results
Using the partitioning technique, we have the following main results.
Theorem 1.
Given nonnegative integer scalars
τ
M
and
τ
m
, the system
(1)
x
(
k
+
1
)
=
A
x
(
k
)
+
A
h
x
(
k
-
h
(
k
)
)
,
x
(
k
)
=
φ
(
k
)
,
k
∈
[
-
τ
M
,
0
]
is asymptotically stable, where
x
(
k
)
∈
R
n
is the state vector;
A
,
A
h
are constant matrices with appropriate dimensions;
φ
(
k
)
is the initial condition of
k
∈
[
-
τ
M
,
0
]
; for any time varying delay
h
(
k
)
satisfying
0
≤
τ
m
≤
h
(
k
)
≤
τ
M
, there exist real matrices
P
=
P
T
>
0
,
Q
i
=
Q
i
T
>
0
(
i
=
1,2
,
3,4
)
,
R
=
R
T
>
0
,
Z
=
Z
T
>
0
, and any appropriately dimensioned matrices
N
=
[
N
1
T
,
N
2
T
]
T
,
M
=
[
M
1
T
,
M
2
T
]
T
,
S
=
[
S
1
T
,
S
2
T
]
T
, and
T
=
[
T
1
T
,
T
2
T
]
T
such that the following two sets of LMIs hold.
Case I (
0
≤
τ
m
≤
h
(
k
)
≤
τ
avg
).
Consider
(2)
Θ
1
=
[
X
11
X
12
N
1
X
12
T
X
22
N
2
N
1
T
N
2
T
Z
]
≥
0
,
Θ
2
=
[
X
11
+
Y
11
X
12
+
Y
12
M
1
X
12
+
Y
12
T
X
22
+
Y
22
M
2
M
1
T
M
2
T
R
+
Z
]
≥
0
,
Θ
3
=
[
X
11
+
Y
11
X
12
+
Y
12
S
1
X
12
T
+
Y
12
T
X
22
+
Y
22
S
2
S
1
T
S
2
T
R
+
Z
]
≥
0
,
Θ
4
=
[
X
11
X
12
T
1
X
12
T
X
22
T
2
T
1
T
T
2
T
Z
]
≥
0
,
Φ
=
[
Φ
11
Φ
12
-
N
1
+
M
1
-
T
1
-
S
1
+
T
1
Φ
12
T
Φ
22
-
N
2
+
M
2
-
T
2
-
S
2
+
T
2
-
N
1
T
+
M
1
T
-
N
2
T
+
M
2
T
-
Q
2
0
0
-
T
1
T
-
T
2
T
0
-
Q
3
0
-
S
1
T
+
T
1
T
-
S
2
T
+
T
2
T
0
0
-
Q
4
]
<
0
,
where
(3)
Φ
11
=
-
P
+
Q
1
+
Q
2
+
Q
3
+
Q
4
+
(
τ
M
-
τ
m
)
Q
1
+
N
1
+
N
1
T
+
τ
M
X
11
+
δ
1
Y
11
+
A
T
P
A
+
(
A
-
I
)
T
Θ
(
A
-
I
)
,
Φ
12
=
N
2
T
-
M
1
+
S
1
+
τ
M
X
12
+
δ
1
Y
12
+
A
T
P
A
h
+
(
A
-
I
)
T
Θ
A
h
,
Φ
22
=
-
M
2
-
M
2
T
+
S
2
+
S
2
T
-
Q
1
+
τ
M
X
22
+
δ
1
Y
22
+
A
h
T
P
A
h
+
A
h
T
Θ
A
h
,
Θ
=
δ
1
R
+
τ
M
Z
.
Case II (
τ
avg
≤
h
(
k
)
≤
τ
M
).
