Research ArticleFinite Frequency𝐻
∞Filtering for Time-Delayed Singularly
Perturbed Systems
Ping Mei,1,2 Jingzhi Fu,1,2 and Yunping Liu1
1Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China2Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology, Nanjing, Jiangsu 210044, China
Correspondence should be addressed to Ping Mei; pmei [email protected]
Received 18 September 2014; Revised 29 January 2015; Accepted 16 February 2015
Academic Editor: Thomas Hanne
Copyright © 2015 Ping Mei et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates the problem of finite frequency (FF) 𝐻∞
filtering for time-delayed singularly perturbed systems. Ourattention 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 systematicmethods are proposed for designing 𝐻
∞filters for delayed singularly perturbed systems.
1. Introduction
Several physical processes are on one hand of high order andon the other hand complex, what returns their analysis andespecially their control, with the aim of certain objectives,very delicate. However, knowing that these systems possessvariables evolving in various speeds (temperature, pressure,intensity, voltage. . .), it could be possible to model thesesystems by singularly perturbed technique [1, 2]. They arisein many physical systems such as electrical power systemsand 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 morecomplicated than regular systems. Nevertheless, they havebeen studied extensively [3–5].
As the dual of control problem, the filtering problemsof dynamic systems are of great theoretical and practicalmeaning in the field of control and signal processing and thefiltering problem has always been a concern in the controltheory [6–8]. The state estimation of singularly perturbedsystems also has attracted considerable attention over the pastdecades 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 singularlyperturbed systems can also contain a delay, which has beenstudied in many references such as [12–15]. For example,Fridman [12] has considered the effect of small delay onstability of the singularly perturbed systems. In [13, 14], thecontrollability problem of nonstandard singularly perturbedsystems with small state delay and stabilization problem ofnonstandard singularly perturbed systems with small delaysboth in state and control have been studied, respectively.In [15] a composite control law for singularly perturbedbilinear systems via successive Galerkin approximation waspresented. However, there is seldom literature dealing withthe synthesis design for the delayed singularly perturbedsystems, which is the main motivation of this paper.
With the fundamental theory-generalized KYP lemmaproposed by Iwasaki and Hara [16], the applications usingthe 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 con-servative due to overdesign. Consequently, the FF approachhas a wide application range. In future work, we can use thisapproach to Markovain jump systems [24, 25].
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 456768, 7 pageshttp://dx.doi.org/10.1155/2015/456768
2 Mathematical Problems in Engineering
Lately, using FF approach to analyze and design controlproblems becomes a new interesting in singularly perturbedsystems [19–21]. In [20] the author has studied the 𝐻
∞
control problem for singularly perturbed systems within thefinite frequency. In [21] the positive control problem forsingularly perturbed systems has been studied based on thegeneralized KYP lemma. The idea is that design in the finitefrequency is critical to singularly perturbed systems sincethe transfer function of singularly perturbed systems has twodifferent frequencies, that is, the high frequency and lowfrequency, which are corresponding to the fast subsystem andthe low subsystem separately. Hence the idea based on thegeneralized KYP lemma actually is constructing the designproblem in separate time scale and also separate frequencyscale. Obviously, it could be seen that the conservativenessis much less than existing results. As far as the author’sknowledge, there is seldom literature referring to the filteringdesign problem within the separate frequencies for delayedsingularly perturbed systems, which is also the main motiva-tion of this paper.
In this paper we are concerned with FF 𝐻∞
filtering forsingularly perturbed systems with time-delay. The frequen-cies of the exogenous noises are assumed to reside in a knownrectangular region, which is the most remarkable differenceof our results compared with existing ones. Based on thegeneralized KYP lemma, we first obtained an FF boundedreal lemma in the parameter-independent sense. Filter designmethods will be derived by a simple procedure. The maincontribution of this paper is summarized as follows: thestandard 𝐻
∞filtering for singularly perturbed systems has
been extended to the FF𝐻∞filtering for singularly perturbed
systemswith time-delay, and systematic filter designmethodshave been proposed.
The paper is organized as follows. Section 2 gives theproblem formulation and preliminaries used in the next sec-tions. The main results are given in Section 3. An illustrativeexample is given in Section 4. And conclusions are given inthe last section.Notation. For a matrix 𝑋, its transpose is denoted by 𝑋𝑇; 𝑁
𝑋
is an arbitrary matrix whose column forms a basis of the nullspace of 𝑋. Sym(𝑋) indicates 𝑋𝑇 + 𝑋. 𝜎max(⋅) denotes maxi-mum singular value of transfer function. diag{⋅ ⋅ ⋅ } stands fora block-diagonal matrix.
