Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 378376 8 pageshttpdxdoiorg1011552013378376

Research ArticleFinite-Time Synchronization of Singular HybridCoupled Networks

Cong Zheng1 and Jinde Cao12

1 Research Center for Complex Systems and Network Sciences and Department of Mathematics Southeast UniversityNanjing 210096 China

2Department of Mathematics King Abdulaziz University Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Jinde Cao jdcaoseueducn

Received 20 December 2012 Accepted 11 February 2013

Academic Editor Jong Hae Kim

Copyright copy 2013 C Zheng and J CaoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates finite-time synchronization of the singular hybrid coupled networks The singular systems studied in thispaper are assumed to be regular and impulse-free Some sufficient conditions are derived to ensure finite-time synchronization ofthe singular hybrid coupled networks under a state feedback controller by using finite-time stability theory A numerical exampleis finally exploited to show the effectiveness of the obtained results

1 Introduction

In recent years singular systems also known as descrip-tor systems generalized state-space systems differential-algebraic systems or semistate systems are attracting moreand more attentions from many fields of scientific researchbecause they can better describe a larger class of dynamicsystems than the regular onesMany results of regular systemshave been extended to the area about singular systems suchas [1ndash21] For example stability (robust stability or quadraticstability) and stabilization for singular systems have beenstudied via LMI approach in [2ndash8] robust control (or 119867

2

119867infin

control and robust dissipative filtering) for singularsystems has been discussed in [9ndash16] synchronization (orstate estimation) for singular complex networks has beenconsidered in [17ndash21]

Synchronization is an interesting and important charac-teristic in the coupled networks There are a lot of resultsin regular coupled networks Recently some authors studysynchronization of the singular systems such as [17ndash21] andthe references therein In [17] Xiong et al introduced thesingular hybrid coupled systems to describe complex networkwith a special class of constrains They gave a sufficientcondition for global synchronization of singular hybridcoupled system with time-varying nonlinear perturbation

based on Lyapunov stability theory Synchronization issuesare studied for singular systems with delays by using LinearMatrix Inequality (LMI) approach [18] Koo et al consideredsynchronization of singular complex dynamical networkwith time-varying delays [19] Li et al in [20] investigatedsynchronization and state estimation for singular complexdynamical networks with time-varying delays Li et al in[21] investigated robust 119867

infincontrol of synchronization for

uncertain singular complex delayed networks with stochasticswitched coupling

Finite-time synchronization or finite-time control isinteresting topic for its practical application There are someresults on finite-time stability [22ndash26] finite-time synchro-nization [27ndash33] finite-time consensus or agreement [34ndash37] and finite-time observers [38] However these results areobtained for regular systems Up to now to the best of ourknowledge few authors studied finite-time synchronizationof singular hybrid coupled systemswhose structures aremorecomplex than those in [27ndash33] Considering the importantrole of synchronization of complex networks the finite-timesynchronization of singular hybrid coupled networks isworthstudying

Motivated by the previous discussions in this paperwe investigate finite-time synchronization of singular hybridcomplex systems Some sufficient conditions for it are

2 Journal of Applied Mathematics

obtained by the state feedback controller based on the finite-time stability theory Finally a numerical example is exploitedto illustrate the effectiveness of the obtained result

The rest of this paper is organized as follows In Section 2a singular hybrid coupled system is given and some pre-liminaries are briefly outlined In Section 3 some sufficientcriteria are derived for the finite-time synchronization ofthe proposed singular system by the feedback controller InSection 4 an example is provided to show the effectiveness ofthe obtained results Some conclusions are finally drawn inSection 5

2 Model Formulation and Some Preliminaries

Consider a singular hybrid coupled system as follows

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

119873

sum

119895=1

119887119894119895(119905) Γ119909

119895(119905)

119894 = 1 2 119873

(1)

where 119909119894(119905) = (119909

1198941(119905) 1199091198942(119905) 119909

119894119899(119905))119879

isin R119899 represents thestate vector of the 119894th node119860 119864 isin R119899times119899 are constantmatricesand 119864 may be singular Without loss of generality we willassume that 0 lt rank(119864) = 119903 lt 119899 119891(119909

119894(119905) 119905) is a vector-value

function The constant 119888 gt 0 denotes the coupling strengthand Γ = diag(120574

1 1205742 120574

119899) isin R119899times119899 is inner-coupling matrix

between nodes 119861 = (119887119894119895)119873times119873

describes the linear couplingconfiguration of the network which satisfies

119887119894119895

= 119887119895119894 for 119894 = 119895

119887119894119894= minus

119873

sum

119895=1119895 = 119894

119887119894119895 119894 = 1 2 119873

(2)

Remark 1 If rank(119864) = 119899 then system (1) is a general nonsin-gular coupled networkWewill also give a sufficient conditionof the finite-time synchronization for this circumstance SeeCorollary 10

Definition 2 The singular system (1) is said to be synchro-nized in the finite time if for a suitable designed feedbackcontroller there exists a constant 119905lowast gt 0 (which depends onthe initial vector value 119909(0) = (119909

119879

1(0) 119909119879

2(0) 119909

119879

119873(0))119879)

such that lim119905rarr 119905lowast 119909119894(119905) minus 119909

119895(119905) = 0 and 119909

119894(119905) minus 119909

119895(119905) equiv 0

for 119905 gt 119905lowast 119894 119895 = 1 2 119873

Assumption 3 Assume that the singular system (1) is con-nected in the sense that there are no isolated clusters that isthe matrix 119861 is an irreducible matrix

With Assumption 3 we obtain that zero is an eigenvalueof 119861 with multiplicity 1 and all the other eigenvalues of119861 are strictly negative which are denoted by 0 = 120582

1gt

1205822

ge sdot sdot sdot ge 120582119873 At the same time since 119861 is a symmetric

matrix there exists a unitarymatrix119882 = (11988211198822 119882

119873) isin

R119899times119899 such that 119861 = 119882Λ119882119879 with 119882119882

119879= 119868 and Λ =

diag(1205821 1205822 120582

119873)

Let 119904(119905) be a function to which all 119909119894(119905) are expected to

synchronize in the finite time That is the synchronizationstate is 119904(119905) Suppose that 119904(119905) satisfies the equation 119864 119904(119905) =

119860119904(119905) + 119891(119904(119905) 119905) Let 119890119894(119905) = 119909

119894(119905) minus 119904(119905) 119894 = 1 2 119873 We

can obtain the following singular error system

119864 119890119894(119905) = 119860119890

119894(119905) + 119891 (119909

119894(119905) 119905) minus 119891 (119904 (119905) 119905)

+ 119888

119873

sum

119895=1

119887119894119895(119905) Γ119890119895(119905) 119894 = 1 2 119873

(3)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119873(119905)) 119910(119905) = 119890(119905)119882 then

system (3) can be written as

119864 119890 (119905) = 119860119890 (119905) + 119865 (119890 (119905) 119905) + 119888Γ119890 (119905) 119861119879

119864 119910 (119905) = 119860119910 (119905) + 119865 (119890 (119905) 119905)119882 + 119888Γ119910 (119905) Λ

(4)

where 119865(119890(119905) 119905) = (119891(1199091(119905) 119905) minus 119891(119904(119905) 119905) 119891(119909

2(119905) 119905) minus

119891(119904(119905) 119905) 119891(119909119873(119905) 119905)minus119891(119904(119905) 119905)) 119910(119905) = (119910

1(119905) 1199102(119905)

119910119873(119905)) and 119910

119894(119905) = 119890(119905)119882

119894isin R119899 119894 = 1 2 119873 Then system

(4) can be written as

119864 119910119894(119905) = 119860119910

119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ 119888120582119894Γ119910119894(119905)

= (119860 + 119888120582119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894

(5)

Therefore the finite-time synchronization problem ofsystem (1) is equivalent to the finite-time stabilization ofsystem (5) at the origin under the suitable controllers 119906

119894

119894 = 1 2 119873

Assumption 4 Assume that there exist nonnegative constants119871119894such that

1003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119904 (119905) 119905)

1003817100381710038171003817 le 119871119894

1003817100381710038171003817119909119894 (119905) minus 119904 (119905)1003817100381710038171003817

119894 = 1 2 119873

(6)

Assumption 5 There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873 (7)

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120578

119894119868 119894 = 2 119873

(8)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894

Lemma 6 (see [26]) Suppose that the function 119881(119905) [1199050

infin) rarr [0infin) is differentiable (the derivative of 119881(119905) at 1199050

is in fact its right derivative) and (119905) le minus119870(119881(119905))120572 forall119905 ge 0

119881(1199050) ge 0 where 119870 gt 0 0 lt 120572 lt 1 are two constants Then

for any given 1199050 119881(119905) satisfies the following inequality

1198811minus120572

(119905) le 1198811minus120572

(1199050) minus 119870 (1 minus 120572) (119905 minus 119905

0) 119905

0le 119905 le 119905

lowast

119881 (119905) equiv 0 forall119905 gt 119905lowast

(9)

with 119905lowast given by 119905

lowast= 1199050+ 1198811minus120572

(1199050)119870(1 minus 120572)

Journal of Applied Mathematics 3

Lemma 7 (Jensenrsquos Inequality) If 1198861 1198862 119886

119899are positive

numbers and 0 lt 119903 lt 119901 then

(

119899

sum

119894=1

119886119901

119894)

(1119901)

le (

119899

sum

119894=1

119886119903

119894)

(1119903)

(10)

3 Main Results

In this section we consider the finite-time synchronizationof the singular coupled network (1) under the appropriatecontrollers In order to control the states of all nodes to thesynchronization state 119904(119905) in finite time we apply some simplecontrollers 119906

119894(119905) isin R119899 119894 = 1 2 119873 to system (1) Then

the controlled system can be written as

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

119873

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 119873

(11)

Then we have

119864 119890119894(119905) = 119860119890

119894(119905) + 119891 (119909

119894(119905) 119905) minus 119891 (119904 (119905) 119905)

+ 119888

119873

sum

119895=1

119887119894119895Γ119890119895(119905) + 119906

119894

(12)

119864 119910119894(119905) = (119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894 (13)

where V119894= 119906119882119894 119906 = (119906

1 1199062 119906

119873)

With Assumption 5 it follows from the proof of Theo-rem 1 in [2] and Lemma 22 in [3] that the pair (119864 119860 + 119888120582

