CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-43-3; ISSN 2283-9216 

A Finite-time Synchronization Scheme for Complex Networks 
Yingying Sun 
Beihua University; Jilin Jilin 132013, China 
wsyy_9@126.com 

This paper presents a method of finite-time synchronization, which can identify the topological structure of 
complex networks and unknown parameters. According to the theory of the finite-time stability, the feedback 
controller is designed to realize finite-time synchronization and exponential synchronization. Not only the 
unknown structure complex network topological is identified but also the parameters of the complex networks 
are identified. Numerical simulations are presented to illustrate the effectiveness of the theoretical analysis. 

1.  Introduction 

The complex networks consist of the mutual coupling nodes. The types and structure of node of complex 
networks may be the same or different, which were confirmed (Tung and Chen, 1986; Hoang et al, 1994; 
Chan and Chau, 1997). Synchronization means that two or more systems, which are either periodic or chaotic, 
adjust to each other and eventually reach common dynamical behavior, which were confirmed (Zhang and 
Mao, 2002; Solsona, 1996; Yahyazadeh et al., 2011; Wu et al., 2010; Zhuang, 2007). The synchronization of 
complex networks may be defined as the node state is consistent with some node or the slave complex 
network state consistent with the master complex network state, which was confirmed (Mahmoud and 
Mahmoud, 2010). The exponential synchronization has fast convergence rate which have good effect for 
synchronization.  
Finite-time synchronization has good convergence time which can be calculated in advanced. Finite-time 
synchronization will be more reasonable than other synchronization due to its synchronization in a given time, 
finite-time synchronization is more important than asymptotic synchronization. This implies the optimality in 
settling time. To realize faster convergent time in a control system, finite-time control way is a very important 
tool. Moreover, the finite-time synchronization with better robustness.   
Up to now, few paper research finite-time synchronization, parameter identification and topology weight 
identification and exponential synchronization in complex dynamical networks. How one realizes these 
methods in dynamical networks is an urgent issue to tackle. In this paper, we research how to realize them in 
time delay networks with disturbance. 
This paper is organized as follows. In the second section, finite-time synchronization, parameter identification 
and exponential synchronization between two complex networks with delay and disturbance are researched. 
In Section 3, we present how to realize finite-time synchronization, parameter identification and exponential 
synchronization. In Section 4, some numerical simulations are showed to test the effectiveness of theoretical 
analysis. Finally, some conclusions are drawn in the last part. 

2. Preliminaries 

A dynamical system with uncertainty can be showed as follows, 

( ) ( , ( )) ( , ( ))
i i i i i

x t f t x t g t x t α  ,                                                                                                                 (1) 

Where xi(t)=(xi1(t), xi2(t), …, xin(t)) T∈Rn is the ith node state vector, αi∈Rni are unknown parameter vectors, 

i=1, 2, …, N, ni is nonnegative integers. Fi (t, xi(t)) is an n×1 trix, gi (t, xi(t)) is an n×ni matrix.  
The dynamical system with controlled can be described as follows, 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651132

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Sun Y.Y., 2016, A finite-time synchronization scheme for complex networks, Chemical Engineering Transactions, 51, 
787-792  DOI:10.3303/CET1651132   

787



ˆ( ) ( , ( )) ( , ( )) ( ),
i i i i i i

y t f t y t g t y t α u t                                                                                                (2) 

where yi(t)=(yi1(t), yi2(t, …, yin(t))T ∈Rn is the ith node state vector, ˆ i
n

i
α R  are unknown parameter 

estimated vectors, i=1, 2, …, N, ni is nonnegative integers. Fi (t, yi(t)) is an n×1 matrix, gi (t, yi(t)) is an n×ni 
matrix. 
This paper will use the following definition and lemmas. 
Definition 1. The drive－response system are said to be finite-time topology identification and parameter 
estimated if there is a constant T>0 so that  

lim( ) lim( ) 0
ij ij i i

t T x T
c c α α

 
    ,                                                                                                           (3) 

where i,j=1,2,…,N, , topology identification and parameters identification is realize in a finite-time. 
Definition 2. The system (2) and the system (1) realize exponentially synchronization if there exists a 
controller u(t) and positive constants h and α such that the synchronous error satisfies  

( ) exp( ),    0,e t h αt t                                                                                                                        (4) 

where α is called the exponential convergence rate. 
Definition 3. Supposing that W(t) is a system’s Lyapunov function. If W(t) satisfies the following inequality, 

