synchronization of chaotic system1

8/10/2019 Synchronization of Chaotic System1

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Synchronization of chaotic

system


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Contents

Motivation

Introduction

Literature review Work proposed

Work done

Results References


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Motivation

Chaotic phenomena was first observed by Lorentz while working

on weather model.

Pecora & Carroll first proposed chaotic system synchronization

and potential possibility of its use in secure communication [1].

Survey of different application areas of chaotic system is

presented in [2]. Application areas are

Mechanical System [3][4]

Chemical System [5]

Biological System [6]

Economics [7]

Electrical System [8]


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Motivation

Keeping in view different applications it is pertinent to

explore chaotic behavior of different systems, their control

and synchronization.


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Introduction

As the proposed work will revolve around nonlinear

systems and specifically chaotic/hyperchaotic systems so it

is important to understand their typical behavior.


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Linear ode

if y is a function of x then general form of a Linear ordinary

differential equation of order n is

where each ai as well as f depends on the independent variable x

alone and does not have the dependent variable y or any of its

derivatives in it.


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Nonlinear ode

Nonlinear because of exp term

system of nonlinear equations because of the terms xz and xy


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Linear System

Behavior of the linear system depends on its parameters


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Nonlinear System

Similarly let us study the effect of variation of parameter on the

behavior of Chua System

[ ]z c y pz

[ ( ) ]y s a y x z

( ) ( )x a y x G x , 1

( ) [1 ( 1)]*sgn( ), 1 10

[10( 10) (9b 10)]*sgn( ), 10

x x

G x b x x x

x x x


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Bifurcation Diagram


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Phase Plot for various values of c


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Phase Plot for various values of c


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Literature Review

Various aspects of chaos synchronization has been

discussed in a review paper [9]

Uncertainty in parameters has been addressed in literature

using adaptive estimation of parameters. [10-12]

A review paper [13] has considered different work [14-21]

and has reached to the conclusion that all these

methodologies can be derived form the work given in [22]


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Literature Review

Observer based synchronization: Observers are used toestimate the states of the system. Theory of Non-linearobserver has been discussed in [23].

Different types of observer based synchronization [24] likeLMI (linear matrix Inequality) approach [25-27], slidingmode [28-30], Adaptive sliding mode[31], Adaptiveobserver [32], Differential mean value theorem [33] basedobserver, Nonlinear unknown input observer (NUIO) [34],have been reported in literature.

Synchronization of chaotic system in the presence of noise[35] and in systems driven by common noise [36] has beenobserved


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Research Gaps

On the basis of Literature review following gaps are identified

Projective synchronization of Time delayed systems

Nonlinear Unknown Input Observer based synchronization

using Differential Mean Value Theorem Adaptive Sliding Mode Observers are not explored to much

depth


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Proposed Work

1. Synchronization with uncertain parameters

a) Adaptive Synchronization of chaotic/ hyperchaotic

systems with uncertainty in parameters

b) Extension to time delayed systemsc) Synchronization of different order systems


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Proposed Work

2. Observer Based synchronization scheme

a) Observer design for NUIO (Nonlinear Unknown Input

Observer) case and extension to synchronization

b) Reduced order synchronization3. Contraction Theory based synchronization scheme with

and without uncertainty.


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Proposed Work

4. Robust Observer design and application to

synchronization

a) Sliding Mode based approach

b) Robust adaptive synchronization schemec) Synchronization in the presence of noise


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Work done so far

Course WorkSubjects AdaptiveSignal

Processing

Chaos Control

and

Synchronizatio

n

Nonlinear

Control

Research

Methodology

Grades BC AB AB A

Utility Wiener filter

Steepest

descent algo

LMS algo

RLS algo

Features of

chaos

Parameter

dependent

Synchronizationadaptive,

- observer

based, phase,

communicati

on

Feedback

Linearization

Sliding

Control

BacksteppingAdaptive

Control


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Work done so far

Synchronization of chaotic systems with uncertainty in

parameters

Adaptive Synchronization of time delayed chaotic systems

with parameter uncertainty

Nonlinear unknown input observer design

Sliding mode based observer design


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Synchronization of Chaotic System

Unidirectional: Master-Slave

Synchronization in which typically two systems

are synchronized such that slave system

mimics the motion of master system.

Bidirectional: may involve several systems

synchronizing without prescribed hierarchy.


