# Reliable Control for Uncertain Singular Systems with Randomly Occurring Time-Varying Delay and Actuator Faults

Authors: Yu-Lin Li; Lin-Sheng Li; Zhi-Cheng Zhao; Jing-Gang Zhang
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

The problem of reliable control is investigated for uncertain continuous singular systems with randomly occurring time-varying delay and actuator faults in this work. The delay occurs in a random way, and such randomly occurring delay obeys certain mutually uncorrelated Bernoulli distributed white noise sequences. The uncertainties under consideration are norm-bounded, and may vary with time. Then, with the constructed Lyapunov function, a sufficient condition is given to ensure the unforced system is mean-square exponentially stable and the corresponding controller can be derived from such condition, and the actuator faults problem is guaranteed. A numerical example is provided to show the effectiveness of the methods.


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## Body

## 1. Introduction

During the past three decades, the studies of singular systems have been an active field of research in many scientific and technical disciplines. Dynamic input-output model, electronic network, constrained robots, nuclear reactors, and other noncausal systems all belong to the typical singular systems. Recently, the problems of robust stability analysis and robust stabilization for singular systems have been studied. It is worth noticing that the robust stability problem for singular systems is much more complicated than that for regular systems because it requires considering not only stability robustness, but also regularity and absence of impulses (for continuous singular systems) and causality (for discrete-singular systems) at the same time, and the latter two need not be considered in regular systems. There are lots of papers that have studied these subjects [1–3].On the other hand, singular systems with time-delays arise in a variety of practical systems such as chemical processes and lossless transmission lines. Since singular systems with time-delays are matrix delay differential equations coupled with matrix difference equations, the study of such systems is much more complicated than that of standard state-space time-delay systems or singular systems. The existence and uniqueness of a solution to a given singular time-delay system is not always guaranteed. In accordance with the advance of robust control theory, a number of robust stabilization methods have been proposed for uncertain time-delay systems [4–9].For continuous singular systems, a few of the studies have mentioned the robust stabilization of the uncertain time-delay system. In this paper, we address the problems of robust stability and stabilization for uncertain singular systems with randomly occurring time-varying delay (ROTD). Although the randomly occurring delay has appeared in some papers [1, 10, 11], few related results have been established for uncertain singular systems.Recently, much effort has been devoted to the reliable control with unexpected failures which were often found in the real world. Therefore, designing a controller which could tolerate some actuator failures has been investigated for dynamical systems, discrete-time fuzzy system, networked control system, and so forth. Up to now, the issue of reliable control for uncertain singular systems with randomly occurring time-varying delay and actuator failures has not been fully investigated.In this paper, we deal with the problem of reliable control and exponential stability analysis for uncertain singular systems with ROTD and actuator faults. A random variable, which obeys Bernoulli distribution, is introduced to account for ROTD. By the LMI approach, a state feedback controller is established to guarantee the resultant closed-loop system is delay-dependent exponentially admissible. Finally, a numerical example is given to show the usefulness of the result derived.Notations. R
n denotes the n-dimensional Euclidean space and R
m
×
n is the set of all m
×
n real matrices. The notation X
>
Y
(
X
≥
Y
), where X and Y are symmetric matrices, means that X
-
Y is positive definite (positive semidefinite). I and 0 represent the identity matrix and a zero matrix, respectively; and diag{
⋯
} stands for a block-diagonal matrix. The superscript “T” represents the transpose and the asterisk “*” in a matrix is used to represent the term which is induced by symmetry. E
{
x
} means the expectation of the stochastic variable x.


## 2. Problem Formulation and Preliminaries

Consider the uncertain singular system with randomly occurring time-varying delay:(1)
E
x
˙
t
=
A
+
Δ
A
x
t
+
α
t
A
d
+
Δ
A
d
x
t
-
d
t
+
B
+
Δ
B
u
F
t
x
t
=
ϕ
t
,
t
∈
-
d
,
0
,
where x
(
t
)
∈
R
n is the state vector and u
t
∈
R
m is the control input vector. ϕ
(
t
) is a compatible vector valued initial function. The matrix E
∈
R
n
×
n may be singular and it is assumed that rank
⁡
(
E
)
=
r
≤
n. d
(
t
) is a time-varying continuous function that satisfies 0
<
d
(
t
)
≤
d and d
˙
(
t
)
≤
μ, where 0 and d are the lower and upper bounds of the time-delay d
(
t
). A, A
d, and B are known real constant matrices with appropriate dimensions. Δ
A, Δ
A
d, and Δ
B are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form
(2)
Δ
A
Δ
A
d
Δ
B
=
M
F
t
N
1
N
2
N
3
,
where M, N
1, N
2, and N
3 are known real constant matrices with appropriate dimensions and F
(
t
)
∈
R
q
×
k is an unknown real and possibly time-varying matrix satisfying
(3)
F
t
T
F
t
≤
I
.The parametric uncertaintiesΔ
A, Δ
A
d, and Δ
B are said to be admissible if both 2 and 3 hold.To account for the phenomena of ROTD, we introduce the stochastic variableα
t, which is a Bernoulli-distributed white sequence. The natural assumption on α
t is as follows:
(4)
Prob
⁡
α
t
=
1
=
E
α
t
=
α
,
Prob
⁡
α
t
=
0
=
1
-
E
α
t
=
1
-
α
,
where the constant α
∈
[
0,1
].When the actuators experience failures, we useu
F
t to describe the control signal sent from actuators. Consider the actuator failure model with failure parameter
(5)
u
F
t
=
W
u
t
.
