# RobustH

Authors: Li-li Liu; Bao-wei Wu; Yue-E Wang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

A new version of delay-dependent bounded real lemma for singular systems with state delay is established by parameterized Lyapunov-Krasovskii functional approach. In order to avoid generating nonconvex problem formulations in control design, a strategy that introduces slack matrices and decouples the system matrices from the Lyapunov-Krasovskii parameter matrices is used. Examples are provided to demonstrate that the results in this paper are less conservative than the existing corresponding ones in the literature.


---

## Body

## 1. Introduction

During the last decade, a considerable amount of attention has been paid to stability and control of singular systems with delay, since they can better describe and analyze physical systems than the state-space time-delay ones [1, 2]. It should be pointed out that the control problems for singular systems are much more complicated than that for state-space systems, because singular systems usually have three types of modes, namely, finite dynamic modes, impulse modes, and nondynamic modes, whereas the latter two do not appear in state-space systems [1]. Recently, the problem of H
∞ control for singular systems with delay has been investigated by many researchers and various approaches have been adopted [3–15]. There are two performance indexes which are used to judge the conservatism of the derived conditions. One is the H
∞ performance index while the other is admissible upper bound of the delay. For a given time-delay, the smaller the H
∞ performance index, the better the conditions; for a prescribed H
∞ performance level, the larger the admissible upper bound of the time-delay, the less conservative the conditions.It is well-known that Lyapunov-Krasovskii theorems are basic theories for the study of all types of time-delay system [3–22]. The choice of appropriate Lyapunov-Krasovskii functional (LKF) is crucial for obtaining stability criteria and bounded real lemmas (BRLs) and, as a result, for obtaining solutions to various control problems. In [3–5], the simple form LKFs V
(
t
,
x
t
) containing definition integral terms were used to obtain the delay-independent bounded real lemma for singular systems with state delay. Generally speaking, delay-dependent conditions are less conservative than the delay-independent ones, especially when the size of delay is small. To obtain delay-dependent conditions, many efforts have been made in the literature [6–14]. Improved LKFs V
(
t
,
x
t
) containing double integral terms were given in [6–13]. Along the system trajectory, there will be definition integral term in V
˙
(
t
,
x
t
), which causes an unwieldy question: how to deal with it to get less conservative conditions. Some strategies are usually adopted to solve the problem such as different classes of model transformations and over-bounding cross terms [6–8]; free weighting matrix approach [9–12]; delay fractioning technique [13, 14]; integral inequalities approach [15]. However, there are no obvious ways to obtain less conservative results even if one is willing to commit more computational effort to the problem and to find a more tighter bounding for cross terms. This is the serious limitation for these criteria. To overcome the limitation, one has to find some more general LKFs to handle the stability problem for singular systems.On the other hand, LMI stability conditions via complete quadratic LKF and discretization were introduced by Gu in [17] for time delay system and appeared to be very efficient, leading in some examples to results close to analytical ones. The discretized LKF method has been extended to singular time-delay systems in [16]. An improved bounded real lemma (BRL) is presented by discretization LKF method which greatly lowers the conservatism, but the initial parameter is needed to introduce controller design. To further improve the conclusions, the H
∞ control for singular time-delay systems is proceeded with studies by parameterized LKF approach.The contribution of this paper lies in three aspects. First, a singular-type complete quadratic Lyapunov-Krasovskii functional is introduced in which the quadratic weighting matrices are defined using matrix polynomial functions. This class of LKFV
(
t
,
x
t
) covers those considered in [3–15] as special cases, where the quadratic weighting matrices are defined by constant matrices. Second, a delay-dependent BRL is presented to ensure the system to be regular, impulse free, and stable with prescribed H
∞ performance level, which greatly lower the conservatism. Third, the free weighting matrix approach is employed to decouple the system matrices from the weighting matrices of LKF, which urge that the robust H
∞ control problem is solved and an explicit expression of the desired state-feedback control law is also derived.Notation. Throughout this note, the superscript “T” stands for matrix transposition, R
n denotes the n-dimensional Euclidean space with vector norm ∥
·
∥, R
n
×
m is the set of all n
×
m real matrices, and the notation P
>
0 means that P is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by *. L
2
n
[
0
,
∞
] is the space of square integrable functions f
:
[
0
,
∞
]
→
R
n with the norm ∥
f
∥
L
2
=
[
∫
0
∞
‍
∥
f
(
t
)
∥
2
d
t
]
1
/
2.


## 2. Preliminaries and Problem Formulation

Consider the following linear singular system with state delay and parameter uncertainties described by(1)
Σ
1
:
E
x
˙
t
=
A
0
+
Δ
A
0
x
t
+
A
1
+
Δ
A
1
x
t
-
r
+
B
+
Δ
B
u
t
+
D
ω
t
,
w
w
i
z
t
=
C
0
+
Δ
C
0
x
t
+
C
1
+
Δ
C
1
x
t
-
r
+
H
+
Δ
H
u
t
,
w
w
w
w
w
w
w
i
x
t
=
ϕ
t
,
t
∈
-
r
,
0
,
where x
(
t
)
∈
R
n is the state vector; u
(
t
)
∈
R
m is the control input vector; ω
(
t
)
∈
R
p is the disturbance input which belongs to L
2
[
0
,
∞
), and z
(
t
)
∈
R
l is the controlled output. The matrix E
∈
R
n
×
n may be singular. We will assume that rank
⁡
(
E
)
=
q
≤
n. The matrices A
0, A
1, B, C
0, C
1, D, and H are known real constant matrices with appropriate dimensions. r
>
0 is a constant time delay. ϕ
(
t
) is a compatible vector valued initial function. Δ
A
0, Δ
A
1, Δ
B, Δ
C
0, Δ
C
1, and Δ
H are time-invariant matrices representing norm-bounded parameter uncertainties and are assumed to be of the form
(2)
Δ
A
0
Δ
A
1
Δ
B
Δ
C
0
Δ
C
1
Δ
H
=
M
1
M
2
F
N
1
N
2
N
3
,
where M
1, M
2, N
1, N
2, and N
3 are constant matrices and F is an unknown real matrix satisfying
(3)
F
T
F
≤
I
.
The parameter uncertainties Δ
A
0, Δ
A
1, Δ
B, Δ
C
0, Δ
C
1, and Δ
H are said to be admissible if both (2) and (3) hold.The nominal singular time-delay system ofΣ
1 with u
(
t
)
=
0 can be written as
(4)
Σ
0
:
E
x
˙
t
=
A
0
x
t
+
A
1
x
t
-
r
+
D
ω
t
,
z
t
=
C
0
x
t
+
C
1
x
t
-
r
,
x
t
=
ϕ
t
,
t
∈
-
r
,
0
.
Throughout the paper, we will adopt the following definition.Definition 1 (see [1]).
(
1
) The pair (
E
,
A
0
) is said to be regular if det
⁡
(
s
E
-
A
0
) is not identically zero.
(
2
) The pair (
E
,
A
0
) is said to be impulse free if deg
⁡
(
det
⁡
(
s
E
-
A
0
)
)
=
rank
⁡
E.Lemma 2 (see [3]).
Suppose the pair(
E
,
A
0
) is regular and impulsive free, then the solution to unforced systems Σ
0 exists and is impulse free and unique on [
0
,
∞
).Remark 3.
The regularity of pair(
E
,
A
0
) guarantees the existence and uniqueness of solution for system Σ
0; impulse terms are generally not expected to appear since strong impulse behavior may stop the system from work or even destroy it.In view of this, we introduce the following definition for singular time-delay systemΣ
0.Definition 4 (see [3]).
(
1
) The singular time-delay system Σ
0 is said to be regular and impulse free, if the pair (
E
,
A
0
) is regular and impulse free.
(
2
) The singular time-delay system Σ
0 is said to be stable if, for any ɛ
>
0, there exists a scalar δ
(
ɛ
)
>
0, such that for any compatible initial function ϕ
(
t
) satisfies sup
⁡
-
r
≤
t
≤
0
∥
ϕ
(
t
)
∥
≤
δ
(
ɛ
), and the solutions x
(
t
) of system Σ
0 satisfy ∥
x
(
t
)
∥
≤
ɛ for t
≥
0. Furthermore, x
(
t
)
→
0 as t
→
∞.Definition 5.
The uncertain singular time-delay systemΣ
1 is said to be robustly stable if the unforced system of Σ
1 is regular, impulse free, and stable for all admissible uncertainties.Definition 6.
For a given scalarγ
>
0, the uncertain singular time-delay system Σ
1 with u
(
t
)
=
0 is said to be robustly stable with H
∞ performance γ, if it is robustly stable and under zero initial condition ∥
z
(
t
)
∥
2
<
γ
∥
ω
(
t
)
∥
2 is satisfied for any nonzero ω
(
t
)
∈
L
2
p
[
0
,
∞
) and all admissible uncertainties.The aim of this paper is for prescribed scalarsr
>
0 and γ
>
0 to develop a state feedback controller
(5)
u
t
=
K
0
x
t
+
K
1
x
t
-
r
such that closed-loop system is robustly stable with H
∞ performance γ, where K
0 and K
1 are matrices to be determined.We conclude this section by presenting the following lemma, which is extensively used in uncertain system research.Lemma 7 (see [23]).
For appropriate dimensional matricesΓ, Ξ and symmetric matrix Ω, all the F
(
t
) satisfied F
T
(
t
)
F
(
t
)
≤
I,
(6)
Ω
+
Γ
F
t
Ξ
+
Ξ
T
F
T
t
Γ
T
<
0
if and only if there exists a constant ɛ
>
0 such that
(7)
Ω
+
ɛ
Γ
Γ
T
+
ɛ
-
1
Ξ
T
Ξ
<
0
.