( ) ( ) ( ),   0,  (0) 0,W t λW t βW t t W


                                                                                              (5) 
then the system (5) can realize exponentially finite-time stable, where α,β>0 and 0<ω<1 are constants. In the 
inequality (5), λ is the exponential convergence rates and β is the finite-time convergence rates. 
Lemmas 1, which was confirmed (Chuang et al., 2012). According to Definition 3, if the Lyapunov function 
W(t) satisfies Eq. (5), the following inequality holds: 

  1 1( ) ( ) exp ln( (0) ) (1 )( 0) ,  0 ,W t β λ λW β α t λ t T                             
(6) 

and w(T)=0 , t T  where the finite time T can be illustrated in Eq. (7), 

  1ln (0) 1 (1 )T λW β λ     
.                                                                                                  (7) 

It is clear that finite time T is given by the positive values of α, β and ω. The following Lemmas 2 extend from 
the paper (Chuang et al., 2012) claimed is used to proved the main theorem. 
Lemmas 2. There exists inequality (8), 

1 2 1 2
( )

c c c c

n n
a a a a a a       ,                                                                                                      (8) 

where a1, a2,…, an are all positive numbers and 0<c<1 

3. Finite-time adaptive synchronization  

In this section, we will research finite-time adaptive synchronization. 
Now, we consider the complex networks consisting of N linearly coupled nodes, every node is n-dimensional 
dynamical system which can be showed by the following Eq.(9): 

1

( ) ( ( )) ( ( )) ( ),   1, 2, , ,
N

i i i i i i ij j

j

x t f x t g x t α c x t i N


                                                                   (9) 

where xi(t)=(xi1(t), xi2(t), …, xin(t))T∈Rn is the ith node state vector, C=(Cij)N×N∈R is an unknown matrix, if 
node i and node j(i≠j) with connection, then Cij≠0. Otherwise, Cij=0(i≠j). τ is time delay and τ≥0. 
If the model (9) is the drive networks, then the response networks as follows, 

1

ˆ ˆ( ) ( ( )) ( ( )) ( ) ( ),   1, 2, , ,
N

i i i i i i ij j i

j

y t f y t g y t α c y t u t i N


                                                     (10) 

Where yi(t)=(yi1(t), yi2(t), …, yin(t))T∈Rn is the I the node state vector, ui(t) (i=1,2,…,N) are controllers to be 

designed, ˆ ˆ( )
ij N N

c


C  is estimated value of weight C. ˆ
i
α  is the unknown parameter vector αi, estimated 

value. Τ is time delay and τ≥0.  

788



Let ei(t)=yi(t)-xi(t) according to Eq. (9) and Eq. (10), Eq. (11) is the error dynamical network, 

1 1

ˆ ˆ( ) ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( )
N N

i i i i i i i i i i i ij i ij i i

j j

e t f t y t f t x t g t y t α g t x t α c y c x u t
 

         
,          (11) 

The finite-time topology and parameter identification can be regarded as the finite-time stabilization of the 
error system. The controller ui(t) can guarantee finite-time synchronization. 
Theorem 1. The error system (11) can realize finite-time adaptive synchronization and exponential 
synchronization via apply the following controller and updated rule, 

1 1

ˆ ˆ ˆ ˆ( ) ( , ( )) ( , ( )) ( , ( )) ( , ( ))

            ( ) ( ) ( , ( )) ,
2 2

N N

i i i i i i ij i i i i i i ij i

j j

p

q

i i i i i

u t f t y t g t y t α c y f t x t g t x t α c x

λ β
e t e t g t x t α

 

        

  

                (12) 

ˆ ˆ( ) ( ), ,
2 2 2 2

p p

q q

ij ij i j i i i

β β
c c c e t x t α α α

 
                                                                                (13) 

where λ>0, β>0, (0,1)
p

q
 ,p and q are two positive odd integers. The finite-time adaptive synchronization in 

a finite time 

  1ln (0) 1 (1 )
p

T W β
q


 

 
   
 

,                                                                                               (14) 

where 
2 2 2

1 1 1 1

1 1 1
(0) (0) (0) (0)

2 2 2

N N N N

i i ij

i i i j

W e α c
   

     , (0)ie , (0)iα , (0)ijc  are the initial values of 

( )
i

e t , 
i
α , 

ij
c . 

