# Finite FrequencyH

Authors: Ping Mei; Jingzhi Fu; Yunping Liu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper investigates the problem of finite frequency (FF)H
∞ filtering for time-delayed singularly perturbed systems. Our attention is focused on designing filters guaranteeing asymptotic stability and FF H
∞ disturbance attenuation level of the filtering error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF H
∞ filters are obtained in terms of solving an optimization problem, which is delay-independent. The main contribution of this paper is that systematic methods are proposed for designing H
∞ filters for delayed singularly perturbed systems.


---

## Body

## 1. Introduction

Several physical processes are on one hand of high order and on the other hand complex, what returns their analysis and especially their control, with the aim of certain objectives, very delicate. However, knowing that these systems possess variables evolving in various speeds (temperature, pressure, intensity, voltage…), it could be possible to model these systems by singularly perturbed technique [1, 2]. They arise in many physical systems such as electrical power systems and electrical machines (e.g., an asynchronous generator, a DC motor, and electrical converters), electronic systems (e.g., oscillators), mechanical systems (e.g., fighter aircrafts), biological systems (e.g., bacterial-yeast-virus cultures, heart), and also economic systems with various competing sectors. This class of systems has two time scales, namely, “fast” and “slow” dynamics. This makes their analysis and control more complicated than regular systems. Nevertheless, they have been studied extensively [3–5].As the dual of control problem, the filtering problems of dynamic systems are of great theoretical and practical meaning in the field of control and signal processing and the filtering problem has always been a concern in the control theory [6–8]. The state estimation of singularly perturbed systems also has attracted considerable attention over the past decades and a great number of results have been proposed in various schemes, such as Kalman filtering [9, 10] and H
∞ filtering [11].Like all kinds of systems which can contain a time-delay in their dynamic or in their control, the singularly perturbed systems can also contain a delay, which has been studied in many references such as [12–15]. For example, Fridman [12] has considered the effect of small delay on stability of the singularly perturbed systems. In [13, 14], the controllability problem of nonstandard singularly perturbed systems with small state delay and stabilization problem of nonstandard singularly perturbed systems with small delays both in state and control have been studied, respectively. In [15] a composite control law for singularly perturbed bilinear systems via successive Galerkin approximation was presented. However, there is seldom literature dealing with the synthesis design for the delayed singularly perturbed systems, which is the main motivation of this paper.With the fundamental theory-generalized KYP lemma proposed by Iwasaki and Hara [16], the applications using the GKYP lemma have been sprung up in recent years [17–23]. Actually, if noise belongs to a finite frequency (FF) range, more accurately, low/middle/high frequency (LF/MF/HF) range, design methods for the entire range will be much conservative due to overdesign. Consequently, the FF approach has a wide application range. In future work, we can use this approach to Markovain jump systems [24, 25].Lately, using FF approach to analyze and design control problems becomes a new interesting in singularly perturbed systems [19–21]. In [20] the author has studied the H
∞ control problem for singularly perturbed systems within the finite frequency. In [21] the positive control problem for singularly perturbed systems has been studied based on the generalized KYP lemma. The idea is that design in the finite frequency is critical to singularly perturbed systems since the transfer function of singularly perturbed systems has two different frequencies, that is, the high frequency and low frequency, which are corresponding to the fast subsystem and the low subsystem separately. Hence the idea based on the generalized KYP lemma actually is constructing the design problem in separate time scale and also separate frequency scale. Obviously, it could be seen that the conservativeness is much less than existing results. As far as the author’s knowledge, there is seldom literature referring to the filtering design problem within the separate frequencies for delayed singularly perturbed systems, which is also the main motivation of this paper.In this paper we are concerned with FFH
∞ filtering for singularly perturbed systems with time-delay. The frequencies of the exogenous noises are assumed to reside in a known rectangular region, which is the most remarkable difference of our results compared with existing ones. Based on the generalized KYP lemma, we first obtained an FF bounded real lemma in the parameter-independent sense. Filter design methods will be derived by a simple procedure. The main contribution of this paper is summarized as follows: the standard H
∞ filtering for singularly perturbed systems has been extended to the FF H
∞ filtering for singularly perturbed systems with time-delay, and systematic filter design methods have been proposed.The paper is organized as follows. Section2 gives the problem formulation and preliminaries used in the next sections. The main results are given in Section 3. An illustrative example is given in Section 4. And conclusions are given in the last section.Notation. For a matrix X, its transpose is denoted by X
T; N
X is an arbitrary matrix whose column forms a basis of the null space of X. Sym
(
X
) indicates X
T
+
X. σ
max
⁡
· denotes maximum singular value of transfer function. diag
⋯ stands for a block-diagonal matrix.


