UNIVERSITÀ DEGLI STUDI DI CATANIA

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FACOLTÀ DI INGEGNERIA

DIPARTIMENTO DI INGEGNERIA ELETTRICA ELETTRONICA E DEI SISTEMI

XXII BRAINSTORMING DAY

Catania, May 20, 2011PHAM VIET THANH

Ph.D. Student in INGEGNERIA ELETTRONICA, AUTOMATICA E DEL CONTROLLO DI SISTEMI COMPLESSI

XXV CICLO

Coordinatore: Prof. Ing. L. Fortuna

Tutor: Prof. Ing. R. Caponetto

Outline

Realization of synchronization of coupled multiple1

Fractional-order differential equation of2

delay systems

phase-locked loops

Memristor-based Cellular Neural Network33

Realization of synchronization of coupled multiple1

Fractional-order differential equation of2

delay systems [1]

phase-locked loops

[1] V.-T. Pham, L. Fortuna, M. Frasca, T.T. Anh and T.M. Hoang, “Realization of synchronization of coupled multiple delay systems on FPGA platform”, 3rd International workshop on nonlinear dynamics and synchronization INDS’11, Klagenfurt, Austria, July 25-27, 2011, accepted.

Memristor-based Cellular Neural Network3

Motivation

Confirm the theoretical results

Potential applications in secure communication

Investigate novel features

Realization of synchronization

of coupled multiple delay

systems

Main steps

Experimental results

FPGAModelSim

Matlab

Models

Simulationsresults

Model of two-delay systemDynamical equation: ( )

2

1ii

i

dx x m f xdt τα

=

= − +∑

Synchronization

1

Complete synchronization refers to the phenomenon in which the state of master is equal to one of slave

2

Projective–lag synchronization corresponds to the proportion of retarded state of master to the state of slave is a constant

3

Projective–anticipating synchronizationthe slave anticipates the master’s motion (in contrast to the projective-lag synchronization)

We consider:

SynchronizationChaos synchronization schemes in

coupled multiple time delay systems MS: Master DRV: Driving signal SL: Slave

SynchronizationScheme of complete synchronization MS:

DRV:

SL: Synchronization condition (by using

Krasovskii-Lyapunov functional approach)

( )2

1,

iii

dx x m f xdt τα

=

= − +∑

( ) ( )2

1,

iii

DRV t k f xτ=

= ∑

( )2

1.

iii

dy y n f ydt τα

=

= − +∑

( )2

1

,

sup ,i

i i i

ii

n m k

n f xτα=

= − ′>

∑

SynchronizationProjective-lag synchronization scheme

MS:

DRV:

SL: Synchronization condition (by using

Krasovskii-Lyapunov functional approach)

( )2

1,

iii

dx x m f xdt τα

=

= − +∑

( )2

1.

iii

dy y n f ydt τα

=

= − +∑

( ) ( )2

2

1,

iii

DRV t k f xτ +=

= ∑

( )2

1

,

sup ,i d

i i i

ii

an bm ak

an f xτ τα +=

= − ′>

∑

( ) ( )day t bx t τ= −

SynchronizationProjective-anticipating synchronization scheme:

MS:

DRV:

SL: Synchronization condition (by using Krasovskii-

Lyapunov functional approach)

( )2

1,

iii

dx x m f xdt τα

=

= − +∑

( ) ( )2

1,

iii

DRV t k f xτ=

= ∑

( )2

1.

iii

dy y n f ydt τα

=

= − +∑

( ) ( )day t bx t τ= +

( )2

1

,

sup ,i d

i i i

ii

an bm ak

an f xτ τα −=

= − ′>

∑ 2 .i i dτ τ τ+ = −

SimulationsSimulation results by Matlab

Chaotic signal

Complete synch.Phase portrait

SimulationsSimulation results by ModelSim

Complete synch. Projective-lag synch.

Projective-anticipate synch.Schematic of the complete synch.

Experimental results

CompleteSynch.

Proj.-LagSynch.

Proj.-Anti.Synch.

Logic elements 3.25% 3.37% 3.38%

Registers 824 848 850

Memory bits 23% 28% 21%

Utilized resources for implementation of three synchronization schemes

Experimental results

Complete synch. Projective-lag synch.

Projective-anticipate synch.

