1. Introduction
Network control systems (NCSs) are the feedback control systems in which the control loops are closed via realtime networks [1]. NCSs have great advantages compared to the traditional pointtopoint control systems, including high reliability, ease of installation and maintenance, and low cost, and they have been applied in many areas, such as computer integrated manufacturing systems, intelligent traffic systems, aircraft control, and teleoperation. However, due to the insertion of communication network in feedback control loops, time delay caused by data transmission and packet dropout in NCSs is always inevitable, which may degrade system performance or even lead to the potential system instability. Thus it is of significance to cope with the adverse influences of induced delay and packet dropout. Recently, the problem of robust stability analysis for NCSs has attained considerable attention, and a great number of research results have been reported [1–9]. In order to reduce the conservatism, Yu et al. [5] obtained the sufficient condition on the stabilization of NCSs; the admissible upper bounds of induced delay and packet dropout can be computed by using LyapunovRazumikhi function techniques and the quasiconvex optimization algorithm. The free weighting matrices method was proposed in [3] to solve the problem of networkbased control. However, using too many free weighting matrices makes the system analysis complex; what is more, some terms were neglected directly, which brings the conservative. Peng et al. [9] indicated that if more information of induced delays in NCSs were utilized, conservatism in system analysis could be reduced. Moreover, it can build a bridge to connect quality of control (QoC). In [4], a stability criterion based on maximum allowable networkinduced delay rate is proposed for choosing a reasonable sampling period.
Inspired by the above research results, in the present paper, the robust stabilization problem for a class of NCSs with networkinduced delay and packet dropout is addressed. The NCSs are modeled as continuoustime linear systems on the basis of the input delay approach. By exploiting the information of lower and upper bounds of the delay, a new LyapunovKrasovskii functional is constructed. Moreover, reciprocally convex combination technique is introduced such that less conservative results are obtained. Two numerical examples are given to illustrate the validity and feasibility of the proposed results.
2. System Description
In this paper, the sensor module is assumed to act in a clockdriven fashion with transmission period h; the controller and actuator modules are assumed to act in an eventdriven fashion. A new packet will be used by the controller immediately after its arrival. Single packet transmission is considered, where all the data sent or received over the network are sampled at the same sampling instant and assembled together into one network packet.
The plant is described by the following linear plant model proposed in [7]:
(1)
x
˙
(
t
)
=
A
x
(
t
)
+
B
u
(
t
)
,
where
x
(
t
)
∈
R
n
,
u
(
t
)
∈
R
m
are the plant’s state and input vectors, respectively.
A
and
B
are known to be real constant matrices with proper dimensions. This control signal is based on the plant’s state at the instant
i
k
h
. So, the control law can be described as
(2)
u
(
t
+
)
=
K
x
(
i
k
h
)
,
t
+
∈
[
i
k
h
+
τ
k
,
i
k
+
1
h
+
τ
k
+
1
)
,
k
=
0,1
,
2,3
,
…
,
where
K
is the state feedback gain matrix,
τ
k
denotes the networkinduced delay, and
h
is the sampling period,
{
i
1
,
i
2
,
i
3
,
…
}
⊂
{
1,2
,
3
,
…
}
; due to the introduction of logical zeroorder holder, the actuator will use the latest available control input, so
i
k
+
1
>
i
k
. If
{
i
1
,
i
2
,
i
3
,
…
}
=
{
0,1
,
2,3
,
…
}
, then no packet dropout occurred in the transmission. And the numbers of consecutive packet dropouts during the time interval
(
i
k
h
,
i
k
+
1
h
)
can be described as follows:
(3)
i
k
+
1

i
k
=
1
,
0
packet
is
lost
,
i
k
+
1

i
k
=
2
,
1
packet
is
lost
,
⋮
i
k
+
1

i
k
=
p
,
p

1
packets
are
lost
.
Assume the existence of constants
τ
m
>
0
,
τ
M
>
0
. One has
(4)
τ
m
≤
τ
k
,
(
i
k
+
1