Proof.  Lyapunov function as follows, 

2 2 2

1 1 1 1

1 1 1

2 2 2

N N N N

i i ij

i i i j

W e α c
   

   
,                                                                                                           (15) 

where ˆ ˆ, .
i i i ij ij ij
α α α c c c    ˆ ˆ, .

i i ij ij
α α c c   

The time derivative of Eq. (15) along the Eq. (11),  

1 1 1 1

1 1 1

1 1 1

ˆ ˆ

ˆ ˆ  [ ( , ( )) ( , ( )) ( , ( )) ( , ( )) ( )]

     [ ] [
2 2 2 2

N N N N

i i i i ij ij

i i i j

N N N

i i i i i i i i i i i ij i ij i i

i j j

pN N N
q

i i i ij ij i

i i j

W e e α α c c

e f t y t f t x t g t y t α g t x t α c y c x u t

β β
α α α c c c

 

   

  

  

  

        

     

  

  

 

2 2 2

1 1 1 1 1 1 1 1

2 2 2 2 22 2

1 1 1 1 1 1

( ) ( )]

  [ ( ) ] [ ( ) ( ) ]
2 2

  [ ( ) ] [ ( ( )) ( )
2 2

p

q

j i j

p p pN N N N N N N N
q q q

i i ij i i ij ij

i i i j i i i j

p q p qN N N N N N
q q

i i ij i

i i i j i i

e t x t

β
e t α c e t e t αα c c

β
e t α c e t α





       

 

     

 

      

      

     

    
2 2

1 1

2 2 2 2 2 2 2

1 1 1 1 1 1 1 1

2

( ) ]

  [ ( ) ] [ ( ) ]
2 2

 .
2 2

p qN N
q

ij

i j

p qN N N N N N N N
q

i i ij i ij

i i i j i i i j

p q

q

c

β
e t α c e t α c

β
W W







 



       



      

  



     

       

(16)

 

789



So, under the controller (12) and updated laws (13), the complex networks (9) and (10) are synchronized in a 
finite time  

  1ln (0) 1 (1 )
p

T W β
q


 

 
   
 

.  

From Eq. (16), we can deduced that 

exp(2 ) ( ) exp(2 ) 2 ( ) [exp(2 ) 2 ( )] 0,   0.
d

λt W t λt λW t λt W t t
dt

                                      (17) 

0 0
[exp(2 ) ( )] exp(2 ) ( ) (0) 0 0,   0.

t td
λt W t dt λt W t W dt t

dt
                                            (18) 

From Eqs. (15) and (18),  

2 2 2 2

1 1 1 1

2

1

( ) 2 ( )

            2 ( ) 2 exp( 2 ) (0),   0.

N N N N

i i ij

i i i j

N
t

i
t

i

e t W t e α c

e d λt W t


 

   




   

     

  



                                                            (19) 

( ) 2 (0) exp( ),   0e t W λt t    .                                                                                                      (20) 

So the controller (12) and updated (13) realize exponential synchronization between two complex networks (9) 
and (10) in a finite time. 
Remark 1. To realize the finite-time stability, the parameters p/q have to satisfy the condition that (p/q) ∈(0,1) 
is an irreducible fraction with two positive odd integers p and q. If p is even and q is odd, then the state of 

( ) ( 1, 2, , )
p

q

i
e t i N is always positive which can cause the system to be unstable when ei(t) is negative. In 

contrast, if p is odd and q  is even, then the state of ( ) ( 1, 2, , )
p

q

i
e t i N  will become complex when ei(t) 

is negative. The parameter p/q can be assigned arbitrarily under the condition mentioned above, then p+q/2q
∈(0,1).  

4. The simulation research 

In this section, taking a chaotic Loenz system is as nodes of network to test the effectiveness of the raised 
scheme. 
 The Lorenz system is as follows, 

1 2 1

2 3 1 2

3 1 2 3

( )

( )

x a x x

x b x x x

x x x cx

 


  
  

,                                                                                                                             

(21) 

where x1, x2, x3 are state variables. When parameters a=10, b=28, c=8/3 the system (21) is chaos. 
The coupling configuration matrix C=(Cij)N×N is chosen be as follows 

1 0 1

1 1 0

0 1 1

 
 

 
 
  

C
. 

