Abstract—This paper presents a general uncertain complex singular dynamical network model. Since the nodes of the network are singular systems, the overall network can only be described by a singular model. Due to the flexible of singular systems in describing practical applications, the complex singular dynamical network has more applications than the regular complex network in which the nodes are described by the state-space dynamical systems. Robust stabilization of such uncertain complex singular dynamical network is investigated via impulsive control. Based on the Lyapunov stability theory, sufficient conditions for the stabilization of the uncertain complex singular dynamical network are obtained under impulsive control. A systematic-design procedure is presented, and a numerical example is provided for illustration.

I. INTRODUCTION N recent years, stability and stabilization of complex dynamical networks (CDNs) have become important research subjects with the rapidly increasing research in

CDNs. Some research efforts [1-3] focus on the inherent stability properties of CDNs, not subject to any controllers. Other efforts [4-7] have addressed the stabilization problem of CDNs by utilizing different control strategies, such as the pinning control strategy [4, 5], the decentralized control strategy [6], the hybrid control strategy [7], and so on.

On the other hand, singular systems have attracted particular attention in the scientific community largely due to the important applications in different fields, such as circuits, biological and mechanical systems and so on [8]. In particular, singular systems can still model physical systems in some cases when state-space ones are not applicable [13, 14]. In situations when the nodes of CDNs can only be described by singular systems, the CDNs under consideration become complex singular dynamical networks (CSDNs). Due to the flexible of singular systems in describing practical applications, CSDNs have more applications than the regular CDNs in which the nodes are described by the state-space dynamical systems.

Manuscript received Feb, 2009. This work was supported in part by the

National Natural Science Foundation of China under Grant 60704035, 60834002, 60802002, Program for New Century Excellent Talents in University, and Hubei Province Foundation under Grant 2007ABA222.

M. Yang, Y.-W. Wang, and Z.-H. Guan are with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R.China (e-mail: wangyw@mail.hust.edu.cn ).

Y.-W. Wang and H.O. Wang are with the Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA (e-mail: wangh@bu.edu).

Y.-W. Wang is the corresponding author (phone & fax: 86-27-87543130; e-mail: wangyw@mail.hust.edu.cn).

Note that lots of efforts focus on the CDNs of which the coupling topologies are known exactly [1-7]. However, it is usually hard to obtain the exact coupling topologies for most of the practical networks, such as the internet networks, the social networks, and so on. Therefore, it is interesting to study the CDNs of which the topologies are partially or even completely unknown but bounded.

Inspired by [9, 10], in this paper, a general uncertain complex singular dynamical network (UCSDN) model is presented. The uncertain coupling topology function is presented in the model instead of the linearly coupling term. The impulsive control strategy is applied to stabilize the UCSDN, due to its advantages in, for instance, utilizing small control efforts, using less information, simple implementation, and providing effective mechanisms to cope with complex systems [16-19]. Guan et al. [19] studied an isolated singular-impulsive system with perturbations. To our knowledge, however, there are no results on the CSDN under impulsive control, which consists of an amount of interconnected singular-impulsive systems. Based on the Lyapunov stability theory, robust stabilization conditions for the UCSDN under impulsive control are obtained. In addition, a systematic-design procedure for the impulsive control is presented, and a numerical example is carried out to demonstrate the effectiveness of the proposed robust stabilization strategy. For simplicity, the systems of all nodes are assumed to be identical.

Notations: The norms of a matrix nnRQ ×∈ and a vector

( ) nTn Rxxx ∈= ,,1 are ( )( ) 2/1

max: QQQ Tλ= and

2/1

1

2: ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

=

n

jjxx respectively.