Consider
(4)
Θ
¯
1
=
[
X
11
X
12
N
1
X
12
T
X
22
N
2
N
1
T
N
2
T
Z
]
≥
0
,
Θ
¯
2
=
[
X
11
X
12
M
1
X
12
T
X
22
M
2
M
1
T
M
2
T
Z
]
≥
0
,
Θ
¯
3
=
[
X
11
+
Y
11
X
12
+
Y
12
S
1
X
12
+
Y
12
T
X
22
+
Y
22
S
2
S
1
T
S
2
T
R
+
Z
]
≥
0
,
Θ
¯
4
=
[
X
11
+
Y
11
X
12
+
Y
12
T
1
X
12
T
+
Y
12
T
X
22
+
Y
22
T
2
T
1
T
T
2
T
R
+
Z
]
≥
0
,
Φ
¯
=
[
Φ
11
Φ
¯
12
-
N
1
+
M
1
-
T
1
-
M
1
+
S
1
Φ
¯
12
T
Φ
¯
22
-
N
2
+
M
2
-
T
2
-
M
2
+
S
2
-
N
1
T
+
M
1
T
-
N
2
T
+
M
2
T
-
Q
2
0
0
-
T
1
T
-
T
2
T
0
-
Q
3
0
-
M
1
T
+
S
1
T
-
M
2
T
+
S
2
T
0
0
-
Q
4
]
<
0
,
where
(5)
Φ
¯
12
=
N
2
T
-
S
1
+
T
1
+
τ
M
X
12
+
δ
2
Y
12
+
A
T
P
A
h
+
(
A
-
I
)
T
Θ
A
h
,
Φ
¯
22
=
-
S
2
-
S
2
T
+
T
2
+
T
2
T
-
Q
1
+
τ
M
X
22
+
δ
2
Y
22
+
A
h
T
P
A
h
+
A
h
T
Θ
A
h
.
Proof.
First, denote
τ
avg
=
[
(
1
/
2
)
(
τ
M
+
τ
m
)
]
, that is, the floor function of
(
1
/
2
)
(
τ
M
+
τ
m
)
,
δ
1
=
τ
avg
-
τ
m
, and
δ
2
=
τ
M
-
τ
avg
. Let
(6)
y
(
l
)
=
x
(
l
+
1
)
-
x
(
l
)
.
Then, it is clear that
(7)
y
(
k
)
=
x
(
k
+
1
)
-
x
(
k
)
=
(
A
-
I
)
x
(
k
)
+
A
h
(
k
-
h
(
k
)
)
.
We divide the proof into two cases.
Case I. We choose the Lyapunov function candidate
(8)
V
(
k
)
=
∑
i
=
1
5
V
i
(
k
)
,
where
(9)
V
1
(
k
)
=
x
T
(
k
)
P
x
(
k
)
,
V
2
(
k
)
=
∑
i
=
k
-
h
(
k
)
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
,
V
3
(
k
)
=
∑
i
=
k
-
τ
m
k
-
1
x
T
(
i
)
Q
2
x
(
i
)
+
∑
i
=
k
-
τ
M
k
-
1
x
T
(
i
)
Q
3
x
(
i
)
+
∑
i
=
k
-
τ
avg
k
-
1
x
T
(
i
)
Q
4
x
(
i
)
,
V
4
(
k
)
=
∑
j
=
-
τ
M
+
1
-
τ
m
∑
i
=
k
+
j
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
,
V
5
(
k
)
=
∑
j
=
-
τ
avg
-
τ
m
-
1
∑
i
=
k
+
j
k
-
1
y
T
(
i
)
R
y
(
i
)
+
∑
j
=
-
τ
M
-
1
∑
i
=
k
+
j
k
-
1
y
T
(
i
)
Z
y
(
i
)
,
and
P
,
Q
1
,
Q
2
,
Q
3
,
Q
4
,
R
, and
Z
are positive definite matrices to be determined. Define
Δ
V
i
(
k
)
=
V
i
(
k
+
1
)
-
V
i
(
k
)
which leads to
(10)
Δ
V
1
(
k
)
=
x
T
(
k
)
A
T
P
A
x
(
k
)
+
x
T
(
k
)
A
T
P
A
h
x
(
k
-
h
(
k
)
)
+
x
T
(
k
-
h
(
k
)
)
A
h
T
P
A
x
(
k
)
+
x
T
(
k
-
h
(
k
)
)
×
A
h
T
P
A
h
x
(
k
-
h
(
k
)
)
-
x
T
(
k
)
P
x
(
k
)
,
Δ
V
2
(
k
)
=
∑
i
=
k
+
1
-
h
(
k
+
1
)
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
+
x
T
(
k
)
Q
1
x
(
k
)
-
∑
i
=
k
+
1
-
h
(
k
)
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
-
x
T
(
k
-
h
(
k
)
)
Q
1
x
(
k
-
h
(
k
)
)
≤
∑
i
=
k
+
1
-
τ
m
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
+
∑
i
=
k
+
1
-
τ
M
k
-
τ
m
x
T
(
i
)
Q
1
x
(
i
)
+
x
T
(
k
)
Q
1
x
(
k
)
-
∑
i
=
k
+
1
-
τ
m
k
-
1
x
T
(
i
)
Q
1
x
(
i
)
-
x
T
(
k
-
h
(
k
)