i
k
)
h
+
τ
k
+
1
≤
τ
M
,
k
=
0,1
,
2,3
,
…
,
where
τ
m
and
τ
M
indicate the lower and the upper bounds of the total delay involving both transmission delays and packet dropouts, respectively.
From a straightforward combination of (1)–(4), the system can be rewritten as follows:
(5)
x
˙
(
t
)
=
A
x
(
t
)
+
A
d
x
(
t

d
(
t
)
)
,
x
(
t
)
=
φ
(
t
)
,
t
∈
(

τ
M
,
0
)
,
where
A
d
=
B
K
, the function
d
(
t
)
=
t

i
k
h
which satisfies
τ
m
≤
d
(
t
)
≤
τ
M
denotes the timevarying delay in the control signal, and
φ
(
t
)
is the state’s initial function.
The following technical lemmas are introduced, which are indispensable for the proof of the main result.
Lemma 1 (see [<xref reftype="bibr" rid="B18">13</xref>]).
For any positive matrix
M
>
0
, scalar
r
>
0
, and a vector function
w
:
[
0
,
r
]
→
R
n
such that the integration
∫
0
r
w
(
s
)
T
M
w
(
s
)
d
s
is well defined,
(6)
r
(
∫
0
r
w
(
s
)
T
M
w
(
s
)
d
s
)
≥
(
∫
0
r
w
(
s
)
d
s
)
T
M
(
∫
0
r
w
(
s
)
d
s
)
.
Lemma 2 (see [<xref reftype="bibr" rid="B13">14</xref>]).
For any positive matrix
R
, scalars
a
and
b
satisfying
a
>
b
, and
a
vector function
x
, the following inequality holds:
(7)
(
a

b
)
2
2
∫
b
a
∫
s
a
x
T
(
u
)
R
x
(
u
)
d
u
d
s
≥
(
∫
b
a
∫
s
a
x
(
u
)
d
u
d
s
)
T
R
(
∫
b
a
∫
s
a
x
(
u
)
d
u
d
s
)
.
Lemma 3 (see [<xref reftype="bibr" rid="B11">15</xref>]).
Let
F
1
,
F
2
,
F
3
,
…
F
N
:
R
m
↦
R
have positive values for arbitrary value of independent variable in an open subset
W
of
R
m
. The reciprocally convex combination of
F
i
(
i
=
1,2
,
…
,
N
)
in
W
satisfies
(8)
min
∑
i
=
1
N
1
η
i
F
i
(
t
)
=
∑
i
=
1
N
F
i
(
t
)
+
max
∑
i
=
1
N
∑
j
=
1
,
j
≠
i
N
W
i
,
j
(
t
)
subject to
(9)
{
[
F
i
(
t
)
W
i
,
j
(
t
)
W
i
,
j
(
t
)
F
j
(
t
)
]
η
i
>
0
,
∑
i
=
1
N
η
i
=
1
,
W
i
,
j
(
t
)
:
R
m
⟼
R
,
W
j
,
i
(
t
)
=
W
i
,
j
(
t
)
,
[
F
i
(
t
)
W
i
,
j
(
t
)
W
i
,
j
(
t
)
F
j
(
t
)
]
≥
0
∑
i
=
1
N
η
i
=
1
,
W
i
,
j
(
t
)
:
R
m
⟼
R
}
.
3. Main Results
Theorem 4.
For a given scalar
τ
M
>
τ
m
≥
0
, NCS (5) is asymptotically stable if there exist symmetric matrices
P
=
[
P
i
j
]
4
×
4
>
0
,
Q
=
[
Q
11
Q
12
*
Q
22
]
>
0
,
R
=
[
R
11
R
12
*
R
22
]
>
0
,
Z
i
(
i
=
1,2
,
3
)
>
0
,
R
i
(
i
=
1,2
,
3
)
>
0
, and proper dimensions matrice
S
12
such that
(10)
[
Z
2
S
12
*
Z
2
]
>
0
,
Θ
=
[
Θ
11
Θ
12
*
Θ
22
]
<
0
,
where
(11)
Θ
12
=
[
τ
m
2
8
Ψ
R
1
(
τ
M

