# Stability Analysis andH

Authors: K. H. Kim; M. J. Park; O. M. Kwon; E. J. Cha
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

The problem of stability analysis andH
∞ output tracking control for linear systems with time-varying delays is studied. First, by construction of a newly augmented Lyapunov-Krasovskii functional, a delay-dependent stability criterion for nominal systems with time-varying delays is established in terms of linear matrix inequalities (LMIs). Second, based on the H
∞ sense, the proposed method is extended to solve the problem of designing an H
∞ output tracking controller to track the output of a given reference model. Finally, three examples are included to show the validity and effectiveness of the presented delay-dependent stability and the H
∞ output tracking controller design.


---

## Body

## 1. Introduction

The problem of output tracking control has received a great deal of attention since this issue is an important requirement in many systems [1–5]. In detail, output tracking controller design methods are utilized in robot systems [1, 2], flight systems [3, 4], and so on. Moreover, in [5], the problem of output tracking control is derived in H
∞ sense for time-delayed systems with nonlinear perturbations. The main purpose of output tracking control is to design a closed-loop feedback controller such that the output of plant tracks the output of a given reference model as close as possible. Moreover, disturbances should be considered in designing output tracking control since disturbances can lead to adverse effects on the performance of systems. To minimize the effects of the disturbances on systems, one possible approach is to design a tracking controller in H
∞ sense. Since H
∞ control which has been used to minimize the effects of the disturbances was firstly introduced by Zames [6], the study on H
∞ controller design has been attracted by many researchers [7–10]. Similarly, H
∞ output tracking controller has an objective of designing a controller such that the closed-loop system is asymptotically stable and the tracking error by the effects of disturbances does not exceed a prescribed level.On the other hand, time-delay is one of the sources of instability and poor performance in various systems such as physical and chemical systems and industrial and engineering systems. The reason is that time-delays frequently occur in various systems due to the finite capabilities of information processing and transmission. For this reason, during the past several decades, stability analysis for time-delayed systems has been widely studied by many researchers [11–23]. One of the aims of analysis for time-delayed systems is to establish a less conservative stability criterion which can find maximum upper bounds of time-delays for guaranteeing the asymptotic stability of systems. Stability analysis for time-delayed systems can be classified into delay-independent criteria and delay-dependent ones. Since delay-dependent stability criteria use information of time-delays such as lower bounds and upper bounds of delays while delay-independent ones do not have them, delay-dependent stability criteria are generally less conservative than delay-independent ones especially when the sizes of time-delays are small.In line with this thinking, according to Zhang and Yu [5], it has been well acknowledged that output tracking control design for time-delayed system is more general and more difficult than stabilization. However, most existing results on output tracking control have focused on systems without time-delays. From the practical point of view, it is worth designing output tracking controllers for time-delayed systems since it is well known that there exist time-delays in many practical systems. Therefore, to establish less conservative stability criteria and design output tracking controllers for time-delayed systems are still challenging.Motivated by the matters mentioned above, this paper investigates the problem of stability analysis andH
∞ tracking controller designing for linear systems with time-varying delays and disturbances. First, in Theorem 5, by constructing a newly augmented Lyapunov-Krasovskii functional, an improved delay-dependent stability criterion is derived by utilizing reciprocally convex approach [15] and some new zero equalities with LMI framework [24] which can be formulated as convex optimization algorithms. Second, based on the results of Theorem 5, an H
∞ tracking controller design method for linear systems with time-varying delays and disturbances will be proposed in Theorem 7. Finally, through three examples, validity and effectiveness of the proposed theorem will be verified.Notation.
R
n is the n-dimensional Euclidean space and R
m
×
n denotes the set of m
×
n real matrix. For symmetric matrices X and Y, X
>
Y (resp., X
≥
Y) means that the matrix X
-
Y is positive definite (resp., nonnegative). X
T denotes the transposition of X. I
n denotes the n
×
n identity matrix. 0
n and 0
n
×
m denote the n
×
n zero matrix and the n
×
m zero matrix, respectively. ∥
·
∥ refers to the Euclidean vector norm and the induced matrix norm. diag
⁡
{
⋯
} denotes the block diagonal matrix, respectively. * represents the elements below the main diagonal of a symmetric matrix. L
2
[
0
,
∞
) is the space of square integrable vector. X
[
f
(
t
)
]
∈
R
m
×
n means that the elements of the matrix X
[
f
(
t
)
] include the value of f
(
t
); for example, X
[
f
0
]
≡
X
[
f
(
t
)
=
f
0
].


## 2. Problem Statements

Consider the following system with time-varying delays:(1)
x
˙
(
t
)
=
A
x
(
t
)
+
A
d
(
t
)
x
(
t
-
h
(
t
)
)
+
B
u
(
t
)
+
B
w
w
(
t
)
,
z
(
t
)
=
C
1
x
(
t
)
+
C
2
x
(
t
-
h
(
t
)
)
+
B
1
u
(
t
)
,
where x
(
t
)
∈
R
n is the state vector, u
(
t
)
∈
R
m is the control input, w
(
t
)
∈
R
r is the disturbance input which belongs to L
2
[
0
,
∞
), z
(
t
)
∈
R
s is the vector of controlled output, and A
∈
R
n
×
n, A
d
∈
R
n
×
n, B
∈
R
n
×
m, B
w
∈
R
n
×
r, C
1
∈
R
s
×
n, C
2
∈
R
s
×
n, and B
1
∈
R
s
×
m are known real constant matrices.  h
(
t
) means a time-delay satisfying time-varying continuous function as follows:
(2)
0
≤
h
(
t
)
≤
h
M
,
|
h
˙
(
t
)
|
≤
h
d
,
where h
M
>
0 and 0
≤
h
d
<
1 are known scalars.Designing the output tracking control has an objective to design a controller such that the output of the system tracks a reference signal. Therefore, the reference signalz
r
(
t
) is assumed to be generated by reference model as follows:
(3)
x
˙
r
(
t
)
=
F
x
r
(
t
)
+
r
(
t
)
,
z
r
(
t
)
=
H
x
r
(
t
)
,
where z
r
(
t
)
∈
R
s is the reference signal which has the same dimension as z
(
t
), x
r
(
t
)
∈
R
p is the reference state, and r
(
t
)
∈
R
p is the energy bounded reference input. G
∈
R
p
×
p and H
∈
R
s
×
p are known real constant matrices with the assumption that the matrix G is Hurwitz. The state-feedback controller is considered as the following form:
(4)
u
(
t
)
=
K
1
x
(
t
)
+
K
2
x
r
(
t
)
,
where K
1
∈
R
m
×
n and K
2
∈
R
m
×
p are gain matrices of the state-feedback controller. Let us define ξ
(
t
)
=
[
x
T
(
t
)
,
x
r
T
(
t
)
]
T
∈
R
q, e
(
t
)
=
z
(
t
)
-
z
r
(
t
), and v
(
t
)
=
[
w
T
(
t
)
,
r
T
(
t
)
]
T. Then the following augmented systems can be obtained as
(5)
ξ
˙
(
t
)
=
(
A
¯
+
B
-
K
-
)
ξ
(
t
)
+
A
-
d
(
t
)
ξ
(
t
-
h
(
t
)
)
+
B
-
w
v
(
t
)
,
e
(
t
)
=
C
-
1
ξ
(
t
)
+
C
-
2
ξ
(
t
-
h
(
t
)
)
,
where
(6)
A
¯
=
[
A
0
n
×
p
0
p
×
n
F
]
,
B
¯
=
[
B
0
p
×
m
]
,
K
¯
=
[
K
1
,
K
2
]
,
A
¯
d
=
[
A
d
0
n
×
p
0
p
×
n
0
p
×
p
]
,
B
¯
w
=
[
B
w
0
n
×
p
0
p
×
r
I
p
]
,
C
¯
1
=
[
C
1
+
B
1
K
1
,
-
H
+
B
1
K
2
]
,
C
¯
2
=
[
C
2
,
0
s
×
p
]
.From theH
∞ sense, let us consider the two requirements of the output tracking controller as follows.(i)
Withv
(
t
)
=
0, the closed-loop system (5) with control input u
(
t
) is asymptotically stable.
(ii)
The effects ofw
(
t
) and r
(
t
) on the tracking error e
(
t
) are attenuated below a desired level in the H
∞ sense; that is,
(7)
J
=
{
∫
0
∞
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
d
s
}
⩽
0
,
(
i
.
e
.
sup
⁡
v
(
t
)
≠
0
,
v
(
t
)
∈
L
2
[
0
,
∞
)
∥
e
(
t
)
∥
2
∥
v
(
t
)
∥
2
≤
γ
)
,where γ
>
0 is a prescribed scalar. Then, if the obtained controller u
(
t
) is satisfied with two requirements, then it is said to be an H
∞ output tracking controller.The objective of this paper is to design a state-feedback controller (4) such that system (5) achieves the H
∞ output tracking below a prescribed level and the H
∞ output tracking performance γ is minimized.Before proceeding further, the following lemmas will be utilized in deriving main results.Lemma 1 (see [17]).
For a positive matrixZ, scalars h
2
>
h
1
>
0 such that the integrations are well defined; then
(8)
-
(
h
2
-
h
1
)
∫
t
-
h
2
t
-
h
1
‍
x
T
(
s
)
Z
x
(
s
)
d
s
≤
-
(
∫
t
-
h
2
t
-
h
1
‍
x
(
s
)
d
s
)
T
Z
(
∫
t
-
h
2
t
-
h
1
‍
x
(
s
)
d
s
)
,
-
(
h
2
2
-
h
1
2
)
2
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
x
T
(
s
)
Z
x
(
s
)
d
s
d
θ
≤
-
(
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
x
(
s
)
d
s
d
θ
)
T
Z
(
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
x
(
s
)
d
s
d
θ
)
,
-
(
h
2
3
-
h
1
3
)
6
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
∫
λ
t
‍
x
T
(
s
)
Z
x
(
s
)
d
s
d
λ
d
θ
≤
-
(
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
∫
λ
t
‍
x
(
s
)
d
s
d
λ
d
θ
)
T
×
Z
(
∫
t
-
h
2
t
-
h
1
‍
∫
θ
t
‍
∫
λ
t
‍
x
(
s
)
d
s
d
λ
d
θ
)
.Lemma 2 (see [15]).
For a scalarα in the interval (
0,1
), a given matrix R
∈
R
n
×
n
>
0, two matrices W
1
∈
R
n
×
m and W
2
∈
R
n
×
m, for all vector ξ
∈
R
m, let one define a function Θ
(
α
,
R
) given by
(9)
Θ
(
α
,
R
)
=
1
α
ξ
T
W
1
T
R
W
1
ξ
+
1
1
-
α
ξ
T
W
2
T
R
W
2
ξ
.
Then, if there exists a matrix X in R
n
×
n such that [
R
X
*
R
]
>
0, then the following inequality holds
(10)
min
⁡
α
∈
(
0,1
)
Θ
(
α
,
R
)
≥
[
W
1
ξ
W
2
ξ
]
T
[
R
X
*
R
]
[
W
1
ξ
W
2
ξ
]
.Lemma 3 (Finsler’s lemma [25]).
Letζ
∈
R
n, Φ
=
Φ
T
∈
R
n
×
n, and B
∈
R
m
×
n such that rank
⁡
(
B
)
<
n. The following statements are equivalent:(1)
ζ
T
Φ
ζ
<
0
,
∀
B
ζ
=
0
,
ζ
≠
0.
(2)
(
B
⊥
)
T
Φ
B
⊥
<
0, where B
⊥ is a right orthogonal complement of B.Lemma 4 (see [21]).
For a positive matrixΛ, a symmetric matrix Ξ, and a matrix Γ, the following two statements are equivalent:(1)
Ξ
-
Γ
T
Λ
Γ
<
0;
(2)
There exists a matrix of appropriate dimensionU such that
(11)
[
Ξ
+
U
T
Γ
+
U
T
Γ
U
T
U
-
Λ
]
≤
0
.


