# Finite-Time Boundedness Analysis for a Class of Switched Linear Systems with Time-Varying Delay

Authors: Yanke Zhong; Tefang Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The problem of finite-time boundedness for a class of switched linear systems with time-varying delay and external disturbance is investigated. First of all, the multiply Lyapunov function of the system is constructed. Then, based on the Jensen inequality approach and the average dwell time method, the sufficient conditions which guarantee the system is finite-time bounded are given. Finally, an example is employed to verify the validity of the proposed method.


---

## Body

## 1. Introduction

The switched system is a special kind of hybrid dynamic system, composed of a family of subsystems and a switching law specifying the switches between subsystems [1, 2]. The fact that the structure and working mechanism of the switched system are more complex than general systems leads to that the switched system possesses much richer dynamic characteristics. The switched systems are widely applied in engineering practice, such as power system control, robot control, network control, and so forth [3–9].In practice, switched systems are commonly subjected to time-delay and external disturbance. Due to their significant impact on the performances of switched systems, many scholars have been attracted to investigate the problem. Sun et al. analyzed the asymptotic stability of the switched linear system with time-delay perturbation by using common Lyapunov function and multiple Lyapunov function [3]. Lu and Zhao also investigated the asymptotic stability for switched linear systems with time-delay and proposed an effective method which can direct researchers to choose an appropriate switching law to make sure the system is asymptotic stable [10]. Zhao and Zhang studied the stability of the switched system with time-varying delays based on the average dwell time and time-delay decomposition approaches [11]. For switched systems with time-varying delay, Lian et al. utilized the Lyapunov-Krasovskii function method to design H infinity filter [12]. For switched systems affected by the nonlinear impact and disturbance, Sun used transfer matrix estimation and Gronwall inequality methods to design a feedback law stabilizing system [13]. For the switched system with fixed time-delay and norm bounded disturbance, Lin et al. proposed the finite-time boundedness concept and a method to judge whether the system is finite-time bounded [14].Up to now, to the best of the authors’ knowledge, there are a few papers concerning the finite-time boundedness problem of switched system. For switched systems with time-varying delay and external disturbance, the problem has not yet been discussed by any literature. However, in practical engineering, the time-delays are generally changeable over time, not fixed. In addition, many practical systems are just required that their state trajectories are bounded over a fixed interval. In other words, those systems may be unstable. On the contrary, although some systems are asymptotically stable, they cannot meet the application requirements because of their large transient state amplitudes. Considering the wide application of switched systems with time-varying delay and the requirements for transient behaviors in engineering fields, it is a significant task to investigate finite-time boundedness for switched linear systems with time-varying delay and external disturbance. The main contributions in this paper are listed as follows. (1) For the convenience of processing, a concise definition on the finite-time boundedness is proposed for the switched system. (2) Sufficient conditions of finite-time boundedness for switched linear systems with time-varying delay and external disturbance are given.


