Research ArticleOn Homogeneous Parameter-Dependent QuadraticLyapunov Function for Robust119867

infinFiltering Design in Switched

Linear Discrete-Time Systems with Polytopic Uncertainties

Guochen Pang and Kanjian Zhang

Key Laboratory of Measurement and Control of CSE Ministry of Education School of Automation Southeast UniversityNanjing 210096 China

Correspondence should be addressed to Guochen Pang guochenpang163com

Received 23 November 2014 Revised 30 January 2015 Accepted 30 January 2015

Academic Editor Pavel Kurasov

Copyright copy 2015 G Pang and K Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is concerned with the problem of robust 119867infin

filter design for switched linear discrete-time systems with polytopicuncertaintiesThe condition of being robustly asymptotically stable for uncertain switched system and less conservative119867

infinnoise-

attenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function Moreover a morefeasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the givenarbitrary switching signal Lastly simulation results are used to illustrate the effectiveness of our method

1 Introduction

Switched systems are a class of hybrid systems that consist ofa finite number of subsystems and a logical rule orchestratingswitching between the subsystems Since this class of systemshas numerous applications in the control of mechanicalsystems the automotive industry aircraft and air trafficcontrol switching power converters and many other fieldsthe problems of stability analysis and control design forswitched systems have receivedwide attention during the pasttwo decades [1ndash15] Reference [2] proposed the 119867

infinweight

learning law for switched Hopfield neural networks withtime-delay under parametric uncertainty Reference [8] dealtwith the delay-dependent exponentially convergent state esti-mation problem for delayed switched neural networks Somecriteria for exponential stability and asymptotic stability ofa class of nonlinear hybrid impulsive and switching systemshave been established using switched Lyapunov functions in[9] Reference [14] investigated the problem of designing aswitching compensator for a plant switching amongst a familyof given configurations

On the other hand within robust control theory schemethe 119867

infinnoise-attenuation level is an important index for

the influence of external disturbance on system stability [16ndash19] However the uncertainties which generally exist in manypractical plants and environments may result in significantchanges in robust 119867

infinnoise-attenuation level In order to

suppress the conservativeness many newmethods have beenconsidered Among these methods homogeneous polyno-mial parameter-dependent quadratic Lyapunov function isone of the most effective methods The main feature ofthese functions is that they are quadratic Lyapunov functionswhose dependence on the uncertain parameters is expressedas a polynomial homogeneous form It is firstly introducedto study robust stability of polynomial systems in [20] Mostresults have been presented in [21ndash25] In [19] homogeneousparameter-dependent quadratic Lyapunov functions wereused to establish tightness in robust 119867

infinanalysis Reference

[21] presented some general results concerning the exis-tence of homogeneous polynomial solutions to parameter-dependent linear matrix inequalities whose coefficients arecontinuous functions of parameters lying in the unit simplexReference [23] investigated the problems of checking robuststability and evaluating robust 119867

2performance of uncertain

continuous-time linear systems with time-invariant parame-ters lying in polytropic domains Reference [25] introduced

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 271765 11 pageshttpdxdoiorg1011552015271765

2 Advances in Mathematical Physics

Gram-tight forms that is forms whose minimum coincideswith the lower bound provided by LMI optimizations basedon SOS (sum of squares of polynomials) relaxations

Thus in order to suppress the influence of uncertain-ties on the systemrsquos robust 119867

infincontrol the homogeneous

parameter-dependent quadratic Lyapunov functions are usedto design robust119867

infinfilter in this paper We first consider the

systemrsquos anti-interference of external disturbance Along thisdirection the robust 119867

infinfilters for switched linear discrete-

time systems are designed Lastly through the comparisonwe have the fact that homogeneous parameter-dependentquadratic Lyapunov functions can suppress conservativenesswhich the uncertainties bring

The rest of this paper is organised as follows We statethe problem formulation in Section 2 The main results arepresented in Section 3 Section 4 illustrates the obtainedresult by numerical examples which is followed by theconclusion in Section 5

Notation R119899 denotes the 119899-dimension Euclidean space andR119899times119898 is the real matrices with dimension 119899 times 119898 R119899

0means

R1198990 the notation119883 ge 119884 (resp119883 gt 119884) where119883 and119884 aresymmetric matrices represents the fact that the matrix119883minus119884is positive semidefinite (resp positive definite) 119860119879 denotesthe transposedmatrix of119860 119904119902(119909) represents (1199092

1 119909

2

119899)with

119909 isin R119899 ℎ119890(119883)means119883+119883119879 with119883 isin R119899times119899 119909otimes119910 denotes theKronecker product of vectors 119909 and 119910 sdot denotes Euclideannorm for vector or the spectral norm of matrices

2 Problem Statement

Consider a class of uncertain switched linear discrete-timesystems which were given in [1]

119909 (119896 + 1) = 119860119894(120582) 119909 (119896) + 119861

119894(120582) 120596 (119896)

119910 (119896) = 119862119894(120582) 119909 (119896) + 119863

119894(120582) 120596 (119896)

119911 (119896) = 119867119894(120582) 119909 (119896) + 119871

119894(120582) 120596 (119896)

(1)

where 119909(119896) isin R119899 is state vector120596(119896) isin R119897 is disturbance inputwhich belongs to 119897

2[0 +infin) 119910(119896) is the measurement output

119911(119896) is objective signal to be attenuated and 119894 is switchingrule which takes its value in the finite set Π = 1 119873As in [1] the switching signal 119894 is unknown a priori but itsinstantaneous value is available in real time As an arbitrarydiscrete time 119896 the switching signal 119894 is dependent on 119896or 119909(119896) or both or other switching rules 120582 is an uncertainparameter vector supposed to satisfy 120582 isin Λ = 120582

119895ge 0sum

119904

119895=1=

1 The vector 120582 represents the time-invariant parametricuncertainty which affects linearly the system dynamics Thevector 120582 can take any value inΛ but it is known to be constantin time

The matrices of each subsystem have appropriate dimen-sions and are assumed to belong to a given convex-boundedpolyhedral domain described by 119904 vertices in the 119894th subsys-tem that is

(119860119894(120582) 119861

119894(120582) 119862

119894(120582) 119863

119894(120582) 119867

119894(120582) 119871

119894(120582)) isin Γ

119894 (2)

where

Γ119894=

119860119894(120582) 119861

119894(120582) 119862

119894(120582) 119863

119894(120582) 119867

119894(120582) 119871

119894(120582)

=

119904

sum

119895=1

120582119895(119860119894119895119861119894119895119862119894119895119863119894119895119867119894119895119871119894119895)

120582119895ge 0

119904

sum

119895=1

120582119895= 1

(3)

Hence we are interested in designing an estimator or filterof the form

119909119891(119896 + 1) = 119860

119891119894119909119891(119896) + 119861

119891119894119910 (119896)

119911119891(119896) = 119862

119891119894119909119891(119896) + 119863

119891119894119910 (119896)

(4)

where 119909119891(119896) isin R119899 119911

119891(119896) isin R119902 119860

119894119895 119861119894119895 119862119894119895and 119863

119894119895are

the parameterized filter matrices to be determined The filterwith the above structure may be called switched linear filterin which the switching signal 119894 is also assumed unknown apriori but available in real-time and homogeneous with theswitching signal in system (1)

Augmenting the model of (1) to include the state of filter(4) we obtain the filtering error system

119909119890(119896 + 1) = 119860

119890119894(120582) 119909119890(119896) + 119861

119890119894(120582) 120596 (119896)

119890 (119896) = 119862119890119894(120582) 119909119890(119896) + 119863

119890119894(120582) 120596 (119896)

(5)

where

119909119890(119896) = (119909 (119896) 119909119891 (

119896))

119879

119890 (119896) = 119911 (119896) minus 119911119891(119896)

119860119890119894(120582) = (

119860119894(120582) 0

119861119891119894119862119894(120582) 119860

119891119894

)

119861119890119894(120582) = (119861119894

(120582) 119861119891119894119863119894(120582))

119879

119862119890119894(120582) = (119867119894 (

120582) minus 119863119891119894119862119894(120582) minus119862

119891119894)

119863119890119894(120582) = 119871

119894(120582) minus 119863

119891119894119863119894(120582)

(6)

Based on [1] the robust 119867infin

filtering problem addressedin this paper can be formulated as follows finding a pre-scribed level of noise attention 120574 gt 0 and determining arobust switched linear filter (4) such that the filtering errorsystem is robustly asymptotically stable and

1198902lt 1205742

1205962 (7)

This problem has received a great deal of attention Inorder to get a better level of noise attention 120574 gt 0 wewill use homogeneous polynomial functions which havedemonstrated nonconservative result for serval problemsIn this paper we extend these methods to design robustswitched linear filter

The following preliminaries are given which are essentialfor later developments We firstly recall the homogeneouspolynomial function from Chesi et al [26]

Advances in Mathematical Physics 3

Definition 1 The function ℎ R119899 rarr R is a form of degree 119889in 119899 scalar variables if

ℎ (119909) = sum

119902isin119871119899119904

119886119902119909119902

(8)

where 119871119899119904= 119902 isin 119873

119899

sum119899

119894=1119902119894= 119904 and 119909 isin R119899 and 119886

119902isin R is

coefficient of the monomial 119909119902

The set of forms of degree 119904 in 119899 scalar variables is definedas

Ξ119899119904= ℎ R119899 997888rarr R (8) holds (9)

Definition 2 The function 119891 R119899 rarr R is a polynomial ofdegree less than or equal to 119904 in 119899 scalar variables if

119891 (119909) =

119904

sum

119894=0

ℎ119894(119909) (10)

where 119909 isin R119899 and ℎ119894isin Ξ119899119894 119894 = 1 119904

Definition 3 Consider the vector 119909 isin R119899 119909 = [1199091 119909

119899]119879

The power transformation of degree119898 is a nonlinear changeof coordinates that forms a new vector 119909119898 of all integerpowered monomials of degree 119898 that can be made from theoriginal 119909 vector

119909119898

119895= 119888119895119909

1198981198951

1119909

1198981198952

2sdot sdot sdot 119909

119898119895119899

119899 119898119895119894isin 1 119899

119899

sum

119894=1

119898119895119894= 119898 119895 = 1 119889

(119899119898) 119889(119899119898)

=

(119899 + 119898 minus 1)

(119899 minus 1)119898

(11)

Usually we take 119888119895= 1 then with119898 gt 0

(

1199091

1199092

119909119899

)