τ
m
)
2
2
Ψ
R
2
τ
M
2
8
Ψ
R
3
τ
m
2
Ψ
Z
1
(
τ
M

τ
m
)
Ψ
Z
2
τ
M
2
Ψ
Z
3
]
,
Θ
11
=
diag
{

R
1

R
2

R
3

Z
1

Z
2

Z
3
}
,
Θ
22
=
[
Ω
11
P
11
A
d
P
12

P
12
Ω
15
Ω
16
Ω
17
Ω
18
Ω
19
*
Ω
22
Ω
23
Ω
24
0
0
A
d
T
P
12
A
d
T
P
13
A
d
T
P
14
*
*
Ω
33
S
12

Q
12
T
0
P
22
P
23
P
24
*
*
*
Ω
44
0

R
12
T

P
22

P
23

P
24
*
*
*
*
Ω
55
0

P
23
T

P
33

P
34
*
*
*
*
*
Ω
66

P
24
T

P
34
T

P
44
*
*
*
*
*
*

R
2
0
0
*
*
*
*
*
*
*

R
1
0
*
*
*
*
*
*
*
*

R
3
]
,
Ψ
=
[
A
A
d
0
0
0
0
0
0
0
]
with
(12)
Ω
11
=
A
T
P
11
+
P
11
A
+
P
13
+
P
13
T
+
P
14
+
P
14
T
+
R
11
+
Q
11

Z
1

Z
3

τ
m
2
4
R
1

(
τ
M

τ
m
)
2
R
2

τ
M
2
4
R
3
,
Ω
15
=

P
13
+
Z
1
+
Q
12
,
Ω
16
=
R
12
+
Z
3

P
14
,
Ω
17
=
A
T
P
12
+
P
23
T
+
P
24
T
+
(
τ
M

τ
m
)
R
2
,
Ω
18
=
A
T
P
13
+
P
33
+
P
34
T
+
τ
m
2
R
1
,
Ω
19
=
A
T
P
14
+
P
34
+
P
44
+
τ
M
2
R
3
,
Ω
22
=

2
Z
2
+
S
12
+
S
12
T
,
Ω
23
=
Z
2

S
12
,
Ω
24
=
Z
2

S
12
T
,
Ω
33
=

Q
22

Z
2
,
Ω
44
=

R
22

Z
2
,
Ω
55
=

Z
1

Q
11
+
Q
22
,
Ω
66
=

Z
3

R
11
+
R
22
.
Proof.
Let us choose a LyapunovKrasovskii functional candidate as
(13)
V
(
x
t
)
=
∑
i
=
1
4
V
i
(
x
t
)
,
where
(14)
V
1
(
x
t
)
=
[
x
(
t
)
∫
t

τ
M
t

τ
m
x
(
s
)
d
s
∫
t

(
τ
m
/
2
)
t
x
(
s
)
d
s
∫
t

(
τ
M
/
2
)
t
x
(
s
)
d
s
]
T
P
[
x
(
t
)
∫
t

τ
M
t

τ
m
x
(
s
)
d
s
∫
t

(
τ
m
/
2
)
t
x
(
s
)
d
s
∫
t

(
τ
M
/
2
)
t
x
(
s
)
d
s
]
,
V
2
(
x
t
)
=
∫
t

(
τ
m
/
2
)
t
[
x
(
s
)
x
(
s

τ
m
2
)
]
T
[
Q
11
Q
12
*
Q
22
]
[
x
(
s
)
x
(
s

τ
m
2
)
]
d
s
+
∫
t

(
τ
M
/
2
)
t
[
x
(
s
)
x
(
s

τ
M
2
)
]
T
[
R
11
R
12
*
R
22
]
[
x
(
s
)
x
(
s

τ
M
2
)
]
d
s
,
V
3
(
x
t
)
=
τ
m
2
∫

τ
m
/
2
0
∫
t
+
s
t
x
˙
T
(
u
)
Z
1
x
˙
(
u
)
d
u
d
s
+
(
τ
M

τ
m
)
∫

τ
M

τ
m
∫
t
+
s
t
x
˙
T
(
u
)
Z
2
x
˙
(
u
)
d
u
d
s
+
τ
M
2
∫

τ
M
/
2
0
∫
t
+
s
t
x
˙
T
(
u
)
Z
3
x
˙
(
u
)
d
u
d
s
,
V
4
(
x
t
)
=
τ
m
2
8
∫
t

