research article finite frequency filtering for time-delayed...
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Research ArticleFinite Frequency๐ป
โFiltering 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 ยฉ 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) ๐ปโ
filtering for time-delayed singularly perturbed systems. Ourattention is focused on designing filters guaranteeing asymptotic stability and FF ๐ป
โdisturbance attenuation level of the filtering
error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF ๐ปโfilters 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 ๐ป
โfilters 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, โfastโ andโslowโ dynamics. This makes their analysis and control morecomplicated than regular systems. Nevertheless, they havebeen studied extensively [3โ5].
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 [6โ8]. 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 ๐ปโ
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 [12โ15]. 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 [17โ23]. 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
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2 Mathematical Problems in Engineering
Lately, using FF approach to analyze and design controlproblems becomes a new interesting in singularly perturbedsystems [19โ21]. In [20] the author has studied the ๐ป
โ
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 authorโsknowledge, 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 ๐ปโ
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 ๐ป
โfiltering for singularly perturbed systems has
been extended to the FF๐ปโfiltering 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 ๐, its transpose is denoted by ๐๐; ๐
๐
is an arbitrary matrix whose column forms a basis of the nullspace of ๐. Sym(๐) indicates ๐๐ + ๐. ๐max(โ ) denotes maxi-mum singular value of transfer function. diag{โ โ โ } stands fora block-diagonal matrix.
2. Problem Formulation and Preliminaries
The following time-delayed singularly perturbed systems willbe considered in this paper:
๏ฟฝฬ๏ฟฝ1(๐ก) = ๐ด
01๐ฅ1(๐ก) + ๐ด
02๐ฅ2(๐ก) + ๐ด
11๐ฅ1(๐ก โ ๐)
+ ๐ด12๐ฅ2(๐ก โ ๐) + ๐ต
1๐ (๐ก) ,
๐๏ฟฝฬ๏ฟฝ2(๐ก) = ๐ด
03๐ฅ1(๐ก) + ๐ด
04๐ฅ2(๐ก) + ๐ด
13๐ฅ1(๐ก โ ๐)
+ ๐ด14๐ฅ2(๐ก โ ๐) + ๐ต
2๐ (๐ก) ,
๐ฆ (๐ก) = ๐ถ1๐ฅ1(๐ก) + ๐ถ
2๐ฅ2(๐ก) + ๐ถ
3๐ฅ1(๐ก โ ๐) + ๐ถ
4๐ฅ2(๐ก โ ๐) ,
๐ (๐ก) = ๐บ1๐ฅ1(๐ก) + ๐บ
2๐ฅ2(๐ก) + ๐บ
3๐ฅ1(๐ก โ ๐) + ๐บ
4๐ฅ2(๐ก โ ๐) ,
(1)
where ๐ฅ1(๐ก) โ ๐
๐1 is โslowโ state and ๐ฅ
2(๐ก) โ ๐
๐2 is โfastโ
state. Denote ๐๐ฅ
= ๐1+ ๐2; ๐ฅ(๐ก) = [๐ฅ
1(๐ก)๐
๐ฅ2(๐ก)๐
]๐
โ ๐ ๐๐ฅ
is the state vector; ๐ฆ(๐ก) โ ๐ ๐๐ฆ is the measured output signal,๐ง(๐ก) โ ๐
๐๐ง is the signal to be estimated, and ๐(๐ก) โ ๐ ๐๐ is the
noise input signal in the ๐ฟ2[0, +โ) functional space domain.
