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Athens nov 20121 Benjamin Busam, Julius Huijts, Edoardo Martino ATHENS - Nov 2012 Control of Chaos -...
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Transcript of Athens nov 20121 Benjamin Busam, Julius Huijts, Edoardo Martino ATHENS - Nov 2012 Control of Chaos -...
Athens nov 2012 1
Benjamin Busam, Julius Huijts, Edoardo Martino
ATHENS - Nov 2012
Control of Chaos-
Stabilising chaotic behaviour
Athens nov 2012 2
Chaos in a nutshell
Small change in initial condition
Huge difference in results
Deterministic systems, impossible to predict
See: [CT]
Athens nov 2012 3
Control of Chaos
• Stabilisation
– Suppression– Synchronisation
See: [Fe], [BG]
Athens nov 2012 4
Control of Chaos
• Stabilisation– Suppression
– Synchronisation
See: [Fe], [SA]
?
Athens nov 2012 5
Controlling Methods
1. Pyragas MethodDelayed Feedback Control
2. OGY-MethodShort explanation
See: [AF]
Athens nov 2012 6
Pyragas
See: [Fe], [SA]
Desired Orbit
Athens nov 2012 7
Pyragas
See: [Fe], [SA]
SystemX(t) Y(t)
u
u(t)=G[Y0-Y(t)]
Athens nov 2012 8
Pyragas
See: [Fe], [SA]
SystemX(t) Y(t)
u
u(t)=G[Y(t-T)-Y(t)]
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
Athens nov 2012 10
Controlling Methods
1. Pyragas MethodDelayed Feedback Control
2. OGY-MethodShort explanation
See: [AF]
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
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.
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
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
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]
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]
Athens nov 2012 17
Duffing Oscillator
See: [We], [YT]
30 cosx x x x f t
driving force
damping
restoring force
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
Athens nov 2012 19
D.O. - Phase Portrait
See: [SA]
30 cosx x x x f t
Athens nov 2012 20
D.O. - Control
3 20 cos 1x x x x f t x u
control term
Athens nov 2012 21
D.O. - Control
3 20 cos 1x x x x f t x u
control term
Athens nov 2012 22
D.O. - Noise
See: [SA]
3 20 cos 1x x x x f t x u kv
noise
Athens nov 2012 23
D.O. - Noise
See: [SA]
3 20 cos 1x x x x f t x u kv
noise
Athens nov 2012 24
D.O. - Noise
3 20 cos 1x x x x f t x u kv
See: [SA]
noise
Athens nov 2012 25
Control of laser chaos
See: [HH]
Athens nov 2012 26
Control of laser chaos
)'cos('0 tRRR
See: [HH]
Athens nov 2012 27
Control of laser chaos
See: [HH]
Athens nov 2012 28
Control of laser chaos
See: [HH]
Athens nov 2012 29
Conclusion
1. Pyragas Method
2. OGY-Method
3. Applications
Athens nov 2012 30
Any questions?
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]
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]
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]
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.
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.