W
=
diag
⁡
w
1
,
w
2
,
…
,
w
m is the actuator failure matrix with the following property:
(6)
0
≤
w
i
min
⁡
≤
w
i
≤
w
i
max
⁡
≤
1
,
i
=
1,2
,
…
,
m
,
where
(7)
w
i
min
⁡
=
w
i
max
⁡
=
1
,
i
=
1,2
,
…
,
m
⟹
i
th
actuator
has
no
failure
w
i
min
⁡
=
w
i
max
⁡
=
0
,
i
=
1,2
,
…
,
m
⟹
i
th
actuator
is
outage
0
<
w
i
<
1
,
i
=
1,2
,
…
,
m
⟹
i
th
actuator
has
partial
failure
.Define the following notations:(8a)
W
0
=
diag
⁡
w
01
,
w
02
,
…
,
w
0
m
,
(8b)
H
=
diag
⁡
h
1
,
h
2
,
…
,
h
m
,
(8c)
G
=
diag
⁡
g
1
,
g
2
,
…
,
g
m
,
(8d)
G
=
diag
⁡
g
1
,
g
2
,
…
,
g
m
,where
(9)
w
0
i
=
w
i
max
⁡
+
w
i
min
⁡
2
,
h
i
=
w
i
max
⁡
-
w
i
min
⁡
w
i
max
⁡
+
w
i
min
⁡
,
g
i
=
w
i
-
w
0
i
w
0
i
,
i
=
1,2
,
…
,
m
.From notations8a-8b, we have
(10)
W
=
W
0
I
+
G
,
G
≤
H
≤
I
.
Because actuator failure that might occur is unknown, the matrix M is not well known in advance. The matrix M satisfies 6 in which the matrix M
0 is known in advance.The nominal unforced singular systems with randomly occurring time-varying delay of1 can be described as follows:
(11)
E
x
˙
t
=
A
x
t
+
α
t
A
d
x
t
-
d
t
,
x
t
=
ϕ
t
,
t
∈
-
d
,
0
.Definition 1.
System11 is said to be regular and impulse-free, if the pair (
E
,
A
) is regular and impulse-free.Definition 2.
System11 is said to be exponentially stable, if there exist scalars α
>
0 and β
>
0 such that x
(
t
)
≤
α
e
-
β
t
ϕ
(
t
)
d.Definition 3.
System11 is said to be exponentially admissible, if it is regular, impulse-free, and exponentially stable.Definition 4.
The uncertain singular system1 is said to be exponentially admissible, if the unforced system is regular, impulse-free, and exponentially stable for all admissible uncertainties Δ
A and Δ
A
d.Let us consider the following controller:(12)
u
t
=
K
x
t
,
where K
∈
R
m
×
n is the controller gain matrix to be designed.Substitutingu
F
t for u
t in 1 and considering 5 and 12, the closed-loop system can be described by
(13)
E
x
˙
t
=
A
+
B
W
K
+
Δ
A
+
Δ
B
W
K
x
t
+
α
t
A
d
+
Δ
A
d
x
t
-
d
t
,
x
t
=
ϕ
t
,
t
∈
-
d
,
0
.Lemma 5 (Lu et al. [12]).
For any matrixM
>
0, integers γ
1 and γ
2 satisfying γ
2
>
γ
1, and vector function ω
:
N
[
γ
1
,
γ
2
]
→
R
n, such that the sums concerned are well defined,
(14)
γ
2
-
γ
1
∫
γ
1
γ
2
ω
α
T
M
ω
α
d
α
≥
∫
γ
1
γ
2
ω
α
d
α
T
M
∫
γ
1
γ
2
ω
α
d
α
.Lemma 6 (Wu and Zheng [13]).
