International Journal of Modern Nonlinear Theory and Application, 2012, 1, 1-5
http://dx.doi.org/10.4236/ijmnta.2012.11001 Published Online March 2012 (http://www.SciRP.org/journal/ijmnta)
State Reconstruction for Complex Dynamical Networks
with Noises*
Chunxia Fan, Guoping Jiang
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, China
Received January 30, 2012; revised February 25, 2012; accepted March 12, 2012
ABSTRACT
The state reconstruction problem is addressed for complex dynamical networks coupled with states and outputs respec-
tively, in a noisy transmission channel. By using Lyapunov stability theory and H∞ performance, two schemes of state
reconstruction are proposed for the complex dynamical networks with the nodes coupled by states and outputs respec-
tively, and the estimation errors are convergent to zeros with H∞ performance index. A numerical simulation demon-
strates the effectiveness of the proposed observers.
Keywords: State Reconstruction; Complex Dynamical Networks; Noisy Circumstance; H∞ Performance
1. Introduction
Complex dynamical networks have recently been a hot
topic in science and engineering fields because it can
describe many phenomena in nature and engineering
[1-4]. For instance, power grid is a complex network if
the electrical equipments are treated as nodes and the
interactions between the equipments as edges; the indi-
viduals are treated as nodes and the interactions between
the individuals as edges in a community then the com-
munity can be described as a network.
The synchronization of a complex dynamical network
has been reported in the latest decade including inner
synchronization and outer synchronization [5-12]. All
state variables are required to construct the synchroniza-
tion controllers in [5-7,11]. In [8,10,12], partial state va-
riables are needed to construct the synchronization con-
trollers. For outer synchronization, the circumstance
noise is considered in [9].
Recently, topology identification, fault diagnosis and
parameter identification [13-20] of complex networks
have become hot topic in complex network applications,
and network synchronization has found applications in
these fields. For topology identification, it is assumed
that all of the states are available for a complex network
in [15,17-20]. For monitoring topology change of net-
work, it is assumed that the partial state variables are
required for the network in [14,16].
In the above study and other fields of complex net-
work, all or partial state variables are needed for design.
However, for a large scale network, measuring all state
variables is not easy or even impossible in practice, and
locating many sensors costs much. Therefore, it is very
important to estimate or reconstruct all state variables
based on some limited available network information.
For discrete complex networks, state reconstruction has
been reported in [21]. For continuous time complex dy-
namical networks with transmission noise, there has been
little theoretical work on state estimation in the literature.
Motivated by the above observations, in this paper we
study the state reconstruction or state estimation problem
for a complex network with transmission channel noise.
By using Lyapunov stability theory and H∞ control the-
ory, some state reconstruction schemes are derived for
complex dynamical networks including state coupling
and outputs coupling. For suppressing noise in the chan-
nel, the integral observers [22,23] are applied and the
estimation errors are bounded with H∞ performance.
Some numerical examples are given to shown the effect-
tiveness of the proposed schemes.
The rest of this paper is organized as follows. In Section
2, the state reconstruction of state coupling networks is
studied and some estimation criteria are derived in the form
of linear matrix inequality. In Section 3, the state recon-
struction of output coupling networks is studied and some
estimation criteria are given. Some examples are given in
Section 4 and conclusions are drawn in Section 5.
2. State Reconstruction of Networks Coupled
with States
*This work was supported in part by the National Natural Science
Foundation of China under 60874091 and 61104103, the Natural Sci-
ence Foundation for Colleges and Universities in Jiangsu Province,
China under 10KJB120001. Consider a general complex dynamical network consist-
C
opyright © 2012 SciRes. IJMNTA

C. X. FAN ET AL.
2
ing of N identical nodes with states couplings, which is
described as follows

1
N
iiiijj i
j
i
x
Axf xcxyHx

 


Tn
(1)
where , ii
1iN

12
,,,
i in
x
xx xR
m
i
is the state
vector of the ith node,
y
R
nn
is the output vector of
the ith node,
A
R 
nn
is the system matrix of node i,
:
f
RR is a nonlinear smooth vector field, mn
H
R


is the output matrix of node i, is the coupling
matrix of node i, node dynamics is
nn
R 


x
Axf x
,

N
N

ij NN
Cc R is the coupling configuration matrix.
If there is a link from node i to node , then
; otherwise . Assume that C is a diffusive

ji j

1
ij
c0
ij
c
N
matrix satisfying . It is noted that the con-
1
ij ii
jji
cc



figuration matrix C does not need to be irreducible and
symmetric.
Hypothesis 1: (H1) Suppose that

f
x is Lipschitz
continuous. That is, there exists a positive number con-
stant α satisfying
 
ˆˆ
f
xfx  xx for n
x
R,
ˆn
x
R, where represents the Euclidean normal.
For most networks, all of the states are generally not
available. To reconstruct the states of network (1), out-
puts i are transmitted from (1) to the observer through
the transmission channel. In the practical engineering,
there exists noise in the transmission channel. Therefore,
the measurements received by the observer are charac-
terized by
y

