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Identification and Monitoring of Identification and Monitoring of Complex Networks Based on Complex Networks Based on Synchronization Synchronization

Wallace Tang Wallace Tang

Department of Electronic Engineering Department of Electronic Engineering City University of Hong Kong City University of Hong Kong

International Workshop on Complex Systems and Networks 2007 July 19-21, 2007 Guilin, China

Outline Outline

• Introduction

Introduction

• The Target

The Target

• Get Ready: Global Adaptive Observer Design

Get Ready: Global Adaptive Observer Design

• Move Forward: A Proposal of Design

Move Forward: A Proposal of Design

• Another Direction: Evolutionary Algorithm

Another Direction: Evolutionary Algorithm

• Conclusions

Conclusions

Introduction Introduction

• Complex networks: biological neural network, power system,

Complex networks: biological neural network, power system, network of chaotic oscillators, … network of chaotic oscillators, …

• Statistical models: Small

Statistical models: Small-

• world networks, scale

world networks, scale-

• free networks,

free networks, random networks random networks

• However, uncovering the exact topology and model of a

However, uncovering the exact topology and model of a targeted network is difficult. targeted network is difficult.

• Useful for understanding the behaviour over the network,

Useful for understanding the behaviour over the network, reporting any connection failure, … reporting any connection failure, …

Example: Biological Neural Network Example: Biological Neural Network

• A neuron is a nerve cell.

A neuron is a nerve cell.

• Neurons have specialized projections

Neurons have specialized projections called dendrites and axons. Dendrites called dendrites and axons. Dendrites bring information to the cell body and bring information to the cell body and axons take information away from the axons take information away from the cell body. cell body.

• Neurons communicate with each other

Neurons communicate with each other through through

• synaptic coupling

synaptic coupling (an electrochemical (an electrochemical process) process)

• electrical coupling

electrical coupling

presynaptic terminal dendrite of receiving neuron axon

HR Neuron Model HR Neuron Model

• H

Hindmarsh indmarsh-

• Rose (HR) neuron model

Rose (HR) neuron model

• 3

3rd

rd order dynamical e

• rder dynamical equations

quations

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

2 3 2

z c bx z y x f t z y x a z y x f t y z y x ax z y x f t x

z y x

− + = = − + = = − − − = = µ α & & &

a=2.8, b=9, c=5, α=1.6, µ=0.001

Tonic bursting

where x(t) is the membrane potential y(t) and z(t) are the recovery variables w.r.t. fast and slow current, respectively

E.M. Izhikevich, “Which model to use for cortical spiking neurons,” IEEE Trans. Neural Network, 15(5), pp. 1063-1070, Sept 2004.

Model of BNN Model of BNN

• Biological Neural Network: A set of connected neurons

Biological Neural Network: A set of connected neurons

• E

Evolution of its nodes volution of its nodes with coupling with coupling: :

) ( ) ( ) ( ) ( ) (

2 1 3 2 i i i i i i i i i i i ij N j ij i i i i i i

z c x b t z y x a t y g z y x x a t x − + = − + = − − − − =

∑

=

µ α γ & & &

where for electrical coupling, or for synaptic coupling. If neurons i and j are connected to each other, and otherwise. Note:

i j ij

x x − = γ

>

ij

g =

ij

g

ji ij

g g =

) (

1

s j

x v s i ij

e V x

θ

γ

− −

+ − =

Example: Network of Chaotic Oscillators Example: Network of Chaotic Oscillators

• Collection of chaotic oscillators

Collection of chaotic oscillators

1 2 3

The Target The Target

Estimate the states and parameters of Estimate the states and parameters of a complex network a complex network

Imprecise models Imprecise models (Unknown parameters, Unknown topology) (Unknown parameters, Unknown topology) Limited observable states Limited observable states

Problem Formulation Problem Formulation

• A complex network,

A complex network, Ω Ω

• Design another system

Design another system Φ Φ s.t. s.t. Φ Φ and and Ω Ω are synchronized. are synchronized.

