Synchronization in sensor networks Synchronization in sensor - PowerPoint PPT Presentation

Synchronization in sensor networks Synchronization in sensor networks Jie Gao Computer Science Department Stony Brook University

Papers Papers • [Mirollo90] M. Mirollo and S. Strogatz. Synchronization of pulse-coupled biological oscillators , SIAM J. Applied Math., 50(6):1645-1662, 1990. • [Lucarelli04] D. Lucarelli and I. Wang, Decentralized synchronization protocols with nearest neighbor communication , Sensys’04. • [Werner05] G. Werner-Allen, G. Tewari, A. Patel, M. Welsh, R. Nagpal, Firefly-inspired sensor network synchronicity with realistic radio effects , Sensys’05. • Many slides are from G. Werner-Allen’s talk in Sensys 2005. • http://www.eecs.harvard.edu/~werner/

What is synchronicity? synchronicity? What is Synchronicity: the ability to organize simultaneous collective action ...contrast with... Time Synchronization: the ability to establish a common time base allowing events to be time-stamped in a meaningful way borrowed from G. Werner-Allen’s talk in Sensys 2005

Natural Synchronicity Natural Synchronicity Fireflies! Cardiac Cells borrowed from G. Werner-Allen’s talk in Sensys 2005

Fireflies Fireflies Imagine a tree 35 or 40 feet high…, apparently with a firefly on every leaf and all the fireflies flashing in perfect unison at the rate of about three times in two seconds, the tree being in complete darkness between flashes… From H. M. Smith, Science 82 (1935), p.151.

Pulse Coupling Pulse Coupling Each node: 1) Is an oscillator 2) Periodically emits a pulse 3) Adjusts the phase of its pulse by observing other pulses borrowed from G. Werner-Allen’s talk in Sensys 2005

Nice Properties Nice Properties Each node observes the others’ actions and try to align itself. No leaders No “absolute clock” No global information No routing Very simple borrowed from G. Werner-Allen’s talk in Sensys 2005

Synchronicity in sensornet sensornet Synchronicity in Network Timer Useful for: � coordinated sampling; � network-level duty cycling; � coordinate transmission to avoid interference; � wake up schedule; � etc... borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T Example of system dynamics: 1) Nodes move together at a fixed rate from 0 to T 2) When nodes reach T they “fire”, return to 0 3) Each nodes base period is T 4) Overhearing a “fire” moves a node forward borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T t A A hears B fire at t A borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A A B 0 T t A t A ' � ( t A ) A hears B fire at t A A jumps to t A '=t A + � ( t A ) borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A A B 0 T t A t A ' � ( t A ) A hears B fire at t A A jumps to t A '=t A + � ( t A ) “Jump Function” borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T A hears B fire at t A A jumps to t A '=t A + � ( t A ) B returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T A hears B fire at t A A jumps to t A '=t A + � ( t A ) B returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T A hears B fire at t A A jumps to t A '=t A + � ( t A ) B returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T A hears B fire at t A A jumps to t A '=t A + � ( t A ) B returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : B A 0 T A hears B fire at t A A jumps to t A '=t A + � ( t A ) B returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T t B B hears A fire at t B borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B B 0 T t B t B ' � ( t B ) B hears A fire at t B B jumps to t B '=t B + � ( t B ) borrowed from G. Werner-Allen’s talk in Sensys 2005

Coupled Oscillators : Strogatz Strogatz Coupled Oscillators : A B 0 T B hears A fire at t B B jumps to t B '=t B + � ( t B ) A returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity t=0 A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity t=0 A B 0 T T t=T A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity t=0 A B 0 T T t=T A B 0 T t=2T A B 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

Goal: Synchronicity Goal: Synchronicity t=0 A B 0 T T t=T A B 0 T t=2T A B 0 T B t=3T A 0 T borrowed from G. Werner-Allen’s talk in Sensys 2005

The Jump Function The Jump Function Call � ( t ) the Jump Function Synchronicity emerges when � ( t ) is monotonically increasing : If t 1 > t 2 then � ( t 1 ) > � ( t 2 ) Intuitively, as a node gets closer to firing other firing events affect it more strongly Note that a node cannot jump past T ! If t' = t' + � (t) > T the node fires and returns to 0 borrowed from G. Werner-Allen’s talk in Sensys 2005

Theoretical Results Theoretical Results We will prove the simple case of 2 oscillators. Theorem: the phase difference converges to 0 under some favourable conditions. First let’s start from the oscillator model.