## 3. Main Results

First of all, we present delay-dependent result that assures the nominal unforced singular systemΣ
0 to be regular, impulse free, and stable with H
∞ performance γ, which will play a key role in solving the H
∞ control problem.


### 3.1. A New Version of Delay-Dependent Bounded Real Lemma

We introduce a singular-type complete quadratic Lyapunov-Krasovskii functional(8)
V
t
,
x
t
=
x
T
t
E
T
P
E
x
t
+
2
x
T
t
E
T
∫
-
r
0
‍
Q
ξ
x
t
+
ξ
d
ξ
+
∫
-
r
0
‍
∫
-
r
0
‍
x
T
t
+
ξ
R
ξ
,
η
x
t
+
η
d
ξ
d
η
+
∫
-
r
0
‍
x
T
t
+
ξ
S
ξ
x
t
+
ξ
d
ξ
,
where Q
(
ξ
)
,
S
(
ξ
)
=
S
T
(
ξ
) and R
(
ξ
,
η
)
=
R
T
(
η
,
ξ
) are matrix polynomial functions in the form of
(9)
Q
ξ
=
Q
W
ξ
,
S
ξ
=
S
+
r
+
ξ
W
T
ξ
T
W
ξ
,
R
ξ
,
η
=
W
T
ξ
R
W
η
,
where
(10)
W
T
(
ξ
)
=
I
n
ξ
I
n
⋯
ξ
N
-
1
I
n
,
Q
=
Q
1
Q
2
⋯
Q
N
,
T
=
T
11
T
12
⋯
T
1
N
*
T
22
⋯
T
2
N
⋮
⋮
⋱
⋮
*
*
⋯
T
N
N
,
R
=
R
11
R
12
⋯
R
1
N
*
R
22
⋯
R
2
N
⋮
⋮
⋱
⋮
*
*
⋯
R
N
N
.Noting that the differentiation of the partitioned matrixW
˙
(
ξ
)
=
D
W
(
ξ
), where
(11)
D
=
D
⊗
I
n
,
D
=
0
0
⋯
0
0
1
0
⋯
0
0
⋮
⋱
⋱
⋮
⋮
0
0
⋯
0
0
0
0
⋯
N
-
1
0
.Based on it, the differentiation of the functionsQ
(
ξ
), S
(
ξ
), and R
(
ξ
,
η
) is given by
(12)
Q
˙
ξ
=
Q
D
W
ξ
,
S
˙
ξ
=
W
T
ξ
T
+
r
+
ξ
D
T
T
+
T
D
W
ξ
,
∂
R
ξ
,
η
∂
ξ
=
W
T
ξ
D
T
R
W
η
,
∂
R
ξ
,
η
∂
η
=
W
T
ξ
R
D
W
η
.
A new version of bounded real lemma is obtained via the singular-type complete quadratic Lyapunov-Krasovskii functional (8) with polynomial parameter (9).Theorem 8.
For a given scalarγ
>
0, the system Σ
0 is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p such that
(13)
Φ
=
E
T
P
E
E
T
Q
*
R
≥
0
,
(14)
Π
=
T
+
r
D
T
T
+
T
D
+
T
D
+
D
T
>
0
,
(15)
Ψ
=
Ψ
11
Ψ
12
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
C
0
T
*
-
S
A
1
T
P
2
Ψ
24
A
1
T
P
4
C
1
T
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
0
*
*
*
Ψ
44
P
3
T
D
0
*
*
*
*
Ψ
55
0
*
*
*
*
*
-
I
<
0
hold, where V
∈
R
n
×
(
n
-
q
) is any matrix with full column rank and satisfies E
T
V
=
0 and
(16)
Ψ
11
=
P
1
T
A
0
+
A
0
T
P
1
+
E
T
Q
W
0
+
W
0
T
Q
T
E
+
S
+
r
W
0
T
T
W
0
,
Ψ
12
=
P
1
T
A
1
-
E
T
Q
W
(
-
r
)
,
Ψ
14
=
A
0
T
P
3
-
E
T
Q
D
+
W
0
T
R
,
Ψ
24
=
A
1
T
P
3
-
W
T
-
r
R
,
Ψ
44
=
-
Π
r
-
D
T
R
-
R
D
,
Ψ
55
=
P
4
T
D
+
D
T
P
4
-
γ
2
I
,
D
=
diag
⁡
0,1
,
…
,
N
-
1
⊗
I
n
,
P
1
=
P
E
+
V
U
.Proof.
Sincerank
⁡
(
E
)
=
q
≤
n, there exist nonsingular matrices G
¯ and H
¯ such that E
¯
=
G
¯
E
H
¯
=
I
q
0
0
0. Noting this and E
T
V
=
0, rank
⁡
(
V
)
=
n
-
q, we can show that V can be parameterized as V
=
G
¯
T
0
V
¯, where V
¯
∈
R
(
n
-
q
)
×
(
n
-
q
) is any nonsingular matrix. Accordingly, we define the following transformations:
(17)
P
¯
=
G
¯
-
T
P
G
¯
-
1
=
P
¯
11
P
¯
12
P
¯
12
T
P
¯
22
,
A
¯
0
=
G
¯
A
0
H
¯
=
A
¯
01
A
¯
02
A
¯
03
A
¯
04
,
S
¯
=
H
¯
T
S
H
¯
=
S
¯
11
S
¯
12
S
¯
12
T
S
¯
22
,
T
¯
11
=
H
¯
T
T
11
H
¯
=
T
¯
111
T
¯
112
T
¯
113
T
¯
114
,
Q
¯
1
=
G
¯
-
T
Q
1
H
¯
=
Q
¯
11
Q
¯
12
Q
¯
13
Q
¯
14
,
U
¯
=
U
H
¯
=
U
¯
1
U
¯
2
.
Then, pre- and postmultiplying Ψ
11 by H
¯
T and H
¯ get
(18)
⋆
⋆
⋆
A
¯
04
T
V
¯
U
¯
2
+
U
¯
2
T
V
¯
T
A
¯
04
+
S
¯
22
+
r
T
¯
114
<
0
,
which obviously implies that A
¯
04 is nonsingular. Then we can deduce system Σ
0 is regular and impulse free. Here, the terms denoted ⋆ are irrelevant to the results of the above discussion, so the real expression of these variables is omitted.
Next, we will show the nominal singular systemΣ
0 is stable and has the H
∞ performance γ. Consider the functional (8) with the functions Q
(
ξ
), R
(
ξ
,
η
), and S
(
ξ
) as in (9), and define the vector ψ
(
t
)
=
∫
-
r
0
‍
W
(
ξ
)
x
(
t
+
ξ
)
d
ξ; then
(19)
V
t
,
x
t
=
x
T
t
E
T
P
E
x
t
+
2
x
T
t
E
T
Q
ψ
t
+
∫
-
r
0
‍
x
T
t
+
ξ
S
x
t
+
ξ
d
ξ
+
ψ
T
t
R
ψ
t
+
∫
-
r
0
‍
r
+
ξ
x
T
t
+
ξ
W
T
ξ
T
W
ξ
x
t
+
ξ
d
ξ
.
Denote ξ
T
(
t
)
=
x
T
(
t
)
ψ
T
(
t
); the functional (8) satisfies
(20)
V
t
,
x
t
≥
ξ
T
t
Φ
ξ
t
+
∫
-
r
0
‍
x
T
t
+
ξ
S
x
t
+
ξ
d
ξ
+
∫
-
r
0
‍
r
+
ξ
x
T