Given matricesΞ
,
Γ, and Φ with appropriate dimensions and with Ξ symmetrical, then Ξ
+
H
F
Φ
+
Φ
T
F
T
H
T
<
0 for any F satisfying F
T
F
≤
I, if and only if there exists a scalar ε
>
0 such that Ξ
+
ε
H
H
T
+
ε
-
1
Φ
T
Φ
<
0.Lemma 7 (Mao et al. [9]).
Suppose that a positive continuous functionf
(
t
) satisfies
(15)
f
t
≤
ζ
1
sup
⁡
t
-
d
≤
s
≤
t
f
t
+
ζ
2
e
-
ε
t
,
where ε
>
0, 0
<
ζ
1
<
1, 0
<
ζ
1
e
ε
d
<
1, ζ
2
>
0, and d
>
0; then,
(16)
f
t
≤
e
-
ε
t
f
s
d
+
ζ
2
e
-
ε
t
1
-
ζ
1
e
ε
d
.


## 3. Main Result

In this section, we first derive a condition to guarantee nominal unforced system11 to be mean-square exponential stability.Theorem 8.
Singular time-delay system11 is exponentially robustly stable, if there exist matrices P
,
Y
,
Q
>
0, R
>
0, and Z
>
0 such that(17a)
E
T
P
=
P
T
E
≥
0
,
(17b)
Ξ
=
Ξ
11
Ξ
12
E
T
Y
E
d
A
T
Z
*
Ξ
22
-
E
T
Y
E
+
E
T
Z
E
d
α
A
d
T
Z
*
*
-
R
0
*
*
*
-
Z
<
0
,
(17c)
Z
Y
*
Z
>
0
,where
(18)
Ξ
11
=
P
T
A
+
A
T
P
+
Q
+
R
-
E
T
Z
E
,
Ξ
12
=
α
P
T
A
d
+
E
T
Z
E
-
E
T
Y
E
,
Ξ
22
=
-
1
-
μ
Q
+
E
T
Y
E
+
E
T
Y
T
E
-
2
E
T
Z
E
.Proof.
Firstly, the regularity and absence of impulses of system8a, 8b, 8c, and 8d are given. For singular system, we know that there exist nonsingular matrices G and H such that
(19)
G
E
H
=
I
r
0
0
0
.
Denote
(20)
G
A
H
=
A
1
A
2
A
3
A
4
,
G
-
T
P
H
=
P
1
P
2
P
3
P
4
.
From 17a and using the expressions in 19 and 20, it can be found that P
2
=
0. Then, premultiplying and postmultiplying Ξ
11
<
0 by H
T and H, respectively, we have
(21)
A
4
T
P
4
+
P
4
T
A
4
<
0
which implies A
4 is nonsingular and thus the pair (
E
,
A
) is regular and impulse-free. Hence, by Definition 1, system 11 is regular and impulse-free.
Then, the problem for the unforced system11 to be exponentially stable is given. Consider the following Lyapunov function for system 11:
(22)
V
x
t
=
V
1
x
t
+
V
2
x
t
+
V
3
x
t
,
where
(23)
V
1
x
t
=
x
t
T
E
T
P
x
t
,
V
2
x
t
=
∫
t
-
d
t
t
x
s
T
Q
x
s
d
s
+
∫
t
-
d
t
x
s
T
R
x
s
d
s
,
V
3
x
t
=
d
∫
-
d
0
∫
t
+
θ
t
x
˙
s
E
T
Z
E
x
˙
s
d
s
d
θ
.
Define the infinitesimal operatorL of V
(
x
t
) as follows:
(24)
L
V
x
t
=
lim
⁡
h
→
0
+
⁡
1
h
E
V
x
t
+
h
∣
x
t
-
V
x
t
.
We obtain
(25)
E
L
V
x
t
=
E
L
V
1
x
t
+
E
L
V
2
x
t
+
E
L
V
3
x
t
,
where
(26)
E
L
V
1
x
t
=
E
2
x
t
T
E
T
P
x
˙
t
=
E
2
x
t
T
E
T
P
A
x
t
+
α
t
A
d
x
t
-
d
t
=
2
x
t
T
E
T
P
A
x
t
+
α
A
d
x
t
-
d
t
E
L
V
2
x
t
≤
E
x
t
T
Q
x
t
-
1
-
μ
x
t
-
d
t
T
h
h
h
h
·
Q
x
t
-
d
t
+
x
t
T
R
x
t
-
x
t
-
d
T
R
x
t
-
d
x
t
-
d
t
T
=
x
t
T
Q
x
t
-
1
-
μ
x
t
-
d
t
T
Q
x
t
-
d
t
+
x
t
T
R
x
t
-
x
t
-
d
T
R
x
t
-
d
,
E
L
V
3
x
t
=
E
d
2
x
˙
t
T
E
T
Z
E
x
˙
t
-
d
∫
t
-
d
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
=
d
2
A
x
t
+
α
A
d
x
t
-
d
t
T
·
Z
A
x
t
+
α
A
d
x
t
-
d
t
-
d
∫
t
-
d
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
.