⎩ ⎨ ⎧ = = Ω Cx p) F(x, x y & :

model

• bservable
• utput

n

R x∈

y the state vector the observable output

⎪ ⎩ ⎪ ⎨ ⎧ = = Φ ) p , x R( p ) p , x F( x y y , ˆ ˆ ˆ , ˆ ˆ ˆ : & &

p p x x → → ˆ ˆ when ∞ → t

F

nonlinear smooth function a constant matrix C

Possible Ways to Go Possible Ways to Go

• Known p

Known p Synchronization problem Synchronization problem

• Unknown p

Unknown p

• adaptive observer

adaptive observer design design

• Error minimization problem

Error minimization problem

• Gradient descent methods (Quasi

Gradient descent methods (Quasi-

• Newton, Powell minimization

Newton, Powell minimization algorithm) algorithm) ⇒ ⇒ Local optimal Local optimal

• Optimization:

Optimization: Evolutionary Algorithms Evolutionary Algorithms, Particle Swarm Optimization , Particle Swarm Optimization

Get Ready … Get Ready …

Adaptive Observer Adaptive Observer

for joint state for joint state-

• parameter estimation of a targeted system

parameter estimation of a targeted system

• Consider the following system

Consider the following system where where i=1,2,…,n

i=1,2,…,n and are smooth functions

and are smooth functions

• Assumptions

Assumptions

• is the output of the syst

is the output of the system em

• are unknown paramete

are unknown parameters rs

• s.t. for the system , we have

s.t. for the system , we have , , evaluated along the solution of error dynamics, evaluated along the solution of error dynamics, and and is negative definite. is negative definite.

A Global Adaptive Observer Design A Global Adaptive Observer Design

( ) ( ) ( )

∑

=

≡ + =

i

m j i ij ij i i

f p c x

1

p x, F x x &

( ) ( )

i ij m j ij ij

m j n i r n i t f r

i

L L L , 1 , , 1 , , 1 ,

1

= = = ⇒ = =

∑

=

x

i

F

u ∃

( ) ( )

y , x u p , x F x ˆ ˆ ˆ + = &

( )

n k x x x y

k

< = , , , ,

2 1

L

i ij

m j k i p , , 2 , 1 ; , , 2 , 1 , L L = =

( )

e e e

T

2 1

1

= V

( )

e

1

V & x x e − = ˆ

• Adaptive Observer:

Adaptive Observer:

• Let where

Let where By numerical cancellation, it can be proved that By numerical cancellation, it can be proved that

• Based on the equation for the error dynamics, and

Based on the equation for the error dynamics, and it follows that it follows that ⇒ ⇒

A Global Adaptive Observer Design A Global Adaptive Observer Design

( ) ( ) ( ) ( ) ( )

k i f x x p y u f p c x

ij i i ij ij m j i ij ij i i

i

, , 2 , 1 ˆ ˆ ˆ , ˆ ˆ ˆ ˆ ˆ

1

L & & = − − = + + =

∑

=

x x x x δ

2 1 1 1 2

1 2 1 2 1

ij k i m j ij n i i

r e V

i

∑∑ ∑

= = =

+ = δ

1

V V & & =

ij ij ij i i i

p p r x x e − = − = ˆ , ˆ

( ) ( )

∑

=

=

i

m j ij ij

t f r

1

x

→

i

e

ij ij

p p = ˆ

[ ]

1 1 1 1 1 1 1 1 1

) ˆ ( ) ˆ ( ) ( ) ˆ ( ) ˆ ( ) ˆ ( V f r e f r e u e f r e x x e V

i i i

m j ij ij k i i m j ij ij k i i i i i n i i m j ij ij k i i n i i i i

& & & & = − + − + = − − =

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

= = = = = = = =

x x p x, F p , x F x

Identify Identify and monitor

and monitor the topology of a BNN the topology of a BNN with limited observable states with limited observable states

• Consider the entire network of N

Consider the entire network of N-

• neurons as a dynamical system

neurons as a dynamical system where where

• Let be the observable output

Let be the observable output

• The coupling

The coupling g g is sufficient so that is sufficient so that synchronizes the network. synchronizes the network.

Formulation Formulation

) , , (

2 1 N

ξ ξ ξ L = ξ

) , ( ) ( ) (

1

g ξ ξ ξ

i N j ij ij i i

F f g c ≡ + =

∑

=

ξ &

⎩ ⎨ ⎧ ≤ < ≤ ≤ = N 3 i N if N i 1 if ) ( ) ( ) (

, j v i V ij

s s

f ξ γ ξ σ

θ

ξ

) ( ) (

s i i V

V

s

− − = ξ ξ σ

) ( ,

1 1 ) (

s j s

v j v

e

θ ξ θ ξ

γ

− −

+ =

) ( ) ( ) ( ) ( ) (

2 1 3 2 i i i i i i i i i i i ij N j ij i i i i i i

z c x b t z y x a t y g z y x x a t x − + = − + = − − − − =

∑

=

µ α γ & & &

[ ] [ ]

T N N N T N

z z y y x x ... ... ... ...