Time-delay ๐ is known and time-invariantฮฆ(๐ก) is the knowninitial condition in the domain [โ๐, 0]; let ๐ธ
๐= [
๐ผ๐10
0 ๐๐ผ๐2
],๐ด = [
๐ด01๐ด02
๐ด03๐ด04
], ๐ด๐= [๐ด11๐ด12
๐ด13๐ด14
], ๐ต = [ ๐ต1๐ต2
], ๐ถ = [๐ถ1
๐ถ2], ๐ถ๐=
[๐ถ3
๐ถ4], ๐บ = [๐บ
1๐บ2], ๐บ๐
= [๐บ3
๐บ4], ๐ด๐๐
(๐ = 0, 1; ๐ =
1, . . . , 4), ๐ต๐, ๐ถ๐, ๐บ๐, ๐ = 1, 2, ๐ถ
๐, ๐บ๐, ๐ = 3, 4, be appropriate
constant matrices.Then the above system can be regulated as
๐ธ๐๏ฟฝฬ๏ฟฝ = ๐ด๐ฅ (๐ก) + ๐ด
๐๐ฅ (๐ก โ ๐) + ๐ต๐ (๐ก) ,
๐ฆ (๐ก) = ๐ถ๐ฅ (๐ก) + ๐ถ๐๐ฅ (๐ก โ ๐) ,
๐ง (๐ก) = ๐บ๐ฅ (๐ก) + ๐บ๐๐ฅ (๐ก โ ๐) ,
(2)
๐ฅ (๐ก) = ฮฆ (๐ก) โ๐ก = [โ๐, 0] . (3)
First, we give the following assumption on noise signal ๐(๐ก).
Assumption 1. Noise signal ๐(๐ก) is only defined in the low,medium, and high frequency domains
ฮฉ
=ฬ
{{{{
{{{{
{
{๐ โ ๐ | |๐| โฅ ๐โ, ๐โโฅ 0} (high frequency)
{๐ โ ๐ | ๐1โค |๐| โค ๐
2, ๐1โค ๐2} , (media frequency)
{๐ โ ๐ | |๐| โค ๐๐, ๐๐โฅ 0} (low frequency) .
(4)
Remark 2. By the generalized KYP lemma in [16] and by anappropriate choice ฮฆ and ฮจ, the set ๐ can be specialized todefine a certain range of the frequency variable ๐. For thecontinuous time setting, we have ฮฆ = [ 0 1
1 0], ๐ = {๐๐ : ๐ โ
๐ }, whereฮฉ is defined in (4); in this situation,ฮจ can be chosenas
ฮจ =ฬ
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
[
[
1 0
0 ๐2
โ
]
]
,
when ๐ โ {๐ โ ๐ | |๐| โฅ ๐โ, ๐โโฅ 0}
(high frequency)
[
[
โ1 ๐๐๐
โ๐๐๐
โ๐1๐2
]
]
,
when ๐ โ {๐ โ ๐ | ๐1โค |๐| โค ๐
2, ๐1โค ๐2} ,
(media frequency)
[
[
โ1 0
0 ๐2
๐
]
]
,
when ๐ โ {๐ โ ๐ | |๐| โค ๐๐, ๐๐โฅ 0}
(low frequency) ,(5)
where ๐๐:= (๐1+ ๐2)/2.
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Mathematical Problems in Engineering 3
Themain objective of this paper is to design the followingfull-order linear filtering:
๏ฟฝฬ๏ฟฝ๐น(๐ก) = ๐ด
๐น๐ฅ๐น(๐ก) + ๐ต
๐น๐ฆ (๐ก) , ๐ฅ
๐น(0) = 0,
๐ง๐น(๐ก) = ๐ถ
๐น๐ฅ๐น(๐ก) + ๐ท
๐น๐ฆ (๐ก) ,
(6)
where ๐ฅ๐น(๐ก) โ ๐
๐๐ฅ is the state vector, ๐ฆ(๐ก) โ ๐ ๐๐ฆ is the mea-
sured output signal, ๐ง๐น(๐ก) โ ๐
๐๐ง is the output for the filtering
systems, and ๐ด๐น, ๐ต๐น, ๐ถ๐น, ๐ท๐นare the filtering matrices to be
solved. Combine (2) and (6) and let ๐ฅ(๐ก) = [๐ฅ(๐ก)๐ ๐ฅ๐น(๐ก)๐
]๐;