II. PROBLEM FORMULATION AND PRELIMINARIES Consider an uncertain complex singular dynamical

network (UCSDN), which consists of N identical coupled nodes and thus is described by

( ) ( ) ( )Niiii xxthxtgtAxtxE ,,,),( 1++=( ) ,,,1, Nitu =+ (1)

where ( ) ( ) ( ) ( ) nTiniii Rtxtxtxtx ∈= ),,,( 21 is the

state vector of the node i , 0tt ≥ . E may be a singular

matrix, and ( ) nrErank ≤=<0 . A is a constant nn ×

Robust Stabilization of Uncertain Complex Singular Dynamical Networks via Impulsive Control

Meng Yang, Yan-Wu Wang, Hua O. Wang, and Zhi-Hong Guan

I

Joint 48th IEEE Conference on Decision and Control and28th Chinese Control ConferenceShanghai, P.R. China, December 16-18, 2009

ThAIn2.7

978-1-4244-3872-3/09/$25.00 ©2009 IEEE 3891

matrix. nn RRRg →×+: is a time-varying nonlinear

vector- valued function with 0)0,( ≡tg , and nN

i RRh ×+: nR→ are unknown nonlinear diffusively

coupling functions with 0)0,,0,( ≡thi . Remark 1: The UCSDN (1) is a general form of the

complex linearly coupled networks studies in [1-7, 9, 10],

when IE = or/and ( ) ( )txbxxth jN

j ijNi ∑ ==

11 ,,, .

For (1), an impulsive controller is designed as follows:

[ ],)()()(1∑∞

=−=

k kik tttExctu δ (2)

where kc is a gain constant, )(•δ is the Dirac impulse

function, with discontinuous points <<<< kttt 21 ,

∞=∞→ kk

tlim where 01 tt > .

From (1)-(2), the controlled UCSDN can be derived as: ( ) ( ) ( )Niiii xxthxtgtAxtxE ,,,),( 1++=

[ ] ,,,1,)()(1

NitttExck kik =−+∑∞

=δ (3)

The following lemmas are needed in the subsequent

discussion. Lemma 1 [11]: If X and Y are real matrices with

appropriate dimensions, there exists a constant 0>ε , such

that YYXXXYYX TTTT 1−+≤+ εε . Lemma 2 [15]: If ( )( )tztL , and ( )( )tzU satisfy the

Lipschiz condition, there exists a uniqueness of solution to nonlinear impulsive differential equation which is written as:

( ) ( )( )( ) ( )( )

( )⎪⎩

⎪⎨

⎧

==

==Δ≠=

+ .,2,1,

,,,

00 kztz

tttzUtztttztLtz

kk

k

,

where ( ) nRtz ∈ , nn RRRL →×+: , nnk RRU →: .

III. ROBUST STABILIZATION OF THE UCSDN In this section, sufficient conditions for the global robust

stabilization of the UCSDN are derived. The following assumptions are made for the results.

Assumption 1: For 0tt ≥ , nii Rxx ∈~, , there exist

nonnegative real constants ijγ ( )Nji ≤≤ ,1 , such that

( ) ( )NiNi xxthxxth ~,,~,,,, 11 −

( ) ( )txtx jjN

j ij~

1−≤∑ =

γ .

Remark 2: Assumption 1 is naturally held for the complex linear coupled networks [1-7, 9, 10].

Assumption 2: For 0tt ≥ , nii Rxx ∈~, , Ni ,,1…= ,

there exists a nonnegative real constant K , such that ( ) ( ) ( ) ( )txtxKxtgxtg iiii

~~,, −≤− .

Assumption 3: There exist a matrix P and a constant λ such that

0≥= EPPE TT , (4)

( ) PEPPAPPA TTTT λεε ++++=Φ −− 12

11

( ) 022

21 <Γ++ ∗ nIK εε , (5)

where ( ) 2/1

12

1 112max ∑∑ ∑ == =≤≤∗ −=Γ N

j jiN

j

N

m jmjiNiγγγ .

Using the Schur complement, (5) can be rewritten as:

[ ]( )

011

21

1

22

21 <

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−

−Γ++++

−−−∗

n

T

n

TTT

IP

PIK

PEAPPA

εεεε

λ

Assumption 4: The pair ( )AE, is regular and impulse free.

