Synchronization of chaotic system1.ppt

8/10/2019 Synchronization of chaotic system1.ppt

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Synchronization of chaotic

systemVivek Sharma

2k13-PhD-EE-212

SupervisorsDr. B. B. Sharma

Dr. R. Nath


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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 toexplore chaotic behavior of different systems, their controland synchronization.


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Introduction

As the proposed work will revolve around nonlinearsystems and specifically chaotic/hyperchaotic systems so itis important to understand their typical behavior.


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Linear odeif 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 xalone and does not have the dependent variable y or any of itsderivatives 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 thebehavior 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) (9 b 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 beendiscussed in a review paper [9]

Uncertainty in parameters has been addressed in literatureusing adaptive estimation of parameters. [10-12]

A review paper [13] has considered different work [14-21]and has reached to the conclusion that all thesemethodologies can be derived form the work given in [22]


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Literature Review Observer based synchronization: Observers are used to

estimate 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 parametersa) 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 schemea) 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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Work done so far

Course WorkSubjects AdaptiveSignal

Processing

Chaos Controland

Synchronization

NonlinearControl

ResearchMethodology

Grades BC AB AB A

Utility Wiener filter Steepestdescent algo LMS algo RLS algo

Features ofchaos Parameterdependent Synchronization adaptive,- observerbased, phase, communication

FeedbackLinearization SlidingControl Backstepping AdaptiveControl


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Work done so far

Synchronization of chaotic systems with uncertainty inparameters

Adaptive Synchronization of time delayed chaotic systemswith parameter uncertainty

Nonlinear unknown input observer design Sliding mode based observer design


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Synchronization of Chaotic System

Unidirectional: Master -SlaveSynchronization in which typically two systemsare synchronized such that slave systemmimics the motion of master system.

Bidirectional: may involve several systemssynchronizing 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 R F 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( , ,..., )

0m

i

e y x

diag

1 Complete Synchronization

1 Anti Synchronization

Projective Synchronization

i

i

i


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Error vector

1 2( , ,..., )

0m

i

e y x

diag

Choose a suitablecontroller 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( , ,..., )

0m

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 parameters10, 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 parameters36, 3, 20

y a y y

y c y y y

y y y b y

y y y

a b ca b c


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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 parameters10, 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 u y 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

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 ParameterUncertainty

Error vector

1 2( , ,..., )

0m

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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Then 0limt

y x

Theorem: If the adaptive controller and theadaptive 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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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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Synchronization of Chaotic Systemusing 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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Synchronization of Chaotic Systemusing Observer Master System

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

where ; ;

( ) : nonlinear vector function

is the number of nonlinearities

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 ChaoticSystem using Sliding Mode Observer Master System

1

2

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

where ; ;

( ) : nonlinear vector

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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Synchronization of uncertain ChaoticSystem using Sliding Mode Observer Master System

1

2

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

where ; ;

( ) : nonlinear vector

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 Bv A A GC


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Synchronization of uncertain ChaoticSystem using Sliding Mode Observer

1 2 1 2

011 012 10

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 B A 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) getsynchronized


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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 t V 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


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Microeconomical Chaos, Chaos, Solitons, Fractals, 1997, vol. 8,1489-1505. 8. Chen, J.H., Chau, K.T., Siu, S.M., and Chan, C.C., Experimental Stabilization of Chaos in a Voltage-

Mode Dc Drive System, IEEE Trans. Circ. Syst. I , 2000, no. 47, 1093-10959. Boccaletti, Stefano, et al. "The synchronization of chaotic systems." Physics Reports 366.1 (2002): 1-

101. 10. Xu, Jiang, Guoliang Cai, and Song Zheng. "Adaptive synchronization for an uncertain new

hyperchaotic Lorenz system." International Journal of Nonlinear Science 8.1 (2009), 117-123.


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References11. Wang, Zuo-Lei. "Projective synchronization of hyperchaotic L system and Liu

system." Nonlinear Dynamics 59.3 (2010): 455-462.12. Wenwen, Ranchao , Modified projective synchronization of different -order chaotic systems

with uncertain parameters IEEE Fourth International Workshop on Chaos -Fractals Theoriesand Applications, 2011, 221-224.

13. Adloo, Hassan, and Mehdi Roopaei. "Review article on adaptive synchronization of chaoticsystems with unknown parameters." Nonlinear Dynamics 65.1-2 (2011): 141-159.

14. Huang, J.: Adaptive synchronization between different hyperchaotic systems with fullyuncertain parameters. Phys. Lett. A 372.27, 4799 4804, 2008

15. Huang, Jian. "Chaos synchronization between two novel different hyperchaotic systems withunknown parameters." Nonlinear Analysis: Theory, Methods & Applications 69.11 (2008):4174-4181.

16. Yassen, M. T. "Adaptive synchronization of two different uncertain chaotic systems." Physics

Letters A 337.4 (2005): 335-341.17. Lu, Jianquan, and Jinde Cao. "Adaptive complete synchronization of two identical or different

chaotic (hyperchaotic) systems with fully unknown parameters." Chaos: An Interdisciplinary Journal of Nonlinear Science 15.4 (2005): 043901.

18. Chen, Xiaoyun, and Jianfeng Lu. "Adaptive synchronization of different chaotic systems withfully unknown parameters." Physics letters A 364.2 (2007): 123-128.


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References27. Chen, Fengxiang, and Weidong Zhang. "LMI criteria for robust chaos synchronization of a class of

chaotic systems." Nonlinear Analysis: Theory, Methods & Applications 67.12 (2007): 3384-3393.28. Feng, Yong, Jianfei Zheng, and Lixia Sun. "Chaos synchronization based on sliding mode observer."

Systems and Control in Aerospace and Astronautics, 2006. ISSCAA 2006. 1st InternationalSymposium on . IEEE, 2006.

29. Chen, Maoyin, Donghua Zhou, and Yun Shang. "A sliding mode observer based secure

communication scheme." Chaos, Solitons & Fractals 25.3 (2005): 573-578.30. Wang, Hua, et al. "Sliding mode control for chaotic systems based on LMI." Communications in

Nonlinear Science and Numerical Simulation 14.4 (2009): 1410-1417.31. Dimassi, Habib, Antonio Loria, and Safya Belghith. "An adaptive sliding-mode observer for nonlinear

systems with unknown inputs and noisy measurement." IFAC World Congress, Milan, Italy . 2011.32. Bowong, Samuel, and Jean Jules Tewa . "Unknown inputs adaptive observer for a class of chaotic

systems with uncertainties." Mathematical and Computer Modelling 48.11 (2008): 1826-1839.33. Zemouche, Ali, Mohamed Boutayeb, and G. Iulia Bara. "Observer Design for Nonlinear Systems: An

Approach Based on the Differential Mean Value Theorem." Decision and Control, and EuropeanControl Conference , 2005.

34. Chen, Weitian, and Mehrdad Saif. "Unknown input observer design for a class of nonlinear systems:an LMI approach." IEEE American Control Conference, Minneapolis USA, June 2006, pp 834-38


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References35. Millerioux, Gilles, and Jamal Daafouz. "Global chaos synchronization and robust filtering in

noisy context." Circuits and Systems I: Fundamental Theory and Applications, IEEETransactions on 48.10 (2001): 1170-1176.

36. Senthilkumar, D. V., and J. Kurths. "Characteristics and synchronization of time-delay systemsdriven by a common noise." The European Physical Journal Special Topics 187.1 (2010): 87-93.

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Synchronization of chaotic system1.ppt