119894Γ)

is regular and impulse-free that is there exist nonsingularmatrices 119872

119894 119876119894isin R119899times119899 satisfying that

119872119894119864119876119894= diag 119868

119903 0

119872119894(119860 + 119888120582

119894Γ)119876119894= diag 119860

119894 119868119899minus119903

(14)

where 119860119894isin R119903times119903 119894 = 1 2 119873 So system (13) is equivalent

to

1199101

119894(119905) = 119860

1198941199101

119894(119905) + 119872

1

119894119865 (119890 (119905) 119905)119882

119894+ 1198721

119894V119894 (15)

0 = 1199102

119894(119905) + 119872

2

119894119865 (119890 (119905) 119905)119882

119894+ 1198722

119894V119894 (16)

where 119876minus1

119894119910119894(119905) = (

1199101

119894(119905)

1199102

119894(119905)

) 1199101119894(119905) isin R119903 and 119910

2

119894(119905) isin R119899minus119903 And

119872119894= (1198721

119894

1198722

119894

) 1198721119894

isin R119903times119899 1198722119894

isin R(119899minus119903)times119899 119876119894= ( 119876

1

1198941198762

119894) 1198761119894isin

R119899times119903 and 1198762

119894isin R119899times(119899minus119903)

In order to achieve our aim we design the followingcontrollers

V119894= minus119896119872

minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

(17)

where119872119894119864119890 (119905)119882

119894= 119872119894119864119910119894(119905) = 119872

119894119864119876119894119876minus1

119894119910119894(119905)

= (119868119903

0

0 0)(

1199101

119894

1199102

119894

) = (1199101

119894

0)

1003816100381610038161003816119872119894119864119910119894 (119905)1003816100381610038161003816

120573

= (100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

119879

sign (119872119894119864119910119894(119905)) = diag( sign (119910

1

1198941(119905)) sign (119910

1

1198942(119905))

sign (1199101

119894119903(119905)) 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

(18)

119896 gt 0 is a tunable constant and the real number 120573 satisfies0 lt 120573 lt 1 So we obtain 119906 = (V

1 V2 V

119873)119882minus1

=

V(1205851 1205852 120585

119873) That is 119906

119894= V120585119894

Remark 8 From (17) the controllers 119906119894are dependent not

only on the coupled matrix 119861 but also on the singular matrix119864 And from the shape of controllers we only use the states1199101

119894of slow subsystems (15) in controllers V

119894 but we do not

consider the states 1199102119894of fast subsystems (16) It is very special

It is interesting for our future research to designmore generalcontroller whichmakes the singular hybrid coupled networkssynchronize in finite time

Theorem 9 Suppose that Assumptions 3 4 and 5 holdUnder the controllers (17) the singular system (1) is syn-chronized in a finite time 119905

lowast= 119905

0+ (119881

(1minus120573)2(1199050)

119886119896119887minus(1+120573)2

(1 minus 120573)) where 119881(1199050) = sum

119873

119894=1119910119879

119894(1199050)119864119879119875119894119910119894(1199050) =

sum119873

119894=1(119890(1199050)119882119894)119879119864119879119875119894(119890(1199050)119882119894) 119890(1199050) is the initial condition of

119890(119905) and 119886 and 119887 are defined as (25)

Proof Consider the following Lyapunov function

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) (19)

The derivative of 119881(119905) along the trajectory of system (13) is

(119905)

=

119873

sum

119894=1

[119910119879

119894(119905) 119875119879

119894((119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)

+((119860 + 119888120582119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)119879

119875119894119910119894(119905) ]

=

119873

sum

119894=1

[ 119910119879

119894(119905) (119875

119879

119894(119860 + 119888120582

119894Γ) + (119860 + 119888120582

119894Γ)119879

119875119894) 119910119894(119905)

+ 2119910119879

119894(119905) 119875119879

119894119865 (119890 (119905) 119905)119882

119894

minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

]

(20)

4 Journal of Applied Mathematics

By using Assumption 4 one can get the following inequality

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 =

1003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119896=1

[119891 (119909119896(119905) 119905) minus 119891 (119904 (119905) 119905)]119882

119894119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119873

sum

119896=1

119871119896

1003817100381710038171003817119890119896 (119905)1003817100381710038171003817

=

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905) 120585119896

1003817100381710038171003817 le

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905)1003817100381710038171003817 = 119871

1003817100381710038171003817119910 (119905)1003817100381710038171003817

le 119871

119873

sum

119896=1

1003817100381710038171003817119910119896 (119905)1003817100381710038171003817

(21)

where119882119894= (11988211989411198821198942 119882

119894119899)119879 and (120585

1 1205852 120585

119873) = 119882

minus1=

119882119879Define 119872

minus119879

119894119875119894119876119894

= (1198751

1198941198752

119894

1198753

1198941198754

119894

) where 1198751

119894isin R119903times119903 119875

2

119894isin

R119903times(119899minus119903) 1198753119894

isin R(119899minus119903)times119903 and 1198754

119894isin R(119899minus119903)times(119899minus119903) Using (7) and

(8) (see [2]) one can obtain that 1198751119894

= (1198751

119894)119879

gt 0 and 1198752

119894= 0

then

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) =

119873

sum

119894=1

(1199101

119894(119905))119879

1198751

1198941199101

119894(119905) (22)

minus 2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

= minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894(sign (119910

1

1198941(119905))

100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

sign (1199101

119894119903(119905))

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0)

= minus2119896119910119879

119894(119905) 119876minus119879

119894(

1198751

1198940

1198753

1198941198754

119894

)

119879

119872119894119872minus1

119894

times diag (sign (1199101

119894(119905)) 0) (

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

0)

= minus21198961199101119879

119894(119905) 1198751119879

119894sign (119910

1

119894(119905))

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

le minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(23)

Substituting (8) (21) and (23) into (20) and letting 1205781= 0

while 120578119894ge 2119871(119873 minus 1) 119875

119894 119894 = 2 119873 one has

(119905) le

119873

sum

119894=1

[

[

minus120578119894

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

+ 21198711003817100381710038171003817119875119894

1003817100381710038171003817

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573]

]

le

119873

sum

119894=1

minus 2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(24)

Let

119886 ≜ min119894

120582min (1198751119879

119894) 119887 ≜ max

119894

120582max (1198751119879

119894) (25)

From (22) we get

119886

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

le 119881 (119905) le 119887

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

(26)

By the use of (24)ndash(26) and Lemma 7 we can obtain that

(119905) le minus 2119886119896

119873

sum

119894=1

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

le minus2119886119896(

119873

sum

119894=1

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

)

(1+120573)2

le minus 2119886119896(1

119887119881 (119905))

(1+120573)2

= minus2119886119896119887minus(1+120573)2

(119881 (119905))(1+120573)2

(27)

FromLemma 6 we have that the solutions1199101119894(119905) of system

(15) are globally asymptotically stable with respect to 1199101

119894(119905) =

0 in the finite time 119905lowast that is

lim119905rarr 119905lowast

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0 for 119905 ge 119905

lowast (28)

where 119905lowast

= 1199050+ (119881(1minus120573)2

(1199050)119886119896119887minus(1+120573)2

(1 minus 120573))In the following we show that 1199102

119894(119905) are globally asymp-

totically stable with respect to 1199102

119894(119905) = 0 in the finite time 119905

lowastFrom (16) one has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817V1198941003817100381710038171003817 (29)

Similar to the proof of Lemma 22 in [3] and the proof ofTheorem 1 in [17] let 119872

2

119894(1198722

119894)119879

= 119868119899minus119903

which implies that 1198722

119894= 1 One has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +1003817100381710038171003817119906119894119882119894

1003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171003817minus 119896119872

minus1

119894sign (119872

119894119864119910119894(119905))

times1003816100381610038161003816119872119894119864119910119894 (119905)

1003816100381610038161003816

120573100381710038171003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817) + 119896

10038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

times100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

(30)

Then

119873

sum

119894=1

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

[

[

119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817)

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573]

]

(31)

Journal of Applied Mathematics 5

that is

119873

sum

119894=1

(1 minus 1198731198711003817100381710038171003817119876119894

1003817100381710038171003817)100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

(1198731198711003817100381710038171003817119876119894

1003817100381710038171003817

100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

)

(32)

With Assumption 4 there must exist nonsingular matri-ces 119872

119894 119876119894

isin R119899times119899 satisfying the equalities 119872119894119864119876119894

=

diag119868119903 0119872

119894(119860+119888120582

119894Γ)119876119894= diag119860

119894 119868119899minus119903

where119860119894isin R119903times119903

119894 = 1 2 119873 Moreover nonsingular matrices 119876119894can be

suitably chosen to satisfy 1 minus 119873119871 119876119894

gt 0 for forall119894 isin

1 2 119873 Therefore one can obtain lim119905rarr 119905lowast 1199102

119894(119905) = 0

and 1199102

119894(119905) = 0 for 119905 ge 119905

lowast from (28) and (32) 119894 = 1 2 119873Consequently lim

119905rarr 119905lowast 119890

119894(119905) = 0 and 119890

119894(119905) = 0 for

119905 ge 119905lowast 119894 = 1 2 119873 The proof is completed

If rank(119864) = 119899 system (1) is a general nonsingularcoupled network By using the controllers 119906

119894similar to V

119894in

(17) we can derive the finite-time synchronization of system(1) For simplicity let 119864 = 119868

119899 Then we have the following

Corollary 10 When 119864 = 119868119899 under Assumptions 3 and 4 let

the controllers 119906119894(119905) be as follows

119906119894(119905) = minus119896 sign (119890

119894(119905))

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816

120573

119894 = 1 2 119873 (33)

system (1) is synchronized in a finite time

Remark 11 Since the conditions in Assumption 5 are notstrict LMIs problems they cannot be solved directly bythe LMI Matlab Toolbox According to Lemma 1 in [17]Lemma 1 in [9] and Remark 3 in [18] if matrix 119864 has thedecomposition as

119864 = 119880(119868119903

0

0 0)(

Ξ119903

0

0 119868119899minus119903

)119881119879 (34)

where 119880 = (1198801 1198802) 119881 = (119881

1 1198812) and Ξ

119903= diag984858

1 984858

119903

with 984858119894

gt 0 for 119894 = 1 2 119903 then Assumption 5 can betransformed into a strict LMIs problem

Corollary 12 Suppose that Assumptions 3 and 4 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) the singular hybrid coupled network (1) can besynchronized in the finite time in the sense of Definition 2 ifthere exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and 119878