The chaotic Lorenz system is the nodes of networks (1) and (2) which can be described as follows, 

790



                      

1 2 1 3

2 1 3 2 1

1

33 1 2

ˆ( )        0 0 0

ˆ ˆ( ) 0 0 ( ) ( ),     1, 2, 3,

ˆ0 0( )       

ii i i

i i i i i i ij j i

j

ii i i i

ay t y y

y t y y y y b c y t u t i

yy t y y c




      
      

             
            


             

where τ is time delay, the controller ui(t) and updated laws as follows 
3 3

1 1

ˆ ˆ ˆ ˆ( ) ( , ( )) ( , ( )) ( , ( )) ( , ( ))

            ( ) ( ) ( , ( )) ,
2 2

i i i i i i ij i i i i i i ij i

j j

p

q

i i i i i

u t f t y t g t y t α c y f t x t g t x t α c x

λ β
e t e t g t x t α

 

        

  

 
 

ˆ ˆ( ) ( ), ,     1, 2, 3.
2 2 2 2

p p

q q

ij ij i j i i i

β β
c c c e t x t α α α i

 
          

In the numerical simulation, taking the initial states as: xi(0)=(1,2,3)T, yi(0)=(4,5,5)T, ˆ (0) 1ijc  ,

ˆˆ ˆ( (0), (0), (0)) ( 1 ,1 2 , 3 ),
i i i

a b c i i i      where 1≤I, j≤3. selecting λ=β=1. Figure 1 shows the 

synchronization errors, Figure 2 shows the parameters identification, and Figure 3 shows the identification of 
networks weight. 

 
t/s                                             t/s                                             t/s 

Figure 1: The synchronization errors between two complex networks 

 
t/s                               t/s                                     t/s 

Figure 2: Parameters identification between two complex networks 

 
t/s                                        t/s                                      t/s 

Figure 3: Identification of networks weight between two complex networks 

1 2 1 3

2 1 3 2 1

1

33 1 2

( )        0 0 0

( ) 0 0 ( ),   1, 2, 3,

0 0( )       

i i i

i i i i i ij j

j

ii i i

x t x x a

x t x x x x b c x t i

x cx t x x




      
      

      
      

           



791



5.   Conclusions 

This paper presents an approach for complex networks with delay and disturbance to realize finite-time 
synchronization, parameter identification and exponential synchronization. When complex networks with delay 
exist parameters disturbance, exotic and topology weight disturbance, based on the theory of finite-time 
stability and Lyapunov theory, a feedback controller with updated law is designed to not only realize finite-time 
synchronization and exponential synchronization but also realize parameter and topology identification. 
Numerical simulation is provided to show the effectiveness of the proposed method. 

References 

Chan C. C., Chau K. T., 1997, An overview of power electronics in electric vehicles. IEEE Transaction on 
Industrial Electronics, 44, 3-13. 

Chuang C. F., Sun Y. J., Wang W. J., 2012, A novel synchronization scheme with a simple linear control and 
guaranteed convergence time for generalized Lorenz chaotic systems, 22, 043108. 

Hoang L. H., Slimani K., Viarouge P., 1994, Analysis and implementation of a real-time predictive current 
controller for permanent- magnet synchronous servo drives, IEEE Transactions on Industrial Electronics, 
41, 110-117. 

Mahmoud G. M., Mahmoud E. E., 2010, Synchronization and control of hyperchaotic complex Lorenz system. 
Mathematics and Computers in Simulation, 80, 2286-2296. DOI: 10.1016/j.matcom.2010.03.012. 

Solsona J., 1996, A nonlinear reduced order observer for permanent magnet synchronous motors. IEEE 
Transactions on Industrial Electronics, 43, 492-497.    

Tung P. C., Chen S. C., 1993, Experiment and analytical studies of the sinusoidal dither signal in a DC motor 
system. Dynamics and Control, 1, 53-69. 

Wu J., Singla M., Olmi C., 2010, Digital controller design for absolute value function constrained nonlinear 
systems via scalar sign function approach, ISA transactions, 49,302-310. DOI: 
10.1016/j.isatra.2010.03.005. 

Yahyazadeh M., Noei A. R., Ghaderi R., 2011, Synchronization of chaotic systems with known and unknown 
parameters using a modified active sliding mode control. ISA transactions, 50, 262-267. DOI: 
10.1016/j.isatra.2010.10.009. 

Zhang B., Mao Z. Y., 2002, Entrainment and migration control of Permanent-magnet synchronous motor 
system. Control Theory and Application, 19, 53-56. 

Zhuang X., 2007, Direct torque and flux regulation of an IPM Synchronous motor drive using variable structure 
control approach. IEEE Transactions on Power Electronics, 22, 2487-2498. 

 
 
 
 

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