Athens nov 2012 1

Benjamin Busam, Julius Huijts, Edoardo Martino

ATHENS - Nov 2012

Control of Chaos-

Stabilising chaotic behaviour

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Chaos in a nutshell

Small change in initial condition

Huge difference in results

Deterministic systems, impossible to predict

See: [CT]

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Control of Chaos

• Stabilisation

– Suppression– Synchronisation

See: [Fe], [BG]

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Control of Chaos

• Stabilisation– Suppression

– Synchronisation

See: [Fe], [SA]

?

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Controlling Methods

1. Pyragas MethodDelayed Feedback Control

2. OGY-MethodShort explanation

See: [AF]

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Pyragas

See: [Fe], [SA]

Desired Orbit

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Pyragas

See: [Fe], [SA]

SystemX(t) Y(t)

u

u(t)=G[Y0-Y(t)]

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Pyragas

See: [Fe], [SA]

SystemX(t) Y(t)

u

u(t)=G[Y(t-T)-Y(t)]

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Pyragas

See: [Fe], [SA]

SystemX(t) Y(t)

u

u(t)=G[Y(t-T)-Y(t)]

Only need to know T

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Controlling Methods

1. Pyragas MethodDelayed Feedback Control

2. OGY-MethodShort explanation

See: [AF]

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OGY method

Objective

Reach equilibrium by small perturbation.

Why it will work

•Large number of low-period orbits•Ergodicity : trajectory visits neighborhood.

•Chaotical system is sensible to small perturbation

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OGY methodSteps

• Determinate the low period orbit embedded in the chaotic set.• Determinate the stable orbit or point embedded in the

attractor.• Apply small perturbation to stabilize the system.

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OGY method

System: x(t +1) = f (x(t),u(t))

When u(t)= u` (constant)

x(t) passes by x` infinite times.

Equilibrium point x` in the attractor.

Problem: Find a control law u(t)=h(x(t)) that stabilizes the system.

x: analyzed parameter

u: tunable parameter

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OGY method

1. Restriction on u:• Small perturbation [u-δ;u+δ] δ«|u|

2. Approximation of x(t +1) = f (x(t),u(t)):

• Linear approximation: dx(t +1) = Adx(t) + bdu(t) Where A=∂f/ ∂x|x`,u` b=∂f/ ∂u|

x`,u`

Control law: du(t) = kdx(t)→ dx(t +1) = (A+bk)dx(t)

k depends on the physics of the system

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OGY methodOGY control law:

u(t)=h(x(t))=u’ If |x(t) – x’|>ε

u’ + k(x(t)-x’) If |x(t) – x’|≤ ε{

•Far from the stable point (curve) •Near the stable point (curve)

See: [1], [2]

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OGY method

‹t›: transient time γ>0 γ: depends on dimension

)(

)()]([2)]([)(

tx

txtxdxtxP

Probability curve moves to neighbors:

→‹t›=1/P(ε)≈ε-1≈δ-1‹t›≈δ-γ

How long will it take?

See: [BG]

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Duffing Oscillator

See: [We], [YT]

30 cosx x x x f t

driving force

damping

restoring force

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Duffing Oscillator

See: [We], [Ka]

30 cosx x x x f t

driving force

damping

restoring force

Poincaré section of the duffing oscillator

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D.O. - Phase Portrait

See: [SA]

30 cosx x x x f t

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D.O. - Control

3 20 cos 1x x x x f t x u

control term

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D.O. - Control

3 20 cos 1x x x x f t x u

control term

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D.O. - Noise

See: [SA]

3 20 cos 1x x x x f t x u kv

noise

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D.O. - Noise

See: [SA]

3 20 cos 1x x x x f t x u kv

noise

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D.O. - Noise

3 20 cos 1x x x x f t x u kv

See: [SA]

noise

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Control of laser chaos

See: [HH]

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Control of laser chaos

)'cos('0 tRRR

See: [HH]

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Control of laser chaos

See: [HH]

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Control of laser chaos

See: [HH]

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Conclusion

1. Pyragas Method

2. OGY-Method

3. Applications

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Any questions?

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Practical Chaos control

Situation:•Toroidal cell in vertical position full of liquid

•Lower half in heater

•Two thermometer at 3 and 9 o’clock

Chaos in the fluid: Situation

Chaos:ΔT changes chaotically

→Fluid dynamics equation

Convective flux

See: [BG]

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Practical Chaos control

Control by feedback:Controlling the ΔT (decreasing oscillation amplitude) by applying perturbation to heater proportional to ΔT.

Chaos in the fluid: Control

See: [BG]

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From chaos to orderChaotical systems can become non chaotical:

Fireflies

http://www.youtube.com/watch?gl=IT&hl=it&v=sROKYelaWbo

Rules:

•Fireflies have their own clock

•Try to synchronize with ones next to it

Result:

Up to the parameter synchronization is possible

See: [YT2]

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Bibliography[AF] B.R. Andrievskii, A.L. Fradkov,

Control of Chaos: Methods and Applications, I. Methods,Automation and Remote Control, Vol. 64, No.5, 2003, pp. 673-713.

[BG] S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza,The control of chaos: theory and applicationsPhysics Report 329 2000, pp. 103-197.

[CM] Fireflies, INFNhttp://oldweb.ct.infn.it/~cactus/laboratorio/Fireflies.html,2012-11-22.

[CT] Chaos theory and global warming: can climate be predicted?http://www.skepticalscience.com/print.php?r=134,2012-11-22.

[Fe] R. Femat, G. Solis-Perales,Robust Synchronization of Chaotic Systems via Feedback,LNCIS, Springer 2008, pp.1-3.

[HH] H. Haken,light , volume 2, laser light dynamicsNorth-Holland 1985, chapter 8.

[Ka] T. Kanamaru (2008),Duffing oscillator, Scholarpedia, 3(3):6327 http://www.scholarpedia.org/article/Duffing_oscillator,2012-11-22.

[Py] K. Pyragas,Continuous control of chaos by self-controlling feedback, Physics LettersA 170, North-Holland 1992, pp. 421-428.

[SA] H. Salarieh, A. Alasty,Control of stochastic chaos using sliding mode method,Journal of Computational and Applied Mathematics,Vol. 225, Elsevier 2009, pp. 135-145.

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Bibliography[We] E.W. Weisstein,

Duffing Differential Equation,MathWorld – A Wolfram Web Resource,http://mathworld.wolfram.com/DuffingDifferentialEquation.html,2012-11-22.

[YT2] Youtube,fireflies synchttp://www.youtube.com/watch?gl=IT&hl=it&v=sROKYelaWbo2012-11-22.

[1] People waiting at bus stop http://worldteamjourney.files.wordpress.com/2012/06/people_waiting_at_bus_stop_42-16795068.jpg2012-11-22.

[2] Autostop http://www.digi.to.it/public/autostop%281%29.jpg2012-11-22.