Model of oscillator Model of oscillator X=f( φ ): smooth, monotonically increasing, and concave down. f ’ >0, f ’’ <0. X=f( φ ) 1.0 B Energy function x A φ 1.0 Define phase of A: φ A , as the distance to origin. We start with ( φ A , φ B )=(0, φ ).

Model of oscillator Model of oscillator X=f( φ ): smooth, monotonically increasing, and concave down. f’>0, f’’<0. X=f( φ ) B 1.0 A x 1.0 1- φ

Model of oscillator Model of oscillator X=f( φ ): smooth, monotonically increasing, and concave down. f’>0, f’’<0. X=f( φ ) B 1.0 ε A Boost-up energy x 1- φ 1.0 A jumps to g( ε +f(1- φ )) where g=f -1

Model of oscillator Model of oscillator Firing map: h( φ )=g( ε +f(1- φ )). After B fires, the system moves from ( φ A , φ B )=(0, φ ) to a current state (h( φ ), 0). X=f( φ ) B 1.0 ε A Boost-up energy x 1- φ 1.0 A jumps to g( ε +f(1- φ )) where g=f -1

Model of oscillator Model of oscillator Firing map: h( φ )=g( ε +f(1- φ )). After B fires, the system moves from ( φ A , φ B )=(0, φ ) to a current state (h( φ ), 0). After A fires, the system moves to (0, h(h( φ ))). Now we finish a full loop. Return map: R( φ )=h(h( φ )). R( φ ) is the new phase difference after 2 firings. Theorem: R( φ ) has a fixed point which is a repeller.

Model of oscillator Model of oscillator Theorem: R( φ ) has a fixed point which is a repeller. R( φ *)= φ *. When φ < φ *, R( φ )< φ . When φ > φ *, R( φ )> φ . “Repeller”: no matter where you start you are always pushed away from the fixed point --- not a stable fixed point. Whenever φ is pushed to 0 or 1, then it’s done!

Dynamics Dynamics Goal: prove that R( φ )= h(h( φ )) has a fixed point, with h( φ )=g( ε +f(1- φ )). We will prove that h( φ ) has a fixed point φ *, I.e., h( φ *)= φ *. It’s obvious that R( φ *)= φ *. Now take F( φ )= φ -h( φ ); we argue F( φ )=0 for a value φ *.

Dynamics Dynamics Observation 1: F( δ )= δ -g( ε +f(1- δ )), with δ very small. Thus ε +f(1- δ )>1, I.e., A will fire, F( δ )= δ -1 <0. X=f( φ ) B 1.0 ε A Boost-up energy x 1- φ 1.0

Dynamics Dynamics Observation 1: F( δ )= δ -g( ε +f(1- δ )), with δ very small. Thus ε +f(1- δ )>1, I.e., A will fire, F( δ )= δ -1 <0. Observation 2: F( δ )= δ -g( ε +f(1- δ )), with δ very close to 1. Now F( δ ) >0. Then there must be a point φ * in (0, 1) such that F( φ *)=0 F( φ ) δ φ h -1 ( δ )

Dynamics Dynamics First goal: prove h’<-1. Now h( φ )=g( ε +f(1- φ )). Just do calculus. Replace f(1- φ ) by u. Since g is the inverse of f, then g’>0 and g’’>0. So g’( ε +u)>g’(u). Thus h’<-1. QED