1 1 1 3 2 1

= = ξ ξ ξ ξ ) , ˆ ( ) , ˆ ( ˆ ξ ξ u g ξ F ξ + = &

Adaptive Observer Design Adaptive Observer Design

• Observer:

Observer:

• Lyapunov Function:

Lyapunov Function: where where

• , when t

, when t →∞ →∞ for any initial values of for any initial values of

→

i

e

( ) ( )

g (0), ξ , g ξ(0), ˆ ˆ

∑∑ ∑

= = =

+ =

N i N j ij ij N i i

r e V

1 1 2 3 1 2

1 2 1 2 1 δ

i i i

e ξ ξ − = ˆ

ij ij ij

g g r − = ˆ

N i f g u f g c

ij i i ij ij i N j ij ij i i

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

1

= − − = + + =

∑

=

ξ ) ξ , ξ ξ ξ ξ ξ δ ξ & &

→

ij

r

Simulation I Simulation I

• An illustrative example

An illustrative example

• a network of seven neurons

a network of seven neurons

• divided into two subnets

divided into two subnets

Case I: Entire Network Topology Case I: Entire Network Topology

• Identified the entire network topology

Identified the entire network topology

• where

where g g is a constant and c is a constant and cij

ij=0 or 1.

=0 or 1.

Estimation of cij of the entire network, (for clarity, only j+ c1j where j= 2,… 7, are shown).

5 0 0 s

ij ij

gc g =

Synchronization Error Synchronization Error

• Synchronization error

Synchronization error

Neurons # 1 & # 3 and neurons # 1 & # 7 are disconnected at t= 500s

Case II: A Subnet Case II: A Subnet

• Focus on the connections in the Subnet I.

Focus on the connections in the Subnet I.

• Only the state

Only the state x x of

• f neurons #1

neurons #1-

• 4

4 are measured. are measured.

5 0 0 s

Synchronization Error Synchronization Error

• Synchronization error

Synchronization error

Subnet I is isolated at t= 500s

Connections of Neurons Connections of Neurons

Identification of External Connection Identification of External Connection

• Due to the existence of external connection, the

Due to the existence of external connection, the estimation result will not be exact. estimation result will not be exact.

• Now

Now

• If there

If there’ ’s no external connection: s no external connection:

• Since

Since

• Neuron

Neuron # #1 is also connected with an external neuron. 1 is also connected with an external neuron.

, 1 , 1 , , 5 . 1 , 5 . 1

41 31 21 14 13 12

≈ ≈ ≈ ≈ ≈ ≈ c c c c c c

∑ ∑

≠ = ≠ =

=

N i j j ji N i j j ij

c c

, 1 , 1

, 1 ) (

2 1 1

= −

∑

= N j j j

c c

Case III: A Neuron Case III: A Neuron

• Objective:

Objective: Identify the connectivity of a neuron Identify the connectivity of a neuron

• It is observed that the pulsing of neurons looks very similar

It is observed that the pulsing of neurons looks very similar (but not identical) under coupling. (but not identical) under coupling.

• Only the state

Only the state x x of the

• f the neuron #1

neuron #1 is monitored is monitored

• Construct an adaptive observer

Construct an adaptive observer based on based on two connected two connected neurons neurons

5 0 0 s

Synchronization Error Synchronization Error

Synchronization error Measured x1(t)

Connectivity of a Neuron Connectivity of a Neuron

• Taking the time instance

when the state estimation errors small

• The connectivity of

neuron #1 is around 2.7 ≈ 3 connections

• Disconnection of neurons

#1 & #7 at 500s, connectivity of neuron #1 is around 1.9 ≈ 2 connections

Simulation II Simulation II

• 15 subnets, total 98

15 subnets, total 98 neurons with neurons with synaptic coupling . synaptic coupling .

• Neuron #1 of

Neuron #1 of Subnets #2 Subnets #2-

• 15 are

15 are connected to Neuron connected to Neuron #1 of Subnet #1 #1 of Subnet #1

• Neuron #5 of

Neuron #5 of Subnets #3 Subnets #3-

• 12 are

12 are connected one connected one-

• by

by-

• ne in sequence.
• ne in sequence.
• The neurons are

The neurons are about synchronized. about synchronized.

1 1 1

Connectivity Connectivity

Note: There’re 2 connections for Neuron #1 in the Subnet #1.

Simulation III Simulation III

• 15 subnets, total 98

15 subnets, total 98 neurons with neurons with electrically coupling electrically coupling

• Neuron #2 of

Neuron #2 of Subnets #2 Subnets #2-

• 15 are

15 are connected to Neuron connected to Neuron #2 of Subnet #1. #2 of Subnet #1.

• Neuron #5 of

Neuron #5 of Subnets #3 Subnets #3-

• 15 are

15 are connected one connected one-

• by

by-

• ne in sequence.
• ne in sequence.