2. Problem Formulation and Preliminaries
The following time-delayed singularly perturbed systems willbe considered in this paper:
�̇�1(𝑡) = 𝐴
01𝑥1(𝑡) + 𝐴
02𝑥2(𝑡) + 𝐴
11𝑥1(𝑡 − 𝑑)
+ 𝐴12𝑥2(𝑡 − 𝑑) + 𝐵
1𝜔 (𝑡) ,
𝜀�̇�2(𝑡) = 𝐴
03𝑥1(𝑡) + 𝐴
04𝑥2(𝑡) + 𝐴
13𝑥1(𝑡 − 𝑑)
+ 𝐴14𝑥2(𝑡 − 𝑑) + 𝐵
2𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶1𝑥1(𝑡) + 𝐶
2𝑥2(𝑡) + 𝐶
3𝑥1(𝑡 − 𝑑) + 𝐶
4𝑥2(𝑡 − 𝑑) ,
𝑍 (𝑡) = 𝐺1𝑥1(𝑡) + 𝐺
2𝑥2(𝑡) + 𝐺
3𝑥1(𝑡 − 𝑑) + 𝐺
4𝑥2(𝑡 − 𝑑) ,
(1)
where 𝑥1(𝑡) ∈ 𝑅
𝑛1 is “slow” state and 𝑥
2(𝑡) ∈ 𝑅
𝑛2 is “fast”
state. Denote 𝑛𝑥
= 𝑛1+ 𝑛2; 𝑥(𝑡) = [𝑥
1(𝑡)𝑇
𝑥2(𝑡)𝑇
]𝑇
∈ 𝑅𝑛𝑥
is the state vector; 𝑦(𝑡) ∈ 𝑅𝑛𝑦 is the measured output signal,𝑧(𝑡) ∈ 𝑅
𝑛𝑧 is the signal to be estimated, and 𝜔(𝑡) ∈ 𝑅𝑛𝜔 is the
noise input signal in the 𝐿2[0, +∞) functional space domain.
Time-delay 𝑑 is known and time-invariantΦ(𝑡) is the knowninitial condition in the domain [−𝑑, 0]; let 𝐸
𝜀= [
𝐼𝑛10
0 𝜀𝐼𝑛2
],𝐴 = [
𝐴01𝐴02
𝐴03𝐴04
], 𝐴𝑑= [𝐴11𝐴12
𝐴13𝐴14
], 𝐵 = [ 𝐵1𝐵2
], 𝐶 = [𝐶1
𝐶2], 𝐶𝑑=
[𝐶3
𝐶4], 𝐺 = [𝐺
1𝐺2], 𝐺𝑑
= [𝐺3
𝐺4], 𝐴𝑖𝑗
(𝑖 = 0, 1; 𝑗 =
1, . . . , 4), 𝐵𝑘, 𝐶𝑘, 𝐺𝑘, 𝑘 = 1, 2, 𝐶
𝑙, 𝐺𝑙, 𝑙 = 3, 4, be appropriate
constant matrices.Then the above system can be regulated as
𝐸𝜀�̇� = 𝐴𝑥 (𝑡) + 𝐴
𝑑𝑥 (𝑡 − 𝑑) + 𝐵𝜔 (𝑡) ,
𝑦 (𝑡) = 𝐶𝑥 (𝑡) + 𝐶𝑑𝑥 (𝑡 − 𝑑) ,
𝑧 (𝑡) = 𝐺𝑥 (𝑡) + 𝐺𝑑𝑥 (𝑡 − 𝑑) ,
(2)
𝑥 (𝑡) = Φ (𝑡) ∀𝑡 = [−𝑑, 0] . (3)
First, we give the following assumption on noise signal 𝜔(𝑡).
Assumption 1. Noise signal 𝜔(𝑡) is only defined in the low,medium, and high frequency domains
Ω
=̂
{{{{
{{{{
{
{𝜔 ∈ 𝑅 | |𝜔| ≥ 𝜔ℎ, 𝜔ℎ≥ 0} (high frequency)
{𝜔 ∈ 𝑅 | 𝜔1≤ |𝜔| ≤ 𝜔
2, 𝜔1≤ 𝜔2} , (media frequency)
{𝜔 ∈ 𝑅 | |𝜔| ≤ 𝜔𝑙, 𝜔𝑙≥ 0} (low frequency) .
(4)
Remark 2. By the generalized KYP lemma in [16] and by anappropriate choice Φ and Ψ, the set 𝑍 can be specialized todefine a certain range of the frequency variable 𝜔. For thecontinuous time setting, we have Φ = [ 0 1
1 0], 𝑍 = {𝑗𝜔 : 𝜔 ∈
𝑅}, whereΩ is defined in (4); in this situation,Ψ can be chosenas
Ψ =̂
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
[
[
1 0
0 𝜔2
ℎ
]
]
,
when 𝜔 ∈ {𝜔 ∈ 𝑅 | |𝜔| ≥ 𝜔ℎ, 𝜔ℎ≥ 0}
(high frequency)
[
[
−1 𝑗𝜔𝑐
−𝑗𝜔𝑐
−𝜔1𝜔2
]
]
,
when 𝜔 ∈ {𝜔 ∈ 𝑅 | 𝜔1≤ |𝜔| ≤ 𝜔
2, 𝜔1≤ 𝜔2} ,
(media frequency)
[
[
−1 0
0 𝜔2
𝑙
]
]
,
when 𝜔 ∈ {𝜔 ∈ 𝑅 | |𝜔| ≤ 𝜔𝑙, 𝜔𝑙≥ 0}
(low frequency) ,(5)
where 𝜔𝑐:= (𝜔1+ 𝜔2)/2.