1, 2,,
iii

y
Hx wiN  (2)
where m
i
y
R is the actual measurement outputs and
are the noises in the transmission channel.
i
w
Hypothesis 2: (H2) Suppose that the disturbances in
the transmission channel are bounded, i.e., there exists a
positive constant d such that i wd.
To reconstruct all the states of (1), the following ob-
server is presented
 

1
ˆˆˆ ˆˆ
ˆˆ ˆ
N
ii iijjiii
j
ii iii iiii
xAxfx cxKzz
zyHxw zyHxlzz

 
  



ˆ
(3)
where ,
1, 2,,iNˆn
i
x
R
ˆi
is the state estimated for
ith node in the network (1), m
y
R is the output vector
of the ith in (3), nm
i
K
R and are the observer
gain matrices to be determined.
mm
i
lR
Remark 1: The observer (3) is different from the tradi-
ional proportional observer because its controller is the t
integral of the measurements [22,23]. The states recon-
structed by (3) can better converge to the states of (1)
since the disturbance is not amplified if a large propor-
tional gain i
K
is used [8].
The aim is to determine appropriate observer gain ma-
trices nm
i
K
R

 and such that the recon-
structed states
mm
i
lR


ˆi
x
approach the network states i
x
.
Define the state errors
ˆ1, 2,,
ˆ
iii
zii i
exxi
ezz
 
 N (4)
Then it follows from (1) and (3) that
 
1
ˆ
1,2,,
N
ii iiijji
j
ziiii zi
eAefx fxceKe
eHewleiN

 
 



zi
(5)
Define 0
0
A
BH






, ,
0
00
G



i
i
i
K
Kl



,
T
TT
zi
e
ii
Ee






Fx
T
,,
 
ˆ
ii i
xf x


ˆ
,0
T
T
i
f x




0DI and . Then (5) can be
rewritten as

,,
N
diag PP
 
N


1
ˆ ,T
ii iiijjii
j
EBEFxx cGEDwKDE

 

i
(5)
Then we will derive i
K
to guarantee that i are con-
vergent to zeros when
E
0
i
w

, and i are convergent
to zeros with
E
H

performance

, characterized by the
following inequality, when and 0
i
wi wd

.
 
2
2 2
00
dd00
iii
Et twttE 



 (6)
Theorem 1: Suppose that H1 and H2 hold. If there ex-
ist matrices 0
T
PP

,

1, 2,,
i

M
iNand a con-
stant 0

 such that the following inequality holds
  
0CP CP
T

  (7)
where



22TT
diagPBB PIPIDDPI



 


22
11
,,
TT TT
D
MMDPBBP IPIDDPI


 
TT

NN
D
MMD
0
i
w
, then the error dynamical system (6)
will converge to zeros with H∞ performance γ when

. Consequently, network (3) can estimate the state
of network (1) with H∞ performance γ when 0
i
w

and
1
ii
K
PM

.
Proof: Define a Lyapunov function .
1
Differentiating V along the error dynamical system (6)
nd using Hypothesis 1, one obtains
NT
i
i
VEP

i E
i
i

a


 
111
2
111
ˆ 2, 2
T
NNN
TTTT TTT
iiiiiiiiijjijjii
ijj
T
NNN
TTT TTTT
iii iiiiijjijj
ijj
VEPBBPDKPPKDEEPFxxE PcGEcGEPEEPDw
EPB BP DKPPKDIPIDDPEwwEPc GEcGEPE



 

 
 

 

 
 

 


 

 



(8)

C. X. FAN ET AL. 3
Define . Using (8), one obtains

1,, T
TT
N
EE E

T

E
  


  


2
1
2
1
T
NTT
ii ii
i
T
T
NTT
ii ii
i
VECP CPE
EE ww
ECPCP
EE ww













(9)
From (10) and Lyapunov stability theorem, are ex-
ponentially convergent to zeros when i. Under
, integrating (10) from 0 to ∞ yields that
i
E
0w

00
i
E
H

performance

. The proof is completed.
To easily solve the matrix inequality, Schur comple-
ments lemma [9] is used here. Then (8) is transformed
into the following linear matrix inequality
  

10
T
T
CP CP
IDD

  





 
T
(10)
where

22
11
,
TT
diagPBBPIIDMMD 

 
22TTT

,NN PBBPIIDMMD   

,,diag PP 
, and
.
3. State Reconstruction of Networks Coupled
with Outputs
Next, we consider a complex dynamical network consist-
ing N identical nodes coupled with the outputs charac-
terized by

1
N
iiiijj i
j
i
x
AxfxcLyyHx

 

 (11)
where .
nm
LR


The outputs i are disturbed by noise when they are
transmitted from network (12) to the observer. Therefore
information received by the observer is characterized by
(2). The observer is designed as the following form
y
 

1
ˆˆˆˆˆ
ˆˆ ˆ
N
ii ijjiii
j
ii iii iiii
xAx fxcLy Kzz
zyHxwzyHxlzz

 
  