the following filtering error system is obtained:
๐ธ๐
ฬ๐ฅ (๐ก) = ๐ด๐ฅ (๐ก) + ๐ด
๐๐ฅ (๐ก โ ๐) + ๐ต๐ (๐ก) ,
๐ (๐ก) = ๐ถ๐ฅ (๐ก) + ๐ถ๐๐ฅ (๐ก โ ๐) ,
(7)
๐ฅ (๐ก) = [ฮฆ๐
(๐ก) , 0]
๐
โ๐ก = [โ๐, 0] , (8)
where ๐(๐ก) = ๐ง(๐ก) โ ๐ง๐น(๐ก) is the filtering error and ๐ธ
๐=
[
๐ผ๐10 0
0 ๐๐ผ๐20
0 0 ๐ผ๐๐ฅ
]; the filtering system matrices are
๐ด =[
[
[
๐ด01
๐ด02
0
๐ด03
๐ด04
0
๐ต๐น๐ถ1
๐ต๐น๐ถ2
0
]
]
]
,
๐ด๐=
[
[
[
๐ด11
๐ด12
0
๐ด13
๐ด14
0
๐ต๐น๐ถ3
๐ต๐น๐ถ4
0
]
]
]
, ๐ต =[
[
[
๐ต1
๐ต2
0
]
]
]
,
๐ถ = [๐บ1โ ๐ท๐น๐ถ1
๐บ2โ ๐ท๐น๐ถ2
โ๐ถ๐น] ,
๐ถ๐= [๐บ3โ ๐ท๐น๐ถ3
๐บ4โ ๐ท๐น๐ถ4
0] .
(9)
The transfer function of the filtering error system in (7) from๐ to ๐ง is given by
๐บ๐(๐ ) = (๐ถ + ๐
โ๐๐
๐ถ๐๐พ) (๐ธ
๐๐ ๐ผ โ ๐ด โ ๐
โ๐๐
๐ด๐๐พ)
โ1
๐ต, (10)
where โ๐ โ 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 ๐พ, under the zeroinitial condition, the following finite frequency indexis satisfied:
sup๐max [๐บ๐ (๐๐)] < ๐พ, โ๐ โ ฮฉ. (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 ๐, ๐, and ฮฃ begiven. There exists a matrix ๐ satisfying
Sym(๐๐๐๐) + ฮฃ < 0 (12)
if and only if the following projection inequalities are satisfied:
๐๐
๐ฮฃ๐๐
< 0, ๐๐
๐ฮฃ๐๐
< 0. (13)
3. Main Results
3.1. FF ๐ปโ
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 ๐พ > 0, ๐ > 0,๐ > 0, the filtering error system in (7) is asymptotically stablefor all ๐ โ ฮฉ and satisfies the specifications in (11) if there existmatrices ๐
๐โ ๐ 2๐๐ฅร2๐๐ฅ with the form of (16), 0 < ๐
11โ ๐ ๐1ร๐1 ,
0 < ๐13
โ ๐ ๐2ร๐2 , ๐2
โ ๐ ๐๐ฅร๐๐ฅ , ๐3
โ ๐ ๐๐ฅร๐๐ฅ , ๐๐
โ ๐ 2๐๐ฅร2๐๐ฅ
with the form of (17), 0 < ๐11
โ ๐ ๐1ร๐1 , 0 < ๐
13โ ๐ ๐2ร๐2 ,
๐2โ ๐ ๐๐ฅร๐๐ฅ , ๐3โ ๐ ๐๐ฅร๐๐ฅ such that ๐ธ
๐๐1๐
> 0, ๐ธ๐๐1๐
> 0, andmatrices 0 < ๐ โ ๐ 2๐๐ฅร2๐๐ฅ , ๐
1โ ๐ ๐๐ฅร๐๐ฅ , and 0 โค ๐
2โ ๐ ๐๐ฅร๐๐ฅ
that satisfy
๐น๐
0ฮ0๐น0+ ๐น๐
1ฮ1๐น1+ ๐น๐
2(ฮฆ โ ๐
๐+ ฮจ โ ๐)๐น
2< 0, (14)
๐น๐
3ฮ2๐น3+ ๐น๐
4(ฮฆ โ ๐
๐) ๐น4< 0, (15)
where
๐๐= [
๐1๐
0
๐2
๐3
] , ๐1๐
= [
๐11
๐๐๐
12
๐12
๐13
] , (16)
๐๐= [
๐1๐
0
๐2
๐3
] , ๐1๐
= [
๐11
๐๐๐
12
๐12
๐13
] , (17)
๐น0= [
๐ถ ๐ถ๐
0
0 0 ๐ผ
] , ๐น1=
[
[
[
[
๐ด ๐ด๐
๐ต
๐ผ 0 0
0 ๐ผ 0
]
]
]
]
,
๐น2= [
๐ด ๐ด๐
๐ต
๐ผ 0 0
] , ๐น3=
[
[
[
[
๐ด ๐ด๐
๐ผ 0
0 ๐ผ
]
]
]
]
,
๐น4= [
๐ด ๐ด๐
๐ผ 0
] , ฮ0โ [
๐ผ 0
0 โ๐พ2
๐ผ
] ,
ฮ๐= diag {0 ๐
๐โ๐ ๐} (๐ = 1, 2) .
