# Improved Robust Stability Criterion of Networked Control Systems with Transmission Delays and Packet Loss

Authors: Shenping Xiao; Liyan Wang; Hongbing Zeng; Wubin Cheng
Journal: Journal of Control Science and Engineering
 (2014)

---

## Abstract

The problem of stability analysis for a class of networked control systems (NCSs) with network-induced delay and packet dropout is investigated in this paper. Based on the working mechanism of zero-order holder, the closed-loop NCS is modeled as a continuous-time linear system with input delay. By introducing a novel Lyapunov-Krasovskii functional which splits both the lower and upper bounds of the delay into two subintervals, respectively, and utilizes reciprocally convex combination technique, a new stability criterion is derived in terms of linear matrix inequalities. Compared with previous results in the literature, the obtained stability criterion is less conservative. Numerical examples demonstrate the validity and feasibility of the proposed method.


---

## Body

## 1. Introduction

Network control systems (NCSs) are the feedback control systems in which the control loops are closed via real-time networks [1]. NCSs have great advantages compared to the traditional point-to-point 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 Lyapunov-Razumikhi function techniques and the quasiconvex optimization algorithm. The free weighting matrices method was proposed in [3] to solve the problem of network-based 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 network-induced 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 network-induced delay and packet dropout is addressed. The NCSs are modeled as continuous-time linear systems on the basis of the input delay approach. By exploiting the information of lower and upper bounds of the delay, a new Lyapunov-Krasovskii 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 clock-driven fashion with transmission period h; the controller and actuator modules are assumed to act in an event-driven 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 network-induced delay, and h is the sampling period, {
i
1
,
i
2
,
i
3
,
…
}
⊂
{
1,2
,
3
,
…
}; due to the introduction of logical zero-order 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 constantsτ
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 time-varying 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 [13]).
For any positive matrixM
>
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 [14]).
For any positive matrixR, 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 [15]).
LetF
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 scalarτ
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 Lyapunov-Krasovskii 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 extended-state 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 Theorem4 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 Lyapunov-Krasovskii functional contains some triple-integral terms in [12]. However, the new Lyapunov-Krasovskii functional in our paper which not only contains some triple-integral 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 time-varying 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 Lyapunov-Krasovskii functional.Remark 9.
The delay-departioning 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.


## 4. Numerical Example

Example 1.
Consider the following NCS [10]:
(26)
x
˙
(
t
)
=
[
0
1
0
-
0.1
]
x
(
t
)
+
[
0
0.1
]
u
(
t
)
.
Assume that the state feedback gain matrix K
=
[
-
3.75
-
11.5
]; when we do not consider the lower bound of the delay, that is, τ
m
=
0.00, by applying Theorem 4, the maximum upper bound of delay obtained is τ
M
=
1.113, while in [10], τ
M
=
0.941, and the maximum allowable value is τ
M
=
1.003 in [8]. A more detailed comparison for different values of τ
m is provided in Table 1. As shown in the table, it can be seen that our results are less conservative than the ones in [8, 10].Table 1
Admissible upper boundτ
M for various τ
m.
τ
m
0.00
0.01
0.05
0.10
Jiang and Han [10]
0.941
0.942
0.948
0.952
Zhang et al. [8]
1.003
1.024
1.026
1.027
Theorem4
1.113
1.114
1.116
1.119Example 2.
Consider the NCS (5) with the following parameters [11]:
(27)
A
=
[
0
1
-
1
-
2
]
,
A
d
=
B
K
=
[
0
0
-
1
1
]
.
Table 2 lists the maximum allowable upper bound of τ
M with respect to different conditions of τ
m along with some existing results from the literature. From Table 2, we can conclude that the criterion derived in this paper presents superior results.Table 2
Admissible upper boundτ
M for various τ
m.
τ
m
1
2
3
4
Shao [11]
1.617
2.480
3.389
4.325
Sun et al. [12]
1.620
2.488
3.403
4.342
Theorem4
1.765
2.606
3.502
4.429


## 5. Conclusion

The problem of robust stability is addressed for a class of NCSs with network-induced delay and packet dropout. Improved and simplified stability criterion is established without involving any model transformation or free weighting matrices. The simulation results indicate that the criterion derived in this paper can exhibit better performance compared to that in the existing literature.



---

*Source: 936856-2014-02-12.xml*