## 2. Problem Formulation and Preliminaries

The following time-delayed singularly perturbed systems will be considered in this paper:(1)
x
˙
1
t
=
A
01
x
1
t
+
A
02
x
2
t
+
A
11
x
1
t
-
d
+
A
12
x
2
t
-
d
+
B
1
ω
t
,
ε
x
˙
2
t
=
A
03
x
1
t
+
A
04
x
2
t
+
A
13
x
1
t
-
d
+
A
14
x
2
t
-
d
+
B
2
ω
t
,
y
t
=
C
1
x
1
t
+
C
2
x
2
t
+
C
3
x
1
t
-
d
+
C
4
x
2
t
-
d
,
Z
t
=
G
1
x
1
t
+
G
2
x
2
t
+
G
3
x
1
t
-
d
+
G
4
x
2
t
-
d
,
where x
1
(
t
)
∈
R
n
1 is “slow” state and x
2
(
t
)
∈
R
n
2 is “fast” state. Denote n
x
=
n
1
+
n
2; x
t
=
H
H
H
x
1
(
t
)
T
x
2
(
t
)
T
H
H
H
T
∈
R
n
x is the state vector; y
(
t
)
∈
R
n
y is the measured output signal, z
(
t
)
∈
R
n
z is the signal to be estimated, and ω
(
t
)
∈
R
n
ω is the noise input signal in the L
2
[
0
,
+
∞
) functional space domain. Time-delay d is known and time-invariant Φ
(
t
) is the known initial condition in the domain [
-
d
,
0
]; let E
ε
=
I
n
1
0
0
ε
I
n
2, A
=
A
01
A
02
A
03
A
04, A
d
=
A
11
A
12
A
13
A
14, B
=
B
1
B
2, C
=
[
C
1
C
2
], C
d
=
C
3
C
4, G
=
[
G
1
G
2
], G
d
=
[
G
3
G
4
], A
i
j
(
i
=
0
,
1
;
j
=
1
,
…
,
4
), B
k, C
k, G
k, k
=
1
,
2, C
l, G
l, l
=
3,4, be appropriate constant matrices. Then the above system can be regulated as
(2)
E
ε
x
˙
=
A
x
t
+
A
d
x
t
-
d
+
B
ω
t
,
y
t
=
C
x
t
+
C
d
x
t
-
d
,
z
t
=
G
x
t
+
G
d
x
t
-
d
,
(3)
x
t
=
Φ
t
∀
t
=
-
d
,
0
.
First, we give the following assumption on noise signal ω
(
t
).Assumption 1.
Noise signalω
(
t
) is only defined in the low, medium, and high frequency domains
(4)
Ω
=
^
ω
∈
R
∣
ω
≥
ω
h
,
ω
h
≥
0
high
frequency
ω
∈
R
∣
ω
1
≤
ω
≤
ω
2
,
ω
1
≤
ω
2
,
media
frequency
ω
∈
R
∣
ω
≤
ω
l
,
ω
l
≥
0
low
frequency
.Remark 2.
By the generalized KYP lemma in [16] and by an appropriate choice Φ and Ψ, the set Z can be specialized to define a certain range of the frequency variable ω. For the continuous time setting, we have Φ
=
0
1
1
0, Z
=
j
ω
:
ω
∈
R, where Ω is defined in (4); in this situation, Ψ can be chosen as
(5)
Ψ
=
^
1
0
0
ω
h
2
,
when
ω
∈
ω
∈
R
∣
ω
≥
ω
h
,
ω
h
≥
0
(
high
frequency
)
-
1
j
ω
c
-
j
ω
c
-
ω
1
ω
2
,
when
ω
∈
ω
∈
R
∣
ω
1
≤
ω
≤
ω
2
,
ω
1
≤
ω
2
,
(
media
frequency
)
-
1
0
0
ω
l
2
,
when
ω
∈
ω
l
ω
∈
R
∣
ω
≤
ω
l
,
ω
l
≥
0
low
frequency
,
where ω
c
∶
=
(
ω
1
+
ω
2
)
/
2.
The main objective of this paper is to design the following full-order linear filtering:(6)
x
˙
F
(
t
)
=
A
F
x
F
(
t
)
+
B
F
y
(
t
)
,
x
F
(
0
)
=
0
,
z
F
t
=
C
F
x
F
t
+
D
F
y
t
,
where x
F
(
t
)
∈
R
n
x is the state vector, y
(
t
)
∈
R
n
y is the measured output signal, z
F
(
t
)
∈
R
n
z is the output for the filtering systems, and A
F, B
F, C
F, D
F are the filtering matrices to be solved. Combine (2) and (6) and let x
-
t
=
x
(
t
)
T
x
F
(
t
)
T
T; the following filtering error system is obtained:
(7)
E
-
ε
x
-
˙
t
=
A
-
x
-
t
+
A
-
d
x
-
t
-
d
+
B
-
ω
t
,
e
t
=
C
-
x
-
t
+
C
-
d
x
-
t
-
d
,
(8)
x
-
t
=
Φ
T
t
,
0
T
∀
t
=
-
d
,
0
,
where e
(
t
)
=
z
(
t
)
-
z
F
(
t
) is the filtering error and E
-
ε
=
I
n
1
0
0
0
ε
I
n
2
0
0
0
I
n
x; the filtering system matrices are
(9)
A
-
=
A
01
A
02
0
A
03
A
04
0
B
F
C
1
B
F
C
2
0
,
A
-
d
=
A
11
A
12
0
A
13
A
14
0
B
F
C
3
B
F
C
4
0
,
B
-
=
B
1
B
2
0
,
C
-
=
G
1
-
D
F
C
1
G
2
-
D
F
C
2
-
C
F
,
C
-
d
=
G
3
-
D
F
C
3
G
4
-
D
F
C
4
0
.
The transfer function of the filtering error system in (7) from ω to z is given by
(10)
G
e
s
=
C
-
+
e
-
d
s
C
-
d
K
E
-
ε
s
I
-
A
-
-
e
-
d
s
A
-
d
K
-
1
B
-
,
where “s” is the Laplace operator.
Due to the asymptotic stability of the filtering error system (7) depends on system (2), while delayed system (2) does not include input channel for the input signal, so the following assumption is given.Assumption 3.
Time-delayed singularly perturbed system in (2) is asymptotically stable.Now the problems to be solved can be summarized as follows.Problem 4.
For the continuous time-delay system in (2), find a full-order linear filtering (6), such that the filtering error system in (7) satisfies the following conditions.(1)
The filtering error system in (7) is asymptotically stable.
(2)
Given the appropriate positive realγ, under the zero initial condition, the following finite frequency index is satisfied:
(11)
sup
⁡
σ
max
⁡
G
e
j
ω
<
γ
,
∀
ω
∈
Ω
.To conclude this section, we give the following technical lemma that plays an instrumental role in deriving our results.Lemma 5 ((projection lemma) [26]).
LetX, Z, and Σ be given. There exists a matrix Y satisfying
(12)
Sym
X
T
Y
Z
+
Σ
<
0
if and only if the following projection inequalities are satisfied:
(13)
N
X
T
Σ
N
X
<
0
,
N
Z
T
Σ
N
Z
<
0
.