Mathematical Problems in Engineering 3
Themain objective of this paper is to design the followingfull-order linear filtering:
�̇�𝐹(𝑡) = 𝐴
𝐹𝑥𝐹(𝑡) + 𝐵
𝐹𝑦 (𝑡) , 𝑥
𝐹(0) = 0,
𝑧𝐹(𝑡) = 𝐶
𝐹𝑥𝐹(𝑡) + 𝐷
𝐹𝑦 (𝑡) ,
(6)
where 𝑥𝐹(𝑡) ∈ 𝑅
𝑛𝑥 is the state vector, 𝑦(𝑡) ∈ 𝑅𝑛𝑦 is the mea-
sured output signal, 𝑧𝐹(𝑡) ∈ 𝑅
𝑛𝑧 is the output for the filtering
systems, and 𝐴𝐹, 𝐵𝐹, 𝐶𝐹, 𝐷𝐹are the filtering matrices to be
solved. Combine (2) and (6) and let 𝑥(𝑡) = [𝑥(𝑡)𝑇 𝑥𝐹(𝑡)𝑇
]𝑇;
the following filtering error system is obtained:
𝐸𝜀
̇𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐴
𝑑𝑥 (𝑡 − 𝑑) + 𝐵𝜔 (𝑡) ,
𝑒 (𝑡) = 𝐶𝑥 (𝑡) + 𝐶𝑑𝑥 (𝑡 − 𝑑) ,
(7)
𝑥 (𝑡) = [Φ𝑇
(𝑡) , 0]
𝑇
∀𝑡 = [−𝑑, 0] , (8)
where 𝑒(𝑡) = 𝑧(𝑡) − 𝑧𝐹(𝑡) is the filtering error and 𝐸
𝜀=
[
𝐼𝑛10 0
0 𝜀𝐼𝑛20
0 0 𝐼𝑛𝑥
]; the filtering system matrices are
𝐴 =[
[
[
𝐴01
𝐴02
0
𝐴03
𝐴04
0
𝐵𝐹𝐶1
𝐵𝐹𝐶2
0
]
]
]
,
𝐴𝑑=
[
[
[
𝐴11
𝐴12
0
𝐴13
𝐴14
0
𝐵𝐹𝐶3
𝐵𝐹𝐶4
0
]
]
]
, 𝐵 =[
[
[
𝐵1
𝐵2
0
]
]
]
,
𝐶 = [𝐺1− 𝐷𝐹𝐶1
𝐺2− 𝐷𝐹𝐶2
−𝐶𝐹] ,
𝐶𝑑= [𝐺3− 𝐷𝐹𝐶3
𝐺4− 𝐷𝐹𝐶4
0] .
(9)
The transfer function of the filtering error system in (7) from𝜔 to 𝑧 is given by
𝐺𝑒(𝑠) = (𝐶 + 𝑒
−𝑑𝑠
𝐶𝑑𝐾) (𝐸
𝜀𝑠𝐼 − 𝐴 − 𝑒
−𝑑𝑠
𝐴𝑑𝐾)
−1
𝐵, (10)
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 thefollowing assumption is given.
Assumption 3. Time-delayed singularly perturbed system in(2) is asymptotically stable.
Now the problems to be solved can be summarized asfollows.
Problem 4. For the continuous time-delay system in (2), finda full-order linear filtering (6), such that the filtering errorsystem in (7) satisfies the following conditions.
(1) The filtering error system in (7) is asymptoticallystable.
(2) Given the appropriate positive real 𝛾, under the zeroinitial condition, the following finite frequency indexis satisfied:
sup𝜎max [𝐺𝑒 (𝑗𝜔)] < 𝛾, ∀𝜔 ∈ Ω. (11)
To conclude this section, we give the following technicallemma that plays an instrumental role in deriving our results.
Lemma 5 ((projection lemma) [26]). Let 𝑋, 𝑍, and Σ begiven. There exists a matrix 𝑌 satisfying
Sym(𝑋𝑇𝑌𝑍) + Σ < 0 (12)
if and only if the following projection inequalities are satisfied:
𝑁𝑇
𝑋Σ𝑁𝑋
< 0, 𝑁𝑇
𝑍Σ𝑁𝑍
< 0. (13)
3. Main Results
3.1. FF 𝐻∞
Filtering Performance Analysis. To ensure theasymptotic stability and FF specification in (11) for thefiltering error system, we need to resort to the generalizedKYP lemma in [16]. Based on this, the following lemma canbe obtained.