Assumption 4 implies that there exist nonsingular

matrices nnRQW ×∈, satisfying that

( )0,rIdiagWEQ = , ( )rnIAdiagWAQ −= ,1 , (6)

where rrRA ×∈1 . Then the UCSDN (3) is equivalent to

( )( )

( ) ( ) ( ) ( )( ) ( )⎪

⎪

⎩

⎪⎪

⎨

⎧

===

===−=Δ

++=

≠++=

++

+

.,,1,,

,2,1,,

,,,,),()(0

,,,,),()()(

200

2100

1

1111122

2111

11

1

Niytyyty

ktttyctytyty

xxthWxtgWty

ttxxthWxtgWtyAty

iiii

kikkikii

Niii

kNiiii

(7)

where ( ) ( ) ( )( ) ( )( )( )TTi

Tiii tytytxQty 211 ,== − , ( )tyi

1

rR∈ , ( ) rni Rty −∈2 , ( )TTT WWW 21 ,= , ( )21,QQQ = ,

nrRW ×∈1 , ( ) nrnRW ×−∈2 , rnRQ ×∈1 , ( )rnnRQ −×∈2 . Equation (7) can be written as

( ) ( )( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

===

===−=Δ

⊗+⊗+=≠⊗+

⊗+⊗=

++

+

.,,1,,

,2,1,,

,,),()(0,,

),()()(

10021001

1111

222

1

1111

NiYtYYtY

ktttYctYtYtY

XtHWIXtGWItYttXtHWI

XtGWItYAItY

kkkk

NN

kN

NN

(8)

where ( ) ( )( ) ( )( )( )TTN

T tytytY 1111 ,,= , ( ) rNRtY ∈1 ,

ThAIn2.7

3892

( ) ( )( ) ( )( )( )TTN

T tytytY 2212 ,,= , ( ) ( )NrnRtY −∈2 ,

( ) ( )( ) nNTTT RtYtYQX ∈= 21 , , ( ),,(),( 1xtgXtG T=

) nNTN

T Rxtg ∈,(, , (( ,,(),( 11 xthXtH =

) ( )( ) ) nNTTNN

TN Rxxthx ∈,,,,,), 1 .

Theorem 1: The solution to (3) exists and is unique on

[ )∞∈ ,0tt for any given compatible initial condition if

( ) 01 21 >Γ+−=Ξ ∗ QK holds.

Proof. Similar to [12], rn

T IWW −=22 ,. Then

( )( )TNN WIWI 22 ⊗⊗ ( )( )TT

NN WIWI 22 ⊗⊗=

( ) ( ) ( )rnNTT

NN IWWII −=⊗= 22 , thus 12 =⊗WI N .

First, for any given ( ) ( ) ( )NrnRtYtY −∈22~, and any

[ )∞∈ ,0tt , it can be obtained that 2

22112211 )~,(),( YQYQtGYQYQtG +−+

∑ =+−+= N

i iiii yQyQtgyQyQtg1

222

11

22

11 )~,(),(

222221

2 ~ii

N

iyyQK −≤∑ =

2

222

22 ~YYQK −= ,

and, 2

22112211 )~,(),( YQYQtHYQYQtH +−+

222

11

212

111

12

21

1212

111

)~,,~,(

),,,(

NNi

N

i NNi

yQyQyQyQth

yQyQyQyQth

++−

++=∑ =

( )( )∑

∑ ∑

≠=

= =

−−+

⎜⎝⎛ −≤

N

jmm mmjjimij

jjijN

i

N

j

yyyy

yyQ

,12222

22221 1

22

~~2

~

γγ

γ

⎜⎝⎛ −≤∑ ∑= =

22221 1

22

~jjij

N

i

N

jyyQ γ

∑ ≠= ⎟⎠⎞⎟⎠⎞⎜

⎝⎛ −+−+ N

jmm mmjjimij yyyy,1

222222 ~~γγ

2

2222

2~YYQ −Γ≤ ∗ .

Consequently, ( ) ( )[ ]221122112 ,),( YQYQtHYQYQtGWI N +++⊗

( ) ( )( )221122112~,)~,( YQYQtHYQYQtGWIN +++⊗−

{})~,(),(

)~,(),(

22112211

221122112

YQYQtHYQYQtH

YQYQtGYQYQtGWIN

+−++

+−+⊗≤

[ ] 222~YYQK −Γ+≤ ∗ ( ) 22

2/1 ~1 YY −+≤ −ξ ,

where 0>ξ is a sufficiently small constant. Then, for any

[ )∞∈ ,0tt and any given ( )tY2 , following the fixed-point

principle, there exists a unique solution ( )12 ,YtY ϕ= .