## 3. Main Results

### 3.1. FFH
∞ Filtering Performance Analysis

To ensure the asymptotic stability and FF specification in (11) for the filtering error system, we need to resort to the generalized KYP lemma in [16]. Based on this, the following lemma can be obtained.Lemma 6.
(i) Given system in (2) and scalars γ
>
0, ε
>
0, d
>
0, the filtering error system in (7) is asymptotically stable for all ω
∈
Ω and satisfies the specifications in (11) if there exist matrices P
ε
∈
R
2
n
x
×
2
n
x with the form of (16), 0
<
P
11
∈
R
n
1
×
n
1, 0
<
P
13
∈
R
n
2
×
n
2, P
2
∈
R
n
x
×
n
x, P
3
∈
R
n
x
×
n
x, O
ε
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
O
11
∈
R
n
1
×
n
1, 0
<
O
13
∈
R
n
2
×
n
2, O
2
∈
R
n
x
×
n
x, O
3
∈
R
n
x
×
n
x such that E
ε
P
1
ε
>
0, E
ε
O
1
ε
>
0, and matrices 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x that satisfy
(14)
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
(
Φ
⊗
P
ε
+
Ψ
⊗
Q
)
F
2
<
0
,
(15)
F
3
T
Ξ
2
F
3
+
F
4
T
(
Φ
⊗
O
ε
)
F
4
<
0
,
where
(16)
P
ε
=
P
1
ε
0
P
2
P
3
,
P
1
ε
=
P
11
ε
P
12
T
P
12
P
13
,
(17)
O
ε
=
O
1
ε
0
O
2
O
3
,
O
1
ε
=
O
11
ε
O
12
T
O
12
O
13
,
(18)
F
0
=
C
-
C
-
d
0
0
0
I
,
F
1
=
A
-
A
-
d
B
-
I
0
0
0
I
0
,
F
2
=
A
-
A
-
d
B
-
I
0
0
,
F
3
=
A
-
A
-
d
I
0
0
I
,
F
4
=
A
-
A
-
d
I
0
,
Ξ
0
≅
I
0
0
-
γ
2
I
,
Ξ
i
=
diag
0
R
i
-
R
i
i
=
1
,
2
.
(ii) Givend
>
0, if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16) when ε
=
0, 0
<
P
11
∈
R
n
1
×
n
1, 0
<
P
13
∈
R
n
2
×
n
2, P
2
∈
R
n
x
×
n
x, P
3
∈
R
n
x
×
n
x, O
0
∈
R
2
n
x
×
2
n
x with the form of (17) when ε
=
0, 0
<
O
11
∈
R
n
1
×
n
1, 0
<
O
13
∈
R
n
2
×
n
2, O
2
∈
R
n
x
×
n
x, O
3
∈
R
n
x
×
n
x, 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x such that (14), (15) are feasible for ε
=
0 then filtering error system (7) is asymptotic stable and satisfies the specifications in (11) for all small enough ε
>
0 and 0
≤
d
-
≤
d.Proof.
(i) Letξ
(
t
)
=
^
x
-
(
t
)
x
-
(
t
-
d
)
ω
(
t
).
Give the following Lyapunov-Krasovskii functional(19)
V
1
(
t
)
=
^
V
1,1
(
t
)
+
V
1,2
(
t
)
,
where
(20)
V
1,1
(
t
)
=
x
-
T
(
t
)
E
-
ε
P
ε
x
-
(
t
)
,
V
1,2
t
=
∫
t
-
d
t
x
-
T
η
R
1
x
-
η
d
η
,
V
˙
1,1
=
x
-
˙
T
E
-
ε
P
ε
x
-
t
+
x
-
T
t
P
ε
T
E
-
ε
x
-
˙
t
=
A
-
x
-
t
+
A
-
d
K
x
-
t
-
d
+
B
-
ω
t
T
·
P
ε
x
-
t
+
x
-
T
t
·
P
ε
T
A
-
x
-
t
+
A
-
d
K
x
-
t
-
d
+
B
-
ω
t
,
V
˙
1,2
=
x
-
T
t
R
1
x
t
-
x
-
T
t
-
d
R
1
x
-
t
-
d
.
Then
(21)
V
˙
1
=
V
˙
1,1
t
+
V
˙
1,2
t
=
ξ
t
T
F
2
T
Φ
⊗
P
ε
F
2
+
F
1
T
Ξ
1
F
1
ξ
t
.
Define the following performance index:(22)
J
=
∫
0
∞
e
T
(
t
)
e
(
t
)
-
γ
2
ω
T
(
t
)
ω
(
t
)
d
t
.
Using the zero initial conditions, we could get(23)
J
≤
∫
0
∞
e
T
t
e
t
-
γ
2
ω
T
t
ω
t
d
t
+
V
1
∞
-
V
1
0
=
∫
0
∞
ξ
T
t
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
Φ
⊗
P
ε
F
2
ξ
t
.
Let Θ
=
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
(
Φ
⊗
P
ε
)
F
2 and use the Parseval equality to get
(24)
∫
0
∞
ξ
T
(
t
)
Θ
ξ
(
t
)
d
t
=
1
2
π
∫
-
∞
∞
ξ
s
T
Θ
ξ
s
d
ω
.
Considering frequency ω
∈
Ω and combining (23) and (24), we know that ξ
s
T
Θ
ξ
s
<
0 is a sufficient condition of J
<
0, for all ω
∈
Ω. Additionally, due to Γ
λ
F
2
ξ
s
=
0, we know that ξ
s is a zero space of Γ
λ
F
2.
By the zero space theory, we know that(25)
N
Γ
λ
F
2
T
Θ
N
Γ
λ
F
2
<
0
⟹
ξ
s
T
Θ
ξ
s
<
0
.
Then by generalized KYP lemma in [16], the sufficient condition for N
Γ
λ
F
2
T
Θ
N
Γ
λ
F
2
<
0 is existing P
∈
R
2
n
x
×
2
n
x, 0
<
Q
∈
R
2
n
x
×
2
n