## 3. Main Results

This section consists of two subsections. The first section introduces an improved stability criterion for system (1) with u
(
t
)
=
0 and w
(
t
)
=
0 as Theorem 5. The second subsection will investigate the problem of an H
∞ output tracking controller design method for the augmented system (5) based on the results of Theorem 5.


### 3.1. Stability Analysis

Let us consider the system (1) with w
(
t
)
=
0 and u
(
t
)
=
0 given by
(12)
x
˙
(
t
)
=
A
x
(
t
)
+
A
d
x
(
t
-
h
(
t
)
)
.
In this subsection, a delay-dependent stability criterion for system (12) is derived. For simplicity of matrix and vector representation, e
i
(
i
=
1
,
…
,
10
)
∈
R
10
n
×
n are defined as block entry matrices which will be used in Theorem 5. For example, e
1
=
[
I
n
,
0
n
×
9
n
]
T and e
5
=
[
0
n
×
4
n
,
I
n
,
0
n
×
5
n
]
T. The other notations are defined as
(13)
ζ
T
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
x
˙
T
(
t
)
,
f
d
s
f
f
s
d
f
x
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
f
d
s
f
s
d
d
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
f
d
s
f
d
s
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
x
˙
T
(
t
-
h
(
t
)
)
]
,
Θ
=
(
A
e
1
T
+
A
d
e
2
T
-
e
4
T
)
,
Ψ
11
=
[
e
1
,
e
2
,
e
3
,
e
6
,
e
7
,
e
8
,
e
9
]
,
Ψ
12
=
[
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
,
e
7
,
e
10
]
,
Ψ
21
=
[
e
1
,
e
4
,
0
10
n
×
n
,
0
10
n
×
n
]
,
Ψ
22
=
[
e
3
,
e
5
,
e
1
-
e
3
,
e
6
+
e
7
]
,
Ψ
23
=
[
e
6
+
e
7
,
e
1
-
e
3
,
h
M
e
1
-
e
6
-
e
7
,
e
8
+
e
9
]
,
Ψ
24
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
31
=
[
e
1
,
e
4
,
0
10
n
×
n
,
0
10
n
×
n
]
,
Ψ
32
=
[
e
2
,
e
10
,
e
1
-
e
2
,
e
6
]
,
Ψ
33
[
h
(
t
)
]
=
[
e
6
,
e
1
-
e
3
,
h
(
t
)
e
1
-
e
6
,
e
8
]
,
Ψ
34
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
35
=
[
e
2
,
e
10
,
e
1
-
e
2
,
e
6
]
,
Ψ
36
=
[
e
3
,
e
5
,
e
1
-
e
3
,
e
6
+
e
7
]
,
Ψ
37
[
h
(
t
)
]
=
[
e
7
,
e
2
-
e
3
,
(
h
M
-
h
(
t
)
)
e
1
-
e
7
,
e
9
]
,
Ψ
38
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
41
=
[
e
1
,
e
4
]
,
Ψ
42
=
[
e
6
,
e
1
-
e
2
,
e
7
,
e
2
-
e
3
]
,
Ψ
5
=
[
e
1
,
e
4
]
,
Ψ
6
=
(
(
h
M
2
2
)
e
1
-
e
8
-
e
9
)
,
Υ
1
=
[
Q
1
+
[
P
3
P
1
P
1
0
n
]
S
1
S
1
T
Q
1
+
[
P
4
P
2
P
2
0
n
]
]
,
Υ
2
=
[
Q
2
+
(
4
h
M
)
[
0
n
P
3
P
3
0
n
]
S
2
S
2
T
Q
2
+
(
4
h
M
)
[
0
n
P
4
P
4
0
n
]
]
,
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
0
n
×
3
n
,
I
n
,
0
n
×
4
n
0
n
×
7
n
,
(
1
-
h
˙
(
t
)
)
I
n
0
n
×
4
n
,
I
n
,
0
n
×
3
n
I
n
,
-
(
1
-
h
˙
(
t
)
)
I
n
,
0
n
×
6
n
0
n
,
(
1
-
h
˙
(
t
)
)
I
n
,
-
I
n
,
0
n
×
5
n
h
(
t
)
I
n
,
0
n
×
4
n
,
-
(
1
-
h
˙
(
t
)
)
I
n
,
0
n
×
2
n
(
h
M
-
h
(
t
)
)
I
n
,
0
n
×
4
n
,
-
h
˙
(
t
)
I
n
,
0
n
×
2
n
]
,
Γ
2
[
h
(
t
)
]
=
[
0
n
×
3
n
,
I
n
,
0
n
h
(
t
)
I
n
,
-
I
n
,
0
n
×
3
n
0
n
×
4
n
,
I
n
(
h
M
-
h
(
t
)
)
I
n
,
0
n
,
-
I
n
,
0
n
×
2
n
]
[
e
1
T
e
6
T
e
7
T
e
8
T
e
9
T
]
,
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
11
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
R
Ψ
12
T
+
Ψ
12
R
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
T
Ψ
11
T
,
Ξ
2
=
Ψ
21
N
Ψ
21
T
-
Ψ
22
N
Ψ
22
T
+
Ψ
23
N
Ψ
24
T
+
Ψ
24
N
Ψ
23
T
,
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
31
G
Ψ
31
T
-
(
1
-
h
˙
(
t
)
)
Ψ
32
G
Ψ
32
T
d
s
a
d
d
s
d
s
a
d
+
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
G
Ψ
34
T
+
Ψ
34
G
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
T
d
s
a
d
d
s
d
s
a
d
+
(
1
-
h
˙
(
t
)
)
Ψ
35
M
Ψ
35
T
-
Ψ
36
M
Ψ
36
T
d
s
a
d
d
s
d
s
a
d
+
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
M
Ψ
38
T
+
Ψ
38
M
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
T
,
Ξ
zero
[
h
(
t
)
]
=
h
M
(
e
1
P
1
e
1
T
-
e
2
P
1
e
2
T
+
e
2
P
2
e
2
T
-
e
3
P
2
e
3
T
)
d
s
a
d
d
s
d
s
a
d
+
h
M
(
h
(
t
)
e
1
P
3
e
1
T
-
(
h
M
-
h
(
t
)
)
e
1
P
4
e
1
T
)
,
Ξ
4
[
h
(
t
)
]
=
h
M
2
Ψ
41
Q
1
Ψ
41
T
-
Ψ
42
Υ
1
Ψ
42
T
+
Ξ
zero
[
h
(
t
)
]
,
Ξ
5
=
(
h
M
2
2
)
2
Ψ
5
Q
2
Ψ
5
T
,
Ξ
6
=
(
h
M
3
6
)
2
e
4
Q
3
e
4
T
-
Ψ
6
Q
3
Ψ
6
T
,
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
2
+
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
d
s
a
d
d
s
d
s
a
d
+
Ξ
4
[
h
(
t
)
]
+
Ξ
5
+
Ξ
6
,
L
[
h
(
t
)
,
h
˙
(
t
)
]
=
Θ
⊥
T
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
Θ
⊥
+
Θ
⊥
T
Γ
2
[
h
(
t
)
]
T
Λ
d
s
a
d
d
s
d
s
a
d
+
Λ
T
Γ
2
[
h
(
t
)
]
Θ
⊥
,
Φ
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
[
h
(
t
)
,
h
˙
(
t
)
]
Λ
T
Λ
-
Υ
2
]
,
where Θ
⊥ is the right orthogonal complement of Θ.Now, a delay-dependent stability of the system (12) is given as Theorem 5.Theorem 5.
For given scalars0
≤
h
M and 0
≤
h
d
<
1, the system (12) is asymptotically stable for 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d, if there exist positive definite matrices R
∈
R
7
n
×
7
n, N
∈
R
4
n
×
4
n, G
∈
R
4
n
×
4
n, M
∈
R
4
n
×
4
n, Q
1
∈
R
2
n
×
2
n, Q
2
∈
R
2
n
×
2
n, and Q
3
∈
R
n
×
n, symmetric matrices P
i
∈
R
n
×
n
(
i
=
1
,
…
,
4
), and any matrices S
1
∈
R
2
n
×
2
n, S
2
∈
R
2
n
×
2
n, and Λ
∈
R
4
n
×
9
n, satisfying the following LMIs:
(14)
Φ
j
,
k
<
0
,
(
j
,
k
=
1,2
)
,
Υ
i
≥
0
,
(
i
=
1,2
)
,
where Φ
j
,
k are the four vertices of Φ
[
h
(
t
)
,
h
˙
(
t
)
] with the bounds of h
(
t
) and h
˙
(
t
), that is, h
M and h
D when j
=
k
=
1, h
M and -
h
D when j
<
k, 0 and h
D when j
>
k, and 0 and -
h
D when j
=
k
=
2.Proof.
For positive definite matricesR
∈
R
7
n
×
7
n, N
∈
R
4
n
×
4
n, G
∈
R
4
n
×
4
n, Q
1
∈
R
2
n
×
2
n, Q
2
∈
R
2
n
×
2
n, and Q
3
∈
R
n
×
n, let us consider the following Lyapunov-Krasovskii functional candidate as
(15)
V
(
t
)
=
∑
i
=
1
6
‍
V
i
(
t
)
,
where
(16)
V
1
(
t
)
=
Ω
1
T
(
t
)
R
Ω
1
(
t
)
,
V
2
(
t
)
=
∫
t
-
h
M
t
‍
Ω
2
T
(
t
,
s
)
N
Ω
2
(
t
,
s
)
d
s
,
V
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
Ω
2
T
(
t
,
s
)
G
Ω
2
(
t
,
s
)
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
Ω
2
T
(
t
,
s
)
M
Ω
2
(
t
,
s
)
d
s
,
V
4
(
t
)
=
h
M
∫
t
-
h
M
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
T
Q
1
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
,
V
5
(
t
)
=
(
h
M
2
2
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
[
x
(
v
)
x
˙
(
v
)
]
T
Q
2
[
x
(
v
)
x
˙
(
v
)
]
d
v
d
u
d
s
,
V
6
(
t
)
=
(
h
M
3
6
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
∫
v
t
‍
x
˙
T
(
λ
)
Q
3
x
˙
(
λ
)
d
λ
d
v
d
u
d
s
,
with Ω
1
(
t
)
=
[
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T and Ω
2
(
t
,
s
)
=
[
x
T
(
s
)
,
x
˙
T
(
s
)
,
∫
s
t
‍
x
˙
T
(
u
)
d
u
,
∫
s
t
‍
x
T
(
u
)
d
u
]
T.
Now, calculating the time-derivative ofV
1
(
t
) yields
(17)
V
˙
1
(
t
)
=
2
Ω
1
(
t
)
R
×
[
x
˙
(
t
)
(
1
-
h
˙
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
(
1
-
h
˙
(
t
)
)
x
(
t
-
h
M
)
(
1
-
h
˙
(
t
)
)
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
h
(
t
)
x
(
t
)
-
(
1
-
h
˙
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
(
(
h
M
-
h
(
t
)
)
x
(
t
)
+
(
1
-
h
˙
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
-
∫
t
-
h
M
t
‍
x
(
s
)
d
s
)
]
=
ζ
T
(
t
)
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
ζ
(
t
)
.
Also, V
˙
2
(
t
) can be represented as
(18)
V
˙
2
(
t
)
=
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
T
N
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
-
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
u
)
d
u
]
T
×
N
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
x
(
u
)
d
u
]
+
2
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
x
(
t
)
-
x
(
s
)
∫
s
t
‍
x
(
u
)
d
u
]
T
d
s
N
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
=
ζ
T
(
t
)
Ξ
2
ζ
(
t
)
.
The time-derivative of V
3
(
t
) can be given as follows:
(19)
V
˙
3
(
t
)
=
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
T
G
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
-
(
1
-
h
˙
(
t
)
)
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
T
×
G
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
+
2
∫
t
-
h
(
t
)
t
‍
[
x
(
s
)
x
˙
(
t
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
d
s
G
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
+
(
1
-
h
˙
(
t
)
)
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
T
×
M
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
-
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
s
)
d
s
]
T
M
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
s
)
d
s
]
+
2
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
(
h
M
-
h
(
t
)