(
τ
m
/
2
)
t
∫
s
t
∫
u
t
x
˙
T
(
v
)
R
1
x
˙
(
v
)
d
v
d
u
d
s
+
(
τ
M

τ
m
)
2
2
∫
t

τ
M
t

τ
m
∫
s
t
∫
u
t
x
˙
T
(
v
)
R
2
x
˙
(
v
)
d
v
d
u
d
s
+
τ
M
2
8
∫
t

(
τ
M
/
2
)
t
∫
s
t
∫
u
t
x
˙
T
(
v
)
R
3
x
˙
(
v
)
d
v
d
u
d
s
.
Define an extendedstate vector as
(15)
ξ
T
(
x
t
)
=
Δ
[
x
T
(
t
)
x
T
(
t

d
(
t
)
)
x
T
(
t

τ
m
)
x
T
(
t

τ
M
)
x
T
(
t

τ
m
2
)
x
T
(
t

τ
M
2
)
∫
t

τ
M
t

τ
m
x
T
(
s
)
d
s
∫
t

(
τ
m
/
2
)
t
x
T
(
s
)
d
s
∫
t

(
τ
M
/
2
)
t
x
T
(
s
)
d
s
]
,
and then NCS (5) is simplified as
(16)
x
˙
(
x
t
)
=
ψ
ξ
(
x
t
)
.
Calculating the derivative of
V
1
(
x
t
)
along the trajectory of NCS (5) yields
(17)
V
˙
1
(
x
t
)
=
2
[
x
(
t
)
∫
t

τ
M
t

τ
m
x
(
s
)
d
s
∫
t

(
τ
m
/
2
)
t
x
(
s
)
d
s
∫
t

(
τ
M
/
2
)
t
x
(
s
)
d
s
]
T
P
[
ψ
ξ
(
t
)
x
(
t

τ
m
)

x
(
t

τ
M
)
x
(
t
)

x
(
t

τ
m
2
)
x
(
t
)

x
(
t

τ
M
2
)
]
.
It is easy to get
(18)
V
˙
2
(
x
t
)
=
[
x
(
t
)
x
(
t

τ
m
2
)
]
T
[
Q
11
Q
12
*
Q
22
]
[
x
(
t
)
x
(
t

τ
m
2
)
]

[
x
(
t

τ
m
2
)
x
(
t

τ
m
)
]
T
[
Q
11
Q
12
*
Q
22
]
[
x
(
t

τ
m
2
)
x
(
t

τ
m
)
]
+
[
x
(
t
)
x
(
t

τ
M
2
)
]
T
[
R
11
R
12
*
R
22
]
[
x
(
t
)
x
(
t

τ
M
2
)
]

[
x
(
t

τ
M
2
)
x
(
t

τ
M
)
]
T
[
R
11
R
12
*
R
22
]
[
x
(
t

τ
M
2
)
x
(
t

τ
M
)
]
.
The time derivative of
V
3
(
x
t
)
can be represented as
(19)
V
˙
3
(
x
t
)
=
x
˙
T
(
t
)
(
(
τ
m
2
4
)
Z
1
+
(
τ
M

τ
m
)
2
Z
2
+
(
τ
M
2
4
)
Z
3
)
×
x
˙
(
t
)

τ
m
2
∫

τ
m
/
2
0
x
˙
T
(
s
)
Z
1
x
˙
(
s
)
d
s

(
τ
M

τ
m
)
×
∫

τ
M

τ
m
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s

τ
M
2
∫

τ
M
/
2
0
x
˙
T
(
s
)
Z
3
x
˙
(
s
)
d
s
.
By utilizing Lemma 1, we obtain
(20)