x, such that the following inequality is satisfied:
(26)
Θ
+
F
2
T
(
Φ
⊗
P
+
Ψ
⊗
Q
)
F
2
<
0
.
By redefining P
ε
+
P as P
ε, we obtain inequality (14) that completes the first part of (i).
As for the asymptotic stability, the Lyapunov-Krasovskii functional can be reselected as follows:(27)
V
2
t
=
^
V
2,1
t
+
V
2,2
t
,
where V
2,1
(
t
)
=
x
-
T
(
t
)
E
-
ε
O
ε
x
-
(
t
) and V
2,2
(
t
)
=
∫
t
-
d
t
x
-
T
(
η
)
R
2
x
-
(
η
)
d
η.
Then similar to the proof of the first part of (i), it could be shown that inequality (15) can guarantee the asymptotic stability of (7).
(ii) If (14), (15) are feasible for ε
=
0, then they are feasible for all small enough ε
>
0 and thus, due to (i), filtering error system in (7) is asymptotic stable and satisfies the specifications in (11) for these values ε
>
0. Furthermore, linear matrix inequalities (14), (15) are convex with respect to d; hence they are feasible for some d; then they are feasible for all 0
≤
d
-
≤
d.Remark 7.
Though the essence of the proof of Lemma6 follows from that of Theorem 1 in [6], there are also some differences in Lemma 6. First, in Lemma 6, time-delayed singularly perturbed systems are considered, which are totally different from the regular systems. Since the singularly perturbed systems have perturbed parameter ε, the existence of ε can lead to the ill-conditioned numerical problems, so here comes part (ii) of Lemma 6. Furthermore, considering the special structure of singularly perturbed systems, note that, in the proof of part (i), the selected Lyapunov-Krasovskii functionals are different from the regular systems.To facilitate and reduce the conservatism of the filter design using the projection lemma, we present an alternative of Lemma6.Lemma 8.
Given delayed systems in (2) and scalars γ
>
0, d
>
0, filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16), O
0
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x and Y
i
∈
R
4
n
x
×
2
n
x
(
i
=
1
,
2
) satisfying
(28)
-
I
n
z
0
n
z
×
2
n
x
C
-
C
-
d
0
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
Sym
X
T
Y
1
Z
1
<
0
,
(29)
Ξ
2
+
Ξ
O
0
+
Sym
(
X
2
T
Y
2
Z
2
)
<
0
,
with
(30)
Ξ
P
0
=
Φ
c
⊗
P
0
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
Ξ
O
0
=
Φ
c
⊗
O
0
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
Ξ
Q
=
Ψ
c
⊗
Q
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
X
1
=
I
4
n
x
0
4
n
x
×
(
n
x
+
n
ω
)
,
Z
1
=
-
I
2
n
x
A
-
A
-
d
B
-
,
X
2
=
I
4
n
x
0
4
n
x
×
n
x
,
Z
2
=
-
I
2
n
x
A
-
A
-
d
;
the other notations are defined in (14) and (15).Proof.
DefineM
=
0
n
x
×
2
n
x
C
-
C
-
d
0. By the Schur complement, (28) is equivalent to
(31)
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
M
T
M
+
Sym
X
1
T
Y
1
Z
1
<
0
.
By the definition of matrices X
1 and Z
1, one can choose N
X
1
=
0 and N
Z
1
=
A
-
A
-
d
B
-
I
2
n
x
0
0
0
I
n
x
0
0
0
I
n
ω; thus the first inequality in (13) vanishes and then by Lemma 5, (31) is equivalent to
(32)
N
Z
1
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
M
*
M
N
Z
1
<
0
.
By calculation, we can obtain (32) is equivalent to (14) when ε
=
0.
Similarly, by introducing the following null space,(33)
N
X
2
=
0
,
N
Z
2
=
A
-
A
-
d
I
2
n
x
0
0
I
n
x
.
Using projection lemma, (29) is equivalent to the following inequality:
(34)
N
Z
2
T
Ξ
2
+
Ξ
(
O
0
)
+
Sym
(
X
2
T
Y
2
Z
2
)
N
Z
2
<
0
.
By calculation, (34) is equivalent to (15) when ε
=
0.
Thus, Lemma8 is equivalent to Lemma 6.