119894isin R(119899minus119903)times119903

119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(35)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894) 119894 =

2 119873 and 119871 = sum119873

119894=1119871119894

Suppose that we choose the average state of all node statesas synchronized state that is 119904(119905) = (1119873)sum

119873

119896=1119909119896(119905) We

have similar results Before giving these results we need someassumptions as follows

Assumption 21015840 Assume that there exist nonnegative con-stants 119871

119894119895such that

10038171003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119909

119895(119905) 119905)

10038171003817100381710038171003817le 119871119894119895

10038171003817100381710038171003817119909119894(119905) minus 119909

119895(119905)

10038171003817100381710038171003817

119894 119895 = 1 2 119873

(36)

Assumption 31015840There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879

119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120589

119894119868

119894 = 2 119873

(37)

where 120589119894gt (4119871119873)(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894 and 119871

119894=

sum119873

119896=1119896 = 119894119871119894119896

Theorem 13 Suppose that Assumptions 3 21015840 and 31015840 hold Bythe controllers (17) the singular hybrid coupled network (1) canbe synchronized to the average state of all node states in thefinite time in the sense of Definition 2

Corollary 14 Suppose that Assumptions 3 and 21015840 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) if there exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and

119878119894isin R(119899minus119903)times119903 119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(38)

where 120578119894gt (4119871119873)(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894)

119894 = 2 119873 119871 = sum119873

119894=1119871119894 and 119871

119894= sum119873

119896=1119896 = 119894119871119894119896 the singular

hybrid coupled network (1) can be synchronized to the averagestate of all node states in the finite time

Remark 15 In this paper we study finite-time synchroniza-tion of the singular hybrid coupled networks when the sin-gular systems studied in this paper are assumed to be regularand impulse-freeHowever itmay bemore complicatedwhenwe do not assume in advance that the systems are regular andimpulse free Synchronization or finite-time synchronizationof singular coupled systems is worth discussing withoutthe assumption that the considered systems are regular andimpulsive free

4 An Illustrative Example

In this section a numerical example will be given to verify thetheoretical results obtained earlier

6 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

obtained by the state feedback controller based on the finite-time stability theory Finally a numerical example is exploitedto illustrate the effectiveness of the obtained result

The rest of this paper is organized as follows In Section 2a singular hybrid coupled system is given and some pre-liminaries are briefly outlined In Section 3 some sufficientcriteria are derived for the finite-time synchronization ofthe proposed singular system by the feedback controller InSection 4 an example is provided to show the effectiveness ofthe obtained results Some conclusions are finally drawn inSection 5

2 Model Formulation and Some Preliminaries

Consider a singular hybrid coupled system as follows

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

119873

sum

119895=1

119887119894119895(119905) Γ119909

119895(119905)

119894 = 1 2 119873

(1)

where 119909119894(119905) = (119909

1198941(119905) 1199091198942(119905) 119909

119894119899(119905))119879

isin R119899 represents thestate vector of the 119894th node119860 119864 isin R119899times119899 are constantmatricesand 119864 may be singular Without loss of generality we willassume that 0 lt rank(119864) = 119903 lt 119899 119891(119909

119894(119905) 119905) is a vector-value

function The constant 119888 gt 0 denotes the coupling strengthand Γ = diag(120574

1 1205742 120574

119899) isin R119899times119899 is inner-coupling matrix

between nodes 119861 = (119887119894119895)119873times119873

describes the linear couplingconfiguration of the network which satisfies

119887119894119895

= 119887119895119894 for 119894 = 119895

119887119894119894= minus

119873

sum

119895=1119895 = 119894

119887119894119895 119894 = 1 2 119873

(2)

Remark 1 If rank(119864) = 119899 then system (1) is a general nonsin-gular coupled networkWewill also give a sufficient conditionof the finite-time synchronization for this circumstance SeeCorollary 10

Definition 2 The singular system (1) is said to be synchro-nized in the finite time if for a suitable designed feedbackcontroller there exists a constant 119905lowast gt 0 (which depends onthe initial vector value 119909(0) = (119909

119879

1(0) 119909119879

2(0) 119909

119879

119873(0))119879)

such that lim119905rarr 119905lowast 119909119894(119905) minus 119909

119895(119905) = 0 and 119909

119894(119905) minus 119909

119895(119905) equiv 0

for 119905 gt 119905lowast 119894 119895 = 1 2 119873

Assumption 3 Assume that the singular system (1) is con-nected in the sense that there are no isolated clusters that isthe matrix 119861 is an irreducible matrix

With Assumption 3 we obtain that zero is an eigenvalueof 119861 with multiplicity 1 and all the other eigenvalues of119861 are strictly negative which are denoted by 0 = 120582

1gt

1205822

ge sdot sdot sdot ge 120582119873 At the same time since 119861 is a symmetric

matrix there exists a unitarymatrix119882 = (11988211198822 119882

119873) isin

R119899times119899 such that 119861 = 119882Λ119882119879 with 119882119882

119879= 119868 and Λ =

diag(1205821 1205822 120582

119873)

Let 119904(119905) be a function to which all 119909119894(119905) are expected to

synchronize in the finite time That is the synchronizationstate is 119904(119905) Suppose that 119904(119905) satisfies the equation 119864 119904(119905) =

119860119904(119905) + 119891(119904(119905) 119905) Let 119890119894(119905) = 119909

119894(119905) minus 119904(119905) 119894 = 1 2 119873 We

can obtain the following singular error system

119864 119890119894(119905) = 119860119890

119894(119905) + 119891 (119909

119894(119905) 119905) minus 119891 (119904 (119905) 119905)

+ 119888

119873

sum

119895=1

119887119894119895(119905) Γ119890119895(119905) 119894 = 1 2 119873

(3)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119873(119905)) 119910(119905) = 119890(119905)119882 then

system (3) can be written as

119864 119890 (119905) = 119860119890 (119905) + 119865 (119890 (119905) 119905) + 119888Γ119890 (119905) 119861119879

119864 119910 (119905) = 119860119910 (119905) + 119865 (119890 (119905) 119905)119882 + 119888Γ119910 (119905) Λ

(4)

where 119865(119890(119905) 119905) = (119891(1199091(119905) 119905) minus 119891(119904(119905) 119905) 119891(119909

2(119905) 119905) minus

119891(119904(119905) 119905) 119891(119909119873(119905) 119905)minus119891(119904(119905) 119905)) 119910(119905) = (119910

1(119905) 1199102(119905)

119910119873(119905)) and 119910

119894(119905) = 119890(119905)119882

119894isin R119899 119894 = 1 2 119873 Then system

(4) can be written as

119864 119910119894(119905) = 119860119910

119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ 119888120582119894Γ119910119894(119905)

= (119860 + 119888120582119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894

(5)

Therefore the finite-time synchronization problem ofsystem (1) is equivalent to the finite-time stabilization ofsystem (5) at the origin under the suitable controllers 119906

119894

119894 = 1 2 119873

Assumption 4 Assume that there exist nonnegative constants119871119894such that

1003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119904 (119905) 119905)

1003817100381710038171003817 le 119871119894

1003817100381710038171003817119909119894 (119905) minus 119904 (119905)1003817100381710038171003817

119894 = 1 2 119873

(6)

Assumption 5 There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873 (7)

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120578

119894119868 119894 = 2 119873

(8)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894

Lemma 6 (see [26]) Suppose that the function 119881(119905) [1199050

infin) rarr [0infin) is differentiable (the derivative of 119881(119905) at 1199050

is in fact its right derivative) and (119905) le minus119870(119881(119905))120572 forall119905 ge 0

119881(1199050) ge 0 where 119870 gt 0 0 lt 120572 lt 1 are two constants Then

for any given 1199050 119881(119905) satisfies the following inequality

1198811minus120572

(119905) le 1198811minus120572

(1199050) minus 119870 (1 minus 120572) (119905 minus 119905

0) 119905

0le 119905 le 119905

lowast

119881 (119905) equiv 0 forall119905 gt 119905lowast

(9)

with 119905lowast given by 119905

lowast= 1199050+ 1198811minus120572

(1199050)119870(1 minus 120572)

Journal of Applied Mathematics 3

Lemma 7 (Jensenrsquos Inequality) If 1198861 1198862 119886

119899are positive

numbers and 0 lt 119903 lt 119901 then

(

119899

sum

119894=1

119886119901

119894)

(1119901)

le (

119899

sum

119894=1

119886119903

119894)

(1119903)

(10)

3 Main Results

In this section we consider the finite-time synchronizationof the singular coupled network (1) under the appropriatecontrollers In order to control the states of all nodes to thesynchronization state 119904(119905) in finite time we apply some simplecontrollers 119906

119894(119905) isin R119899 119894 = 1 2 119873 to system (1) Then

the controlled system can be written as

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

119873

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 119873

(11)

Then we have

119864 119890119894(119905) = 119860119890

119894(119905) + 119891 (119909

119894(119905) 119905) minus 119891 (119904 (119905) 119905)

+ 119888

119873

sum

119895=1

119887119894119895Γ119890119895(119905) + 119906

119894

(12)

119864 119910119894(119905) = (119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894 (13)

where V119894= 119906119882119894 119906 = (119906

1 1199062 119906

119873)

With Assumption 5 it follows from the proof of Theo-rem 1 in [2] and Lemma 22 in [3] that the pair (119864 119860 + 119888120582

119894Γ)

is regular and impulse-free that is there exist nonsingularmatrices 119872

119894 119876119894isin R119899times119899 satisfying that

119872119894119864119876119894= diag 119868

119903 0

119872119894(119860 + 119888120582

119894Γ)119876119894= diag 119860

119894 119868119899minus119903

(14)

where 119860119894isin R119903times119903 119894 = 1 2 119873 So system (13) is equivalent

to

1199101

119894(119905) = 119860

1198941199101

119894(119905) + 119872

1

119894119865 (119890 (119905) 119905)119882

119894+ 1198721

119894V119894 (15)

0 = 1199102

119894(119905) + 119872

2

119894119865 (119890 (119905) 119905)119882

119894+ 1198722

119894V119894 (16)

where 119876minus1

119894119910119894(119905) = (

1199101

119894(119905)

1199102

119894(119905)

