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Athens nov 2012 1
Benjamin Busam, Julius Huijts, Edoardo Martino
ATHENS - Nov 2012
Control of Chaos-
Stabilising chaotic behaviour
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Athens nov 2012 2
Chaos in a nutshell
Small change in initial condition
Huge difference in results
Deterministic systems, impossible to predict
See: [CT]
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Athens nov 2012 3
Control of Chaos
• Stabilisation
– Suppression– Synchronisation
See: [Fe], [BG]
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Athens nov 2012 4
Control of Chaos
• Stabilisation– Suppression
– Synchronisation
See: [Fe], [SA]
?
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Athens nov 2012 5
Controlling Methods
1. Pyragas MethodDelayed Feedback Control
2. OGY-MethodShort explanation
See: [AF]
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Athens nov 2012 6
Pyragas
See: [Fe], [SA]
Desired Orbit
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Athens nov 2012 7
Pyragas
See: [Fe], [SA]
SystemX(t) Y(t)
u
u(t)=G[Y0-Y(t)]
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Athens nov 2012 8
Pyragas
See: [Fe], [SA]
SystemX(t) Y(t)
u
u(t)=G[Y(t-T)-Y(t)]
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Athens nov 2012 9
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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Athens nov 2012 10
Controlling Methods
1. Pyragas MethodDelayed Feedback Control
2. OGY-MethodShort explanation
See: [AF]
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Athens nov 2012 11
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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Athens nov 2012 12
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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Athens nov 2012 13
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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Athens nov 2012 14
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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Athens nov 2012 15
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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Athens nov 2012 16
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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Athens nov 2012 17
Duffing Oscillator
See: [We], [YT]
30 cosx x x x f t
driving force
damping
restoring force
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Athens nov 2012 18
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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Athens nov 2012 19
D.O. - Phase Portrait
See: [SA]
30 cosx x x x f t
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Athens nov 2012 20
D.O. - Control
3 20 cos 1x x x x f t x u
control term
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Athens nov 2012 21
D.O. - Control
3 20 cos 1x x x x f t x u
control term
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Athens nov 2012 22
D.O. - Noise
See: [SA]
3 20 cos 1x x x x f t x u kv
noise
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Athens nov 2012 23
D.O. - Noise
See: [SA]
3 20 cos 1x x x x f t x u kv
noise
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Athens nov 2012 24
D.O. - Noise
3 20 cos 1x x x x f t x u kv
See: [SA]
noise
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Athens nov 2012 25
Control of laser chaos
See: [HH]
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Athens nov 2012 26
Control of laser chaos
)'cos('0 tRRR
See: [HH]
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Athens nov 2012 27
Control of laser chaos
See: [HH]
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Athens nov 2012 28
Control of laser chaos
See: [HH]
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Athens nov 2012 29
Conclusion
1. Pyragas Method
2. OGY-Method
3. Applications
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Athens nov 2012 30
Any questions?
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Athens nov 2012 31
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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Athens nov 2012 32
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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Athens nov 2012 33
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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Athens nov 2012 34
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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Athens nov 2012 35
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.