)
Q
1
x
(
k
-
h
(
k
)
)
=
∑
i
=
k
+
1
-
τ
M
k
-
τ
m
x
T
(
i
)
Q
1
x
(
i
)
+
x
T
(
k
)
Q
1
x
(
k
)
-
x
T
(
k
-
h
(
k
)
)
Q
1
x
(
k
-
h
(
k
)
)
,
Δ
V
3
(
k
)
=
x
T
(
k
)
(
Q
2
+
Q
3
+
Q
4
)
x
(
k
)
-
x
T
(
k
-
τ
m
)
×
Q
2
x
(
k
-
τ
m
)
-
x
T
(
k
-
τ
M
)
Q
3
x
(
k
-
τ
M
)
-
x
T
(
k
-
τ
avg
)
Q
4
x
(
k
-
τ
avg
)
,
Δ
V
4
(
k
)
=
(
τ
M
-
τ
m
)
x
T
(
k
)
Q
1
x
(
k
)
-
∑
i
=
k
+
1
-
τ
M
k
-
τ
m
x
T
(
i
)
Q
1
x
(
i
)
,
Δ
V
5
(
k
)
=
(
τ
avg
-
τ
m
)
y
T
(
k
)
R
y
(
k
)
-
∑
i
=
k
-
τ
avg
k
-
τ
m
-
1
y
T
(
i
)
R
y
(
i
)
+
τ
M
y
T
(
k
)
Z
y
(
k
)
-
∑
i
=
k
-
τ
M
k
-
1
y
T
(
i
)
Z
y
(
i
)
.
From (6), the following zero equations
(11)
2
ζ
1
T
(
k
)
N
(
x
(
k
)
-
x
(
k
-
τ
m
)
-
∑
i
=
k
-
τ
m
k
-
1
y
(
i
)
)
=
0
,
(12)
2
ζ
1
T
(
k
)
M
(
x
(
k
-
τ
m
)
-
x
(
k
-
h
(
k
)
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
m
-
1
y
(
i
)
)
=
0
,
(13)
2
ζ
1
T
(
k
)
S
(
x
(
k
-
h
(
k
)
)
-
x
(
k
-
τ
avg
)
-
∑
i
=
k
-
τ
avg
k
-
h
(
k
)
-
1
y
(
i
)
)
=
0
,
(14)
2
ζ
1
T
(
k
)
T
(
x
(
k
-
τ
avg
)
-
x
(
k
-
τ
M
)
-
∑
i
=
k
-
τ
M
k
-
τ
avg
-
1
y
(
i
)
)
=
0
,
(15)
τ
M
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
m
k
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
m
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
avg
k
-
h
(
k
)
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
M
k
-
τ
avg
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
=
0
,
(16)
δ
1
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
m
-
1
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
-
∑
i
=
k
-
τ
avg
k
-
h
(
k
)
-
1
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
=
0
hold for any appropriate dimensions matrices
N
,
M
,
S
, and
T
, where
ζ
1
T
(
k
)
=
[
x
T
(
k
)
x
T
(
k
-
h
(
k
)
)
]
, and semi-positive-definite matrices
(17)
X
=
[
X
11
X
12
X
12
T
X
22
]
,
Y
=
[
Y
11
Y
12
Y
12
T
Y
22
]
.
Taking the forward difference of
V
(
k
)
and adding all the terms of (11)–(16) to
Δ
V
(
k
)
leads to
(18)
Δ
V
(
k
)
≤
ζ
2
T
(
k
)
Φ
ζ
2
(
k
)
-
∑
i
=
k
-
τ
m
k
-
1
ζ
3
T
(
k
,
i
)
Θ
1
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
m
-
1
ζ
3
T
(
k
,
i
)
Θ
2
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
τ
avg
k
-
h
(
k
)
-
1
ζ
3
T
(
k
,
i
)
Θ
3
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
τ
M
k
-
τ
avg
-
1
ζ
3
T
(
k
,
i
)
Θ
4
ζ
3
(
k
,
i
)
,
where
ζ
2
T
(
k
)
=
[
ζ
1
T
(
k
)
x
T
(
k
-
τ
m
)
x
T
(
k
-
τ
M
)
x
T
(
k
-
τ
avg
)
]
and
ζ
3
(
k
,
i
)
T
(
k
)
=
[
ζ
2
T
(
k
)
y
(
i
)
T
]
. Thus, LMIs (2) assure that
Δ
V
(
k
)
is negative definite.