(18)
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4 Mathematical Problems in Engineering
(ii) Given ๐ > 0, if there exist matrices ๐0โ ๐ 2๐๐ฅร2๐๐ฅ with
the form of (16) when ๐ = 0, 0 < ๐11
โ ๐ ๐1ร๐1 , 0 < ๐
13โ
๐ ๐2ร๐2 , ๐2โ ๐ ๐๐ฅร๐๐ฅ , ๐3โ ๐ ๐๐ฅร๐๐ฅ , ๐0โ ๐ 2๐๐ฅร2๐๐ฅ with the form
of (17) when ๐ = 0, 0 < ๐11
โ ๐ ๐1ร๐1 , 0 < ๐
13โ ๐ ๐2ร๐2 ,
๐2โ ๐ ๐๐ฅร๐๐ฅ , ๐3โ ๐ ๐๐ฅร๐๐ฅ , 0 < ๐ โ ๐ 2๐๐ฅร2๐๐ฅ , ๐
1โ ๐ ๐๐ฅร๐๐ฅ , and
0 โค ๐ 2โ ๐ ๐๐ฅร๐๐ฅ such that (14), (15) are feasible for ๐ = 0 then
filtering error system (7) is asymptotic stable and satisfies thespecifications in (11) for all small enough ๐ > 0 and 0 โค ๐ โค ๐.
Proof. (i) Let ๐(๐ก) =ฬ [๐ฅ(๐ก)
๐ฅ(๐กโ๐)
๐(๐ก)
].Give the following Lyapunov-Krasovskii functional
๐1(๐ก) =ฬ ๐
1,1(๐ก) + ๐
1,2(๐ก) , (19)
where
๐1,1
(๐ก) = ๐ฅ๐
(๐ก) ๐ธ๐๐๐๐ฅ (๐ก) ,
๐1,2
(๐ก) = โซ
๐ก
๐กโ๐
๐ฅ๐
(๐) ๐ 1๐ฅ (๐) ๐๐,
๏ฟฝฬ๏ฟฝ1,1
=ฬ
๐ฅ
๐
๐ธ๐๐๐๐ฅ (๐ก) + ๐ฅ
๐
(๐ก) ๐๐
๐๐ธ๐
ฬ๐ฅ (๐ก)
= (๐ด๐ฅ (๐ก) + ๐ด๐๐พ๐ฅ (๐ก โ ๐) + ๐ต๐ (๐ก))
๐
โ ๐๐๐ฅ (๐ก) + ๐ฅ
๐
(๐ก)
โ ๐๐
๐(๐ด๐ฅ (๐ก) + ๐ด
๐๐พ๐ฅ (๐ก โ ๐) + ๐ต๐ (๐ก)) ,
๏ฟฝฬ๏ฟฝ1,2
= ๐ฅ๐
(๐ก) ๐ 1๐ฅ (๐ก) โ ๐ฅ
๐
(๐ก โ ๐) ๐ 1๐ฅ (๐ก โ ๐) .