Lemma 6. (i) Given system in (2) and scalars 𝛾 > 0, 𝜀 > 0,𝑑 > 0, the filtering error system in (7) is asymptotically stablefor all 𝜔 ∈ Ω and satisfies the specifications in (11) if there existmatrices 𝑃
𝜀∈ 𝑅2𝑛𝑥×2𝑛𝑥 with the form of (16), 0 < 𝑃
11∈ 𝑅𝑛1×𝑛1 ,
0 < 𝑃13
∈ 𝑅𝑛2×𝑛2 , 𝑃2
∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑃3
∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑂𝜀
∈ 𝑅2𝑛𝑥×2𝑛𝑥
with the form of (17), 0 < 𝑂11
∈ 𝑅𝑛1×𝑛1 , 0 < 𝑂
13∈ 𝑅𝑛2×𝑛2 ,
𝑂2∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑂3∈ 𝑅𝑛𝑥×𝑛𝑥 such that 𝐸
𝜀𝑃1𝜀
> 0, 𝐸𝜀𝑂1𝜀
> 0, andmatrices 0 < 𝑄 ∈ 𝑅2𝑛𝑥×2𝑛𝑥 , 𝑅
1∈ 𝑅𝑛𝑥×𝑛𝑥 , and 0 ≤ 𝑅
2∈ 𝑅𝑛𝑥×𝑛𝑥
that satisfy
𝐹𝑇
0Ξ0𝐹0+ 𝐹𝑇
1Ξ1𝐹1+ 𝐹𝑇
2(Φ ⊗ 𝑃
𝜀+ Ψ ⊗ 𝑄)𝐹
2< 0, (14)
𝐹𝑇
3Ξ2𝐹3+ 𝐹𝑇
4(Φ ⊗ 𝑂
𝜀) 𝐹4< 0, (15)
where
𝑃𝜀= [
𝑃1𝜀
0
𝑃2
𝑃3
] , 𝑃1𝜀
= [
𝑃11
𝜀𝑃𝑇
12
𝑃12
𝑃13
] , (16)
𝑂𝜀= [
𝑂1𝜀
0
𝑂2
𝑂3
] , 𝑂1𝜀
= [
𝑂11
𝜀𝑂𝑇
12
𝑂12
𝑂13
] , (17)
𝐹0= [
𝐶 𝐶𝑑
0
0 0 𝐼
] , 𝐹1=
[
[
[
[
𝐴 𝐴𝑑
𝐵
𝐼 0 0
0 𝐼 0
]
]
]
]
,
𝐹2= [
𝐴 𝐴𝑑
𝐵
𝐼 0 0
] , 𝐹3=
[
[
[
[
𝐴 𝐴𝑑
𝐼 0
0 𝐼
]
]
]
]
,
𝐹4= [
𝐴 𝐴𝑑
𝐼 0
] , Ξ0≅ [
𝐼 0
0 −𝛾2
𝐼
] ,
Ξ𝑖= diag {0 𝑅
𝑖−𝑅𝑖} (𝑖 = 1, 2) .
(18)
4 Mathematical Problems in Engineering
(ii) Given 𝑑 > 0, if there exist matrices 𝑃0∈ 𝑅2𝑛𝑥×2𝑛𝑥 with
the form of (16) when 𝜀 = 0, 0 < 𝑃11
∈ 𝑅𝑛1×𝑛1 , 0 < 𝑃
13∈
𝑅𝑛2×𝑛2 , 𝑃2∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑃3∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑂0∈ 𝑅2𝑛𝑥×2𝑛𝑥 with the form
of (17) when 𝜀 = 0, 0 < 𝑂11
∈ 𝑅𝑛1×𝑛1 , 0 < 𝑂
13∈ 𝑅𝑛2×𝑛2 ,
𝑂2∈ 𝑅𝑛𝑥×𝑛𝑥 , 𝑂3∈ 𝑅𝑛𝑥×𝑛𝑥 , 0 < 𝑄 ∈ 𝑅2𝑛𝑥×2𝑛𝑥 , 𝑅
1∈ 𝑅𝑛𝑥×𝑛𝑥 , and
0 ≤ 𝑅2∈ 𝑅𝑛𝑥×𝑛𝑥 such that (14), (15) are feasible for 𝜀 = 0 then
filtering error system (7) is asymptotic stable and satisfies thespecifications in (11) for all small enough 𝜀 > 0 and 0 ≤ 𝑑 ≤ 𝑑.
