Phase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion

Transcription

1 Phase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion Ariën J. van der Wal Netherlands Defence Academy (NLDA) 15 th ICCRTS Santa Monica, CA, June 22-24, 2010 Copyright 2010 A.J. van der Wal 1

2 Contents 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 Copyright 2010 A.J. van der Wal 2

3 Introduction 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 oscillators 3. Linear coupling of 2 non-linear oscillators Copyright 2010 A.J. van der Wal 3

4 Examples 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 Copyright 2010 A.J. van der Wal 4

5 Synchronization 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 lock in antiphase! Constant phase difference between 2 oscillations: 0 constant Only possible if both have the same frequency: 0 f1 f2 Amplitude of oscillator can be chaotic Copyright 2010 A.J. van der Wal 5

6 The 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 x xt () x 1 x( ) yt () Hx P d t i () t zt () xt () iyt () rte () z yt () tan( ( t)) xt () Copyright 2010 A.J. van der Wal 6

7 Two coupled linear oscillators (1) 1,2 2 oscillators, each with its own eigenfrequency : d 1 1U12( 12) dt d 2 2 U 21( 2 1) dt with nonlinear interaction U12( ) dependent on the phase difference 1 2 so that 1 2 d u( ) with 12 dt u( ) U ( ) U ( ) Copyright 2010 A.J. van der Wal 7

8 Two coupled linear oscillators (2) If synchronization occurs, we have: d 0 u( ) dt So if real roots for this algebraic equation exist, we have found synchronous solutions! A an example we take u( ) sin and find the graphical solutions of s d sin 0 dt u is stable and unstable u q s q u -Δω q Copyright 2010 A.J. van der Wal 8

9 Two coupled linear oscillators (3) Synchronization occurs when d 0 u( ) dt This algebraic equation only has real roots iff the difference in eigenfrequencies lies within the interval of values of the function u( ) : u min u max ε synchronized 1 2 In our example we have: u( ) sin 0 ω 1 ω 2 Arnold tongue Copyright 2010 A.J. van der Wal 9

10 Two coupled linear oscillators (4) The common synchronization frequency of the two coupled oscillators follows from: 0 U ( ) U ( ) where is the phase difference from the stable graphical solution Outside the entrainment region the motions are not synchronous, but they can still influence each other significantly. (-> phase slips) 1 As an example take U ( ) U ( ) sin d so that sin dt with solution ( t) 2arctan 1 tan( 2 2 t) Copyright 2010 A.J. van der Wal 10

11 Two coupled linear oscillators (5) Time dependence of phase difference 1 2 outside the region of synchronization θ 2π phase slip time Copyright 2010 A.J. van der Wal 11

12 NEC in inhomogeneous networks 9 feb 10 Air coordination centre Remote HQ Tactical Sensor Air effectors Sensors Coordination & Fusion centre HQ Image interpretation system UAV Ground Station Remote Calculation Station Copyright 2010 A.J. van der Wal 12

13 Modeling 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

14 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, small complexity 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

15 N linear oscillators Kuramoto model ensemble of N nearly identical oscillators symmetric distribution of eigenfrequences global coupling strength K 0 evolution of oscillator phase given by N dk t K sin( t t ) ( k 1,..., N) dt k j k N j1 Stationary synchronization (mean-field approximation) complex order parameter r: d k dt t re i N 1 ik N k 1 Krsin( ) ( k 1,..., N) k k e g( ) g( )

16 Sensors acting as detectors Distributed, dense sensor network Detection as a stochastic process: i 1 if an event is detected if no event is detected Probability of detection p 0 If the network is sufficiently large the phase rate * converges to : * t t j i () (0) 0 N c * k 1 k k N c k 1 k * d p (1 p ) dt * 0 0 arcsin with probability 1 p0 Kr j (0) Kr j * 1 1 arcsin with probability p

17 Synchronisation phase diagram N=400 oscillators synchronized fraction Synchronized fraction = oscillators UNSYNCHRONIZED (1-2/) SYNCHRONIZED interaction strength K

18 How fast is synchronization for N=400? ntegration time step = K= ordered fraction ordered fraction time t

19 Results of simulations N=400 presentatie\freq N=400.avi presentatie\phase N=400.avi 5 K= E K= E deviation from central frequency oscillator phase mod oscillator # oscillator # frequency phase

20 Synergy The basic notion of synergy is: or at least: g( AB) g( A) g( B) g( AB) max( g( A), g( B)) 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 1 g( AB) g( A) g( B) g( A) g( B)

21 Conclusions onlinear coupling of 2 linear oscillators results in very fast hase synchronization, provided that the interaction is strong nough. ynchronization of 2 nonlinear oscillators occurs already at very eak coupling. n a non-linear globally interacting many-particle system we bserve spontaneous (partial) synchronization above a critical nteraction strength. he fast and spontaneous synchronization of globally interacting ystems is a form of emergent behavior and may be exploited as mechanism for military smart sensor networks.

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