119898

=

(

(

(

(

1199091(1199091 1199092 1199093 119909

119899)119898minus1119879

1199092(1199092 1199093 119909

119899)119898minus1119879

119909119899(119909119899)119898minus1119879

)

)

)

)

(12)

otherwise

(

1199091

1199092

119909119899

)

119898

= 1 (13)

For example 119899 = 2119898 = 2rArr 119889(119899119898)

= 3 and

119909 = (

1199091

1199092

) 997904rArr 1199092

= (

1199092

1

11990911199092

1199092

2

) (14)

Definition 4 The function119872 R119899 rarr R119903times119903 is a homogeneousparameter-dependentmatrix of degree119898 in 119899 scalar variablesif

119872119894119895isin Ξ119899119898 forall119894 119895 = 1 119903 (15)

We denote the set of 119903 times 119903 homogeneous parameter-dependent matrices of degree119898 in 119899 scalar variables as

Ξ119899119898119903

= 119872 R119899 997888rarr R119903times119903 (15) holds (16)

and the set of symmetric matrix forms as

Ξ119899119898119903

= 119872 isin Ξ119899119898119903

119872 (119909) = 119872119879

(119909) forall119909 isin R119899 (17)

Definition 5 Let119872 isin Ξ1198992119898119903

and119867 isin S119903119889(119899119898) such that

119872(119909) = Φ (119867 119909119898

119903) (18)

whereΦ(119867 119909119898 119903) = (119909119898otimes119868119903)119879

119867(119909119898

otimes119868119903)Then (18) is called

a squarematricial representation (SMR) of119872(119909)with respectto 119909119898 otimes 119868

119903 Moreover119867 is called a SMR matrix of119872(119909) with

respect to 119909119898 otimes 119868119903

Lemma 6 (Chesi et al [26]) Let119872 isin Ξ1198992119898119903

ThenH = 119867 isin

S119903119889(119899119898) (18) ℎ119900119897119889119904 is an affine space Moreover

119867(119872) = 119867 + 119871 119867 isin S119903119889(119899119898) 119904119886119905119894119904119891119894119890119904 (18) 119871 isin L119899119898119903

(19)

where L119899119898119903

is linear space

L119899119898119903

= 119871 isin S119903119889(119899119898) Φ (119871 119909119898 119903) = 0119903times119903

forall119909 isin R119899 (20)

whose dimension is given by

120596 (119899119898 119903) =

119903

2

((119889(119899119898)

(119903119889(119899119898)

+ 1)) minus (119903 + 1) 119889(1198992119898)

)

(21)

Lemma 7 (Chesi et al [26]) Let119872 isin Ξ119899119904119903

Then

119872(119909) gt 0 forall119909 isin 120574119902= 119909 isin R119902 119909

119894ge 0

119902

sum

119894=1

119909119894= 1

(22)

holds if and only if

119872(119904119902 (119909)) gt 0 forall119909 isin R1198990 (23)

Lemma 8 (Boyd et al [27]) The linear matrix inequality

119878 = (

1198781111987812

119878119879

1211987822

) lt 0 (24)

where 11987811= 119878119879

11and 11987822= 119878119879

22 is equivalent to

11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (25)

4 Advances in Mathematical Physics

3 Main Result

In this section the sufficient condition for existence of robust119867infin

filter for uncertain switched systems is formulated Forthis purpose we firstly consider the anti-interference ofsystem (1) for disturbance

Theorem 9 For a given scalar 120574 gt 0 Consider system (1) ifthere exist matrices 119875

119894119895gt 0 and matrices 119877

119894119895such that

(

Ψ11(P119894119895 119877119894119895) 0 Ψ

13(119877119894119895 119860119894119895) Ψ14(119877119894119895 119861119894119895)

lowast minus119868119888

Ψ23(119862119894119895) Ψ

24(119863119894119895)

lowast lowast Ψ33(119875119894119895) 0

lowast lowast lowast minus1205742

119868119888

)

+(

L11

L12

L13

L14

lowast L22

L23

L24

lowast lowast L33

L34

lowast lowast lowast L44

) lt 0 119894 119894 isin prod 119895 isin [1 119904]

(26)

with

Φ119875119894119877119894

(120582119898+1

) =

119904

sum

119895=1

120582119895(119875119894(120582119898

) minus R119894(120582119898

) minus 119877119879

119894(120582119898

))

Φ119877119894119860119894

(120582119898+1

) = 119877119894(120582119898

) 119860119894(120582)

Φ119877119894119861119894

(120582119898+1

) = 119877119894(120582119898

) 119861119894(120582)

Φ119862119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119862119894(120582)

Φ119863119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119863119894(120582)

Φ119868(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898+1

119868

Φ119875119894

(120582119898+1

) =

119904

sum

119895=1

120582119895119875119894(120582119898

)

Φ119875119894119877119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ11(119875119894119895 119877119894119895) 120603 (120582

119898+1

)

Φ119877119894119860119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ13(119877119894119895 119860119894119895) 120603 (120582

119898+1

)

Φ119877119894119861119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ14(119877119894119895 119861119894119895) 120603 (120582

119898+1

)

Φ119862119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ23(119862119894119895) 120603 (120582

119898+1

)

Φ119863119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ24(119863119894119895) 120603 (120582

119898+1

)

Φ119875119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ33(119875119894119895) 120603 (120582

119898+1

)

Φ119868(119904119902 (120582

119898+1

)) = 120603119879

(120582119898+1

) 119868119888120603 (120582119898+1

)

120603 (120582119898+1

) = 120582119898+1

otimes 119868

L11sdot sdot sdotL44isin L119904119898119899

(27)

whereΨ(sdot) is a matrix which is made up of119860119894119895 119861119894119895119862119894119895119863119894119895119875119894119895

119877119894119895and the set L

119904119898119899is defined in Lemma 6 then system (1) is

robustly asymptotically stable with119867infin

performance 120574 for anyswitching signal

Proof Consider the following homogeneous parameter-depend-ent quadratic Lyapunov function

119881 (119896 119909 (119896)) = 119909119879

(119896) 119875119894(120582119898

) 119909 (119896) (28)

Then along the trajectory of system (1) we have

Δ119881 = 119881 (119896 + 1 119909 (119896 + 1)) minus 119881 (119896 119909 (119896))

= 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

+ 2119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

+ 120596119879

(119896) [119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

(29)

When 119894 = 119894 the switched system is described by the 119894thmode When 119894 = 119894 it represents the switched system being atthe switching times from mode 119894 to mode 119894

Before and aftermultiplying inequality (26) by 120582119898+1otimes119868119879

and 120582119898+1 otimes 119868 we have

(

Φ119875119894119877119894

(119904119902 (120582119898+1

)) 0 Φ119877119894119860119894

(119904119902 (120582119898+1

)) Φ119877119894119861119894

(119904119902 (120582119898+1

))

lowast minusΦ119868(119904119902 (120582

119898+1

)) Φ119862119894

(119904119902 (120582119898+1

)) Φ119863119894

(119904119902 (120582119898+1

))

lowast lowast Φ119875119894

(119904119902 (120582119898+1

)) 0

lowast lowast lowast minus1205742

Φ119868(119904119902 (120582

119898+1

))

) lt 0 (30)

From Lemma 7 one has

(

Φ119875119894119877119894

(120582119898+1

) 0 Φ119877119894119860119894

(120582119898+1

) Φ119877119894119861119894

(120582119898+1

)

lowast minusΦ119868(120582119898+1

) Φ119862119894

(120582119898+1

) Φ119863119894

(120582119898+1

)

lowast lowast Φ119875119894

(120582119898+1

) 0

lowast lowast lowast minus1205742

Φ119868(120582119898+1

)

) lt 0 (31)

Advances in Mathematical Physics 5

which implies

(

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582) 119877

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast 119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0 (32)

Under the disturbance 120596(119896) = 0 we get

Δ119881 = 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

(33)

If

119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) lt 0 forall119894 119894 isin Π (34)

then Δ119881 lt 0 It implies system (1) is robust asymptotic stableIn terms of Lemma 8 condition (34) is equivalent to

(

minus119875119894(120582119898

) 119875119894(120582119898

) 119860119894(120582)

lowast 119875119894(120582119898

)

) lt 0 (35)

Since (26) it follows that

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) lt 0 (36)

which implies the matrices 119877119894(120582119898

) are nonsingular for each 119894Then we have

(119875119894(120582119898

) minus 119877119894(120582119898

)) 119875minus1

119894(120582119898

) (119875119894(120582119898

) minus 119877119894(120582119898

))119879

ge 0

(37)

which is equivalent to

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) ge minus119877119894(120582119898

) 119875minus1

119894(120582119898

) 119877119879

119894(120582119898

)

(38)

Hence it can be readily established that (32) is equivalent to

(

minus119877119894(120582119898

) 119875minus1

119894

(120582119898

) 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582)

lowast minus119868 119862119894(120582)

lowast lowast minus119875119894(120582119898

)

lowast lowast lowast

119877119894(120582119898

) 119861119894(120582)

119863119894(120582)

0

minus1205742

119868

) lt 0 (39)

Before and after multiplying the above inequality bydiagminus119877minus119879

119894(120582119898

)119875119894(120582119898

) 119868 119868 119868 we obtain

(

minus119875119894(120582119898

) 0 119875119894(120582119898

) 119860119894(120582) 119875

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast minus119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0

(40)

which implies that (35) holdsThen the stability of system (1)can be deduced

On the other hand let

119869 =

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896)] (41)

be performance index to establish the 119867infin

performance ofsystem (1) When the initial condition 119909(0) = 0 we have119881(119896 119909(119896))|

119896=0= 0 which implies

119869 lt

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896) + Δ119881]

=

infin

sum

119896=0

(119909119879

(119896) 120596119879

(119896))(

Θ11Θ12

Θ119879

12Θ22

)(

119909 (119896)

120596 (119896)

)

(42)

where

Θ11= 119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) + 119862119879

119894(120582) 119862119894(120582)

Θ12= 119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119862

119879

119894(120582)119863119894(120582)

Θ22= minus1205742

119868 + 119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119863

119879

119894(120582)119863119894(120582)

(43)

In terms of Lemma 8 we have 119869 lt 0whichmeans that 1199102lt

1205742

1205962 Then the proof is completed

Remark 10 Within robustly 119867infin

performance scheme thebasic idea is to get an upper bound of119867

infinnoise-attenuation

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

Gram-tight forms that is forms whose minimum coincideswith the lower bound provided by LMI optimizations basedon SOS (sum of squares of polynomials) relaxations