Next, from Lemma 2, if it can be shown that ( )1Yf

( )( ) ( )( )12111211 ,,,, YtQYQtGYtQYQtH ϕϕ +++= is

Lipschitz with respect to 1Y , it can be concluded that 1Y exists and is unique.

For any given ( ) ( ) rNRtYtY ∈11~, ,

( ) ( )( )( ) ( )( )( )( ) ( )( )12111211

12111211

11

~,~,,,

~,~,,,

~

YtQYQtHYtQYQtH

YtQYQtGYtQYQtG

YfYf

ϕϕ

ϕϕ

+−++

+−+=

−

[ ] ( ) ( )112~,, YtYtQK ϕϕ −Γ+≤ ∗

[ ] 111~YYQK −Γ++ ∗ , (9)

which implies that if ( )12 ,YtY ϕ= is Lipschitz with respect

to 1Y , ( )1Yf is Lipschitz with respect to 1Y . Consider the second equation of (8), it can be obtained

( ) ( )( ) ( ) ( )1111~~,, YfYfYtYt −=− ϕϕ . (10)

From (9) and (10),

( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( ) ,~~,,

1~,,11

11111

2112/1

YYQKYtYt

QKYtYt

−Γ+≤−⋅

Γ+−≤−+−∗

∗−

ϕϕ

ϕϕξ

then, ( ) ( )( )11~,, YtYt ϕϕ −

( ) ( ) ,~111 1111 YYQK −Γ++++≤ ∗

− ξξξ

which implies that both ( )12 ,YtY ϕ= and ( )1Yf is

Lipschitz with respect to 1Y . The proof is thus completed. For convenience, define

2)1( kk c+≥η , ,2,1=k , (11) Theorem 2: Suppose that Assumptions 1-4 are satisfied,

and ( ){ } 01min112 >+−=Ξ ∑ =≤≤

QK N

j jiNiγ .

i) If 0>λ and there exists a constants β satisfying

0≥≥ βλ such that

0)(ln 1 ≤−− −kkk ttβη , ,2,1=k (12)

ii) If 0≤λ , and there exists a constant 1≥α such that

ThAIn2.7

3893

( ) ( ) 0ln 1 ≤−− + kkk ttλαη , ,2,1=k (13) then the UCSDN (1) is asymptotically stable under the control of (2), and 1=α implies that the controlled UCSDN (3) is stable.

Proof. Construct a Lyapunov candidate as

( ) ( ) ( )tPxEtxtV N

i iTT

i∑ ==

1.

Define ⎟⎟⎠

⎞⎜⎜⎝

⎛=−

42

31

PPPP

PQW T . According to (4) and (6),

it can be deduced that 011 >= TPP and 03 =P , then

( ) ( ) ( ) ( )( ) ( )tyPtytPxEtxtV N

i iT

iN

i iTT

i ∑∑ ====

11

11

1.

For any ],( 1 kk ttt −∈ , the total derivative of )(tV with respect to (3) is

( ) ( )( ){ ( ) ( ) ( )tPxxtgtxAPPAtxtV iiT

iN

iTTT

i ,1

++=∑ =

( ) ( ) ( ) ( )tPxxxhxtgPtx iNTii

TTi ,,, 1++

( ) ( )}NiTT

i xxhPtx ,,1+

( )( ){ ( ) ( ) ( )iiT

iN

iTTT

i xtgxtgtxAPPAtx ,,11ε++≤∑ =

( ) ( ) ( )tPxPtx iTT

i1

21

1−− ++ εε

( ) ( )}NiNTi xxhxxh ,,,, 112ε+

According to Assumption 1-2, for ( ) 0~ =txi , ( )ixtg ,

( )txK i≤ and ( ) ( )txxxth jN

j ijNi ∑ =≤

11 ,,, γ .