t
+
ξ
W
T
ξ
T
W
ξ
x
t
+
ξ
d
ξ
.
If T
>
0, S
>
0, and Φ
≥
0, then V
(
t
,
x
t
) is positive definite. The derivation of V
(
t
,
x
t
) along system Σ
0 gives
(21)
V
˙
t
,
x
t
+
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
=
x
T
t
E
T
Q
W
0
+
W
T
0
Q
T
E
+
S
w
w
w
w
w
w
w
+
r
W
T
0
T
W
0
+
C
0
T
C
0
x
t
w
w
w
i
i
+
2
x
T
t
-
E
T
Q
W
-
r
+
C
0
T
C
1
x
t
-
r
w
w
w
i
i
+
2
x
T
t
E
T
P
+
U
T
V
T
E
x
˙
t
w
w
w
i
i
+
2
x
T
t
W
T
0
R
-
E
T
Q
D
ψ
t
w
w
w
i
i
+
x
T
t
-
r
-
S
+
C
1
T
C
1
x
t
-
r
w
w
w
i
i
-
2
x
T
t
-
r
W
T
-
r
R
ψ
t
+
2
x
˙
T
t
E
T
Q
ψ
t
w
w
w
i
i
-
ψ
T
t
D
T
R
+
R
D
ψ
t
w
w
w
i
i
-
∫
-
r
0
‍
x
T
t
+
ξ
W
T
ξ
Π
W
ξ
x
t
+
ξ
d
ξ
w
w
w
i
i
-
γ
2
ω
T
t
ω
t
.
As Π
>
0, Jensen’s inequality ensures that
(22)
-
∫
-
r
0
‍
x
T
t
+
ξ
W
T
ξ
Π
W
ξ
x
t
+
ξ
d
ξ
≤
-
1
r
ψ
T
t
Π
ψ
t
.
On the other hand, for any appropriate dimensional matricesP
2, P
3, and P
4, the following equation is true:
(23)
2
x
T
t
P
1
T
+
x
˙
T
t
E
T
P
2
T
+
ψ
T
t
P
3
T
+
ω
T
t
P
4
T
×
A
0
x
t
+
A
1
x
t
-
r
-
E
x
˙
t
+
D
ω
t
=
0
.
Hence(24)
V
˙
t
,
x
t
+
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
≤
ζ
T
t
Ψ
ζ
t
,
where ζ
T
(
t
)
=
x
T
(
t
)
x
T
(
t
-
r
)
x
˙
T
(
t
)
E
T
ψ
T
(
t
)
ω
T
(
t
). Under the zero initial condition, it can be shown that, for any nonzero ω
(
t
)
∈
L
2
[
0
,
∞
), the following index is(25)
J
z
ω
=
∫
0
∞
‍
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
d
t
≤
∫
0
∞
‍
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
+
V
˙
t
,
x
t
d
t
≤
∫
0
∞
‍
ζ
T
t
Ω
ζ
t
d
t
,
where
(26)
Ω
=
Ψ
11
+
C
0
T
C
0
Ψ
12
+
C
0
T
C
1
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
*
-
S
+
C
1
T
C
1
A
1
T
P
2
Ψ
24
A
1
T
P
4
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
*
*
*
Ψ
44
P
3
T
D
*
*
*
*
Ψ
55
.
Thus,Ψ
<
0 implies that J
z
ω
<
0; that is, ∥
z
(
t
)
∥
2
<
γ
∥
ω
(
t
)
∥
2 for any nonzero ω
(
t
)
∈
L
2
[
0
,
∞
). Moreover, similar to [16], the feasibility of LMIs (13), (14), and
(27)
Ψ
11
Ψ
12
A
0
T
P
2
Ψ
14
*
-
S
A
1
T
P
2
Ψ
24
*
*
-
P
2
-
P
2
T
Q
-
P
3
*
*
*
Ψ
44
<
0
implies V
˙
(
t
,
x
t
)
<
0; then the system Σ
0 with ω
(
t
)
=
0 is stable. This completes the proof.Remark 9.
In Theorem8, matrices P
i
(
i
=
1,2
,
3,4
) are introduced to decouple the system matrices A
0, A
1 from the LKF weighting matrices P and Q
i
,
(
i
=
1,2
,
…
,
N
), which provides convenience for controller design.Remark 10.
WhenE is nonsingular, the singular system Σ
0 reduces to a state-space system. In this case, if we choose V
=
0 in the proof of Theorem 8, we can derive a new version of bounded real lemma for state-space system.Based on Theorem8, we obtain the following result on H
∞ performance analysis for the uncertain singular system Σ
1 with u
(
t
)
=
0.Theorem 11.
For a given scalarγ
>
0, the system Σ
1 is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p and scalar ɛ
>
0 such that (13), (14), and
(28)
Ψ
~
11
Ψ
~
12
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
C
0
T
P
1
T
M
1
*
Ψ
~
22
A
1
T
P
2
Ψ
24
A
1
T
P
4
C
1
T
0
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
0
P
2
T
M
1
*
*
*
Ψ
44
P
3
T
D
0
P
3
T
M
1
*
*
*
*
Ψ
55
0
P
4
T
M
1
*
*
*
*
*
-
I
M
2
*
*
*
*
*
*
-
ɛ
I
<
0
hold, where Ψ
~
11
=
Ψ
11
+
ɛ
N
1
T
N
1, Ψ
~
12
=
Ψ
12
+
ɛ
N
1
T
N
2, and Ψ
~
22
=
-
S
+
ɛ
N
2
T
N
2.Proof.
Suppose that there exist symmetric positive definite matricesP
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p and scalar ɛ such that (13), (14), and (28) hold. Then, by Schur complement, it follows from (28) that
(29)
Ψ
+
ɛ
Γ
T
Γ
+
1
ɛ
Ξ
Ξ
T
<
0
,
where
(30)
Ξ
T
=
M
1
T
P
1
0
M
1
T
P
2
M
1
T
P
3
M
1
T
P
4
M
2
T
,
Γ
=
N
1
N
2
0
0
0
0
.
By Lemma 7, Ψ
+
Ξ
F
Γ
+
(
Ξ
F
Γ
)
T
<
0, which is in the form of Ψ by replacing A
0, A
1, C
0, and C
1 with A
0
+
M
1
F
N
1, A
1
+
M
1
F
N
2, C
0
+
M
2
F
N
1, and C
1
+
M
2
F
N
2, respectively. Thus, by Theorem 8, we have that the uncertain singular system Σ
1 with u
(
t
)
=
0 is robustly stable with H
∞ performance γ.