(20)
Then
๏ฟฝฬ๏ฟฝ1= ๏ฟฝฬ๏ฟฝ1,1
(๐ก) + ๏ฟฝฬ๏ฟฝ1,2
(๐ก)
= ๐ (๐ก)๐
[๐น๐
2(ฮฆ โ ๐
๐) ๐น2+ ๐น๐
1ฮ1๐น1] ๐ (๐ก) .
(21)
Define the following performance index:
๐ฝ = โซ
โ
0
[๐๐
(๐ก) ๐ (๐ก) โ ๐พ2
๐๐
(๐ก) ๐ (๐ก)] ๐๐ก. (22)
Using the zero initial conditions, we could get
๐ฝ โค โซ
โ
0
[๐๐
(๐ก) ๐ (๐ก) โ ๐พ2
๐๐
(๐ก) ๐ (๐ก)] ๐๐ก + ๐1(โ) โ ๐
1(0)
= โซ
โ
0
๐๐
(๐ก) [๐น๐
0ฮ0๐น0+ ๐น๐
1ฮ1๐น1+ ๐น๐
2(ฮฆ โ ๐
๐) ๐น2] ๐ (๐ก) .
(23)
Letฮ = ๐น๐0ฮ0๐น0+๐น๐
1ฮ1๐น1+๐น๐
2(ฮฆโ๐
๐)๐น2and use the Parseval
equality to get
โซ
โ
0
๐๐
(๐ก) ฮ๐ (๐ก) ๐๐ก =
1
2๐
โซ
โ
โโ
๐๐
๐ ฮ๐๐ ๐๐. (24)
Considering frequency ๐ โ ฮฉ and combining (23) and (24),we know that ๐๐
๐ ฮ๐๐ < 0 is a sufficient condition of ๐ฝ < 0, for
all ๐ โ ฮฉ. Additionally, due to ฮ๐๐น2๐๐ = 0, we know that ๐
๐ is
a zero space of ฮ๐๐น2.
By the zero space theory, we know that
๐๐
ฮ๐๐น2
ฮ๐ฮ๐๐น2
< 0 โ ๐๐
๐ ฮ๐๐ < 0. (25)
Then by generalized KYP lemma in [16], the sufficientcondition for ๐๐
ฮ๐๐น2
ฮ๐ฮ๐๐น2
< 0 is existing ๐ โ ๐ 2๐๐ฅร2๐๐ฅ , 0 <๐ โ ๐
2๐๐ฅร2๐๐ฅ , such that the following inequality is satisfied:
ฮ + ๐น๐
2(ฮฆ โ ๐ + ฮจ โ ๐)๐น
2< 0. (26)
By redefining ๐๐+ ๐ as ๐
๐, 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:
๐2(๐ก) =ฬ ๐
2,1(๐ก) + ๐
2,2(๐ก) , (27)
where ๐2,1
(๐ก) = ๐ฅ๐
(๐ก)๐ธ๐๐๐๐ฅ(๐ก) and ๐
2,2(๐ก) =
โซ
๐ก
๐กโ๐
๐ฅ๐
(๐)๐ 2๐ฅ(๐)๐๐.
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 ๐ = 0, then they arefeasible for all small enough ๐ > 0 and thus, due to (i),filtering error system in (7) is asymptotic stable and satisfiesthe specifications in (11) for these values ๐ > 0. Furthermore,linear matrix inequalities (14), (15) are convex with respect to๐; hence they are feasible for some ๐; then they are feasiblefor all 0 โค ๐ โค ๐.
