nanjing university of posts & telecommunications synchronization and fault diagnosis of complex...

NanJing University of Posts & NanJing University of Posts & TelecommunicationsTelecommunications

Synchronization and Fault Diagnosis of

Complex Dynamical Networks

Guo-Ping Jiang

College of Automation, Nanjing University of Posts & Telecommunications

Email:[email protected]


outline1. Motivation and Background

2. Synchronization of Network

3. Our Research Results

4. Conclusions


1. Motivation and Background

Network Synchronization Inner syn. and outer Syn. Identification

Network topology is uncertain in real engineering Network topological identification

Monitoring Fault diagnosis of networks


2. Synchronization of Network -Inner Synchronization: a collective behaviour within a network

Coupling with all state variables Coupling with output variable


Coupling with all state variables The model of a dynamical complex network:

State variables of the node: Inner coupling matrix: Coupling matrix: (connected)

(network topology) (otherwise)

1

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

i i ij jj

x f x c Ax i N

1 2[ ... ]Ti i i inx x x x

( )ij N NC c 0ij jic c

0ij jic c

1

N

ii ijjj i

c c

A

[1]. X. Wang and G. Chen, “Complex network: Small-world, scale-free, and beyond,” IEEE Circuits Syst. Mag., vol. 3, no. 2, pp. 6-20, 2003.[2]. J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 841-846, 2005.


The model of a dynamical complex network:

Outer coupling variable: Outer coupling matrix:

(connected) (network topology)

(otherwise) Observer gain matrix:

1

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

i i ij jj

x f x c Ly i N

i iy Hx 1 2[ ... ]nH h h h

1 2[ , ,..., ]TnL l l l


0ij jic c

1

N

ii ijjj i

c c

Coupling with output variable

[1] G. –P. Jiang, W. K. -S. Tang, G. Chen, “A state-observer-based approach for synchronization in complex dynamical networks,” IEEE Trans. on Circuits &Systems-I, vol. 53, pp. 2739-2745, 2006


[1] C. Li, W. Sun, J. Kurths, “Synchronization between two coupled complex networks ,” Phys Rev. E vol. 76, 046204, 2007[2] H. Tang, L. Chen, J.-A. Lu, C. K. Tse, “Adaptive synchronization between two complex networks with nonlidentical topological structure,” Physica A, vol. 387, pp. 5623-5630, 2008.[3] C.-X. Fan, G.-P. Jiang, F.-H. Jiang, “Synchronization between two complex networks using scalar signals under pinning control,” IEEE Transaction on Circuits and Systems-I, vol. 57, 2010

Outer Synchronization: between two or more networks

Driving Network Response Network

x1

x2

x3

x4

x1

x4

x4

x1

x2

x3

状态耦合复杂动态网络Driving Network Response Network

y1

y2

y3

y4

y1

y4

y4

y1

y2

y3

y2

y3

输出耦合复杂动态网络


The drive network model:

The response network model:

Control law:

1

( ) ( ( )) ( ), 1N

i i ij jj

x t f x t c Ax t i N

1

ˆ ˆ ˆ( ) ( ( )) ( ) , 1N

i i ij j ij

x t f x t d Ax t u i N

2

1

: , ,N

Ti ij j i i ij i j i i i

j

C D u b Ay g e b e Ay g k e

2: , .ˆ .

i i i i i i

i i i

C D u g e g k e

e x x


Identification of network topology using outer synchronization of network

L. Zhu et al. assume that the dynamics of the network

can be described by a linear stochastic model,

But if a more complex network is considered, it may not

be true.

[1] L. Zhu, Y. C. Lai, F. C. Hoppensteadt, J. He, “Characterization of neural interaction during learning and adaptation from spike-train data,” Mathematical Biosciences and Engineering, vol. 2, pp. 1-23, Jan.2005.


W. K. -S. Tang et al. develop an adaptive observer approach

that using a state variable to identify and monitor the

topology of neural network with each ode being a HR model

Effective for special dynamics of nodes, but difficult to be

extended to a general case, where the node dynamics is a

general nonlinear system

[1] W. K. -S. Tang, Y. Mao, L. Kocarev. “Identification and monitoring of biological neural network,” IEEE International Synposium on Circuits and Systems, pp. 2646-2649, May. 2007.