Realization of synchronization of coupled multiple1

Fractional-order differential equation of2

delay systems

phase-locked loops

Memristor-based Cellular Neural Network33

Motivation

Practical applications of PLL

Chaotic behavior of PLL

Fractional calculus

Fractional-order differential equation of

PLLs

Phase-locked loop

( ) ( )1 2

2 2 1sin sin cosx xx x x M t M tβ βσ β=

= − − + + Ω + Ω Ω

PLL is described as

, , , Mβ σ Ω normalized natural frequency, normalized frequency detuning, normalized modulation frequency and normalized maximum frequency derivation respectively

Fractional-order PLLThe fractional-order PLL is expressed as

where is the derivative orderThe fractional-order PLL is chaotic when changing the derivative order and keeping other parameters (as the

circumstance of chaotic PLL)

( ) ( )0 1 2

0 2 2 1sin sin cost

t

D x x

D x x x M t M t

α

α β βσ β

=

= − − + + Ω + Ω Ωα

α

Fractional-order PLL Lyapunov exponents of FOPLL

Fractional-order PLLChaotic behavior of FOPLL

0.98, 0.056, 0.2, 0.8, 0.7Mα β σ= = = = Ω =

Fractional-order PLLThe locking range of FOPLL is similar to the one of PLL

Chaos control in FOPLLFOPLL in the state-space form

Controlled FOPLL

With control term selected as( )0 ,t cDα = + + +x Ax Bf x u u

( ) ( ) ( )1 2

0.

sin cosc k x x M t M tβσ β

= + − − Ω − Ω Ω u

( )0 ,tDα = + +x Ax Bf x u0 10 β

= − A

0 00 1

= − B 1

2

xx

=

x

( )1

0sin x

=

f x ( ) ( )0

.sin cosM t M tβσ β

= + Ω + Ω Ω

u

Chaos control in FOPLLThe reduced form of FOPLL system

The Jacobian matrix

The equilibrium points of system are asymptotically stable if the following condition is satisfied

( )0 ,tDα =x g x ( ) ( )2

1 1 2

.sin

xx kx k xβ

= − + + −

g x

1

0 1.

cosk x k β

= − − J

( )( ) ( )arg eig arg ,2iπλ α= >J

Chaos control in FOPLLThe parameter is chosen as 16k = −

Synch. of chaos in FOPLL

Master system

Synchronization

Slave system

( )0 ,t m m mDα = + +x Ax Bf x u ( )0 ,t s s mDα = + + +x Ax Bf x u Ke

1

2

mm

m

xx

=

x

( )1

0sinm

mx

=

f x

1

2

ss

s

xx

=

x

2 2R ×∈K

1 11

2 22

m s

m s

x xex xe

− = = −

e

0 10 β

= − A

0 00 1

= − B

Synch. of chaos in FOPLL

Stable condition Synch. error Feedback gain matrix

( )0 tDα = −e A K e( )( )arg eig2πα− >A K

1 10 2 β

= − K

How to find the feedback gain matrix

Synch. of chaos in FOPLLThe synchronization occurs when choosing

1 1.

0 2 β

= − K

lim lim 0m st t→∞ →∞− = =x x e

Realization of synchronization of coupled multiple1

Fractional-order differential equation of2

delay systems

phase-locked loops

Memristor-based Cellular Neural Network 33

Memristor-based CNNConsider the combination of memristor

and CNN

Combine memristors with CNN?CNN [2] Memristor [3]

ij ij kl klx x Ay Bu I= − + + +∑ ∑

( ) ( )1 1 12ij ij ij ijy f x x x= = + − −

ij ijx u=

( )Mij ij ijy x u=

[2] L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Trans. Cir. Sys., vol, 35, no. 10, pp. 1257-1272, 1988.[3] L. O. Chua, “Memristor-The missing circuit element,” IEEE Trans. Circuit Theory,vol. 18, no.5, pp. 507–519, 1971.

, ,x q u i y v= = =

ConclusionExperimental synchronization in FPGA–

based multiple delay systems is implemented Experimental results agreed with the numerical simulations The first step to achieve one advanced chaotic secure

communication

Introduce model of fractional-order differential equation-based FOPLL Consider the chaotic behavior of FOPLL Propose the methods to synchronize and control chaos in FOPLLs

Consider the ability to realize CCN using memristors

Publication and CoursesPublication:

V.-T. Pham, L. Fortuna, M. Frasca, T.T. Anh and T.M. Hoang, “Realization of synchronization of coupled multiple delay systems on FPGA platform”, 3rd International workshop on nonlinear dynamics and synchronization INDS’11, Klagenfurt, Austria, July 25-27, 2011, accepted.

Summer/Professional school: PhD School: Electronic, Automation and

Control of Complex Systems, Oct. 4-28, 2010. Professional School: 3rd Euro-Mediterranean

UNIversity Summer Semester Catania, Aug. 20-Sept. 10, 2010.

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