Applying Lemma 5, when 0
<
d
(
t
)
<
d, we have that
(27)
-
d
∫
t
-
d
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
=
-
d
∫
t
-
d
t
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
-
d
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
≤
-
d
d
t
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
d
-
d
t
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
=
-
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
-
d
-
d
t
d
t
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
-
d
t
d
t
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
-
d
t
d
t
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
t
d
-
d
t
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
.
It can be from 17c that 29
(28)
d
-
d
t
d
t
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
t
d
-
d
t
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
T
Z
Y
*
Z
·
d
-
d
t
d
t
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
t
d
-
d
t
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
>
0
which implies
(29)
-
d
-
d
t
d
t
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
-
d
t
d
-
d
t
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
d
s
Z
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
≤
-
∫
t
-
d
t
t
x
˙
t
T
E
T
d
s
Y
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
-
∫
t
-
d
t
-
d
t
x
˙
t
T
E
T
d
s
Y
T
∫
t
-
d
t
t
E
x
˙
t
d
s
.
We can get from 27 and 29 that
(30)
-
d
∫
t
-
d
t
x
˙
t
T
E
T
Z
E
x
˙
t
d
s
≤
-
∫
t
-
d
t
t
E
x
˙
t
d
s
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
T
Z
Y
*
Z
∫
t
-
d
t
t
E
x
˙
t
d
s
∫
t
-
d
t
-
d
t
E
x
˙
t
d
s
=
x
t
x
t
-
d
t
x
t
-
d
T
×
-
E
T
Z
E
E
T
Z
E
-
E
T
Y
E
E
T
Y
E
*
-
2
E
T
Z
E
+
E
T
Y
E
+
E
T
Y
T
E
-
E
T
Y
E
+
E
T
Z
E
*
*
-
E
T
Z
E
×
x
t
x
t
-
d
t
x
t
-
d
.
Then,
(31)
E
L
V
3
x
t
≤
d
2
A
x
t
+
α
A
d
x
t
-
d
t
T
Z
·
A
x
t
+
α
A
d
x
t
-
d
t
-
x
t
T
E
T
Z
E
x
t
+
2
x
t
T
E
T
Z
E
-
E
T
Y
E
·
Z
x
t
-
d
t
+
2
x
t
T
E
T
Y
E
x
t
-
d
+
x
t
-
d
t
T
-
2
E
T
Z
E
+
E
T
Y
E
+
E
T
Y
T
E
x
t
-
d
t
+
2
x
t
-
d
t
T
-
E
T
Y
E
+
E
T
Z
E
x
t
-
x
t
-
d
T
E
T
Z
E
x
t
-
d
=
∇
.
So
(32)
E
L
V
x
t
=
E
L
V
1
x
t
+
E
L
V
2
x
t
+
E
L
V
3
x
t
≤
2
x
t
T
E
T
P
A
x
t
+
α
A
d
x
t
-
d
t
+
x
t
T
Q
x
t
-
1
-
μ
x
t
-
d
t
T
Q
x
t
-
d
t
+
x
t
T
R
x
t
-
x
t
-
d
T
R
x
t
-
d
+
∇
=
E
ζ
t
T
Υ
ζ
t
,
where
(33)
ζ
t
=
x
t
x
t
-
d
t
x
t
-
d
,
Υ
=
Ξ
11
Ξ
12
E
T
Y
E
*
Ξ
22
-
E
T
Y
E
+
E
T
Z
E
*
*
-
R
+
A
T
A
d
T
0
d
2
Z
A
T
A
d
T
0
T
.
By application of Schur complement and 17b, we have Υ
<
0, which guarantees there exists a scalar λ
>
0 such that
(34)
E
L
V
x
t
<
0
.
Set
(35)
G
^
=
I
r
-
A
2
A
4
-
1
0
A
4
-
1
G
.
It is easy to get
(36)
G
^
E
H
=
I
r
0
0
0
,
G
^
A
H
=
A
^
1
0
A
^
3
I
,
where A
^
1
=
A
1
-
A
2
A
4
-
1
A
3 and A
^
3
=
A
4
-
1
A
3.
Consider(37)
G
^
A
d
H
=
A
d
1
A
d
2
A
d
3
A
d
4
,
G
^
-
1
P
H
=
P
1
P
2
P
3
P
4
,
H
T
Q
H
=
Q
11
Q
21
*
Q
22
.
It can be seen from 17b that
(38)
I
I
I
T
Ξ
11
Ξ
12
E
T
Y
E
*
Ξ
22
-
E
T
Y
E
+
E
T
Z
E
*
*
-
R
I
I
I
<
0
,
which implies
(39)
P
T
A
+
α
A
d
+
A
+
α
A
d
T
P
<
0
.