Thus,

( ) ( )( ){ ( ) )()(211

txtxKtxAPPAtxtV iTii

N

iTTT

i ε++≤∑ =

( ) ( ) ( )tPxPtx iTT

i1

21

1−− ++ εε

( )}∑ ∑= ≠=++ N

j

N

jmm mjimijjij txtxtx1 ,1

222 )()(2)( γγγε

( )( ){ ( ) )()(211

txtxKtxAPPAtx iTii

N

iTTT

i ε++≤∑ =

( ) ( ) ( ) (∑ =−− +++ N

j jijiTT

i txtPxPtx1

222

12

11 )(γεεε

( ))}∑ ≠=++ N

jmm mjimij txtx,1

22)()(γγ

( ){ [ 21

12

111

)( KPPAPPAtx TTTN

iTi εεε ++++= −−

=∑( )] } )(2

12

1 12 txI inN

j jiN

j

N

m jmji ∑∑ ∑ == =−+ γγγε

∑ =−< N

i iTT

i tPxEtx1

)()(λ )(tVλ−= ,

where λ is a constant satisfying (5). Then,

( )[ ] ],(,exp)()( 111 kkkk ttttttVtV −−+− ∈−−≤ λ . (14)

On the other hand, when += ktt ,

( )( ) ( )+=

++ ∑= kN

i iT

kik tyPtytV1

11

1)(

( ) ( )∑ =++= N

i kikT

kik tycPtyc1

11

1 ])1[(])1[(

)( kk tVη≤ , ,2,1=k . (15)

From (14) and (15), for any ],( 10 ttt ∈ ,

( )[ ]00 exp)()( tttVtV −−≤ + λ , which leads to

( )[ ]0101 exp)()( tttVtV −−≤ + λ , and )()( 111 tVtV η≤+

( )[ ]0101 exp)( tttV −−≤ + λη . Similarly, for ],( 21 ttt ∈ ,

( )[ ]11 exp)()( tttVtV −−≤ + λ

( )[ ]001 exp)( tttV −−≤ + λη .

In general, for ],( 10 +∈ kttt ,

( )[ ]010 exp)()( tttVtV k −−≤ + ληη . (16)

i) If 0>λ and there exists a constants β satisfying

0≥≥ βλ , for ],( 1 kk ttt −∈ , it follows from (16) that

)](exp[)()( 0210 tttVtV k −−≤ + ληηη

)](exp[)( 01210 tttV kk −−≤ −+ βηηη

)].)((exp[ 0tt −−−⋅ βλ

If (12) holds, [ ] 1)(exp 1 ≤−− −kkk ttβη , ,2,1=k .

Obviously, for ],( 1+∈ kk ttt ,

)],)((exp[)()( 00 tttVtV k −−−≤ + βλη (17) which implies that the solutions of controlled UCSDN (3) are globally exponentially stable about ( ) 01 =tyi . ii) If 0≤λ , and there exists a constant 1≥α , from (16), it follows that

)](exp[)()( 0210 tttVtV k −−≤ + ληηη

( )[ ]01210 exp)( tttV kk −−≤ ++ ληηη .

If (12) holds, ( )[ ] 1exp 1 ≤−− + kkk ttλαη ,

,2,1=k , therefore, for ],( 1+∈ kk ttt ,

( )[ ]kkk tttVtV −−≤ +

−+10 exp)()( λα . (18)

Consequently, 1=α implies that the trivial solution of (3)

is stable about ( ) 01 =tyi , and 1>α implies that the trivial

solution of (3) is asymptotically stable about ( ) 01 =tyi . From (17) and (18), the solutions of controlled UCSDN

(3) are globally stable about ( ) 01 =tyi , i.e.,

( ) 0lim 1 =∞→

tyit, Ni ,,1= . In the following, it is shown

ThAIn2.7

3894

that ( )tyi2 is also globally uniformly asymptotically stable.

From (7),

( )Niii xxhWxtgWty ,,),()( 1222 +≤ . Similar

to [12], rnT IWW −=22 , which implies 12 =W . Then,

( )∑∑∑ ===+≤ N

i NiN

i iN

i i xxhxtgty1 111

2 ,,),()(

( ) ( )∑∑ ==≤≤++≤ N

i iiN

j jiNitytyQK

121

11)()(max γ .