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 ๐, the existenceof ๐ 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 ๐พ > 0,๐ > 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 ๐
0โ ๐ 2๐๐ฅร2๐๐ฅ with the form of (16), ๐
0โ
๐ 2๐๐ฅร2๐๐ฅ with the form of (17), 0 < ๐ โ ๐ 2๐๐ฅร2๐๐ฅ , ๐
1โ ๐ ๐๐ฅร๐๐ฅ ,
and 0 โค ๐ 2โ ๐ ๐๐ฅร๐๐ฅ and ๐
๐โ ๐ 4๐๐ฅร2๐๐ฅ(๐ = 1, 2) satisfying
[
[
โ๐ผ๐๐ง
[0๐๐งร2๐๐ฅ
๐ถ ๐ถ๐
0]
โ diag {ฮ1+ ฮ (๐
0) + ฮ (๐) , โ๐พ
2
๐ผ๐๐
} + Sym(๐๐๐1๐1)
]
]
< 0,
(28)
ฮ2+ ฮ (๐
0) + Sym(๐๐
2๐2๐2) < 0, (29)
-
Mathematical Problems in Engineering 5
with
ฮ (๐0) = [
ฮฆ๐โ ๐0
04๐๐ฅร๐๐ฅ
0๐๐ฅร4๐๐ฅ
0๐๐ฅร๐๐ฅ
] ,
ฮ (๐0) = [
ฮฆ๐โ ๐0
04๐๐ฅร๐๐ฅ
0๐๐ฅร4๐๐ฅ
0๐๐ฅร๐๐ฅ
] ,
ฮ (๐) = [
ฮจ๐โ ๐ 0
4๐๐ฅร๐๐ฅ
0๐๐ฅร4๐๐ฅ
0๐๐ฅร๐๐ฅ
] ,
๐1= โ๐ผ4๐๐ฅ
04๐๐ฅร(๐๐ฅ+๐๐)โ , ๐
1= [โ๐ผ2๐๐ฅ
๐ด ๐ด๐
๐ต] ,
๐2= โ๐ผ4๐๐ฅ
04๐๐ฅร๐๐ฅโ , ๐
2= [โ๐ผ2๐๐ฅ
๐ด ๐ด๐] ;
(30)
the other notations are defined in (14) and (15).
Proof. Define๐ = [0๐๐ฅร2๐๐ฅ
๐ถ ๐ถ๐
0]. By the Schur comple-ment, (28) is equivalent to
diag {ฮ1+ ฮ (๐
0) + ฮ (๐) , โ๐พ
2
๐ผ๐๐
}
+ ๐๐
๐ + Sym (๐๐1๐1๐1) < 0.
(31)
By the definition of matrices ๐1and ๐
1, one can choose
๐๐1
= 0 and ๐๐1
=[
[
๐ด ๐ด๐๐ต
๐ผ2๐๐ฅ0 0
0 ๐ผ๐๐ฅ0
0 0 ๐ผ๐๐
]
]
; thus the first inequality
in (13) vanishes and then by Lemma 5, (31) is equivalent to
๐โ
๐1
(diag {ฮ1+ ฮ (๐
0) + ฮ (๐) , โ๐พ
2
๐ผ๐๐
} + ๐โ
๐)๐๐1
< 0.
(32)
By calculation, we can obtain (32) is equivalent to (14) when๐ = 0.
Similarly, by introducing the following null space,
๐๐2
= 0, ๐๐2
=
[
[
[
[
๐ด ๐ด๐
๐ผ2๐๐ฅ
0
0 ๐ผ๐๐ฅ
]
]
]
]
. (33)
Using projection lemma, (29) is equivalent to the followinginequality:
๐๐
๐2
{ฮ2+ ฮ (๐
0) + Sym (๐๐
2๐2๐2)}๐๐2
< 0. (34)
By calculation, (34) is equivalent to (15) when ๐ = 0.Thus, Lemma 8 is equivalent to Lemma 6.