Thus in order to suppress the influence of uncertain-ties on the systemrsquos robust 119867

infincontrol the homogeneous

parameter-dependent quadratic Lyapunov functions are usedto design robust119867

infinfilter in this paper We first consider the

systemrsquos anti-interference of external disturbance Along thisdirection the robust 119867

infinfilters for switched linear discrete-

time systems are designed Lastly through the comparisonwe have the fact that homogeneous parameter-dependentquadratic Lyapunov functions can suppress conservativenesswhich the uncertainties bring

The rest of this paper is organised as follows We statethe problem formulation in Section 2 The main results arepresented in Section 3 Section 4 illustrates the obtainedresult by numerical examples which is followed by theconclusion in Section 5

Notation R119899 denotes the 119899-dimension Euclidean space andR119899times119898 is the real matrices with dimension 119899 times 119898 R119899

0means

R1198990 the notation119883 ge 119884 (resp119883 gt 119884) where119883 and119884 aresymmetric matrices represents the fact that the matrix119883minus119884is positive semidefinite (resp positive definite) 119860119879 denotesthe transposedmatrix of119860 119904119902(119909) represents (1199092

1 119909

2

119899)with

119909 isin R119899 ℎ119890(119883)means119883+119883119879 with119883 isin R119899times119899 119909otimes119910 denotes theKronecker product of vectors 119909 and 119910 sdot denotes Euclideannorm for vector or the spectral norm of matrices

2 Problem Statement

Consider a class of uncertain switched linear discrete-timesystems which were given in [1]

119909 (119896 + 1) = 119860119894(120582) 119909 (119896) + 119861

119894(120582) 120596 (119896)

119910 (119896) = 119862119894(120582) 119909 (119896) + 119863

119894(120582) 120596 (119896)

119911 (119896) = 119867119894(120582) 119909 (119896) + 119871

119894(120582) 120596 (119896)

(1)

where 119909(119896) isin R119899 is state vector120596(119896) isin R119897 is disturbance inputwhich belongs to 119897

2[0 +infin) 119910(119896) is the measurement output

119911(119896) is objective signal to be attenuated and 119894 is switchingrule which takes its value in the finite set Π = 1 119873As in [1] the switching signal 119894 is unknown a priori but itsinstantaneous value is available in real time As an arbitrarydiscrete time 119896 the switching signal 119894 is dependent on 119896or 119909(119896) or both or other switching rules 120582 is an uncertainparameter vector supposed to satisfy 120582 isin Λ = 120582

119895ge 0sum

119904

119895=1=

1 The vector 120582 represents the time-invariant parametricuncertainty which affects linearly the system dynamics Thevector 120582 can take any value inΛ but it is known to be constantin time

The matrices of each subsystem have appropriate dimen-sions and are assumed to belong to a given convex-boundedpolyhedral domain described by 119904 vertices in the 119894th subsys-tem that is

(119860119894(120582) 119861

119894(120582) 119862

119894(120582) 119863

119894(120582) 119867

119894(120582) 119871

119894(120582)) isin Γ

119894 (2)

where

Γ119894=

119860119894(120582) 119861

119894(120582) 119862

119894(120582) 119863

119894(120582) 119867

119894(120582) 119871

119894(120582)

=

119904

sum

119895=1

120582119895(119860119894119895119861119894119895119862119894119895119863119894119895119867119894119895119871119894119895)

120582119895ge 0

119904

sum

119895=1

120582119895= 1

(3)

Hence we are interested in designing an estimator or filterof the form

119909119891(119896 + 1) = 119860

119891119894119909119891(119896) + 119861

119891119894119910 (119896)

119911119891(119896) = 119862

119891119894119909119891(119896) + 119863

119891119894119910 (119896)

(4)

where 119909119891(119896) isin R119899 119911

119891(119896) isin R119902 119860

119894119895 119861119894119895 119862119894119895and 119863

119894119895are

the parameterized filter matrices to be determined The filterwith the above structure may be called switched linear filterin which the switching signal 119894 is also assumed unknown apriori but available in real-time and homogeneous with theswitching signal in system (1)

Augmenting the model of (1) to include the state of filter(4) we obtain the filtering error system

119909119890(119896 + 1) = 119860

119890119894(120582) 119909119890(119896) + 119861

119890119894(120582) 120596 (119896)

119890 (119896) = 119862119890119894(120582) 119909119890(119896) + 119863

119890119894(120582) 120596 (119896)

(5)

where

119909119890(119896) = (119909 (119896) 119909119891 (

119896))

119879

119890 (119896) = 119911 (119896) minus 119911119891(119896)

119860119890119894(120582) = (

119860119894(120582) 0

119861119891119894119862119894(120582) 119860

119891119894

)

119861119890119894(120582) = (119861119894

(120582) 119861119891119894119863119894(120582))

119879

119862119890119894(120582) = (119867119894 (

120582) minus 119863119891119894119862119894(120582) minus119862

119891119894)

119863119890119894(120582) = 119871

119894(120582) minus 119863

119891119894119863119894(120582)

(6)

Based on [1] the robust 119867infin

filtering problem addressedin this paper can be formulated as follows finding a pre-scribed level of noise attention 120574 gt 0 and determining arobust switched linear filter (4) such that the filtering errorsystem is robustly asymptotically stable and

1198902lt 1205742

1205962 (7)

This problem has received a great deal of attention Inorder to get a better level of noise attention 120574 gt 0 wewill use homogeneous polynomial functions which havedemonstrated nonconservative result for serval problemsIn this paper we extend these methods to design robustswitched linear filter

The following preliminaries are given which are essentialfor later developments We firstly recall the homogeneouspolynomial function from Chesi et al [26]

Advances in Mathematical Physics 3

Definition 1 The function ℎ R119899 rarr R is a form of degree 119889in 119899 scalar variables if

ℎ (119909) = sum

119902isin119871119899119904

119886119902119909119902

(8)

where 119871119899119904= 119902 isin 119873

119899

sum119899

119894=1119902119894= 119904 and 119909 isin R119899 and 119886

119902isin R is

coefficient of the monomial 119909119902

The set of forms of degree 119904 in 119899 scalar variables is definedas

Ξ119899119904= ℎ R119899 997888rarr R (8) holds (9)

Definition 2 The function 119891 R119899 rarr R is a polynomial ofdegree less than or equal to 119904 in 119899 scalar variables if

119891 (119909) =

119904

sum

119894=0

ℎ119894(119909) (10)

where 119909 isin R119899 and ℎ119894isin Ξ119899119894 119894 = 1 119904

Definition 3 Consider the vector 119909 isin R119899 119909 = [1199091 119909

119899]119879

The power transformation of degree119898 is a nonlinear changeof coordinates that forms a new vector 119909119898 of all integerpowered monomials of degree 119898 that can be made from theoriginal 119909 vector

119909119898

119895= 119888119895119909

1198981198951

1119909

1198981198952

2sdot sdot sdot 119909

119898119895119899

119899 119898119895119894isin 1 119899

119899

sum

119894=1

119898119895119894= 119898 119895 = 1 119889

(119899119898) 119889(119899119898)

=

(119899 + 119898 minus 1)

(119899 minus 1)119898

(11)

Usually we take 119888119895= 1 then with119898 gt 0

(

1199091

1199092

119909119899

)

119898

=

(

(

(

(

1199091(1199091 1199092 1199093 119909

119899)119898minus1119879

1199092(1199092 1199093 119909

119899)119898minus1119879

119909119899(119909119899)119898minus1119879

)

)

)

)

(12)

otherwise

(

1199091

1199092

119909119899

)

119898

= 1 (13)

For example 119899 = 2119898 = 2rArr 119889(119899119898)

= 3 and

119909 = (

1199091

1199092

) 997904rArr 1199092

= (

1199092

1

11990911199092

1199092

2

) (14)

Definition 4 The function119872 R119899 rarr R119903times119903 is a homogeneousparameter-dependentmatrix of degree119898 in 119899 scalar variablesif

119872119894119895isin Ξ119899119898 forall119894 119895 = 1 119903 (15)

We denote the set of 119903 times 119903 homogeneous parameter-dependent matrices of degree119898 in 119899 scalar variables as

Ξ119899119898119903

= 119872 R119899 997888rarr R119903times119903 (15) holds (16)

and the set of symmetric matrix forms as

Ξ119899119898119903

= 119872 isin Ξ119899119898119903

119872 (119909) = 119872119879

(119909) forall119909 isin R119899 (17)

Definition 5 Let119872 isin Ξ1198992119898119903

and119867 isin S119903119889(119899119898) such that

119872(119909) = Φ (119867 119909119898

119903) (18)

whereΦ(119867 119909119898 119903) = (119909119898otimes119868119903)119879

119867(119909119898

otimes119868119903)Then (18) is called

a squarematricial representation (SMR) of119872(119909)with respectto 119909119898 otimes 119868

119903 Moreover119867 is called a SMR matrix of119872(119909) with

respect to 119909119898 otimes 119868119903

Lemma 6 (Chesi et al [26]) Let119872 isin Ξ1198992119898119903

ThenH = 119867 isin

S119903119889(119899119898) (18) ℎ119900119897119889119904 is an affine space Moreover

119867(119872) = 119867 + 119871 119867 isin S119903119889(119899119898) 119904119886119905119894119904119891119894119890119904 (18) 119871 isin L119899119898119903

(19)

where L119899119898119903

is linear space

L119899119898119903

= 119871 isin S119903119889(119899119898) Φ (119871 119909119898 119903) = 0119903times119903

forall119909 isin R119899 (20)

whose dimension is given by

120596 (119899119898 119903) =

119903

2

((119889(119899119898)

(119903119889(119899119898)

+ 1)) minus (119903 + 1) 119889(1198992119898)

)

(21)

Lemma 7 (Chesi et al [26]) Let119872 isin Ξ119899119904119903

Then

119872(119909) gt 0 forall119909 isin 120574119902= 119909 isin R119902 119909

119894ge 0

119902

sum

119894=1

119909119894= 1

(22)

holds if and only if

119872(119904119902 (119909)) gt 0 forall119909 isin R1198990 (23)

Lemma 8 (Boyd et al [27]) The linear matrix inequality

119878 = (

1198781111987812

119878119879

1211987822

) lt 0 (24)

where 11987811= 119878119879

11and 11987822= 119878119879

22 is equivalent to

11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (25)