Case II. We choose the Lyapunov function candidate
(19)
V
(
k
)
=
∑
i
=
1
5
V
¯
i
(
k
)
,
where the
V
¯
i
(
k
)
=
V
i
(
k
)
for
i
=
1,2
,
3,4
(are same as Case I) except for
i
=
5
where
(20)
V
¯
5
(
k
)
=
∑
j
=
-
τ
M
-
τ
avg
-
1
∑
i
=
k
+
j
k
-
1
y
T
(
i
)
R
y
(
i
)
+
∑
j
=
-
τ
M
-
1
∑
i
=
k
+
j
k
-
1
y
T
(
i
)
Z
y
(
i
)
,
which yields
(21)
Δ
V
¯
5
(
k
)
=
(
τ
M
-
τ
avg
)
y
T
(
k
)
R
y
(
k
)
-
∑
i
=
k
-
τ
M
k
-
τ
avg
-
1
y
T
(
i
)
R
y
(
i
)
+
τ
M
y
T
(
k
)
Z
y
(
k
)
-
∑
i
=
k
-
τ
M
k
-
1
y
T
(
i
)
Z
y
(
i
)
.
Similar to (11)–(16), (11) and the following zero equations
(22)
2
ζ
1
T
(
k
)
M
(
x
(
k
-
τ
m
)
-
x
(
k
-
τ
avg
)
-
∑
i
=
k
-
τ
avg
k
-
τ
m
-
1
y
(
i
)
)
=
0
,
2
ζ
1
T
(
k
)
S
(
x
(
k
-
τ
avg
)
-
x
(
k
-
h
(
k
)
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
avg
-
1
y
(
i
)
)
=
0
,
2
ζ
1
T
(
k
)
T
(
x
(
k
-
h
(
k
)
)
-
x
(
k
-
τ
M
)
-
∑
i
=
k
-
τ
M
k
-
h
(
k
)
-
1
y
(
i
)
)
=
0
,
τ
M
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
m
k
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
avg
k
-
τ
m
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
avg
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
-
∑
i
=
k
-
τ
M
k
-
h
(
k
)
-
1
ζ
1
T
(
k
)
X
ζ
1
(
k
)
=
0
,
δ
2
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
avg
-
1
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
-
∑
i
=
k
-
τ
M
k
-
h
(
k
)
-
1
ζ
1
T
(
k
)
Y
ζ
1
(
k
)
=
0
.
Taking the forward difference of
V
(
k
)
and adding all the terms of (11) and (22) to
Δ
V
(
k
)
allow us to write
Δ
V
(
k
)
as
(23)
Δ
V
(
k
)
≤
ζ
2
T
(
k
)
Φ
¯
ζ
2
(
k
)
-
∑
i
=
k
-
τ
m
k
-
1
ζ
3
T
(
k
,
i
)
Θ
¯
1
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
τ
avg
k
-
τ
m
-
1
ζ
3
T
(
k
,
i
)
Θ
¯
2
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
h
(
k
)
k
-
τ
avg
-
1
ζ
3
T
(
k
,
i
)
Θ
¯
3
ζ
3
(
k
,
i
)
-
∑
i
=
k
-
τ
M
k
-
h
(
k
)
-
1
ζ
3
T
(
k
,
i
)
Θ
¯
4
ζ
3
(
k
,
i
)
.
Thus, LMIs (4) assure that
Δ
V
(
k
)
is negative definite.
Remark 2.
The main goal of the partitioning technique is to divide the summation interval
[
0
,
τ
M
]
of the Lyapunov function difference to as many subintervals (e.g.,
[
0
,
τ
m
]
,
[
τ
m
,
τ
avg
]
,
[
τ
avg
,
h
(
k
)
]
,
[
h
(
k
)
,
τ
M
]
,
[
0
,
τ
m
]
,
[
τ
m
,
h
(
k
)
]
,
[
h
(
k
)
,
τ
avg
]
, and
[
τ
avg
,
τ
M
]
in the analysis) as possible. Using this technique along with the zero equations (11)–(16) and (22), we introduce free weighting matrices which increase feasibility of the resulting LMIs.