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
]
T
×
M
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
=
ζ
T
(
t
)
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
ζ
(
t
)
.
Calculating V
˙
4
(
t
) gives
(20)
V
˙
4
(
t
)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
Q
1
[
x
(
s
)
x
˙
(
s
)
]
d
s
.
Inspired by the work in [22], the following four zero equalities with symmetric matrices P
1, P
2, P
3, and P
4 are considered:
(21)
0
=
x
T
(
t
)
(
h
M
P
1
)
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
(
h
M
P
1
)
x
(
t
-
h
(
t
)
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
1
x
˙
(
s
)
d
s
,
0
=
x
T
(
t
-
h
(
t
)
)
(
h
M
P
2
)
x
(
t
-
h
(
t
)
)
-
x
T
(
t
-
h
M
)
(
h
M
P
2
)
x
(
t
-
h
M
)
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
P
2
x
˙
(
s
)
d
s
,
0
=
x
T
(
t
)
(
h
M
h
(
t
)
P
3
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
3
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
,
0
=
x
T
(
t
)
(
h
M
(
h
M
-
h
(
t
)
)
P
4
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
4
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
u
˙
(
s
)
d
u
d
s
.
By adding (21) into (20), V
˙
4
(
t
) can be represented as follows:
(22)
V
˙
4
(
t
)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
Q
1
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
x
T
(
t
)
(
h
M
P
1
)
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
×
(
h
M
P
1
)
x
(
t
-
h
(
t
)
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
1
x
˙
(
s
)
d
s
+
x
T
(
t
-
h
(
t
)
)
(
h
M
P
2
)
x
(
t
-
h
(
t
)
)
-
x
T
(
t
-
h
M
)
(
h
M
P
2
)
x
(
t
-
h
M
)
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
P
2
x
˙
(
s
)
d
s
+
x
T
(
t
)
(
h
M
h
(
t
)
P
3
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
3
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
x
T
(
t
)
(
h
M
(
h
M
-
h
(
t
)
)
P
4
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
4
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
u
˙
(
s
)
d
u
d
s
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
-
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
From Lemma 1, an upper-bound of V
˙
4
(
t
) can be calculated as
(23)
V
˙
4
(
t
)
≤
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
h
(
t
)
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
-
h
M
h
M
-
h
(
t
)
×
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
(24)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
+
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
T
×
[
1
ϕ
1
(
t
)
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
0
2
n
0
2
n
1
1
-
ϕ
1
(
t
)
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
]
×
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
,
where ϕ
1
(
t
)
=
h
(
t
)
/
h
M which satisfies 0
≤
ϕ
1
(
t
)
≤
1. It should be noted that if h
(
t
) is zero, then the term of ∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s and x
(
t
)
-
x
(
t
-
h
(
t
)
) are zero. Also, if h
(
t
) is h
M, then the term of ∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s and x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
) are zero. Thus, inequality (24) still holds.
By using Lemma2 with Υ
1
≥
0 which is defined in (13), a new upper-bound of V
˙
4
(
t
) can be obtained as
(25)
V
˙
4
(
t
)
≤
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
T
×
Υ
1
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
×
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
ζ
T
(
t
)
(
h
M
2
Ψ
41
Q
1
Ψ
41
T
-
Ψ
42
Υ
1
Ψ
42
T
+
Ξ
zero
[
h
(
t
)
]
)
︸
Ξ
4
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
TheV
˙
5
(
t
) is calculated as(26)
V
˙
5
(
t
)
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
2
2
∫
t
-
h
M
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
Q
2
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
︸
0
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
︸
0
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
2
2
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
-
h
M
2
2
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.Applying Lemma 1 to (26), the upper-bound of V
˙
5
(
t
) can be derived as
(27)
V
˙
5
(
t
)
≤
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
(
h
M
h
(
t
)
)
2
(
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
T
×
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
(
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
-
(
h
M
2
h
M
2
-
h
(
t
)
2
)
(
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
T
×
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
(
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s
]
T
×
[
1
ϕ
2
(
t
)
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
0
2
n
0
2
n
1
1
-
ϕ
2
(
t
)
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
]
×
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s
]
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
,
where ϕ
2
(
t
)
=
(
h
(
t
)
/
h
M
)
2 satisfies 0
≤
ϕ
2
(
t
)
≤
1. Note that when h
(
t
) is zero, the terms of ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s are zero, and when h
(
t
) is h
M, the terms of ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s are zero. Thus, inequality (27) still holds.
By the use of Lemma2 to (27) with Υ
2
≥
0 defined in (13), the following inequality can be obtained:
(28)
V
˙
5
(
t
)
≤
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
h
(
t
)
x
(
t
)
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
(
h
(
t
)
-
h
M
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
]
T
Υ
2
×
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
h
(
t
)
x
(
t
)
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
(
h
(
t
)
-
h
M
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
]
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
ζ
T
(
t
)
(
Ξ
5
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
[
2
h
(
t
)
]
)
ζ
(
t
)
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
The time-derivative of V
6
(
t
) is obtained as follows:
(29)
V
˙
6
(
t
)
=
(
h
M
3
6
)
2
x
˙
T
(
t
)
Q
3
x
˙
(
t
)
-
(
h
M
3
6
)
×
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
x
˙
T
(
v
)
Q
3
x
˙
(
v
)
d
v
d
u
d
s
.
Using Lemma 1 to (29) yields a new upper-bound of V
˙
6
(
t
) as
(30)
V
˙
6
(
t
)
≤
(
h
M
3
6
)
2
x
˙
T
(
t
)
Q
3
x
˙
(
t
)
-
(
h
M
2
2
x
(
t
)
-
∫
t
-
h
M
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
)
T
×
Q
3
(
h
M
2
2
x
(
t
)
-
∫
t
-
h
M
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
)
=
ζ
T
(
t
)
Ξ
6
ζ
(
t
)
.
By summation of (17)–(30), an upper-bound of V
˙
(
t
) is obtained as follows:
(31)
V
˙
(
t
)
≤
ζ
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
(
t
)
.
From the dynamic equation of the system (12), zero equality 0
=
Θ
ζ
(
t
) can be obtained. Therefore, from Lemma 3, a stability criterion for the system (12) can be obtained as follows:
(32)
Θ
⊥
T
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
Θ
⊥
<
0
.
By using Lemma 4 with any matrix U
∈
R
4
n
×
9
n, inequality (32) is equivalent to
(33)
Φ
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
Θ
⊥
T
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
Θ
⊥
+
Θ
⊥
T
Γ
2
[
h
(
t
)
]
T
Λ
+
Λ
T
Γ
2
[
h
(
t
)
]
Θ
⊥
Λ
T
Λ
-
Υ
2
]
=
[
L
[
h
(
t
)
,
h
˙
(
t
)
]
Λ
T
Λ
-
Υ
2
]
<
0
,
where L
[
h
(
t
)
,
h
˙
(
t
)
] is defined in (13). Note that inequality (33) is affinely dependent on h
(
t
) and h
˙
(
t
) where 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d
<
1. So, Φ
[
h
(
t
)
,
h
˙
(
t
)
]
<
0 is satisfied if
(34)
Φ
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
h
d
]
<
0
,
Φ
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
h
d
]
<
0
.
This completes our proof.Remark 6.
Compared with the existing works, the main differences of Lyapunov-Krasovskii functionals areV
2
(
t
) and V
3
(
t
). In detail, V
2
(
t
) and V
3
(
t
) have augmented vectors ∫
t
-
h
(
t
)
t
‍
x
˙
(
s
)
d
s, respectively. With this consideration, some new cross terms are calculated and utilized in stability condition. Furthermore, the time-varying delayed state x
(
t
-
h
(
t
)
) and double integral terms such as ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s are utilized as an element of the augmented vectors of V
1
(
t
). In next section, the results for Theorem 5 will show larger delay bounds than the works in other literature to confirm the less conservatism of proposed criterion.