1 1 1

Connection of Neurons Connection of Neurons

Connectivity of Neuron #2 Connectivity of Neuron #2

Note: There’re 3 connections for Neuron #1 in the Subnet #1.

Simulation IV Simulation IV

• 3 neurons with synaptic coupling

3 neurons with synaptic coupling

1 5 0 0 0 s

1 2 3 1 2 3

Topology Topology

Problem Formulation Problem Formulation

Reformulate the BNN into Reformulate the BNN into where where and and v v are the state and the output of the system, respectively; are the state and the output of the system, respectively; A A and and C C are known matrices; are known matrices; is the unknown constant parameter vector, and is the unknown constant parameter vector, and are are matrix and vector of observable states, matrix and vector of observable states, v v, respectively. , respectively.

ψ

Cξ ν ψθ Aξ ξ = + + = ϕ &

ξ

θ

ϕ

Assumptions Assumptions

• a constant matrix

a constant matrix K K to stabilize to stabilize A A-

• KC

KC. .

• is a matrix of signals generated by following ODE:

is a matrix of signals generated by following ODE:

• is bounded and persistently exciting such that

is bounded and persistently exciting such that for all t for all t ≥ ≥ 0, some 0, some δ δ > > 0, finite T 0, finite T > > 0 and a positive 0 and a positive definite matrix definite matrix Σ Σ γ ψ KC)γ (A γ + − = & ψ

∫

+

≥

T t t

d I ΣCγ C γ

T T

δ τ

∃

• Adaptive observer:

Adaptive observer:

• It can be deduced that

It can be deduced that where and . where and .

• By considering the positive definite Lyapunov function

By considering the positive definite Lyapunov function it can be proved that , eventually giving it can be proved that , eventually giving and tend to zero when and tend to zero when

Adaptive Observer Design Adaptive Observer Design

η KC) (A η − = &

[ ][

] [ ]

ξ C v Σ C γ θ ξ C v Σ C γγ K θ ψ ξ A ξ

T T T T

ˆ ˆ ˆ ˆ ˆ ˆ − = − + + + + = & & ϕ ~ → θ

∞ → t

• Q. Zhang and B. Delyon, “A new approach to adaptive observer design for MIMO systems,”
• Proc. American Control Conference, June 2001.

) θ γ ΣC(η C γ θ

T T

~ ~ + − = &

θ θ Pη η

T T

~ ~ ) ( + = t V

~ → ξ θ γ ξ η ~ ~ − =

θ θ θ ξ, ξ ξ − = − = ˆ ~ ˆ ~

) ( ≤ t V &

→ η

Synchronization Error Synchronization Error

1 2 3

Neurons 1 and 2 are disconnected

Topology and Parameter Estimation Topology and Parameter Estimation

Neurons 1 and 2 are disconnected

Conclusive Remarks Conclusive Remarks

• Adaptive Observer Design

Adaptive Observer Design

• Global stability can be assured by the Lyapunov stability

Global stability can be assured by the Lyapunov stability proof proof

• Conditions:

Conditions:

• unknown parameter (connection) must be resided in the dynamical

unknown parameter (connection) must be resided in the dynamical equation of the observable state OR equation of the observable state OR

• it is directly related with the observable state OR

it is directly related with the observable state OR

• Special forms of

Special forms of A A in , such as in , such as

• Dynamical transformation may also help in some cases

Dynamical transformation may also help in some cases

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

22

A A

12

a

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = I A

with an asymptotically stable matrix and ξ1 is the observable state.

22

A

• r

ϕ + + = ψθ Aξ ξ &

What if What if the conditions cannot be satisfied? the conditions cannot be satisfied?

Moving Forward … Moving Forward …

• Consider the following nonlinear system

Consider the following nonlinear system where where

⎩ ⎨ ⎧ = + = = Cx G(x) A(p)x p) F(x, x y &

n

R x∈

A(p)

G(x)

y

is the nonlinearity of the system is a fixed matrix, contributing the linear part of the system with p depends linearly on x is the state vector is the observable output.