Proof. (i) Let 𝜉(𝑡) =̂ [𝑥(𝑡)
𝑥(𝑡−𝑑)
𝜔(𝑡)
].Give the following Lyapunov-Krasovskii functional
𝑉1(𝑡) =̂ 𝑉
1,1(𝑡) + 𝑉
1,2(𝑡) , (19)
where
𝑉1,1
(𝑡) = 𝑥𝑇
(𝑡) 𝐸𝜀𝑃𝜀𝑥 (𝑡) ,
𝑉1,2
(𝑡) = ∫
𝑡
𝑡−𝑑
𝑥𝑇
(𝜂) 𝑅1𝑥 (𝜂) 𝑑𝜂,
�̇�1,1
=̇
𝑥
𝑇
𝐸𝜀𝑃𝜀𝑥 (𝑡) + 𝑥
𝑇
(𝑡) 𝑃𝑇
𝜀𝐸𝜀
̇𝑥 (𝑡)
= (𝐴𝑥 (𝑡) + 𝐴𝑑𝐾𝑥 (𝑡 − 𝑑) + 𝐵𝜔 (𝑡))
𝑇
⋅ 𝑃𝜀𝑥 (𝑡) + 𝑥
𝑇
(𝑡)
⋅ 𝑃𝑇
𝜀(𝐴𝑥 (𝑡) + 𝐴
𝑑𝐾𝑥 (𝑡 − 𝑑) + 𝐵𝜔 (𝑡)) ,
�̇�1,2
= 𝑥𝑇
(𝑡) 𝑅1𝑥 (𝑡) − 𝑥
𝑇
(𝑡 − 𝑑) 𝑅1𝑥 (𝑡 − 𝑑) .
(20)
Then
�̇�1= �̇�1,1
(𝑡) + �̇�1,2
(𝑡)
= 𝜉 (𝑡)𝑇
[𝐹𝑇
2(Φ ⊗ 𝑃
𝜀) 𝐹2+ 𝐹𝑇
1Ξ1𝐹1] 𝜉 (𝑡) .
(21)
Define the following performance index:
𝐽 = ∫
∞
0
[𝑒𝑇
(𝑡) 𝑒 (𝑡) − 𝛾2
𝜔𝑇
(𝑡) 𝜔 (𝑡)] 𝑑𝑡. (22)
Using the zero initial conditions, we could get
𝐽 ≤ ∫
∞
0
[𝑒𝑇
(𝑡) 𝑒 (𝑡) − 𝛾2
𝜔𝑇
(𝑡) 𝜔 (𝑡)] 𝑑𝑡 + 𝑉1(∞) − 𝑉
1(0)
= ∫
∞
0
𝜉𝑇
(𝑡) [𝐹𝑇
0Ξ0𝐹0+ 𝐹𝑇
1Ξ1𝐹1+ 𝐹𝑇
2(Φ ⊗ 𝑃
𝜀) 𝐹2] 𝜉 (𝑡) .
(23)
LetΘ = 𝐹𝑇0Ξ0𝐹0+𝐹𝑇
1Ξ1𝐹1+𝐹𝑇
2(Φ⊗𝑃
𝜀)𝐹2and use the Parseval
equality to get
∫
∞
0
𝜉𝑇
(𝑡) Θ𝜉 (𝑡) 𝑑𝑡 =
1
2𝜋
∫
∞
−∞
𝜉𝑇
𝑠Θ𝜉𝑠𝑑𝜔. (24)
Considering frequency 𝜔 ∈ Ω and combining (23) and (24),we know that 𝜉𝑇
𝑠Θ𝜉𝑠< 0 is a sufficient condition of 𝐽 < 0, for
all 𝜔 ∈ Ω. Additionally, due to Γ𝜆𝐹2𝜉𝑠= 0, we know that 𝜉
𝑠is
a zero space of Γ𝜆𝐹2.
By the zero space theory, we know that
𝑁𝑇
Γ𝜆𝐹2
Θ𝑁Γ𝜆𝐹2
< 0 ⇒ 𝜉𝑇
𝑠Θ𝜉𝑠< 0. (25)
Then by generalized KYP lemma in [16], the sufficientcondition for 𝑁𝑇
Γ𝜆𝐹2
Θ𝑁Γ𝜆𝐹2
< 0 is existing 𝑃 ∈ 𝑅2𝑛𝑥×2𝑛𝑥 , 0 <𝑄 ∈ 𝑅
2𝑛𝑥×2𝑛𝑥 , such that the following inequality is satisfied:
Θ + 𝐹𝑇
2(Φ ⊗ 𝑃 + Ψ ⊗ 𝑄)𝐹
2< 0. (26)
By redefining 𝑃𝜀+ 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:
𝑉2(𝑡) =̂ 𝑉
2,1(𝑡) + 𝑉
2,2(𝑡) , (27)
where 𝑉2,1
(𝑡) = 𝑥𝑇
(𝑡)𝐸𝜀𝑂𝜀𝑥(𝑡) and 𝑉
2,2(𝑡) =
∫
𝑡
𝑡−𝑑
𝑥𝑇
(𝜂)𝑅2𝑥(𝜂)𝑑𝜂.
Then similar to the proof of the first part of (i), it couldbe shown that inequality (15) can guarantee the asymptoticstability of (7).
(ii) If (14), (15) are feasible for 𝜀 = 0, then they arefeasible for all small enough 𝜀 > 0 and thus, due to (i),filtering error system in (7) is asymptotic stable and satisfiesthe specifications in (11) for these values 𝜀 > 0. Furthermore,linear matrix inequalities (14), (15) are convex with respect to𝑑; hence they are feasible for some 𝑑; then they are feasiblefor all 0 ≤ 𝑑 ≤ 𝑑.