3.2. Design of FF ๐ปโ
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 ๐ป
โfilters. The following result can be derived
via specifying the structure of the slack matrices ๐1and ๐
2in
Lemma 8.Theorem9. Given time-delayed systems in (2) and scalars ๐พ >0, ๐ > 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 ๐
0โ ๐ 2๐๐ฅร2๐๐ฅ with the form of (16),
๐0
โ ๐ 2๐๐ฅร2๐๐ฅ with the form of (17), 0 < ๐ โ ๐ 2๐๐ฅร2๐๐ฅ , ๐
1โ
๐ ๐๐ฅร๐๐ฅ , 0 โค ๐
2โ ๐ ๐๐ฅร๐๐ฅ , ฮ๐,๐
โ ๐ ๐๐ฅร๐๐ฅ(๐ = 1, 2, ๐ = 1, . . . , 4),
ฮ5
โ ๐ ๐๐ฅร๐๐ฅ , ฮ1
โ ๐ ๐๐ฅร๐๐ฅ , ฮ2
โ ๐ ๐๐ฅร๐๐ฆ , ฮ3
โ ๐ ๐๐งร๐๐ฅ , ฮ4
โ
๐ ๐๐งร๐๐ฆ such that the followingmatrices are satisfied for the given
scalars ๐ ๐(๐ = 1, . . . , 4):
[
โ๐ผ๐๐ง
ฮฃ
โ diag {ฮ1+ ฮ (๐
0) + ฮ (๐) , โ๐พ
2
๐ผ๐๐
} + Sym(ฮ1)
]
< 0,
ฮ2+ ฮ (๐
0) + Sym(ฮ
2) < 0,
(35)
where
ฮฃ =ฬ โ0๐๐งร2๐๐ฅ
๐บ โ ฮ4๐ถ โฮ
3๐บ๐โ ฮ4๐ถ๐
0โ ,
ฮ1=ฬ
[
[
[
[
[
[
[
[
[
โฮ1,1
โฮ5
ฮ1,1
๐ด + ฮ2๐ถ ฮ
1ฮ1,1
๐ด๐+ ฮ2๐ถ๐
ฮ1,1
๐ต
โฮ1,2
โฮ5
ฮ1,2
๐ด + ฮ2๐ถ ฮ
1ฮ1,2
๐ด๐+ ฮ2๐ถ๐
ฮ1,2
๐ต
โฮ1,3
โ๐ 1ฮ5
ฮ1,3
๐ด + ๐ 1ฮ2๐ถ ๐ 1ฮ1
ฮ1,3
๐ด๐+ ๐ 1ฮ2๐ถ๐
ฮ1,3
๐ต
โฮ1,4
โ๐ 2ฮ5
ฮ1,4
๐ด + ๐ 2ฮ2๐ถ ๐ 2ฮ1
ฮ1,4
๐ด๐+ ๐ 2ฮ2๐ถ๐
ฮ1,4
๐ต
0 0 0 0 0 0
]
]
]
]
]
]
]
]
]
,
ฮ2=ฬ
[
[
[
[
[
[
[
[
[
โฮ2,1
โฮ5
ฮ2,1
๐ด + ฮ2๐ถ ฮ
1ฮ2,1
๐ด๐+ ฮ2๐ถ๐
โฮ2,2
โฮ5
ฮ2,2
๐ด + ฮ2๐ถ ฮ
1ฮ2,2
๐ด๐+ ฮ2๐ถ๐
โฮ2,3
โ๐ 3ฮ5
ฮ2,3
๐ด + ๐ 3ฮ2๐ถ ๐ 3ฮ1
ฮ2,3
๐ด๐+ ๐ 3ฮ2๐ถ๐
โฮ2,4
โ๐ 4ฮ5
ฮ2,4
๐ด + ๐ 4ฮ2๐ถ ๐ 4ฮ1
ฮ2,4
๐ด๐+ ๐ 4ฮ2๐ถ๐
0 0 0 0 0
]
]
]
]
]
]
]
]
]
.