## 2. Preliminaries and Problem Formulation

Consider the following switched linear system with time-varying delay and external disturbance:(1)
x
˙
(
t
)
=
A
σ
(
t
)
x
(
t
)
+
B
σ
(
t
)
x
(
t
-
h
(
t
)
)
+
G
σ
(
t
)
w
(
t
)
,
h
(
t
)
≥
0
,
t
≥
0
,
x
(
t
)
=
φ
(
t
)
,
max
⁡
|
φ
˙
(
t
)
|
≤
ρ
,
ρ
≥
0
,
h
h
h
h
h
h
h
t
∈
[
-
d
,
0
)
,
d
≥
h
(
0
)
,
where x
(
t
) is state variable and σ
(
t
) is the switching law which is a piecewise continuous function with σ
(
t
)
∈
M
=
[
1
,
2
,
…
,
m
] which means the switched system is consisted of m subsystems. The ith subsystem is activated when σ
(
t
)
=
i
·
A
σ
(
t
), B
σ
(
t
), and G
σ
(
t
) are constant matrices. h
(
t
) represents time-varying delay. w
(
t
) stands for external disturbance. φ
(
t
) is the continuous vector-valued initial function on t
∈
[
-
d
,
0
)
·
φ
˙
(
t
) denotes the derivative of φ
(
t
)
·
ρ is a positive constant.For the convenience of subsequent processing, assume that the system (1) satisfies the following assumptions.Assumption 1 (see [14]).
The value of external disturbance changes over time, but it satisfies(2)
∫
0
+
∞
w
T
(
t
)
w
(
t
)
d
t
≤
γ
,
γ
≥
0
,
∀
t
>
0
.Assumption 2.
For the time-varying delay, the following inequalities hold:(3)
h
(
t
)
≥
0
,
h
˙
(
t
)
≤
k
,
k
<
1
,
h
(
t
)
≤
h
max
⁡
,
where k and h
max
⁡ are positive constants.Assumption 3.
The system state variable does not “jump” at switching instant, that is to say the state trajectory is continuous. In addition, the switching number ofσ
(
t
) is finite in a limited time interval which implies that the frequency of switching signal is not infinite.Definition 4 (see [15]).
ForT
≥
t
≥
0, let N
σ
(
t
,
T
) denote the switching number of σ
(
t
) over (
t
,
T
]. If
(4)
N
σ
(
t
,
T
)
≤
N
0
+
T
-
t
τ
a
holds for τ
a
≥
0 and an integer N
0
≥
0, then τ
a is called average dwell time.Definition 5.
For a given four positive constantsc
1, c
2, T
f, γ, and a switching signal σ
(
t
), if
(5)
x
T
(
t
-
0
)
x
(
t
-
0
)
≤
c
1
⟹
x
T
(
t
)
x
(
t
)
<
c
2
,
h
h
l
c
1
<
c
2
,
∀
t
∈
[
0
,
T
f
]
,
∀
w
(
t
)
:
∫
0
T
f
w
T
(
s
)
w
(
s
)
d
t
≤
γ
,
then the system (1) is said to be finite-time bounded. Where x
T
(
t
-
0
)
x
(
t
-
0
)
=
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
)
x
(
t
)
}, without loss of generality, specify c
1
=
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
)
x
(
t
)
}.Remark 6.
Definition5 implies that if the system (1) is finite-time bounded, the state remains within the prescribed bound in the fixed interval. Notice that finite-time boundedness is different from asymptotic stability. The system which is finite-time bounded may not be asymptotically stable while a system is asymptotically stable does not mean it is finite-time bounded either. In a word, there is no necessary relation between them.Remark 7.
The definition of finite-time boundedness in this paper is much more concise than that in [14]. However, they are consistent in essence. By using the definition in this paper, some complex matrix transformations can be avoided in the subsequent mathematical processing.