### 3.2.H
∞ Output Tracking Controller Design

In this subsection, anH
∞ output tracking controller design method for the system (5) will be introduced based on the result of Theorem 5. For simplicity of matrix and vector representation, the scalar q which will be used in representing dimensions of matrices and vectors is defined as q
=
n
+
r. For example, block entry matrices e
i
(
i
=
1
,
…
,
10
)
∈
R
10
q
×
q will be used in Theorem 7. The other notations are defined as
(35)
ζ
¯
T
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
ξ
T
(
t
)
,
ξ
T
(
t
-
h
(
t
)
)
,
ξ
T
(
t
-
h
M
)
,
ξ
˙
T
(
t
)
,
f
d
s
f
d
s
f
f
ξ
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
f
s
d
f
d
s
f
∫
t
-
h
M
t
-
h
(
t
)
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
,
f
d
s
f
f
d
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
,
ξ
˙
T
(
t
-
h
(
t
)
)
]
,
Θ
¯
=
(
A
¯
+
B
¯
K
¯
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
,
Υ
¯
1
=
[
Q
¯
1
+
[
P
¯
3
P
¯
1
P
¯
1
0
q
]
S
¯
1
S
¯
1
T
Q
¯
1
+
[
P
¯
4
P
¯
2
P
¯
2
0
q
]
]
,
Υ
¯
2
=
[
Q
¯
2
+
(
4
h
M
)
[
0
q
P
¯
3
P
¯
3
0
q
]
S
¯
2
S
¯
2
T
Q
¯
2
+
(
4
h
M
)
[
0
q
P
¯
4
P
¯
4
0
q
]
]
,
Ξ
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
11
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
R
¯
Ψ
12
T
+
Ψ
12
R
¯
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
T
Ψ
11
T
,
Ξ
¯
2
=
Ψ
21
N
¯
Ψ
21
T
-
Ψ
22
N
¯
Ψ
22
T
+
Ψ
23
N
¯
Ψ
24
T
+
Ψ
24
N
¯
Ψ
23
T
,
Ξ
¯
3
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
31
G
¯
Ψ
31
T
-
(
1
-
h
˙
(
t
)
)
Ψ
32
G
¯
Ψ
32
T
f
s
d
f
s
d
f
f
d
+
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
G
¯
Ψ
34
T
+
Ψ
34
G
¯
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
T
dasdsaddfs
+
(
1
-
h
˙
(
t
)
)
Ψ
¯
35
M
¯
Ψ
35
T
-
Ψ
36
M
¯
Ψ
36
T
f
d
s
f
f
d
s
f
d
+
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
G
¯
Ψ
38
T
+
Ψ
38
G
¯
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
T
,
Ξ
¯
4
[
h
(
t
)
]
=
h
M
2
Ψ
41
Q
¯
1
Ψ
41
T
Ψ
42
Υ
1
Ψ
42
T
gfdggd
+
h
M
(
e
1
P
¯
1
e
1
T
-
e
2
P
¯
1
e
2
T
+
e
2
P
¯
2
e
2
T
-
e
3
P
¯
2
e
3
T
)
fsdfdsf
+
h
M
(
h
(
t
)
e
1
P
¯
3
e
1
T
-
(
h
M
-
h
(
t
)
)
e
1
P
¯
4
e
1
T
)
,
Ξ
¯
5
=
(
h
M
2
2
)
2
Ψ
5
Q
¯
2
Ψ
5
T
,
Ξ
¯
6
=
(
h
M
3
6
)
2
e
4
Q
¯
3
e
4
T
-
Ψ
6
Q
¯
3
Ψ
6
T
,
Ξ
¯
7
=
α
(
(
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
)
T
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
gff
+
(
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
)
T
)
,
Ξ
¯
=
Ξ
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
¯
2
+
Ξ
¯
3
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
¯
4
[
h
(
t
)
]
+
Ξ
¯
5
+
Ξ
¯
6
+
Ξ
¯
7
,
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ξ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
+
Γ
2
[
h
(
t
)
]
T
Λ
¯
+
Λ
¯
T
Γ
2
[
h
(
t
)
]
,
L
¯
2
=
[
C
1
,
-
H
]
X
e
1
T
+
B
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
,
Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
B
¯
w
Λ
¯
T
L
¯
2
T
⋆
-
γ
2
I
p
+
r
0
0
⋆
⋆
-
Υ
¯
2
0
⋆
⋆
⋆
-
I
s
]
,
and other notations are the same ones as (13). Now, the following theorem is given by the second result.Theorem 7.
For given scalars0
≤
h
M, 0
≤
h
d
<
1, and α
>
0, the augmented system (5) achieves the H
∞ output tracking performance γ for 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d, if there exist positive definite matrices R
¯
=
[
R
¯
i
j
]
7
×
7
∈
R
7
q
×
7
q, X
=
R
¯
11
∈
R
q
×
q
N
¯
∈
R
4
q
×
4
q, G
¯
∈
R
4
q
×
4
q, Q
¯
1
∈
R
2
q
×
2
q, Q
¯
2
∈
R
2
q
×
2
q, and Q
¯
3
∈
R
q
×
q, symmetric matrices P
¯
i
∈
R
q
×
q
(
i
=
1
,
…
,
4
), and any matrices S
¯
1
∈
R
2
q
×
2
q, S
¯
2
∈
R
2
q
×
2
q, Λ
¯
∈
R
4
q
×
9
q, and Y
∈
R
m
×
q satisfying the following LMIs:
(36)
Φ
¯
j
,
k
<
0
,
(
j
,
k
=
1,2
)
,
Υ
¯
i
≥
0
,
(
i
=
1,2
)
,
where Φ
¯
j
,
k are the four vertices of Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
] with the bounds of h
(
t
) and h
˙
(
t
), that is h
M and h
D when j
=
k
=
1, h
M and -
h
D when j
<
k, 0 and h
D when j
>
k, and 0 and -
h
D when j
=
k
=
2. Moreover, if the above conditions are feasible, a desired H
∞ tracking controller gain matrix is obtained by K
¯
=
[
K
1
,
K
2
]
=
Y
X
-
1.Proof.
For positive definite matricesR
∈
R
7
q
×
7
q, N
∈
R
4
q
×
4
q, G
∈
R
4
q
×
4
q, Q
1
∈
R
2
q
×
2
q, Q
2
∈
R
2
q
×
2
q, and Q
3
∈
R
q
×
q, let us consider the following Lyapunov-Krasovskii functional candidate as
(37)
V
^
(
t
)
=
∑
i
=
1
6
‍
V
^
i
(
t
)
,
where
(38)
V
^
1
(
t
)
=
Ω
^
1
T
(
t
)
R
Ω
^
1
(
t
)
,
V
^
2
(
t
)
=
∫
t
-
h
M
t
‍
Ω
^
2
T
(
t
,
s
)
N
Ω
^
2
(
t
,
s
)
d
s
,
V
^
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
Ω
^
2
T
(
t
,
s
)
G
Ω
^
2
(
t
,
s
)
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
Ω
^
2
T
(
t
,
s
)
M
Ω
^
2
(
t
,
s
)
d
s
,
V
^
4
(
t
)
=
h
M
∫
t
-
h
M
t
‍
∫
s
t
‍
[
ξ
(
u
)
ξ
˙
(
u
)
]
T
Q
1
[
ξ
(
u
)
ξ
˙
(
u
)
]
d
u
d
s
,
V
^
5
(
t
)
=
(
h
M
2
2
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
[
ξ
(
v
)
ξ
˙
(
v
)
]
T
Q
2
[
ξ
(
v
)
ξ
˙
(
v
)
]
d
v
d
u
d
s
,
V
^
6
(
t
)
=
(
h
M
3
6
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
∫
v
t
‍
ξ
˙
T
(
λ
)
Q
3
ξ
˙
(
λ
)
d
λ
d
v
d
u
d
s
,
with Ω
^
1
(
t
)
=
[
ξ
T
(
t
)
,
ξ
T
(
t
-
h
(
t
)
)
,
ξ
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
]
T, and Ω
^
2
(
t
,
s
)
=
[
ξ
T
(
s
)
,
ξ
˙
T
(
s
)
,
∫
s
t
‍
ξ
˙
T
(
u
)
d
u
,
∫
s
t
‍
ξ
T
(
u
)
d
u
]
T. By using similar method in the proof of Theorem 5, an upper-bound of V
^
˙
(
t
) can be obtained as follows:
(39)
V
^
˙
(
t
)
≤
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
,
with
(40)
Υ
1
≥
0
,
Υ
2
≥
0
.
Since zero equality Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
=
0 can be obtained from the dynamic equation of the augmented system (5), the following zero equality for any matrix Z
∈
R
q
×
q is considered:
(41)
0
=
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
Z
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
.
For R
=
[
R
i
j
]
7
×
7
∈
R
7
q
×
7
q and a given scalar α
>
0, let us define Z
=
α
R
11. By adding (41) into (39), a new upper-bound of V
^
˙
(
t
) can be
(42)
V
^
˙
(
t
)
≤
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
+
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
α
R
11
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
.
Since V
^
(
t
)
|
t
=
0
=
0 and V
^
(
t
)
|
t
=
∞
≥
0 are satisfied under the zero initial condition, H
∞ performance index in (7) leads to the following inequality:
(43)
J
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
]
d
s
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
-
∫
0
∞
‍
V
^
˙
(
s
)
d
s
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
-
V
^
(
t
)
|
t
=
∞
+
V
^
(
t
)
|
t
=
0
≤
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
<
0
.
If inequality (43) is satisfied, the augmented system (5) achieves the H
∞ output tracking performance γ under the obtained H
∞ tracking controller (4). With e
T
(
t
)
e
(
t
)
=
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
T
[
C
¯
1
T
C
¯
2
T
]
[
C
¯
1
T
C
¯
2
T
]
T
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
], (43) is equivalent to the following inequality:
(44)
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
+
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
α
R
11
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
+
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
T
[
C
¯
1
T
C
¯
2
T
]
[
C
¯
1
T
C
¯
2
T
]
T
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
-
γ
2
v
T
(
t
)
v
(
t
)
=
[
ζ
¯
(
t
)
v
(
t
)
]
T
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
[
ζ
¯
(
t
)
v
(
t
)
]
<
0
,
where
(45)
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
+
α
(
e
1
T
+
e
4
T
)
T
R
11
Θ
¯
+
α
Θ
¯
T
R
11
(
e
1
T
+
e
4
T
)
+
(
C
¯
1
e
1
T
+
C
¯
2
e
2
T
)
T
(
C
¯
1
e
1
T
+
C
¯
2
e
2
T
)
,
Π
2
=
α
(
e
1
T
+
e
4
T
)
T
R
11
B
w
,
Π
3
=
-
γ
2
I
p
+
r
.
Therefore, inequality (44) is equivalent to the following condition:
(46)
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
<
0
.
Let us define X
=
R
11
-
1
>
0 and Σ
i
=
diag
⁡
{
X
,
…
,
X
︸
i
}
∈
R
i
q
×
i
q. For example, Σ
1
=
X and Σ
2
=
diag
⁡
{
X
,
X
}. And other notations are defined as R
¯
=
Σ
7
T
R
Σ
7, N
¯
=
Σ
4
T
N
Σ
4, G
¯
=
Σ
4
T
G
Σ
4, M
¯
=
Σ
4
T
M
Σ
4, Q
¯
1
=
Σ
2
T
Q
1
Σ
2, Q
¯
2
=
Σ
2
T
Q
2
Σ
2, Q
¯
3
=
Σ
1
T
Q
3
Σ
1,P
¯
1
=
Σ
1
T
P
1
Σ
1, P
¯
2
=
Σ
1
T
P
1
Σ
1, P
¯
3
=
Σ
1
T
P
3
Σ
1, P
¯
4
=
Σ
1
T
P
4
Σ
1, S
¯
1
=
Σ
2
T
S
1
Σ
2, S
¯
2
=
Σ
2
T
S
1
Σ
2, and Y
=
K
¯
X. By pre- and post-multiplying (40) and (46), respectively, by Σ
4 and diag
⁡
{
Σ
10
,
I
p
+
r
}, the following inequalities can be obtained:
(47)
0
≤
Σ
4
Υ
1
Σ
4
=
Υ
¯
1
,
0
≤
Σ
4
Υ
2
Σ
4
=
Υ
¯
2
,
(48)
[
Σ
10
0
10
q
×
(
p
+
r
)
0
(
p
+
r
)
×
10
q
I
p
+
r
]
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
×
[
Σ
10
0
10
q
×
(
p
+
r
)
0
(
p
+
r
)
×
10
q
I
p
+
r
]
=
[
Π
-
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
-
2
Π
-
2
T
Π
-
3
]
<
0
,
where
(49)
Π
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
(
Ξ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
[
h
(
t
)
]
T
Υ
¯
2
Γ
[
h
(
t
)
]
)
+
(
[
C
1
,
-
H
]
X
e
1
T
+
D
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
)
T
×
(
[
C
1
,
-
H
]
X
e
1
T
+
D
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
)
Π
¯
2
=
α
(
e
1
T
+
e
4
T
)
T
B
w
,
Π
¯
3
=
-
γ
2
I
p
+
r
.
Also, by using Schur complements and Lemma 4 with any matrix Λ
¯
∈
R
4
q
×
(
10
q
+
p
+
r
), inequality (47) is equivalent to
(50)
Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
B
¯
w
Λ
¯
T
L
¯
2
T
⋆
-
γ
2
I
p
+
r
0
0
⋆
⋆
-
Υ
¯
2
0
⋆
⋆
⋆
-
I
s
]
<
0
,
where L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
] and L
¯
2 are defined in (35). Note that inequality (50) is affinely dependent on h
(
t
) and h
˙
(
t
). Therefore, Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
<
0 holds if
(51)
Φ
¯
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
h
d
]
<
0
.
This completes our proof.Remark 8.
This subsection concludes the criterion ofH
∞ output tracking controller design for linear systems with time-varying delays and disturbances. To obtain an H
∞ output tracking controller gain which provides enhanced feasible region, the authors introduce the improved delay-dependent stability criterion as shown in Theorem 5. Then, based on the results of Theorem 5, the designing method of H
∞ output tracking controller gain was introduced in Theorem 7.Remark 9.
Theorems5 and 7 have some restriction such as h
d
<
1. In order to remove the restrictive condition h
d
<
1 in Theorem 5, V
1
(
t
) and V
3
(
t
) of Lyapunov-Krasovskii functional (15) should be changed as follows:
(52)
V
1
(
t
)
=
Ω
1
T
(
t
)
R
Ω
1
(
t
)
,
V
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
G
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
M
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
d
s
,
where R
∈
R
6
n
×
6
n
>
0, G
∈
R
3
n
×
3
n
>
0, M
∈
R
3
n
×
3
n
>
0, and Ω
1
(
t
)
=
[
x
T
(
t
)
,
x
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T. Then, by utilizing the following vector,
(53)
ζ
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
x
˙
T
(
t
)
,
f
d
s
x
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
f
d
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
f
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T
,
as an augmented vector in deriving a stability condition, condition h
d
<
1 can be removed. Similarly, in Theorem 7, the restrictive condition h
d
<
1 can be also removed by using the following V
^
1
(
t
) and V
^
3
(
t
) of Lyapunov-Krasovskii functional (37) with positive matrices R
∈
R
6
q
×
6
q, G
∈
R
3
q
×
3
q, and M
∈
R
3
q
×
3
q as
(54)
V
^
1
(
t
)
=
Ω
^
1
T
(
t
)
R
Ω
^
1
(
t
)
,
V
^
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
T
G
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
T
M
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
d
s
,
where Ω
^
1
(
t
)
=
[
ξ
T
(
t
)
,
ξ
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
].