It is easy to find that matrix P satisfying the above inequality is nonsingular. Thus, considering 17a, we can deduce that P
1
>
0 and P
2
=
0.
Define(40)
ς
^
t
=
ς
1
t
ς
2
t
=
H
-
1
x
t
.
Then, system 11 is equivalent to
(41)
ς
˙
t
=
A
^
1
ς
1
t
+
α
t
A
d
1
ς
1
t
-
d
t
+
α
t
A
d
2
ς
2
t
-
d
t
-
ς
2
t
=
A
^
3
ς
1
t
+
α
t
A
d
3
ς
1
t
-
d
t
+
α
t
A
d
4
ς
2
t
-
d
t
ς
^
t
=
Ψ
t
=
H
-
1
ϕ
t
,
t
∈
-
d
,
0
.
To prove the exponential stability of system11, we define a function as
(42)
W
x
t
,
t
=
e
ε
t
V
x
t
,
t
,
where the scalar ε
>
0.
Taking its time derivative yields(43)
W
˙
x
t
,
t
=
ε
e
ε
t
V
x
t
,
t
+
e
ε
t
V
˙
x
t
,
t
≤
ε
e
ε
t
V
x
t
,
t
-
λ
e
ε
t
x
t
2
.
Integrating both sides of 39 from 0 to t, we get that
(44)
W
x
t
,
t
≤
W
x
0
,
0
+
∫
0
t
e
ε
s
ε
E
L
V
x
s
-
λ
x
t
2
d
s
.
By using the similar analysis method of [14], it can be seen from 22, 42, and 44 that if the scalar ε is chosen small enough, a scalar k
>
0 can be found such that, for any t
>
0,
(45)
V
x
t
,
t
≤
k
e
-
ε
t
ϕ
t
d
2
.
Since λ
min
⁡
P
1
ς
1
(
t
)
2
≤
x
t
T
E
T
P
x
(
t
)
≤
V
(
x
t
,
t
), it can be shown from 43 that for any t
>
0
(46)
ς
1
t
2
≤
β
e
-
ε
t
ϕ
t
d
2
,
where β
=
λ
min
⁡
P
1
k. Define
(47)
e
t
=
A
^
3
ς
1
t
+
α
t
A
d
3
ς
1
t
-
d
t
;
then, from 46, a scalar m
>
0 can be found such that, for any t
>
0,
(48)
e
t
2
≤
m
e
-
ε
t
ϕ
t
d
2
.
To study the exponential stability of ς
2
t, we construct a function as
(49)
J
t
=
ς
2
t
T
Q
22
ς
2
t
-
ς
2
t
-
d
t
T
Q
22
ς
2
t
-
d
t
.
By premultiplying the second equation of 41 with ς
2
t
T
P
4
T, we get that
(50)
0
=
2
ς
2
t
T
P
4
T
ς
2
t
+
ς
2
t
T
P
4
T
α
t
A
d
4
ς
2
t
-
d
t
hhhhhh
+
ς
2
t
T
P
4
T
e
t
.
Adding 50 to 49 yields that
(51)
J
t
=
ς
2
t
T
P
4
T
+
P
4
+
Q
22
ς
2
t
+
2
ς
2
t
T
P
4
T
α
t
A
d
4
ς
2
t
-
d
t
-
ς
2
t
-
d
t
T
Q
22
ς
2
t
-
d
t
+
2
ς
2
t
T
P
4
T
e
t
≤
ς
2
t
ς
2
t
-
d
t
T
×
P
4
T
+
P
4
+
Q
22
α
P
4
T
A
d
4
*
-
Q
22
ς
2
t
ς
2
t
-
d
t
+
η
1
ς
2
t
T
ς
2
t
+
η
1
-
1
e
t
T
P
4
P
4
T
e
t
,
where η
1 is any positive scalar.
On the other hand, we can get from17b that
(52)
Q
-
E
T
Z
E
+
P
T
A
+
A
T
P
α
P
T
A
d
+
E
T
Z
E
-
E
T
Y
E
*
-
Q
+
E
T
Y
E
+
E
T
Y
T
E
-
2
E
T
Z
E
<
0
.
Premultiplying and postmultiplying 52 by H
0
0
H
T and H
0
0
H, respectively, a scalar η
2
>
0 can be found such that
(53)
P
4
T
+
P
4
+
Q
22
α
P
4
T
A
d
4
*
-
Q
22
≤
-
η
2
I
0
0
0
.
On the other hand, since η
1 can be chosen arbitrarily, η
1 can be chosen small enough such that η
2
-
η
1
>
0. Then, a scalar η
3
>
1 can always be found such that
(54)
Q
22
-
η
2
-
η
1
I
≥
η
3
Q
22
.