(36)
-
6 Mathematical Problems in Engineering
ฮ๐, ฮ(๐0), ฮ(๐), and ฮ(๐
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
[
๐ด๐น
๐ต๐น
๐ถ๐น
๐ท๐น
] = [
ฮโ1
50
0 ๐ผ
] [
ฮ1
ฮ2
ฮ3
ฮ4
] . (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 ๐
2has the
following form:
๐2=
[
[
[
[
[
[
ฮ2,1
ฮ5
ฮ2,2
ฮ6
ฮ2,3
ฮ7
ฮ2,4
ฮ8
]
]
]
]
]
]
. (38)
Here, ฮ2,๐
(๐ = 1, . . . , 4), ฮ๐(๐ = 5, . . . , 8) are complexmatrices
with dimension ๐๐ฅร ๐๐ฅ. In fact, ฮ
6is nonsingular and can be
implied from (24); then by multiplying ๐2from the left side
and the right side, respectively, with ๐ผ1= diag {๐ผ ฮ
5ฮโ1
6๐ผ ๐ผ}
and ๐ผ2= diag {๐ผ (ฮ
5ฮโ1
6)๐
}, we could get
๐ผ1๐2๐ผ2=
[
[
[
[
[
[
[
[
[
ฮ2,1
ฮ5(ฮ5ฮโ1
6)
๐
ฮ5ฮโ1
6ฮ2,2
ฮ6(ฮ5ฮโ1
6)
๐
ฮ2,3
ฮ7(ฮ5ฮโ1
6)
๐
ฮ2,4
ฮ8(ฮ5ฮโ1
6)
๐
]
]
]
]
]
]
]
]
]
. (39)
Without loss of generality, we could restrict ฮ5โก ฮ6.
On the other hand, to overcome the difficulty of filteringdesign, more work should be done. Next, we will consider ฮ
7
and ฮ8to be linearly ฮ
5-dependent; that is, ฮ
7= ๐ 3ฮ5, ฮ8
=
๐ 4ฮ5, ๐ 3and ๐
4are scalars, respectively. Similarly, the slack
matrix ๐1has the same structure restriction; that is,
๐1=
[
[
[
[
[
[
ฮ1,1
ฮ5
ฮ1,2
ฮ5
ฮ1,3
๐ 1ฮ5
ฮ1,4
๐ 2ฮ5
]
]
]
]
]
]
, ๐2=
[
[
[
[
[
[
ฮ2,1
ฮ5
ฮ2,2
ฮ5
ฮ2,3
๐ 3ฮ5
ฮ2,4
๐ 4ฮ5
]
]
]
]
]
]
. (40)
Define
[
ฮ1
ฮ2
ฮ3
ฮ4
] = [
ฮ5
0
0 ๐ผ
][
๐ด๐
๐ต๐
๐ถ๐
๐ท๐
] . (41)
By substituting ๐๐, ๐๐in (28) and (29) and ๐
๐of (40) into
๐๐
๐๐๐๐๐, one can obtain ฮ
๐= ๐๐
๐๐๐๐๐(๐ = 1, 2). Moreover,
by using (41), one can also obtain ฮฃ โก ๐. 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
๐ด = [
โ1 0
0 โ1
] , ๐ด๐= [
โ1 0
1 โ1
] , ๐ต = [
โ0.5
2
] ,
๐ถ = [0 1] , ๐ถ๐= [1 2] , ๐บ = [2 1] ,
๐บ๐= [0 0] , ๐ = 0.1, ๐ = 0.1.
(42)
Suppose the frequencies ๐๐= 1, ๐
โ= 100, we calculate
the achieved minimum performance ๐พโ by using Theorem 9in this paper. For brevity, the scalar parameters inTheorem 9are given by ๐
1= ๐ 2= 5, ๐ 3= ๐ 4= 1.The obtainedminimum
performance is ๐พโ = 0.9976, when ๐ = 0; the problembecomes a standard ๐ป
โfiltering problem and the minimum
performance of the nominal ๐ปโ
filtering is ๐พโ = 1.4533.And the obtained state-space matrices of๐ป
โfiltering are
[
๐ด๐น
๐ต๐น
๐ถ๐น
๐ท๐น
] =[
[
[
โ1.5861 โ0.0552 0
โ0.2522 โ1.1106 0
0 0 0
]
]
]
. (43)
5. Conclusions
This paper has studied the problem of FF ๐ปโ
filtering fortime-delayed singularly perturbed systems. The frequenciesof the exogenous noise are assumed to reside in a known rect-angular region and the standard ๐ป
โfiltering for singularly
perturbed systems has been extended to the FF๐ปโcase.The
generalized KYP lemma for singularly perturbed systems hasbeen further developed to derive conditions that are moresuitable for FF ๐ป
โperformance 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 ๐ป
โdisturbance
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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