## 3.1. Stability Analysis

Let us consider the system (1) with w
(
t
)
=
0 and u
(
t
)
=
0 given by
(12)
x
˙
(
t
)
=
A
x
(
t
)
+
A
d
x
(
t
-
h
(
t
)
)
.
In this subsection, a delay-dependent stability criterion for system (12) is derived. For simplicity of matrix and vector representation, e
i
(
i
=
1
,
…
,
10
)
∈
R
10
n
×
n are defined as block entry matrices which will be used in Theorem 5. For example, e
1
=
[
I
n
,
0
n
×
9
n
]
T and e
5
=
[
0
n
×
4
n
,
I
n
,
0
n
×
5
n
]
T. The other notations are defined as
(13)
ζ
T
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
x
˙
T
(
t
)
,
f
d
s
f
f
s
d
f
x
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
f
d
s
f
s
d
d
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
f
d
s
f
d
s
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
x
˙
T
(
t
-
h
(
t
)
)
]
,
Θ
=
(
A
e
1
T
+
A
d
e
2
T
-
e
4
T
)
,
Ψ
11
=
[
e
1
,
e
2
,
e
3
,
e
6
,
e
7
,
e
8
,
e
9
]
,
Ψ
12
=
[
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
,
e
7
,
e
10
]
,
Ψ
21
=
[
e
1
,
e
4
,
0
10
n
×
n
,
0
10
n
×
n
]
,
Ψ
22
=
[
e
3
,
e
5
,
e
1
-
e
3
,
e
6
+
e
7
]
,
Ψ
23
=
[
e
6
+
e
7
,
e
1
-
e
3
,
h
M
e
1
-
e
6
-
e
7
,
e
8
+
e
9
]
,
Ψ
24
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
31
=
[
e
1
,
e
4
,
0
10
n
×
n
,
0
10
n
×
n
]
,
Ψ
32
=
[
e
2
,
e
10
,
e
1
-
e
2
,
e
6
]
,
Ψ
33
[
h
(
t
)
]
=
[
e
6
,
e
1
-
e
3
,
h
(
t
)
e
1
-
e
6
,
e
8
]
,
Ψ
34
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
35
=
[
e
2
,
e
10
,
e
1
-
e
2
,
e
6
]
,
Ψ
36
=
[
e
3
,
e
5
,
e
1
-
e
3
,
e
6
+
e
7
]
,
Ψ
37
[
h
(
t
)
]
=
[
e
7
,
e
2
-
e
3
,
(
h
M
-
h
(
t
)
)
e
1
-
e
7
,
e
9
]
,
Ψ
38
=
[
0
10
n
×
n
,
0
10
n
×
n
,
e
4
,
e
1
]
,
Ψ
41
=
[
e
1
,
e
4
]
,
Ψ
42
=
[
e
6
,
e
1
-
e
2
,
e
7
,
e
2
-
e
3
]
,
Ψ
5
=
[
e
1
,
e
4
]
,
Ψ
6
=
(
(
h
M
2
2
)
e
1
-
e
8
-
e
9
)
,
Υ
1
=
[
Q
1
+
[
P
3
P
1
P
1
0
n
]
S
1
S
1
T
Q
1
+
[
P
4
P
2
P
2
0
n
]
]
,
Υ
2
=
[
Q
2
+
(
4
h
M
)
[
0
n
P
3
P
3
0
n
]
S
2
S
2
T
Q
2
+
(
4
h
M
)
[
0
n
P
4
P
4
0
n
]
]
,
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
0
n
×
3
n
,
I
n
,
0
n
×
4
n
0
n
×
7
n
,
(
1
-
h
˙
(
t
)
)
I
n
0
n
×
4
n
,
I
n
,
0
n
×
3
n
I
n
,
-
(
1
-
h
˙
(
t
)
)
I
n
,
0
n
×
6
n
0
n
,
(
1
-
h
˙
(
t
)
)
I
n
,
-
I
n
,
0
n
×
5
n
h
(
t
)
I
n
,
0
n
×
4
n
,
-
(
1
-
h
˙
(
t
)
)
I
n
,
0
n
×
2
n
(
h
M
-
h
(
t
)
)
I
n
,
0
n
×
4
n
,
-
h
˙
(
t
)
I
n
,
0
n
×
2
n
]
,
Γ
2
[
h
(
t
)
]
=
[
0
n
×
3
n
,
I
n
,
0
n
h
(
t
)
I
n
,
-
I
n
,
0
n
×
3
n
0
n
×
4
n
,
I
n
(
h
M
-
h
(
t
)
)
I
n
,
0
n
,
-
I
n
,
0
n
×
2
n
]
[
e
1
T
e
6
T
e
7
T
e
8
T
e
9
T
]
,
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
11
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
R
Ψ
12
T
+
Ψ
12
R
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
T
Ψ
11
T
,
Ξ
2
=
Ψ
21
N
Ψ
21
T
-
Ψ
22
N
Ψ
22
T
+
Ψ
23
N
Ψ
24
T
+
Ψ
24
N
Ψ
23
T
,
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
31
G
Ψ
31
T
-
(
1
-
h
˙
(
t
)
)
Ψ
32
G
Ψ
32
T
d
s
a
d
d
s
d
s
a
d
+
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
G
Ψ
34
T
+
Ψ
34
G
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
T
d
s
a
d
d
s
d
s
a
d
+
(
1
-
h
˙
(
t
)
)
Ψ
35
M
Ψ
35
T
-
Ψ
36
M
Ψ
36
T
d
s
a
d
d
s
d
s
a
d
+
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
M
Ψ
38
T
+
Ψ
38
M
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
T
,
Ξ
zero
[
h
(
t
)
]
=
h
M
(
e
1
P
1
e
1
T
-
e
2
P
1
e
2
T
+
e
2
P
2
e
2
T
-
e
3
P
2
e
3
T
)
d
s
a
d
d
s
d
s
a
d
+
h
M
(
h
(
t
)
e
1
P
3
e
1
T
-
(
h
M
-
h
(
t
)
)
e
1
P
4
e
1
T
)
,
Ξ
4
[
h
(
t
)
]
=
h
M
2
Ψ
41
Q
1
Ψ
41
T
-
Ψ
42
Υ
1
Ψ
42
T
+
Ξ
zero
[
h
(
t
)
]
,
Ξ
5
=
(
h
M
2
2
)
2
Ψ
5
Q
2
Ψ
5
T
,
Ξ
6
=
(
h
M
3
6
)
2
e
4
Q
3
e
4
T
-
Ψ
6
Q
3
Ψ
6
T
,
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
2
+
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
d
s
a
d
d
s
d
s
a
d
+
Ξ
4
[
h
(
t
)
]
+
Ξ
5
+
Ξ
6
,
L
[
h
(
t
)
,
h
˙
(
t
)
]
=
Θ
⊥
T
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
Θ
⊥
+
Θ
⊥
T
Γ
2
[
h
(
t
)
]
T
Λ
d
s
a
d
d
s
d
s
a
d
+
Λ
T
Γ
2
[
h
(
t
)
]
Θ
⊥
,
Φ
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
[
h
(
t
)
,
h
˙
(
t
)
]
Λ
T
Λ
-
Υ
2
]
,
where Θ
⊥ is the right orthogonal complement of Θ.Now, a delay-dependent stability of the system (12) is given as Theorem 5.Theorem 5.
For given scalars0
≤
h
M and 0
≤
h
d
<
1, the system (12) is asymptotically stable for 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d, if there exist positive definite matrices R
∈
R
7
n
×
7
n, N
∈
R
4
n
×
4
n, G
∈
R
4
n
×
4
n, M
∈
R
4
n
×
4
n, Q
1
∈
R
2
n
×
2
n, Q
2
∈
R
2
n
×
2
n, and Q
3
∈
R
n
×
n, symmetric matrices P
i
∈
R
n
×
n
(
i
=
1
,
…
,
4
), and any matrices S
1
∈
R
2
n
×
2
n, S
2
∈
R
2
n
×
2
n, and Λ
∈
R
4
n
×
9
n, satisfying the following LMIs:
(14)
Φ
j
,
k
<
0
,
(
j
,
k
=
1,2
)
,
Υ
i
≥
0
,
(
i
=
1,2
)
,
where Φ
j
,
k are the four vertices of Φ
[
h
(
t
)
,
h
˙
(
t
)
] with the bounds of h
(
t
) and h
˙
(
t
), that is, h
M and h
D when j
=
k
=
1, h
M and -
h
D when j
<
k, 0 and h
D when j
>
k, and 0 and -
h
D when j
=
k
=
2.Proof.
For positive definite matricesR
∈
R
7
n
×
7
n, N
∈
R
4
n
×
4
n, G
∈
R
4
n
×
4
n, Q
1
∈
R
2
n
×
2
n, Q
2
∈
R
2
n
×
2
n, and Q
3
∈
R
n
×
n, let us consider the following Lyapunov-Krasovskii functional candidate as
(15)
V
(
t
)
=
∑
i
=
1
6
‍
V
i
(
t
)
,
where
(16)
V
1
(
t
)
=
Ω
1
T
(
t
)
R
Ω
1
(
t
)
,
V
2
(
t
)
=
∫
t
-
h
M
t
‍
Ω
2
T
(
t
,
s
)
N
Ω
2
(
t
,
s
)
d
s
,
V
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
Ω
2
T
(
t
,
s
)
G
Ω
2
(
t
,
s
)
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
Ω
2
T
(
t
,
s
)
M
Ω
2
(
t
,
s
)
d
s
,
V
4
(
t
)
=
h
M
∫
t
-
h
M
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
T
Q
1
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
,
V
5
(
t
)
=
(
h
M
2
2
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
[
x
(
v
)
x
˙
(
v
)
]
T
Q
2
[
x
(
v
)
x
˙
(
v
)
]
d
v
d
u
d
s
,
V
6
(
t
)
=
(
h
M
3
6
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
∫
v
t
‍
x
˙
T
(
λ
)
Q
3
x
˙
(
λ
)
d
λ
d
v
d
u
d
s
,
with Ω
1
(
t
)
=
[
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T and Ω
2
(
t
,
s
)
=
[
x
T
(
s
)
,
x
˙
T
(
s
)
,
∫
s
t
‍
x
˙
T
(
u
)
d
u
,
∫
s
t
‍
x
T
(
u
)
d
u
]
T.
Now, calculating the time-derivative ofV
1
(
t
) yields
(17)
V
˙
1
(
t
)
=
2
Ω
1
(
t
)
R
×
[
x
˙
(
t
)
(
1
-
h
˙
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
(
1
-
h
˙
(
t
)
)
x
(
t
-
h
M
)
(
1
-
h
˙
(
t
)
)
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
h
(
t
)
x
(
t
)
-
(
1
-
h
˙
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
(
(
h
M
-
h
(
t
)
)
x
(
t
)
+
(
1
-
h
˙
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
-
∫
t
-
h
M
t
‍
x
(
s
)
d
s
)
]
=
ζ
T
(
t
)
Ξ
1
[
h
(
t
)
,
h
˙
(
t
)
]
ζ
(
t
)
.
Also, V
˙
2
(
t
) can be represented as
(18)
V
˙
2
(
t
)
=
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
T
N
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
-
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
u
)
d
u
]
T
×
N
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
x
(
u
)
d
u
]
+
2
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
x
(
t
)
-
x
(
s
)
∫
s
t
‍
x
(
u
)
d
u
]
T
d
s
N
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
=
ζ
T
(
t
)
Ξ
2
ζ
(
t
)
.
The time-derivative of V
3
(
t
) can be given as follows:
(19)
V
˙
3
(
t
)
=
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
T
G
[
x
(
t
)
x
˙
(
t
)
0
n
×
1
0
n
×
1
]
-
(
1
-
h
˙
(
t
)
)
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
T
×
G
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
+
2
∫
t
-
h
(
t
)
t
‍
[
x
(
s
)
x
˙
(
t
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
d
s
G
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
+
(
1
-
h
˙
(
t
)
)
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
T
×
M
[
x
(
t
-
h
(
t
)
)
x
˙
(
t
-
h
(
t
)
)
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
]
-
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
s
)
d
s
]
T
M
[
x
(
t
-
h
M
)
x
˙
(
t
-
h
M
)
x
(
t
)
-
x
(
t
-
h
M
)
∫
t
-
h
M
t
‍
x
(
s
)
d
s
]
+
2
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
(
h
M
-
h
(
t
)
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
]
T
×
M
[
0
n
×
1
0
n
×
1
x
˙
(
t
)
x
(
t
)
]
=
ζ
T
(
t
)
Ξ
3
[
h
(
t
)
,
h
˙
(
t
)
]
ζ
(
t
)
.
Calculating V
˙
4
(
t
) gives
(20)
V
˙
4
(
t
)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
Q
1
[
x
(
s
)
x
˙
(
s
)
]
d
s
.
Inspired by the work in [22], the following four zero equalities with symmetric matrices P
1, P
2, P
3, and P
4 are considered:
(21)
0
=
x
T
(
t
)
(
h
M
P
1
)
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
(
h
M
P
1
)
x
(
t
-
h
(
t
)
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
1
x
˙
(
s
)