4 Advances in Mathematical Physics

3 Main Result

In this section the sufficient condition for existence of robust119867infin

filter for uncertain switched systems is formulated Forthis purpose we firstly consider the anti-interference ofsystem (1) for disturbance

Theorem 9 For a given scalar 120574 gt 0 Consider system (1) ifthere exist matrices 119875

119894119895gt 0 and matrices 119877

119894119895such that

(

Ψ11(P119894119895 119877119894119895) 0 Ψ

13(119877119894119895 119860119894119895) Ψ14(119877119894119895 119861119894119895)

lowast minus119868119888

Ψ23(119862119894119895) Ψ

24(119863119894119895)

lowast lowast Ψ33(119875119894119895) 0

lowast lowast lowast minus1205742

119868119888

)

+(

L11

L12

L13

L14

lowast L22

L23

L24

lowast lowast L33

L34

lowast lowast lowast L44

) lt 0 119894 119894 isin prod 119895 isin [1 119904]

(26)

with

Φ119875119894119877119894

(120582119898+1

) =

119904

sum

119895=1

120582119895(119875119894(120582119898

) minus R119894(120582119898

) minus 119877119879

119894(120582119898

))

Φ119877119894119860119894

(120582119898+1

) = 119877119894(120582119898

) 119860119894(120582)

Φ119877119894119861119894

(120582119898+1

) = 119877119894(120582119898

) 119861119894(120582)

Φ119862119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119862119894(120582)

Φ119863119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119863119894(120582)

Φ119868(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898+1

119868

Φ119875119894

(120582119898+1

) =

119904

sum

119895=1

120582119895119875119894(120582119898

)

Φ119875119894119877119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ11(119875119894119895 119877119894119895) 120603 (120582

119898+1

)

Φ119877119894119860119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ13(119877119894119895 119860119894119895) 120603 (120582

119898+1

)

Φ119877119894119861119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ14(119877119894119895 119861119894119895) 120603 (120582

119898+1

)

Φ119862119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ23(119862119894119895) 120603 (120582

119898+1

)

Φ119863119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ24(119863119894119895) 120603 (120582

119898+1

)

Φ119875119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ33(119875119894119895) 120603 (120582

119898+1

)

Φ119868(119904119902 (120582

119898+1

)) = 120603119879

(120582119898+1

) 119868119888120603 (120582119898+1

)

120603 (120582119898+1

) = 120582119898+1

otimes 119868

L11sdot sdot sdotL44isin L119904119898119899

(27)

whereΨ(sdot) is a matrix which is made up of119860119894119895 119861119894119895119862119894119895119863119894119895119875119894119895

119877119894119895and the set L

119904119898119899is defined in Lemma 6 then system (1) is

robustly asymptotically stable with119867infin

performance 120574 for anyswitching signal

Proof Consider the following homogeneous parameter-depend-ent quadratic Lyapunov function

119881 (119896 119909 (119896)) = 119909119879

(119896) 119875119894(120582119898

) 119909 (119896) (28)

Then along the trajectory of system (1) we have

Δ119881 = 119881 (119896 + 1 119909 (119896 + 1)) minus 119881 (119896 119909 (119896))

= 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

+ 2119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

+ 120596119879

(119896) [119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

(29)

When 119894 = 119894 the switched system is described by the 119894thmode When 119894 = 119894 it represents the switched system being atthe switching times from mode 119894 to mode 119894

Before and aftermultiplying inequality (26) by 120582119898+1otimes119868119879

and 120582119898+1 otimes 119868 we have

(

Φ119875119894119877119894

(119904119902 (120582119898+1

)) 0 Φ119877119894119860119894

(119904119902 (120582119898+1

)) Φ119877119894119861119894

(119904119902 (120582119898+1

))

lowast minusΦ119868(119904119902 (120582

119898+1

)) Φ119862119894

(119904119902 (120582119898+1

)) Φ119863119894

(119904119902 (120582119898+1

))

lowast lowast Φ119875119894

(119904119902 (120582119898+1

)) 0

lowast lowast lowast minus1205742

Φ119868(119904119902 (120582

119898+1

))

) lt 0 (30)

From Lemma 7 one has

(

Φ119875119894119877119894

(120582119898+1

) 0 Φ119877119894119860119894

(120582119898+1

) Φ119877119894119861119894

(120582119898+1

)

lowast minusΦ119868(120582119898+1

) Φ119862119894

(120582119898+1

) Φ119863119894

(120582119898+1

)

lowast lowast Φ119875119894

(120582119898+1

) 0

lowast lowast lowast minus1205742

Φ119868(120582119898+1

)

) lt 0 (31)

Advances in Mathematical Physics 5

which implies

(

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582) 119877

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast 119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0 (32)

Under the disturbance 120596(119896) = 0 we get

Δ119881 = 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

(33)

If

119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) lt 0 forall119894 119894 isin Π (34)

then Δ119881 lt 0 It implies system (1) is robust asymptotic stableIn terms of Lemma 8 condition (34) is equivalent to

(

minus119875119894(120582119898

) 119875119894(120582119898

) 119860119894(120582)

lowast 119875119894(120582119898

)

) lt 0 (35)

Since (26) it follows that

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) lt 0 (36)

which implies the matrices 119877119894(120582119898

) are nonsingular for each 119894Then we have

(119875119894(120582119898

) minus 119877119894(120582119898

)) 119875minus1

119894(120582119898

) (119875119894(120582119898

) minus 119877119894(120582119898

))119879

ge 0

(37)

which is equivalent to

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) ge minus119877119894(120582119898

) 119875minus1

119894(120582119898

) 119877119879

119894(120582119898

)

(38)

Hence it can be readily established that (32) is equivalent to

(

minus119877119894(120582119898

) 119875minus1

119894

(120582119898

) 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582)

lowast minus119868 119862119894(120582)

lowast lowast minus119875119894(120582119898

)

lowast lowast lowast

119877119894(120582119898

) 119861119894(120582)

119863119894(120582)

0

minus1205742

119868

) lt 0 (39)

Before and after multiplying the above inequality bydiagminus119877minus119879

119894(120582119898

)119875119894(120582119898

) 119868 119868 119868 we obtain

(

minus119875119894(120582119898

) 0 119875119894(120582119898

) 119860119894(120582) 119875

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast minus119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0

(40)

which implies that (35) holdsThen the stability of system (1)can be deduced

On the other hand let

119869 =

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896)] (41)

be performance index to establish the 119867infin

performance ofsystem (1) When the initial condition 119909(0) = 0 we have119881(119896 119909(119896))|

119896=0= 0 which implies

119869 lt

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896) + Δ119881]

=

infin

sum

119896=0

(119909119879

(119896) 120596119879

(119896))(

Θ11Θ12

Θ119879

12Θ22

)(

119909 (119896)

120596 (119896)

)

(42)

where

Θ11= 119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) + 119862119879

119894(120582) 119862119894(120582)

Θ12= 119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119862

119879

119894(120582)119863119894(120582)

Θ22= minus1205742

119868 + 119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119863

119879

119894(120582)119863119894(120582)

(43)

In terms of Lemma 8 we have 119869 lt 0whichmeans that 1199102lt

1205742

1205962 Then the proof is completed

Remark 10 Within robustly 119867infin

performance scheme thebasic idea is to get an upper bound of119867

infinnoise-attenuation

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

Definition 1 The function ℎ R119899 rarr R is a form of degree 119889in 119899 scalar variables if

ℎ (119909) = sum

119902isin119871119899119904

119886119902119909119902

(8)

where 119871119899119904= 119902 isin 119873

119899

sum119899

119894=1119902119894= 119904 and 119909 isin R119899 and 119886

119902isin R is

coefficient of the monomial 119909119902

The set of forms of degree 119904 in 119899 scalar variables is definedas

Ξ119899119904= ℎ R119899 997888rarr R (8) holds (9)

Definition 2 The function 119891 R119899 rarr R is a polynomial ofdegree less than or equal to 119904 in 119899 scalar variables if

119891 (119909) =

119904

sum

119894=0

ℎ119894(119909) (10)

where 119909 isin R119899 and ℎ119894isin Ξ119899119894 119894 = 1 119904

Definition 3 Consider the vector 119909 isin R119899 119909 = [1199091 119909

119899]119879

The power transformation of degree119898 is a nonlinear changeof coordinates that forms a new vector 119909119898 of all integerpowered monomials of degree 119898 that can be made from theoriginal 119909 vector

119909119898

119895= 119888119895119909

1198981198951

1119909

1198981198952

2sdot sdot sdot 119909

119898119895119899

119899 119898119895119894isin 1 119899

119899

sum

119894=1

119898119895119894= 119898 119895 = 1 119889

(119899119898) 119889(119899119898)

=

(119899 + 119898 minus 1)

(119899 minus 1)119898

(11)

Usually we take 119888119895= 1 then with119898 gt 0

(

1199091

1199092

119909119899

)

119898

=

(

(

(

(

1199091(1199091 1199092 1199093 119909

119899)119898minus1119879

1199092(1199092 1199093 119909

119899)119898minus1119879

119909119899(119909119899)119898minus1119879

)

)

)

)

(12)

otherwise

(

1199091

1199092

119909119899

)

119898

= 1 (13)

For example 119899 = 2119898 = 2rArr 119889(119899119898)

= 3 and

119909 = (

1199091

1199092

) 997904rArr 1199092

= (

1199092

1

11990911199092

1199092

2

) (14)

Definition 4 The function119872 R119899 rarr R119903times119903 is a homogeneousparameter-dependentmatrix of degree119898 in 119899 scalar variablesif

119872119894119895isin Ξ119899119898 forall119894 119895 = 1 119903 (15)

We denote the set of 119903 times 119903 homogeneous parameter-dependent matrices of degree119898 in 119899 scalar variables as

Ξ119899119898119903

= 119872 R119899 997888rarr R119903times119903 (15) holds (16)

and the set of symmetric matrix forms as

Ξ119899119898119903

= 119872 isin Ξ119899119898119903

119872 (119909) = 119872119879

(119909) forall119909 isin R119899 (17)

Definition 5 Let119872 isin Ξ1198992119898119903

and119867 isin S119903119889(119899119898) such that

119872(119909) = Φ (119867 119909119898

119903) (18)

whereΦ(119867 119909119898 119903) = (119909119898otimes119868119903)119879

119867(119909119898

otimes119868119903)Then (18) is called

a squarematricial representation (SMR) of119872(119909)with respectto 119909119898 otimes 119868

