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Research ArticleFinite Frequencyπ»
βFiltering 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 njist@sina.com
Received 18 September 2014; Revised 29 January 2015; Accepted 16 February 2015
Academic Editor: Thomas Hanne
Copyright Β© 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) π»β
filtering for time-delayed singularly perturbed systems. Ourattention is focused on designing filters guaranteeing asymptotic stability and FF π»
βdisturbance attenuation level of the filtering
error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF π»βfilters 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 π»
βfilters 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, βfastβ andβslowβ dynamics. This makes their analysis and control morecomplicated than regular systems. Nevertheless, they havebeen studied extensively [3β5].
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 [6β8]. 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 π»β
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 [12β15]. 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 [17β23]. 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 [19β21]. In [20] the author has studied the π»
β
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 authorβsknowledge, 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 π»β
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 π»
βfiltering for singularly perturbed systems has
been extended to the FFπ»βfiltering 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 π, its transpose is denoted by ππ; π
π
is an arbitrary matrix whose column forms a basis of the nullspace of π. Sym(π) indicates ππ + π. πmax(β ) denotes maxi-mum singular value of transfer function. diag{β β β } stands fora block-diagonal matrix.
2. Problem Formulation and Preliminaries
The following time-delayed singularly perturbed systems willbe considered in this paper:
οΏ½ΜοΏ½1(π‘) = π΄
01π₯1(π‘) + π΄
02π₯2(π‘) + π΄
11π₯1(π‘ β π)
+ π΄12π₯2(π‘ β π) + π΅
1π (π‘) ,
ποΏ½ΜοΏ½2(π‘) = π΄
03π₯1(π‘) + π΄
04π₯2(π‘) + π΄
13π₯1(π‘ β π)
+ π΄14π₯2(π‘ β π) + π΅
2π (π‘) ,
π¦ (π‘) = πΆ1π₯1(π‘) + πΆ
2π₯2(π‘) + πΆ
3π₯1(π‘ β π) + πΆ
4π₯2(π‘ β π) ,
π (π‘) = πΊ1π₯1(π‘) + πΊ
2π₯2(π‘) + πΊ
3π₯1(π‘ β π) + πΊ
4π₯2(π‘ β π) ,
(1)
where π₯1(π‘) β π
π1 is βslowβ state and π₯
2(π‘) β π
π2 is βfastβ
state. Denote ππ₯
= π1+ π2; π₯(π‘) = [π₯
1(π‘)π
π₯2(π‘)π
]π
β π ππ₯
is the state vector; π¦(π‘) β π ππ¦ is the measured output signal,π§(π‘) β π
ππ§ is the signal to be estimated, and π(π‘) β π ππ is the
noise input signal in the πΏ2[0, +β) functional space domain.
Time-delay π is known and time-invariantΞ¦(π‘) is the knowninitial condition in the domain [βπ, 0]; let πΈ
π= [
πΌπ10
0 ππΌπ2
],π΄ = [
π΄01π΄02
π΄03π΄04
], π΄π= [π΄11π΄12
π΄13π΄14
], π΅ = [ π΅1π΅2
], πΆ = [πΆ1
πΆ2], πΆπ=
[πΆ3
πΆ4], πΊ = [πΊ
1πΊ2], πΊπ
= [πΊ3
πΊ4], π΄ππ
(π = 0, 1; π =
1, . . . , 4), π΅π, πΆπ, πΊπ, π = 1, 2, πΆ
π, πΊπ, π = 3, 4, be appropriate
constant matrices.Then the above system can be regulated as
πΈποΏ½ΜοΏ½ = π΄π₯ (π‘) + π΄
ππ₯ (π‘ β π) + π΅π (π‘) ,
π¦ (π‘) = πΆπ₯ (π‘) + πΆππ₯ (π‘ β π) ,
π§ (π‘) = πΊπ₯ (π‘) + πΊππ₯ (π‘ β π) ,
(2)
π₯ (π‘) = Ξ¦ (π‘) βπ‘ = [βπ, 0] . (3)
First, we give the following assumption on noise signal π(π‘).
Assumption 1. Noise signal π(π‘) is only defined in the low,medium, and high frequency domains
Ξ©
=Μ
{{{{
{{{{
{
{π β π | |π| β₯ πβ, πββ₯ 0} (high frequency)
{π β π | π1β€ |π| β€ π
2, π1β€ π2} , (media frequency)
{π β π | |π| β€ ππ, ππβ₯ 0} (low frequency) .