### 3.2. Design of FFH
∞ Filters

Lemma8 does not give a solution to filter realization explicitly. Based on the result in the following section we focus on developing methods for designing FF H
∞ filters. The following result can be derived via specifying the structure of the slack matrices Y
1 and Y
2 in Lemma 8.Theorem 9.
Given time-delayed systems in (2) and scalars γ
>
0, d
>
0, filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16), O
0
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, 0
≤
R
2
∈
R
n
x
×
n
x, Γ
i
,
j
∈
R
n
x
×
n
x
(
i
=
1
,
2
,
j
=
1
,
…
,
4
), Γ
5
∈
R
n
x
×
n
x, Δ
1
∈
R
n
x
×
n
x, Δ
2
∈
R
n
x
×
n
y, Δ
3
∈
R
n
z
×
n
x, Δ
4
∈
R
n
z
×
n
y such that the following matrices are satisfied for the given scalars κ
i
(
i
=
1
,
…
,
4
):
(35)
-
I
n
z
Σ
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
Sym
Λ
1
<
0
,
Ξ
2
+
Ξ
(
O
0
)
+
Sym
Λ
2
<
0
,
where(36)
Σ
=
^
0
n
z
×
2
n
x
G
-
Δ
4
C
-
Δ
3
G
d
-
Δ
4
C
d
0
,
Λ
1
=
^
-
Γ
1,1
-
Γ
5
Γ
1,1
A
+
Δ
2
C
Δ
1
Γ
1,1
A
d
+
Δ
2
C
d
Γ
1,1
B
-
Γ
1,2
-
Γ
5
Γ
1,2
A
+
Δ
2
C
Δ
1
Γ
1,2
A
d
+
Δ
2
C
d
Γ
1,2
B
-
Γ
1,3
-
κ
1
Γ
5
Γ
1,3
A
+
κ
1
Δ
2
C
κ
1
Δ
1
Γ
1,3
A
d
+
κ
1
Δ
2
C
d
Γ
1,3
B
-
Γ
1,4
-
κ
2
Γ
5
Γ
1,4
A
+
κ
2
Δ
2
C
κ
2
Δ
1
Γ
1,4
A
d
+
κ
2
Δ
2
C
d
Γ
1,4
B
0
0
0
0
0
0
,
Λ
2
=
^
-
Γ
2,1
-
Γ
5
Γ
2,1
A
+
Δ
2
C
Δ
1
Γ
2,1
A
d
+
Δ
2
C
d
-
Γ
2,2
-
Γ
5
Γ
2,2
A
+
Δ
2
C
Δ
1
Γ
2,2
A
d
+
Δ
2
C
d
-
Γ
2,3
-
κ
3
Γ
5
Γ
2,3
A
+
κ
3
Δ
2
C
κ
3
Δ
1
Γ
2,3
A
d
+
κ
3
Δ
2
C
d
-
Γ
2,4
-
κ
4
Γ
5
Γ
2,4
A
+
κ
4
Δ
2
C
κ
4
Δ
1
Γ
2,4
A
d
+
κ
4
Δ
2
C
d
0
0
0
0
0
.
Ξ
i, Ξ
(
P
0
), Ξ
(
Q
), and Ξ
(
O
0
) are defined in (28) and (29). Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the filter in (6) is given by
(37)
A
F
B
F
C
F
D
F
=
Γ
5
-
1
0
0
I
Δ
1
Δ
2
Δ
3
Δ
4
.Proof.
First, in order to prove Theorem9, we just need to prove that (28) and (29) in Lemma 8 can be deduced from (35) in Theorem 9. It is noted that the slack matrix Y
2 has the following form:
(38)
Y
2
=
Γ
2,1
Γ
5
Γ
2,2
Γ
6
Γ
2,3
Γ
7
Γ
2,4
Γ
8
.
Here, Γ
2
,
i
(
i
=
1
,
…
,
4
), Γ
j
(
j
=
5
,
…
,
8
) are complex matrices with dimension n
x
×
n
x. In fact, Γ
6 is nonsingular and can be implied from (24); then by multiplying Y
2 from the left side and the right side, respectively, with I
1
=
diag
I
Γ
5
Γ
6
-
1
I
I and I
2
=
diag
I
Γ
5
Γ
6
-
1
T, we could get
(39)
I
1
Y
2
I
2
=
Γ
2,1
Γ
5
Γ
5
Γ
6
-
1
T
Γ
5
Γ
6
-
1
Γ
2,2
Γ
6
Γ
5
Γ
6
-
1
T
Γ
2,3
Γ
7
Γ
5
Γ
6
-
1
T
Γ
2,4
Γ
8
Γ
5
Γ
6
-
1
T
.
Without loss of generality, we could restrict Γ
5
≡
Γ
6.
On the other hand, to overcome the difficulty of filtering design, more work should be done. Next, we will considerΓ
7 and Γ
8 to be linearly Γ
5-dependent; that is, Γ
7
=
κ
3
Γ
5, Γ
8
=
κ
4
Γ
5, κ
3 and κ
4 are scalars, respectively. Similarly, the slack matrix Y
1 has the same structure restriction; that is,
(40)
Y
1
=
Γ
1,1
Γ
5
Γ
1,2
Γ
5
Γ
1,3
κ
1
Γ
5
Γ
1,4
κ
2
Γ
5
,
Y
2
=
Γ
2,1
Γ
5
Γ
2,2
Γ
5
Γ
2,3
κ
3
Γ
5
Γ
2,4
κ
4
Γ
5
.
Define
(41)
Δ
1
Δ
2
Δ
3
Δ
4
=
Γ
5
0
0
I
A
f
B
f
C
f
D
f
.
By substituting X
i, Z
i in (28) and (29) and Y
i of (40) into X
i
T
Y
i
Z
i, one can obtain Λ
i
=
X
i
T
Y
i
Z
i  (
i
=
1
,
2
). Moreover, by using (41), one can also obtain Σ
≡
M. The proof is complete.