Thus, ( )[ ] QtyQK N

i iN

j jiNi≤+− ∑∑ ==≤≤ 1

211

)(1min γ

( ) .)(1

11 ∑∑ ==

+⋅ N

i iN

j ji tyK γ Nonsingular matrices Q

can be suitably chosen to satisfy 02 >Ξ , for any

Ni ,,1= . Therefore, ( ) 0lim 2 =∞→

tyit, Ni ,,1= .

Consequently, ( ) 0lim =∞→

txit, Ni ,,1= .

The proof is thus completed. Remark 3: Theorem 2 can also be applied to the complex

linear coupled networks [1-7, 9, 10].

IV. ILLUSTRATIVE EXAMPLE In this section, a systematic-design procedure is presented

for the impulsive control on the basis of Theorem 2 given in Section III. An example is also given to illustrate the effectiveness of the proposed method.

Based on Theorem 2, the suggested design procedure to determine the control gain and the impulsive intervals to globally stabilize the UCSDN (1) is presented as follows.

Step 1) Calculate the parameters K and ( )

NNij ×=Γ γ .

Step 2) Choose nnRQR ×∈, satisfying (6), 01 >Ξ and

02 >Ξ .

Step 3) For the UCSDN (1), select the gain constants kc

firstly. For convenience, one may choose cck = . Then,

calculate kη given in (11).

Step 4) Select a matrix P satisfying (4). Step 5) Calculate the constants 0, 21 >εε and λ

satisfying (5). If 0>λ , go to Step 6; otherwise, go to Step 7).

Step 6) Select a constant β satisfying 0≥≥ βλ and (12). If the impulsive intervals satisfy

kkk tt ηβ ln11

−− ≥− , then according to Theorem 2, the

impulsive controller u defined by (2) can globally stabilize UCSDN (1); otherwise, go back to Step 3).

Step 7) Select a constant 1≥α , then the impulsive intervals

is ( )kkk tt αηλ ln11

−− ≤− , then the impulsive controller

u defined by (2) can globally stabilize UCSDN (1); otherwise, go back to Step 3). For illustration, consider the UCSDN (1) with parameters

as follows: 100=N , 2=n , Tiii xxx ),( 21= ,

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00010

E , ⎟⎟⎠

⎞⎜⎜⎝

⎛=

101010

A , ( )( txxtg ii 1tanh),( =

)T0, , ( ) ∑ == N

j jijNj xbxxth11 ,,, with ( )

NNijbB×

=

( )

100100200011121110

012000112

sin

×⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

= t .

The initial conditions of nodes are randomly chosen from 0 to 100.

Following the presented design procedure, one can design

the impulsive controller such that the UCSDN specified above is globally stabilized. Step 1) Calculate the parameters 1=K and B=Γ .

Step 2) Choose 2IR = , ⎥⎦

⎤⎢⎣

⎡−

=1.001.0

01Q satisfying

(6), 05.01 >=Ξ and 06846.02 >=Ξ .

Step 3) Choose 4.0−=c . Then, calculate 36.0=kη .

Step 4) Select ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=10

01P satisfying (4).

Step 5) Calculate 121 == εε , 95.3−=λ and

01115.0

<⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−

=Φ satisfying (5). Then, go to Step 7).

Step 7) Select a constants 4.2=α satisfying (12). If the

impulsive intervals satisfy ( )kkk tt αηλ ln11

−− ≤−

037.0= , then the impulsive controller u defined by (2) can globally stabilize UCSDN (1). In the simulation, let

0.0371 =− −kk tt , the variations of the two states of 100 nodes are shown in Fig.1 as (a) and (b) respectively.

ThAIn2.7

3895

0 0.5 1 1.5 2 2.5 30

50

100

150

t(a)

x i1(t

)(i=

1,2,

...,

100)

0 0.5 1 1.5 2 2.5 3-15

-10

-5

0

5

t(b)

x i2(t

)(i=

1,2,

...,

100)

Fig.1. States of the controlled UCSDN with random initial values.