### 3.2.  RobustH
∞ Controller Design

In this subsection, we will apply the bounded real lemma obtained above to design the state feedback controller (5) such that the resultant closed-loop system is regular, impulse free, and stable with H
∞ performance γ.The nominal closed-loop singular system ofΣ
0 can be written as
(31)
Σ
0
c
:
E
x
˙
t
=
A
0
+
B
K
0
x
t
w
w
w
w
w
w
w
i
+
A
1
+
B
K
1
x
t
-
r
+
D
ω
(
t
)
,
z
t
=
C
0
+
H
K
0
x
t
+
C
1
+
H
K
1
x
t
-
r
,
x
t
=
ϕ
t
,
t
∈
-
r
,
0
.Theorem 12.
For prescribed scalarγ
>
0, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices S
¯
∈
R
n
×
n, T
¯
∈
R
n
N
×
n
N, and R
¯
∈
R
n
N
×
n
N and matrices P
¯
∈
R
n
×
n, Q
¯
∈
R
n
×
n
N, and L
i
∈
R
m
×
n, i
=
0,1 and scalars δ
j
>
0, j
=
1,2 such that
(32)
Φ
¯
=
E
P
¯
E
T
E
Q
¯
*
R
¯
≥
0
,
(33)
Π
¯
=
T
¯
+
r
(
D
T
T
¯
+
T
¯
D
)
+
T
¯
D
+
D
T
¯
>
0
,
(34)
Ψ
¯
=
Ψ
¯
11
Ψ
¯
12
Ψ
¯
13
Ψ
¯
14
D
Ψ
¯
16
*
-
S
¯
Ψ
¯
23
Ψ
¯
24
0
Ψ
¯
26
*
*
Ψ
¯
33
Ψ
¯
34
δ
1
D
0
*
*
*
Ψ
¯
44
Ψ
¯
45
0
*
*
*
*
-
γ
2
I
0
*
*
*
*
*
-
I
<
0
hold. Then a suitable state-feedback control law is given by
(35)
u
t
=
L
0
P
¯
-
1
x
t
+
L
1
P
¯
-
1
x
t
-
r
,
where
(36)
Ψ
¯
11
=
A
0
P
¯
+
B
L
0
+
A
0
P
¯
+
B
L
0
T
+
E
Q
¯
W
0
+
W
0
T
Q
¯
T
E
T
+
S
¯
+
r
T
¯
11
,
Ψ
¯
12
=
A
1
P
¯
+
B
L
1
-
E
Q
¯
W
-
r
,
Ψ
¯
13
=
δ
1
P
¯
T
A
0
T
+
L
0
T
B
T
,
Ψ
¯
14
=
δ
2
P
¯
T
A
0
T
+
L
0
T
B
T
⋯
δ
2
P
¯
T
A
0
T
+
L
0
T
B
T
-
E
Q
¯
D
+
W
0
T
R
¯
,
Ψ
¯
16
=
P
¯
T
C
0
T
+
L
0
T
H
T
,
Ψ
¯
26
=
P
¯
T
C
1
T
+
L
1
T
H
T
,
Ψ
¯
23
=
δ
1
P
¯
T
A
1
T
+
L
1
T
B
T
,
Ψ
¯
24
=
δ
2
P
¯
T
A
1
T
+
L
1
T
B
T
⋯
δ
2
P
¯
T
A
1
T
+
L
1
T
B
T
-
W
T
-
r
R
¯
,
Ψ
¯
33
=
-
δ
1
P
¯
+
P
¯
T
,
Ψ
¯
34
=
Q
¯
-
δ
2
P
¯
T
⋯
δ
2
P
¯
T
,
Ψ
¯
44
=
-
Π
¯
r
-
D
T
R
¯
-
R
¯
D
,
Ψ
¯
45
T
=
δ
2
D
T
⋯
D
T
.Proof.
According to Theorem8, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ, if there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, and T
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, R
∈
R
n
N
×
n
N, Q
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p such that (13), (14), and Ψ
^
<
0 hold, where Ψ
^ is in the form of Ψ with A
0
,
A
1
,
C
0
,
C
1 replaced by A
0
+
B
K
0, A
1
+
B
K
1, C
0
+
H
K
0, and C
1
+
H
K
1. Let P
2
=
δ
1
P
1, P
3
=
δ
2
P
1
⋯
P
1, P
4
=
[
0
]
n
×
p, and P
1
-
1
=
P
¯. Pre- and postmultiply Φ by diag
⁡
{
P
¯
T
,
P
T
} and its transpose; pre- and postmultiply Π by P
T and its transpose; pre- and postmultiply Ψ
^ by diag
⁡
{
P
¯
T
,
P
¯
T
,
P
¯
T
,
P
T
,
I
,
I
} and its transpose. We introduce the following: (37)
Q
¯
=
P
¯
Q
P
,
R
¯
=
P
T
R
P
,
S
¯
=
P
¯
T
S
P
¯
,
T
¯
=
P
T
T
P
,
P
=
diag
⁡
P
¯
,
P
¯
,
…
,
P
¯
∈
R
n
N
×
n
N
,
L
0
=
K
0
P
¯
,
L
1
=
K
1
P
¯
.
After some manipulation, we can obtain (32), (33), and (34), and the desired controller gains are given by K
0
=
L
0
P
¯
-
1, K
1
=
L
1
P
¯
-
1.Remark 13.
In this case, we can assume that the matrixP
1 is nonsingular. If this is not the case, we can choose some θ
∈
(
0,1
) such that P
~
1
=
P
1
+
θ
P
^ is nonsingular and satisfies (32) and (34), in which P
^ is any nonsingular matrix satisfying E
T
P
^
=
P
^
T
E.Remark 14.
In the case wherex
(
t
-
r
) is available for feedback, the use of K
0 and K
1 can lead to a significative reduction on the values of γ. On the other hand, when x
(
t
-
r
) is not available for feedback, a memoryless control law is required; it is enough to set L
1
=
0 in LMI (34).Now we are in a position to present the result on the problem of delay-dependent robustH
∞ control for the uncertain singular system Σ
1. The closed-loop singular system of Σ
1 can be written as
(38)
Σ
1
c
:
E
x
˙
t
=
A
0
+
Δ
A
0
+
B
+
Δ
B
K
0
x
t
w
w
+
A
1
+
Δ
A
1
+
B
+
Δ
B
K
1
×
x
t
-
r
+
D
ω
t
,
w
w
w
z
(
t
)
=
C
0
+
Δ
C
0
+
H
+
Δ
H
K
0
x
t
i
i
w
+
C
1
+
Δ
C
1
+
H
+
Δ
H
K
1
×
x
t
-
r
,
w
w
w
w
w
w
w
o
i
x
t
=
ϕ
t
,
t
∈
-
r
,
0
.Theorem 15.
For prescribed scalarγ
>
0, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices S
¯
∈
R
n
×
n, T
¯
∈
R
n
N
×
n
N, R
¯
∈
R
n
N
×
n
N and matrices P
¯
∈
R
n
×
n, Q
¯
∈
R
n
×
n
N, L
i
∈
R
m
×
n, i
=
0,1 and scalars δ
j
>
0, (
j
=
1,2
), ɛ
>
0 such that (32), (33) and
(39)
Ψ
^
=
Ψ
^
11
Ψ
¯
12
Ψ
^
13
Ψ
^
14
D
Ψ
^
16
Ψ
^
17
*
-
S
¯
Ψ
¯
23
Ψ
¯
24
0
Ψ
¯
26
Ψ
^
27
*
*
Ψ
^
33
Ψ
^
34
δ
1
D
ɛ
δ
1
M
1
M
2
T
0
*
*
*
Ψ
^
44
Ψ
¯
45
Ψ
^
46
0
*
*
*
*
-
γ
2
I
0
0
*
*
*
*
*
-
I
+
ɛ
M
2
M
2
T
0
*
*
*
*
*
*
-
ɛ
I
<
0
hold. Then a suitable state-feedback control law is given by
(40)
u
t
=
L
0
P
¯
-
1
x
t
+
L
1
P
¯
-
1
x
t
-
r
,
where
(41)
Ψ
^
11
=
Ψ
¯
11
+
ɛ
M
1
M
1
T
,
Ψ
^
13
=
Ψ
¯
13
+
ɛ
δ
1
M
1
M
1
T
,
Ψ
^
14
=
Ψ
¯
14
+
ɛ
δ
2
M
1
M
1
T
⋯
M
1
T
,
Ψ
^
16
=
Ψ
¯
16
+
ɛ
M
1
M
2
T
,
Ψ
^
33
=
Ψ
¯
33
+
ɛ
δ
1
2
M
1
M
1
T
,
Ψ
^
34
=
Ψ
¯
34
+
ɛ
δ
1
δ
2
M
1
M
1
T
⋯
M
1
T
,
Ψ
^
44
=
Ψ
¯
44
+
ɛ
δ
2
2
M
1
M
1
T
⋯
M
1
M
1
T
⋮
⋱
⋮
M
1
M
1
T
⋯
M
1
M
1
T
,
Ψ
^
46
=
ɛ
δ
2
M
1
M
2
T
⋮
M
2
T
,
Ψ
^
17
=
P
¯
T
N
1
T
+
L
0
T
N
3
T
,
Ψ
^
27
=
P
¯
T
N
2
T
+
L
1
T
N
3
T
.The proof can be carried out by resorting to Theorem12 and following a similar line as in the proof of Theorem 11 and is thus omitted.