Remark 7. Though the essence of the proof of Lemma 6follows from that of Theorem 1 in [6], there are also somedifferences in Lemma 6. First, in Lemma 6, time-delayedsingularly perturbed systems are considered,which are totallydifferent from the regular systems. Since the singularlyperturbed systems have perturbed parameter 𝜀, the existenceof 𝜀 can lead to the ill-conditioned numerical problems, sohere comes part (ii) of Lemma 6. Furthermore, consideringthe special structure of singularly perturbed systems, notethat, in the proof of part (i), the selected Lyapunov-Krasovskiifunctionals are different from the regular systems.
To facilitate and reduce the conservatism of the filterdesign using the projection lemma, we present an alternativeof Lemma 6.
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∈ 𝑅2𝑛𝑥×2𝑛𝑥 with the form of (16), 𝑂
0∈
𝑅2𝑛𝑥×2𝑛𝑥 with the form of (17), 0 < 𝑄 ∈ 𝑅2𝑛𝑥×2𝑛𝑥 , 𝑅
1∈ 𝑅𝑛𝑥×𝑛𝑥 ,
and 0 ≤ 𝑅2∈ 𝑅𝑛𝑥×𝑛𝑥 and 𝑌
𝑖∈ 𝑅4𝑛𝑥×2𝑛𝑥(𝑖 = 1, 2) satisfying
[
[
−𝐼𝑛𝑧
[0𝑛𝑧×2𝑛𝑥
𝐶 𝐶𝑑
0]
∗ diag {Ξ1+ Ξ (𝑃
0) + Ξ (𝑄) , −𝛾
2
𝐼𝑛𝜔
} + Sym(𝑋𝑇𝑌1𝑍1)
]
]
< 0,
(28)
Ξ2+ Ξ (𝑂
0) + Sym(𝑋𝑇
2𝑌2𝑍2) < 0, (29)
Mathematical Problems in Engineering 5
with
Ξ (𝑃0) = [
Φ𝑐⊗ 𝑃0
04𝑛𝑥×𝑛𝑥
0𝑛𝑥×4𝑛𝑥
0𝑛𝑥×𝑛𝑥
] ,
Ξ (𝑂0) = [
Φ𝑐⊗ 𝑂0
04𝑛𝑥×𝑛𝑥
0𝑛𝑥×4𝑛𝑥
0𝑛𝑥×𝑛𝑥
] ,
Ξ (𝑄) = [
Ψ𝑐⊗ 𝑄 0
4𝑛𝑥×𝑛𝑥
0𝑛𝑥×4𝑛𝑥
0𝑛𝑥×𝑛𝑥
] ,
𝑋1= ⌊𝐼4𝑛𝑥
04𝑛𝑥×(𝑛𝑥+𝑛𝜔)⌋ , 𝑍
1= [−𝐼2𝑛𝑥
𝐴 𝐴𝑑
𝐵] ,
𝑋2= ⌊𝐼4𝑛𝑥
04𝑛𝑥×𝑛𝑥⌋ , 𝑍
2= [−𝐼2𝑛𝑥
𝐴 𝐴𝑑] ;
(30)
the other notations are defined in (14) and (15).
Proof. Define𝑀 = [0𝑛𝑥×2𝑛𝑥
𝐶 𝐶𝑑
0]. By the Schur comple-ment, (28) is equivalent to
diag {Ξ1+ Ξ (𝑃
0) + Ξ (𝑄) , −𝛾
2
𝐼𝑛𝜔
}
+ 𝑀𝑇
𝑀 + Sym (𝑋𝑇1𝑌1𝑍1) < 0.
(31)
By the definition of matrices 𝑋1and 𝑍
1, one can choose
𝑁𝑋1
= 0 and 𝑁𝑍1
=[
[
𝐴 𝐴𝑑𝐵
𝐼2𝑛𝑥0 0
0 𝐼𝑛𝑥0
0 0 𝐼𝑛𝜔
]
]
; thus the first inequality
in (13) vanishes and then by Lemma 5, (31) is equivalent to
𝑁∗
𝑍1
(diag {Ξ1+ Ξ (𝑃
0) + Ξ (𝑄) , −𝛾
2
𝐼𝑛𝜔
} + 𝑀∗
𝑀)𝑁𝑍1
< 0.
(32)
By calculation, we can obtain (32) is equivalent to (14) when𝜀 = 0.
Similarly, by introducing the following null space,
𝑁𝑋2
= 0, 𝑁𝑍2
=
[
[
[
[
𝐴 𝐴𝑑
𝐼2𝑛𝑥
0
0 𝐼𝑛𝑥
]
]
]
]
. (33)
Using projection lemma, (29) is equivalent to the followinginequality:
𝑁𝑇
𝑍2
{Ξ2+ Ξ (𝑂
0) + Sym (𝑋𝑇
2𝑌2𝑍2)}𝑁𝑍2
< 0. (34)
By calculation, (34) is equivalent to (15) when 𝜀 = 0.Thus, Lemma 8 is equivalent to Lemma 6.