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Master-Slave Synchronization

Master System Slave System

( ) ( )

; ::

is a parameter vector

of the sytem

m m m

m mxk k

x f x F x

x R f R RF R R R

( ) ( ) U

; :

:

is a parameter vector

of the sytem

U is the controller

m m m

m mxl l

y g y G y

y R g R R

G R R R


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Error vector

1 2( , ,..., )

0

m

i

e y x

diag

1 Complete Synchronization

1 Anti Synchronization

Projective Synchronization

i

i

i


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Error vector

1 2( , ,..., )

0

m

i

e y x

diag

Choose a suitable

controller such that

0

( ) ( )

( ( ) ( ) )

( ) ( ) ( )

( )

limt y xe y x

e g y G y

f x F x U

U g y G y f x

F x ke


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Error vector

1 2( , ,..., )

0

m

i

e y x

diag

0

( ) ( )

( ( ) ( ) )

( ) ( ) ( )

( )

limt

y x

e y x

e g y G y

f x F x U

U g y G y f x

F x ke

e ke


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Example

Hyper-Chaotic Lorenz Lu System

1 1 2 1 4

2 1 1 2 1 3

3 1 2 1 3

4 2 3 1 4

1 2 3 4

1 1 1 1

1 1

1 1

( )

, , , are state variables

, , , are parameters

10, 8 / 3

12, 1

x a x x x

x c x x x x

x x x b x

x x x d x

x x x x

a b c d

a b

c d

1 2 2 1

2 2 2 1 3

3 1 2 2 3

1 2 3

2 2 2

2 2 2

( )

, , are state variables

, , are parameters

36, 3, 20

y a y y

y c y y y

y y y b y

y y y

a b c

a b c


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Example



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Example


1 1 2 1 4

2 1 1 2 1 3

3 1 2 1 3

4 2 3 1 4

1 2 3 4

1 1 1 1

1 1

1 1

( )

, , , are state variables

, , , are parameters

10, 8 / 3

12, 1

x a x x x

x c x x x x

x x x b x

x x x d x

x x x x

a b c d

a b

c d

1 2 2 1 1

2 2 2 1 3 2

3 1 2 2 3 3

4 4

1 2 3

2 2 2

2 2 2

( )

0

, , are state variables

, , are parameters

36, 3, 20

y a y y u

y c y y y uy y y b y u

y u

y y y

a b c

a b c


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Example

Error dynamics e y x

1 2 2 1 1 1 2 1 1 4 1

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

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

4 4 2 3 4 1 4 4

( ) ( )e a y y a x x x u

e c y y y c x x x x u

e y y b y x x b x ue x x d x u

e ke


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Example

Error dynamics e y x

1 2 2 1 1 1 2 1 1 4 1

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

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

4 4 2 3 4 1 4 4

( ) ( )e a y y a x x x u

e c y y y c x x x x u

e y y b y x x b x ue x x d x u

1 2 2 1 1 1 2 1 1 4 1 1

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

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

4 4 2 3 4 1 4 4 4

( ) ( )u a y y a x x x k e

u c y y y c x x x x k e

u y y b y x x b x k e

u x x d x k e


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Example


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M-S Synchronization with Parameter

Uncertainty

Error vector

1 2( , ,..., )

0

m

i

e y x

diag

0

( ) ( )

( ( ) ( ) )

( ) ( ) ( )

( )

limt

y x

e y x

e g y G y

f x F x U

U g y G y f x

F x ke

e ke


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Uncertainty

Then 0limt

y x

Theorem:If the adaptive controller and the

adaptive laws are chosen as

U

( ) ( ) ( ) ( )

( )

( )

T T

T

U g y G y f x F x ke

F x e

G y e


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Uncertainty

1( )

2

0

T T T

T T T

T

V e e

V e e

V ke e

Proof:Let us choose the Lyapunov function as

where ; )


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Uncertainty


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Uncertainty


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Observer Design


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using Observer

Master System( ) ( ) ( ) ...(1)

where ; ;

( ) : nonlinear vector

function

is the number of nonlinearities

n nxn nxm

n m

x t Ax t Bf x

x R A R B R

f x R R

m


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using Observer

Master System( ) ( ) ( ) ...(1)

where ; ;


function


n n n n m

n m

x t Ax t Bf x

x R A R B R

f x R R

m

Observer

1

if ( ) is scaler fucnction i.e. 1

( ) ( ) ( ) where

Observer for the master system( ) ( ) ( ( ) ( )) ...(2)

where ( ) ( ) ( )