τ
m
2
∫
t

(
τ
m
/
2
)
t
x
˙
T
(
s
)
Z
1
x
˙
(
s
)
d
s
≤
[
x
(
t
)
x
(
t

τ
m
2
)
]
T
[

Z
1
Z
1
*

Z
1
]
[
x
(
t
)
x
(
t

τ
m
2
)
]
,

τ
M
2
∫
t

(
τ
M
/
2
)
t
x
˙
T
(
s
)
Z
3
x
˙
(
s
)
d
s
≤
[
x
(
t
)
x
(
t

τ
M
2
)
]
T
[

Z
3
Z
3
*

Z
3
]
[
x
(
t
)
x
(
t

τ
M
2
)
]
.
Define
α
=
(
d
(
t
)

τ
m
)
/
(
τ
M

τ
m
)
,
β
=
(
τ
M

d
(
t
)
)
/
(
τ
M

τ
m
)
; by the reciprocally convex combination in Lemma 3, the following inequality holds:
(21)

[
β
α
(
x
(
t

τ
m
)

x
(
t

d
(
t
)
)
)

α
β
(
x
(
t

d
(
t
)
)

x
(
t

τ
M
)
)
]
T
[
Z
2
S
12
*
Z
2
]
×
[
β
α
(
x
(
t

τ
m
)

x
(
t

d
(
t
)
)
)

α
β
(
x
(
t

d
(
t
)
)

x
(
t

τ
M
)
)
]
<
0
.
Due to
τ
m
≤
d
(
t
)
≤
τ
M
, according to Lemma 1 and inequalities (21), we have
(22)

(
τ
M

τ
m
)
∫
t

τ
M
t

τ
m
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s
=

(
τ
M

τ
m
)
∫
t

d
(
t
)
t

τ
m
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s

(
τ
M

τ
m
)
∫
t

τ
M
t

d
(
t
)
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s
≤

τ
M

τ
m
d
(
t
)

τ
m
[
x
(
t

τ
m
)
x
(
t

d
(
t
)
)
]
T
[
Z
2

Z
2
*
Z
2
]
[
x
(
t

τ
m
)
x
(
t

d
(
t
)
)
]

τ
M

τ
m
τ
M

d
(
t
)
[
x
(
t

d
(
t
)
)
x
(
t

τ
M
)
]
T
[
Z
2

Z
2
*
Z
2
]
[
x
(
t

d
(
t
)
)
x
(
t

τ
M
)
]
≤

[
x
(
t

τ
m
)

x
(
t

d
(
t
)
)
x
(
t

d
(
t
)
)

x
(
t

τ
M
)
]
T
[
Z
2
S
12
*
Z
2
]
×
[
x
(
t

τ
m
)

x
(
t

d
(
t
)
)
x
(
t

d
(
t
)
)

x
(
t

τ
M
)
]
=
ζ
T
(
x
t
)
[

2
Z
2
+
S
12
+
S
12
T
Z
2

S
12
Z
2

S
12
*

Z
2
S
12
*
*

Z
2
]
ζ
(
x
t
)
,
where
(23)
ζ
T
(
x
t
)
=
[
x
T
(
t

d
(
t
)
)
x
T
(
t

τ
m
)
x
T
(
t

τ
M
)
]
.
Based on Lemma 2, an upper bound of
V
˙
4
(
x
t
)
can be estimated as
(24)
V
˙
4
(
x
t
)
=
τ
m
4
64
x
˙
T
(
t
)
R
1
x
˙
(
t
)

τ
m
2
8
×
∫
t

(
τ
m
/
2
)
t
∫
s
t
x
˙
T
(
u
)
R
1
x
˙
(
u
)
d
u
d
s
+
τ
M
4
64
x
˙
T
(
t
)
R
3
x
˙
(
t
)

τ
M
2
8
×
∫
t

(
τ
M
/
2
)
t
∫
s
t
x
˙
T
(
u
)
R
3
x
˙
(
u
)
d
u
d
s
+
(
τ
M

τ
m
)
4
4
x
˙
T
(
t
)
R
2
x
˙
(
t
)

