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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Phase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusionPhase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion
Ariën J. van der Wal
Netherlands Defence Academy (NLDA) 15th ICCRTS
Santa Monica, CA, June 22-24, 2010
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
ContentsContents
Introduction Synchronization as a paradigm for sensor data fusion
What is synchronization ? Phase Frequency
Examples of synchronization How do two systems synchronize: phase transition analogy Multi- oscillator synchronization in practice: Preliminary
results
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
IntroductionIntroduction
Why are we interested in synchronization ? Motivation: try to understand mechanisms of sensor fusion Emergent behavior: What is it and when does it occur? Simple oscillators provide a good model for studying these We focus on phase synchronization as a basic mechanism for
inducing co-operative behavior Is it possible to extend the paradigm to real applications,
e.g. in modelling military sensor networks ?
Systems studied: 1. Non-linear coupling of 2 linear oscillators 2. Non-linear coupling between N linear oscillators3. Linear coupling of 2 non-linear oscillators
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Examples of synchronization processesExamples of synchronization processes
Biology: fireflies, yeast, algae, crickets Physiology: heart, brain, biological clocks, ovulation cycle Chemistry: chemical clocks Engineering: Power grids, distribution of time (UTC) Communication requires synchronisation at all OSI layers Physics: coherence of lasers and masers, phase transitions,
ferromagnetism, superconductivity, spin waves
SHOW physics demo: 3 metronomes synchronization presentatie\Synchronization of Three Metronomes.MP4
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
SynchronizationSynchronization
History: Christiaan Huygens (Feb 1665) “Sympathie des horloges” 2 pendulum clocks suspended from the same beam will in a
relatively short period assume the same rythm if they are initially out-of-phase; they will eventually synchronize and lockin antiphase !
Constant phase difference between 2 oscillations:
Only possible if both have the same frequency:
Amplitude of oscillator can be chaotic
0 constant
1 20 f f
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
The definition of phaseThe definition of phase
How can “phase” be defined for an arbitrary, periodic signal?
Different ways to define momentary phase: For a simple sine fœ[0,2p] For a periodic function phase plane; Poincaré map For a complex oscillator Hilbert transform
( )
( )
1 ( )( )
( ) ( ) ( ) ( )( )tan( ( ))( )
i t
x x t x
xy t P dt
z t x t iy t r t e zy ttx t
H x
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Two coupled linear oscillators (1)Two coupled linear oscillators (1)
2 oscillators, each with its own eigenfrequency :
with nonlinear interaction dependent on the phase differenceso that
with
1 1 12 1 2
2 2 21 2 1
( )
( )
d Udtd Udt
( )d udt
1 2
1 2
12 21( ) ( ) ( )u U U
1 2 12 ( )U
1,2
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Two coupled linear oscillators (2) Two coupled linear oscillators (2)
If synchronization occurs, we have:
So if real roots for this algebraic equation exist, we have found synchronous solutions !
A an example we take and find the graphical solutions of
is stable and unstable
sin 0ddt
0 ( )d udt
( ) sinu
-Δω
quqs
u
q
us
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Two coupled linear oscillators (3) Two coupled linear oscillators (3)
Synchronization occurs when
This algebraic equation only has real roots iff the difference in eigenfrequencies lies within the interval of values of the function :
In our example we have:
“Arnold tongue”
( ) sinu
0 ( )d udt
( )u
1 2 synchronized
ω1 ω2
ε
0
min maxu u
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Two coupled linear oscillators (4) Two coupled linear oscillators (4)
The common synchronization frequency of the two coupled oscillators follows from:
where is the phase difference from the stable graphical solution
Outside the entrainment region the motions are notsynchronous, but they can still influence each other significantly. (-> phase slips)As an example take
so that
with solution
1 12 0 2 21 0( ) ( )U U
sinddt
0
112 21 2( ) ( ) sinU U
22 21
22( ) 2arctan 1 tan( )t t
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
Two coupled linear oscillators (5) Two coupled linear oscillators (5)
Time dependence of phase difference outside the region of synchronization.