## 3.1. A New Version of Delay-Dependent Bounded Real Lemma

We introduce a singular-type complete quadratic Lyapunov-Krasovskii functional(8)
V
t
,
x
t
=
x
T
t
E
T
P
E
x
t
+
2
x
T
t
E
T
∫
-
r
0
‍
Q
ξ
x
t
+
ξ
d
ξ
+
∫
-
r
0
‍
∫
-
r
0
‍
x
T
t
+
ξ
R
ξ
,
η
x
t
+
η
d
ξ
d
η
+
∫
-
r
0
‍
x
T
t
+
ξ
S
ξ
x
t
+
ξ
d
ξ
,
where Q
(
ξ
)
,
S
(
ξ
)
=
S
T
(
ξ
) and R
(
ξ
,
η
)
=
R
T
(
η
,
ξ
) are matrix polynomial functions in the form of
(9)
Q
ξ
=
Q
W
ξ
,
S
ξ
=
S
+
r
+
ξ
W
T
ξ
T
W
ξ
,
R
ξ
,
η
=
W
T
ξ
R
W
η
,
where
(10)
W
T
(
ξ
)
=
I
n
ξ
I
n
⋯
ξ
N
-
1
I
n
,
Q
=
Q
1
Q
2
⋯
Q
N
,
T
=
T
11
T
12
⋯
T
1
N
*
T
22
⋯
T
2
N
⋮
⋮
⋱
⋮
*
*
⋯
T
N
N
,
R
=
R
11
R
12
⋯
R
1
N
*
R
22
⋯
R
2
N
⋮
⋮
⋱
⋮
*
*
⋯
R
N
N
.Noting that the differentiation of the partitioned matrixW
˙
(
ξ
)
=
D
W
(
ξ
), where
(11)
D
=
D
⊗
I
n
,
D
=
0
0
⋯
0
0
1
0
⋯
0
0
⋮
⋱
⋱
⋮
⋮
0
0
⋯
0
0
0
0
⋯
N
-
1
0
.Based on it, the differentiation of the functionsQ
(
ξ
), S
(
ξ
), and R
(
ξ
,
η
) is given by
(12)
Q
˙
ξ
=
Q
D
W
ξ
,
S
˙
ξ
=
W
T
ξ
T
+
r
+
ξ
D
T
T
+
T
D
W
ξ
,
∂
R
ξ
,
η
∂
ξ
=
W
T
ξ
D
T
R
W
η
,
∂
R
ξ
,
η
∂
η
=
W
T
ξ
R
D
W
η
.
A new version of bounded real lemma is obtained via the singular-type complete quadratic Lyapunov-Krasovskii functional (8) with polynomial parameter (9).Theorem 8.
For a given scalarγ
>
0, the system Σ
0 is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p such that
(13)
Φ
=
E
T
P
E
E
T
Q
*
R
≥
0
,
(14)
Π
=
T
+
r
D
T
T
+
T
D
+
T
D
+
D
T
>
0
,
(15)
Ψ
=
Ψ
11
Ψ
12
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
C
0
T
*
-
S
A
1
T
P
2
Ψ
24
A
1
T
P
4
C
1
T
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
0
*
*
*
Ψ
44
P
3
T
D
0
*
*
*
*
Ψ
55
0
*
*
*
*
*
-
I
<
0
hold, where V
∈
R
n
×
(
n
-
q
) is any matrix with full column rank and satisfies E
T
V
=
0 and
(16)
Ψ
11
=
P
1
T
A
0
+
A
0
T
P
1
+
E
T
Q
W
0
+
W
0
T
Q
T
E
+
S
+
r
W
0
T
T
W
0
,
Ψ
12
=
P
1
T
A
1
-
E
T
Q
W
(
-
r
)
,
Ψ
14
=
A
0
T
P
3
-
E
T
Q
D
+
W
0
T
R
,
Ψ
24
=
A
1
T
P
3
-
W
T
-
r
R
,
Ψ
44
=
-
Π
r
-
D
T
R
-
R
D
,
Ψ
55
=
P
4
T
D
+
D
T
P
4
-
γ
2
I
,
D
=
diag
⁡
0,1
,
…
,
N
-
1
⊗
I
n
,
P
1
=
P
E
+
V
U
.Proof.
Sincerank
⁡
(
E
)
=
q
≤
n, there exist nonsingular matrices G
¯ and H
¯ such that E
¯
=
G
¯
E
H
¯
=
I
q
0
0
0. Noting this and E
T
V
=
0, rank
⁡
(
V
)
=
n
-
q, we can show that V can be parameterized as V
=
G
¯
T
0
V
¯, where V
¯
∈
R
(
n
-
q
)
×
(
n
-
q
) is any nonsingular matrix. Accordingly, we define the following transformations:
(17)
P
¯
=
G
¯
-
T
P
G
¯
-
1
=
P
¯
11
P
¯
12
P
¯
12
T
P
¯
22
,
A
¯
0
=
G
¯
A
0
H
¯
=
A
¯
01
A
¯
02
A
¯
03
A
¯
04
,
S
¯
=
H
¯
T
S
H
¯
=
S
¯
11
S
¯
12
S
¯
12
T
S
¯
22
,
T
¯
11
=
H
¯
T
T
11
H
¯
=
T
¯
111
T
¯
112
T
¯
113
T
¯
114
,
Q
¯
1
=
G
¯
-
T
Q
1
H
¯
=
Q
¯
11
Q
¯
12
Q
¯
13
Q
¯
14
,
U
¯
=
U
H
¯
=
U
¯
1
U
¯
2
.
Then, pre- and postmultiplying Ψ
11 by H
¯
T and H
¯ get
(18)
⋆
⋆
⋆
A
¯
04
T
V
¯
U
¯
2
+
U
¯
2
T
V
¯
T
A
¯
04
+
S
¯
22
+
r
T
¯
114
<
0
,
which obviously implies that A
¯
04 is nonsingular. Then we can deduce system Σ
0 is regular and impulse free. Here, the terms denoted ⋆ are irrelevant to the results of the above discussion, so the real expression of these variables is omitted.
Next, we will show the nominal singular systemΣ
0 is stable and has the H
∞ performance γ. Consider the functional (8) with the functions Q
(
ξ
), R
(
ξ
,
η
), and S
(
ξ
) as in (9), and define the vector ψ
(
t
)
=
∫
-
r
0
‍
W
(
ξ
)
x
(
t
+
ξ
)
d
ξ; then
(19)
V
t
,
x
t
=
x
T
t
E
T
P
E
x
t
+
2
x
T
t
E
T
Q
ψ
t
+
∫
-
r
0
‍
x
T
t
+
ξ
S
x
t
+
ξ
d
ξ
+
ψ
T
t
R
ψ
t
+
∫
-
r
0
‍
r
+
ξ
x
T
t
+
ξ
W
T
ξ
T
W
ξ
x
t
+
ξ
d
ξ
.
Denote ξ
T
(
t
)
=
x
T
(
t
)
ψ
T
(
t
); the functional (8) satisfies
(20)
V
t
,
x
t
≥
ξ
T
t
Φ
ξ
t
+
∫
-
r
0
‍
x
T
t
+
ξ
S
x
t
+
ξ
d
ξ
+
∫
-
r
0
‍
r
+
ξ
x
T
t
+
ξ
W
T
ξ
T
W
ξ
x
t
+
ξ
d
ξ
.
If T
>
0, S
>
0, and Φ
≥
0, then V
(
t
,
x
t
) is positive definite. The derivation of V
(
t
,
x
t
) along system Σ