V. CONCLUSION This paper has presented a general uncertain complex

singular dynamical network model. Due to the flexible of singular systems in modeling practical applications, the proposed complex singular dynamical network model has more applications than the regular complex network in which the nodes are described by the state-space dynamical systems. Based on the Lyapunov stability theory, the robust stabilization conditions of the uncertain complex singular dynamical network under impulsive control are obtained theoretically. A systematic-design procedure is presented, and a numerical example is provided to demonstrate the effectiveness.

REFERENCES [1] Z. Li and G. Chen, “Global synchronization and asymptotic stability

of complex dynamical networks,” IEEE Trans. Cir. & Sys.- II, vol. 53, pp. 28 – 33, 2006.

[2] I.C. Lestas and G. Vinnicombe, “Scalable robust stability for nonsymmetric heterogeneous networks,” Automatica, vol. 43, pp. 714 – 723, 2007.

[3] W.-J. Yuan, X.-S. Luo, P.-Q. Jiang et al, “Stability of a complex dynamical network model,” Physica A, vol. 374, pp. 478 – 482, 2007.

[4] J. Xiang and G. Chen, “On the V-stability of complex dynamical networks,” Automatica, vol. 43, pp. 1049 – 1057, 2007.

[5] T. Chen, X. Liu, and W. Lu, “Pinning complex networks by a single controller,” IEEE Trans. Cir. & Sys. -I, vol. 54, pp. 1317 – 1326, 2007.

[6] Z. Duan, J. Wang, G. Chen, and L. Huang, “Stability analysis and decentralized control of a class of complex dynamical networks,” Automatica, vol. 44, pp. 1028 – 1035, 2008.

[7] Z.-H. Guan, and H. Zhang, “Stabilization of complex network with hybrid impulsive and switching control,” Chaos, Solitons & Fractals, vol. 37, pp. 1372 – 1382, 2008.

[8] J. Yao, Z.-H. Guan, G. Chen et al, “Stability, robust stabilization and

∞H control of singular-impulsive systems via switching control,” Syst. Contr. Lett., vol. 55, pp. 879 – 886, 2006.

[9] K. Li and C.H. Lai, “Adaptive-impulsive synchronization of uncertain complex dynamical networks,” Physics Letters A, vol. 372, pp. 1601 – 1606, 2008.

[10] W. Xiong, D.W.C. Ho, and J. Cao, “Synchronization analysis of singular hybrid coupled networks,” Physics Letters A , vol. 32, pp. 6633 – 6637, 2008.

[11] Y. Wang, L. Xie, and C. E. de Souza , “Robust control of a class of uncertain nonlinear systems,” Syst. Contr. Lett., vol. 19, pp. 139 – 149,1992.

[12] G. Lu and D.W.C. Ho, “Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation”, IEEE Trans. Automat. Control, vol. 51, pp. 818 – 823, 2006.

[13] S. Xu, P. Van Dooren, R. Stefan, and J. Lam, “Robust stability and stabilization for singular systems with state delay and parameter uncertainty”, IEEE Trans Automat. Control, vol. 47, pp. 1122 – 1128, 2002.

[14] M. Darouach and L. Boutat-Baddas, “Observers for a class of nonlinear singular systems”, IEEE Trans Automat. Control, vol. 53, pp. 2627 – 2633, 2008.

[15] G. P. Sudhakar and G. D. Sadashiv, Differential Systems Involving Impulses, Heidelberg: Springer Berlin, 1982, ch. 2.

[16] J. Dong, Y. Yu, and H. Zhang, “Impulsive modeling of nonlinear delay system and its stability,” in Proc. International Conference on Communications, Circuits and Systems, pp. 1287 – 1291, 2008

[17] Z. Yang and D. Xu, “Robust stability of uncertain impulsive control systems with time-varying delay,” Computers and Mathematics with Application, vol. 53, pp. 760 – 769, 2007.

[18] Z. Yang and D. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Trans. Automat. Control, vol. 52, pp. 1448 – 1454, 2007.

[19] Z. Guan, C.W. Chan, A.Y.T. Leung, and G. Chen, “Robust stabilization of singular-impulsive-delay systems with nonlinear perturbation,” IEEE Trans. Cir. & Sys. –I, vol. 48, pp. 1011 – 1019, 2001.

ThAIn2.7

3896