(4)
Remark 2. By the generalized KYP lemma in [16] and by anappropriate choice Ξ¦ and Ξ¨, the set π can be specialized todefine a certain range of the frequency variable π. For thecontinuous time setting, we have Ξ¦ = [ 0 1
1 0], π = {ππ : π β
π }, whereΞ© is defined in (4); in this situation,Ξ¨ can be chosenas
Ξ¨ =Μ
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
[
[
1 0
0 π2
β
]
]
,
when π β {π β π | |π| β₯ πβ, πββ₯ 0}
(high frequency)
[
[
β1 πππ
βπππ
βπ1π2
]
]
,
when π β {π β π | π1β€ |π| β€ π
2, π1β€ π2} ,
(media frequency)
[
[
β1 0
0 π2
π
]
]
,
when π β {π β π | |π| β€ ππ, ππβ₯ 0}
(low frequency) ,(5)
where ππ:= (π1+ π2)/2.
Mathematical Problems in Engineering 3
Themain objective of this paper is to design the followingfull-order linear filtering:
οΏ½ΜοΏ½πΉ(π‘) = π΄
πΉπ₯πΉ(π‘) + π΅
πΉπ¦ (π‘) , π₯
πΉ(0) = 0,
π§πΉ(π‘) = πΆ
πΉπ₯πΉ(π‘) + π·
πΉπ¦ (π‘) ,
(6)
where π₯πΉ(π‘) β π
ππ₯ is the state vector, π¦(π‘) β π ππ¦ is the mea-
sured output signal, π§πΉ(π‘) β π
ππ§ is the output for the filtering
systems, and π΄πΉ, π΅πΉ, πΆπΉ, π·πΉare the filtering matrices to be
solved. Combine (2) and (6) and let π₯(π‘) = [π₯(π‘)π π₯πΉ(π‘)π
]π;
the following filtering error system is obtained:
πΈπ
Μπ₯ (π‘) = π΄π₯ (π‘) + π΄
ππ₯ (π‘ β π) + π΅π (π‘) ,
π (π‘) = πΆπ₯ (π‘) + πΆππ₯ (π‘ β π) ,
(7)
π₯ (π‘) = [Ξ¦π
(π‘) , 0]
π
βπ‘ = [βπ, 0] , (8)
where π(π‘) = π§(π‘) β π§πΉ(π‘) is the filtering error and πΈ
π=
[
πΌπ10 0
0 ππΌπ20
0 0 πΌππ₯
]; the filtering system matrices are
π΄ =[
[
[
π΄01
π΄02
0
π΄03
π΄04
0
π΅πΉπΆ1
π΅πΉπΆ2
0
]
]
]
,
π΄π=
[
[
[
π΄11
π΄12
0
π΄13
π΄14
0
π΅πΉπΆ3
π΅πΉπΆ4
0
]
]
]
, π΅ =[
[
[
π΅1
π΅2
0
]
]
]
,
πΆ = [πΊ1β π·πΉπΆ1
πΊ2β π·πΉπΆ2
βπΆπΉ] ,
πΆπ= [πΊ3β π·πΉπΆ3
πΊ4β π·πΉπΆ4
0] .