It follows from 48, 49, 51, 53, and 54 that
(55)
ς
2
t
T
Q
22
ς
2
t
≤
η
3
-
1
ς
2
t
-
d
t
T
Q
22
ς
2
t
-
d
t
+
η
1
η
3
-
1
e
t
T
P
4
P
4
T
e
t
which infers
(56)
f
t
≤
η
3
-
1
sup
⁡
t
-
d
≤
s
≤
t
f
t
+
Ξ
e
-
δ
t
,
where 0
<
δ
<
min
⁡
⁡
{
ε
,
d
-
1
ln
⁡
⁡
η
3
}, f
t
=
ς
2
t
T
Q
22
ς
2
t, and ξ
=
η
1
η
3
-
1
m
P
4
2
ϕ
(
t
)
d
2.
Applying Lemma7 to 56 yields that
(57)
ς
2
t
2
≤
λ
min
⁡
-
1
Q
22
λ
max
⁡
Q
22
e
-
δ
t
ς
2
t
d
2
+
λ
min
⁡
-
1
Q
22
ξ
e
-
δ
t
1
-
η
3
-
1
e
δ
d
.
We can find from 46 and 57 that system 11 is exponentially stable. This completes the proof.Next, we will present the robust stability criterion for the following system:(58)
E
x
˙
t
=
A
+
Δ
A
x
t
+
α
t
A
d
+
Δ
A
d
x
t
-
d
t
x
t
=
ϕ
t
,
t
∈
-
τ
,
0
,
via Theorem 8.Theorem 9.
Singular time-delay system58 is robustly stable, if there exist a scalar ε
>
0, matrices P, Y, Q
>
0, R
>
0, and Z
>
0 such that
(59)
Ξ
11
Ξ
12
E
T
Y
E
d
A
T
Z
P
T
M
*
Ξ
22
-
E
T
Y
E
+
E
T
Z
E
d
α
A
d
T
Z
0
*
*
-
R
0
0
*
*
*
-
Z
d
M
*
*
*
*
-
ε
I
<
0
,
where
(60)
Ξ
11
=
P
T
A
+
A
T
P
+
Q
+
R
-
E
T
Z
E
+
ε
N
1
T
N
1
,
Ξ
12
=
α
P
T
A
d
+
E
T
Z
E
-
E
T
Y
E
+
ε
α
N
1
T
N
2
,
Ξ
22
=
-
1
-
μ
Q
+
E
T
Y
E
+
E
T
Y
T
E
-
2
E
T
Z
E
+
ε
α
2
N
2
T
N
2
.Proof.
Condition17b with A and A
d replaced by A
+
M
F
(
t
)
N
1 and A
d
+
M
F
(
t
)
N
2, respectively, can be written as
(61)
Ξ
+
H
F
t
Φ
+
Φ
T
F
t
T
H
T
<
0
,
where Ξ follows the same definition as in Theorem 8 and
(62)
H
=
M
T
P
0
0
d
M
T
,
Φ
=
N
1
α
N
2
0
0
.
By Lemma 6, 61 holds if there exists a scalar ε
>
0 such that
(63)
Ξ
+
ε
H
H
T
+
ε
-
1
Φ
T
Φ
<
0
.
By Schur complement, 63 is equivalent to 59. Hence, according to Theorem 8 and Definition 4, system 58 is robustly stable. This completes the proof.Theorem 10.
Singular system13 is robustly stable, if there exist a scalar ε
>
0, matrices L, Y
^, Q
^
>
0, R
^
>
0, and Z
^
>
0, and matrices V with appropriate dimensions such that(64a)
L
T
E
T
=
E
L
≥
0
,
(64b)
Ξ
^
=
Ξ
^
11
Ξ
^
12
E
Y
^
E
T
Ξ
^
14
V
T
W
0
H
L
T
N
1
T
+
V
T
W
0
+
W
0
H
T
N
3
T
*
Ξ
^
22
-
E
Y
^
E
T
+
E
Z
^
E
T
d
α
L
T
A
d
T
0
α
L
T
N
2
T
*
*
-
R
^
0
0
0
*
*
*
-
Z
^
-
L
T
-
L
+
d
2
+
ε
2
d
2
M
M
T
0
0
*
*
*
*
-
ε
1
I
0
*
*
*
*
*
-
ε
2
I
(64c)
Z
^
Y
^
*
Z
^
>
0
,where
(65)
Ξ
^
11
=
A
L
+
L
T
A
T
+
B
W
0
V
+
V
T
W
0
T
B
T
+
Q
^
+
R
^
-
E
Z
^
E
T
+
ε
1
B
B
T
+
ε
2
M
T
M
,
Ξ
^
12
=
α
A
d
L
+
E
Z
^
E
T
-
E
Y
^
E
T
,
Ξ
^
14
=
d
L
T
A
T
+
V
T
W
0
T
B
T
+
d
B
+
ε
2
d
M
M
T
,
Ξ
^
22
=
-
1
-
μ
Q
^
+
E
Y
^
E
T
+
E
Y
^
T
E
T
-
2
E
Z
^
E
T
.