Assumptions Assumptions

• The nonlinearity

The nonlinearity G(x) G(x) is Lipshitzian: is Lipshitzian:

• There exists an observer gain

There exists an observer gain K K, such that , such that (A (A-

• KC)

KC) is a strictly is a strictly stable matrix stable matrix

• If

If p p is known, the system can be synchronized with is known, the system can be synchronized with

• Let

Let be a solution of chaotic system, if be a solution of chaotic system, if then . It is generally true for a chaotic / complex tra then . It is generally true for a chaotic / complex trajectory jectory based on based on persistently excitation persistently excitation. .

x x ) x G( G(x) ˆ ˆ − ≤ − L ) (t x

) ( ] ˆ [ = − t x ) p A( A(p)

p p ˆ = ) (t x

⎪ ⎩ ⎪ ⎨ ⎧ = + + = x C K ) x G( x A(p) x ˆ ˆ ˆ ˆ ˆ y ey &

A Proposal of Design A Proposal of Design

• Adaptive observer can be designed

Adaptive observer can be designed where where

• K stabilize the linear part of the system function

K stabilize the linear part of the system function

• h

hij

ij is a dynamical function to minimize the output errors

is a dynamical function to minimize the output errors

• µ

µij

ij is auxiliary function to ensure the error convergence

is auxiliary function to ensure the error convergence rate is within the similar order rate is within the similar order

• are some constants

are some constants

• Then, when

Then, when

⎪ ⎩ ⎪ ⎨ ⎧ = = = + + = m j n i e h p e

ij y ij ij ij y

, , 2 , 1 ; , , 2 , 1 ) ˆ ( ) , ˆ ( ˆ ˆ ˆ ˆ ˆ K K & & x x K ) x G( x ) p A( x µ δ

ij

δ

p p x x → → ˆ , ˆ ∞ → t

Example 1: Chaotic Oscillator Example 1: Chaotic Oscillator

• Consider the Lorenz system:

Consider the Lorenz system: where where are assume are assumed, exhibiting d, exhibiting the chaotic dynamics, and they are unknown the chaotic dynamics, and they are unknown is the only observable output. is the only observable output.

• It is impossible to synchronize the system and estimate p

It is impossible to synchronize the system and estimate p1,2,3

1,2,3

based on x based on x1

1, using all the reported techniques.

, using all the reported techniques.

Cx G(x) A(p)x p) F(x, x = + = = y &

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ =

3 2 1

x x x x ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − =

3 2 1 1

1 p p p p A(p)

] [1 C =

} 667 . 2 , 28 , 10 { } , , {

3 2 1

= = p p p p

1

x y =

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − =

2 1 3 1

x x x x G(x)

Step 1: Design of K Step 1: Design of K

• Determine K to stabilize the linear part of system

Determine K to stabilize the linear part of system

• is unknown

is unknown

• Select K for stabilizing

Select K for stabilizing

• Let the initial values as , wit

Let the initial values as , with h the eigenvalues are the eigenvalues are

KC ) p A( − ˆ

} 5 . 2 , 26 , 12 { ) ( ˆ = p

33 . 37 , 50 . 2 , 67 .

3 , 2 , 1

− − − = λ

[ ]

T

6 25 25 = K A(p)

Step 2: Design of h Step 2: Design of hij

ij

• Design the function based on minimization

Design the function based on minimization

• Concept: minimize the error of observable output (state)

Concept: minimize the error of observable output (state)

• Since exists in the state equation of , whose

Since exists in the state equation of , whose x xl

l is available,

is available,

• If the evolution of the state , , depend on

If the evolution of the state , , depend on , then , then

• If does not depend on explicitly, a further

If does not depend on explicitly, a further dependence dependence according to the dynamical evolution of the system should be according to the dynamical evolution of the system should be considered: considered:

3 , 2 , 1 , = i hi

• A. Maybhate and R.E. Amritkar, “Use of synchronization and adaptive control in parameter estimation

from a time series,” Phys. Rev. E 59, 284-293, 1999.

[ ]

2

) ˆ ( min ) ˆ ( y y E − = p (say, y=x1)

ij

p ˆ

1

ˆ x

i

x ˆ 1 , ≠ i Fi

1

ˆ x 1 , ≠ i Fi

1

ˆ x

1 1 1 1

ˆ ) ˆ , ˆ ( e p F h

j j

∂ ∂ ∝ p x

1 1

ˆ ) ˆ , ˆ ( ˆ ) ˆ , ˆ ( e p F x F h

ij i i ij

∂ ∂ ∂ ∂ ∝ p x p x

1 1

ˆ ) ˆ , ˆ ( ˆ ) ˆ , ˆ ( ˆ ) ˆ , ˆ ( e p F x F x F h

ij i k i k k ij

∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∝ ∑ p x p x p x

• From Lorenz system

From Lorenz system

• Type I: Type II:

Type I: Type II:

• Both types share the same updating trend and workable.

Both types share the same updating trend and workable.