The network model:

Where:

2 31

2

( ) ( , , ) ( )( )

( ) ( ) , ( ) ( , , )( ) ( )

( ) ( , , )

Nxi i i i i i ij ijj

i i i i i y

i i i i i iz

x t F x y z xx t a x x y z g

y t a x y y t F x y zz t b x c z

z t F x y z

11 12 1

21 22 2

1 2

... 0... 00...0 ... 0...0

.

. ... ... ... ... ...

.0...0 0...0 ...

N

N

N N NN

11, 12 1 21

1 2

1 2

1 2

,..., , ,...,

, 1.3,

( ) ( ), ( ),..., ( )

( ) ( ), ( ),..., ( )

( ) ( ), ( ),..., ( )

T

N NN

ij s ij s ij j i

TN

TN

TN

g g g g g

g g c g x x

x t x t x t x t

y t y t y t y t

z t z t z t z t


The observer model:

Where:

is the condition constant we want to get.

ˆˆ ˆ ˆˆ( ) ( , , ) ( ) ( )

ˆ ˆ( ) ( , , )

ˆ ˆ( ) ( , , )

ˆ ˆ( ) ( )( )

x x

y

z

T

x t F x y z x K x x

y t F x y z

z t F x y z

t K x x x

11, 12 1 21

1 2 1 2

1 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,..., , ,..., , , 1.3

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

ˆ ˆ ˆ ˆ( ) ( ), ( ),..., ( )

T

N NN ij s ij s

T TN N

TN

g g g g g g g c g

x t x t x t x t y t y t y t y t

z t z t z t z t

xK


J. Zhou et al. construct a state observer and use all the state variables

to get network synchronization for topological identification.

X. Q. Wu extends to time-delay networks.

But if some state variables are not measurable, it may not be

practical.

[1] J. Zhou, J. A. Lu, “Topology identification of weighted complex dynamical networks,” Physica A, vol. 386, pp. 481-491, 2007.[2] X. Q. Wu, “Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,” Physica A, vol. 387, pp. 997-1008, 2008.


1

( ) , 1N

i i ij jj

x f x c Ax i N

1

2

ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ( )

ˆ( )

N

i i ij j i i ij

Tij i i j

i i i i

x f x c Ax d x x

c x x Ax

d k x x

The drive network model:

The response network model:

where is any positive constant

ik


1

( ) ( ( )), 1N

i i ij jj

x f x c Ax t t i N

1

2

ˆ ˆ ˆ ˆ ˆ( ) ( ( )) ( )

ˆ ˆ ˆ( ) ( ( ))

ˆ( )

N

i i ij j i i ij

Tij ij i i j

i i i i

x f x c Ax t t d x x

c x x Ax t t

d k x x

ik

Networks with time-varying coupling delay


3.Our research results:

Inner synchronization Outer synchronization Topological identification Fault diagnosis.


Inner synchronization: The model of a dynamical complex network:

Outer coupling variable: Outer coupling matrix:

(connected) (network topology) (otherwise) Observer gain matrix:

1

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

i i ij jj

x f x c Ly i N

i iy Hx 1 2[ ... ]nH h h h

1 2[ , ,..., ]TnL l l l


0ij jic c

1

N

ii ijjj i

c c



Nkk

TTk

T

,,2，022

PIPPPLHPLHPAPA

2( )

2, ,X = PL

T T Tk k

k N

PA A P P XH H X P I0

P I I


Outer synchronization: (State-coupling Model)


1

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

i i ij jj

i i

x f x c Ax i N

y Hx

state variables of the node 1 2[ ... ]Ti i i inx x x x

outer coupling variable 1 2[ ... ]i i ny Hx H h h h

inner coupling matrix(network topology)

1

( ) 0 (conneted)

0 (otherwise)ij N N ij ji

ij ji

N

ii ijjj i

C c c c

c c

c c


Outer synchronization: The response network:

1

ˆ ˆ ˆ ˆ( ) ( ), , 1,2,...,

ˆ ˆ

N

i i ij j i ij

i i

x f x c Ax kB y y i N

y Hx

Hypothesis1 (H1): ˆ ˆ( ) ( ) 0f x f x x x

Hypothesis2 (H2): min min2 ( )2

TA AA

Hypothesis3 (H3):min is a modifying matrix of by replacing the diagnoal elements by iiC C c