0 gives
(21)
V
˙
t
,
x
t
+
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
=
x
T
t
E
T
Q
W
0
+
W
T
0
Q
T
E
+
S
w
w
w
w
w
w
w
+
r
W
T
0
T
W
0
+
C
0
T
C
0
x
t
w
w
w
i
i
+
2
x
T
t
-
E
T
Q
W
-
r
+
C
0
T
C
1
x
t
-
r
w
w
w
i
i
+
2
x
T
t
E
T
P
+
U
T
V
T
E
x
˙
t
w
w
w
i
i
+
2
x
T
t
W
T
0
R
-
E
T
Q
D
ψ
t
w
w
w
i
i
+
x
T
t
-
r
-
S
+
C
1
T
C
1
x
t
-
r
w
w
w
i
i
-
2
x
T
t
-
r
W
T
-
r
R
ψ
t
+
2
x
˙
T
t
E
T
Q
ψ
t
w
w
w
i
i
-
ψ
T
t
D
T
R
+
R
D
ψ
t
w
w
w
i
i
-
∫
-
r
0
‍
x
T
t
+
ξ
W
T
ξ
Π
W
ξ
x
t
+
ξ
d
ξ
w
w
w
i
i
-
γ
2
ω
T
t
ω
t
.
As Π
>
0, Jensen’s inequality ensures that
(22)
-
∫
-
r
0
‍
x
T
t
+
ξ
W
T
ξ
Π
W
ξ
x
t
+
ξ
d
ξ
≤
-
1
r
ψ
T
t
Π
ψ
t
.
On the other hand, for any appropriate dimensional matricesP
2, P
3, and P
4, the following equation is true:
(23)
2
x
T
t
P
1
T
+
x
˙
T
t
E
T
P
2
T
+
ψ
T
t
P
3
T
+
ω
T
t
P
4
T
×
A
0
x
t
+
A
1
x
t
-
r
-
E
x
˙
t
+
D
ω
t
=
0
.
Hence(24)
V
˙
t
,
x
t
+
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
≤
ζ
T
t
Ψ
ζ
t
,
where ζ
T
(
t
)
=
x
T
(
t
)
x
T
(
t
-
r
)
x
˙
T
(
t
)
E
T
ψ
T
(
t
)
ω
T
(
t
). Under the zero initial condition, it can be shown that, for any nonzero ω
(
t
)
∈
L
2
[
0
,
∞
), the following index is(25)
J
z
ω
=
∫
0
∞
‍
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
d
t
≤
∫
0
∞
‍
z
T
t
z
t
-
γ
2
ω
T
t
ω
t
+
V
˙
t
,
x
t
d
t
≤
∫
0
∞
‍
ζ
T
t
Ω
ζ
t
d
t
,
where
(26)
Ω
=
Ψ
11
+
C
0
T
C
0
Ψ
12
+
C
0
T
C
1
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
*
-
S
+
C
1
T
C
1
A
1
T
P
2
Ψ
24
A
1
T
P
4
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
*
*
*
Ψ
44
P
3
T
D
*
*
*
*
Ψ
55
.
Thus,Ψ
<
0 implies that J
z
ω
<
0; that is, ∥
z
(
t
)
∥
2
<
γ
∥
ω
(
t
)
∥
2 for any nonzero ω
(
t
)
∈
L
2
[
0
,
∞
). Moreover, similar to [16], the feasibility of LMIs (13), (14), and
(27)
Ψ
11
Ψ
12
A
0
T
P
2
Ψ
14
*
-
S
A
1
T
P
2
Ψ
24
*
*
-
P
2
-
P
2
T
Q
-
P
3
*
*
*
Ψ
44
<
0
implies V
˙
(
t
,
x
t
)
<
0; then the system Σ
0 with ω
(
t
)
=
0 is stable. This completes the proof.Remark 9.
In Theorem8, matrices P
i
(
i
=
1,2
,
3,4
) are introduced to decouple the system matrices A
0, A
1 from the LKF weighting matrices P and Q
i
,
(
i
=
1,2
,
…
,
N
), which provides convenience for controller design.Remark 10.
WhenE is nonsingular, the singular system Σ
0 reduces to a state-space system. In this case, if we choose V
=
0 in the proof of Theorem 8, we can derive a new version of bounded real lemma for state-space system.Based on Theorem8, we obtain the following result on H
∞ performance analysis for the uncertain singular system Σ
1 with u
(
t
)
=
0.Theorem 11.
For a given scalarγ
>
0, the system Σ
1 is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p and scalar ɛ
>
0 such that (13), (14), and
(28)
Ψ
~
11
Ψ
~
12
A
0
T
P
2
Ψ
14
A
0
T
P
4
+
P
1
T
D
C
0
T
P
1
T
M
1
*
Ψ
~
22
A
1
T
P
2
Ψ
24
A
1
T
P
4
C
1
T
0
*
*
-
P
2
-
P
2
T
Q
-
P
3
P
2
T
D
-
P
4
0
P
2
T
M
1
*
*
*
Ψ
44
P
3
T
D
0
P
3
T
M
1
*
*
*
*
Ψ
55
0
P
4
T
M
1
*
*
*
*
*
-
I
M
2
*
*
*
*
*
*
-
ɛ
I
<
0
hold, where Ψ
~
11
=
Ψ
11
+
ɛ
N
1
T
N
1, Ψ
~
12
=
Ψ
12
+
ɛ
N
1
T
N
2, and Ψ
~
22
=
-
S
+
ɛ
N
2
T
N
2.Proof.
Suppose that there exist symmetric positive definite matricesP
∈
R
n
×
n, S
∈
R
n
×
n, T
∈
R
n
N
×
n
N, and R
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, U
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p and scalar ɛ such that (13), (14), and (28) hold. Then, by Schur complement, it follows from (28) that
(29)
Ψ
+
ɛ
Γ
T
Γ
+
1
ɛ
Ξ
Ξ
T
<
0
,
where
(30)
Ξ
T
=
M
1
T
P
1
0
M
1
T
P
2
M
1
T
P
3
M
1
T
P
4
M
2
T
,
Γ
=
N
1
N
2
0
0
0
0
.
By Lemma 7, Ψ
+
Ξ
F
Γ
+
(
Ξ
F
Γ
)
T
<
0, which is in the form of Ψ by replacing A
0, A
1, C
0, and C
1 with A
0
+
M
1
F
N
1, A
1
+
M
1
F
N
2, C
0
+
M
2
F
N
1, and C
1
+
M
2
F
N
2, respectively. Thus, by Theorem 8, we have that the uncertain singular system Σ
1 with u
(
t
)
=
0 is robustly stable with H
∞ performance γ.