## 3. Main Result

Theorem 8.
For system (1), for all i
∈
M and for all t
∈
[
0
,
T
f
], assume there exists symmetric positive matrixes P
i, R
i
1, R
i
2, Q
i, Z
i
1, Z
i
2, and H and positive constants α, β
≥
1 such that(6)
[
ξ
11
P
i
B
i
+
d
2
A
i
T
Z
i
B
i
2
d
e
α
d
/
2
Z
i
,
1
0
P
i
G
i
+
d
2
A
i
T
Z
i
G
i
*
d
2
B
i
T
Z
i
B
i
-
(
1
-
k
)
Q
0
0
d
2
B
i
T
Z
i
G
i
*
*
e
α
d
/
2
R
i
-
2
d
e
α
d
/
2
Z
i
2
d
e
α
d
Z
i
,
2
0
*
*
*
-
e
α
d
R
i
,
2
-
2
d
e
α
d
Z
i
,
2
0
*
*
*
*
d
2
(
G
i
T
Z
i
G
i
-
H
)
]
<
0
.
If the average dwell time satisfies(7)
τ
a
<
T
f
ln
⁡
β
ln
⁡
C
1
+
ln
⁡
λ
8
-
ln
⁡
(
η
1
C
1
+
η
2
)
-
α
T
f
,
then system (1) is finite-time bounded, where
(8)
ξ
11
=
A
i
T
P
i
+
P
i
A
i
+
R
i
,
1
+
Q
i
+
d
2
A
i
T
(
Z
i
,
1
+
Z
i
,
2
)
A
i
-
2
d
e
α
d
/
2
-
α
P
i
,
Z
i
=
Z
i
,
1
+
Z
i
,
2
,
R
i
=
R
i
,
1
+
R
i
,
2
,
P
i
≤
β
P
j
,
R
i
,
1
≤
β
R
j
,
1
,
R
i
,
2
≤
β
R
j
,
2
,
Q
i
≤
β
Q
j
Z
i
,
1
≤
β
Z
j
,
1
,
Z
i
,
2
≤
β
Z
j
,
2
,
i
,
j
∈
[
1,2
,
…
,
m
]
,
λ
1
=
max
⁡
i
∈
M
{
λ
max
⁡
(
P
i
)
}
,
λ
2
=
max
⁡
i
∈
M
{
λ
max
⁡
(
R
i
,
1
)
}
,
λ
3
=
max
⁡
i
∈
M
{
λ
max
⁡
(
R
i
,
2
)
}
,
λ
4
=
max
⁡
i
∈
M
{
λ
max
⁡
(
Q
i
)
}
,
λ
5
=
max
⁡
i
∈
M
{
λ
max
⁡
(
Z
i
,
1
)
}
,
λ
6
=
max
⁡
i
∈
M
{
λ
max
⁡
(
Z
i
,
2
)
}
,
λ
7
=
λ
max
⁡
(
H
)
,
λ
8
=
min
⁡
i
∈
M
{
λ
min
⁡
(
P
i
)
}
,
η
1
=
λ
1
+
d
2
e
α
d
/
2
λ
2
+
d
2
e
α
d
λ
3
+
h
max
⁡
e
α
h
max
⁡
λ
4
,
η
2
=
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
+
d
2
λ
7
γ
,
C
1
=
sup
⁡
-
d
≤
t
0
≤
0
{
x
T
(
t
¯
0
)
x
(
t
¯
0
)
}
,
C
2
=
(
β
N
e
α
T
f
η
1
C
1
+
β
N
e
α
T
f
d
2
4
+
d
2
e
α
T
f
λ
7
γ
)
f
l
×
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
f
f
f
l
f
l
d
2
4
+
d
2
e
α
T
f
λ
7
γ
)
)
(
λ
8
)
-
1
.The left of inequality (6) is a symmetric matrix. Thus, the symmetric terms are denoted by “*”. λ
max
⁡
(
P
i
) represents the maximum eigenvalue of P
i.Proof.
Construct the multiply Lyapunov function as follows:(9)
V
(
t
)
=
V
i
(
t
)
=
V
i
,
1
(
t
)
+
V
i
,
2
(
t
)
+
V
i
,
3
(
t
)
+
V
i
,
4
(
t
)
,
V
i
,
1
(
t
)
=
x
T
(
t
)
P
i
x
(
t
)
,
V
i
,
2
(
t
)
=
∫
t
-
(
d
/
2
)
t
x
T
(
s
)
e
-
α
(
s
-
t
)
R
i
,
1
x
(
s
)
d
s
+
∫
t
-
d
t
-
(
d
/
2
)
x
T
(
s
)
e
-
α
(
s
-
t
)
R
i
,
2
x
(
s
)
d
s
,
V
i
,
3
(
t
)
=
∫
t
-
h
(
t
)
t
x
T
(
s
)
e
-
α
(
s
-
t
)
Q
i
x
(
s
)
d
s
,
V
i
,
4
(
t
)
=
∫
-
d
/
2
0
∫
t
+
θ
t
x
˙
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
1
x
˙
(
s
)
d
s
d
θ
+
∫
-
d
-
d
/
2
∫
t
+
θ
t
x
˙
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
2
x
˙
(
s
)
d
s
d
θ
.
Calculate the derivatives ofV
i
,
1
(
t
)
,
V
i
,
2
(
t
)
,
V