1 3 1 3 1 1 2 1 1 2 1

) ˆ ˆ ( ) ˆ ( ) ˆ ˆ ( e x x h e x h e x x h = = − =

1 3 1 3 1 1 2 1 1 2 1

) ˆ ˆ sgn( ) ˆ sgn( ) ˆ ˆ sgn( e x x h e x h e x x h = = − =

Two Types of Designs Two Types of Designs

1 2 1 1

ˆ ˆ ˆ x x p F − = ∂ ∂

1 1 2 2 2 1

ˆ ˆ ˆ ˆ x p p F x F = ∂ ∂ ∂ ∂

3 1 1 3 3 3 1

ˆ ˆ ˆ ˆ ˆ ˆ x x p p F x F x F

i i i

= ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂

∑

• Design such that the error convergence rate

Design such that the error convergence rate of is

• f is

approximately the same. approximately the same.

• Approximated by the convergence rate along the surface with one

Approximated by the convergence rate along the surface with one single unknown parameter single unknown parameter

• The followings are based on Type II, while similar procedures

The followings are based on Type II, while similar procedures can be derived for Type I. can be derived for Type I.

3 , 2 , 1 , = i

i

µ

Step 3: Design of Step 3: Design of µ µij

ij

Assume there is only one unknown, say and define Assume there is only one unknown, say and define Linearize the error dynamical system evaluated on a typical Linearize the error dynamical system evaluated on a typical trajectory, we have trajectory, we have where where

1

p

1 1

ˆ , ˆ p p r − = − = x x e

⎪ ⎩ ⎪ ⎨ ⎧ = + = e e e

) 1 ( ) 1 ( ) 1 (

21 12 11

J r r J J & &

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − =

3 1 3 2 1 2 3 2 1 1 1 11

ˆ ˆ ˆ 1 ˆ ˆ ˆ

) 1 (

p x k x x k x p p k p J

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = ˆ ˆ

1 2 12

) 1 (

x x J

[ ]

) ˆ ˆ sgn( ) ˆ (

1 2 1 1 21

) 1 (

x x J − − = x µ δ

Convergence Rate Convergence Rate

The convergence of The convergence of e e and and r r is governed by: is governed by: Similarly, Similarly, where where

1 2 2 1 3 1 11 1 12 1 11 21 1

ˆ ˆ ) ˆ ( ) (

) 1 ( ) 1 ( ) 1 ( ) 1 (

x x x p J J J J t − + − = − = Γ

−

µ δ

3 1 1 3 11 3 3 1 3 1 2 11 2 2

ˆ ˆ ) ( ˆ ) (

) 3 ( ) 2 (

x x p J t x p p J t µ δ µ δ − = Γ − = Γ

) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) ˆ ˆ )( ( ) ˆ ˆ ˆ ˆ ˆ ( ) ˆ )( ( ) ˆ ˆ ˆ ˆ ( ˆ ) ˆ )( ˆ (

1 3 2 1 2 3 3 3 2 3 1 2 1 3 1 1 11 1 3 2 3 2 1 3 3 3 2 1 2 1 3 1 1 11 1 3 2 3 2 1 3 3 3 2 1 2 1 3 1 1 11

) 3 ( ) 2 ( ) 1 (

x k x x k p x p p p p x p k p J x k k p x x x p p p p x p k p J x k k p x x x p p p p x p k p J + − − − − + + = + − − − − + + = + − − − − + + =

Convergence Rate Convergence Rate

3

Γ

• and are dependent on higher order term and the

and are dependent on higher order term and the dynamical evolution is dependent on the corresponding state, dynamical evolution is dependent on the corresponding state, while is of first order and dependent only on . while is of first order and dependent only on .

• To make all the with similar dynamics and convergence

To make all the with similar dynamics and convergence rate, rate,

• Note: The choice of K should also make , so

Note: The choice of K should also make , so that are all stable. It implies that k that are all stable. It implies that k1

1 and k

and k2

2 are large while

are large while k k3

3 is small.

is small.

1

Γ

3 , 2 , 1 11 )

(

>

= i

i

J

2

Γ

i

Γ

2 2 3 1

, 1 x = = = µ µ µ

1

ˆ x

Choice of Choice of µ µij

ij

i

Γ

Largest CLE with Different K Largest CLE with Different K

• Let

Let

• Point A:

Point A:

with with CLE = CLE = -

• 0.1435

0.1435

K

[ ]

T

6 25 25 = K

[ ]

T

10 2 10 = δ

5 10 15 20 10 20

• 0.04
• 0.02

0.02 0.04 0.06

δ3 δ2=0.5 δ1

CLE 5 10 15 20 10 20

• 0.06
• 0.04
• 0.02

0.02 0.04

δ3 δ2=1 δ1

CLE 5 10 15 20 10 20

• 0.2
• 0.1

0.1 0.2

δ3 δ2=2 δ1

CLE 5 10 15 20 10 20

• 0.15
• 0.1
• 0.05

0.05 0.1

δ3 δ2=4 δ1

CLE

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06

Largest CLE with Different Largest CLE with Different δ δ

Simulation Results Simulation Results

i

p ˆ

vs time

[ ]