Define ( ) ( ) ( )

( ) ( ) ( )

n

f x m

y t f x Kx t K R

x Ax t Bf x B y t y t

y t f x Kx t

e t x t x t

e t A Bk e t

...(3)

by choosing suitable error system (3)

can be made stable asymptotically

so ( ) ( ) as

k

x t x t t


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using Observer

Master System( ) ( ) ( ) ...(1)

where ; ;


function


n n n n m

n m

x t Ax t Bf x

x R A R B R

f x R R

m

Observer

1

if ( ) is nonlinear vector fucnction

( ) ( ) ( ) where

Observer for the master system

( ) ( ) (t)( ( ) ( )) ...(4)

( ) ( (t) ( )) ( ) ( ( ) ( )) ...(5)

n

T

T

f x

y t K t x t K R

x Ax t Bf x BK y t y t

e t A BK K t e t B f x f x


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Synchronization of uncertain Chaotic

System using Sliding Mode Observer

Master System

1

2

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

where ; ;


function

( , ) : denotes system

uncertainties

( ) ( , ) ( , )

( , )

n n n n m

n m

n n

x t Ax t Bf x t x

x R A R B R

f x R R

t x R R R

f x r t x B t x

t x r

Observer


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Master System

1

2

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

where ; ;


function

( , ) : denotes system

uncertainties

( ) ( , ) ( , )

( , )

n n n n m

n m

n n

x t Ax t Bf x t x

x R A R B R

f x R R

t x R R R

f x r t x B t x

t x r

Observer

1

0

( ) ( ) where

( ) ,

Robust Sliding Mode Observer

( ) ( ) ( ) ...(2)

constant design parameter matrix

( , ) is control input

( ) ( ) ( ( ) ( ) (

p n

p

n

m

y t Cx t C R

y t R p m

x Ax t Bf x G Cx y Bv

G R

v x y R

e t A e t B f x f x t

0

, )) ...(3)where

x BvA A GC


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1 2 1 2

011 012 1

0

021 022

1 011 1 012 2 1 1

Sliding Surface is designed as

( ) ...(4)

,

[ ] ,

0

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

y

m n m p

T T m n m

s Me FCe Fe F Cx y

M R F R

e e e e R e R

A A BA B

A A

e t A e t A e t B f x f x t x B v

2 021 1 022 2

1 1 2 2

( )

1 2

..(5 )

( ) ( ) ( ) ...(5 )

So can be rewritten as

...(6)

,

a

b

m m m n m

e t A e t A e t

s

s M e M e

M R M R


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Theorem: If the sliding mode manifold is chosen as (4)

and the controller is designed as follows

( )s t

1 2

...(7a)

( ) ...(7b)

( )( ) ...(7c)

l n

l

T T

n T

v v v

v f x

s MBv r r

s MB

Then master system (1) and slave system (2) get

synchronized

f


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Proof: consider the Lypunov function

0

0 0

1 1 1( ) ( )

2 2 2

the derivative of ( ) along the error system (3) is( )

= ( ( ) ( ( ) ( ) ( , )) )

= (( ) / 2) ( )

(

T T T T

T T T

T T

T T T T

T

V t s s Me Me e M Me

V tV t s s e M Me

e M M A e t B f x f x t x Bv

e M MA A M M e t

s MBf x

0 0

max

) ( ) ( , )

1( )

2

( ) 0 ...(c1)

T T T

T T T

s

s

s MBf x s MB t x s MBv

A M MA A M M

A

h f h


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( ) ( ) ( ) ( , )

( ) ( ) ( , )

( )

( ) 0The error system (3) will reach 0 in finite time

(0) /

on the surface 0 error dynamics will b

T T T T

T T T

n

T

r

V t s MBf x s MBf x s MB t x s MBv

V t s MBf x s MB t x s MBv

V t MB s

V t s s ss

t s

s

1 1 2 2

1

1 1 2 2

1

2 022 021 1 2 2 2

e

0

( ) ( ) ( )

Design M s.t. is hurwitz ...(c2)

M

M

s Me M e M e

e M M e

e t A A M M e A e t

A

h i i f i h i


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48/52

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synchronization of chaotic system1

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