## 3.1. FFH
∞ Filtering Performance Analysis

To ensure the asymptotic stability and FF specification in (11) for the filtering error system, we need to resort to the generalized KYP lemma in [16]. Based on this, the following lemma can be obtained.Lemma 6.
(i) Given system in (2) and scalars γ
>
0, ε
>
0, d
>
0, the filtering error system in (7) is asymptotically stable for all ω
∈
Ω and satisfies the specifications in (11) if there exist matrices P
ε
∈
R
2
n
x
×
2
n
x with the form of (16), 0
<
P
11
∈
R
n
1
×
n
1, 0
<
P
13
∈
R
n
2
×
n
2, P
2
∈
R
n
x
×
n
x, P
3
∈
R
n
x
×
n
x, O
ε
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
O
11
∈
R
n
1
×
n
1, 0
<
O
13
∈
R
n
2
×
n
2, O
2
∈
R
n
x
×
n
x, O
3
∈
R
n
x
×
n
x such that E
ε
P
1
ε
>
0, E
ε
O
1
ε
>
0, and matrices 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x that satisfy
(14)
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
(
Φ
⊗
P
ε
+
Ψ
⊗
Q
)
F
2
<
0
,
(15)
F
3
T
Ξ
2
F
3
+
F
4
T
(
Φ
⊗
O
ε
)
F
4
<
0
,
where
(16)
P
ε
=
P
1
ε
0
P
2
P
3
,
P
1
ε
=
P
11
ε
P
12
T
P
12
P
13
,
(17)
O
ε
=
O
1
ε
0
O
2
O
3
,
O
1
ε
=
O
11
ε
O
12
T
O
12
O
13
,
(18)
F
0
=
C
-
C
-
d
0
0
0
I
,
F
1
=
A
-
A
-
d
B
-
I
0
0
0
I
0
,
F
2
=
A
-
A
-
d
B
-
I
0
0
,
F
3
=
A
-
A
-
d
I
0
0
I
,
F
4
=
A
-
A
-
d
I
0
,
Ξ
0
≅
I
0
0
-
γ
2
I
,
Ξ
i
=
diag
0
R
i
-
R
i
i
=
1
,
2
.
(ii) Givend
>
0, if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16) when ε
=
0, 0
<
P
11
∈
R
n
1
×
n
1, 0
<
P
13
∈
R
n
2
×
n
2, P
2
∈
R
n
x
×
n
x, P
3
∈
R
n
x
×
n
x, O
0
∈
R
2
n
x
×
2
n
x with the form of (17) when ε
=
0, 0
<
O
11
∈
R
n
1
×
n
1, 0
<
O
13
∈
R
n
2
×
n
2, O
2
∈
R
n
x
×
n
x, O
3
∈
R
n
x
×
n
x, 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x such that (14), (15) are feasible for ε
=
0 then filtering error system (7) is asymptotic stable and satisfies the specifications in (11) for all small enough ε
>
0 and 0
≤
d
-
≤
d.Proof.
(i) Letξ
(
t
)
=
^
x
-
(
t
)
x
-
(
t
-
d
)
ω
(
t
).
Give the following Lyapunov-Krasovskii functional(19)
V
1
(
t
)
=
^
V
1,1
(
t
)
+
V
1,2
(
t
)
,
where
(20)
V
1,1
(
t
)
=
x
-
T
(
t
)
E
-
ε
P
ε
x
-
(
t
)
,
V
1,2
t
=
∫
t
-
d
t
x
-
T
η
R
1
x
-
η
d
η
,
V
˙
1,1
=
x
-
˙
T
E
-
ε
P
ε
x
-
t
+
x
-
T
t
P
ε
T
E
-
ε
x
-
˙
t
=
A
-
x
-
t
+
A
-
d
K
x
-
t
-
d
+
B
-
ω
t
T
·
P
ε
x
-
t
+
x
-
T
t
·
P
ε
T
A
-
x
-
t
+
A
-
d
K
x
-
t
-
d
+
B
-
ω
t
,
V
˙
1,2
=
x
-
T
t
R
1
x
t
-
x
-
T
t
-
d
R
1
x
-
t
-
d
.
Then
(21)
V
˙
1
=
V
˙
1,1
t
+
V
˙
1,2
t
=
ξ
t
T
F
2
T
Φ
⊗
P
ε
F
2
+
F
1
T
Ξ
1
F
1
ξ
t
.
Define the following performance index:(22)
J
=
∫
0
∞
e
T
(
t
)
e
(
t
)
-
γ
2
ω
T
(
t
)
ω
(
t
)
d
t
.
Using the zero initial conditions, we could get(23)
J
≤
∫
0
∞
e
T
t
e
t
-
γ
2
ω
T
t
ω
t
d
t
+
V
1
∞
-
V
1
0
=
∫
0
∞
ξ
T
t
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
Φ
⊗
P
ε
F
2
ξ
t
.
Let Θ
=
F
0
T
Ξ
0
F
0
+
F
1
T
Ξ
1
F
1
+
F
2
T
(
Φ
⊗
P
ε
)
F
2 and use the Parseval equality to get
(24)
∫
0
∞
ξ
T
(
t
)
Θ
ξ
(
t
)
d
t
=
1
2
π
∫
-
∞
∞
ξ
s
T
Θ
ξ
s
d
ω
.
Considering frequency ω
∈
Ω and combining (23) and (24), we know that ξ
s
T
Θ
ξ
s
<
0 is a sufficient condition of J
<
0, for all ω
∈
Ω. Additionally, due to Γ
λ
F
2
ξ
s
=
0, we know that ξ
s is a zero space of Γ
λ
F
2.
By the zero space theory, we know that(25)
N
Γ
λ
F
2
T
Θ
N
Γ
λ
F
2
<
0
⟹
ξ
s
T
Θ
ξ
s
<
0
.
Then by generalized KYP lemma in [16], the sufficient condition for N
Γ
λ
F
2
T
Θ
N
Γ
λ
F
2
<
0 is existing P
∈
R
2
n
x
×
2
n
x, 0
<
Q
∈
R
2
n
x
×
2
n
x, such that the following inequality is satisfied:
(26)
Θ
+
F
2
T
(
Φ
⊗
P
+
Ψ
⊗
Q
)
F
2
<
0
.
By redefining P
ε
+
P as P
ε, we obtain inequality (14) that completes the first part of (i).
As for the asymptotic stability, the Lyapunov-Krasovskii functional can be reselected as follows:(27)
V
2
t
=
^
V
2,1
t
+
V
2,2
t
,
where V
2,1
(
t
)
=
x
-
T
(
t
)
E
-
ε
O
ε
x
-
(
t
) and V
2,2
(
t
)
=
∫
t
-
d
t
x
-
T
(
η
)
R
2
x
-
(
η
)
d
η.
Then similar to the proof of the first part of (i), it could be shown that inequality (15) can guarantee the asymptotic stability of (7).
(ii) If (14), (15) are feasible for ε
=
0, then they are feasible for all small enough ε
>
0 and thus, due to (i), filtering error system in (7) is asymptotic stable and satisfies the specifications in (11) for these values ε
>
0. Furthermore, linear matrix inequalities (14), (15) are convex with respect to d; hence they are feasible for some d; then they are feasible for all 0
≤
d
-
≤
d.Remark 7.
Though the essence of the proof of Lemma6 follows from that of Theorem 1 in [6], there are also some differences in Lemma 6. First, in Lemma 6, time-delayed singularly perturbed systems are considered, which are totally different from the regular systems. Since the singularly perturbed systems have perturbed parameter ε, the existence of ε can lead to the ill-conditioned numerical problems, so here comes part (ii) of Lemma 6. Furthermore, considering the special structure of singularly perturbed systems, note that, in the proof of part (i), the selected Lyapunov-Krasovskii functionals are different from the regular systems.To facilitate and reduce the conservatism of the filter design using the projection lemma, we present an alternative of Lemma6.Lemma 8.
Given delayed systems in (2) and scalars γ
>
0, d
>
0, filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16), O
0
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, and 0
≤
R
2
∈
R
n
x
×
n
x and Y
i
∈
R
4
n
x
×
2
n
x
(
i
=
1
,
2
) satisfying
(28)
-
I
n
z
0
n
z
×
2
n
x
C
-
C
-
d
0
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
Sym
X
T
Y
1
Z
1
<
0
,
(29)
Ξ
2
+
Ξ
O
0
+
Sym
(
X
2
T
Y
2
Z
2
)
<
0
,
with
(30)
Ξ
P
0
=
Φ
c
⊗
P
0
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
Ξ
O
0
=
Φ
c
⊗
O
0
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
Ξ
Q
=
Ψ
c
⊗
Q
0
4
n
x
×
n
x
0
n
x
×
4
n
x
0
n
x
×
n
x
,
X
1
=
I
4
n
x
0
4
n
x
×
(
n
x
+
n
ω
)
,
Z
1
=
-
I
2
n
x
A
-
A
-
d
B
-
,
X
2
=
I
4
n
x
0
4
n
x
×
n
x
,
Z
2
=
-
I
2
n
x
A
-
A
-
d
;
the other notations are defined in (14) and (15).Proof.
DefineM
=
0
n
x
×
2
n
x
C
-
C
-
d
0. By the Schur complement, (28) is equivalent to
(31)
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
M
T
M
+
Sym
X
1
T
Y
1
Z
1
<
0
.
By the definition of matrices X
1 and Z
1, one can choose N
X
1
=
0 and N
Z
1
=
A
-
A
-
d
B
-
I
2
n
x
0
0
0
I
n
x
0
0
0
I
n
ω; thus the first inequality in (13) vanishes and then by Lemma 5, (31) is equivalent to
(32)
N
Z
1
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
M
*
M
N
Z
1
<
0
.
By calculation, we can obtain (32) is equivalent to (14) when ε
=
0.
Similarly, by introducing the following null space,(33)
N
X
2
=
0
,
N
Z
2
=
A
-
A
-
d
I
2
n
x
0
0
I
n
x
.
Using projection lemma, (29) is equivalent to the following inequality:
(34)
N
Z
2
T
Ξ
2
+
Ξ
(
O
0
)
+
Sym
(
X
2
T
Y
2
Z
2
)
N
Z
2
<
0
.
By calculation, (34) is equivalent to (15) when ε
=
0.
Thus, Lemma8 is equivalent to Lemma 6.