d
s
,
0
=
x
T
(
t
-
h
(
t
)
)
(
h
M
P
2
)
x
(
t
-
h
(
t
)
)
-
x
T
(
t
-
h
M
)
(
h
M
P
2
)
x
(
t
-
h
M
)
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
P
2
x
˙
(
s
)
d
s
,
0
=
x
T
(
t
)
(
h
M
h
(
t
)
P
3
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
3
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
,
0
=
x
T
(
t
)
(
h
M
(
h
M
-
h
(
t
)
)
P
4
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
4
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
u
˙
(
s
)
d
u
d
s
.
By adding (21) into (20), V
˙
4
(
t
) can be represented as follows:
(22)
V
˙
4
(
t
)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
Q
1
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
x
T
(
t
)
(
h
M
P
1
)
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
×
(
h
M
P
1
)
x
(
t
-
h
(
t
)
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
1
x
˙
(
s
)
d
s
+
x
T
(
t
-
h
(
t
)
)
(
h
M
P
2
)
x
(
t
-
h
(
t
)
)
-
x
T
(
t
-
h
M
)
(
h
M
P
2
)
x
(
t
-
h
M
)
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
P
2
x
˙
(
s
)
d
s
+
x
T
(
t
)
(
h
M
h
(
t
)
P
3
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
3
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
x
T
(
t
)
(
h
M
(
h
M
-
h
(
t
)
)
P
4
)
x
(
t
)
-
h
M
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
P
4
x
(
s
)
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
u
˙
(
s
)
d
u
d
s
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
-
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
From Lemma 1, an upper-bound of V
˙
4
(
t
) can be calculated as
(23)
V
˙
4
(
t
)
≤
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
h
(
t
)
∫
t
-
h
M
t
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
-
h
M
h
M
-
h
(
t
)
×
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
x
˙
(
s
)
]
T
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
[
x
(
s
)
x
˙
(
s
)
]
d
s
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
(24)
=
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
+
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
T
×
[
1
ϕ
1
(
t
)
(
Q
1
+
[
P
3
P
1
P
1
0
n
]
)
0
2
n
0
2
n
1
1
-
ϕ
1
(
t
)
(
Q
1
+
[
P
4
P
2
P
2
0
n
]
)
]
×
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
,
where ϕ
1
(
t
)
=
h
(
t
)
/
h
M which satisfies 0
≤
ϕ
1
(
t
)
≤
1. It should be noted that if h
(
t
) is zero, then the term of ∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s and x
(
t
)
-
x
(
t
-
h
(
t
)
) are zero. Also, if h
(
t
) is h
M, then the term of ∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s and x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
) are zero. Thus, inequality (24) still holds.
By using Lemma2 with Υ
1
≥
0 which is defined in (13), a new upper-bound of V
˙
4
(
t
) can be obtained as
(25)
V
˙
4
(
t
)
≤
h
M
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
1
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
T
×
Υ
1
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
×
[
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
x
(
t
)
-
x
(
t
-
h
(
t
)
)
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
x
(
t
-
h
(
t
)
)
-
x
(
t
-
h
M
)
]
+
ζ
T
(
t
)
Ξ
zero
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
ζ
T
(
t
)
(
h
M
2
Ψ
41
Q
1
Ψ
41
T
-
Ψ
42
Υ
1
Ψ
42
T
+
Ξ
zero
[
h
(
t
)
]
)
︸
Ξ
4
[
h
(
t
)
]
ζ
(
t
)
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
TheV
˙
5
(
t
) is calculated as(26)
V
˙
5
(
t
)
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
2
2
∫
t
-
h
M
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
Q
2
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
-
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
︸
0
-
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
︸
0
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
h
M
2
2
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
-
h
M
2
2
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.Applying Lemma 1 to (26), the upper-bound of V
˙
5
(
t
) can be derived as
(27)
V
˙
5
(
t
)
≤
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
(
h
M
h
(
t
)
)
2
(
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
T
×
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
(
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
-
(
h
M
2
h
M
2
-
h
(
t
)
2
)
(
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
T
×
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
(
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
[
x
(
u
)
x
˙
(
u
)
]
d
u
d
s
)
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s
]
T
×
[
1
ϕ
2
(
t
)
(
Q
2
+
4
h
M
[
0
n
P
3
P
3
0
n
]
)
0
2
n
0
2
n
1
1
-
ϕ
2
(
t
)
(
Q
2
+
4
h
M
[
0
n
P
4
P
4
0
n
]
)
]
×
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s
]
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
,
where ϕ
2
(
t
)
=
(
h
(
t
)
/
h
M
)
2 satisfies 0
≤
ϕ
2
(
t
)
≤
1. Note that when h
(
t
) is zero, the terms of ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
˙
d
u
d
s are zero, and when h
(
t
) is h
M, the terms of ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
˙
(
u
)
d
u
d
s are zero. Thus, inequality (27) still holds.
By the use of Lemma2 to (27) with Υ
2
≥
0 defined in (13), the following inequality can be obtained:
(28)
V
˙
5
(
t
)
≤
(
h
M
2
2
)
2
[
x
(
t
)
x
˙
(
t
)
]
T
Q
2
[
x
(
t
)
x
˙
(
t
)
]
-
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
h
(
t
)
x
(
t
)
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
(
h
(
t
)
-
h
M
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
]
T
Υ
2
×
[
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
h
(
t
)
x
(
t
)
-
∫
t
-
h
(
t
)
t
‍
x
(
s
)
d
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
(
h
(
t
)
-
h
M
)
x
(
t
)
-
∫
t
-
h
M
t
-
h
(
t
)
‍
x
(
s
)
d
s
]
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
=
ζ
T
(
t
)
(
Ξ
5
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
[
2
h
(
t
)
]
)
ζ
(
t
)
+
2
h
M
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
P
3
x
˙
(
u
)
d
u
d
s
+
2
h
M
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
P
4
x
˙
(
s
)
d
u
d
s
.
The time-derivative of V
6
(
t
) is obtained as follows:
(29)
V
˙
6
(
t
)
=
(
h
M
3
6
)
2
x
˙
T
(
t
)
Q
3
x
˙
(
t
)
-
(
h
M
3
6
)
×
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
x
˙
T
(
v
)
Q
3
x
˙
(
v
)
d
v
d
u
d
s
.
Using Lemma 1 to (29) yields a new upper-bound of V
˙
6
(
t
) as
(30)
V
˙
6
(
t
)
≤
(
h
M
3
6
)
2
x
˙
T
(
t
)
Q
3
x
˙
(
t
)
-
(
h
M
2
2
x
(
t
)
-
∫
t
-
h
M
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
)
T
×
Q
3
(
h
M
2
2
x
(
t
)
-
∫
t
-
h
M
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s
)
=
ζ
T
(
t
)
Ξ
6
ζ
(
t
)
.
By summation of (17)–(30), an upper-bound of V
˙
(
t
) is obtained as follows:
(31)
V
˙
(
t
)
≤
ζ
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
(
t
)
.
From the dynamic equation of the system (12), zero equality 0
=
Θ
ζ
(
t
) can be obtained. Therefore, from Lemma 3, a stability criterion for the system (12) can be obtained as follows:
(32)
Θ
⊥
T
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
Θ
⊥
<
0
.
By using Lemma 4 with any matrix U
∈
R
4
n
×
9
n, inequality (32) is equivalent to
(33)
Φ
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
Θ
⊥
T
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
Θ
⊥
+
Θ
⊥
T
Γ
2
[
h
(
t
)
]
T
Λ
+
Λ
T
Γ
2
[
h
(
t
)
]
Θ
⊥
Λ
T
Λ
-
Υ
2
]
=
[
L
[
h
(
t
)
,
h
˙
(
t
)
]
Λ
T
Λ
-
Υ
2
]
<
0
,
where L
[
h
(
t
)
,
h
˙
(
t
)
] is defined in (13). Note that inequality (33) is affinely dependent on h
(
t
) and h
˙
(
t
) where 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d
<
1. So, Φ
[
h
(
t
)
,
h
˙
(
t
)
]
<
0 is satisfied if
(34)
Φ
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
h
d
]
<
0
,
Φ
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
h
d
]
<
0
.
This completes our proof.Remark 6.
Compared with the existing works, the main differences of Lyapunov-Krasovskii functionals areV
2
(
t
) and V
3
(
t
). In detail, V
2
(
t
) and V
3
(
t
) have augmented vectors ∫
t
-
h
(
t
)
t
‍
x
˙
(
s
)
d
s, respectively. With this consideration, some new cross terms are calculated and utilized in stability condition. Furthermore, the time-varying delayed state x
(
t
-
h
(
t
)
) and double integral terms such as ∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
(
u
)
d
u
d
s and ∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
(
u
)
d
u
d
s are utilized as an element of the augmented vectors of V
1
(
t
). In next section, the results for Theorem 5 will show larger delay bounds than the works in other literature to confirm the less conservatism of proposed criterion.