(9)
The transfer function of the filtering error system in (7) fromπ to π§ is given by
πΊπ(π ) = (πΆ + π
βππ
πΆππΎ) (πΈ
ππ πΌ β π΄ β π
βππ
π΄ππΎ)
β1
π΅, (10)
where βπ β 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 πΎ, under the zeroinitial condition, the following finite frequency indexis satisfied:
supπmax [πΊπ (ππ)] < πΎ, βπ β Ξ©. (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 π, π, and Ξ£ begiven. There exists a matrix π satisfying
Sym(ππππ) + Ξ£ < 0 (12)
if and only if the following projection inequalities are satisfied:
ππ
πΞ£ππ
< 0, ππ
πΞ£ππ
< 0. (13)
3. Main Results
3.1. FF π»β
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 πΎ > 0, π > 0,π > 0, the filtering error system in (7) is asymptotically stablefor all π β Ξ© and satisfies the specifications in (11) if there existmatrices π
πβ π 2ππ₯Γ2ππ₯ with the form of (16), 0 < π
11β π π1Γπ1 ,
0 < π13
β π π2Γπ2 , π2
β π ππ₯Γππ₯ , π3
β π ππ₯Γππ₯ , ππ
β π 2ππ₯Γ2ππ₯
with the form of (17), 0 < π11
β π π1Γπ1 , 0 < π
13β π π2Γπ2 ,
π2β π ππ₯Γππ₯ , π3β π ππ₯Γππ₯ such that πΈ
ππ1π
> 0, πΈππ1π
> 0, andmatrices 0 < π β π 2ππ₯Γ2ππ₯ , π
1β π ππ₯Γππ₯ , and 0 β€ π
2β π ππ₯Γππ₯
that satisfy
πΉπ
0Ξ0πΉ0+ πΉπ
1Ξ1πΉ1+ πΉπ
2(Ξ¦ β π
π+ Ξ¨ β π)πΉ
2< 0, (14)
πΉπ
3Ξ2πΉ3+ πΉπ
4(Ξ¦ β π
π) πΉ4< 0, (15)
where
ππ= [
π1π
0
π2
π3
] , π1π
= [
π11
πππ
12
π12
π13
] , (16)
ππ= [
π1π
0
π2
π3
] , π1π
= [
π11
πππ
12
π12
π13
] , (17)
πΉ0= [
πΆ πΆπ
0
0 0 πΌ
] , πΉ1=
[
[
[
[
π΄ π΄π
π΅
πΌ 0 0
0 πΌ 0
]
]
]
]
,
πΉ2= [
π΄ π΄π
π΅
πΌ 0 0
] , πΉ3=
[
[
[
[
π΄ π΄π
πΌ 0
0 πΌ
]
]
]
]
,
πΉ4= [
π΄ π΄π
πΌ 0
] , Ξ0β [
πΌ 0
0 βπΎ2
πΌ
] ,
Ξπ= diag {0 π
πβπ π} (π = 1, 2) .
(18)
4 Mathematical Problems in Engineering
(ii) Given π > 0, if there exist matrices π0β π 2ππ₯Γ2ππ₯ with
the form of (16) when π = 0, 0 < π11
β π π1Γπ1 , 0 < π
13β
π π2Γπ2 , π2β π ππ₯Γππ₯ , π3β π ππ₯Γππ₯ , π0β π 2ππ₯Γ2ππ₯ with the form
of (17) when π = 0, 0 < π11
β π π1Γπ1 , 0 < π
13β π π2Γπ2 ,
π2β π ππ₯Γππ₯ , π3β π ππ₯Γππ₯ , 0 < π β π 2ππ₯Γ2ππ₯ , π
1β π ππ₯Γππ₯ , and
0 β€ π 2β π ππ₯Γππ₯ such that (14), (15) are feasible for π = 0 then
filtering error system (7) is asymptotic stable and satisfies thespecifications in (11) for all small enough π > 0 and 0 β€ π β€ π.
Proof. (i) Let π(π‘) =Μ [π₯(π‘)
π₯(π‘βπ)
π(π‘)
].Give the following Lyapunov-Krasovskii functional
π1(π‘) =Μ π
1,1(π‘) + π
1,2(π‘) , (19)
where
π1,1
(π‘) = π₯π
(π‘) πΈππππ₯ (π‘) ,
π1,2
(π‘) = β«
π‘
π‘βπ
π₯π
(π) π 1π₯ (π) ππ,
οΏ½ΜοΏ½1,1
=Μ
π₯
π
πΈππππ₯ (π‘) + π₯
π
(π‘) ππ
ππΈπ
Μπ₯ (π‘)
= (π΄π₯ (π‘) + π΄ππΎπ₯ (π‘ β π) + π΅π (π‘))
π
β πππ₯ (π‘) + π₯
π
(π‘)
β ππ
π(π΄π₯ (π‘) + π΄
ππΎπ₯ (π‘ β π) + π΅π (π‘)) ,
οΏ½ΜοΏ½1,2
= π₯π
(π‘) π 1π₯ (π‘) β π₯
π
(π‘ β π) π 1π₯ (π‘ β π) .
(20)
Then
οΏ½ΜοΏ½1= οΏ½ΜοΏ½1,1
(π‘) + οΏ½ΜοΏ½1,2
(π‘)
= π (π‘)π
[πΉπ
2(Ξ¦ β π
π) πΉ2+ πΉπ
1Ξ1πΉ1] π (π‘) .