In this case, the desired controller gain is given as K
=
V
L
-
1.Proof.
By Condition17b with A and A
d replaced by A
+
B
W
K
+
M
F
(
t
)
N
1
+
M
F
(
t
)
N
3
W
K and A
d
+
M
F
(
t
)
N
2, respectively, we have
(66)
Ξ
-
11
Ξ
-
12
E
T
Y
E
d
A
+
B
W
K
+
M
F
t
N
1
+
M
F
t
N
3
W
K
T
Z
*
Ξ
22
-
E
T
Y
E
+
E
T
Z
E
d
α
A
d
+
M
F
t
N
2
T
Z
*
*
-
R
0
*
*
*
-
Z
<
0
,
where
(67)
Ξ
-
11
=
P
T
A
+
B
W
K
+
M
F
t
N
1
+
M
F
t
N
3
W
K
+
A
+
B
W
K
+
M
F
t
N
1
+
M
F
t
N
3
W
K
T
P
+
Q
+
R
-
E
T
Z
E
,
Ξ
-
12
=
α
P
T
A
d
+
M
F
t
N
2
+
E
T
Z
E
-
E
T
Y
E
.
Define matrix J
=
diag
⁡
{
L
,
L
,
L
,
Z
-
1
}, L
=
P
-
1, Q
^
=
L
T
Q
L, R
^
=
L
T
R
L, Y
^
=
L
Y
L
T, Z
^
=
L
Z
L
T, and V
=
K
L. Then, pre- and postmultiplying the left side of 63 by J
T and J, respectively, we can find that
(68)
Σ
+
H
F
t
Φ
+
Φ
T
F
t
T
H
T
<
0
,
where
(69)
Σ
=
Ξ
̿
11
Ξ
^
12
E
Y
^
E
T
d
L
T
A
T
+
V
T
W
T
B
T
*
Ξ
^
22
-
E
Y
^
E
T
+
E
Z
^
E
T
d
α
L
T
A
d
T
*
*
-
R
^
0
*
*
*
-
L
T
Z
^
-
1
L
<
0
Ξ
̿
11
=
A
L
+
L
T
A
T
+
B
W
V
+
V
T
W
T
B
T
+
Q
^
+
R
^
-
E
Z
^
E
T
H
=
L
T
N
1
T
+
V
T
W
T
N
3
T
α
L
T
N
2
T
0
0
T
Φ
=
M
T
0
0
d
M
T
.
Noting Z
>
0, we have -
L
T
Z
^
-
1
L
≤
Z
^
-
L
T
-
L.
By Lemma6, 68 holds if there exists a scalar ε
>
0 such that
(70)
Ψ
=
Σ
+
ε
2
H
H
T
+
ε
2
-
1
Φ
T
Φ
<
0
.
By Condition 70 with 10, we have
(71)
Ψ
^
=
Ψ
W
=
W
0
I
+
G
=
Σ
^
+
B
0
0
d
W
0
G
V
0
0
0
+
V
T
G
T
W
0
T
0
0
0
B
T
0
0
d
+
ε
2
H
^
H
^
T
+
ε
2
-
1
Φ
T
Φ
≤
Σ
~
+
ε
1
B
0
0
d
B
0
0
d
T
+
ε
1
-
1
V
T
W
0
H
0
0
0
V
T
W
0
H
0
0
0
T
+
ε
2
H
~
H
~
T
+
ε
2
-
1
Φ
T
Φ
=
Σ
ˇ
+
ε
2
H
~
H
~
T
+
ε
2
-
1
Φ
~
T
Φ
~
<
0
,
where(72)
Σ
^
=
Ξ
¨
11
Ξ
^
12
E
Y
^
E
T
d
L
T
A
T
+
V
T
W
0
+
W
0
G
T
B
T
*
Ξ
^
22
-
E
Y
^
E
T
+
E
Z
^
E
T
d
α
L
T
A
d
T
*
*
-
R
^
0
*
*
*
Z
^
-
L
T
-
L
Ξ
¨
11
=
A
L
+
L
T
A
T
+
B
W
0
+
W
0
G
V
+
V
T
W
0
+
W
0
G
T
B
T
+
Q
^
+
R
^
-
E
Z
^
E
T
Σ
~
=
Ξ
˙
11
Ξ
^
12
E
Y
^
E
T
d
L
T
A
T
+
V
T
W
0
T
B
T
*
Ξ
^
22
-
E
Y
^
E
T
+
E
Z
^
E
T
d
α
L
T
A
d
T
*
*
-
R
^
0
*
*
*
Z
^
-
L
T
-
L
Ξ
˙
11
=
A
L
+
L
T
A
T
+
B
W
0
V
+
V
T
W
0
T
B
T
+
Q
^
+
R
^
-
E
Z
^
E
T
Σ
ˇ
=
Ξ
-
~
11
Ξ
^
12
E
Y
^
E
T
d
L
T
A
T
+
V
T
W
0
T
B
T
+
d
B
V
T
W
0
H
*
Ξ
^
22
-
E
Y
^
E
T
+
E
Z
^
E
T
d
α
L
T
A
d
T
0
*
*
-
R
^
0
0
*
*
*
Z
^
-
L
T
-
L
+
d
2
0
*
*
*
*
ε
1
I
Ξ
-
~
11
=
A
L
+
L
T
A
T
+
B
W
0
V
+
V
T
W
0
T
B
T
+
Q
^
+
R
^
-
E
Z
^
E
T
+
ε
1
B
B
T
H
^
=
L
T
N
1
T
+
V
T
W
0
+
W
0
G
T
N
3
T
α
L
T
N
2
T
0
0
0
T
H
~
=
L
T
N
1
T
+
V
T
W
0
+
W
0
H
T
N
3
T
α
L
T
N
2
T
0
0
0
T
Φ
~
=
M
T
0
0
d
M
T
0
.By Schur complement, 71 is equivalent to 64a, 64b, and 64c. Hence, according to Theorem 8 and Definition 4, the system controlled by u
(
t
)
=
V
L
-
1
x
(
t
) is robustly stable. This completes the proof.