) 1199101119894(119905) isin R119903 and 119910

2

119894(119905) isin R119899minus119903 And

119872119894= (1198721

119894

1198722

119894

) 1198721119894

isin R119903times119899 1198722119894

isin R(119899minus119903)times119899 119876119894= ( 119876

1

1198941198762

119894) 1198761119894isin

R119899times119903 and 1198762

119894isin R119899times(119899minus119903)

In order to achieve our aim we design the followingcontrollers

V119894= minus119896119872

minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

(17)

where119872119894119864119890 (119905)119882

119894= 119872119894119864119910119894(119905) = 119872

119894119864119876119894119876minus1

119894119910119894(119905)

= (119868119903

0

0 0)(

1199101

119894

1199102

119894

) = (1199101

119894

0)

1003816100381610038161003816119872119894119864119910119894 (119905)1003816100381610038161003816

120573

= (100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

119879

sign (119872119894119864119910119894(119905)) = diag( sign (119910

1

1198941(119905)) sign (119910

1

1198942(119905))

sign (1199101

119894119903(119905)) 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

(18)

119896 gt 0 is a tunable constant and the real number 120573 satisfies0 lt 120573 lt 1 So we obtain 119906 = (V

1 V2 V

119873)119882minus1

=

V(1205851 1205852 120585

119873) That is 119906

119894= V120585119894

Remark 8 From (17) the controllers 119906119894are dependent not

only on the coupled matrix 119861 but also on the singular matrix119864 And from the shape of controllers we only use the states1199101

119894of slow subsystems (15) in controllers V

119894 but we do not

consider the states 1199102119894of fast subsystems (16) It is very special

It is interesting for our future research to designmore generalcontroller whichmakes the singular hybrid coupled networkssynchronize in finite time

Theorem 9 Suppose that Assumptions 3 4 and 5 holdUnder the controllers (17) the singular system (1) is syn-chronized in a finite time 119905

lowast= 119905

0+ (119881

(1minus120573)2(1199050)

119886119896119887minus(1+120573)2

(1 minus 120573)) where 119881(1199050) = sum

119873

119894=1119910119879

119894(1199050)119864119879119875119894119910119894(1199050) =

sum119873

119894=1(119890(1199050)119882119894)119879119864119879119875119894(119890(1199050)119882119894) 119890(1199050) is the initial condition of

119890(119905) and 119886 and 119887 are defined as (25)

Proof Consider the following Lyapunov function

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) (19)

The derivative of 119881(119905) along the trajectory of system (13) is

(119905)

=

119873

sum

119894=1

[119910119879

119894(119905) 119875119879

119894((119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)

+((119860 + 119888120582119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)119879

119875119894119910119894(119905) ]

=

119873

sum

119894=1

[ 119910119879

119894(119905) (119875

119879

119894(119860 + 119888120582

119894Γ) + (119860 + 119888120582

119894Γ)119879

119875119894) 119910119894(119905)

+ 2119910119879

119894(119905) 119875119879

119894119865 (119890 (119905) 119905)119882

119894

minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

]

(20)

4 Journal of Applied Mathematics

By using Assumption 4 one can get the following inequality

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 =

1003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119896=1

[119891 (119909119896(119905) 119905) minus 119891 (119904 (119905) 119905)]119882

119894119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119873

sum

119896=1

119871119896

1003817100381710038171003817119890119896 (119905)1003817100381710038171003817

=

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905) 120585119896

1003817100381710038171003817 le

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905)1003817100381710038171003817 = 119871

1003817100381710038171003817119910 (119905)1003817100381710038171003817

le 119871

119873

sum

119896=1

1003817100381710038171003817119910119896 (119905)1003817100381710038171003817

(21)

where119882119894= (11988211989411198821198942 119882

119894119899)119879 and (120585

1 1205852 120585

119873) = 119882

minus1=

119882119879Define 119872

minus119879

119894119875119894119876119894

= (1198751

1198941198752

119894

1198753

1198941198754

119894

) where 1198751

119894isin R119903times119903 119875

2

119894isin

R119903times(119899minus119903) 1198753119894

isin R(119899minus119903)times119903 and 1198754

119894isin R(119899minus119903)times(119899minus119903) Using (7) and

(8) (see [2]) one can obtain that 1198751119894

= (1198751

119894)119879

gt 0 and 1198752

119894= 0

then

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) =

119873

sum

119894=1

(1199101

119894(119905))119879

1198751

1198941199101

119894(119905) (22)

minus 2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

= minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894(sign (119910

1

1198941(119905))

100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

sign (1199101

119894119903(119905))

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0)

= minus2119896119910119879

119894(119905) 119876minus119879

119894(

1198751

1198940

1198753

1198941198754

119894

)

119879

119872119894119872minus1

119894

times diag (sign (1199101

119894(119905)) 0) (

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

0)

= minus21198961199101119879

119894(119905) 1198751119879

119894sign (119910

1

119894(119905))

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

le minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(23)

Substituting (8) (21) and (23) into (20) and letting 1205781= 0

while 120578119894ge 2119871(119873 minus 1) 119875

119894 119894 = 2 119873 one has

(119905) le

119873

sum

119894=1

[

[

minus120578119894

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

+ 21198711003817100381710038171003817119875119894

1003817100381710038171003817

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573]

]

le

119873

sum

119894=1

minus 2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(24)

Let

119886 ≜ min119894

120582min (1198751119879

119894) 119887 ≜ max

119894

120582max (1198751119879

119894) (25)

From (22) we get

119886

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

le 119881 (119905) le 119887

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

(26)

By the use of (24)ndash(26) and Lemma 7 we can obtain that

(119905) le minus 2119886119896

119873

sum

119894=1

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

le minus2119886119896(

119873

sum

119894=1

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

)

(1+120573)2

le minus 2119886119896(1

119887119881 (119905))

(1+120573)2

= minus2119886119896119887minus(1+120573)2

(119881 (119905))(1+120573)2

(27)

FromLemma 6 we have that the solutions1199101119894(119905) of system

(15) are globally asymptotically stable with respect to 1199101

119894(119905) =

0 in the finite time 119905lowast that is

lim119905rarr 119905lowast

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0 for 119905 ge 119905

lowast (28)

where 119905lowast

= 1199050+ (119881(1minus120573)2

(1199050)119886119896119887minus(1+120573)2

(1 minus 120573))In the following we show that 1199102

119894(119905) are globally asymp-

totically stable with respect to 1199102

119894(119905) = 0 in the finite time 119905

lowastFrom (16) one has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817V1198941003817100381710038171003817 (29)

Similar to the proof of Lemma 22 in [3] and the proof ofTheorem 1 in [17] let 119872

2

119894(1198722

119894)119879

= 119868119899minus119903

which implies that 1198722

119894= 1 One has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +1003817100381710038171003817119906119894119882119894

1003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171003817minus 119896119872

minus1

119894sign (119872

119894119864119910119894(119905))

times1003816100381610038161003816119872119894119864119910119894 (119905)

1003816100381610038161003816

120573100381710038171003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817) + 119896

10038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

times100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

(30)

Then

119873

sum

119894=1

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

[

[

119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817)

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573]

]

(31)

Journal of Applied Mathematics 5

that is

119873

sum

119894=1

(1 minus 1198731198711003817100381710038171003817119876119894

1003817100381710038171003817)100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

(1198731198711003817100381710038171003817119876119894

1003817100381710038171003817

100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

)

(32)

With Assumption 4 there must exist nonsingular matri-ces 119872

119894 119876119894

isin R119899times119899 satisfying the equalities 119872119894119864119876119894

=

diag119868119903 0119872

119894(119860+119888120582

119894Γ)119876119894= diag119860

119894 119868119899minus119903

where119860119894isin R119903times119903

119894 = 1 2 119873 Moreover nonsingular matrices 119876119894can be

suitably chosen to satisfy 1 minus 119873119871 119876119894

gt 0 for forall119894 isin

1 2 119873 Therefore one can obtain lim119905rarr 119905lowast 1199102

119894(119905) = 0

and 1199102

119894(119905) = 0 for 119905 ge 119905

lowast from (28) and (32) 119894 = 1 2 119873Consequently lim

119905rarr 119905lowast 119890

119894(119905) = 0 and 119890

119894(119905) = 0 for

119905 ge 119905lowast 119894 = 1 2 119873 The proof is completed

If rank(119864) = 119899 system (1) is a general nonsingularcoupled network By using the controllers 119906

119894similar to V

119894in

(17) we can derive the finite-time synchronization of system(1) For simplicity let 119864 = 119868

119899 Then we have the following

Corollary 10 When 119864 = 119868119899 under Assumptions 3 and 4 let

the controllers 119906119894(119905) be as follows

119906119894(119905) = minus119896 sign (119890

119894(119905))

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816

120573

119894 = 1 2 119873 (33)

system (1) is synchronized in a finite time

Remark 11 Since the conditions in Assumption 5 are notstrict LMIs problems they cannot be solved directly bythe LMI Matlab Toolbox According to Lemma 1 in [17]Lemma 1 in [9] and Remark 3 in [18] if matrix 119864 has thedecomposition as

119864 = 119880(119868119903

0

0 0)(

Ξ119903

0

0 119868119899minus119903

)119881119879 (34)

where 119880 = (1198801 1198802) 119881 = (119881

1 1198812) and Ξ

119903= diag984858

1 984858

119903

with 984858119894

gt 0 for 119894 = 1 2 119903 then Assumption 5 can betransformed into a strict LMIs problem

Corollary 12 Suppose that Assumptions 3 and 4 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) the singular hybrid coupled network (1) can besynchronized in the finite time in the sense of Definition 2 ifthere exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and 119878

119894isin R(119899minus119903)times119903

119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(35)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894) 119894 =

2 119873 and 119871 = sum119873

119894=1119871119894

Suppose that we choose the average state of all node statesas synchronized state that is 119904(119905) = (1119873)sum

119873

119896=1119909119896(119905) We

have similar results Before giving these results we need someassumptions as follows

Assumption 21015840 Assume that there exist nonnegative con-stants 119871

119894119895such that

10038171003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119909

119895(119905) 119905)

10038171003817100381710038171003817le 119871119894119895

10038171003817100381710038171003817119909119894(119905) minus 119909

119895(119905)

10038171003817100381710038171003817

119894 119895 = 1 2 119873

(36)

Assumption 31015840There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879

119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120589

119894119868

119894 = 2 119873

(37)

where 120589119894gt (4119871119873)(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894 and 119871