ˆ
(12)
where nm
i
K
R  and mm
i
lR

 are the observer gain
matrices to be determined. Then one can obtain the error
dynamics
 
1
ˆ1, 2,,
N
ii iiijjizi
j
ziiii zi
eAefx fxcLHeKeiN
eHewle

  




(13)
Let , then (14) can be rewritten as
T
TT
iizi
Eee



1
ˆ ,
NT
i ijjiii
E cLHEDwKDE 


iii
j
BEFxx

(14)
where 0
00
L
L






 and


0HH. Consequently,
atrices determine mi
K
such that error dynamics (15) is
H

stable with peance, that is, observer (13) re- rform
f construct the states onetwork (12) with
H
 perform-
ance. The foing theorem gives the criteria of deter-
mining matrices i
llow
K
.
Theorem 2: Suppose that H1 and H2 hold. If there ex-
ist matrices PP 0
T

,


1, 2,,
i
M
iN and a con-
stant 0

 such that the following inequality holds



0
T
CPLHCPLH

  (15)
where


22TT
diagPBBPI PIDDPI



  


22
1
,,
T TT
MDPBBP IPIDDPI
1
T
DM


 
TT

)

NN
DMM D
will converge to
then the error dynamical system (15
the zeros with
H
 performance γ
when 0
i
w

. Consequently, observer (13) can estimate
the state of network (12) with
H
 performance γ when
0
i
w

and 1
ii
K
PM

.
The proof of Theorem 2 is simr to that of Theorem 1,
omitted
ila
so it is here.
Using Schur complement lemma, one obtains the fol-
lowing linear matrix inequality
  
T
CPLHCPLH

  



10
T
ID
D







(16)
where

22
11
,
TT
diagPBBPIIDMM D 

 
22TTT
T
and

NN
B PIIDMMD   ,PB


,,diag PP .
Remark 2: In this section, the complex dynamical
netwed ork couplwith the outputs is considered because
this kind of networks is practical in engineering for sav-
ing communications and sensors. When the transmission
channel is ideal, the observer (13) can reconstruct the
states of network (12) with exponential convergence.
When the transmission channel is noisy, the observer can
reconstruct the states of network (12) with
H

per-
formance.
4. Numerical Simulations
ples are given to dem-
sed state reconstruct-
In this section, some numerical exam
onstrate the effectiveness of the propo
tion scheme for complex dynamical networks. In the net-
work, chaotic Lorenz system is taken as the node dynamics.
Lorenz chaotic system is a well-known typical bench-
mark chaos, which can be described by the following [10]

C. X. FAN ET AL.
4
00.5 11.5 22.5 3
-10
0
10
e
i 1
, i = 1, 2, ... , 10
00.5 11.5 22.5 3
-10
0
10
simulation time
e
i 2
, i = 1, 2, ... , 10
00.5 11.522.5 3
-40
-30
-20
-10
0
10
simulation time
e
i 3
, i = 1, 2, ... , 10
Figure 1. Errors between small world network (1) and ob-
server (3).
13
12
ii
ii
xx
(17)
where a, b and c are parameters. When a = 10, b = 28
and
1i
x



1
22
33
00
10
00
i
ii
ii
ii
x aa
xbx xx
c
xx
Axf x
 

  
 
  

 

 



83c, the system (18) is chaotic.
For any two state vectors i
x
and
j
x
of the Lorenz
syste chaotic attractor is bounded m, since in a certain re-
gion, there exists a constant

satisfyi ngik
x


and
jk
x

 for 1, 2, 3k. Then one gets the following

 

2
2
1212
2
ij
ii j j
ij ij
fx
xxx x
xx xx 


 
(18)
Then the Lorenz system satisfies the Hypothesis H1.
The complex dynamical network is assumed to contain
by
G
13 13jj ii
xx xx
fx


10 nodes and transmission noise is characterized
uass stochastic noise with mean 0 and magnitude 0.1.
The other parameters of networks are


100H,

111diag , and

111
T
L. The initial val-
ues of networks are randomly evaluated in
network c states is consid-
ered. Using MATLAB LMI toolbox, one obtains
[0, 1].
A small world oupled with
2


10.61674.86270.0003 0.1234
4.8627 2.22810.00080.0354




0.00030.0008 0.59950.0014
0.12340.03540.0014 20.6716
P 
 



16230.5230
35426.0276
56.9691
322.2331
K

















The simulation results are shown in Figure 1. From Fig-
ure 1, one can see that the error dynamics converge to
zeros although there is transmission noise.
5. Conclusion
nd the output
re both considered. To attenuate noise
channel, integral observers are used
In this paper, we study the problem of state reconstruct-
tion for a complex dynamical network under noisy cir-
cumstances. The state coupling network a
coupling network a
in the transmission
and estimation errors with H∞ performance index are
obtained. Some examples are given to demonstrate the
effectiveness of the proposed scheme.
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