T

6 25 25 = K

[ ]

T

1 1 1

0 =

x

[ ]

T

2 2 2 ˆ 0 = x

[ ]

T

5 . 2 26 12 ˆ 0 = p

ei vs time

Parameter Setting: Initial Conditions:

[ ]

T

10 2 10 = δ

Simulation Results Simulation Results

i

p ˆ

vs time

[ ]

T

6 25 25 = K

[ ]

T

1 1 1

0 =

x

[ ]

T

6 5 4 ˆ 0 = x

[ ]

T

2 10 20 ˆ 0 = p

ei vs time

Parameter Setting: Initial Conditions:

[ ]

T

10 2 10 = δ

Simulation Results Simulation Results

[ ]

T

10 2 10 = δ

[ ]

T

10 2 10 = δ

[ ]

T

20 20 = K

[ ]

T

2 2 2 ˆ 0 = x

[ ]

T

2 10 20 ˆ 0 = p

Parameter Settings:

[ ]

T

6 5 4 ˆ 0 = x

[ ]

T

5 . 2 26 12 ˆ 0 = p

Case I: Case II:

[ ]

T

2 10 20 ˆ 0 = p

Convergence Speed Convergence Speed

Synchronization time vs K

τ

5 10 15 20 5 10 15 20 200 400 600 800

δ3 δ1

synchronization time(τ) 100 200 300 400 500 600 700 2 4 6 8 20 40 60 80 100 100 200 300 400 k3 k2 synchronization time(τ) 50 100 150 200 250 B

Synchronization time vs δ

τ

τ is defined as the time required for

the estimated parameters converged to the true values within the error of 10-7

Comparison with Type I Comparison with Type I

) ˆ ˆ sgn( ) ˆ sgn( ) ˆ ˆ sgn(

3 1 3 1 2 1 2 1

x x h x h x x h = = − =

20 40 60 80 100 120 140 160 180 200

• 5

5

time(s) e1

20 40 60 80 100 120 140 160 180 200

• 10

10

time(s) e2

20 40 60 80 100 120 140 160 180 200

• 20

20

time(s) e3

Black line: Type I Red line: Type II (using sign function) t=23s t=60s

y y y

e x x x p e x p e x x p

2 3 3 1 3 3 1 2 2 1 2 1 1

ˆ 1 ˆ ˆ ˆ ˆ ˆ ) ˆ ˆ ( ˆ + = = − = δ δ δ & & & Type I Type II

y y y

e x x p e x x p e x x p ) ˆ ˆ sgn( ˆ ) ˆ sgn( ˆ ˆ ) ˆ ˆ sgn( ˆ

3 1 3 3 1 2 2 2 1 2 1 1

δ δ δ = = − = & & &

Example 2: Biological NN Model Example 2: Biological NN Model

• A BNN of 4 neurons with synaptic coupling

A BNN of 4 neurons with synaptic coupling

) 5 ( 001 . 45 . 4 8 . 2 ) 5 9 ( 001 . 4 . 4 8 . 2

2 2 2 2 2 2 2 2 4 1 2 2 2 2 3 2 2 2 2 1 1 1 1 1 2 1 1 4 1 1 1 1 1 3 1 2 1 1

z x r z y x y g c z y x x x z x z y r x y g c z y x x x

j j j j j j

− + = − = − − − − = − + = − = − − − − =

∑ ∑

= =

& & & & & & γ γ 1 2 3 4 ) 5 9 ( 001 . 46 . 4 8 . 2 ) 5 9 ( 001 . 8 . 2

4 4 4 4 4 2 4 4 4 1 4 4 4 4 3 4 2 4 4 3 3 3 3 2 3 3 3 4 1 3 3 3 3 3 3 2 3 3

z r x z y x y g c z y x x x z x z y x r y g c z y x x x

j j j j j j

− + = − = − − − − = − + = − = − − − − =

∑ ∑

= =

& & & & & & γ γ Note: r1 = 1; r2 = 9; r3 = 4.43; r4 = 1

Connection of Neurons Connection of Neurons

1 2 3 4

Synchronization and Estimation Errors Synchronization and Estimation Errors

Initial guess of r : [0.3 3 2 0.3]

A Totally Different Way … A Totally Different Way …

Evolutionary Algorithms Evolutionary Algorithms

• Viewed as a nonlinear system modeling problem, to find a

Viewed as a nonlinear system modeling problem, to find a model to minimize the observable output (state) error model to minimize the observable output (state) error