i
,
3
(
t
), and V
i
,
4
(
t
) as
(10)
V
˙
i
,
1
(
t
)
=
x
T
(
t
)
[
A
i
T
P
i
+
P
i
A
i
]
x
(
t
)
+
x
T
(
t
-
h
(
t
)
)
B
i
T
P
i
x
(
t
)
+
w
T
(
t
)
G
i
T
P
i
x
(
t
)
+
x
T
(
t
)
P
i
B
i
x
(
t
-
h
(
t
)
)
+
x
T
(
t
)
P
i
G
i
w
(
t
)
.
Furthermore, it follows that(11)
V
˙
i
,
1
(
t
)
-
α
V
i
,
1
=
x
T
(
t
)
[
A
i
T
P
i
+
P
i
A
i
]
x
(
t
)
+
x
T
(
t
-
h
(
t
)
)
B
i
T
P
i
x
(
t
)
+
w
T
(
t
)
G
i
T
P
i
x
(
t
)
+
x
T
(
t
)
P
i
B
i
x
(
t
-
h
(
t
)
)
+
x
T
(
t
)
P
i
G
i
w
(
t
)
-
α
x
T
(
t
)
P
i
x
(
t
)
,
(12)
V
˙
i
,
2
(
t
)
=
α
V
i
,
2
(
t
)
+
x
T
(
t
)
R
i
,
1
x
(
t
)
+
x
T
(
t
-
d
2
)
e
α
d
/
2
[
R
i
,
2
-
R
i
,
1
]
x
(
t
-
d
2
)
-
x
T
(
t
-
d
)
e
α
d
R
i
,
2
x
(
t
-
d
)
,
(13)
V
˙
i
,
3
(
t
)
=
α
V
i
,
3
(
t
)
+
x
T
(
t
)
Q
i
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
(
1
-
h
˙
(
t
)
)
×
e
α
h
(
t
)
Q
i
x
(
t
-
h
(
t
)
)
≤
α
V
i
,
3
(
t
)
+
x
T
(
t
)
Q
i
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
(
1
-
k
)
e
α
h
(
t
)
Q
i
x
(
t
-
h
(
t
)
)
≤
α
V
i
,
3
(
t
)
+
x
T
(
t
)
Q
i
x
(
t
)
-
x
T
(
t
-
h
(
t
)
)
(
1
-
k
)
Q
i
x
(
t
-
h
(
t
)
)
,
(14)
V
˙
i
,
4
(
t
)
=
α
V
i
,
4
(
t
)
+
d
2
x
˙
T
(
t
)
[
Z
i
,
1
+
Z
i
,
2
]
x
˙
(
t
)
-
∫
t
-
(
d
/
2
)
t
x
˙
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
1
x
˙
(
s
)
d
s
-
∫
t
-
d
t
-
(
d
/
2
)
x
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
2
x
(
s
)
d
s
=
α
V
i
,
4
(
t
)
+
d
2
[
A
i
x
(
t
)
+
B
i
x
×
(
t
-
h
(
t
)
)
+
G
i
w
(
t
)
h
h
h
h
h
h
h
h
h
h
h
×
(
t
-
h
(
t
)
)
+
G
i
w
(
t
)
]
T
×
[
Z
i
,
1
+
Z
i
,
2
]
×
[
A
i
x
(
t
)
+
B
i
x
(
t
-
h
(
t
)
)
+
G
i
w
(
t
)
]
-
∫
t
-
(
d
/
2
)
t
x
˙
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
1
x
˙
(
s
)
d
s
-
∫
t
-
d
t
-
(
d
/
2
)
x
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
2
x
(
s
)
d
s
.
Due to the Jensen inequality, inequality (15) holds
(15)
∫
t
-
(
d
/
2
)
t
x
˙
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
1
x
˙
(
s
)
d
s
≥
2
d
[
x
(
t
)
-
x
(
t
-
d
2
)
]
T
h
h
×
e
α
d
/
2
Z
i
,
1
[
x
(
t
)
-
x
(
t
-
d
2
)
]
,
∫
t
-
d
t
-
(
d
/
2
)
x
T
(
s
)
e
-
α
(
s
-
t
)
Z
i
,
2
x
(
s
)
d
s
≥
2
d
[
x
(
t
-
d
2
)
-
x
(
t
-
d
)
]
T
h
h
×
e
α
d
Z
i
,
2
[
x
(
t
-
d
2
)
-
x
(
t
-
d
)
]
.
By (14) and (15), we obtain
(16)
V
˙
i
,
4
(
t
)
≤
α
V
i
,
4
(
t
)
+
d
2
[
A
i
x
(
t
)
+
B
i
x
(
t
-
h
(
t
)
)
+
G
i
w
(
t
)
]
T
×
[
Z
i
,
1
+
Z
i
,
2
]
[
A
i
x
(
t
)
+
B
i
x
(
t
-
h
(
t
)
)
+
G
i
w
(
t
)
]
-
2
d
[
x
(
t
)
-
x
(
t
-
d
2
)
]
T
×
e
α
d
/
2
Z
i
,
1
[
x
(
t
)
-
x
(
t
-
d
2
)
]
-
2
d
[
x
(
t
-
d
2
)
-
x
(
t
-
d
)
]
T
×
e
α
d
Z
i
,
2
[
x
(
t
-
d
2
)
-
x
(
t
-
d
)