119894=

sum119873

119896=1119896 = 119894119871119894119896

Theorem 13 Suppose that Assumptions 3 21015840 and 31015840 hold Bythe controllers (17) the singular hybrid coupled network (1) canbe synchronized to the average state of all node states in thefinite time in the sense of Definition 2

Corollary 14 Suppose that Assumptions 3 and 21015840 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) if there exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and

119878119894isin R(119899minus119903)times119903 119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(38)

where 120578119894gt (4119871119873)(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894)

119894 = 2 119873 119871 = sum119873

119894=1119871119894 and 119871

119894= sum119873

119896=1119896 = 119894119871119894119896 the singular

hybrid coupled network (1) can be synchronized to the averagestate of all node states in the finite time

Remark 15 In this paper we study finite-time synchroniza-tion of the singular hybrid coupled networks when the sin-gular systems studied in this paper are assumed to be regularand impulse-freeHowever itmay bemore complicatedwhenwe do not assume in advance that the systems are regular andimpulse free Synchronization or finite-time synchronizationof singular coupled systems is worth discussing withoutthe assumption that the considered systems are regular andimpulsive free

4 An Illustrative Example

In this section a numerical example will be given to verify thetheoretical results obtained earlier

6 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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

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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

Journal of Applied Mathematics 3

Lemma 7 (Jensenrsquos Inequality) If 1198861 1198862 119886

119899are positive

numbers and 0 lt 119903 lt 119901 then

(

119899

sum

119894=1

119886119901

119894)

(1119901)

le (

119899

sum

119894=1

119886119903

119894)

(1119903)

(10)

3 Main Results

In this section we consider the finite-time synchronizationof the singular coupled network (1) under the appropriatecontrollers In order to control the states of all nodes to thesynchronization state 119904(119905) in finite time we apply some simplecontrollers 119906

119894(119905) isin R119899 119894 = 1 2 119873 to system (1) Then

the controlled system can be written as

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

119873

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 119873

(11)

Then we have

119864 119890119894(119905) = 119860119890

119894(119905) + 119891 (119909

119894(119905) 119905) minus 119891 (119904 (119905) 119905)

+ 119888

119873

sum

119895=1

119887119894119895Γ119890119895(119905) + 119906

119894

(12)

119864 119910119894(119905) = (119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894 (13)

where V119894= 119906119882119894 119906 = (119906

1 1199062 119906

119873)

With Assumption 5 it follows from the proof of Theo-rem 1 in [2] and Lemma 22 in [3] that the pair (119864 119860 + 119888120582

119894Γ)

is regular and impulse-free that is there exist nonsingularmatrices 119872

119894 119876119894isin R119899times119899 satisfying that

119872119894119864119876119894= diag 119868

119903 0

119872119894(119860 + 119888120582

119894Γ)119876119894= diag 119860

119894 119868119899minus119903

(14)

where 119860119894isin R119903times119903 119894 = 1 2 119873 So system (13) is equivalent

to

1199101

119894(119905) = 119860

1198941199101

119894(119905) + 119872

1

119894119865 (119890 (119905) 119905)119882

119894+ 1198721

119894V119894 (15)

0 = 1199102

119894(119905) + 119872

2

119894119865 (119890 (119905) 119905)119882

119894+ 1198722

119894V119894 (16)

where 119876minus1

119894119910119894(119905) = (

1199101

119894(119905)

1199102

119894(119905)

) 1199101119894(119905) isin R119903 and 119910

2

119894(119905) isin R119899minus119903 And

119872119894= (1198721

119894

1198722

119894

) 1198721119894

isin R119903times119899 1198722119894

isin R(119899minus119903)times119899 119876119894= ( 119876

1

1198941198762

119894) 1198761119894isin

R119899times119903 and 1198762

119894isin R119899times(119899minus119903)

In order to achieve our aim we design the followingcontrollers

V119894= minus119896119872

minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

(17)

where119872119894119864119890 (119905)119882

119894= 119872119894119864119910119894(119905) = 119872

119894119864119876119894119876minus1

119894119910119894(119905)

= (119868119903

0

0 0)(

1199101

119894

1199102

119894

) = (1199101

119894

0)

1003816100381610038161003816119872119894119864119910119894 (119905)1003816100381610038161003816

120573

= (100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

119879

sign (119872119894119864119910119894(119905)) = diag( sign (119910

1

1198941(119905)) sign (119910

1

1198942(119905))

sign (1199101

119894119903(119905)) 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus119903

)

(18)

119896 gt 0 is a tunable constant and the real number 120573 satisfies0 lt 120573 lt 1 So we obtain 119906 = (V

1 V2 V

119873)119882minus1

=

V(1205851 1205852 120585

119873) That is 119906

119894= V120585119894

Remark 8 From (17) the controllers 119906119894are dependent not

only on the coupled matrix 119861 but also on the singular matrix119864 And from the shape of controllers we only use the states1199101

119894of slow subsystems (15) in controllers V

119894 but we do not

consider the states 1199102119894of fast subsystems (16) It is very special

It is interesting for our future research to designmore generalcontroller whichmakes the singular hybrid coupled networkssynchronize in finite time

Theorem 9 Suppose that Assumptions 3 4 and 5 holdUnder the controllers (17) the singular system (1) is syn-chronized in a finite time 119905

lowast= 119905

0+ (119881

(1minus120573)2(1199050)

119886119896119887minus(1+120573)2

(1 minus 120573)) where 119881(1199050) = sum

119873

119894=1119910119879

119894(1199050)119864119879119875119894119910119894(1199050) =

sum119873

119894=1(119890(1199050)119882119894)119879119864119879119875119894(119890(1199050)119882119894) 119890(1199050) is the initial condition of

119890(119905) and 119886 and 119887 are defined as (25)

Proof Consider the following Lyapunov function

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) (19)

The derivative of 119881(119905) along the trajectory of system (13) is

(119905)

=

119873

sum

119894=1

[119910119879

119894(119905) 119875119879

119894((119860 + 119888120582

119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)

+((119860 + 119888120582119894Γ) 119910119894(119905) + 119865 (119890 (119905) 119905)119882

119894+ V119894)119879

119875119894119910119894(119905) ]

=

119873

sum

119894=1

[ 119910119879

119894(119905) (119875

119879

119894(119860 + 119888120582

119894Γ) + (119860 + 119888120582

119894Γ)119879

119875119894) 119910119894(119905)

+ 2119910119879

119894(119905) 119875119879

119894119865 (119890 (119905) 119905)119882

119894

minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

]

(20)

4 Journal of Applied Mathematics

By using Assumption 4 one can get the following inequality

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 =

1003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119896=1

[119891 (119909119896(119905) 119905) minus 119891 (119904 (119905) 119905)]119882

119894119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119873

sum

119896=1

119871119896

1003817100381710038171003817119890119896 (119905)1003817100381710038171003817

=

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905) 120585119896

1003817100381710038171003817 le

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905)1003817100381710038171003817 = 119871

1003817100381710038171003817119910 (119905)1003817100381710038171003817

le 119871

119873

sum

119896=1

1003817100381710038171003817119910119896 (119905)1003817100381710038171003817

(21)

where119882119894= (11988211989411198821198942 119882

119894119899)119879 and (120585

1 1205852 120585

119873) = 119882

minus1=

119882119879Define 119872

minus119879

119894119875119894119876119894

= (1198751

1198941198752

119894

1198753

1198941198754

119894

) where 1198751

119894isin R119903times119903 119875

2

119894isin

R119903times(119899minus119903) 1198753119894

isin R(119899minus119903)times119903 and 1198754

119894isin R(119899minus119903)times(119899minus119903) Using (7) and

(8) (see [2]) one can obtain that 1198751119894

= (1198751

119894)119879

gt 0 and 1198752

119894= 0

then

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) =

119873

sum

119894=1

(1199101

119894(119905))119879

1198751

1198941199101

119894(119905) (22)

minus 2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

= minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894(sign (119910

1

1198941(119905))

100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

sign (1199101

119894119903(119905))

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0)

= minus2119896119910119879

119894(119905) 119876minus119879

119894(

1198751

1198940

1198753

1198941198754

119894

)

119879

119872119894119872minus1

119894

times diag (sign (1199101

119894(119905)) 0) (

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

0)

= minus21198961199101119879

119894(119905) 1198751119879

119894sign (119910

1

119894(119905))

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

le minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(23)

Substituting (8) (21) and (23) into (20) and letting 1205781= 0

while 120578119894ge 2119871(119873 minus 1) 119875

119894 119894 = 2 119873 one has

(119905) le

119873

sum

119894=1

[

[

minus120578119894

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

+ 21198711003817100381710038171003817119875119894

1003817100381710038171003817

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573]

]

le

119873

sum

119894=1

minus 2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(24)

Let

119886 ≜ min119894

120582min (1198751119879

119894) 119887 ≜ max

119894

120582max (1198751119879

119894) (25)

From (22) we get

119886

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

le 119881 (119905) le 119887

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

(26)

By the use of (24)ndash(26) and Lemma 7 we can obtain that

(119905) le minus 2119886119896

119873

sum

119894=1

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

le minus2119886119896(

119873

sum

119894=1

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

)

(1+120573)2

le minus 2119886119896(1

119887119881 (119905))

(1+120573)2

= minus2119886119896119887minus(1+120573)2

(119881 (119905))(1+120573)2

(27)

FromLemma 6 we have that the solutions1199101119894(119905) of system

(15) are globally asymptotically stable with respect to 1199101

119894(119905) =

0 in the finite time 119905lowast that is

lim119905rarr 119905lowast

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0 for 119905 ge 119905

lowast (28)

where 119905lowast

= 1199050+ (119881(1minus120573)2

(1199050)119886119896119887minus(1+120573)2

(1 minus 120573))In the following we show that 1199102

119894(119905) are globally asymp-

totically stable with respect to 1199102

119894(119905) = 0 in the finite time 119905

lowastFrom (16) one has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817V1198941003817100381710038171003817 (29)

Similar to the proof of Lemma 22 in [3] and the proof ofTheorem 1 in [17] let 119872

2

119894(1198722

119894)119879

= 119868119899minus119903

which implies that 1198722

119894= 1 One has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +1003817100381710038171003817119906119894119882119894

1003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171003817minus 119896119872

minus1

119894sign (119872

119894119864119910119894(119905))

times1003816100381610038161003816119872119894119864119910119894 (119905)

1003816100381610038161003816

120573100381710038171003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817) + 119896

10038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

times100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

(30)