Fan C-X, Jiang G-P, Jiang F-H. Synchronization between two complex networks using scalar signals under pinning control [J]. IEEE Transaction on Circuits and Systems-I: Regular Papers, vol. 57, no. 11, 2010


Outer synchronization: The error system:

1

ˆ( ) ( ) , 1,2,...,N

i i ij j ij

e f x f x c Ae kBHe i N

The synchronization criteria:

Suppose that (H1)-(H4) hold. If there exists a constant ksuch that the following inequality hold

min ( ) 02 2

T T T

N NBH H B C CI k I

then the error system is asymptotically stable.


Simulation result

The network is consisting of 10 nodes, where node dynamical is Lorenz systemK=-10, A=[1 0 0; 0 1 0; 0 0 1], B=[4 5 6], H=[1 1 1]

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

simulation steps

erro

r nor

m (1

0 no

des,

30

erro

r sta

tes)

C=[-5 1 1 1 0 0 1 0 0 1 ; 1 -2 0 1 0 0 0 0 0 0; 1 0 -4 0 0 1 1 1 0 0; 1 1 0 -4 0 0 1 0 0 1; 0 0 0 0 -4 0 1 1 1 1; 0 0 1 0 0 -2 0 0 1 0; 1 0 1 1 1 0 -5 0 0 1; 0 0 1 0 1 0 0 -2 0 0; 0 0 0 0 1 1 0 0 -2 0 ; 1 0 0 1 1 0 1 0 0 -4]


Outer synchronization: (output-coupling model)

Another model of a dynamical complex network:

1

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

i i ij jj

i i

x f x c Ly i N

y Hx

state variables of the node 1 2[ ... ]Ti i i inx x x x

outer coupling variable 1 2[ ... ]i i ny Hx H h h h

inner coupling matrix(network topology)

1

( ) 0 (conneted)

0 (otherwise)ij N N ij ji

ij ji

N

ii ijjj i

C c c c

c c

c c


Outer synchronization: The response network:

1

ˆ ˆ ˆ ˆ( ) ( ), , 1, 2,...,

ˆ ˆ

N

i i ij j i ij

i i

x f x c Ly kB y y i N

y Hx

Hypothesis1 (H1): ˆ ˆ( ) ( ) 0f x f x x x

Hypothesis2 (H2):min min2 ( )

2

T TH L LHLH

Hypothesis3 (H3):min is a modifying matrix of by replacing the diagnoal elements by iiC C c


Outer synchronization: The error system:

1

ˆ( ) ( ) , 1,2,...,N

i i ij j ij

e f x f x c LHe kBHe i N

The synchronization criteria:

Suppose that (H1)-(H3) hold. If there exists a constant ksuch that the following inequality hold

min ( ) 02 2

T T T

N NBH H B C CI k I

then the error system is asymptotically stable.


Simulation result

The network is consisting of 10 nodes, where node dynamics is Lorenz system.K=-10, L=[ 1 2 3]’, H=[1 1 1], B=[ 4 5 6]’

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4

simulation steps

erro

r nor

m (1

0 no

des,

30

erro

r sta

tes) C=[-5 1 1 1 0 0 1 0 0 1 ;

1 -2 0 1 0 0 0 0 0 0; 1 0 -4 0 0 1 1 1 0 0; 1 1 0 -4 0 0 1 0 0 1; 0 0 0 0 -4 0 1 1 1 1; 0 0 1 0 0 -2 0 0 1 0; 1 0 1 1 1 0 -5 0 0 1; 0 0 1 0 1 0 0 -2 0 0; 0 0 0 0 1 1 0 0 -2 0 ; 1 0 0 1 1 0 1 0 0 -4]


Topological identification and fault diagnosis based on outer synchronization

using output variable



state variables of the node: outer coupling variable: inner coupling matrix: (connected) (network topology) (otherwise)

observer gain matrix:


1

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

i i ij jj

x f x c Ly i N

1 2[ ... ]Ti i i inx x x x

i iy Hx 1 2[ ... ]nH h h h

1 2[ , ,..., ]TnL l l l


0ij jic c

1

N

ii ijjj i

c c


Observer design:We can design an observer as follows:

The error system can be written as:

Where , , , is estimation of , is the re-state vector,

1

ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ( )

N

i i ij j i i ij

ij i i j

x f x c Ly k y y

c y y HLy

î i ie x x îj ij ijc c c 1 2[ ... ]T

i i i ink k k k îy i i ie y y He

ijc ijc 1 2ˆ ˆ ˆ ˆ[ ... ]Ti i i inx x x x

ˆ î iy Hx

1 1

ˆ ˆ ˆ( ) ( ) ( )N N

i i i ij j ij j i i ij j

e f x f x c LHe c LHx k y y

[1] H. Liu, G. –P. Jiang, C..-X Fan “State-observer-based approach for identification and monitoring of complex dynamical networks,” IEEE Asia-Pacific Conference on Circuits and Systems, 2008 , Macao, China. [2] Liu H, Song Y-R, Fan C-X, Jiang G-P. Fault diagnosis of time-delay complex dynamical networks using output signals [J]. Chinese Physics B, 2010, 19 (7):070508


Lyapunov function:

Consider a positive Lyapnov function as:

Assuming that

V

2

1 1 1

1 12 2i i

N N NTy y ij

i i j

V e e c

ˆ( , ) ( , ) , 1, 2,...,i i if x t f x t e i N


We have the differential coefficient of as:

Where

V

1 1 1

ˆi i

N N NTy y ij ij

i i j

V e e c c

1 1 1 1 1

ˆ ˆ ˆ ˆ[ ( ) ( ) ( )]N N N N N

T Ti i i ij j ij j i i i ij ij

i j j i j

e H H f x f x c LHe c Ly k y y c c

2

1 1 1 1 1 1 1 1

ˆ ˆi

N N N N N N N NT T T T T T

y i ij j i ij j i i i ij iji i j i j i i j

e e H Hc LHe e H Hc LHx e H Hk He c c

2

1 1 1i

N N NT T T T T

i y i ij j y y y y y yi i j

Hk e e H Hc LHe e Ae e Be e Qe

1 2( , ,..., )NA diag Hk Hk Hk ( ) ( )B C HL HL C

Q A B 1 2

[ ... ]N

Ty y y ye e e e


Theorem 1:

If a suitable is chosen such that , then one

gets and

ik 0Q

lim 0it ye lim 0t ijc


Time-delay case: The network with time-delay can be modelled

as:

where is time-varying delay.

1

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

i i ij jj

x t f t x t c Ly t t

( )t

Liu H, Song Y-R, Fan C-X, Jiang G-P. Fault diagnosis of time-delay complex dynamical networks using output signals [J]. Chinese Physics B, 2010, 19 (7):070508 （ SCI）


First we induce some assumptions and a lemma.Assumption 1: Suppose that there exists a positive

constant satisfying: where are time-varying vectors, represents

2-norm. Assumption 2: is a differential function with:

Lemma 1: For any vectors and positive define matrix , the following matrix inequality holds

( ), ( )y t z t

( , ( )) ( , ( )) ( ) ( )f t y t f t x t y t x t

( )t0 ( ) 1t

Q,x y

12 T T Tx y x Qx y Q y


We can design the observer as follows:

The error system can be written as:

1

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

ˆ ˆ( ) ( ( ))i

N

i i ij j i i ij

ij y j

x t f t x t c Ly t t k y t y t

c e t HLy t t

1 1

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

N N

i i i ij y ij j i yj j

e t f t x t f t x t c Le t t c Ly t t k e t


Consider a positive Lyapnov function as:

According to Assumption 1, we get:

V

ˆ( ( ), ) ( ( ), ) ( ) , 1,2,...,i i if x t t f x t t e t i N

2

( )1 1 1 1

1 1 1( ) ( ) ( ) ( )2 2 2(1 )i i i i

N N N NtT Ty y ij y yt t

i i j i

V e t e t c e e d


We have the differential coefficient of as:

V

1 1 1 1 1

1 1 1

1 1 ( )ˆ( ) ( ) ( ) ( ) ( ( )) ( ( ))2(1 ) 2(1 )