]
.
From (11), (12), (13), and (16), it is easy to get(17)
V
˙
i
(
t
)
-
α
V
i
(
t
)
≤
[
x
(
t
)
x
(
t
-
h
(
t
)
)
x
(
t
-
d
2
)
x
(
t
-
d
)
w
(
t
)
]
T
×
[
ξ
11
P
i
B
i
+
d
2
A
i
T
Z
i
B
i
2
d
e
α
d
/
2
Z
i
,
1
0
P
i
G
i
+
d
2
A
i
T
Z
i
G
i
*
d
2
B
i
T
Z
i
B
i
-
(
1
-
k
)
Q
0
0
d
2
B
i
T
Z
i
G
i
*
*
e
α
d
/
2
R
i
-
2
d
e
α
d
/
2
Z
i
2
d
e
α
d
Z
i
,
2
0
*
*
*
-
e
α
d
R
i
,
2
-
2
d
e
α
d
Z
i
,
2
0
*
*
*
*
d
2
G
i
T
Z
i
G
i
]
×
[
x
(
t
)
x
(
t
-
h
(
t
)
)
x
(
t
-
d
2
)
x
(
t
-
d
)
w
(
t
)
]
.
According to the definition of finite-time boundedness, the rest of the proof will be divided into two steps. Under the given conditions, we need to prove thatx
T
(
t
)
x
(
t
)
<
c
2 and c
1
<
c
2, respectively.
(i) We will prove thatx
T
(
t
)
x
(
t
)
<
C
2 holds for all t on [
0
,
T
f
].
By (6) and (17), inequality (18) holds
(18)
V
˙
i
(
t
)
-
α
V
i
(
t
)
<
d
2
w
T
(
t
)
H
w
(
t
)
.
Since(
d
/
d
t
)
(
e
-
α
t
V
i
(
t
)
)
=
e
-
α
t
[
V
˙
i
(
t
)
-
α
V
i
(
t
)
], inequality (18) can be transformed into
(19)
d
d
t
(
e
-
α
t
V
i
(
t
)
)
<
d
2
e
-
α
t
w
T
(
t
)
H
w
(
t
)
.
Lett
k stand for the instant of the Kth switching.
Integrating fromt
k to t on both sides of (19), it follows that
(20)
V
i
(
t
)
<
e
α
(
t
-
t
k
)
V
i
(
t
k
)
+
d
2
∫
t
k
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
.
Notice thatP
i
≤
β
P
j, R
i
,
1
≤
β
R
j
,
1, R
i
,
2
≤
β
R
j
,
2, Q
i
≤
β
Q
j, Z
i
,
1
≤
β
Z
j
,
1, Z
i
,
2
≤
β
Z
j
,
2, i, and j
∈
[
1,2
,
…
,
m
] and the continuity of x
(
t
), hence (21) holds
(21)
V
i
(
t
)
<
β
e
α
(
t
-
t
k
)
V
i
(
t
k
-
)
+
d
2
∫
t
k
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
,
where t
k
- denotes the instant just before t
k.
It is easy to see(22)
V
i
(
t
k
-
)
<
e
α
(
t
k
-
t
k
-
1
)
V
i
(
t
k
-
1
)
+
d
2
∫
t
k
-
1
t
k
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
.
Then (23) is obtained
(23)
V
i
(
t
)
<
β
2
e
α
(
t
-
t
k
-
1
)
V
i
(
t
(
k
-
1
)
-
)
+
d
2
β
e
α
(
t
-
t
k
)
∫
t
k
-
1
t
k
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
+
d
2
∫
t
k
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
.
Assume the switching number ofσ
(
t
) over [
0
,
T
f
] is N. (24) is obtained via the iterative calculation
(24)
V
i
(
t
)
<
β
N
e
α
t
V
i
(
0
)
+
d
2
β
N
e
α
(
t
-
t
1
)
×
∫
t
0
t
1
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
+
⋯
+
d
2
β
e
α
(
t
-
t
k
)
∫
t
k
-
1
t
k
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
+
d
2
∫
t
k
T
f
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
,
(25)
e
α
T
f
≥
e
α
t
,
e
α
T
f
>
e
α
(
T
f
-
t
1
)
>
e
α
(
T
f
-
t
2
)