• Evolutionary algorithm: a guided

Evolutionary algorithm: a guided-

• random searching

random searching method based on the fitness functions only method based on the fitness functions only

• Good for multimodal, nonlinear, constrained optimization

Good for multimodal, nonlinear, constrained optimization problem problem

• Successfully applied to nonlinear modeling, network

Successfully applied to nonlinear modeling, network topology design, optimal controller design, scheduling in topology design, optimal controller design, scheduling in manufacturing system, asset placement in VOD system, manufacturing system, asset placement in VOD system, … …

• Requirement: a time series of outputs

Requirement: a time series of outputs

Basic Concept Basic Concept

• Natural selection:

Natural selection: Survival of the fittest Survival of the fittest

T DNA Nucleotides Codons Genes A C G C C G A T A

G

A

A

C G C A T A A A A

A A A A T C

A A

C

... ...

DNA structures

Genetic Cycle Genetic Cycle

antennae head wing feet size body body color head color

K.F. Man, K.S. Tang and S. Kwong, “Genetic algorithms: Concepts and designs,” Springer Verlag London, ISBN 1-85233-072-4, 1999.

Crossover Crossover

• A

A major major operation in GA

• peration in GA
• A technique to combine the genes of two parents

A technique to combine the genes of two parents

• A crossover rate to govern the probability of the operation

A crossover rate to govern the probability of the operation (usually very high, 0.6 (usually very high, 0.6-

• 1.0)

1.0)

• Offspring may be the same as their parents

Offspring may be the same as their parents

Single Single-

• point Crossover

point Crossover

Genetic Operations Genetic Operations

• Multi

Multi-

• point Crossover

point Crossover

• Uniform Crossover (mask = 1111000010001111100000000)

Uniform Crossover (mask = 1111000010001111100000000)

Mutation Mutation

• A

A background background operation with low operational rate

• peration with low operational rate

(<0.1) (<0.1)

• A bitwise operation applied to each offspring after

A bitwise operation applied to each offspring after crossover crossover

• Provide the missing gene

Provide the missing gene

• Increase the randomness

Increase the randomness

Jumping Gene Operations Jumping Gene Operations

Cut-and-Paste operation Copy-and-Paste operation

T.M. Chan, K.F. Man, S. Kwong and K.S. Tang, “A jumping gene paradigm for evolutionary multiobjective optimization,” IEEE Trans. Evolutionary Computation, accepted for publication.

Building Block Hypothesis Building Block Hypothesis

• GA seeks near

GA seeks near-

• optimal
• ptimal

performance through the performance through the juxtaposition of short, low juxtaposition of short, low-

• rder, high performance
• rder, high performance

schemata, called the schemata, called the building block building block

P1 P2 Ch1 Ch2

Chromosome Design Chromosome Design

• Topology:

Topology:

• Chromosome Coding

Chromosome Coding

) ( ) ( ) ( ) ( ) (

2 1 3 2 i i i i i i i i i i i ij N j ij i i i i i i

z c x b t z y x a t y g z y x x a t x − + = − + = − − − − =

∑

=

µ α γ & & &

Simulation Results Simulation Results

Conclusions Conclusions

• It is possible to identify the topology and the system parameter

It is possible to identify the topology and the system parameters s

• f a complex network by the design of adaptive observer.
• f a complex network by the design of adaptive observer.
• If the time series of the

If the time series of the data, data, it may also be solved by the it may also be solved by the evolutionary algorithms or some other minimization approaches. evolutionary algorithms or some other minimization approaches.

• Future work

Future work

• A real biological neural network

A real biological neural network

• Examine the experimental data from CPG network of the neurobiolo

Examine the experimental data from CPG network of the neurobiology gy laboratory in INLS UCSD (a rhythm generating biological networks laboratory in INLS UCSD (a rhythm generating biological networks with 14 with 14 neurons in crustaceans) neurons in crustaceans)

• More accurate model, Better understanding of dynamics

More accurate model, Better understanding of dynamics

• Fault detection

Fault detection

• Impacts on synchronization

Impacts on synchronization-

• model based secure communications

model based secure communications

Acknowledgement Acknowledgement

• Professor Ljupco Kocarev, INLS UCSD

Professor Ljupco Kocarev, INLS UCSD

• Professor Kim Man, CityU HK

Professor Kim Man, CityU HK

• Research Students in CityU HK

Research Students in CityU HK

• Ying LIU, Yu MAO, Jay YIN

Ying LIU, Yu MAO, Jay YIN

Thank You! Thank You!