## 3.2.  RobustH
∞ Controller Design

In this subsection, we will apply the bounded real lemma obtained above to design the state feedback controller (5) such that the resultant closed-loop system is regular, impulse free, and stable with H
∞ performance γ.The nominal closed-loop singular system ofΣ
0 can be written as
(31)
Σ
0
c
:
E
x
˙
t
=
A
0
+
B
K
0
x
t
w
w
w
w
w
w
w
i
+
A
1
+
B
K
1
x
t
-
r
+
D
ω
(
t
)
,
z
t
=
C
0
+
H
K
0
x
t
+
C
1
+
H
K
1
x
t
-
r
,
x
t
=
ϕ
t
,
t
∈
-
r
,
0
.Theorem 12.
For prescribed scalarγ
>
0, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices S
¯
∈
R
n
×
n, T
¯
∈
R
n
N
×
n
N, and R
¯
∈
R
n
N
×
n
N and matrices P
¯
∈
R
n
×
n, Q
¯
∈
R
n
×
n
N, and L
i
∈
R
m
×
n, i
=
0,1 and scalars δ
j
>
0, j
=
1,2 such that
(32)
Φ
¯
=
E
P
¯
E
T
E
Q
¯
*
R
¯
≥
0
,
(33)
Π
¯
=
T
¯
+
r
(
D
T
T
¯
+
T
¯
D
)
+
T
¯
D
+
D
T
¯
>
0
,
(34)
Ψ
¯
=
Ψ
¯
11
Ψ
¯
12
Ψ
¯
13
Ψ
¯
14
D
Ψ
¯
16
*
-
S
¯
Ψ
¯
23
Ψ
¯
24
0
Ψ
¯
26
*
*
Ψ
¯
33
Ψ
¯
34
δ
1
D
0
*
*
*
Ψ
¯
44
Ψ
¯
45
0
*
*
*
*
-
γ
2
I
0
*
*
*
*
*
-
I
<
0
hold. Then a suitable state-feedback control law is given by
(35)
u
t
=
L
0
P
¯
-
1
x
t
+
L
1
P
¯
-
1
x
t
-
r
,
where
(36)
Ψ
¯
11
=
A
0
P
¯
+
B
L
0
+
A
0
P
¯
+
B
L
0
T
+
E
Q
¯
W
0
+
W
0
T
Q
¯
T
E
T
+
S
¯
+
r
T
¯
11
,
Ψ
¯
12
=
A
1
P
¯
+
B
L
1
-
E
Q
¯
W
-
r
,
Ψ
¯
13
=
δ
1
P
¯
T
A
0
T
+
L
0
T
B
T
,
Ψ
¯
14
=
δ
2
P
¯
T
A
0
T
+
L
0
T
B
T
⋯
δ
2
P
¯
T
A
0
T
+
L
0
T
B
T
-
E
Q
¯
D
+
W
0
T
R
¯
,
Ψ
¯
16
=
P
¯
T
C
0
T
+
L
0
T
H
T
,
Ψ
¯
26
=
P
¯
T
C
1
T
+
L
1
T
H
T
,
Ψ
¯
23
=
δ
1
P
¯
T
A
1
T
+
L
1
T
B
T
,
Ψ
¯
24
=
δ
2
P
¯
T
A
1
T
+
L
1
T
B
T
⋯
δ
2
P
¯
T
A
1
T
+
L
1
T
B
T
-
W
T
-
r
R
¯
,
Ψ
¯
33
=
-
δ
1
P
¯
+
P
¯
T
,
Ψ
¯
34
=
Q
¯
-
δ
2
P
¯
T
⋯
δ
2
P
¯
T
,
Ψ
¯
44
=
-
Π
¯
r
-
D
T
R
¯
-
R
¯
D
,
Ψ
¯
45
T
=
δ
2
D
T
⋯
D
T
.Proof.
According to Theorem8, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ, if there exist symmetric positive definite matrices P
∈
R
n
×
n, S
∈
R
n
×
n, and T
∈
R
n
N
×
n
N and matrices Q
∈
R
n
×
n
N, R
∈
R
n
N
×
n
N, Q
∈
R
(
n
-
q
)
×
n, P
2
∈
R
n
×
n, P
3
∈
R
n
×
n
N, and P
4
∈
R
n
×
p such that (13), (14), and Ψ
^
<
0 hold, where Ψ
^ is in the form of Ψ with A
0
,
A
1
,
C
0
,
C
1 replaced by A
0
+
B
K
0, A
1
+
B
K
1, C
0
+
H
K
0, and C
1
+
H
K
1. Let P
2
=
δ
1
P
1, P
3
=
δ
2
P
1
⋯
P
1, P
4
=
[
0
]
n
×
p, and P
1
-
1
=
P
¯. Pre- and postmultiply Φ by diag
⁡
{
P
¯
T
,
P
T
} and its transpose; pre- and postmultiply Π by P
T and its transpose; pre- and postmultiply Ψ
^ by diag
⁡
{
P
¯
T
,
P
¯
T
,
P
¯
T
,
P
T
,
I
,
I
} and its transpose. We introduce the following: (37)
Q
¯
=
P
¯
Q
P
,
R
¯
=
P
T
R
P
,
S
¯
=
P
¯
T
S
P
¯
,
T
¯
=
P
T
T
P
,
P
=
diag
⁡
P
¯
,
P
¯
,
…
,
P
¯
∈
R
n
N
×
n
N
,
L
0
=
K
0
P
¯
,
L
1
=
K
1
P
¯
.
After some manipulation, we can obtain (32), (33), and (34), and the desired controller gains are given by K
0
=
L
0
P
¯
-
1, K
1
=
L
1
P
¯
-
1.Remark 13.
In this case, we can assume that the matrixP
1 is nonsingular. If this is not the case, we can choose some θ
∈
(
0,1
) such that P
~
1
=
P
1
+
θ
P
^ is nonsingular and satisfies (32) and (34), in which P
^ is any nonsingular matrix satisfying E
T
P
^
=
P
^
T
E.Remark 14.
In the case wherex
(
t
-
r
) is available for feedback, the use of K
0 and K
1 can lead to a significative reduction on the values of γ. On the other hand, when x
(
t
-
r
) is not available for feedback, a memoryless control law is required; it is enough to set L
1
=
0 in LMI (34).Now we are in a position to present the result on the problem of delay-dependent robustH
∞ control for the uncertain singular system Σ
1. The closed-loop singular system of Σ
1 can be written as
(38)
Σ
1
c
:
E
x
˙
t
=
A
0
+
Δ
A
0
+
B
+
Δ
B
K
0
x
t
w
w
+
A
1
+
Δ
A
1
+
B
+
Δ
B
K
1
×
x
t
-
r
+
D
ω
t
,
w
w
w
z
(
t
)
=
C
0
+
Δ
C
0
+
H
+
Δ
H
K
0
x
t
i
i
w
+
C
1
+
Δ
C
1
+
H
+
Δ
H
K
1
×
x
t
-
r
,
w
w
w
w
w
w
w
o
i
x
t
=
ϕ
t
,
t
∈
-
r
,
0
.Theorem 15.
For prescribed scalarγ
>
0, the closed-loop singular system Σ
0
c is regular, impulse free, and stable with H
∞ performance γ; suppose that there exist symmetric positive definite matrices S
¯
∈
R
n
×
n, T
¯
∈
R
n
N
×
n
N, R
¯
∈
R
n
N
×
n
N and matrices P
¯
∈
R
n
×
n, Q
¯
∈
R
n
×
n
N, L
i
∈
R
m
×
n, i
=
0,1 and scalars δ
j
>
0, (
j
=
1,2
), ɛ
>
0 such that (32), (33) and
(39)
Ψ
^
=
Ψ
^
11
Ψ
¯
12
Ψ
^
13
Ψ
^
14
D
Ψ
^
16
Ψ
^
17
*
-
S
¯
Ψ
¯
23
Ψ
¯
24
0
Ψ
¯
26
Ψ
^
27
*
*
Ψ
^
33
Ψ
^
34
δ
1
D
ɛ
δ
1
M
1
M
2
T
0
*
*
*
Ψ
^
44
Ψ
¯
45
Ψ
^
46
0
*
*
*
*
-
γ
2
I
0
0
*
*
*
*
*
-
I
+
ɛ
M
2
M
2
T
0
*
*
*
*
*
*
-
ɛ
I
<
0
hold. Then a suitable state-feedback control law is given by
(40)
u
t
=
L
0
P
¯
-
1
x
t
+
L
1
P
¯
-
1
x
t
-
r
,
where
(41)
Ψ
^
11
=
Ψ
¯
11
+
ɛ
M
1
M
1
T
,
Ψ
^
13
=
Ψ
¯
13
+
ɛ
δ
1
M
1
M
1
T
,
Ψ
^
14
=
Ψ
¯
14
+
ɛ
δ
2
M
1
M
1
T
⋯
M
1
T
,
Ψ
^
16
=
Ψ
¯
16
+
ɛ
M
1
M
2
T
,
Ψ
^
33
=
Ψ
¯
33
+
ɛ
δ
1
2
M
1
M
1
T
,
Ψ
^
34
=
Ψ
¯
34
+
ɛ
δ
1
δ
2
M
1
M
1
T
⋯
M
1
T
,
Ψ
^
44
=
Ψ
¯
44
+
ɛ
δ
2
2
M
1
M
1
T
⋯
M
1
M
1
T
⋮
⋱
⋮
M
1
M
1
T
⋯
M
1
M
1
T
,
Ψ
^
46
=
ɛ
δ
2
M
1
M
2
T
⋮
M
2
T
,
Ψ
^
17
=
P
¯
T
N
1
T
+
L
0
T
N
3
T
,
Ψ
^
27
=
P
¯
T
N
2
T
+
L
1
T
N
3
T
.The proof can be carried out by resorting to Theorem12 and following a similar line as in the proof of Theorem 11 and is thus omitted.