Then

119873

sum

119894=1

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

[

[

119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817)

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573]

]

(31)

Journal of Applied Mathematics 5

that is

119873

sum

119894=1

(1 minus 1198731198711003817100381710038171003817119876119894

1003817100381710038171003817)100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

(1198731198711003817100381710038171003817119876119894

1003817100381710038171003817

100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

)

(32)

With Assumption 4 there must exist nonsingular matri-ces 119872

119894 119876119894

isin R119899times119899 satisfying the equalities 119872119894119864119876119894

=

diag119868119903 0119872

119894(119860+119888120582

119894Γ)119876119894= diag119860

119894 119868119899minus119903

where119860119894isin R119903times119903

119894 = 1 2 119873 Moreover nonsingular matrices 119876119894can be

suitably chosen to satisfy 1 minus 119873119871 119876119894

gt 0 for forall119894 isin

1 2 119873 Therefore one can obtain lim119905rarr 119905lowast 1199102

119894(119905) = 0

and 1199102

119894(119905) = 0 for 119905 ge 119905

lowast from (28) and (32) 119894 = 1 2 119873Consequently lim

119905rarr 119905lowast 119890

119894(119905) = 0 and 119890

119894(119905) = 0 for

119905 ge 119905lowast 119894 = 1 2 119873 The proof is completed

If rank(119864) = 119899 system (1) is a general nonsingularcoupled network By using the controllers 119906

119894similar to V

119894in

(17) we can derive the finite-time synchronization of system(1) For simplicity let 119864 = 119868

119899 Then we have the following

Corollary 10 When 119864 = 119868119899 under Assumptions 3 and 4 let

the controllers 119906119894(119905) be as follows

119906119894(119905) = minus119896 sign (119890

119894(119905))

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816

120573

119894 = 1 2 119873 (33)

system (1) is synchronized in a finite time

Remark 11 Since the conditions in Assumption 5 are notstrict LMIs problems they cannot be solved directly bythe LMI Matlab Toolbox According to Lemma 1 in [17]Lemma 1 in [9] and Remark 3 in [18] if matrix 119864 has thedecomposition as

119864 = 119880(119868119903

0

0 0)(

Ξ119903

0

0 119868119899minus119903

)119881119879 (34)

where 119880 = (1198801 1198802) 119881 = (119881

1 1198812) and Ξ

119903= diag984858

1 984858

119903

with 984858119894

gt 0 for 119894 = 1 2 119903 then Assumption 5 can betransformed into a strict LMIs problem

Corollary 12 Suppose that Assumptions 3 and 4 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) the singular hybrid coupled network (1) can besynchronized in the finite time in the sense of Definition 2 ifthere exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and 119878

119894isin R(119899minus119903)times119903

119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(35)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894) 119894 =

2 119873 and 119871 = sum119873

119894=1119871119894

Suppose that we choose the average state of all node statesas synchronized state that is 119904(119905) = (1119873)sum

119873

119896=1119909119896(119905) We

have similar results Before giving these results we need someassumptions as follows

Assumption 21015840 Assume that there exist nonnegative con-stants 119871

119894119895such that

10038171003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119909

119895(119905) 119905)

10038171003817100381710038171003817le 119871119894119895

10038171003817100381710038171003817119909119894(119905) minus 119909

119895(119905)

10038171003817100381710038171003817

119894 119895 = 1 2 119873

(36)

Assumption 31015840There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879

119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120589

119894119868

119894 = 2 119873

(37)

where 120589119894gt (4119871119873)(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894 and 119871

119894=

sum119873

119896=1119896 = 119894119871119894119896

Theorem 13 Suppose that Assumptions 3 21015840 and 31015840 hold Bythe controllers (17) the singular hybrid coupled network (1) canbe synchronized to the average state of all node states in thefinite time in the sense of Definition 2

Corollary 14 Suppose that Assumptions 3 and 21015840 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) if there exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and

119878119894isin R(119899minus119903)times119903 119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(38)

where 120578119894gt (4119871119873)(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894)

119894 = 2 119873 119871 = sum119873

119894=1119871119894 and 119871

119894= sum119873

119896=1119896 = 119894119871119894119896 the singular

hybrid coupled network (1) can be synchronized to the averagestate of all node states in the finite time

Remark 15 In this paper we study finite-time synchroniza-tion of the singular hybrid coupled networks when the sin-gular systems studied in this paper are assumed to be regularand impulse-freeHowever itmay bemore complicatedwhenwe do not assume in advance that the systems are regular andimpulse free Synchronization or finite-time synchronizationof singular coupled systems is worth discussing withoutthe assumption that the considered systems are regular andimpulsive free

4 An Illustrative Example

In this section a numerical example will be given to verify thetheoretical results obtained earlier

6 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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 Journal of Applied Mathematics

By using Assumption 4 one can get the following inequality

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 =

1003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119896=1

[119891 (119909119896(119905) 119905) minus 119891 (119904 (119905) 119905)]119882

119894119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119873

sum

119896=1

119871119896

1003817100381710038171003817119890119896 (119905)1003817100381710038171003817

=

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905) 120585119896

1003817100381710038171003817 le

119873

sum

119896=1

119871119896

1003817100381710038171003817119910 (119905)1003817100381710038171003817 = 119871

1003817100381710038171003817119910 (119905)1003817100381710038171003817

le 119871

119873

sum

119896=1

1003817100381710038171003817119910119896 (119905)1003817100381710038171003817

(21)

where119882119894= (11988211989411198821198942 119882

119894119899)119879 and (120585

1 1205852 120585

119873) = 119882

minus1=

119882119879Define 119872

minus119879

119894119875119894119876119894

= (1198751

1198941198752

119894

1198753

1198941198754

119894

) where 1198751

119894isin R119903times119903 119875

2

119894isin

R119903times(119899minus119903) 1198753119894

isin R(119899minus119903)times119903 and 1198754

119894isin R(119899minus119903)times(119899minus119903) Using (7) and

(8) (see [2]) one can obtain that 1198751119894

= (1198751

119894)119879

gt 0 and 1198752

119894= 0

then

119881 (119905) =

119873

sum

119894=1

119910119879

119894(119905) 119864119879119875119894119910119894(119905) =

119873

sum

119894=1

(1199101

119894(119905))119879

1198751

1198941199101

119894(119905) (22)

minus 2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894sign (119872

119894119864119890 (119905)119882

119894)1003816100381610038161003816119872119894119864119890 (119905)119882

119894

1003816100381610038161003816

120573

= minus2119896119910119879

119894(119905) 119875119879

119894119872minus1

119894(sign (119910

1

1198941(119905))

100381610038161003816100381610038161199101

1198941(119905)

10038161003816100381610038161003816

120573

sign (1199101

119894119903(119905))

100381610038161003816100381610038161199101

119894119903(119905)

10038161003816100381610038161003816

120573

0 0)

= minus2119896119910119879

119894(119905) 119876minus119879

119894(

1198751

1198940

1198753

1198941198754

119894

)

119879

119872119894119872minus1

119894

times diag (sign (1199101

119894(119905)) 0) (

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

0)

= minus21198961199101119879

119894(119905) 1198751119879

119894sign (119910

1

119894(119905))

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

le minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(23)

Substituting (8) (21) and (23) into (20) and letting 1205781= 0

while 120578119894ge 2119871(119873 minus 1) 119875

119894 119894 = 2 119873 one has

(119905) le

119873

sum

119894=1

[

[

minus120578119894

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

+ 21198711003817100381710038171003817119875119894

1003817100381710038171003817

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

minus2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573]

]

le

119873

sum

119894=1

minus 2119896120582min (1198751119879

119894)100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

(24)

Let

119886 ≜ min119894

120582min (1198751119879

119894) 119887 ≜ max

119894

120582max (1198751119879

119894) (25)

From (22) we get

119886

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

le 119881 (119905) le 119887

119873

sum

119894=1

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817

2

(26)

By the use of (24)ndash(26) and Lemma 7 we can obtain that

(119905) le minus 2119886119896

119873

sum

119894=1

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

1+120573

le minus2119886119896(

119873

sum

119894=1

1003817100381710038171003817119910119894 (119905)1003817100381710038171003817

2

)

(1+120573)2

le minus 2119886119896(1

119887119881 (119905))

(1+120573)2

= minus2119886119896119887minus(1+120573)2

(119881 (119905))(1+120573)2

(27)

FromLemma 6 we have that the solutions1199101119894(119905) of system

(15) are globally asymptotically stable with respect to 1199101

119894(119905) =

0 in the finite time 119905lowast that is

lim119905rarr 119905lowast

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0

100381710038171003817100381710038171199101

119894(119905)

10038171003817100381710038171003817= 0 for 119905 ge 119905

lowast (28)

where 119905lowast

= 1199050+ (119881(1minus120573)2

(1199050)119886119896119887minus(1+120573)2

(1 minus 120573))In the following we show that 1199102

119894(119905) are globally asymp-

totically stable with respect to 1199102

119894(119905) = 0 in the finite time 119905

lowastFrom (16) one has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +100381710038171003817100381710038171198722

119894

10038171003817100381710038171003817

1003817100381710038171003817V1198941003817100381710038171003817 (29)

Similar to the proof of Lemma 22 in [3] and the proof ofTheorem 1 in [17] let 119872

2

119894(1198722

119894)119879

= 119868119899minus119903

which implies that 1198722

119894= 1 One has

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

1003817100381710038171003817119865 (119890 (119905) 119905)119882119894

1003817100381710038171003817 +1003817100381710038171003817119906119894119882119894

1003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119910119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171003817minus 119896119872

minus1

119894sign (119872

119894119864119910119894(119905))

times1003816100381610038161003816119872119894119864119910119894 (119905)

1003816100381610038161003816

120573100381710038171003817100381710038171003817

le 119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817) + 119896

10038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

times100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

(30)

Then

119873

sum

119894=1

100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

[

[

119871

119873

sum

119895=1

10038171003817100381710038171003817119876119895

10038171003817100381710038171003817(100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817+

100381710038171003817100381710038171199102

119895(119905)

10038171003817100381710038171003817)

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573]

]

(31)

Journal of Applied Mathematics 5

that is

119873

sum

119894=1

(1 minus 1198731198711003817100381710038171003817119876119894

1003817100381710038171003817)100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

(1198731198711003817100381710038171003817119876119894

1003817100381710038171003817

100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

)

(32)