(21)
Define the following performance index:
π½ = β«
β
0
[ππ
(π‘) π (π‘) β πΎ2
ππ
(π‘) π (π‘)] ππ‘. (22)
Using the zero initial conditions, we could get
π½ β€ β«
β
0
[ππ
(π‘) π (π‘) β πΎ2
ππ
(π‘) π (π‘)] ππ‘ + π1(β) β π
1(0)
= β«
β
0
ππ
(π‘) [πΉπ
0Ξ0πΉ0+ πΉπ
1Ξ1πΉ1+ πΉπ
2(Ξ¦ β π
π) πΉ2] π (π‘) .
(23)
LetΞ = πΉπ0Ξ0πΉ0+πΉπ
1Ξ1πΉ1+πΉπ
2(Ξ¦βπ
π)πΉ2and use the Parseval
equality to get
β«
β
0
ππ
(π‘) Ξπ (π‘) ππ‘ =
1
2π
β«
β
ββ
ππ
π Ξππ ππ. (24)
Considering frequency π β Ξ© and combining (23) and (24),we know that ππ
π Ξππ < 0 is a sufficient condition of π½ < 0, for
all π β Ξ©. Additionally, due to ΞππΉ2ππ = 0, we know that π
π is
a zero space of ΞππΉ2.
By the zero space theory, we know that
ππ
ΞππΉ2
ΞπΞππΉ2
< 0 β ππ
π Ξππ < 0. (25)
Then by generalized KYP lemma in [16], the sufficientcondition for ππ
ΞππΉ2
ΞπΞππΉ2
< 0 is existing π β π 2ππ₯Γ2ππ₯ , 0 <π β π
2ππ₯Γ2ππ₯ , such that the following inequality is satisfied:
Ξ + πΉπ
2(Ξ¦ β π + Ξ¨ β π)πΉ
2< 0. (26)
By redefining ππ+ π as π
π, 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:
π2(π‘) =Μ π
2,1(π‘) + π
2,2(π‘) , (27)
where π2,1
(π‘) = π₯π
(π‘)πΈππππ₯(π‘) and π
2,2(π‘) =
β«
π‘
π‘βπ
π₯π
(π)π 2π₯(π)ππ.
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 π = 0, then they arefeasible for all small enough π > 0 and thus, due to (i),filtering error system in (7) is asymptotic stable and satisfiesthe specifications in (11) for these values π > 0. Furthermore,linear matrix inequalities (14), (15) are convex with respect toπ; hence they are feasible for some π; then they are feasiblefor all 0 β€ π β€ π.
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 π, the existenceof π 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 πΎ > 0,π > 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 π
0β π 2ππ₯Γ2ππ₯ with the form of (16), π
0β
π 2ππ₯Γ2ππ₯ with the form of (17), 0 < π β π 2ππ₯Γ2ππ₯ , π
1β π ππ₯Γππ₯ ,
and 0 β€ π 2β π ππ₯Γππ₯ and π
πβ π 4ππ₯Γ2ππ₯(π = 1, 2) satisfying
[
[
βπΌππ§
[0ππ§Γ2ππ₯
πΆ πΆπ
0]
β diag {Ξ1+ Ξ (π
0) + Ξ (π) , βπΎ
2
πΌππ
} + Sym(πππ1π1)
]
]
< 0,
(28)
Ξ2+ Ξ (π
0) + Sym(ππ
2π2π2) < 0, (29)
Mathematical Problems in Engineering 5
with
Ξ (π0) = [
Ξ¦πβ π0
04ππ₯Γππ₯
0ππ₯Γ4ππ₯
0ππ₯Γππ₯
] ,
Ξ (π0) = [
Ξ¦πβ π0
04ππ₯Γππ₯
0ππ₯Γ4ππ₯
0ππ₯Γππ₯
] ,
Ξ (π) = [
Ξ¨πβ π 0
4ππ₯Γππ₯
0ππ₯Γ4ππ₯
0ππ₯Γππ₯
] ,
π1= βπΌ4ππ₯
04ππ₯Γ(ππ₯+ππ)β , π
1= [βπΌ2ππ₯
π΄ π΄π
π΅] ,
π2= βπΌ4ππ₯
04ππ₯Γππ₯β , π
2= [βπΌ2ππ₯
π΄ π΄π] ;
(30)
the other notations are defined in (14) and (15).
Proof. Defineπ = [0ππ₯Γ2ππ₯
πΆ πΆπ
0]. By the Schur comple-ment, (28) is equivalent to
diag {Ξ1+ Ξ (π
0) + Ξ (π) , βπΎ
2
πΌππ
}
+ ππ
π + Sym (ππ1π1π1) < 0.