>

⋯
>
e
α
(
T
f
-
t
k
)
>
1
for
t
∈
[
0
,
T
f
]
,
β
N
≥
β
N
-
1

≥

⋯
≥
β
≥
1
.
Thus, it follows that(26)
V
i
(
t
)
<
β
N
e
α
T
f
V
i
(
0
)
+
d
2
β
N
e
α
T
f
×
∫
t
0
t
1
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
+
⋯
+
d
2
β
N
e
α
T
f
∫
t
k
-
1
t
k
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
+
d
2
e
α
T
f
β
N
∫
t
k
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
,
(27)
V
i
(
t
)
<
β
N
e
α
T
f
V
i
(
0
)
+
d
2
e
α
T
f
β
N
∫
0
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
.
On the other hand, since1
≤
e
α
(
t
-
s
)
≤
e
α
t
≤
e
α
T
f and H
≤
λ
max
⁡
(
H
), we have
(28)
d
2
e
α
T
f
β
N
∫
0
t
w
T
(
s
)
e
α
(
t
-
s
)
H
w
(
s
)
d
s
≤
d
2
e
2
α
T
f
β
N
λ
max
⁡
(
H
)
∫
0
t
w
T
(
s
)
w
(
s
)
d
s
≤
d
2
e
2
α
t
β
N
λ
max
⁡
(
H
)
γ
.
Applying the above inequality to (27), we get
(29)
V
i
(
t
)
<
β
N
e
α
T
f
V
i
(
0
)
+
d
2
e
2
α
T
f
β
N
λ
max
⁡
(
H
)
γ
.
With respect to V
i
(
0
) in (29), it is processed as follows:
(30)
V
i
(
0
)
=
x
T
(
0
)
P
i
x
(
0
)
+
∫
-
d
/
2
0
x
T
(
s
)
e
-
α
s
R
i
,
1
x
(
s
)
d
s
+
∫
-
d
-
d
/
2
x
T
(
s
)
e
-
α
s
R
i
,
2
x
(
s
)
d
s
+
∫
-
h
(
0
)
0
x
T
(
s
)
e
-
α
s
Q
i
x
(
s
)
d
s
+
∫
-
d
/
2
0
∫
θ
0
x
˙
T
(
s
)
e
-
α
s
Z
i
,
1
x
˙
(
s
)
d
s
d
θ
+
∫
-
d
-
d
/
2
∫
θ
0
x
˙
T
(
s
)
e
-
α
s
Z
i
,
2
x
˙
(
s
)
d
s
d
θ
<
λ
max
⁡
(
P
i
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
d
2
e
α
d
/
2
λ
max
⁡
(
R
i
,
1
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
d
2
e
α
d
λ
max
⁡
(
R
i
,
2
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
h
max
⁡
e
α
h
max
⁡
λ
max
⁡
(
Q
i
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
∫
-
d
/
2
0
-
θ
ρ
2
λ
max
⁡
(
Z
i
,
1
)
e
-
α
θ
d
θ
+
∫
-
d
-
d
/
2
-
θ
ρ
2
λ
max
⁡
(
Z
i
,
2
)
e
-
α
θ
d
θ
<
λ
max
⁡
(
P
i
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
d
2
e
α
d
/
2
λ
max
⁡
(
R
i
,
1
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
d
2
e
α
d
λ
max
⁡
(
R
i
,
2
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
)
x
(
t
-
0
)
}
+
h
max
⁡
e
α
h
max
⁡
λ
max
⁡
(
Q
i
)
sup
⁡
-
d
≤
t
≤
0
{
x
T
(
t
-
0
t
)
x
(
t
-
0
)
}
+
d
2
·
d
2
ρ
2
λ
max
⁡
(
Z
i
,
1
)
e
α
d
/
2
+
d
2
·
d
ρ
2
λ
max
⁡
(
Z
i
,
2
)
e
α
d
.
Applying known mathematical relationships to (30), (31) can be obtained as
(31)
V
i
(
0
)
<
λ
1
C
1
+
d
2
e
α
d
/