## 3.2.H
∞ Output Tracking Controller Design

In this subsection, anH
∞ output tracking controller design method for the system (5) will be introduced based on the result of Theorem 5. For simplicity of matrix and vector representation, the scalar q which will be used in representing dimensions of matrices and vectors is defined as q
=
n
+
r. For example, block entry matrices e
i
(
i
=
1
,
…
,
10
)
∈
R
10
q
×
q will be used in Theorem 7. The other notations are defined as
(35)
ζ
¯
T
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
ξ
T
(
t
)
,
ξ
T
(
t
-
h
(
t
)
)
,
ξ
T
(
t
-
h
M
)
,
ξ
˙
T
(
t
)
,
f
d
s
f
d
s
f
f
ξ
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
f
s
d
f
d
s
f
∫
t
-
h
M
t
-
h
(
t
)
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
,
f
d
s
f
f
d
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
,
ξ
˙
T
(
t
-
h
(
t
)
)
]
,
Θ
¯
=
(
A
¯
+
B
¯
K
¯
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
,
Υ
¯
1
=
[
Q
¯
1
+
[
P
¯
3
P
¯
1
P
¯
1
0
q
]
S
¯
1
S
¯
1
T
Q
¯
1
+
[
P
¯
4
P
¯
2
P
¯
2
0
q
]
]
,
Υ
¯
2
=
[
Q
¯
2
+
(
4
h
M
)
[
0
q
P
¯
3
P
¯
3
0
q
]
S
¯
2
S
¯
2
T
Q
¯
2
+
(
4
h
M
)
[
0
q
P
¯
4
P
¯
4
0
q
]
]
,
Ξ
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
11
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
R
¯
Ψ
12
T
+
Ψ
12
R
¯
Γ
1
[
h
(
t
)
,
h
˙
(
t
)
]
T
Ψ
11
T
,
Ξ
¯
2
=
Ψ
21
N
¯
Ψ
21
T
-
Ψ
22
N
¯
Ψ
22
T
+
Ψ
23
N
¯
Ψ
24
T
+
Ψ
24
N
¯
Ψ
23
T
,
Ξ
¯
3
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ψ
31
G
¯
Ψ
31
T
-
(
1
-
h
˙
(
t
)
)
Ψ
32
G
¯
Ψ
32
T
f
s
d
f
s
d
f
f
d
+
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
G
¯
Ψ
34
T
+
Ψ
34
G
¯
Ψ
33
[
h
(
t
)
,
h
˙
(
t
)
]
T
dasdsaddfs
+
(
1
-
h
˙
(
t
)
)
Ψ
¯
35
M
¯
Ψ
35
T
-
Ψ
36
M
¯
Ψ
36
T
f
d
s
f
f
d
s
f
d
+
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
G
¯
Ψ
38
T
+
Ψ
38
G
¯
Ψ
37
[
h
(
t
)
,
h
˙
(
t
)
]
T
,
Ξ
¯
4
[
h
(
t
)
]
=
h
M
2
Ψ
41
Q
¯
1
Ψ
41
T
Ψ
42
Υ
1
Ψ
42
T
gfdggd
+
h
M
(
e
1
P
¯
1
e
1
T
-
e
2
P
¯
1
e
2
T
+
e
2
P
¯
2
e
2
T
-
e
3
P
¯
2
e
3
T
)
fsdfdsf
+
h
M
(
h
(
t
)
e
1
P
¯
3
e
1
T
-
(
h
M
-
h
(
t
)
)
e
1
P
¯
4
e
1
T
)
,
Ξ
¯
5
=
(
h
M
2
2
)
2
Ψ
5
Q
¯
2
Ψ
5
T
,
Ξ
¯
6
=
(
h
M
3
6
)
2
e
4
Q
¯
3
e
4
T
-
Ψ
6
Q
¯
3
Ψ
6
T
,
Ξ
¯
7
=
α
(
(
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
)
T
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
gff
+
(
(
e
1
+
e
¯
4
)
(
(
A
¯
1
X
+
B
¯
Y
)
e
1
T
+
A
¯
d
e
2
T
-
e
4
T
)
)
T
)
,
Ξ
¯
=
Ξ
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
¯
2
+
Ξ
¯
3
[
h
(
t
)
,
h
˙
(
t
)
]
+
Ξ
¯
4
[
h
(
t
)
]
+
Ξ
¯
5
+
Ξ
¯
6
+
Ξ
¯
7
,
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
Ξ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
+
Γ
2
[
h
(
t
)
]
T
Λ
¯
+
Λ
¯
T
Γ
2
[
h
(
t
)
]
,
L
¯
2
=
[
C
1
,
-
H
]
X
e
1
T
+
B
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
,
Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
B
¯
w
Λ
¯
T
L
¯
2
T
⋆
-
γ
2
I
p
+
r
0
0
⋆
⋆
-
Υ
¯
2
0
⋆
⋆
⋆
-
I
s
]
,
and other notations are the same ones as (13). Now, the following theorem is given by the second result.Theorem 7.
For given scalars0
≤
h
M, 0
≤
h
d
<
1, and α
>
0, the augmented system (5) achieves the H
∞ output tracking performance γ for 0
≤
h
(
t
)
≤
h
M and |
h
˙
(
t
)
|
≤
h
d, if there exist positive definite matrices R
¯
=
[
R
¯
i
j
]
7
×
7
∈
R
7
q
×
7
q, X
=
R
¯
11
∈
R
q
×
q
N
¯
∈
R
4
q
×
4
q, G
¯
∈
R
4
q
×
4
q, Q
¯
1
∈
R
2
q
×
2
q, Q
¯
2
∈
R
2
q
×
2
q, and Q
¯
3
∈
R
q
×
q, symmetric matrices P
¯
i
∈
R
q
×
q
(
i
=
1
,
…
,
4
), and any matrices S
¯
1
∈
R
2
q
×
2
q, S
¯
2
∈
R
2
q
×
2
q, Λ
¯
∈
R
4
q
×
9
q, and Y
∈
R
m
×
q satisfying the following LMIs:
(36)
Φ
¯
j
,
k
<
0
,
(
j
,
k
=
1,2
)
,
Υ
¯
i
≥
0
,
(
i
=
1,2
)
,
where Φ
¯
j
,
k are the four vertices of Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
] with the bounds of h
(
t
) and h
˙
(
t
), that is h
M and h
D when j
=
k
=
1, h
M and -
h
D when j
<
k, 0 and h
D when j
>
k, and 0 and -
h
D when j
=
k
=
2. Moreover, if the above conditions are feasible, a desired H
∞ tracking controller gain matrix is obtained by K
¯
=
[
K
1
,
K
2
]
=
Y
X
-
1.Proof.
For positive definite matricesR
∈
R
7
q
×
7
q, N
∈
R
4
q
×
4
q, G
∈
R
4
q
×
4
q, Q
1
∈
R
2
q
×
2
q, Q
2
∈
R
2
q
×
2
q, and Q
3
∈
R
q
×
q, let us consider the following Lyapunov-Krasovskii functional candidate as
(37)
V
^
(
t
)
=
∑
i
=
1
6
‍
V
^
i
(
t
)
,
where
(38)
V
^
1
(
t
)
=
Ω
^
1
T
(
t
)
R
Ω
^
1
(
t
)
,
V
^
2
(
t
)
=
∫
t
-
h
M
t
‍
Ω
^
2
T
(
t
,
s
)
N
Ω
^
2
(
t
,
s
)
d
s
,
V
^
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
Ω
^
2
T
(
t
,
s
)
G
Ω
^
2
(
t
,
s
)
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
Ω
^
2
T
(
t
,
s
)
M
Ω
^
2
(
t
,
s
)
d
s
,
V
^
4
(
t
)
=
h
M
∫
t
-
h
M
t
‍
∫
s
t
‍
[
ξ
(
u
)
ξ
˙
(
u
)
]
T
Q
1
[
ξ
(
u
)
ξ
˙
(
u
)
]
d
u
d
s
,
V
^
5
(
t
)
=
(
h
M
2
2
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
[
ξ
(
v
)
ξ
˙
(
v
)
]
T
Q
2
[
ξ
(
v
)
ξ
˙
(
v
)
]
d
v
d
u
d
s
,
V
^
6
(
t
)
=
(
h
M
3
6
)
∫
t
-
h
M
t
‍
∫
s
t
‍
∫
u
t
‍
∫
v
t
‍
ξ
˙
T
(
λ
)
Q
3
ξ
˙
(
λ
)
d
λ
d
v
d
u
d
s
,
with Ω
^
1
(
t
)
=
[
ξ
T
(
t
)
,
ξ
T
(
t
-
h
(
t
)
)
,
ξ
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
]
T, and Ω
^
2
(
t
,
s
)
=
[
ξ
T
(
s
)
,
ξ
˙
T
(
s
)
,
∫
s
t
‍
ξ
˙
T
(
u
)
d
u
,
∫
s
t
‍
ξ
T
(
u
)
d
u
]
T. By using similar method in the proof of Theorem 5, an upper-bound of V
^
˙
(
t
) can be obtained as follows:
(39)
V
^
˙
(
t
)
≤
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
,
with
(40)
Υ
1
≥
0
,
Υ
2
≥
0
.
Since zero equality Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
=
0 can be obtained from the dynamic equation of the augmented system (5), the following zero equality for any matrix Z
∈
R
q
×
q is considered:
(41)
0
=
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
Z
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
.
For R
=
[
R
i
j
]
7
×
7
∈
R
7
q
×
7
q and a given scalar α
>
0, let us define Z
=
α
R
11. By adding (41) into (39), a new upper-bound of V
^
˙
(
t
) can be
(42)
V
^
˙
(
t
)
≤
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
+
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
α
R
11
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
.
Since V
^
(
t
)
|
t
=
0
=
0 and V
^
(
t
)
|
t
=
∞
≥
0 are satisfied under the zero initial condition, H
∞ performance index in (7) leads to the following inequality:
(43)
J
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
]
d
s
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
-
∫
0
∞
‍
V
^
˙
(
s
)
d
s
=
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
-
V
^
(
t
)
|
t
=
∞
+
V
^
(
t
)
|
t
=
0
≤
∫
0
∞
‍
[
e
T
(
s
)
e
(
s
)
-
γ
2
v
T
(
s
)
v
(
s
)
+
V
^
˙
(
s
)
]
d
s
<
0
.
If inequality (43) is satisfied, the augmented system (5) achieves the H
∞ output tracking performance γ under the obtained H
∞ tracking controller (4). With e
T
(
t
)
e
(
t
)
=
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
T
[
C
¯
1
T
C
¯
2
T
]
[
C
¯
1
T
C
¯
2
T
]
T
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
], (43) is equivalent to the following inequality:
(44)
ζ
¯
T
(
t
)
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
ζ
¯
(
t
)
+
2
ζ
¯
T
(
t
)
(
e
1
T
+
e
4
T
)
T
α
R
11
(
Θ
¯
ζ
¯
(
t
)
+
B
¯
w
v
(
t
)
)
+
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
T
[
C
¯
1
T
C
¯
2
T
]
[
C
¯
1
T
C
¯
2
T
]
T
[
ξ
(
t
)
ξ
(
t
-
h
(
t
)
)
]
-
γ
2
v
T
(
t
)
v
(
t
)
=
[
ζ
¯
(
t
)
v
(
t
)
]
T
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
[
ζ
¯
(
t
)
v
(
t
)
]
<
0
,
where
(45)
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
(
Ξ
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
2
[
h
(
t
)
]
T
Υ
2
Γ
2
[
h
(
t
)
]
)
+
α
(
e
1
T
+
e
4
T
)
T
R
11
Θ
¯
+
α
Θ
¯
T
R
11
(
e
1
T
+
e
4
T
)
+
(
C
¯
1
e
1
T
+
C
¯
2
e
2
T
)
T
(
C
¯
1
e
1
T
+
C
¯
2
e
2
T
)
,
Π
2
=
α
(
e
1
T
+
e
4
T
)
T
R
11
B
w
,
Π
3
=
-
γ
2
I
p
+
r
.
Therefore, inequality (44) is equivalent to the following condition:
(46)
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
<
0
.
Let us define X
=
R
11
-
1
>
0 and Σ
i
=
diag
⁡
{
X
,
…
,
X
︸
i
}
∈
R
i
q
×
i
q. For example, Σ
1
=
X and Σ
2
=
diag
⁡
{
X
,
X
}. And other notations are defined as R
¯
=
Σ
7
T
R
Σ
7, N
¯
=
Σ
4
T
N
Σ
4, G
¯
=
Σ
4
T
G
Σ
4, M
¯
=
Σ
4
T
M
Σ
4, Q
¯
1
=
Σ
2
T
Q
1
Σ
2, Q
¯
2
=
Σ
2
T
Q
2
Σ
2, Q
¯
3
=
Σ
1
T
Q
3
Σ
1,P
¯
1
=
Σ
1
T
P
1
Σ
1, P
¯
2
=
Σ
1
T
P
1
Σ
1, P
¯
3
=
Σ
1
T
P
3
Σ
1, P
¯
4
=
Σ
1
T
P
4
Σ
1, S
¯
1
=
Σ
2
T
S
1
Σ
2, S
¯
2
=
Σ
2
T
S
1
Σ
2, and Y
=
K
¯
X. By pre- and post-multiplying (40) and (46), respectively, by Σ
4 and diag
⁡
{
Σ
10
,
I
p
+
r
}, the following inequalities can be obtained:
(47)
0
≤
Σ
4
Υ
1
Σ
4
=
Υ
¯
1
,
0
≤
Σ
4
Υ
2
Σ
4
=
Υ
¯
2
,
(48)
[
Σ
10
0
10
q
×
(
p
+
r
)
0
(
p
+
r
)
×
10
q
I
p
+
r
]
[
Π
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
2
Π
2
T
Π
3
]
×
[
Σ
10
0
10
q
×
(
p
+
r
)
0
(
p
+
r
)
×
10
q
I
p
+
r
]
=
[
Π
-
1
[
h
(
t
)
,
h
˙
(
t
)
]
Π
-
2
Π
-
2
T
Π
-
3
]
<
0
,
where
(49)
Π
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
=
(
Ξ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
-
Γ
[
h
(
t
)
]
T
Υ
¯
2
Γ
[
h
(
t
)
]
)
+
(
[
C
1
,
-
H
]
X
e
1
T
+
D
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
)
T
×
(
[
C
1
,
-
H
]
X
e
1
T
+
D
1
Y
e
1
T
+
[
C
2
,
0
s
×
p
]
X
e
2
T
)
Π
¯
2
=
α
(
e
1
T
+
e
4
T
)
T
B
w
,
Π
¯
3
=
-
γ
2
I
p
+
r
.
Also, by using Schur complements and Lemma 4 with any matrix Λ
¯
∈
R
4
q
×
(
10
q
+
p
+
r
), inequality (47) is equivalent to
(50)
Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
=
[
L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
]
B
¯
w
Λ
¯
T
L
¯
2
T
⋆
-
γ
2
I
p
+
r
0
0
⋆
⋆
-
Υ
¯
2
0
⋆
⋆
⋆
-
I
s
]
<
0
,
where L
¯
1
[
h
(
t
)
,
h
˙
(
t
)
] and L
¯
2 are defined in (35). Note that inequality (50) is affinely dependent on h
(
t
) and h
˙
(
t
). Therefore, Φ
¯
[
h
(
t
)
,
h
˙
(
t
)
]
<
0 holds if
(51)
Φ
¯
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
0
,
h
˙
(
t
)
=
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
-
h
d
]
<
0
,
Φ
¯
[
h
(
t
)
=
h
M
,
h
˙
(
t
)
=
h
d
]
<
0
.
This completes our proof.Remark 8.
This subsection concludes the criterion ofH
∞ output tracking controller design for linear systems with time-varying delays and disturbances. To obtain an H
∞ output tracking controller gain which provides enhanced feasible region, the authors introduce the improved delay-dependent stability criterion as shown in Theorem 5. Then, based on the results of Theorem 5, the designing method of H
∞ output tracking controller gain was introduced in Theorem 7.Remark 9.
Theorems5 and 7 have some restriction such as h
d
<
1. In order to remove the restrictive condition h
d
<
1 in Theorem 5, V
1
(
t
) and V
3
(
t
) of Lyapunov-Krasovskii functional (15) should be changed as follows:
(52)
V
1
(
t
)
=
Ω
1
T
(
t
)
R
Ω
1
(
t
)
,
V
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
G
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
T
M
[
x
(
s
)
∫
s
t
‍
x
˙
(
u
)
d
u
∫
s
t
‍
x
(
u
)
d
u
]
d
s
,
where R
∈
R
6
n
×
6
n
>
0, G
∈
R
3
n
×
3
n
>
0, M
∈
R
3
n
×
3
n
>
0, and Ω
1
(
t
)
=
[
x
T
(
t
)
,
x
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T. Then, by utilizing the following vector,
(53)
ζ
(
t
)
=
[
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
t
)
,
x
T
(
t
-
h
(
t
)
)
,
x
T
(
t
-
h
M
)
,
x
˙
T
(
t
)
,
f
d
s
x
˙
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
x
T
(
s
)
d
s
,
f
d
∫
t
-
h
M
t
-
h
(
t
)
‍
x
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
,
f
s
∫
t
-
h
M
t
-
h
(
t
)
‍
∫
s
t
‍
x
T
(
u
)
d
u
d
s
]
T
,
as an augmented vector in deriving a stability condition, condition h
d
<
1 can be removed. Similarly, in Theorem 7, the restrictive condition h
d
<
1 can be also removed by using the following V
^
1
(
t
) and V
^
3
(
t
) of Lyapunov-Krasovskii functional (37) with positive matrices R
∈
R
6
q
×
6
q, G
∈
R
3
q
×
3
q, and M
∈
R
3
q
×
3
q as
(54)
V
^
1
(
t
)
=
Ω
^
1
T
(
t
)
R
Ω
^
1
(
t
)
,
V
^
3
(
t
)
=
∫
t
-
h
(
t
)
t
‍
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
T
G
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
d
s
+
∫
t
-
h
M
t
-
h
(
t
)
‍
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
T
M
[
ζ
(
s
)
∫
s
t
‍
ζ
˙
(
u
)
d
u
∫
s
t
‍
ζ
(
u
)
d
u
]
d
s
,
where Ω
^
1
(
t
)
=
[
ξ
T
(
t
)
,
ξ
T
(
t
-
h
M
)
,
∫
t
-
h
(
t
)
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
M
t
‍
ξ
T
(
s
)
d
s
,
∫
t
-
h
(
t
)
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s, ∫
t
-
h
M
t
‍
∫
s
t
‍
ξ
T
(
u
)
d
u
d
s
].