With Assumption 4 there must exist nonsingular matri-ces 119872

119894 119876119894

isin R119899times119899 satisfying the equalities 119872119894119864119876119894

=

diag119868119903 0119872

119894(119860+119888120582

119894Γ)119876119894= diag119860

119894 119868119899minus119903

where119860119894isin R119903times119903

119894 = 1 2 119873 Moreover nonsingular matrices 119876119894can be

suitably chosen to satisfy 1 minus 119873119871 119876119894

gt 0 for forall119894 isin

1 2 119873 Therefore one can obtain lim119905rarr 119905lowast 1199102

119894(119905) = 0

and 1199102

119894(119905) = 0 for 119905 ge 119905

lowast from (28) and (32) 119894 = 1 2 119873Consequently lim

119905rarr 119905lowast 119890

119894(119905) = 0 and 119890

119894(119905) = 0 for

119905 ge 119905lowast 119894 = 1 2 119873 The proof is completed

If rank(119864) = 119899 system (1) is a general nonsingularcoupled network By using the controllers 119906

119894similar to V

119894in

(17) we can derive the finite-time synchronization of system(1) For simplicity let 119864 = 119868

119899 Then we have the following

Corollary 10 When 119864 = 119868119899 under Assumptions 3 and 4 let

the controllers 119906119894(119905) be as follows

119906119894(119905) = minus119896 sign (119890

119894(119905))

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816

120573

119894 = 1 2 119873 (33)

system (1) is synchronized in a finite time

Remark 11 Since the conditions in Assumption 5 are notstrict LMIs problems they cannot be solved directly bythe LMI Matlab Toolbox According to Lemma 1 in [17]Lemma 1 in [9] and Remark 3 in [18] if matrix 119864 has thedecomposition as

119864 = 119880(119868119903

0

0 0)(

Ξ119903

0

0 119868119899minus119903

)119881119879 (34)

where 119880 = (1198801 1198802) 119881 = (119881

1 1198812) and Ξ

119903= diag984858

1 984858

119903

with 984858119894

gt 0 for 119894 = 1 2 119903 then Assumption 5 can betransformed into a strict LMIs problem

Corollary 12 Suppose that Assumptions 3 and 4 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) the singular hybrid coupled network (1) can besynchronized in the finite time in the sense of Definition 2 ifthere exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and 119878

119894isin R(119899minus119903)times119903

119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(35)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894) 119894 =

2 119873 and 119871 = sum119873

119894=1119871119894

Suppose that we choose the average state of all node statesas synchronized state that is 119904(119905) = (1119873)sum

119873

119896=1119909119896(119905) We

have similar results Before giving these results we need someassumptions as follows

Assumption 21015840 Assume that there exist nonnegative con-stants 119871

119894119895such that

10038171003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119909

119895(119905) 119905)

10038171003817100381710038171003817le 119871119894119895

10038171003817100381710038171003817119909119894(119905) minus 119909

119895(119905)

10038171003817100381710038171003817

119894 119895 = 1 2 119873

(36)

Assumption 31015840There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879

119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120589

119894119868

119894 = 2 119873

(37)

where 120589119894gt (4119871119873)(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894 and 119871

119894=

sum119873

119896=1119896 = 119894119871119894119896

Theorem 13 Suppose that Assumptions 3 21015840 and 31015840 hold Bythe controllers (17) the singular hybrid coupled network (1) canbe synchronized to the average state of all node states in thefinite time in the sense of Definition 2

Corollary 14 Suppose that Assumptions 3 and 21015840 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) if there exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and

119878119894isin R(119899minus119903)times119903 119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(38)

where 120578119894gt (4119871119873)(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894)

119894 = 2 119873 119871 = sum119873

119894=1119871119894 and 119871

119894= sum119873

119896=1119896 = 119894119871119894119896 the singular

hybrid coupled network (1) can be synchronized to the averagestate of all node states in the finite time

Remark 15 In this paper we study finite-time synchroniza-tion of the singular hybrid coupled networks when the sin-gular systems studied in this paper are assumed to be regularand impulse-freeHowever itmay bemore complicatedwhenwe do not assume in advance that the systems are regular andimpulse free Synchronization or finite-time synchronizationof singular coupled systems is worth discussing withoutthe assumption that the considered systems are regular andimpulsive free

4 An Illustrative Example

In this section a numerical example will be given to verify thetheoretical results obtained earlier

6 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

Journal of Applied Mathematics 5

that is

119873

sum

119894=1

(1 minus 1198731198711003817100381710038171003817119876119894

1003817100381710038171003817)100381710038171003817100381710038171199102

119894(119905)

10038171003817100381710038171003817le

119873

sum

119894=1

(1198731198711003817100381710038171003817119876119894

1003817100381710038171003817

100381710038171003817100381710038171199101

119895(119905)

10038171003817100381710038171003817

+11989610038171003817100381710038171003817119872minus1

119894

10038171003817100381710038171003817

100381610038161003816100381610038161199101

119894(119905)

10038161003816100381610038161003816

120573

)

(32)

With Assumption 4 there must exist nonsingular matri-ces 119872

119894 119876119894

isin R119899times119899 satisfying the equalities 119872119894119864119876119894

=

diag119868119903 0119872

119894(119860+119888120582

119894Γ)119876119894= diag119860

119894 119868119899minus119903

where119860119894isin R119903times119903

119894 = 1 2 119873 Moreover nonsingular matrices 119876119894can be

suitably chosen to satisfy 1 minus 119873119871 119876119894

gt 0 for forall119894 isin

1 2 119873 Therefore one can obtain lim119905rarr 119905lowast 1199102

119894(119905) = 0

and 1199102

119894(119905) = 0 for 119905 ge 119905

lowast from (28) and (32) 119894 = 1 2 119873Consequently lim

119905rarr 119905lowast 119890

119894(119905) = 0 and 119890

119894(119905) = 0 for

119905 ge 119905lowast 119894 = 1 2 119873 The proof is completed

If rank(119864) = 119899 system (1) is a general nonsingularcoupled network By using the controllers 119906

119894similar to V

119894in

(17) we can derive the finite-time synchronization of system(1) For simplicity let 119864 = 119868

119899 Then we have the following

Corollary 10 When 119864 = 119868119899 under Assumptions 3 and 4 let

the controllers 119906119894(119905) be as follows

119906119894(119905) = minus119896 sign (119890

119894(119905))

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816

120573

119894 = 1 2 119873 (33)

system (1) is synchronized in a finite time

Remark 11 Since the conditions in Assumption 5 are notstrict LMIs problems they cannot be solved directly bythe LMI Matlab Toolbox According to Lemma 1 in [17]Lemma 1 in [9] and Remark 3 in [18] if matrix 119864 has thedecomposition as

119864 = 119880(119868119903

0

0 0)(

Ξ119903

0

0 119868119899minus119903

)119881119879 (34)

where 119880 = (1198801 1198802) 119881 = (119881

1 1198812) and Ξ

119903= diag984858

1 984858

119903

with 984858119894

gt 0 for 119894 = 1 2 119903 then Assumption 5 can betransformed into a strict LMIs problem

Corollary 12 Suppose that Assumptions 3 and 4 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) the singular hybrid coupled network (1) can besynchronized in the finite time in the sense of Definition 2 ifthere exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and 119878

119894isin R(119899minus119903)times119903

119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(35)

where 120578119894gt 2119871(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894) 119894 =

2 119873 and 119871 = sum119873

119894=1119871119894

Suppose that we choose the average state of all node statesas synchronized state that is 119904(119905) = (1119873)sum

119873

119896=1119909119896(119905) We

have similar results Before giving these results we need someassumptions as follows

Assumption 21015840 Assume that there exist nonnegative con-stants 119871

119894119895such that

10038171003817100381710038171003817119891 (119909119894(119905) 119905) minus 119891 (119909

119895(119905) 119905)

10038171003817100381710038171003817le 119871119894119895

10038171003817100381710038171003817119909119894(119905) minus 119909

119895(119905)

10038171003817100381710038171003817

119894 119895 = 1 2 119873

(36)

Assumption 31015840There exist matrices 119875119894such that

119864119879119875119894= 119875119879

119894119864 ge 0 119894 = 1 2 119873

1198601198791198751+ 119875119879

1119860 lt 0

(119860 + 119888120582119894Γ)119879

119875119894+ 119875119879

119894(119860 + 119888120582

119894Γ) le minus120589

119894119868

119894 = 2 119873

(37)

where 120589119894gt (4119871119873)(119873 minus 1) 119875

119894 119871 = sum

119873

119894=1119871119894 and 119871

119894=

sum119873

119896=1119896 = 119894119871119894119896

Theorem 13 Suppose that Assumptions 3 21015840 and 31015840 hold Bythe controllers (17) the singular hybrid coupled network (1) canbe synchronized to the average state of all node states in thefinite time in the sense of Definition 2

Corollary 14 Suppose that Assumptions 3 and 21015840 hold andmatrix 119864 has the decomposition as (34) in Remark 11 By thecontrollers (17) if there exist matrices 119879

119894isin R119903times119903 119879

119894ge 0 and

119878119894isin R(119899minus119903)times119903 119894 = 1 2 119873 such that

119860119879(11988011198791119880119879

1119864 + 119880

21198781) + (119880

11198791119880119879

1119864 + 119880

21198781)119879

119860 lt 0

(119860 + 119888120582119894Γ)119879

(1198801119879119894119880119879

1119864 + 119880

2119878119894)

+ (1198801119879119894119880119879

1119864 + 119880

2119878119894)119879

(119860 + 119888120582119894Γ) le minus120578

119894119868

(38)

where 120578119894gt (4119871119873)(119873 minus 1) 119875

119894 119875119894= (1198801119879119894119880119879

1119864 + 119880

2119878119894)

119894 = 2 119873 119871 = sum119873

119894=1119871119894 and 119871

119894= sum119873

119896=1119896 = 119894119871119894119896 the singular

hybrid coupled network (1) can be synchronized to the averagestate of all node states in the finite time

Remark 15 In this paper we study finite-time synchroniza-tion of the singular hybrid coupled networks when the sin-gular systems studied in this paper are assumed to be regularand impulse-freeHowever itmay bemore complicatedwhenwe do not assume in advance that the systems are regular andimpulse free Synchronization or finite-time synchronizationof singular coupled systems is worth discussing withoutthe assumption that the considered systems are regular andimpulsive free