2
λ
2
C
1
+
d
2
e
α
d
λ
3
C
1
+
h
max
⁡
e
α
h
max
⁡
λ
4
C
1
+
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
.
Inequality (32) is obtained via (29) and (31) as
(32)
V
i
(
t
)
<
β
N
e
α
T
f
×
(
λ
1
+
d
2
e
α
d
/
2
λ
2
+
d
2
e
α
d
λ
3
+
h
max
⁡
e
α
h
max
⁡
λ
4
)
C
1
+
β
N
e
α
T
f
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
h
h
h
h
h
h
h
h
h
h
h
d
2
4
ρ
2
λ
5
+
d
2
e
α
T
f
λ
7
γ
)
=
β
N
e
α
T
f
η
1
C
1
+
β
N
e
α
T
f
×
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
+
d
2
e
α
T
f
λ
7
γ
)
.
According to the definition ofV
i
(
t
), inequality (33) holds
(33)
V
i
(
t
)
>
x
T
(
t
)
P
i
x
(
t
)
≥
λ
min
⁡
(
P
i
)
x
T
(
t
)
x
(
t
)
≥
min
⁡
i
∈
M
{
λ
min
⁡
(
P
i
)
}
x
T
(
t
)
x
(
t
)
.
Then the following holds based on (32) and (33):
(34)
x
T
(
t
)
x
(
t
)
<
(
β
N
e
α
T
f
η
1
C
1
+
β
N
e
α
T
f
d
2
4
ρ
2
d
2
2
d
2
2
+
d
2
e
α
T
f
λ
7
γ
)
h
h
h
h
h
h
h
h
l
×
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
h
h
h
h
h
h
h
h
h
h
h
h
h
l
l
h
h
d
2
2
d
2
2
+
d
2
e
α
T
f
λ
7
γ
)
)
(
λ
8
)
-
1
x
T
(
t
)
x
(
t
)
=
C
2
.
(ii) Next,C
1
<
C
2 will be demonstrated.
By (7), we have
(35)
T
f
τ
a
>
ln
⁡
C
1
+
ln
⁡
λ
8
-
ln
⁡
(
η
1
C
1
+
η
2
)
-
α
T
f
ln
⁡
β
,
(36)
N
>
T
f
τ
a
>
ln
⁡
C
1
+
ln
⁡
λ
8
-
ln
⁡
(
η
1
C
1
+
η
2
)
-
α
T
f
ln
⁡
β
,
(37)
ln
⁡
(
η
1
C
1
+
η
2
)
-
ln
⁡
λ
8
>
ln
⁡
C
1
-
N
ln
⁡
β
-
α
T
f
,
(38)
η
1
C
1
+
η
2
λ
8
e
α
T
f
β
N
>
C
1
.
On the other hand, due toe
α
T
f
≥
1, there exist the following mathematical relations:
(39)
C
2
=
(
β
N
e
α
T
f
η
1
C
1
+
β
N
e
α
T
f
d
2
4
ρ
2
hhllh
×
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
h
h
h
l
h
h
l
h
+
d
2
e
α
T
f
λ
7
γ
)
)
(
λ
8
)
-
1
≥
(
β
N
e
α
T
f
η
1
C
1
+
β
N
e
α
T
f
d
2
4
ρ
2
d
2
2
+
d
2
λ
7
γ
)
h
h
h
h
×
(
d
2
4
ρ
2
λ
5
e
α
d
/
2
+
d
2
2
ρ
2
λ
6
e
α
d
h
h
h
h
h
h
h
l
d
2
2
+
d
2
λ
7
γ
)
)
(
λ
8
)
-
1
=
η
1
C
1
+
η
2
λ
8
e
α
T
f
β
N
.
Combining (38) and (39), we get c
2
>
c
1.
By (i) and (ii), the system (1) satisfies the definition of finite-time boundedness under given conditions. This completes the proof of Theorem 8.Remark 9.
Notice that (6) is not a linear matrix inequality. Thus, it cannot be directly solved via LMI toolbox. Before solving (6), the inequality can be transformed to a linear matrix inequality by specifying the value of α.