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

i i i i i i

i i j i

N N N N NT T Ty y ij ij y y y y

i i j i i

N N NT T Ty i i y ij y y ij j

i i j

tV e t e t c c e t e t e t t e t t

e t H f t x t f t x t e t Hc Le t t e t Hc Ly t

1 1

1 1 1 1

2

1 1 1 1

( ))

1 1 ( )ˆ( ) ( ) ( ) ( ) ( ( )) ( ( ))2(1 ) 2(1 )

1( ) ( ) ( ( )) ( ) ( )2(1 )

i i i i i i

i i j i i

N N

i j

N N N NT T Ty i y ij ij y y y y

i i j i

N N N NT T

y y ij y y i yi i j i

t

te t Hk e t c c e t e t e t t e t t

e t e t Hc Le t t e t Hk e t e

1

1

2

1 1 1 1

( ) ( )

1 ( ) ( ( )) ( ( ))2(1 )

1 1 ( )( ) ( ) ( ( )) ( ( )) ( ( ))2(1 ) 2(1 )

i i

i i

i i j i i

NTy y

i

NTy y

i

N N N NT T

i y y ij y y yi i j i

t e t

t e t t e t t

tHk e t e t Hc Le t t e t t e t t


Using the assumption 2 and lemma 1, one gets:

Where

1

1 1( ) ( ) ( ) ( ) ( ( )) ( ( ))2 2

1 ( ) ( ( )) ( ( ))2(1 )

1( ) ( ) ( ) ( ) ( ) ( )2

i i

T T T Ty y y y y y

NTy y

i

T T T Ty y y y y y

V e t Ae t e t BB e t e t t e t t

t e t t e t t

e t Ae t e t BB e t e t Qe t

1 2

1

2

12(1 )

12(1 )

. ,.

.1

2(1 )

[ ( ) ( ) ... ( )] ,N

N

Ty y y y

Hk

Hk

A

Hk

e e t e t e t B C HL


So, the matrix is negative definite if we get the

suitable . Therefore, base on the Lyapnov

stability theorem, one gets and

converges to a constant when , so the

topology can be approximately identified and

duly monitored.

Q

ik

lim 0it ye ijc

t


Simulation results:

In the simulation, a network of 7 nodes is constructed with each node being a Lorenz model. The Lorenz model can be described as

When a=16,b=4,c=45, the first state variable is depicted in Fig. 1.

1 2 1

2 1 2 1 3

3 1 2 3

i i i

i i i i i

i i i i

x a x xx cx x x xx x x bx


Fig.1 Lorenz model0 10 20 30 40 50 60 70 80 90 100

-40

-30

-20

-10

0

10

20

30x(t)


A dynamical network of seven nodes is constructed, as shown in Fig.2

Fig.

7 7

4 1 0 0 1 1 11 3 1 1 0 0 00 1 1 0 0 0 00 1 0 1 0 0 01 0 0 0 1 0 01 0 0 0 0 1 01 0 0 0 0 0 1

ijC c


The initial values are given as:

1 3 5 7

2 4 6

(0) (0) (0) (0) [ 0.5 0 0] ,

(0) (0) (0) [ 0.6 0 0] ,

ˆ ˆ[1 0 0], [1 1 1] , (0) [0 0 0] , (0) 0

T

T

T Ti ij

x x x x

x x x

H L x c


1 jc

0 200 400 600 800 1000-0.500.511.5 c12

0 200 400 600 800 1000-0.500.51 c13

0 200 400 600 800 1000-0.500.51 c14

0 200 400 600 800 1000-0.500.511.5 c15

0 200 400 600 800 1000-0.500.511.5 c16

0 200 400 600 800 1000-0.500.511.5 c17


Fig.4. Estimation of C12 and the derivative of error . The connection between nodes 1and 2 is broken at t=50s.

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2estimation of c12

0 10 20 30 40 50 60 70 80 90 100-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5differential coefficient of error between estimation of c12 and c12

12C


Fig.5. The synchronization errors ˆ( )iy i ie y y

0 20 40 60 80 100-0.5

0

0.5ey1

0 20 40 60 80 100-1

0

1ey2

0 20 40 60 80 100-1

0

1ey3

0 20 40 60 80 100-1

0

1ey4

0 20 40 60 80 100-0.5

0

0.5ey5

0 20 40 60 80 100-1

0

1ey6

0 20 40 60 80 100-1

0

1ey7


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