4 An Illustrative Example

In this section a numerical example will be given to verify thetheoretical results obtained earlier

6 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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 Journal of Applied Mathematics

Example 16 Consider the following singular hybrid couplednetwork which is similar to one given in [18]

119864119894(119905) = 119860119909

119894(119905) + 119891 (119909

119894(119905) 119905) + 119888

6

sum

119895=1

119887119894119895Γ119909119895(119905) + 119906

119894

119894 = 1 2 6

(39)

where 119909119894(119905) = (119909

1

119894(119905) 1199092

119894(119905))119879 119891(119909

119894(119905) 119905) = ((1

15) tanh(1199091119894(119905)) tanh(1199092

119894(119905)))119879 119904(119905) = (0 0)

119879 119871119894

= 115119871 = 25 119888 = 1 and

119864 = (8 0

0 0) 119860 = (

minus10 1

1 minus10) Γ = (

1 0

0 1)

119861 = (

(

minus5 1 1 1 1 1

1 minus4 1 1 1 0

1 1 minus4 1 0 1

1 1 1 minus4 1 0

1 1 0 1 minus4 1

1 0 1 0 1 minus3

)

)

(40)Since 119861 is symmetric matrix and its six eigenvalues are 120582

1=

0 1205822

= minus3 1205823

= minus4 1205824

= minus5 1205825

= minus6and 1205826

= minus6there exists a unitary matrix

119882 = (1198821 119882

6)

=

(((((((((((((((

(

minus1

radic6

0 0 0 minus1

radic6

minus2

radic6

minus1

radic6

1

radic6

0 minus1

radic2

1

radic6

0

minus1

radic6

0 minus1

radic2

0 minus1

radic6

1

radic6

minus1

radic6

1

radic6

01

radic2

1

radic6

0

minus1

radic6

01

radic2

0 minus1

radic6

1

radic6

minus1

radic6

minus2

radic6

0 01

radic6

0

)))))))))))))))

)

(41)

such that 119861 = 119882Λ119882119879 and Λ = diag(0 minus3 minus4 minus5 minus6 minus6)

Choose

119872119894= (

11

10 minus 120582119894

0 1

) 119876119894= (

1

80

1

8 (10 minus 120582119894)

1

120582119894minus 10

)

119875119894= (

1 0

0 1)

(42)120578119894gt 4 119894 = 1 2 6 and 120573 = 12 satisfying Assumptions

3 4 and 5 Under the controllers 119906119894defined in Theorem 9

the singular hybrid system (39) can be synchronized in thefinite time 119905

lowast= 71858 according toTheorem 9 if 119896 = 1 If the

controller gain 119896 = 5 119905lowast = 14372 Corresponding simulationresults are shown in Figures 1 and 2 with initial conditions119890(0) = (

1 3 2 07 25 03

05 minus1 08 minus2 15 minus05)

0 2 4 6 8 10

0

05

1

15

2

25

minus05

minus1

minus15

119905

1198901 119894(119905)

Figure 1 Error variable 1198901

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

0 2 4 6 8 10

0

005

01

015

02

025

119905

1198902 119894(119905)

minus005

minus1

Figure 2 Error variable 1198902

119894(119905) (119894 = 1 2 6) of system (39) with

119896 = 1

5 Conclusions

In this paper we discuss finite-time synchronization of thesingular hybrid coupled networks with the assumption thatthe considered singular systems are regular and impulsive-free Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled net-works under a state feedback controller by finite-time stabilitytheory A numerical example is finally exploited to show theeffectiveness of the obtained results It will be an interestingtopic for the future researches to extend new methods tostudy synchronization robust control pinning control andfinite-time synchronization of singular hybrid coupled net-works without the assumption that the considered singularsystems are regular and impulsive-free

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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

Journal of Applied Mathematics 7

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China under Grant nos 61272530and 11072059 and the Jiangsu Provincial Natural ScienceFoundation of China under Grant no BK2012741

References

[1] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986

[2] S Xu P VanDooren R Stefan and J Lam ldquoRobust stability andstabilization for singular systemswith state delay and parameteruncertaintyrdquo IEEE Transactions on Automatic Control vol 47no 7 pp 1122ndash1128 2002

[3] G Lu and D W C Ho ldquoGeneralized quadratic stability forcontinuous-time singular systemswith nonlinear perturbationrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 818ndash823 2006

[4] G Grammel ldquoRobustness of exponential stability to singularperturbations and delaysrdquo Systems amp Control Letters vol 57 no6 pp 505ndash510 2008

[5] D M Stipanovic and D D Siljak ldquoRobust stability and stabi-lization of discrete-time non-linear systems the LMI approachrdquoInternational Journal of Control vol 74 no 9 pp 873ndash879 2001

[6] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo

in Proceedings of the 38th IEEE Conference on Decision andControl pp 4092ndash4097 Phoenix Ariz USA 1999

[7] P Wang and J Zhang ldquoStability of solutions for nonlinearsingular systems with delayrdquo Applied Mathematics Letters vol25 no 10 pp 1291ndash1295 2012

[8] Y Zhang C Liu and X Mu ldquoRobust finite-time stabilizationof uncertain singular Markovian jump systemsrdquo Applied Math-ematical Modelling vol 36 no 10 pp 5109ndash5121 2012

[9] L Q Zhang B Huang and J Lam ldquoLMI synthesis of 1198672

and mixed 1198672119867infin

controllers for singular systemsrdquo IEEETransactions on Circuits and Systems II vol 50 pp 615ndash6262003

[10] I Masubuchi Y Kamitane A Ohara and N Suda ldquo119867infin

control for descriptor systems a matrix inequalities approachrdquoAutomatica vol 33 no 4 pp 669ndash673 1997

[11] S Xu and J Lam Robust Control and Filtering of SingularSystems vol 332 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2006

[12] E Fridman ldquo119867infin-control of linear state-delay descriptor sys-

tems an LMI approachrdquo Linear Algebra and Its Applications vol15 pp 271ndash302 2002

[13] Y Xia J Zhang and E-K Boukas ldquoControl of discrete singularhybrid systemsrdquo Automatica vol 44 no 10 pp 2635ndash26412008

[14] S Xu and C Yang ldquo119867infin

state feedback control for discretesingular systemsrdquo IEEE Transactions on Automatic Control vol45 no 7 pp 1405ndash1409 2000

[15] M S Mahmoud and N B Almutairi ldquoStability and imple-mentable119867

infinfilters for singular systems with nonlinear pertur-

bationsrdquo Nonlinear Dynamics vol 57 no 3 pp 401ndash410 2009[16] Z G Feng and J Lam ldquoRobust reliable dissipative filtering for

discrete delay singular systemsrdquo Signal Processing vol 92 pp3010ndash3025 2012

[17] W J Xiong D W C Ho and J D Cao ldquoSynchronizationanalysis of singular hybrid coupled networksrdquo Physics Letters Avol 372 pp 6633ndash6637 2008

[18] J F Zeng and J D Cao ldquoSynchronisation in singgular hybridcomplex networks with delayed couplingrdquo International Journalof Systems Control and Communications vol 3 pp 144ndash1572011

[19] J H Koo D H Ji and S C Won ldquoSynchronization of sin-gular complex dynamical networks with time-varying delaysrdquoAppliedMathematics and Computation vol 217 no 8 pp 3916ndash3923 2010

[20] H J Li Z J Ning Y H Yin and Y Tang ldquoSynchronization andstate estimation for singular complex dynamical networks withtime-varying delaysrdquoCommunications in Nonlinear Science andNumerical Simulation vol 18 pp 194ndash208 2013

[21] S Li W Tang and J Zhang ldquoRobust control for synchro-nization of singular complex delayed networks with stochasticswitched couplingrdquo International Journal of Computer Mathe-matics vol 89 no 10 pp 1332ndash1344 2012

[22] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[23] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[24] LWang S Sun andC Xia ldquoFinite-time stability ofmulti-agentsystem in disturbed environmentrdquoNonlinear Dynamics vol 67no 3 pp 2009ndash2016 2012

[25] V T Haimo ldquoFinite time controllersrdquo SIAM Journal on Controland Optimization vol 24 no 4 pp 760ndash770 1986

[26] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[27] Y Chen and J H Lu ldquoFinite time synchronization of complexdynamical networksrdquo Journal of Systems Science andMathemat-ical Sciences vol 29 no 10 pp 1419ndash1430 2009

[28] H Wang Z Han Q Xie and W Zhang ldquoFinite-time chaoscontrol of unified chaotic systems with uncertain parametersrdquoNonlinear Dynamics vol 55 no 4 pp 323ndash328 2009

[29] Y Z Sun W Li and D H Zhao ldquoFinite-time stochastic outersynchronization between two complex dynamical networkswith different topologiesrdquo Chaos vol 22 no 2 Article ID023152 2012

[30] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011

[31] X Yang and J Cao ldquoFinite-time stochastic synchronization ofcomplex networksrdquo Applied Mathematical Modelling vol 34no 11 pp 3631ndash3641 2010

[32] S Li and Y-P Tian ldquoFinite time synchronization of chaoticsystemsrdquoChaos Solitons and Fractals vol 15 no 2 pp 303ndash3102003

[33] W L Yang X D Xia Y C Dong and S W Zheng ldquoFinitetime synchronization between two different chaotic systemswith uncertain parametersrdquo Computer amp Information Sciencevol 3 pp 174ndash179 2010

[34] L Wang and F Xiao ldquoFinite-time consensus problems fornetworks of dynamic agentsrdquo IEEE Transactions on AutomaticControl vol 55 no 4 pp 950ndash955 2010

[35] S Yang J Cao and J Lu ldquoA new protocol for finite-timeconsensus of detail-balancedmulti-agent networksrdquoChaos vol22 Article ID 043134 2012

8 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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 Journal of Applied Mathematics

[36] L Wang Z Chen Z Liu and Z Yuan ldquoFinite time agreementprotocol design of multi-agent systems with communicationdelaysrdquoAsian Journal of Control vol 11 no 3 pp 281ndash286 2009

[37] Q HuiWM Haddad and S P Bhat ldquoFinite-time semistabilityand consensus for nonlinear dynamical networksrdquo IEEE Trans-actions on Automatic Control vol 53 no 8 pp 1887ndash1900 2008

[38] Y Shen and X Xia ldquoSemi-global finite-time observers fornonlinear systemsrdquo Automatica vol 44 no 12 pp 3152ndash31562008

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