(31)
By the definition of matrices π1and π
1, one can choose
ππ1
= 0 and ππ1
=[
[
π΄ π΄ππ΅
πΌ2ππ₯0 0
0 πΌππ₯0
0 0 πΌππ
]
]
; thus the first inequality
in (13) vanishes and then by Lemma 5, (31) is equivalent to
πβ
π1
(diag {Ξ1+ Ξ (π
0) + Ξ (π) , βπΎ
2
πΌππ
} + πβ
π)ππ1
< 0.
(32)
By calculation, we can obtain (32) is equivalent to (14) whenπ = 0.
Similarly, by introducing the following null space,
ππ2
= 0, ππ2
=
[
[
[
[
π΄ π΄π
πΌ2ππ₯
0
0 πΌππ₯
]
]
]
]
. (33)
Using projection lemma, (29) is equivalent to the followinginequality:
ππ
π2
{Ξ2+ Ξ (π
0) + Sym (ππ
2π2π2)}ππ2
< 0. (34)
By calculation, (34) is equivalent to (15) when π = 0.Thus, Lemma 8 is equivalent to Lemma 6.
3.2. Design of FF π»β
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 π»
βfilters. The following result can be derived
via specifying the structure of the slack matrices π1and π
2in
Lemma 8.Theorem9. Given time-delayed systems in (2) and scalars πΎ >0, π > 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 π
0β π 2ππ₯Γ2ππ₯ with the form of (16),
π0
β π 2ππ₯Γ2ππ₯ with the form of (17), 0 < π β π 2ππ₯Γ2ππ₯ , π
1β
π ππ₯Γππ₯ , 0 β€ π
2β π ππ₯Γππ₯ , Ξπ,π
β π ππ₯Γππ₯(π = 1, 2, π = 1, . . . , 4),
Ξ5
β π ππ₯Γππ₯ , Ξ1
β π ππ₯Γππ₯ , Ξ2
β π ππ₯Γππ¦ , Ξ3
β π ππ§Γππ₯ , Ξ4
β
π ππ§Γππ¦ such that the followingmatrices are satisfied for the given
scalars π π(π = 1, . . . , 4):
[
βπΌππ§
Ξ£
β diag {Ξ1+ Ξ (π
0) + Ξ (π) , βπΎ
2
πΌππ
} + Sym(Ξ1)
]
< 0,
Ξ2+ Ξ (π
0) + Sym(Ξ
2) < 0,
(35)
where
Ξ£ =Μ β0ππ§Γ2ππ₯
πΊ β Ξ4πΆ βΞ
3πΊπβ Ξ4πΆπ
0β ,
Ξ1=Μ
[
[
[
[
[
[
[
[
[
βΞ1,1
βΞ5
Ξ1,1
π΄ + Ξ2πΆ Ξ
1Ξ1,1
π΄π+ Ξ2πΆπ
Ξ1,1
π΅
βΞ1,2
βΞ5
Ξ1,2
π΄ + Ξ2πΆ Ξ
1Ξ1,2
π΄π+ Ξ2πΆπ
Ξ1,2
π΅
βΞ1,3
βπ 1Ξ5
Ξ1,3
π΄ + π 1Ξ2πΆ π 1Ξ1
Ξ1,3
π΄π+ π 1Ξ2πΆπ
Ξ1,3
π΅
βΞ1,4
βπ 2Ξ5
Ξ1,4
π΄ + π 2Ξ2πΆ π 2Ξ1
Ξ1,4
π΄π+ π 2Ξ2πΆπ
Ξ1,4
π΅
0 0 0 0 0 0
]
]
]
]
]
]
]
]
]
,
Ξ2=Μ
[
[
[
[
[
[
[
[
[
βΞ2,1
βΞ5
Ξ2,1
π΄ + Ξ2πΆ Ξ
1Ξ2,1
π΄π+ Ξ2πΆπ
βΞ2,2
βΞ5
Ξ2,2
π΄ + Ξ2πΆ Ξ
1Ξ2,2
π΄π+ Ξ2πΆπ
βΞ2,3
βπ 3Ξ5
Ξ2,3
π΄ + π 3Ξ2πΆ π 3Ξ1
Ξ2,3
π΄π+ π 3Ξ2πΆπ
βΞ2,4
βπ 4Ξ5
Ξ2,4
π΄ + π 4Ξ2πΆ π 4Ξ1
Ξ2,4
π΄π+ π 4Ξ2πΆπ
0 0 0 0 0
]
]
]
]
]
]
]
]
]
.