(
τ
M

τ
m
)
2
2
×
∫
t

τ
M
t

τ
m
∫
s
t
x
˙
T
(
u
)
R
2
x
˙
(
u
)
d
u
d
s
≤
τ
m
4
64
x
˙
T
(
t
)
R
1
x
˙
(
t
)
+
τ
M
4
64
x
˙
T
(
t
)
R
3
x
˙
(
t
)

(
τ
m
2
x
(
t
)

∫
t

(
τ
m
/
2
)
t
x
(
s
)
d
s
)
T
×
R
1
(
τ
m
2
x
(
t
)

∫
t

(
τ
m
/
2
)
t
x
(
s
)
d
s
)

(
τ
M
2
x
(
t
)

∫
t

(
τ
M
/
2
)
t
x
(
s
)
d
s
)
T
×
R
3
(
τ
M
2
x
(
t
)

∫
t

(
τ
M
/
2
)
t
x
(
s
)
d
s
)
+
(
τ
M

τ
m
)
4
4
x
˙
T
(
t
)
R
2
x
˙
(
t
)

(
(
τ
M

τ
m
)
x
(
t
)

∫
t

τ
M
t

τ
m
x
(
s
)
d
s
)
T
×
R
2
(
(
τ
M

τ
m
)
x
(
t
)

∫
t

τ
M
t

τ
m
x
(
s
)
d
s
)
.
From (16)–(24), we have
(25)
V
˙
(
x
t
)
≤
ξ
T
(
x
t
)
(
τ
m
4
64
Ψ
T
R
1
Ψ
+
(
τ
M

τ
m
)
4
4
Ψ
T
R
2
Ψ
+
τ
M
4
64
Ψ
T
R
3
Ψ
+
τ
m
2
4
Ψ
T
Z
1
Ψ
+
(
τ
M

τ
m
)
2
Ψ
T
Z
2
Ψ
+
τ
M
2
4
Ψ
T
Z
3
Ψ
+
Θ
22
)
ξ
(
x
t
)
.
If
Θ
<
0
, then
V
˙
(
x
t
)
is negatively defined, based on the Lyapunov theory, we can conclude that NCS (5) is asymptotically stable.
Remark 5.
The robust stability criteria presented in Theorem 4 are for the nominal system. However, it is easy to further extend Theorem 4 to uncertain systems, where the systems matrices
A
and
A
d
contain parameter uncertainties.
Remark 6.
To reduce the conservatism of stability criteria, the LyapunovKrasovskii functional contains some tripleintegral terms in [12]. However, the new LyapunovKrasovskii functional in our paper which not only contains some tripleintegral terms but also divides the lower and upper bounds of the delay into two equal segments, respectively, is proposed. The results will be less conservative.
Remark 7.
Unlike [2],

(
τ
M

τ
m
)
∫
t

τ
M
t

τ
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s
is enlarged as

(
τ
M

τ
)
∫
t

τ
M
t

τ
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s
which may lead to conservation. In this paper,

(
τ

τ
m
)
∫
t

τ
M
t

τ
x
˙
T
(
s
)
Z
2
x
˙
(
s
)
d
s
is retained as well. Meanwhile, we deal with the integral terms of quadratic quantities by virtue of the reciprocally convex combination in Lemma 3; the obtained results are improved.
Remark 8.
The relationship between the timevarying delay and its lower bound and upper bound is taken into account. As a result, fewer useful information is ignored in the derivative of the LyapunovKrasovskii functional.
Remark 9.
The delaydepartioning method in this paper reduces the conservativeness considerably, which arises from the LMI analysis of the interval
[
τ
m
,
τ
M
]
. But when the interval
[
τ
m
,
τ
M
]
is reduced, that is, when
τ
m
→
τ
M
, the benefits of analysis method are shortened. The new strategy is to further make use of the information of the delay lower bound through departioning of the delay interval
[
0
,
τ
m
]
into
N
equidistant subintervals.