3.2. Design of FF 𝐻∞
Filters. Lemma 8 does not give asolution to filter realization explicitly. Based on the result inthe following section we focus on developing methods fordesigning FF 𝐻
∞filters. The following result can be derived
via specifying the structure of the slack matrices 𝑌1and 𝑌
2in
Lemma 8.Theorem9. Given time-delayed systems in (2) and scalars 𝛾 >0, 𝑑 > 0, filter in (6) exists such that the filtering error systemin (7) is asymptotically stable and satisfies the specifications in(11) if there exist matrices 𝑃
0∈ 𝑅2𝑛𝑥×2𝑛𝑥 with the form of (16),
𝑂0
∈ 𝑅2𝑛𝑥×2𝑛𝑥 with the form of (17), 0 < 𝑄 ∈ 𝑅2𝑛𝑥×2𝑛𝑥 , 𝑅
1∈
𝑅𝑛𝑥×𝑛𝑥 , 0 ≤ 𝑅
2∈ 𝑅𝑛𝑥×𝑛𝑥 , Γ𝑖,𝑗
∈ 𝑅𝑛𝑥×𝑛𝑥(𝑖 = 1, 2, 𝑗 = 1, . . . , 4),
Γ5
∈ 𝑅𝑛𝑥×𝑛𝑥 , Δ1
∈ 𝑅𝑛𝑥×𝑛𝑥 , Δ2
∈ 𝑅𝑛𝑥×𝑛𝑦 , Δ3
∈ 𝑅𝑛𝑧×𝑛𝑥 , Δ4
∈
𝑅𝑛𝑧×𝑛𝑦 such that the followingmatrices are satisfied for the given
scalars 𝜅𝑖(𝑖 = 1, . . . , 4):
[
−𝐼𝑛𝑧
Σ
∗ diag {Ξ1+ Ξ (𝑃
0) + Ξ (𝑄) , −𝛾
2
𝐼𝑛𝜔
} + Sym(Λ1)
]
< 0,
Ξ2+ Ξ (𝑂
0) + Sym(Λ
2) < 0,
(35)
where
Σ =̂ ⌊0𝑛𝑧×2𝑛𝑥
𝐺 − Δ4𝐶 −Δ
3𝐺𝑑− Δ4𝐶𝑑
0⌋ ,
Λ1=̂
[
[
[
[
[
[
[
[
[
−Γ1,1
−Γ5
Γ1,1
𝐴 + Δ2𝐶 Δ
1Γ1,1
𝐴𝑑+ Δ2𝐶𝑑
Γ1,1
𝐵
−Γ1,2
−Γ5
Γ1,2
𝐴 + Δ2𝐶 Δ
1Γ1,2
𝐴𝑑+ Δ2𝐶𝑑
Γ1,2
𝐵
−Γ1,3
−𝜅1Γ5
Γ1,3
𝐴 + 𝜅1Δ2𝐶 𝜅1Δ1
Γ1,3
𝐴𝑑+ 𝜅1Δ2𝐶𝑑
Γ1,3
𝐵
−Γ1,4
−𝜅2Γ5
Γ1,4
𝐴 + 𝜅2Δ2𝐶 𝜅2Δ1
Γ1,4
𝐴𝑑+ 𝜅2Δ2𝐶𝑑
Γ1,4
𝐵
0 0 0 0 0 0
]
]
]
]
]
]
]
]
]
,
Λ2=̂
[
[
[
[
[
[
[
[
[
−Γ2,1
−Γ5
Γ2,1
𝐴 + Δ2𝐶 Δ
1Γ2,1
𝐴𝑑+ Δ2𝐶𝑑
−Γ2,2
−Γ5
Γ2,2
𝐴 + Δ2𝐶 Δ
1Γ2,2
𝐴𝑑+ Δ2𝐶𝑑
−Γ2,3
−𝜅3Γ5
Γ2,3
𝐴 + 𝜅3Δ2𝐶 𝜅3Δ1
Γ2,3
𝐴𝑑+ 𝜅3Δ2𝐶𝑑
−Γ2,4
−𝜅4Γ5
Γ2,4
𝐴 + 𝜅4Δ2𝐶 𝜅4Δ1
Γ2,4
𝐴𝑑+ 𝜅4Δ2𝐶𝑑
0 0 0 0 0
]
]
]
]
]
]
]
]
]
.
(36)
6 Mathematical Problems in Engineering
Ξ𝑖, Ξ(𝑃0), Ξ(𝑄), and Ξ(𝑂
0) are defined in (28) and (29).