(36)
6 Mathematical Problems in Engineering
Ξπ, Ξ(π0), Ξ(π), and Ξ(π
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
[
π΄πΉ
π΅πΉ
πΆπΉ
π·πΉ
] = [
Ξβ1
50
0 πΌ
] [
Ξ1
Ξ2
Ξ3
Ξ4
] . (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 π
2has the
following form:
π2=
[
[
[
[
[
[
Ξ2,1
Ξ5
Ξ2,2
Ξ6
Ξ2,3
Ξ7
Ξ2,4
Ξ8
]
]
]
]
]
]
. (38)
Here, Ξ2,π
(π = 1, . . . , 4), Ξπ(π = 5, . . . , 8) are complexmatrices
with dimension ππ₯Γ ππ₯. In fact, Ξ
6is nonsingular and can be
implied from (24); then by multiplying π2from the left side
and the right side, respectively, with πΌ1= diag {πΌ Ξ
5Ξβ1
6πΌ πΌ}
and πΌ2= diag {πΌ (Ξ
5Ξβ1
6)π
}, we could get
πΌ1π2πΌ2=
[
[
[
[
[
[
[
[
[
Ξ2,1
Ξ5(Ξ5Ξβ1
6)
π
Ξ5Ξβ1
6Ξ2,2
Ξ6(Ξ5Ξβ1
6)
π
Ξ2,3
Ξ7(Ξ5Ξβ1
6)
π
Ξ2,4
Ξ8(Ξ5Ξβ1
6)
π
]
]
]
]
]
]
]
]
]
. (39)
Without loss of generality, we could restrict Ξ5β‘ Ξ6.
On the other hand, to overcome the difficulty of filteringdesign, more work should be done. Next, we will consider Ξ
7
and Ξ8to be linearly Ξ
5-dependent; that is, Ξ
7= π 3Ξ5, Ξ8
=
π 4Ξ5, π 3and π
4are scalars, respectively. Similarly, the slack
matrix π1has the same structure restriction; that is,
π1=
[
[
[
[
[
[
Ξ1,1
Ξ5
Ξ1,2
Ξ5
Ξ1,3
π 1Ξ5
Ξ1,4
π 2Ξ5
]
]
]
]
]
]
, π2=
[
[
[
[
[
[
Ξ2,1
Ξ5
Ξ2,2
Ξ5
Ξ2,3
π 3Ξ5
Ξ2,4
π 4Ξ5
]
]
]
]
]
]
. (40)
Define
[
Ξ1
Ξ2
Ξ3
Ξ4
] = [
Ξ5
0
0 πΌ
][
π΄π
π΅π
πΆπ
π·π
] . (41)
By substituting ππ, ππin (28) and (29) and π
πof (40) into
ππ
πππππ, one can obtain Ξ
π= ππ
πππππ(π = 1, 2). Moreover,
by using (41), one can also obtain Ξ£ β‘ π. 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
π΄ = [
β1 0
0 β1
] , π΄π= [
β1 0
1 β1
] , π΅ = [
β0.5
2
] ,
πΆ = [0 1] , πΆπ= [1 2] , πΊ = [2 1] ,
πΊπ= [0 0] , π = 0.1, π = 0.1.
(42)
Suppose the frequencies ππ= 1, π
β= 100, we calculate
the achieved minimum performance πΎβ by using Theorem 9in this paper. For brevity, the scalar parameters inTheorem 9are given by π
1= π 2= 5, π 3= π 4= 1.The obtainedminimum
performance is πΎβ = 0.9976, when π = 0; the problembecomes a standard π»
βfiltering problem and the minimum
performance of the nominal π»β
filtering is πΎβ = 1.4533.And the obtained state-space matrices ofπ»
βfiltering are
[
π΄πΉ
π΅πΉ
πΆπΉ
π·πΉ
] =[
[
[
β1.5861 β0.0552 0
β0.2522 β1.1106 0
0 0 0
]
]
]
. (43)
5. Conclusions
This paper has studied the problem of FF π»β
filtering fortime-delayed singularly perturbed systems. The frequenciesof the exogenous noise are assumed to reside in a known rect-angular region and the standard π»
βfiltering for singularly
perturbed systems has been extended to the FFπ»βcase.The
generalized KYP lemma for singularly perturbed systems hasbeen further developed to derive conditions that are moresuitable for FF π»
βperformance 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 π»
βdisturbance
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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