## 4. Numerical Examples

In this section, three numerical examples demonstrate the effectiveness of the proposed criteria.Example 1.
Consider the system (12) with following parameters:
(55)
A
=
[
-
2
0
0
-
0.9
]
,
A
d
=
[
-
1
0
-
1
-
1
]
.
The results for maximum upper-bound of h
(
t
) with different h
d are listed in Table 1 which conducts the comparison of the obtained results by Theorem 5 with the previous results. Table 1 shows that when h
d is bigger, the maximum upper-bound of h
(
t
) is smaller. It means that h
d affects the stability region of the system. Moreover, from the results in Table 1, it can be confirmed that Theorem 5 gives less conservative results than the existing results in [11–16].Table 1
The maximumh
M in Example 1.
h
d
0
0.05
0.1
0.5
Wu et al. [11]
4.47
3.98
3.60
2.0
He et al. [12]
4.47
3.98
3.60
2.4
Park and Ko [13]
4.47
4.01
3.66
2.33
Qian and Karimi [14]
4.60
4.12
3.70
2.33
Park et al. [15]
4.66
4.17
3.76
2.12
Kim [16]
4.97
4.35
3.86
2.33
Theorem5
5.2468
4.6423
4.2320
2.8768Example 2.
Consider the system (5) with the following parameters:
(56)
A
=
[
-
0.5
0
0
0.1
]
,
A
d
=
[
-
1
-
0.3
0.2
-
1
]
,
B
=
[
1
0
0
0
]
,
C
1
=
C
2
=
[
1
0
0
1
]
,
B
w
=
0.5
[
1
0
0
1
]
,
B
1
=
[
1
0
0
1
]
.
Also, the reference model is given by
(57)
x
˙
r
(
t
)
=
-
x
r
(
t
)
+
r
(
t
)
,
z
r
(
t
)
=
0.5
x
r
(
t
)
.
Disturbances are assumed to be
(58)
w
1
(
t
)
=
{
sin
(
8
t
)
,
5
≤
t
≤
8
,
0
,
otherwise
,
w
2
(
t
)
=
{
cos
⁡
(
12
t
)
,
6
≤
t
≤
9
,
0
,
otherwise
.
Reference signal r
T
(
t
)
=
[
r
1
T
(
t
)
,
r
2
T
(
t
)
] is defined as
(59)
r
1
(
t
)
=
sin
(
0.1
t
)
,
r
2
(
t
)
=
cos
⁡
(
0.1
t
)
.
In this example, Theorem 7 is used to obtain feedback controller gain K for the system (5) with (56). By applying Theorem 7, the feedback controller gain,
(60)
K
=
[
-
1.8931
-
0.0057
0.4983
0
0.0455
-
1.9317
-
0.0003
0.5000
]
,
can be obtained with minimum γ
=
0.1068 when h
M
=
0.1, h
d
=
0.5, and the tuning parameter α
=
1. Figure 1 illustrates the trajectories of outputs of the system (5) with disturbances (58) and time-delay h
(
t
)
=
0.5
h
M
(
cos
⁡
(
2
h
d
t
/
h
M
)
+
1
) in the 3D plot. The simulation result in Figure 1 shows that the z
(
t
) follows the z
r
(
t
) under the obtained H
∞ output tracking controller. Also, to confirm the results in detail, Figure 2 is added to show the trajectories of z
(
t
) and z
r
(
t
) with disturbances w
(
t
) in 2D plot.Figure 1
Trajectories of outputsz
(
t
) and z
r
(
t
) for Example 2.Figure 2
Trajectories ofz
(
t
), z
r
(
t
), and w
(
t
) in 2D plot for Example 2.Example 3.
Let us consider the system (5) as follows:
(61)
A
=
[
-
0.6
0
0
-
0.5
]
,
A
d
=
[
-
0.1
-
0.1
-
0.3
-
1
]
,
B
=
[
1
-
1
]
,
C
1
=
C
2
=
[
1
1
]
,
B
w
=
[
1
0
]
,
B
1
=
1
.
Moreover, the reference model is given by
(62)
x
˙
r
(
t
)
=
-
x
r
(
t
)
+
r
(
t
)
,
z
r
(
t
)
=
0.5
x
r
(
t
)
.
Disturbances are assumed to be
(63)
w
(
t
)
=
{
1
,
15
≤
t
≤
20
,
0
,
otherwise
.
The reference signal r
(
t
) is as follows:
(64)
r
(
t
)
=
{
1
,
10
≤
t
≤
40
,
0
,
otherwise
.
And assume that x
(
0
)
=
[
0,0
]
T and x
r
(
0
)
=
0. By applying Theorem 7, the feedback controller gain,
(65)
K
=
[
-
1.9454
-
2.0123
0.4929
]
,
can be obtained with minimum γ
=
0.1020 when α
=
1, h
M
=
0.1, and h
d
=
0.2. Figure 3 shows the outputs of the systems when the time-delay is h
(
t
)
=
0.05
(
sin
(
4
t
)
+
1
) which satisfied with 0
≤
h
(
t
)
≤
0.1 and |
h
˙
(
t
)
|
<
0.2. By calculation, ∥
e
(
t
)
∥
2
=
0.0794 and ∥
v
(
t
)
∥
2
=
5.9160, which yields ∥
e
(
t
)
∥
2
/
∥
v
(
t
)
∥
2
=
0.0134
<
γ. This results show the effectiveness of the proposed H
∞ output tacking controller design.Figure 3
Outputsz
(
t
) and z
r
(
t
) with disturbances w
(
t
) for Example 3.


## 5. Conclusions

This paper has investigated the problem of stability analysis andH
∞ output tracking control for linear systems with time-varying delays and disturbances. Main results were made up of two subsections which were stability analysis criterion and H
∞ output tracking controller design criterion. First, in Theorem 5, the stability criterion for the nominal systems with time-varying delays was derived by utilizing the newly augmented Lyapunov-Krasovskii functional and by using reciprocally convex approach. Second, Theorem 5 was extended to the problem of H
∞ output tracking controller design for the augmented system by using the H
∞ sense in Theorem 7. Finally, three examples were given to illustrate the effectiveness of the presented criteria. Based on these results, future research will focus on H
∞ output tracking control for various systems such as Markovian jump systems [26, 27], fault diagnosis problem [28], and discrete-time descriptor systems [29].



---

*Source: 184292-2014-06-29.xml*