Moreover, if the previous conditions are satisfied, an acceptablestate-space realization of the filter in (6) is given by
[
𝐴𝐹
𝐵𝐹
𝐶𝐹
𝐷𝐹
] = [
Γ−1
50
0 𝐼
] [
Δ1
Δ2
Δ3
Δ4
] . (37)
Proof. First, in order to prove Theorem 9, we just need toprove that (28) and (29) in Lemma 8 can be deduced from(35) inTheorem 9. It is noted that the slack matrix 𝑌
2has the
following form:
𝑌2=
[
[
[
[
[
[
Γ2,1
Γ5
Γ2,2
Γ6
Γ2,3
Γ7
Γ2,4
Γ8
]
]
]
]
]
]
. (38)
Here, Γ2,𝑖
(𝑖 = 1, . . . , 4), Γ𝑗(𝑗 = 5, . . . , 8) are complexmatrices
with dimension 𝑛𝑥× 𝑛𝑥. In fact, Γ
6is nonsingular and can be
implied from (24); then by multiplying 𝑌2from the left side
and the right side, respectively, with 𝐼1= diag {𝐼 Γ
5Γ−1
6𝐼 𝐼}
and 𝐼2= diag {𝐼 (Γ
5Γ−1
6)𝑇
}, we could get
𝐼1𝑌2𝐼2=
[
[
[
[
[
[
[
[
[
Γ2,1
Γ5(Γ5Γ−1
6)
𝑇
Γ5Γ−1
6Γ2,2
Γ6(Γ5Γ−1
6)
𝑇
Γ2,3
Γ7(Γ5Γ−1
6)
𝑇
Γ2,4
Γ8(Γ5Γ−1
6)
𝑇
]
]
]
]
]
]
]
]
]
. (39)
Without loss of generality, we could restrict Γ5≡ Γ6.
On the other hand, to overcome the difficulty of filteringdesign, more work should be done. Next, we will consider Γ
7
and Γ8to be linearly Γ
5-dependent; that is, Γ
7= 𝜅3Γ5, Γ8
=
𝜅4Γ5, 𝜅3and 𝜅
4are scalars, respectively. Similarly, the slack
matrix 𝑌1has the same structure restriction; that is,
𝑌1=
[
[
[
[
[
[
Γ1,1
Γ5
Γ1,2
Γ5
Γ1,3
𝜅1Γ5
Γ1,4
𝜅2Γ5
]
]
]
]
]
]
, 𝑌2=
[
[
[
[
[
[
Γ2,1
Γ5
Γ2,2
Γ5
Γ2,3
𝜅3Γ5
Γ2,4
𝜅4Γ5
]
]
]
]
]
]
. (40)
Define
[
Δ1
Δ2
Δ3
Δ4
] = [
Γ5
0
0 𝐼
][
𝐴𝑓
𝐵𝑓
𝐶𝑓
𝐷𝑓
] . (41)
By substituting 𝑋𝑖, 𝑍𝑖in (28) and (29) and 𝑌
𝑖of (40) into
𝑋𝑇
𝑖𝑌𝑖𝑍𝑖, one can obtain Λ
𝑖= 𝑋𝑇
𝑖𝑌𝑖𝑍𝑖(𝑖 = 1, 2). Moreover,
by using (41), one can also obtain Σ ≡ 𝑀. The proof iscomplete.
4. Numerical Example
In this section, we use an example to illustrate the effective-ness 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
𝐴 = [
−1 0
0 −1
] , 𝐴𝑑= [
−1 0
1 −1
] , 𝐵 = [
−0.5
2
] ,
𝐶 = [0 1] , 𝐶𝑑= [1 2] , 𝐺 = [2 1] ,
𝐺𝑑= [0 0] , 𝜀 = 0.1, 𝑑 = 0.1.
(42)
Suppose the frequencies 𝜔𝑙= 1, 𝜔
ℎ= 100, we calculate
the achieved minimum performance 𝛾∗ by using Theorem 9in this paper. For brevity, the scalar parameters inTheorem 9are given by 𝜅
1= 𝜅2= 5, 𝜅3= 𝜅4= 1.The obtainedminimum
performance is 𝛾∗ = 0.9976, when 𝑄 = 0; the problembecomes 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
[
𝐴𝐹
𝐵𝐹
𝐶𝐹
𝐷𝐹
] =[
[
[
−1.5861 −0.0552 0
−0.2522 −1.1106 0
0 0 0
]
]
]
. (43)
5. Conclusions
This paper has studied the problem of FF 𝐻∞
filtering fortime-delayed singularly perturbed systems. The frequenciesof the exogenous noise are assumed to reside in a known rect-angular region and the standard 𝐻
∞filtering for singularly
perturbed systems has been extended to the FF𝐻∞case.The
generalized KYP lemma for singularly perturbed systems hasbeen further developed to derive conditions that are moresuitable for FF 𝐻
∞performance synthesis with time-delay.
Via structural restriction for the slack matrices, systematicmethods have been proposed for the design of the filters thatguarantee the asymptotic stability and FF 𝐻
∞disturbance
attenuation level of the filtering error system.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work was supported in part by National Natural Sci-ence Foundation of China under Grants (no. 61304089,no. 51405243), in part by Natural Science Foundation ofJiangsu Province Universities (no. BK20130999), and in partby Natural Science Foundation of Jiangsu Province (no.BK2011826).
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