## 4. Numerical Example

In this section, we give an example to demonstrate the effectiveness of the proposed method.Consider the singular system1 with parameters as follows:
(73)
E
=
1
0
0
0
A
=
1
-
0.2
3
-
1.1
A
d
=
0
.
2
0
0
.
1
-
0
.
1
B
=
1
-
0.1
0
3
M
=
1
0
0
1
N
1
=
0.1
0
0
0.1
N
2
=
0.1
0
0
0.1
N
3
=
0.1
0
0
0.1
.The state response of open-loop system (u
F
t
=
0) 1 is shown in Figure 1 and it is clear that the open-loop system is unstable.Figure 1
State response of open-loop system.In this example, we choseα
=
0.65, μ
=
0.8, and d
=
0.8. The first actuator failure matrix is W
=
diag
⁡
{
w
1
,
w
2
}, where w
1
min
⁡
=
0.7
≤
w
1
≤
1
=
w
1
max
⁡ and w
2
min
⁡
=
0
≤
w
2
≤
0.8
=
w
2
max
⁡ and the second actuator failure matrix is W
=
diag
⁡
{
w
1
,
w
2
}, where w
1
min
⁡
=
0.1
≤
w
1
≤
0.6
=
w
1
max
⁡ and w
2
min
⁡
=
w
2
=
w
2
max
⁡
=
0.For the first actuator failure, by Theorem10, we have the following controller gain:
(74)
K
=
-
3.9595
0.009
-
0.2146
-
0.1379
.When actuator failures (w
1
=
0.85 and w
2
=
0.4) with the designed feedback controller gain 74, the state response of the closed-loop system and the controller output are shown in Figures 2 and 3. In this example, all controllers are partially damaged. Figure 2 shows the reliable feedback control gain can guarantee that the state responses converge to zero asymptotically.Figure 2
State response of closed-loop system.Figure 3
Controller output.For the second actuator failure, by Theorem10, we have the following controller gain:
(75)
K
=
-
12.3958
0.3135
-
0.3296
-
0.8185
.When actuator failures (w
1
=
0.35 and w
2
=
1) with the designed feedback controller gain 75, the state response of the closed-loop system and the controller output are shown in Figures 4 and 5. In this example, not all controllers are partially damaged. Figure 4 shows the reliable feedback control gain can guarantee that the state responses converge to zero asymptotically.Figure 4
State response of closed-loop system.Figure 5
Controller output.


## 5. Conclusion

The problems of reliable control for uncertain singular systems with randomly occurring time-varying delay and actuator faults have been studied. Based on the LMI approach, a sufficient condition has been established to ensure the considered systems are regular, impulse-free, and exponentially stable. The state feedback controller has been designed to ensure, for all possible actuator failures, the resultant closed-loop system is exponentially admissible. A numerical example has been provided to show the validness and less conservatism of the given result.



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*Source: 238692-2015-04-16.xml*