119903 Moreover119867 is called a SMR matrix of119872(119909) with

respect to 119909119898 otimes 119868119903

Lemma 6 (Chesi et al [26]) Let119872 isin Ξ1198992119898119903

ThenH = 119867 isin

S119903119889(119899119898) (18) ℎ119900119897119889119904 is an affine space Moreover

119867(119872) = 119867 + 119871 119867 isin S119903119889(119899119898) 119904119886119905119894119904119891119894119890119904 (18) 119871 isin L119899119898119903

(19)

where L119899119898119903

is linear space

L119899119898119903

= 119871 isin S119903119889(119899119898) Φ (119871 119909119898 119903) = 0119903times119903

forall119909 isin R119899 (20)

whose dimension is given by

120596 (119899119898 119903) =

119903

2

((119889(119899119898)

(119903119889(119899119898)

+ 1)) minus (119903 + 1) 119889(1198992119898)

)

(21)

Lemma 7 (Chesi et al [26]) Let119872 isin Ξ119899119904119903

Then

119872(119909) gt 0 forall119909 isin 120574119902= 119909 isin R119902 119909

119894ge 0

119902

sum

119894=1

119909119894= 1

(22)

holds if and only if

119872(119904119902 (119909)) gt 0 forall119909 isin R1198990 (23)

Lemma 8 (Boyd et al [27]) The linear matrix inequality

119878 = (

1198781111987812

119878119879

1211987822

) lt 0 (24)

where 11987811= 119878119879

11and 11987822= 119878119879

22 is equivalent to

11987811lt 0 119878

22minus 119878119879

12119878minus1

1111987812lt 0 (25)

4 Advances in Mathematical Physics

3 Main Result

In this section the sufficient condition for existence of robust119867infin

filter for uncertain switched systems is formulated Forthis purpose we firstly consider the anti-interference ofsystem (1) for disturbance

Theorem 9 For a given scalar 120574 gt 0 Consider system (1) ifthere exist matrices 119875

119894119895gt 0 and matrices 119877

119894119895such that

(

Ψ11(P119894119895 119877119894119895) 0 Ψ

13(119877119894119895 119860119894119895) Ψ14(119877119894119895 119861119894119895)

lowast minus119868119888

Ψ23(119862119894119895) Ψ

24(119863119894119895)

lowast lowast Ψ33(119875119894119895) 0

lowast lowast lowast minus1205742

119868119888

)

+(

L11

L12

L13

L14

lowast L22

L23

L24

lowast lowast L33

L34

lowast lowast lowast L44

) lt 0 119894 119894 isin prod 119895 isin [1 119904]

(26)

with

Φ119875119894119877119894

(120582119898+1

) =

119904

sum

119895=1

120582119895(119875119894(120582119898

) minus R119894(120582119898

) minus 119877119879

119894(120582119898

))

Φ119877119894119860119894

(120582119898+1

) = 119877119894(120582119898

) 119860119894(120582)

Φ119877119894119861119894

(120582119898+1

) = 119877119894(120582119898

) 119861119894(120582)

Φ119862119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119862119894(120582)

Φ119863119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119863119894(120582)

Φ119868(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898+1

119868

Φ119875119894

(120582119898+1

) =

119904

sum

119895=1

120582119895119875119894(120582119898

)

Φ119875119894119877119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ11(119875119894119895 119877119894119895) 120603 (120582

119898+1

)

Φ119877119894119860119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ13(119877119894119895 119860119894119895) 120603 (120582

119898+1

)

Φ119877119894119861119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ14(119877119894119895 119861119894119895) 120603 (120582

119898+1

)

Φ119862119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ23(119862119894119895) 120603 (120582

119898+1

)

Φ119863119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ24(119863119894119895) 120603 (120582

119898+1

)

Φ119875119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ33(119875119894119895) 120603 (120582

119898+1

)

Φ119868(119904119902 (120582

119898+1

)) = 120603119879

(120582119898+1

) 119868119888120603 (120582119898+1

)

120603 (120582119898+1

) = 120582119898+1

otimes 119868

L11sdot sdot sdotL44isin L119904119898119899

(27)

whereΨ(sdot) is a matrix which is made up of119860119894119895 119861119894119895119862119894119895119863119894119895119875119894119895

119877119894119895and the set L

119904119898119899is defined in Lemma 6 then system (1) is

robustly asymptotically stable with119867infin

performance 120574 for anyswitching signal

Proof Consider the following homogeneous parameter-depend-ent quadratic Lyapunov function

119881 (119896 119909 (119896)) = 119909119879

(119896) 119875119894(120582119898

) 119909 (119896) (28)

Then along the trajectory of system (1) we have

Δ119881 = 119881 (119896 + 1 119909 (119896 + 1)) minus 119881 (119896 119909 (119896))

= 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

+ 2119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

+ 120596119879

(119896) [119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

(29)

When 119894 = 119894 the switched system is described by the 119894thmode When 119894 = 119894 it represents the switched system being atthe switching times from mode 119894 to mode 119894

Before and aftermultiplying inequality (26) by 120582119898+1otimes119868119879

and 120582119898+1 otimes 119868 we have

(

Φ119875119894119877119894

(119904119902 (120582119898+1

)) 0 Φ119877119894119860119894

(119904119902 (120582119898+1

)) Φ119877119894119861119894

(119904119902 (120582119898+1

))

lowast minusΦ119868(119904119902 (120582

119898+1

)) Φ119862119894

(119904119902 (120582119898+1

)) Φ119863119894

(119904119902 (120582119898+1

))

lowast lowast Φ119875119894

(119904119902 (120582119898+1

)) 0

lowast lowast lowast minus1205742

Φ119868(119904119902 (120582

119898+1

))

) lt 0 (30)

From Lemma 7 one has

(

Φ119875119894119877119894

(120582119898+1

) 0 Φ119877119894119860119894

(120582119898+1

) Φ119877119894119861119894

(120582119898+1

)

lowast minusΦ119868(120582119898+1

) Φ119862119894

(120582119898+1

) Φ119863119894

(120582119898+1

)

lowast lowast Φ119875119894

(120582119898+1

) 0

lowast lowast lowast minus1205742

Φ119868(120582119898+1

)

) lt 0 (31)

Advances in Mathematical Physics 5

which implies

(

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582) 119877

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast 119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0 (32)

Under the disturbance 120596(119896) = 0 we get

Δ119881 = 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

(33)

If

119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) lt 0 forall119894 119894 isin Π (34)

then Δ119881 lt 0 It implies system (1) is robust asymptotic stableIn terms of Lemma 8 condition (34) is equivalent to

(

minus119875119894(120582119898

) 119875119894(120582119898

) 119860119894(120582)

lowast 119875119894(120582119898

)

) lt 0 (35)

Since (26) it follows that

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) lt 0 (36)

which implies the matrices 119877119894(120582119898

) are nonsingular for each 119894Then we have

(119875119894(120582119898

) minus 119877119894(120582119898

)) 119875minus1

119894(120582119898

) (119875119894(120582119898

) minus 119877119894(120582119898

))119879

ge 0

(37)

which is equivalent to

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) ge minus119877119894(120582119898

) 119875minus1

119894(120582119898

) 119877119879

119894(120582119898

)

(38)

Hence it can be readily established that (32) is equivalent to

(

minus119877119894(120582119898

) 119875minus1

119894

(120582119898

) 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582)

lowast minus119868 119862119894(120582)

lowast lowast minus119875119894(120582119898

)

lowast lowast lowast

119877119894(120582119898

) 119861119894(120582)

119863119894(120582)

0

minus1205742

119868

) lt 0 (39)

Before and after multiplying the above inequality bydiagminus119877minus119879

119894(120582119898

)119875119894(120582119898

) 119868 119868 119868 we obtain

(

minus119875119894(120582119898

) 0 119875119894(120582119898

) 119860119894(120582) 119875

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast minus119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0

(40)

which implies that (35) holdsThen the stability of system (1)can be deduced

On the other hand let

119869 =

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896)] (41)

be performance index to establish the 119867infin

performance ofsystem (1) When the initial condition 119909(0) = 0 we have119881(119896 119909(119896))|

119896=0= 0 which implies

119869 lt

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896) + Δ119881]

=

infin

sum

119896=0

(119909119879

(119896) 120596119879

(119896))(

Θ11Θ12

Θ119879

12Θ22

)(

119909 (119896)

120596 (119896)

)

(42)

where

Θ11= 119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) + 119862119879

119894(120582) 119862119894(120582)

Θ12= 119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119862

119879

119894(120582)119863119894(120582)

Θ22= minus1205742

119868 + 119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119863

119879

119894(120582)119863119894(120582)

(43)

In terms of Lemma 8 we have 119869 lt 0whichmeans that 1199102lt

1205742

1205962 Then the proof is completed

Remark 10 Within robustly 119867infin

performance scheme thebasic idea is to get an upper bound of119867

infinnoise-attenuation

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

3 Main Result

In this section the sufficient condition for existence of robust119867infin

filter for uncertain switched systems is formulated Forthis purpose we firstly consider the anti-interference ofsystem (1) for disturbance

Theorem 9 For a given scalar 120574 gt 0 Consider system (1) ifthere exist matrices 119875

119894119895gt 0 and matrices 119877

119894119895such that

(

Ψ11(P119894119895 119877119894119895) 0 Ψ

13(119877119894119895 119860119894119895) Ψ14(119877119894119895 119861119894119895)

lowast minus119868119888

Ψ23(119862119894119895) Ψ

24(119863119894119895)

lowast lowast Ψ33(119875119894119895) 0

lowast lowast lowast minus1205742

119868119888

)

+(

L11

L12

L13

L14

lowast L22

L23

L24

lowast lowast L33

L34

lowast lowast lowast L44

) lt 0 119894 119894 isin prod 119895 isin [1 119904]

(26)

with

Φ119875119894119877119894

(120582119898+1

) =

119904

sum

119895=1

120582119895(119875119894(120582119898

) minus R119894(120582119898

) minus 119877119879

119894(120582119898

))

Φ119877119894119860119894

(120582119898+1

) = 119877119894(120582119898

) 119860119894(120582)

Φ119877119894119861119894

(120582119898+1

) = 119877119894(120582119898

) 119861119894(120582)

Φ119862119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119862119894(120582)

Φ119863119894

(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898

119863119894(120582)

Φ119868(120582119898+1

) = (

119904

sum

119895=1

120582119895)