## 3.2. Design of FFH
∞ Filters

Lemma8 does not give a solution to filter realization explicitly. Based on the result in the following section we focus on developing methods for designing FF H
∞ filters. The following result can be derived via specifying the structure of the slack matrices Y
1 and Y
2 in Lemma 8.Theorem 9.
Given time-delayed systems in (2) and scalars γ
>
0, d
>
0, filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices P
0
∈
R
2
n
x
×
2
n
x with the form of (16), O
0
∈
R
2
n
x
×
2
n
x with the form of (17), 0
<
Q
∈
R
2
n
x
×
2
n
x, R
1
∈
R
n
x
×
n
x, 0
≤
R
2
∈
R
n
x
×
n
x, Γ
i
,
j
∈
R
n
x
×
n
x
(
i
=
1
,
2
,
j
=
1
,
…
,
4
), Γ
5
∈
R
n
x
×
n
x, Δ
1
∈
R
n
x
×
n
x, Δ
2
∈
R
n
x
×
n
y, Δ
3
∈
R
n
z
×
n
x, Δ
4
∈
R
n
z
×
n
y such that the following matrices are satisfied for the given scalars κ
i
(
i
=
1
,
…
,
4
):
(35)
-
I
n
z
Σ
*
diag
Ξ
1
+
Ξ
P
0
+
Ξ
Q
,
-
γ
2
I
n
ω
+
Sym
Λ
1
<
0
,
Ξ
2
+
Ξ
(
O
0
)
+
Sym
Λ
2
<
0
,
where(36)
Σ
=
^
0
n
z
×
2
n
x
G
-
Δ
4
C
-
Δ
3
G
d
-
Δ
4
C
d
0
,
Λ
1
=
^
-
Γ
1,1
-
Γ
5
Γ
1,1
A
+
Δ
2
C
Δ
1
Γ
1,1
A
d
+
Δ
2
C
d
Γ
1,1
B
-
Γ
1,2
-
Γ
5
Γ
1,2
A
+
Δ
2
C
Δ
1
Γ
1,2
A
d
+
Δ
2
C
d
Γ
1,2
B
-
Γ
1,3
-
κ
1
Γ
5
Γ
1,3
A
+
κ
1
Δ
2
C
κ
1
Δ
1
Γ
1,3
A
d
+
κ
1
Δ
2
C
d
Γ
1,3
B
-
Γ
1,4
-
κ
2
Γ
5
Γ
1,4
A
+
κ
2
Δ
2
C
κ
2
Δ
1
Γ
1,4
A
d
+
κ
2
Δ
2
C
d
Γ
1,4
B
0
0
0
0
0
0
,
Λ
2
=
^
-
Γ
2,1
-
Γ
5
Γ
2,1
A
+
Δ
2
C
Δ
1
Γ
2,1
A
d
+
Δ
2
C
d
-
Γ
2,2
-
Γ
5
Γ
2,2
A
+
Δ
2
C
Δ
1
Γ
2,2
A
d
+
Δ
2
C
d
-
Γ
2,3
-
κ
3
Γ
5
Γ
2,3
A
+
κ
3
Δ
2
C
κ
3
Δ
1
Γ
2,3
A
d
+
κ
3
Δ
2
C
d
-
Γ
2,4
-
κ
4
Γ
5
Γ
2,4
A
+
κ
4
Δ
2
C
κ
4
Δ
1
Γ
2,4
A
d
+
κ
4
Δ
2
C
d
0
0
0
0
0
.
Ξ
i, Ξ
(
P
0
), Ξ
(
Q
), and Ξ
(
O
0
) are defined in (28) and (29). Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the filter in (6) is given by
(37)
A
F
B
F
C
F
D
F
=
Γ
5
-
1
0
0
I
Δ
1
Δ
2
Δ
3
Δ
4
.Proof.
First, in order to prove Theorem9, we just need to prove that (28) and (29) in Lemma 8 can be deduced from (35) in Theorem 9. It is noted that the slack matrix Y
2 has the following form:
(38)
Y
2
=
Γ
2,1
Γ
5
Γ
2,2
Γ
6
Γ
2,3
Γ
7
Γ
2,4
Γ
8
.
Here, Γ
2
,
i
(
i
=
1
,
…
,
4
), Γ
j
(
j
=
5
,
…
,
8
) are complex matrices with dimension n
x
×
n
x. In fact, Γ
6 is nonsingular and can be implied from (24); then by multiplying Y
2 from the left side and the right side, respectively, with I
1
=
diag
I
Γ
5
Γ
6
-
1
I
I and I
2
=
diag
I
Γ
5
Γ
6
-
1
T, we could get
(39)
I
1
Y
2
I
2
=
Γ
2,1
Γ
5
Γ
5
Γ
6
-
1
T
Γ
5
Γ
6
-
1
Γ
2,2
Γ
6
Γ
5
Γ
6
-
1
T
Γ
2,3
Γ
7
Γ
5
Γ
6
-
1
T
Γ
2,4
Γ
8
Γ
5
Γ
6
-
1
T
.
Without loss of generality, we could restrict Γ
5
≡
Γ
6.
On the other hand, to overcome the difficulty of filtering design, more work should be done. Next, we will considerΓ
7 and Γ
8 to be linearly Γ
5-dependent; that is, Γ
7
=
κ
3
Γ
5, Γ
8
=
κ
4
Γ
5, κ
3 and κ
4 are scalars, respectively. Similarly, the slack matrix Y
1 has the same structure restriction; that is,
(40)
Y
1
=
Γ
1,1
Γ
5
Γ
1,2
Γ
5
Γ
1,3
κ
1
Γ
5
Γ
1,4
κ
2
Γ
5
,
Y
2
=
Γ
2,1
Γ
5
Γ
2,2
Γ
5
Γ
2,3
κ
3
Γ
5
Γ
2,4
κ
4
Γ
5
.
Define
(41)
Δ
1
Δ
2
Δ
3
Δ
4
=
Γ
5
0
0
I
A
f
B
f
C
f
D
f
.
By substituting X
i, Z
i in (28) and (29) and Y
i of (40) into X
i
T
Y
i
Z
i, one can obtain Λ
i
=
X
i
T
Y
i
Z
i  (
i
=
1
,
2
). Moreover, by using (41), one can also obtain Σ
≡
M. The proof is complete.


## 4. Numerical Example

In this section, we use an example to illustrate the effectiveness and advantages of the design methods developed in this paper. Consider the singularly perturbed system with time-invariant delay in (2) with matrices given by
(42)
A
=
-
1
0
0
-
1
,
A
d
=
-
1
0
1
-
1
,
B
=
-
0.5
2
,
C
=
0
1
,
C
d
=
1
2
,
G
=
2
1
,
G
d
=
0
0
,
ε
=
0.1
,
d
=
0.1
.Suppose the frequenciesω
l
=
1, ω
h
=
100, we calculate the achieved minimum performance γ
* by using Theorem 9 in this paper. For brevity, the scalar parameters in Theorem 9 are given by κ
1
=
κ
2
=
5, κ
3
=
κ
4
=
1. The obtained minimum performance is γ
*
=
0.9976, when Q
=
0; the problem becomes a standard H
∞ filtering problem and the minimum performance of the nominal H
∞ filtering is γ
*
=
1.4533.And the obtained state-space matrices ofH
∞ filtering are
(43)
A
F
B
F
C
F
D
F
=
-
1.5861
-
0.0552
0
-
0.2522
-
1.1106
0
0
0
0
.


## 5. Conclusions

This paper has studied the problem of FFH
∞ filtering for time-delayed singularly perturbed systems. The frequencies of the exogenous noise are assumed to reside in a known rectangular region and the standard H
∞ filtering for singularly perturbed systems has been extended to the FF H
∞ case. The generalized KYP lemma for singularly perturbed systems has been further developed to derive conditions that are more suitable for FF H
∞ performance synthesis with time-delay. Via structural restriction for the slack matrices, systematic methods have been proposed for the design of the filters that guarantee the asymptotic stability and FF H
∞ disturbance attenuation level of the filtering error system.



---

*Source: 456768-2015-03-04.xml*