## 4. Examples

Example 1.
Consider a singular time-delay system in the form ofΣ
0 with
(42)
E
=
1
0
0
0
,
A
0
=
-
0.3012
0.1257
0.2351
-
2.5652
,
A
1
=
-
0.5124
0.9648
0.1023
0.8197
,
D
=
0.2102
-
0.8152
,
C
0
=
1.2321
0.3185
,
C
1
=
0.8765
0.8231
.For givenγ
>
0, we can calculate the maximum allowed delay r
⋆ satisfying the LMIs in Theorem 8. To show the low conservativeness of the result, we compare ours with the criteria of [6, 11, 16] in Table 1. It is clear that the characterization of bounded realness in this paper is an improvement over the previous ones.Table 1
Comparisons of maximum allowed delayr
⋆ for Example 1.
γ
2
2.5
3
3.5
r
⋆ by [6]
1.8116
2.0029
2.1465
2.2587
r
⋆ by [11]
2.2761
2.6131
2.8739
3.0855
r
⋆ by [16]
3.0710
3.4810
3.8030
4.0640
r
⋆ by Theorem 8
3.1303
3.5666
3.9113
4.1860Example 2.
Consider the uncertain singular time-delay systemΣ
1 with u
(
t
)
=
0, where
(43)
E
=
1
0
0
0
,
A
0
=
0.5
0
0
-
1
,
A
1
=
-
1
1
0
-
0.5
,
D
=
-
1
0.5
,
C
0
=
0.5
1
,
C
1
=
0
0
,
M
1
=
0.2
0
0
0.2
,
N
1
=
N
2
=
0.025
0
0
0.025
.Tables2 and 3 give the comparison results on the maximum allowed delay r
⋆ for given γ and the minimum allowed γ for given r
>
0, respectively. It is clear that the results of Theorem 11 are significantly better than those in [10].Table 2
Comparisons of maximum allowed delayr
⋆ for Example 2.
γ
4.0
4.5
5.0
5.5
6.0
6.5
r
⋆ by [10]
0.9425
0.9635
0.9801
0.9938
1.0053
1.0151
r
⋆ by Theorem 11
1.0383
1.0497
1.0589
1.0666
1.0730
1.0784Table 3
Comparisons of minimum allowedγ for Example 2.
r
0.35
0.40
0.45
0.50
0.55
0.60
γ by [10]
1.4160
1.4676
1.5251
1.600
1.6652
1.7546
γ by Theorem 11
1.0392
1.0488
1.0583
1.0677
1.0817
1.0954Example 3.
We consider the problem ofH
∞ control for the singular time-delay system Σ
0 with the following parameters:
(44)
E
=
1
0
0
0
,
A
0
=
0
0
0
0
,
A
1
=
-
1
0
1
-
1
,
B
=
-
0.5
1
,
D
=
1
1
,
C
0
=
1
0.2
,
C
1
=
0
0
,
H
=
0.1
.For a given delayr
=
1.2, Table 4 provides the comparison results on the minimum H
∞ performance index for given delay via the methods in [6, 11, 12, 15, 16] and Theorem 12 in this paper, which shows that Theorem 12 in this paper can lower the H
∞ performance index. The state feedback controller achieving the minimum H
∞ performance level can be obtained as
(45)
u
t
=
-
0.3369
-
0.4316
x
t
+
-
0.8625
1.1541
x
t
-
r
.
The state response of the closed-loop system is shown in Figure 1. We can see that the state responses are converging. The controlled output of the closed-loop system is shown in Figure 2.Table 4
Comparisons of minimumH
∞ performance index γ
⋆ for Example 3.
Methods
[6]
[11]
[12]
[15]
[16]
Theorem12
γ
⋆
21
15.0268
9.9514
9.6754
6.61
4.8610Figure 1
The state response of the closed-loop system in Example3.Figure 2
The controlled output of the closed-loop system in Example3.


## 5. Conclusions

The problem of delay-dependent robustH
∞ control for singular time-delay system with admissible uncertainties has been investigated by parameterized LKF method. A new version of bounded real lemma is presented to ensure the system to be regular, impulse free, and robustly stable with H
∞ performance condition. The slack matrices are suitably introduced to decouple the systems matrices from the LKF parameter, so that the H
∞ controller law can be derived directly. Three examples are given to illustrate the effectiveness of our method and the improvement over some existing ones. As a future research direction, it would be of interest to apply the parameterized LKF method in passivity and H
∞ filtering for neutral systems and singularly perturbed systems [19–22].



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