119898+1

119868

Φ119875119894

(120582119898+1

) =

119904

sum

119895=1

120582119895119875119894(120582119898

)

Φ119875119894119877119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ11(119875119894119895 119877119894119895) 120603 (120582

119898+1

)

Φ119877119894119860119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ13(119877119894119895 119860119894119895) 120603 (120582

119898+1

)

Φ119877119894119861119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ14(119877119894119895 119861119894119895) 120603 (120582

119898+1

)

Φ119862119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ23(119862119894119895) 120603 (120582

119898+1

)

Φ119863119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ24(119863119894119895) 120603 (120582

119898+1

)

Φ119875119894

(119904119902 (120582119898+1

)) = 120603119879

(120582119898+1

)Ψ33(119875119894119895) 120603 (120582

119898+1

)

Φ119868(119904119902 (120582

119898+1

)) = 120603119879

(120582119898+1

) 119868119888120603 (120582119898+1

)

120603 (120582119898+1

) = 120582119898+1

otimes 119868

L11sdot sdot sdotL44isin L119904119898119899

(27)

whereΨ(sdot) is a matrix which is made up of119860119894119895 119861119894119895119862119894119895119863119894119895119875119894119895

119877119894119895and the set L

119904119898119899is defined in Lemma 6 then system (1) is

robustly asymptotically stable with119867infin

performance 120574 for anyswitching signal

Proof Consider the following homogeneous parameter-depend-ent quadratic Lyapunov function

119881 (119896 119909 (119896)) = 119909119879

(119896) 119875119894(120582119898

) 119909 (119896) (28)

Then along the trajectory of system (1) we have

Δ119881 = 119881 (119896 + 1 119909 (119896 + 1)) minus 119881 (119896 119909 (119896))

= 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

+ 2119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

+ 120596119879

(119896) [119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582)] 120596 (119896)

(29)

When 119894 = 119894 the switched system is described by the 119894thmode When 119894 = 119894 it represents the switched system being atthe switching times from mode 119894 to mode 119894

Before and aftermultiplying inequality (26) by 120582119898+1otimes119868119879

and 120582119898+1 otimes 119868 we have

(

Φ119875119894119877119894

(119904119902 (120582119898+1

)) 0 Φ119877119894119860119894

(119904119902 (120582119898+1

)) Φ119877119894119861119894

(119904119902 (120582119898+1

))

lowast minusΦ119868(119904119902 (120582

119898+1

)) Φ119862119894

(119904119902 (120582119898+1

)) Φ119863119894

(119904119902 (120582119898+1

))

lowast lowast Φ119875119894

(119904119902 (120582119898+1

)) 0

lowast lowast lowast minus1205742

Φ119868(119904119902 (120582

119898+1

))

) lt 0 (30)

From Lemma 7 one has

(

Φ119875119894119877119894

(120582119898+1

) 0 Φ119877119894119860119894

(120582119898+1

) Φ119877119894119861119894

(120582119898+1

)

lowast minusΦ119868(120582119898+1

) Φ119862119894

(120582119898+1

) Φ119863119894

(120582119898+1

)

lowast lowast Φ119875119894

(120582119898+1

) 0

lowast lowast lowast minus1205742

Φ119868(120582119898+1

)

) lt 0 (31)

Advances in Mathematical Physics 5

which implies

(

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582) 119877

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast 119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0 (32)

Under the disturbance 120596(119896) = 0 we get

Δ119881 = 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

(33)

If

119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) lt 0 forall119894 119894 isin Π (34)

then Δ119881 lt 0 It implies system (1) is robust asymptotic stableIn terms of Lemma 8 condition (34) is equivalent to

(

minus119875119894(120582119898

) 119875119894(120582119898

) 119860119894(120582)

lowast 119875119894(120582119898

)

) lt 0 (35)

Since (26) it follows that

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) lt 0 (36)

which implies the matrices 119877119894(120582119898

) are nonsingular for each 119894Then we have

(119875119894(120582119898

) minus 119877119894(120582119898

)) 119875minus1

119894(120582119898

) (119875119894(120582119898

) minus 119877119894(120582119898

))119879

ge 0

(37)

which is equivalent to

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) ge minus119877119894(120582119898

) 119875minus1

119894(120582119898

) 119877119879

119894(120582119898

)

(38)

Hence it can be readily established that (32) is equivalent to

(

minus119877119894(120582119898

) 119875minus1

119894

(120582119898

) 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582)

lowast minus119868 119862119894(120582)

lowast lowast minus119875119894(120582119898

)

lowast lowast lowast

119877119894(120582119898

) 119861119894(120582)

119863119894(120582)

0

minus1205742

119868

) lt 0 (39)

Before and after multiplying the above inequality bydiagminus119877minus119879

119894(120582119898

)119875119894(120582119898

) 119868 119868 119868 we obtain

(

minus119875119894(120582119898

) 0 119875119894(120582119898

) 119860119894(120582) 119875

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast minus119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0

(40)

which implies that (35) holdsThen the stability of system (1)can be deduced

On the other hand let

119869 =

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896)] (41)

be performance index to establish the 119867infin

performance ofsystem (1) When the initial condition 119909(0) = 0 we have119881(119896 119909(119896))|

119896=0= 0 which implies

119869 lt

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896) + Δ119881]

=

infin

sum

119896=0

(119909119879

(119896) 120596119879

(119896))(

Θ11Θ12

Θ119879

12Θ22

)(

119909 (119896)

120596 (119896)

)

(42)

where

Θ11= 119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) + 119862119879

119894(120582) 119862119894(120582)

Θ12= 119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119862

119879

119894(120582)119863119894(120582)

Θ22= minus1205742

119868 + 119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119863

119879

119894(120582)119863119894(120582)

(43)

In terms of Lemma 8 we have 119869 lt 0whichmeans that 1199102lt

1205742

1205962 Then the proof is completed

Remark 10 Within robustly 119867infin

performance scheme thebasic idea is to get an upper bound of119867

infinnoise-attenuation

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

which implies

(

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582) 119877

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast 119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0 (32)

Under the disturbance 120596(119896) = 0 we get

Δ119881 = 119909119879

(119896) [119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

)] 119909 (119896)

(33)

If

119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) lt 0 forall119894 119894 isin Π (34)

then Δ119881 lt 0 It implies system (1) is robust asymptotic stableIn terms of Lemma 8 condition (34) is equivalent to

(

minus119875119894(120582119898

) 119875119894(120582119898

) 119860119894(120582)

lowast 119875119894(120582119898

)

) lt 0 (35)

Since (26) it follows that

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) lt 0 (36)

which implies the matrices 119877119894(120582119898

) are nonsingular for each 119894Then we have

(119875119894(120582119898

) minus 119877119894(120582119898

)) 119875minus1

119894(120582119898

) (119875119894(120582119898

) minus 119877119894(120582119898

))119879

ge 0

(37)

which is equivalent to

119875119894(120582119898

) minus 119877119894(120582119898

) minus 119877119879

119894(120582119898

) ge minus119877119894(120582119898

) 119875minus1

119894(120582119898

) 119877119879

119894(120582119898

)

(38)

Hence it can be readily established that (32) is equivalent to

(

minus119877119894(120582119898

) 119875minus1

119894

(120582119898

) 119877119879

119894(120582119898

) 0 119877119894(120582119898

) 119860119894(120582)

lowast minus119868 119862119894(120582)

lowast lowast minus119875119894(120582119898

)

lowast lowast lowast

119877119894(120582119898

) 119861119894(120582)

119863119894(120582)

0

minus1205742

119868

) lt 0 (39)

Before and after multiplying the above inequality bydiagminus119877minus119879

119894(120582119898

)119875119894(120582119898

) 119868 119868 119868 we obtain

(

minus119875119894(120582119898

) 0 119875119894(120582119898

) 119860119894(120582) 119875

119894(120582119898

) 119861119894(120582)

lowast minus119868 119862119894(120582) 119863

119894(120582)

lowast lowast minus119875119894(120582119898

) 0

lowast lowast lowast minus1205742

119868

) lt 0

(40)

which implies that (35) holdsThen the stability of system (1)can be deduced

On the other hand let

119869 =

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896)] (41)

be performance index to establish the 119867infin

performance ofsystem (1) When the initial condition 119909(0) = 0 we have119881(119896 119909(119896))|

119896=0= 0 which implies

119869 lt

infin

sum

119896=0

[119910119879

(119896) 119910 (119896) minus 1205742

120596119879

(119896) 120596 (119896) + Δ119881]

=

infin

sum

119896=0

(119909119879

(119896) 120596119879

(119896))(

Θ11Θ12

Θ119879

12Θ22

)(

119909 (119896)

120596 (119896)

)

(42)

where

Θ11= 119860119879

119894(120582) 119875119894(120582119898

) 119860119894(120582) minus 119875

119894(120582119898

) + 119862119879

119894(120582) 119862119894(120582)

Θ12= 119860119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119862

119879

119894(120582)119863119894(120582)

Θ22= minus1205742

119868 + 119861119879

119894(120582) 119875119894(120582119898

) 119861119894(120582) + 119863

119879

119894(120582)119863119894(120582)

(43)

In terms of Lemma 8 we have 119869 lt 0whichmeans that 1199102lt

1205742

1205962 Then the proof is completed

Remark 10 Within robustly 119867infin

performance scheme thebasic idea is to get an upper bound of119867

infinnoise-attenuation

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

level Since model uncertainties may result in significantchanges in 119867

infinnoise-attenuation level we should effec-

tively reduce this effect For this purpose the homoge-neous parameter-dependent quadratic Lyapunov function isexploited inTheorem 9

Remark 11 InTheorem 9Ψ(sdot) is amatrixwhich ismade up of119860119894119895 119861119894119895 119862119894119895119863119894119895 119875119894119895 119877119894119895 For example when 119899 = 2 and119898 = 1

condition (26) can be written as

Ψ11(119875119894119895 119877119894119895) =

(

(

(

(

(

(

(

(

(

1198751198941minus 119877119879

11989410 0

minus1198771198941

lowast minus1198771198941minus 119877119879

11989410

minus1198771198942minus 119877119879

1198942

+1198751198941+ 1198751198942

lowast lowast 1198751198942minus 1198771198942

minus119877119879

1198942

)

)

)

)

)

)

)

)

)

Ψ13(119877119894119895 119860119894119895) = (

11987711989411198601198941

0 0

lowast 11987711989411198601198942+ 11987711989421198601198941

0

lowast lowast 11987711989421198601198942

)