## 4. A Numerical Example

An example is presented to illustrate Theorem8. Consider
(40)
x
˙
(
t
)
=
A
σ
(
t
)
x
(
t
)
+
B
σ
(
t
)
x
(
t
-
h
(
t
)
)
+
G
σ
(
t
)
w
(
t
)
,
t
≥
0
,
x
(
t
)
=
φ
(
t
)
,
t
∈
[
-
d
,
0
)
,
A
1
=
[
-
1.7
1.7
0
1.3
-
1
0.7
0.7
1
-
0.6
]
,
A
2
=
[
1
-
1
0
0.7
0
-
0.6
1.7
0
-
1.7
]
,
B
1
=
[
1.5
-
1.7
0.1
-
1.3
1
-
0.3
-
0.7
1
0.6
]
,
B
2
=
[
-
1
-
0.3
0.1
1.3
-
0.1
0.6
1.5
0.1
1.8
]
,
G
1
=
G
2
=
[
1
0
0
0
1
0
0
0
1
]
,
w
(
t
)
=
[
0.03
sin
(
t
)
0.02
cos
⁡
(
2
t
)
0.015
(
sin
(
t
+
1
)
+
cos
⁡
(
t
-
2
)
)
]
,
h
(
t
)
=
0.5
t
,
d
=
0.2
,
φ
(
t
)
≡
[
0.5
0.1
0
]
T
,
∀
t
∈
[
-
0.2,0
]
,
max
⁡
|
φ
˙
(
t
)
|
≤
ρ
=
0
,
h
˙
(
t
)
≤
k
=
0.5
,
C
1
=
0.26
.Letα
=
0.02
,
β
=
1.1, and T
f
=
10, then h
(
t
)
≤
h
max
⁡
=
0.5
*
10
=
5 and ∫
0
T
f
w
T
(
s
)
w
(
s
)
d
t
≤
γ
≈
0.022.Solving (6) leads to feasible solutions that
(41)
P
1
=
[
0.8983
-
0.0167
0.1555
-
0.0167
1.0898
-
0.3930
0.1555
-
0.3930
0.9754
]
,
P
2
=
[
0.6101
0.1828
-
0.1480
0.1828
0.8908
-
0.3026
-
0.1480
-
0.3026
0.8153
]
,
R
1,1
=
[
0.7188
-
0.1052
0.0283
-
0.1052
0.7458
-
0.1300
0.0283
-
0.1300
0.7114
]
,
R
1,2
=
[
1.3854
-
0.1336
0.0209
-
0.1336
1.3613
-
0.1532
0.0209
-
0.1532
1.3368
]
,
R
2,1
=
[
0.5289
0.0089
-
0.0566
0.0089
0.6200
-
0.0083
-
0.0566
-
0.0083
0.6743
]
,
R
2,2
=
[
1.1615
0.0282
-
0.0465
0.0282
1.2555
-
0.0175
-
0.0465
-
0.0175
1.3518
]
,
Q
1
=
[
4.2184
-
0.5908
-
0.1106
-
0.5908
4.4575
-
0.3066
-
0.1106
-
0.3066
4.0548
]
,
Q
2
=
[
3.8150
0.0518
0.0675
0.0518
3.7399
0.0356
0.0675
0.0356
4.2709
]
,
Z
1,1
=
[
0.3150
-
0.0492
-
0.0003
-
0.0492
0.2950
-
0.0639
-
0.0003
-
0.0639
0.3082
]
,
Z
1,2
=
[
0.4060
0.0002
0.0006
0.0002
0.4036
0.0013
0.0006
0.0013
0.3934
]
,
Z
2,1
=
[
0.2124
0.0176
-
0.0190
0.0176
0.2812
-
0.0034
-
0.0190
-
0.0034
0.3118
]
,
Z
2,2
=
[
0.3934
0.0070
0.0039
0.0070
0.3920
-
0.0131
0.0039
-
0.0131
0.3961
]
,
H
=
[
12.5148
0.6115
0.1387
0.6115
13.0302
-
0.5248
0.1387
-
0.5248
12.4327
]
,
λ
1
=
1.4539
,
λ
2
=
0.9114
,
λ
3
=
1.5749
,
λ
4
=
4.9735
,
λ
5
=
0.2449
,
λ
6
=
0.1192
,
λ
7
=
13.5367
,
λ
8
=
0.5200
.Further, we get thatC
2
=
2000.6421
>
C
1 and τ
a
<
1.8263. The simulation of the numerical example is performed and its results are shown in Figures 1 and 2. From Figure 1, one can get that τ
a
<
1.8263 holds. From Figure 2, it is easily found that the value of x
T
(
t
)
x
(
t
) remains within C
2 for t
∈
[
0
,
T
f
]. So, the system is indeed finite-time bounded over [
0
,
T
f
].Figure 1
The diagram of switching law.The diagrams ofx
(
t
) and x
T
(
t
)
x
(
t
).
(a)
(b)


## 5. Conclusion

(1) For the switched linear system, a new definition on finite-time boundedness is proposed which can reduce some complex matrix calculations.(2) Under given conditions, the sufficient conditions which guarantee the system is finite-time bounded are given for the switched linear system with time-varying delay and external disturbance.(3) In the future study, a challenging research topic is how to ensure the switched system with time-varying delay remains finite-time bounded for any switching signal.



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*Source: 982414-2014-02-13.xml*