1 2 1 1.05
time
θ
2π phase slip
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Faculty of Military Sciences
15th ICCRTS Conference – The evolution of C2 Copyright 2010 A.J. van der Wal
9 fe
b10
Air coordination centre
Tactical Sensor Sensors
Air effectors
NEC in inhomogeneous networks NEC in inhomogeneous networks
UAV Ground Station
Remote Calculation
Station
Remote HQ
Image interpretationsystem
Coordination &Fusion centre
HQ
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Modeling a homogeneous sensor networkModeling a homogeneous sensor network
Homogeneous network of N nodes (“agents”) acts as a distributed sensor (or detector)
Homogeneous networks are part of typical NEC networks Node composition: (analog) sensor, memory, decision taking Mathematical modelling: Node = oscillator Observable determines the oscillator frequency: Node =
parametric oscillator (or VCO ?) Contact between nodes through non-linear coupling K Study the dynamic behavior of the ensemble of N coupled
oscillators
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Why is synchronization important ?Why is synchronization important ?
Sensor network: each member of the population is represented by a phase oscillator
Synchronization on physical layer, not on protocol layer: faster and more accurate
Greater robustness, fault tolerance, scalability, smallcomplexity self-synchronization
Ultimate goal: local information storage, propagation of information, distributed, “soft” decision taking
Redistribution of mobile sensors to more effectively sample the environment in presence of measurement noise
Propagation and fusion of analog information without a central fusion master
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N linear oscillators N linear oscillators
Kuramoto model ensemble of N nearly identical oscillators symmetric distribution of eigenfrequences global coupling strength evolution of oscillator phase given by
Stationary synchronization (mean-field approximation)
complex order parameter r:
1sin( ) ( 1,..., )
Nk
k j kj
d t K t t k Ndt N
1
1k
Nii
kre e
N
sin( ) ( 1,..., )kk k
d tKr k N
dt
0K ( ) ( )g g
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Sensors acting as detectorsSensors acting as detectors
Distributed, dense sensor network Detection as a stochastic process:
if an event is detectedif no event is detected
Probability of detection p0 If the network is sufficiently large the phase rate
converges to :
0i 1i
** 1
0 1 0 0
1
(1 )N
k kkN
kk
cd p pdt c
*0
0 0
*1
1 0
arcsin 1(0)
arcsinj
with probability pKr
with probability pKr
**( ) (0)j jt t
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Synchronisation phase diagram N=400 oscillatorsSynchronisation phase diagram N=400 oscillators
0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 = 0.5 400 oscillators
Syn
chro
nize
d fr
actio
n
(1-2/)
SYNCHRONIZED
UNSYNCHRONIZED
interaction strength K
sync
hron
ized
fract
ion
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Integration time step = 0.01
How fast is synchronization for N=400 ?How fast is synchronization for N=400 ?
0 10 20 30 40 50 60 70 80 90 1000.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9K=1.700000000000000
time t
orde
red
frac
tion
orde
red
fract
ion
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Results of simulations N=400Results of simulations N=400
presentatie\freq N=400.avi presentatie\phase N=400.avi
frequency phase
0 50 100 150 200 250 300 350 400-5
-4
-3
-2
-1
0
1
2
3
4
5K=5.000000000000000E-001
oscillator #
devi
atio
n fr
om c
entr
al f
requ
ency
0 50 100 150 200 250 300 350 400-8
-6
-4
-2
0
2
4
6
8K=5.000000000000000E-001
oscillator #
osci
llato
r ph
ase
mod
2
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SynergySynergy
The basic notion of synergy is:
or at least:
Non-lineartity is an essential ingredient for understanding sensor data fusion !
Classical approach via Bayesian networks, DS theory and/or fuzzy (belief and plausability) measures
In the present study we focus on a different approach: the paradigm of phase transitions in physics
( ) max( ( ), ( ))g A B g A g B
( ) ( ) ( )g A B g A g B
1 ( ) ( ) ( ) ( ) ( )g A B g A g B g A g B
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ConclusionsConclusions
Nonlinear coupling of 2 linear oscillators results in very fastphase synchronization, provided that the interaction is strongenough. Synchronization of 2 nonlinear oscillators occurs already at veryweak coupling. In a non-linear globally interacting many-particle system we observe spontaneous (partial) synchronization above a critical interaction strength.
The fast and spontaneous synchronization of globally interactingsystems is a form of emergent behavior and may be exploited as a mechanism for military smart sensor networks.
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