Ψ14(119877119894119895 119861119894119895) = (

11987711989411198611198941

0 0

lowast 11987711989411198611198942+ 11987711989421198611198941

0

lowast lowast 11987711989421198611198942

)

Ψ23(119862119894119895) = (

1198621198941

0 0

lowast 1198621198941+ 1198621198942

0

lowast lowast 1198621198942

)

Ψ24(119862119894119895) = (

1198631198941

0 0

lowast 1198631198941+ 1198631198942

0

lowast lowast 1198631198942

)

Ψ33(119875119894119895) = (

minus1198751198941

0 0

lowast minus1198751198941minus 1198751198942

0

lowast lowast minus1198751198942

)

(44)

Remark 12 About the calculation of matrices L11sdot sdot sdotL44

the following algorithm is presented in [26]

(1) choose 119909119898 and 1199092119898 as in Definition 3 where 119909 isin R119899

(2) set 119860 = 0119889(1198992119898)119889(1199032)times3

and 119887 = 0 and define the variable120572 isin R120596(119899119898119903)

(3) for 119894 = 1 119889(119899119898)

and 119895 = 1 119903(4) set 119888 = 119903(119894 minus 1) + 119895(5) for 119896 = 1 119889

(119899119898)and 119897 = max1 119895 + 119903(119894 minus 119896) 119903

(6) set 119889 = 119903(119896 minus 1) + 119897 and 119891 = 119894119899119889((119909119898)119894(119909119898

)119896 1199092119898

)

(7) set 119892 = 119894119899119889(119910119895119910119897 1199102

) and 119886 = 1198911198891199032+ 119892 and 119860

1198861=

1198601198861+ 1

(8) if 1198601198861= 1

(9) set 1198601198862= 119888 and 119860

1198863= 119889

(10) else

(11) set 119887 = 119887 + 1 and 119866 = 0119903119889(119899119898)times119903119889(119899119898)

and 119866119888119889= 1

(12) set ℎ = 1198601198862

and 119901 = 1198601198863

and 119866ℎ119901= 119866ℎ119901minus 1

(13) set 119871 = 119871 + 120572119887119866

(14) endif

(15) endfor

(16) endfor

(17) setL = 05ℎ119890(L)

Next the condition for robust119867infinfilter is formulated For

convenience we define 119899 = 2119898 = 1 and 119904 = 2

Theorem 13 Given a constant 120574 gt 0 if there exist matrices1198752119894119895 119910119894 120577119894119895 119860119891119894 119861119891119894 119862119891119894 119863119891119894 119909119894119895and positive definite matrices

1198751119894119895 1198753119894119895

and scalar 120576119894such that

(

(

(

(

(

(

Λ11Λ12

0 Λ14Λ15Λ16

lowast Λ22

0 Λ24Λ25Λ26

lowast lowast Λ33Λ34Λ35Λ36

lowast lowast lowast Λ44Λ45

0

lowast lowast lowast lowast Λ55

0

lowast lowast lowast lowast lowast Λ66

)

)

)

)

)

)

+

(

(

(

(

(

(

L11

L12

L13

L14

L15

L16

lowast L22

L23

L24

L25

L26

lowast lowast L33

L34

L35

L36

lowast lowast lowast L44

L45

L46

lowast lowast lowast lowast L55

L56

lowast lowast lowast lowast lowast L66

)

)

)

)

)

)

lt 0

(45)

with

Λ11=

(

(

(

(

(

(

(

(

(

11987511198941minus 1199091198941

0 0

minus119909119879

1198941

0 minus1199091198941minus 119909119879

11989410

minus1199091198942minus 119909119879

1198942

+11987511198941+ 11987511198942

0 0 11987511198942minus 1199091198942

minus119909119879

1198942

)

)

)

)

)

)

)

)

)

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

Λ12=

(

(

(

(

(

(

(

(

(

(

(

(

11987521198941minus 120576119894119910119894

0 0

minus1205771198941

0 11987521198941+ 11987521198942

0

minus1205771198941minus 1205771198942

minus2120576119894119910119894

0 0 11987521198941minus 120576119894119910119894

minus1205771198942

)

)

)

)

)

)

)

)

)

)

)

)

Λ14=

(

(

(

(

(

(

(

11990911989411198601198941

0 0

+1205761198941198611198911198941198621198941

0 11990911989411198601198942+ 1205761198941198611198911198941198621198941

0

+11990911989421198601198941+ 1205761198941198611198911198941198621198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198621198942

)

)

)

)

)

)

)

Λ15= (

120576119894119860119891119894

0 0

0 2120576119894119860119891119894

0

0 0 120576119894119860119891119894

)

Λ16=

(

(

(

(

(

(

(

11990911989411198611198941

0 0

+120576119894119861119891i1198631198941

0 11990911989411198611198942+ 1205761198941198611198911198941198631198941

0

+11990911989421198611198941+ 1205761198941198611198911198941198631198942

0 0 11990911989421198601198942

+1205761198941198611198911198941198631198942

)

)

)

)

)

)

)

Λ22=

(

(

(

(

(

(

11987531198941minus 119910119894

0 0

minus119910119879

119894

0 11987531198941+ 11987531198942

0

minus2119910119894minus 2119910119879

119894

0 0 11987531198942minus 119910119894

minus119910119879

119894

)

)

)

)

)

)

Λ24=

(

(

(

(

(

(

(

12057711989411198601198941

0 0

+1198611198911198941198621198941

0 12057711989411198601198942+ 1198611198911198941198621198941

0

+12057711989421198601198941+ 1198611198911198941198621198942

0 0 12057711989421198601198942

+1198611198911198941198621198942

)

)

)

)

)

)

)

Λ25= (

119860119891119894

0 0

0 2119860119891119894

0

0 0 119860119891119894

)

Λ26=

(

(

(

(

(

(

(

12057711989411198611198941

0 0

+1198611198911198941198631198941

0 12057711989411198611198942+ 1198611198911198941198631198941

0

+12057711989421198611198941+ 1198611198911198941198631198942

0 0 12057711989421198601198942

+1198611198911198941198631198942

)

)

)

)

)

)

)

Λ33= (

minus1198681times1

0 0

0 minus21198681times1

0

0 0 minus1198681times1

)

Λ34=(

1198671198941minus 1198631198911198941198621198941

0 0

0 1198671198941minus 1198631198911198941198621198941

0

+1198671198942minus 1198631198911198941198621198942

0 0 1198671198942minus 1198631198911198941198621198942

)

Λ35= (

minus119862119891119894

0 0

0 minus2119862119891119894

0

0 0 minus119862119891119894

)

Λ36=(

1198711198941minus 1198631198911198941198631198941

0 0

0 1198711198941minus 1198631198911198941198631198941

0

+1198711198942minus 1198631198911198941198631198942

0 0 1198711198942minus 1198631198911198941198631198942

)

Λ44= (

minus11987511198941

0 0

0 minus11987511198941minus 11987511198942

0

0 0 minus11987511198942

)

Λ45= (

minus11987521198941

0 0

0 minus11987521198941minus 11987521198942

0

0 0 minus11987521198942

)

Λ55= (

minus11987531198941

0 0

0 minus11987531198941minus 11987531198942

0

0 0 minus11987531198942

)

Λ66= (

minus1205742

1198681times1

0 0

0 minus21205742

1198681times1

0

0 0 minus1205742

1198681times1

)

L11sdot sdot sdotL66isin L222

(46)

then there exists a robust switched linear filter in form of (4)such that for all admissible uncertainties the filter error system(5) is robustly asymptotically stable and performance indexholds for any nonzero 120596 isin 119897

2[0infin) where 119860

119891119894= 119910minus1

119894119860119891119894

119861119891119894= 119910minus1

119894119861119891119894 119862119891119894= 119862119891119894 and 119863

119891119894= 119863119891119894

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Mathematical Physics

Proof Let

119875119894(120582) = (

1198751119894(120582) 119875

2119894(120582)

lowast 1198753119894(120582)

) 119877119894(120582) = (

119909119894(120582) 120576119894119910119894

120577119894(120582) 119910

119894

)

(47)

where

1198751119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987511198941

0

0 11987511198942

) 120582 otimes 119868

1198752119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987521198941

0

0 11987521198942

) 120582 otimes 119868

1198753119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

11987531198941

0

0 11987531198942

) 120582 otimes 119868

119909119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1199091198941

0

0 1199091198942

) 120582 otimes 119868

120577119894(119904119902 (120582)) = 120582 otimes 119868

119879

(

1205771198941

0

0 1205771198942

) 120582 otimes 119868

(48)

From Theorem 9 the filter error system is robustly asymp-totically stable with a prescribed 119867

infinnoise-attenuation level

bound 120574 if the following matrix inequality holds

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

1198751119894(120582) minus 119909

119894(120582) 119875

2119894(120582) minus 120576

1198941199101198940 119909119894(120582) 119860119894(120582) 120576

119894119860119891119894

119909119894(120582) 119861119894(120582)

minus119909119894(120582)119879

minus120577119894(120582) +120576

119894119861119891119894119862119894(120582) +120576

119894119861119891119894119863119894(120582)

lowast 1198753119894minus 119910119894

0 120577119894(120582) 119860119894(120582) 119860

119891119894120577119894(120582) 119861119894(120582)

minus119910119879

119894+119861119891119894119862119894(120582) +119861

119891119894119863119894(120582)

lowast lowast minus119868 119867119894(120582) 119862

119891119894119871119894(120582)

minus119863119891119894119862119894(120582) minus119863

119891119894119863119894(120582)

lowast lowast lowast minus1198751119894(120582) minus119875

2119894(120582) 0

lowast lowast lowast lowast minus1198753119894(120582) 0

lowast lowast lowast lowast lowast minus1205742

119868

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

lt 0 (49)

By (48) and Lemma 7 one can obtain that inequality (45) isequivalent to (49)Thus if (45) holds the filter error system isrobustly asymptotically stable with an119867

infinnoise-attenuation

level bound 120574 gt 0 Then the proof is completed

4 Examples

The following example exhibits the effectiveness and appli-cability of the proposed method for robust 119867

infinfiltering

problems with polytopic uncertaintiesConsider the following uncertain discrete-time switched

linear system (1) consisting of two uncertain subsystemswhich is given in [1] There are two groups of vertex matricesin subsystem 1

11986011= 120588(

082 010

minus006 077

) 11986111= 120588(

0

01

)

11986211= 120588 (1 0) 119863

11= 120588

11986711= 120588 (1 0) 119871

11= 0

11986012= 120588(

082 010

minus006 minus075

) 11986112= 120588(

0

minus01

)

11986212= 120588 (1 02) 119863

12= 08120588

11986712= 120588 (1 0) 119871

12= 0

(50)

and two groups of vertex matrices in subsystem 2

11986021= 120588(

082 006

minus010 077

) 11986121= 120588(

01

0

)

11986221= 120588 (0 minus1) 119863

21= minus120588

11986721= 120588 (1 0) 119871

21= 0

11986022= 120588(

082 006

minus010 minus075

) 11986122= 120588(

minus01

0

)

11986222= 120588 (02 minus1) 119863

22= minus08120588

11986722= 120588 (1 0) 119871

22= 0

(51)

Moreover we define the disturbance

120596 (119896) = 0001119890minus0003119896 sin (0002120587119896) (52)

Firstly consider the problem of being robustly asymptot-ically stable with an 119867

infinnoise-attenuation for the uncertain

switched systemwhich was given inTheorem 9The differentminimum 119867

infinnoise-attenuation level bounds 120574 can be

obtained by different methods Table 1 lists the differentcalculation results From Table 1 it can be clearly seen thatthe results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 9

Table 1 Different minimum 120574 for uncertain switched system

120588 1 11 12[1] 12799 16985 62048Theorem 9 12799 16984 40988

0 50 100 150 200minus02

minus015

minus01

minus005

0

005

01

015

02

Magnitude

tstep

x1(t)

x2(t)

Figure 1 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

0 100 200 300 400 500minus3

minus2

minus1

0

1

2

3

tstep

Magnitude

times10minus3

y(t)

120574w(t)

minus120574w(t)

Figure 2119867infinnoise-attenuation level bound 120574 = 40988

In addition for given 120588 = 12 and initial condition119909(0) = [01 minus03]

119879 Figures 1 and 2 show system (1) is robustlyasymptotically stable with an 119867

infinnoise-attenuation level

bound 120574 = 40988Next we consider the problem of robust119867

infinfiltering In

order to get 119867infin

noise-attenuation level bound 120574 we define

Table 2 Different minimum 120574 for robust switched linear filter

120588 1 11 12[1] 05375 11414 44493Theorem 13 05148 10826 42892

0 50 100 150 200minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

Magnitude

tstep

x1(t)

x2(t)

xf1(t)

xf2(t)

Figure 3 The states responses corresponding to uncertain parame-ters 120582

1= 04 and 120582

2= 06

1205761= 1205762= 1 By solving the corresponding convex optimiza-

tion problems in Theorem 13 we obtain the minimum 119867infin

noise-attenuation level bounds 120574 Table 2 lists our results and[1]rsquos results Moreover when 120588 = 12 the admissible filterparameters can be obtained according toTheorem 13 as

1198601198911= (

08770 00505

minus07334 03356

) 1198611198911= (

minus00852

minus01077

)

1198621198911= (minus12057 00112)

1198601198912= (

11457 06743

minus12492 minus08883

)

1198611198911= (

minus02593

05929

)

1198621198912= (minus08695 minus01525)

1198631198911= minus00073 119863

1198912= minus00045

(53)

By comparison using homogeneous parameter-dependentquadratic Lyapunov function for the existence of a robustswitched linear filter in Theorem 13 is better

By giving 119867infin

noise-attenuation level bound 120574 = 42892and initial condition 119909

119890(0) = [01 minus 03 0 0]

119879 Figure 3shows the filtering error system is robustly asymptoticallystable and Figure 4 shows the error response of the resultingfiltering error system by applying above filter It is clear that

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Mathematical Physics

0 50 100 150 200

minus01

minus005

0

005

01

015

Magnitude

tstep

e(t)

Figure 4 Filtering error response corresponding to uncertainparameters 120582

1= 04 and 120582

2= 06

the method inTheorem 13 is feasible and effective against thevariations of uncertain parameter

5 Conclusions

In this paper the problems of being robustly asymptoticallystable with an 119867

infinnoise-attenuation level bounds 120574 and

switched linear filter design for uncertain switched linearsystem are studied by homogeneous parameter-dependentquadratic Lyapunov functions By using this method theless conservative 119867

infinnoise-attenuation level bounds are

obtained Moreover we also get a more feasible and effectivemethod against the variations of uncertain parameter underthe given arbitrary switching signal Numerical examplesillustrate the effectiveness of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by Major Program of National Nat-ural Science Foundation of China under Grant no 11190015National Natural Science Foundation of China under Grantno 61374006 and Graduate Student Innovation Foundationof Jiangsu province under Grant no 3208004904

References

[1] L Zhang P Shi C Wang and H Gao ldquoRobust 119867infin

filteringfor switched linear discrete-time systems with polytopic uncer-taintiesrdquo International Journal of Adaptive Control and SignalProcessing vol 20 no 6 pp 291ndash304 2006

[2] C K Ahn ldquoAn 119867infin

approach to stability analysis of switchedHopfield neural networks with time-delayrdquo Nonlinear Dynam-ics vol 60 no 4 pp 703ndash711 2010

[3] C K Ahn ldquoPassive learning and input-to-state stability ofswitched Hopfield neural networks with time-delayrdquo Informa-tion Sciences vol 180 no 23 pp 4582ndash4594 2010

[4] C K Ahn and M K Song ldquo1198712minus 119871infinfiltering for time-delayed

switched hopfield neural networksrdquo International Journal ofInnovative Computing Information and Control vol 7 no 4 pp1831ndash1844 2011

[5] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[6] C K Ahn ldquoLinear matrix inequality optimization approachto exponential robust filtering for switched Hopfield neuralnetworksrdquo Journal of OptimizationTheory and Applications vol154 no 2 pp 573ndash587 2012

[7] C K Ahn ldquoSwitched exponential state estimation of neuralnetworks based on passivity theoryrdquo Nonlinear Dynamics vol67 no 1 pp 573ndash586 2012

[8] C K Ahn ldquoExponentially convergent state estimation fordelayed switched recurrent neural networksrdquo The EuropeanPhysical Journal E vol 34 no 11 article 122 2011

[9] Z-H Guan D J Hill and X Shen ldquoOn hybrid impulsive andswitching systems and application to nonlinear controlrdquo IEEETransactions on Automatic Control vol 50 no 7 pp 1058ndash10622005

[10] W Ni and D Cheng ldquoControl of switched linear systems withinput saturationrdquo International Journal of Systems Science vol41 no 9 pp 1057ndash1065 2010

[11] Y Chen S Fei and K Zhang ldquoStabilization of impulsiveswitched linear systems with saturated control inputrdquoNonlinearDynamics vol 69 no 3 pp 793ndash804 2012

[12] Y Chen S Fei K Zhang and Z Fu ldquoControl synthesis ofdiscrete-time switched linear systems with input saturationbased onminimumdwell time approachrdquoCircuits Systems andSignal Processing vol 31 no 2 pp 779ndash795 2012

[13] H Mahboubi B Moshiri and S A Khaki ldquoHybrid modelingand optimal control of a two-tank system as a switched systemrdquoWorld Academy of Science Engineering and Technology vol 11pp 319ndash324 2005

[14] F Blanchini S Miani and F Mesquine ldquoA separation principlefor linear switching systems and parametrization of all stabiliz-ing controllersrdquo IEEE Transactions on Automatic Control vol54 no 2 pp 279ndash292 2009

[15] J Li and G-H Yang ldquoFault detection and isolation for discrete-time switched linear systems based on average dwell-timemethodrdquo International Journal of Systems Science vol 44 no12 pp 2349ndash2364 2013

[16] D Ding and G Yang ldquoFinite frequency 119867infin

filtering foruncertain discrete-time switched linear systemsrdquo Progress inNatural Science vol 19 no 11 pp 1625ndash1633 2009

[17] D-W Ding andG-H Yang ldquo119867infinstatic output feedback control

for discrete-time switched linear systems with average dwelltimerdquo IET Control Theory amp Applications vol 4 no 3 pp 381ndash390 2010

[18] H Zhang X Xie and S Tong ldquoHomogenous polynomiallyparameter-dependent 119867

infinfilter designs of discrete-time fuzzy

systemsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 41 no 5 pp 1313ndash1322 2011

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 11

[19] G Chesi A Garulli A Tesi and A Vicino ldquoPolynomiallyparameter-dependent Lyapunov functions for robust 119867

infinper-

formance analysisrdquo in Proceedings of the 16th Triennial WorldCongress pp 457ndash462 Prague Czech Republic 2005

[20] G Chesi A Garulli A Tesi and A Vicino ldquoRobust stability ofpolytopic systems via polynomially parameter-dependent Lya-punov functionsrdquo in Proceedings of the 42nd IEEE Conferenceon Decision and Control pp 4670ndash4675 Maui Hawaii USADecember 2003

[21] P-A Bliman R C L F Oliveira V F Montagner and P LD Peres ldquoExistence of homogeneous polynomial solutions forparameter-dependent linear matrix inequalities with parame-ters in the simplexrdquo in Proceedings of the 45th IEEE Conferenceon Decision and Control (CDC rsquo06) pp 1486ndash1491 San DiegoCalif USA December 2006

[22] R C L F Oliveira and P L D Peres ldquoParameter-dependentLMIs in robust analysis characterization of homogeneous poly-nomially parameter-dependent solutions via LMI relaxationsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1334ndash1340 2007

[23] R C Oliveira M C de Oliveira and P L Peres ldquoConvergentLMI relaxations for robust analysis of uncertain linear systemsusing lifted polynomial parameter-dependent Lyapunov func-tionsrdquo Systems amp Control Letters vol 57 no 8 pp 680ndash6892008

[24] G Chesi ldquoEstablishing stability and instability of matrix hyper-cubesrdquo Systems and Control Letters vol 54 no 4 pp 381ndash3882005

[25] G Chesi ldquoTightness conditions for semidefinite relaxationsof forms minimizationsrdquo IEEE Transactions on Circuits andSystems II Express Briefs vol 55 no 12 pp 1299ndash1303 2008

[26] G Chesi A Garulli A Tesi and A Vicino HomogneousPolynomial Forms for Robustness Analysis of Uncertain Systemsvol 390 Springer Berlin Germany 2009

[27] S Boyd L EI Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in Systems and Contril Theorey SIAMPhiladelphia Pa USA 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of